Midterm Energy Forecasting using Fuzzy Logic: A comparison of confidence interval estimation techniques CH.N.ELIAS 1, G.J.

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1 Midterm Energy Forecasting using Fuzzy Logic: A comparison of confidence interval estimation techniques CH.N.ELIAS, G.J. TSEKOURAS School of Electrical and Computer Engineering, Department of Electrical & Computer Science, National Technical University of Athens Hellenic Naval Academy 9 Heroon Polytechniou Street, Zografou, Athens Terma Hatzikyriakou, Piraeus GREECE xelia@hlk.forthnet.gr, tsekouras_george_@yahoo.gr Abstract: - The modern methods for power system load and energy prediction are usually based on artificial neural networks and fuzzy logic, which present satisfactory results. However, the estimation of the confidence intervals can not be applied directly, unlike to the classical forecasting methods. The obective of this paper is to present an optimized fuzzy logic method for midterm energy forecasting, which can use different techniques for the estimation of the confidence interval, such as the statistical calculation based on the forecasting method errors of the training set, the re-sampling technique and a novel analytical mathematical method based on the membership functions. Finally, the next annual energy demand of Greek interconnected power system is estimated analytically. Simultaneously, the standard deviations through the aforementioned techniques are calculated and compared. Key-Words: - confidence interval, fuzzy logic, re-sampling technique, midterm energy forecasting Introduction In a liberalized electric energy 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 different time periods: very short-term (for the next few minutes, short-term (for the next few hours to a week, midterm (for the next few weeks to few months and long-term (for the next few months to years. 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. Many forecasting models have been implemented on midterm energy forecasting with different level of success. Electric companies have mainly used simple forecasting models, like linear regression [] and econometric models [], while multiple regression models are used for very large systems [3], large metropolitan areas or small areas. Models that forecast the needs of various types of customers are presented using either physical time series or genetic algorithms [4]. Simple autoregressive and autoregressive integrated moving average models have been also used for monthly and annual energy forecast [5]. Annual time series models have been applied with linear, multinomial and exponential approximation [6], while non stationary time series have been modeled for data with different periodic trends (regular [7] and dynamic [8]. Nowadays, application of artificial intelligence techniques has also been applied to midterm energy forecasting using either neural networks [9], or fuzzy logic [0]. All forecasting models lead to a prediction value which is rarely equal to the real one. The variance between the prediction and the real value should be quantified in advance. In regression and other classical statistical models this is expressed by the confidence interval based on analytical calculations. In case of ANNs, the three commonly used methods are: (a the error output, (b the re-sampling, (c the multi-linear regression adapted to ANN []. In [] and [] the theoretical and practical superiority of the re-sampling technique has been proved. In [3] a new adaptive confidence interval method based on the re-sampling technique has been proposed presenting a full version of the respective statistical background and giving satisfactory results. For methods based on fuzzy logic the standard deviation has been calculated analytically working out at the same time the respective problem [0]. In this paper different techniques of confidence interval estimation for fuzzy logic forecasting models are presented and compared. Specifically, an optimized fuzzy logic method for midterm energy forecasting in interconnected Greek power system is ISBN:

2 presented briefly [0]. The optimization process is based on the determination of the number of the triangular membership functions and their base width, the minimal training set and the proper transformation of the input variables. Afterwards, the theoretical determination of the confidence intervals using (i the statistical calculation based on the forecasting method errors of the training set, (ii the re-sampling technique and (iii a novel mathematical model based on fuzzy logic membership functions are analyzed. The three models are applied for the annual midterm energy forecasting for the Greek intercontinental power system and the results are compared. Fuzzy Logic Midterm Energy Forecasting Model Based on the principles of fuzzy logic theory, a model for annual energy demand midterm forecasting was developed. The outline of the procedure to build the fuzzy model is presented in Fig.. In summary, the main steps of the proposed energy forecasting model are the following: (a The N input variables are selected by the user, based on his engineering knowledge from the respective database. The forecasting model is supplied with the following data: annual energy (E, serial current year ( x, annual energy of previous year (x, number of hot-days (x 3 and of cold-days (x 4, gross national product (x 5, statistical indices of oil and coal products (x 6, of manufacturing (x 7, of basic metal (x 8, of manufacture of final metallic products (x 9, of paper and paper products (x 0, of chemical products (x, of food products-beverages (x, of durable consumption goods (x 3 and of non durable consumption goods (x 4 and number of customers (x 5. (b If necessary, the input variables are transformed to their differences or their relative differences. Specifically, 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 midterm or long-term 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 future values, as it has already explained in [0], section III.A. The -th transformed input variable is symbolized by p. (c The input variables are reduced from N to n through correlation analysis. This happens, because the examination of all possible combinations of the N input variables would require the execution of Ν times of the basic form of the fuzzy model. Since the initial preprocessing leads to an average of 5 variables, 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 the transformed input variables p and output E is greater than a pre-specified value cor, the p 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 cor then both terms are retained; else, only the term with the largest correlation with respect to output y is retained. In this way the input variables decrease from N to n and the combinations are also decreased. (d The combinations of input variables are determined. (e For each input variable the number of membership functions and triangle s base width are determined by combinations. During fuzzification process of this model the triangular membership functions are applied. The odd number of membership functions t 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 the middle triangle c of a variable p and the initial value of the base width b I of each triangle are given by the following expressions: Y c = p Y ( k= k ( (,..., k,..., Y k= Y = b = max p min p t ( I k k where t is the number of membership functions of variable p. Next, the base width of the triangle is modified by ± a % with step s%, while the center of the middle triangle remains constant. Thus, the number of possible triangles h, to be examined per variable, is h= a s +. Therefore, for n input n variables, the total number of combinations is h. (f Fuzzified values of each final form of variables are calculated. (g The rules concerning the years, for which the parameter values are included in the database, are formed. ISBN:

3 Transformation of the input variables Main Procedure Yes Deduction mechanism No Data selection Limitation from N to n input variables through correlations Combinations ( for the optimization of the input variables selection End of combinations (? Fuzzification Final prediction & Standard deviation Fig.. Flowchart of the optimized fuzzy logic method for midterm annual energy forecasting No Yes Combinations ( for the optimization of the number of the membership functions & triangle s base width End of combinations (? No Defuzzification Model s valuation Rule base (h 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. (i Using the rule base, the deduction mechanism based on the generalized modus ponens inference engine, the Larsen-Max product implication with the degree of fulfillment and the border method [4], and the center of the area criterion (COA defuzzification method, a forecast is made concerning the years of the evaluation set. ( Steps (d-(i are repeated for every combination. (k The combination that produces the minimum mean absolute percentage error (MAPE for the evaluation set is selected and the fuzzy model is realized. It is noted that the MAPE index between the measured and the estimated values of annual energy demand for the evaluation set is given by: m ˆ ev E( i E( i MAPEev = 00% m (3 E i ev d= where E(i is the measured value of energy demand for the i-th year of the evaluation set, Ê (i the respective estimated value, m ev the population of the evaluation set. (l 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. (m Finally, the expected amount of energy during the forecasting year is estimated, as in step (i. The standard deviation is calculated. The energy needed during each year of the prediction derives from the difference between the forecasted amount concerning this year of the forecast and the amount concerning its previous year. Practically, for each of the possible combinations of the n input variables, the fuzzy value of each input, which corresponds to the data of each training year, is determined. 