IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 1, FEBRUARY 2009
469
An Annual Midterm Energy Forecasting Model Using Fuzzy Logic Charalambos N. Elias and Nikos D. Hatziargyriou, Fellow, IEEE
Abstract—The objective of this paper is to present a new fuzzy logic method for midterm energy forecasting. The proposed method properly transforms the input variables to differences or relative differences, in order to predict energy values not included in the training set and to use a minimal number of patterns. The input variables, the number of the triangular membership functions and their base widths are simultaneously selected by an optimization process. The standard deviation is calculated analytically by mathematical expressions based on the membership functions. Results from an extensive application of the method to the Greek power system and for different categories of customers are compared to those obtained from the application of standard regression methods and artificial neural networks (ANN). Index Terms—Energy forecasting, fuzzy logic, optimization of membership functions, standard deviation.
I. INTRODUCTION N an open electricity market it is essential to know the values of energy and its growth for future years. Accurate energy forecasting leads to effective scheduling and planning and thus to higher system reliability and lower operational costs. The electricity industry needs load and energy forecasts for short term, i.e., 1–7 days in hourly steps, midterm, i.e., 1–5 years in monthly or longer steps, and long term, i.e., 5–30 years in annual or longer steps. Short term power forecasting is related with economic dispatch of the units and the order of unit commitment. Midterm power and energy forecasting is needed for scheduling the maintenance of the units, the fuel supplies, electrical energy imports/exports and the exploitation of the water reserves for hydrothermal scheduling. Long term energy forecasting is important for planning and expanding the electric system. This paper deals with midterm energy forecasting. Many forecasting models and methods have been implemented on midterm energy forecasting, with different level of success. Electric companies have mainly used simple forecasting models, like linear regression [1] and econometric models [2]. Nowadays, multiple regression models are used for very large systems [3], [4], large metropolitan areas [5], or small areas [6]. Models that forecast the needs of various types of customers are presented using either physical time series [7], or genetic algorithms [8]. Simple autoregressive (AR) and
I
Manuscript received December 12, 2007; revised September 03, 2008. First published January 14, 2009; current version published January 21, 2009. Paper no. TPWRS-00936-2007. The authors are with the School of Electrical and Computer Engineering, National Technical University of Athens, Athens, Greece (e-mail: xelias@hlk. forthnet.gr;
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRS.2008.2009490
autoregressive integrated moving average (ARIMA) models have been also used for monthly and annual energy forecast [9]–[11]. Annual time series models have been presented with linear, exponential and multinomial approximation [7], while non stationary time series have been modeled for data with regular [12] and dynamic periodic trends [13]. Recently, application of artificial intelligence techniques has also been applied to midterm energy forecasting using either neural networks [14], [15], or fuzzy logic [16], [17]. In this paper a new method for midterm energy forecasting is proposed based on fuzzy logic. Its basic features are: • the optimization process to determine the number of the triangular membership functions and their base width; • a minimal training set, since in any application historical data comprise only 14 training patterns of three statistical indices; • a proper transformation of the input variables, so that the fuzzy logic model can successfully deal with the midterm energy forecasting model; • the calculation of standard deviation based on analytical expressions. The proposed model is applied in forecasting the annual energy demand of the Greek power system and for various categories of customers. The results are compared to the ones obtained using the standard regression methods and artificial neural networks (ANNs) [15]. In Section II there is a reference to the basic principles of fuzzy logic. In Section III the fuzzy logic model is presented in detail. In Section IV its application to annual energy demand of the Greek power system is outlined, while in the next section the results of the annual energy demand forecasting for six types of customers are shown. II. BASIC PRINCIPLES OF FUZZY LOGIC The mathematical foundation of fuzzy logic is based on the theory of fuzzy sets, which may be considered as a generalization of the classic theory of sets. The switch from the classic theory, where participation of an object in a set is strictly defined, to the application of fuzzy logic is achieved by the mem, where is the value of the linguistic bership function variable and is the serial number of functions which describe . The membership functions can be the triangular, trapezoidal, Gaussian function, etc. The membership functions and the logical rules comprise the means of realization of the fuzzy logic systems, which consist of four elements: fuzzification, rule base, deduction mechanism, and defuzzification. Fuzzification is the process through which a nonfuzzy set is converted to a fuzzy set. It is based on the definition of the membership function.
0885-8950/$25.00 © 2009 IEEE
470
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 1, FEBRUARY 2009
The rule base is a set of fuzzy rules describing the dependence of one (or more) linguistic variables from another. The rules are described by the following pattern:
(1) are the input variables, where are the respective fuzzy values of the input variables, is the output variable and is the fuzzy value of the output. For each and the respective fuzzy values are of the described by appropriate membership functions. The deduction mechanism comprises three sequential steps: 1) The Generalized modus ponens Inference Engine and the Larsen-Max Product Implication, which for every rule of one input-one output imply the membership function from the input to the output. 2) The degree of fulfillment (DOF), which is the procedure implying the Larsen-Max Product Implication for more than one input variables for each rule. For the th vector the th rule is determined as follows:
(2) 3) The border method, which forms the final function of the output variable. Finally, the defuzzification maps the space of fuzzy values to a space of nonfuzzy values. Unfortunately, there is no systematic procedure for defuzzification. The most common ones are: the maximum, the mean value of the maximum (MOM), and the center of the area criterion (COA). When the DOF method is used [18], the criterion of the center of the area (COA) is the most suitable
(3)
where is the center, is the number of intervals of width dividing the axis of the output variables, is the membership function of the variable , is the value for which the mem. The COA method provides bership function becomes mean square error smaller than the maximum method. Its advantage against (MOM) is that it considers the overall shape of the fuzzy output [18]. is the In the case of the forecasting fuzzy logic system, estimated variable at the time of the forecast for which the input variables are given. The system’s validation is realized through the mean absolute percentage error by the following expression:
(4) where is the actual value of demand at time population of the evaluation set.
and
is the
III. FUZZY MIDTERM FORECASTING MODEL Based on the principles of Section II, a model for midterm forecast of the annual energy demand was developed. The basic notions are the following: • proper transformation of model’s variables (see Section III-A); • selection of the number of membership functions and their characteristics (see Section III-B); • input variables finally obtained through correlation analysis (see Section III-D). The outline of the procedure to build the fuzzy model is presented in Fig. 1. In summary, the main steps of the proposed energy forecasting model are the following. input variables are selected by the user, based on 1) The his engineering knowledge from the respective database. 2) The input variables are reduced from to through correlation analysis. 3) If necessary, the input variables are transformed to their differences or their relative differences. 4) The combinations of input variables are determined, as in Section III-D. 5) For each input variable the number of membership functions and triangle’s base width are determined by combinations. 6) Fuzzified values of each final form of variables are calculated. 7) The rules concerning the years, for which the parameter values are included in the database, are formed. 8) After classifying all the possible combinations of the rules the fuzzy output value is determined via the weight process. Based on these rules the rule base is created. 9) Using the rule base, the deduction mechanism and the COA defuzzification method, a forecast is made concerning the years of the evaluation set. 10) Steps 4–9 are repeated for every combination. 11) The combination that produces the minimum MAPE for the evaluation set is selected and the fuzzy model is realized. 12) The left part of the rule concerning each forecasting year is formed, based on the combination selected after the completion of the previous step and the corresponding rule is determined. 13) Finally, the expected amount of energy during the forecasting year is estimated, as in step 9. The standard deviation is calculated. A. Transformation of Input Variables The input variables comprise parameters such as number and types of customers, energy consumption, temperatures, and various statistical indices. The actual values of the variables are transformed as follows: for values of variables with normal growth the difference of the corresponding variables is used and for exponentially growing values of variables the relative difference. Fuzzy logic systems and artificial neural networks could not make directly forecasting because the future values usually are not in the limit of historical data. The advantage of using difference or relative difference is that the values are in a limited width and it is easier to use them or to predict their
ELIAS AND HATZIARGYRIOU: AN ANNUAL MIDTERM ENERGY FORECASTING MODEL USING FUZZY LOGIC
471
Fig. 2. Transformation of input variables with normal growth: the difference d of variable z is used, in order to bound the respective range of values.
Fig. 3. Transformation of input variables with exponentially growth: the relative difference r of variable z is used, in order to bound the respective range of values.
is used and these variables are denoted by (7)
Fig. 1. Flowchart of the procedure to obtain the fuzzy model.
future values (see Figs. 2 and 3). Thus, instead of the value of the th variable during the th year, either the difference
(5) or the relative difference (6)
where is the number of years for which data are available. If some input attributes of a year are missing, this year will be omitted. B. Fuzzification and Rule Base Fuzzification is realized using triangular membership functions. The odd number of membership functions to be used and the triangle’s base width are selected, in order to optimize the performance of the fuzzy forecasting system. The center of of a variable and the initial value of the middle triangle
472
the base width expressions:
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 1, FEBRUARY 2009
of each triangle are given by the following
(8)
D. Optimization of the Input Variables’ Selection Examination of all possible combinations of the input varitimes of the basic form ables would require the execution of of the fuzzy model. So, after taking into account the optimization of the membership function as to the width of the triangle’s base, the final combinations to be made are
(9) (11) where is the number of membership functions of variable . with Next, the base width of the triangle is modified by , while the center of the middle triangle remains constep stant. Thus, the number of possible triangles , to be examined per variable, is (10) Therefore, for input variables, the total number of combinations is . Following, the fuzzification and the rule base are completed for all alternative of values and of all variables. C. Deduction Mechanism and Results Validation Next, all possible combinations of the rules are classified and the fuzzy output values for every combination of input rules from the training years are obtained. The output is determined by all the weighted average of fuzzy values obtained during training, rather than by the value that occurs more often. For example, let us assume that for each fuzzy output value of the , , 0, 1, 2 for “Very Negsystem a corresponding weight, ative”, “NEgative”, “ZEro”, “PoSitive” and “Big Positive”. In a certain rule the output values may appear with the following frequencies:
Based on the maximum frequency, the fuzzy value “NE” should have been chosen as output, whereas based on the weighted average
The system chooses the closest fuzzy value “ZE”. If a rule does not occur at all, then the fuzzy value “ZE” is selected as output. If a rule occurs only once, the output weight is divided by two, so that it does not affect forecasting excessively. Next, by taking into consideration the input data of the evaluation set, the system sets up the left part of the rules for the respective years of this set. The rules that correspond to the left part of the rules created for each year of the evaluation set are selected from the rule base and their outputs are read. By applying the deduction mechanism and the COA method, the difference between the forecasted amount of energy, which will be required during the current year and the amount of energy of the former year, is calculated. Finally, for each combination the mean absolute percentage error (MAPE) for the evaluation set is found and used as criterion of comparison amongst the results of the various combinations.
Since the initial preprocessing leads to an average of 15 variables, as in [15], it becomes imperative that the preprocessing is repeated with a correlation index between input-output and a correlation amongst the input variables, so that the number of combinations decreases. If the correlation index between and output is greater than a pre-specified value , the transformed input variable is retained for further processing; else, it is not considered any further. Next, for the retained inputs a cross correlation analysis is performed. If the correlation index between any two terms is smaller than a pre-specified value then both terms are retained; else, only the term with the largest correlation with respect to output is retained. In this way the input variables decrease from to and the combinations are also decreased. E. Final Prediction Finally, for each of the possible combinations of the input variables, the fuzzy value of each input, which corresponds to the data of each training year, is determined. Moreover, the membership functions of all fuzzy variables are defined for each combination of the values of membership functions variables and their triangle’s base widths. As a result, rules are created for each year, whose number varies from to . This process is repeated in order to check all possible combinations and the combination with minimum MAPE in the forecast of the evaluation set is selected. This combination is used for midterm energy forecast. The prediction is made in exactly the same way it was done for the evaluation set. Only now the combination of base widths and input data is given. The energy needed during each year of the forecast derives from the difference between the forecasted amount concerning this year of the forecast and the amount concerning its previous year. F. Standard Deviation In order to calculate the standard deviation analytically the respective expression for only one triangular membership function is first found. The mean value and the standard deviation of a triangular membership function, as shown in Fig. 4, are (12)
(13)
ELIAS AND HATZIARGYRIOU: AN ANNUAL MIDTERM ENERGY FORECASTING MODEL USING FUZZY LOGIC
Fig. 4. Formation of a typical triangular membership function.
where
473
Fig. 5. Case of two triangular membership functions.
and the respective integrals are given by
In Fig. 5 the case of two triangular membership functions is and the membership presented, where the relation between functions and is obvious. The respective mean value and standard deviation are (16)
Based on the basic principles of mechanics in the case of more than one triangular membership functions the mean value equals to the abscissa of the gravity center and the standard deviation is equivalent to the moment of the inertia with respect to the axis, which crosses the gravity center and is parallel to the axis of the ordinates. The respective expressions are
(17) The proposed fuzzy system is a type-1 system and the calculated standard deviation is a measure of the uncertainty of the output. Alternatively, a type-2 fuzzy logic system could be used, so that the respective uncertainty could be determined [19], [20]. IV. CASE STUDY I: ANNUAL ENERGY FORECASTING FOR THE GREEK POWER SYSTEM A. Data Preprocessing
(14)
The proposed method is applied for the annual energy forecasting of the Greek Power System. The data are obtained from the Greek National Utility, the Meteorological Service and the National Statistics Service of Greece. Values for the years 1986–2000 are available. The forecasting years are 2001 to 2003. The available data are pre-processed in order to fill in missing data and convert them appropriately so that their dimensionality is reduced. For example, the average hourly temperatures are converted to hot and cold days as follows:
(15) (18) because the following integral is equal to
(19) where is the average daily temperature, is the threshold temperature value under which space heating is necessary (for is the threshold temperature value above Greece 20 ) and which space cooling is required (for Greece 25 ). Based on utility’s engineering knowledge and experience and through data preprocessing, the forecasting model is supplied , serial current year with the following data: annual energy , annual energy of previous year , number of hot-days and of cold-days , gross national product , sta, of manufacturing tistical indices of oil and coal products
474
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 1, FEBRUARY 2009
Fig. 6. Example of the membership function’s structure regarding Cold-Days of base width, basic form, of base width. for three cases:
+20%
020%
, of basic metal , of manufacture of final metallic prod, of paper and paper products , of chemical products , of food products-beverages , of durable conucts and of non durable consumption goods sumption goods and number of customers .
Fig. 7. Membership functions of the finally used three inputs and the difference for annual energy.
B. Fuzzy Logic Model Implementation The serial current year and the number of hot-days and colddays are not transformed. The gross national product is transformed to relative difference and the remaining variables are transformed to differences, as discussed in Section III-A. and are Then, the correlation indices between , or estimated. Indices and result in the . All possible comfollowing input variables: , , and , step , , 5 or binations are examined with a 7, using as training years, the years 1986-2000. The evaluation set is the same with the training set. The final model includes and . variables , The system is finally calibrated for 3, 5, 7 membership func– (i.e., 400tions, width of variable 440-480-520-560-600), of – , of – and of – . The best result for the MAPE is 0.73% and is given for three membership functions for , and five membership functions for , , width of variable , of , and of . The fuzzy logic model of selected the combinations of the evaluation set with the smallest . error. The total number of rules is In Fig. 6, three cases from the set of membership functions concerning cold-days are presented. One is corresponding to the and one corresponding basic form, one corresponding to for . to In Fig. 7, the finally used input functions regarding these three variables and the output one, are indicated. In Fig. 8, the analysis of the Larsen Max-Product, DOF and border methods is presented in the case of year 2003 regarding annual energy. rules, which are activated by the model There are inputs presented in the case of year 2003, for the 3 input vari, , ables , ,
, and
, for . The degree of fulfillment for
is
Rule has output membership function for the annual difference of energy with . Rule has with . Rule has with . Rule has with . Rule has with . Rule has with . Rule has with . Finally rule has with . Using the border method the combined form of the membership function is obtained. The output shown in Fig. 8 is a consequence of the activation of the eight rules. In a more general case the shape of the output could have more than two local maxima. The respective MAPE for the three forecasting years is 0.70%. The analytical results are presented in Table I, while the respective actual and estimated values for the training and forecasting years are shown in Fig. 9. C. Comparison of the Proposed Model The results of the developed system are compared to the results of the following seven widely used forecasting models: 1) Simple regression model expressing the annual energy as . a linear function of current year 2) Second order regression model of .
ELIAS AND HATZIARGYRIOU: AN ANNUAL MIDTERM ENERGY FORECASTING MODEL USING FUZZY LOGIC
475
Fig. 8. Example of the application Larsen Max-Product, DOF and border methods is presented in the case of a year regarding annual energy.
TABLE I ANNUAL MIDTERM ENERGY FORECAST OF YEARS 2001–2003 FOR THE GREEK POWER SYSTEM USING FUZZY LOGIC
TABLE II ANNUAL MIDTERM ENERGY FORECAST OF YEARS 2001–203 FOR THE GREEK POWER SYSTEM USING REGRESSION MODELS AND ADAPTIVE ANN MODEL
3) Third 4) Fourth
order order
regression regression
model . model
of of .
Fig. 9. Actual and estimated values of the annual energy for the Greek power system for the training set (years 1987–2000) and the forecasting set (years 2001–2003).
. 5) Exponential regression model . 6) Logarithmic regression model 7) Multiple regression model, where the annual energy is expressed as a linear combination of , , , , , , , and . These nine variables are selected through correlation analysis.
476
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 1, FEBRUARY 2009
Fig. 10. Annual midterm energy forecast for 2001–2003 for six different categories of customers for the Greek power system using the proposed method.
Results are also compared with the optimized adaptive neural network (ANN), where the selection of the finally used input variables is carried out through a correlation analysis and the ANN parameters (number of neurons, initial value of the learning rate, etc.) are conducted, in order to optimize the respective results (for more information see [15]). All models use the data of years 1986–2000 as a training set. The obtained results are presented in Table II, where MAPE refers to all three forecasting years. It is obvious that the regression models are inferior to the proposed model. Only the second order regression model and the ANN model give satisfactory results. The improvement achieved by the proposed system is in the order of 0.02%–0.2% compared to the ANN and the second order regression model and 1.8%–5.5% for the other six models. The main advantage of the proposed system is the calculation of standard deviation, while the ANN model can only estimate it through the standard deviation of the training years [15] (pop) ulation
(20)
Applying (20) for training years (1987–2000), the respective standard deviation is 447 GWh, while for the three forecasting years the respective standard deviation is 470 GWh. The results
are quite similar to the ANN model [15]. The average standard deviation of the fuzzy logic system for training years is 493 GWh and for the three forecasting years is 459 GWh calculated by (15), which means that (15) gives accurate results. V. CASE STUDY II: ANNUAL ENERGY FORECASTING FOR DIFFERENT CATEGORIES OF CUSTOMERS OF THE GREEK POWER SYSTEM The proposed method is also applied for annual energy midterm forecasting of different categories of customers of the Greek Power system. For each category, a new fuzzy logic model is separately constructed and is compared with the adaptive ANN model and the seven regression models. After proper transformation, the input variables are the first 14 variables of the respective annual forecasting model of the Greek . system and the number of customers of each category . In The output is the difference of annual energy demand Table III, the results of the model are shown together with the selected inputs. In the same table these results are compared with the adaptive ANN model and the best regression model. It is observed that in all cases the fuzzy model provides better or at least equivalent results, when compared with the classical models. The proposed model gives slightly better results than the ANN model in four cases. In Fig. 10 and in Table IV, the respective actual and estimated values from the proposed method of the annual energy of the forecasting years for the six categories of customers are presented.
ELIAS AND HATZIARGYRIOU: AN ANNUAL MIDTERM ENERGY FORECASTING MODEL USING FUZZY LOGIC
477
TABLE III COMPARISON OF ANNUAL MIDTERM ENERGY FORECAST METHODS OF YEARS 2001–2003 FOR DIFFERENT CATEGORIES OF CUSTOMERS FOR THE GREEK POWER SYSTEM
TABLE IV ANNUAL MIDTERM ENERGY FORECAST OF YEARS 2001–203 FOR DIFFERENT CATEGORIES OF CUSTOMERS FOR THE GREEK POWER SYSTEM USING THE PROPOSED MIDTERM ENERGY METHOD
VI. CONCLUSIONS The paper presents a new method for midterm energy forecasting based on fuzzy logic. It properly transforms the candidate input variables to differences or relative differences and selects the finally used ones through correlation analysis. The parameters of the fuzzy logic system (number of membership functions and triangle’s base width) are derived by extensive search, in order to optimize results. Basic advantage of the proposed fuzzy logic system is that the training set requires only few samples (only 14 vectors). Results obtained by the implementation of the proposed system in various cases are presented. The annual energy demand for the Greek system and the annual energy consumption of six categories of customers are forecasted by the proposed method and compared to the results of an adaptive ANN model and seven regression models while the respective standard deviation is calculated. In all cases, the fuzzy logic system provides better results than the regression methods. The basic advantage of the fuzzy model compared to ANN is the calculation of the standard deviation based on the characteristics of the membership functions. ACKNOWLEDGMENT This work is dedicated to the memory of Prof. G. C. Contaxis, who supervised it until he suddenly passed away on November 1, 2004. REFERENCES [1] Z. Mohamed and P. Bodger, “Forecasting electricity consumption in New Zealand using economic and demographic variables,” Energy, vol. 30, pp. 1833–1843, 2005. [2] M. Yang and X. Yu, “China’s rural electricity market-a quantitative analysis,” Energy, vol. 29, pp. 961–977, 2004. [3] T. Haida and S. Muto, “Regression based peak load forecasting using a transformation technique,” IEEE Trans. Power Syst., vol. 9, no. 4, pp. 1788–1794, Nov. 1994.
[4] S. Mirasgedis, Y. Safaridis, E. Georgopoulou, D. P. Lalas, M. Moschovits, F. Karagiannis, and D. Papakonstantinou, “Models for mid-term electricity demand forecasting incorporating weather influences,” Energy, vol. 31, pp. 208–227, 2006. [5] H. L. Willis and J. E. D. N.-Green, “Comparison tests of fourteen distribution load forecasting methods,” IEEE Trans. Power App. Syst., vol. 103, pp. 1190–1197, 1984. [6] H. L. Willis, R. W. Powell, and D. L. Wall, “Load transfer coupling regression curve fitting for distribution load forecasting,” IEEE Trans. Power App. Syst., vol. 103, pp. 1070–1076, 1984. [7] X. Da, Y. Jiangyan, and Y. Jilai, “The physical series algorithm of mid-long term load forecasting of power systems,” Elect. Power Syst. Res., vol. 53, pp. 31–37, 2000. [8] H. K. Ozturk, H. Ceylan, O. E. Canyurt, and A. Hepbasli, “Electricity estimation using genetic algorithm approach: A case study of Turkey,” Energy, vol. 30, pp. 1003–1012, 2005. [9] J. L. Small, “Joint estimation of household and heat-pump electricity demand,” IEEE Trans. Power Syst., vol. 9, no. 1, pp. 413–419, Feb. 1994. [10] S. Saab, E. Badr, and G. Nasr, “Univariate modeling and forecasting of energy consumption: The case of electricity in Lebanon,” Energy, vol. 26, pp. 1–14, 2001. [11] T. N. Goh, S. S. Choi, and S. B. Chen, “Forecasting of electricity demand dy end-use characteristics,” Elect. Power Syst. Res., vol. 6, pp. 177–183, 1983. [12] E. H. Barakat, “Modeling nonstationary time-series data. Part I. Data with regular periodic trends,” Elect. Power Energy Syst., vol. 23, pp. 57–62, 2001. [13] E. H. Barakat, “Modeling nonstationary time-series data. Part II. Data with regular periodic trends,” Elect. Power Energy Syst., vol. 23, pp. 63–68, 2001. [14] E. Doveh, P. Feigin, D. Greig, and L. Hyams, “Experience with FNN models for medium term power demand predictions,” IEEE Trans. Power Syst., vol. 17, no. 2, pp. 538–546, May 2002. [15] G. J. Tsekouras, N. D. Hatziargyriou, and E. N. Dialynas, “An optimized adaptive neural network for annual midterm energy forecasting,” IEEE Trans. Power Syst., vol. 21, no. 1, pp. 385–391, Feb. 2006. [16] M. Y. Chow, J. Zhu, and H. Tram, “Application of fuzzy multi-objective decision making in spatial load forecasting,” IEEE Trans. Power Syst., vol. 13, no. 3, pp. 1185–1190, Aug. 1998. [17] G. J. Chen, K. K. Li, T. S. Chung, H. B. Sun, and G. Q. Tang, “Application of an innovative combined forecasting method in power system load forecasting,” Elect. Power Syst. Res., vol. 59, pp. 131–137, 2001. [18] H. T. Lefteris and E. U. Robert, Fuzzy and Neural Approaches in Engineering. New York: Wiley, 1997. [19] N. N. Karnik, J. M. Mendel, and Q. Liang, “Type-2 fuzzy logic systems,” IEEE Trans. Fuzzy Syst., vol. 7, no. 6, pp. 643–658, Dec. 1999.
478
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 1, FEBRUARY 2009
[20] J. R. Aguero and A. Vargas, “Inference of operative configuration of distribution networks using fuzzy logic techniques-part I: Real-time model,” IEEE Trans. Power Syst., vol. 20, no. 3, pp. 1551–1661, Aug. 2005.
Charalambos N. Elias was born in Athens, Greece, in 1973. He received the diploma in electrical and mechanical engineering from the Electrical Engineering Department at Aristotle University, Thessaloniki, Greece, in 1996 and the M.Sc. degree from the National Technical University of Athens (NTUA), Athens, Greece, in 2002. His research interests include power system analysis, load, and energy forecasting methods. Mr. Elias is a member of the Technical Chamber of Greece.
Nikos D. Hatziargyriou (SM’90–F’09) was born in Athens, Greece, in 1954. He received the diploma in electrical and mechanical engineering from the National Technical University of Athens (NTUA), Athens, Greece, in 1976 and the M.Sc. and Ph.D. degrees from the University of Manchester Institute of Science and Technology, Manchester, U.K., in 1979 and 1982, respectively. He is currently a Professor at the School of Electrical and Computer Engineering of the NTUA and executive Vice-Chair of the Public Power Corporation of Greece. His research interests include modeling and digital techniques for power system analysis and control. Dr. Hatziargyriou is a member of CIGRE SCC6 and the Technical Chamber of Greece.