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. Afterwards, rules are created for each year, whose number varies from 0 to n. This process is repeated in order to check all possible combinations and the combination with minimum MAPE in the prediction of the evaluation set is selected. This combination is going to be used for midterm energy forecast. The prediction is made in exactly the same way it was done for the evaluation set except that the combination of base widths and input data is given. 3 Confidence Interval Estimation The confidence interval can be calculated through the following methods: (i the statistical calculation based on the forecasting method errors of the training set, (ii the re-sampling technique and (iii a novel mathematical model based on fuzzy logic membership functions. Following, each technique is analyzed. ( ISBN:

4 3. Statistical calculation based on the forecasting method errors of training set The errors e t of the training set (for each t-th year during the application of the optimized forecasting method are calculated. Based on the definition of the standard deviation of an independent statistical the respective standard deviation is calculated by: N train et ( Ntrain (4 σ = t= Where N train is the population of the training set. Assuming that the variable of the error follows the normal distribution probability the respective confidence interval can be estimated as [ m -b σ, m + b σ ], where m is the predicted mean value and b is a multiplier parameter, which is determined by the desirable certainty degree / propability (for 99.7% probability parameter b should be equal to Re-sampling technique The estimation of the confidence intervals for nonclassical models is not applied directly. Three techniques have been mentioned for ANN models in []: (a the output error, (b the re-sampling, (c the multi-linear regression adapted to ANN. From these only the second one can be applied on fuzzy logic models. In order to estimate the confidence interval using the re-sampling technique, the prediction and the respective error are calculated for each set and for all available m input vectors. These errors are sorted in ascending order considering the signs and the cumulative sample distribution function of the prediction errors can be estimated by: 0, z< z Sm( z = r m, zr z< zr, zm z + (5 When m is large enough, S m (z is a good approximation of the true cumulative probability distribution F(z. The confidence interval is estimated by keeping the intermediate z r and discarding the extreme values, according to the desired confidence degree. The intervals are computed in order to be symmetrical in probability (not necessarily symmetric in z. The number of cases to discard in each tail of the prediction error distribution is n p, where p is the probability in each tail. If n p is a fractional number, the number of cases to discard in each tail is n p for safety reasons. If the cumulative probability distribution F(Z p equals to p, then there is a probability p that an error is less than or equal to Z p, which indicates that Z p is the lower confidence limit. Consequently, Z -p is the upper limit and there is a (-p confidence interval for future errors. 3.3 Mathematical model based on fuzzy logic membership functions Initially, the standard deviation for only one triangular membership function is analytically found. As shown in Fig., the mean value and the standard deviation of a triangular membership function are: x λ µ µ m= = = µ λ µ 3 ( x ( µ 6 (6 µ λ µ σ = = = (7 λ µ 6 where f(x=m(x and the respective integrals are given by: = λ µ = λ µ m(x λ x = λ µ µ ( x µ f x = λ ( µ 3 ( 6 Fig.. The formation of a typical triangular membership function Using mechanics theory 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 m(x of the ordinates. The respective expressions are: x m ( ( x xf x µ µ µ = = m ( x µ- µ µ µ+ µ µ = m ( µ µ m ( µ x (8 (9 ISBN:

5 σ total = σ total σ total = ( x µ = ( µ x m ( x m ( x m ( µ σ + m ( µ ( µ µ µ m ( µ because the following integral is equal to: µ + µ µ + µ (0 ( ( ( µ ( ( µ ( ( µ m µ ( + ( µ µ ( µ x m x = x m x + µ µ + x + x m ( x It is noted that equations (8 and (0 are the general expressions of the mean value and the variance (the square of the standard deviation. In the case of two triangular membership functions m (x and m (x the respective mean value and standard deviation are: µ λ + µ λ µ = (3 λ + λ λ σ + λ σ + λ ( µ µ + λ ( µ µ σ = (4 λ+ λ The respective total membership function f(x is presented in Fig. 3. m(x λ λ m(x λ λ m (x µ µ µ f(x µ µ µ m (x Fig. 3. The case of two triangular membership functions 0 x x 4 Application of the Confidence Interval Estimation Techniques for Midterm Energy Fuzzy Logic Forecasting Method in Interconnected Greek Power System In [0] the application of the optimized fuzzy logic midterm energy forecasting is described analytically. The serial current year and the number of hot-days and cold-days are not transformed. The gross national product is transformed to relative difference and the remaining variables are transformed to differences, as discussed in section (b. Then, the correlation indices between x, r or d and the difference of energy d E are estimated. Indices cor =0. and cor =0.9 result in the following input variables: the relative difference of the previous annual energy (r, the number of hot-days (x 3 and of cold-days (x 4, and the difference of paper and paper products (d 0. All possible combinations are examined with a=0%, step s=8%, t=3, 5 or 7, using as training years, the years The evaluation set is the same with the training set. The final model includes variables x 3, x 4 and d 0. The best result for the MAPE of the evaluation set is 0.73% and is given for 3 membership functions for x 3, x 4 and 5 membership functions for d 0, d E, width of variable x 3 =600, of x 4 =40, of d 0 =30.8 and of d E =370. The respective MAPE for the three forecasting years is 0.70%. The results are analytically presented in Table. Using the statistical calculation technique based on the forecasting method errors of training set and applying eq. (4 for training years ( , the respective standard deviation is 447 GWh, while for the three forecasting years the respective standard deviation is 470 GWh. Assuming that the error follows the normal distribution probability, the respective confidence interval is estimated in Table for 99.7% probability. Using the re-sampling technique for 4 members of training set and for p=0.5% the number of cases to discard in each tail of the prediction error distribution is practically none. So the lower confidence limit is -.80 TWh and the upper confidence limit is 0.63 TWh. The respective confidence interval for each forecasting year is estimated in Table for 99.7% probability. Using the mathematical model based on fuzzy logic membership functions the standard deviation of each forecasting year is calculated and the respective confidence interval are estimated in Table for 99.7% probability assuming that the ISBN:

6 probability error distribution is normal. 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 eq. (0. TABLE ANNUAL MIDTERM ENERGY FORECASTING IN GREEK INTERCONNECTED POWER SYSTEM USING OPTIMIZED FUZZY LOGIC METHODOLOGY Year Real value TWh Forecasting value TWh Difference TWh Error (% TABLE CONFIDENCE INTERVAL FOR ANNUAL MIDTERM ENERGY FORECASTING IN GREEK INTER- CONNECTED POWER SYSTEM USING TECHNIQUES: ( STATISTICAL, ( RE-SAMPLING, (3 MATHE- MATICAL BASED ON FUZZY LOGIC FUNCTIONS Method Limits (TWh ( Lower Upper ( Lower Upper (3 Standard deviation Lower Upper The confidence intervals of the three techniques of Table for 99.7% probability are also presented in Fig. 4. It is observed that for all forecasting years the real values are inside the respective confidence intervals of all techniques. The lower limits of the confidence intervals for years 00 and 003 are quite similar for all techniques. For year 00 the lower limit of mathematical technique is significantly smaller than both others. The upper limits are quite similar for the st and 3 rd techniques, while the re-sampling technique gives smaller limits. Annual energy (TWh Year Lower limit of ( Lower limit of ( Lower limit of (3 Real value Upper limit of ( Upper limit of ( Upper limit of (3 Fig. 4. Confidence intervals of the following techniques: ( statistical calculation based on the forecasting method errors of training set, ( re-sampling method, (3 mathematical model based on fuzzy logic membership functions for annual midterm energy forecasting in Greek interconnected power system using optimized fuzzy logic methodology The statistical calculation based on the forecasting method errors of training set depends on the errors of the historical data. It is a simple method, which is independent of the fuzzy logic method. The re-sampling technique has the same advantage, but it is more general than the statistical one, because the assumption of the normal distribution of the errors is not necessary. The computational needs are bigger, because the errors of historical data are sorted in ascending order. In this case the population of the evaluation set is not ISBN:

7 large enough, so no conclusion is safe about the technique s performance. The mathematical model based on the membership functions is a special technique for fuzzy logic model. Its main disadvantage is that the assumption of the normal distribution of the errors is necessary. While the main advantages are: ( the simple direct calculation of the standard deviation by eq. (0, ( the dependence of the standard deviation from the fuzzy logic membership functions and the current predicted value and not only by historical data. The second advantage gives a more generalized behavior of the standard deviation than previous techniques. 6 Conclusions This paper presents and compares three different techniques for the confidence interval estimation in case of the midterm energy forecasting based on an optimized fuzzy logic methodology. These techniques are: (i the statistical calculation based on the forecasting method errors of the training set, (ii the re-sampling technique and (iii a novel mathematical model based on fuzzy logic membership functions. From their theoretical comparison the mathematical model based on fuzzy logic is superior to others because of its simplicity and its dependence from the current fuzzy logic values and not only the historical data. The resampling technique has the unique advantage of the direct estimation of the confidence interval without the assumption of the normal distribution of the errors. From their practical comparison the respective results are quite satisfactory, but the test set is of a few members and in future it should be investigated in problems with extended sets, so that more accurate conclusions could be extracted. References: [] Z. Mohamed, P. Bodger. Forecasting electricity consumption in New Zealand using economic and demographic variables. Energy, vol. 30, 005, [] M. Yang, X. Yu. China s rural electricity market-a quantitative analysis. Energy, vol. 9, 004, [3] S. Mirasgedis, Y. Safaridis, E. Georgopoulou, D.P.Lalas, M. Moschovits, F. Karagiannis, D. Papakonstantinou. Models for mid-term electricity demand forecasting incorporating weather influences. Energy, vol. 3, 006, pp [4] H. K. Ozturk, H. Ceylan, O. E. Canyurt, A. Hepbasli. Electricity estimation using genetic algorithm approach: a case study of Turkey. Energy, vol. 30, 005, pp [5] S. Saab, E. Badr, G. Nasr. Univariate modeling and forecasting of energy consumption: the case of electricity in Lebanon. Energy 6, 00, -4. [6] X. Da, Y. Jiangyan, Y. Jilai. The physical series algorithm of mid-long term load forecasting of power systems. Electrical Power Systems Research, vol. 53, 000, pp [7] E.H. Barakat. Modeling nonstationary timeseries data. Part I. Data with regular periodic trends. Electr. Power Energy Syst.3, 00, pp [8] E.H. Barakat. Modeling nonstationary timeseries data. Part II. Data with regular periodic trends. Electr. Power Energy Syst.3 00, pp [9] G.J. Tsekouras, N.D. Hatziargyriou, E.N. Dialynas. An optimized adaptive neural network for annual midterm energy forecasting. IEEE Trans. Power Syst., vol., no., 006, pp [0] Ch. N. Elias, N. D. Hatziargyriou,. An annual midterm energy forecasting model using fuzzy logic. IEEE Trans. on Power Systems, vol. 4, no., January 009, pp [] A.P.A. Silva, L.S. Moulin. Confidence intervals for neural network based short-term load forecasting. IEEE Trans. on Power Systems, vol. 5, no. 4, November 000, pp [] B. Petiau. Confidence interval estimation for short-term load forecasting. 009 IEEE Bucharest Power Tech Conference, June 8- July, Bucharest, Romania. [3] G. J. Tsekouras, N.E. Mastorakis, F.D. Kanellos, V.T. Kontargyri, C.D. Tsirekis, I.S. Karanasiou, Ch.N. Elias, A.D. Salis, P.A. Contaxis, A.A. Gialketsi. Short Term Load Forecasting in Interconnected Greek Power System using ANN: Confidence Interval Estimation using a Novel Re-sampling Technique with Corrective Factor. WSEAS International Conference on Circuits, Systems, Electronics, Control & Signal Processing, (CSECS '0, Vouliagmeni, Athens, Greece, December 9-3, 00. [4] Lefteris H. Tsoukalas, Robert E. Uhrig, Fuzzy and Neural Approaches in Engineering, John Wiley & Sons, New York, 997. ISBN: