Fuzzy neural network and fuzzy expert system for load forecasting - Generation, Transmission and Distribution, IEE Proceedings- k and fuzz P.K. Dash A.C. Liew S.Rahman Indexing terms: Neural networks, Load forecasting, Expert system, Power system planning Abstract: A hybrid neural network fuzzy expert system is developed to forecast short-term electric load accurately. The fuzzy membership values of the load and other weather variables are the inputs to the neural network, and the output comprises the membership values of the predicted load. An adaptive fuzzy correction scheme is used to forecast the final load by using a fuzzy rule base and fuzzy inference mechanism. Extensive studies have been performed for all seasons, and a few examples are presented in the paper, including average, peak and hourly load forecasts. 1 introduction The short-term load forecast is very important to an electric utility. The quality of control of a power sys- tem, and economy of operation, are highly sensitive to forecasting error. A sound basis for load predictions is generalisation of past, known cases. The science of sta- tistics provides a range of tools for this purpose. They are based on the idea of fitting a particular class of models to data and then hypothesising that future events will conform to the fitted model. Many approaches have been applied to electric load forecast- ing, including linear regression, exponential smoothing, stochastic process and state space methods. While each of these methods demonstrates success in forecasting, they have serious disadvantages: reliance on large his- torical databases with possible obsolete and irrelevant data, assumptions about static load shapes and param- eters etc. A detailed comparison of various statistical approaches is found in [l]. One of the most promising application areas of the artificial neural network (ANN) is load forecasting [2- 131. The neural network is able to perform nonlinear modelling and adaptation and does not rely on the explicitly expressed relationship between input varia- 0 IEE, 1996 ZEE Proceedings online no. 19960314 Paper received 26th October 1994 P.K. Dash is with the Department of Electrical Engineering, Regional Engineering College, Rourkela, India A.C. Liew is with the Department of Electrical Engineering, National University of Singapore, Singapore S. Rahman is with the Bradley Department of Electrical Engineering, Vir- ginia Polytechnic Institute & State University, USA 106 bles and forecast load. When using neural networks for load forecasting, one needs to consider only the selec- tion of variables as the network input. The relationship between the input variables and predicted load will be formulated by a training process. Several authors have attempted to apply the backpropagation learning algo- rithm to train the ANNs for forecasting time series. The fuzzy expert system approach [14] has also been applied to forecasting where the advantage of an oper- ator’s expert knowledge is used. However, the fuzzy decision system for load forecasting requires detailed analysis of data and the fuzzy rule base has to be devel- oped heuristically for each season. The rules fixed in this way may not always yield the best forecast. On the other hand, hybrid solutions [15, 161 have been pro- posed for short-term forecasting of electric loads, whereby the functionality of the fuzzy expert system and the learning capabilities of the neural network can be merged to yield a forecasting system more powerful than either of its components alone. The present work is aimed at achieving the said objective of a robust load forecast with improved accu- racy using a fuzzy neural network for initial forecast and a fuzzy expert system (FES) producing load cor- rections to yield the final forecast. For the neural net- work to be called a FNN, the signal and/or the weights should be fuzzified. This type of FNN is based on the multilayer perceptron, using the backpropagation algo- rithm. The fuzzified input vector consists of the mem- bership values of the past load and weather parameters and the output vector is defined in terms of fuzzy class- membership values of the forecast load. A simple fuzzy-inferencing mechanism is used to yield the magni- tude of the forecast load during the initial phase. In the final phase a fuzzy expert system is used to produce load corrections. The input vector to the fuzzy expert system (FES) consists of differences in the weather parameters between the present and the forecasted instant. The output of the FES gives the load correction which, when added to the initial forecast, yields the final fore- cast. Thus, by using a hybrid approach the load-fore- casting errors are expected to reduce considerably. However, as the lead time increases from 1 h to 24h or 48 h, the forecasting error increases, necessitating the use of an adaptive fuzzy load error correction scheme. Several examples presented in this paper include 24h- ahead average, peak and hourly load forecasts using the hybrid approach. The effects of both linear and nonlinear adaptive load-correction schemes are shown. IEE Proc -Genu Transm Distrcb , Vol 143, No 1, January 1996 2 form Fuzzy pattern representation in linguistic The approach used in this paper is aimed at improving the prediction by handling uncertain and ambiguous variations of load patterns and weather parameters using a fuzzy linguistic approach. Since it is easier to convert exact information into linguistic form than vice versa, we consider the major linguistic properties small, medium, and large as the attributes of the input fea- ture. Any input feature value can be described in terms of some combination of membership values for these properties. In traditional two-state classifiers, an ele- ment x either belongs or does not belong to a given class A ; thus, the characteristic function is expressed as 1 i f a : E A 0 otherwise P a ( X ) = In real-life problems such as load forecasting, the classes are often i l l defined, overlapping, or fuzzy and a pattern point may belong to more than one class; in such situation, fuzzy set theoretic techniques can be very useful. We use the modified n-function [17, 181, lying in the range [0,1] to assign membership values for the input features corresponding to the linguistic properties small, medium, and large. The membership function of the input feature otherwise where hi > 0 is the radius of the n-function with ci as the central point at which p ( x i ) = 1. This is shown in Fig. 1. Xirnin Csmoh ‘medium ‘Iorgeximm - ’medium -4 I---- ~~%ar+-- ’ / 2 L o d Fig. 1 n-function representation 2. I Let x,,,, and x,,,, denote the upper and lower bounds of the observed range of feature x, in all L pattern points, considering numeric values only. Then, for the three linguistic property sets, the following are used: Choice of parameters for the .n-function L d t U m ( ~ ) = 0.5(& m a z - G mtn) c m e d z u m ( 2 , ) xz mzn + X m e c i z u m ( X z ) ( 3 ) Xsma11(&) = ( l / f d ) { C m e c l z u m ( & ) - & m z n } csmall ( 2 % ) = cmedzzlm(2t) - O . 5 X S m a 1 ~ (zZ) IEE Pvoc -Gener Trunsm Distvib Vol 143, No 1, January 1996 ;\large (xz) = (l/fci) { 5% m a x - C m e c i z u m ( x z ) } C i a r g e ( G ) = C m e d t u m ( 2 z ) + 0 . 5 h a r g e ( X z ) where 0.5 5 f d 5 1.0 is a parameter controlling the extent of overlapping. The n-function representation permits a more compact and meaningful representation of each pattern point in terms of its linguistic proper- ties, and ensures better handling during both the train- ing and testing phases of the proposed fuzzy neural network model. 3 The present work attempts to build a fuzzy neural net- work model based on the multilayer perceptron using the gradient descent based backpropagation algorithm by incorporating concepts from fuzzy sets at various stages. Fig. 2 shows the fuzzy neural network model for obtaining the initial forecast. The fuzzy sets for load, temperature and humidity parameters are shown in Figs. 3-5. The input to the fuzzy neural network comprises the membership values to the overlapping partitions of linguistic properties small, medium, and large corresponding to each input feature such as past load, temperature, humidity etc. This provides scope for incorporating linguistic information in both the training and testing phases of the said model and increases robustness in tackling imprecise or uncertain input specifications. The components of the output layer consist of the membership values to the overlap- ping partitions of linguistic properties small, medium and large corresponding to the forecast load magni- tude. During training, the network backpropagates the errors with respect to the desired membership values at the output nodes. After a number of cycles, the neural network converges to a minimum error solution by using a gradient descent algorithm as shown in the Appendix. Fuzzy neural network for initial forecast h I th layer j th layer k t h layer I th layer Fig. 2 Fuzzy neural network for initia1,forecast 7 1 .o 0.8 0.6 TJ e Q ._ L 2 0.4 E E 0.2 0 0.116 0.223 0.417 0.611 0.800 1.0 load( normalised) Fig.3 ~ small -0- medium -x- large Fuzzy membership grade for load 107 temperature (normalised) Fig. 4 ~ small -0- medium -x- large Fuzzy membership grade for temperature humidity (normalised) Fig. 5 ~ small -0- medium -x- large Fuzzy membership grade for humidity After the learning phase is over, when separate load and weather patterns are presented at the input layer, the output nodes automatically generate the member- ship values corresponding to the linguistic properties small, medium and large. Thus a centroid defuzzifica- tion technique [I91 is used to obtain the initial load forecast from the membership values and the corre- sponding loads obtained from the n-functions. 4 Selection of training patterns The utility data studied here are susceptible to large and sudden changes in weather and load, so selection of appropriate training cases plays a vital role in train- ing the network. Several techniques for the selection of training patterns have been suggested in [l l-131. The present paper discusses a different training scheme for the selection of training patterns for hourly load fore- cast. To predict hourly loads the following load model is chosen: y ( i , t ) = f { y ( i - m ) , y ( i , t - m ~ I), . . . , y ( i , t - m - n l ) , (4) and where y and z are the load and weather variables, respectively; i and t indicate the day and the hour, respectively, m indicates the lead time for the hourly load forecast, (i.e. m = 1 for 1 h-ahead forecast, m = 24 for 24h-ahead forecast, m = 48 for 48h-ahead forecast, m = 168 for 168h-ahead forecast); nl indicates the data z(2, t - m ) , z ( Z , t ) , . . . , z(2 - t - m ) } m = n2 108 length for load; and n2 indicates the data length for temperature. For hourly load forecasting, eqn. 4 is used to select the training patterns. Various lengths of the past his- torical load and temperature values are used and their effects on the load-forecast accuracy are studied. It is found that, with nl > 0 and n2 > 0, there are no marked improvements in the results for the utility data used in this paper. Also the training time increases con- siderably with larger values of nl and n2. Therefore, n1 = 0 and n2 = 0 are chosen in this paper. Using the above scheme, the network is trained for 14 days (i = 1, ..., 14) at time t and load is predicated for the next 14 days (i = 15, ..., 28) at time t. Hence to predict for all 24h of a given day, 24 different neural networks are used, each one trained separately with the same parameters. This is desirable because the training set is small for each neural network consisting of a few patterns (14 patterns only in this case) with irrelevant data for other hours being discarded. Further, depend- ing upon the difference of the load responses on the day of the week, a day-of-the-week indicator is intro- duced along with the input vector to the network. As the weather variable temperature is the most important parameter in short-term load predictions, an all-temperature model is used to obtain the hourly fore- casts. The training data used for 1 h-ahead predictions are: Input pattern: P(i,t) = power at tth instant of ith day, e(i,t) = temperature at (t+l)th instant of ith day, @(i,t+ 1) = temperature at (t+ 1)th instant of ith day wd(i) = day of the week indicator i. Output pattern: P(i,t+l) = power at (t+l)th instant of ith day. The training data for 24h-ahead, 48h-ahead and 168h- ahead predications are Input patterns: P(i,t) = power at tth instant of ith day, O(i,t) = temperature at tth instant of ith day, B(i+m,t) = temperature at tth instant of (i+m)th day. Output pattern: P(i+m,t) = power at tth instant of (i+m)th day where m = 1, 2, 7 for 24h-, 48h- and 168h-ahead pre- dictions, respectively. However, for average daily and peak-load predic- tions, the following training data are used: Input and output patterns: P(i-1) = average load on (i-1)th day, Qmln(i-I) = minimum temperature of (i-1)th day, OmaX(i-l) = maximum temperature of (i-1)th day, O,,(i) = minimum temperature of ith day, Omax (i) = maximum temperature of ith day, P(i) = average load for ith day. The same data can be used for the peak load forecast. Although an all-temperature model will produce an accurate forecast in most seasons, the other weather parameters like humidity and wind speed affect the forecasting accuracy during summer and winter, respec- tively. Thus, if humidity records are available in a par- ticular season, they may be included in a training pattern. IEE Proc -Gener Transm Distrib Vol 143, No I , January I996 5 Fuzzy expert system for final forecast During training, the neural net produces an initial fore- cast with a set of initial load, temperature and humid- ity data expressed in terms of fuzzy membership values. The error between the actual load A ( i ) and predicted load P(i) for a given hour or a given day (ith hour or day) is expressed as In a similar way, temperature and humidity errors are expressed as aqi) = ~ ( i ) - ~ ( i ) AO(2) = O ( 2 ) - O ( 2 - 1) AH(2) = H ( 2 ) - H ( 2 - 1) ( 5 ) (6) However, if maximum and minimum temperatures are used, ( 7 ) A Q m a z (2) = Q m a z ( 2 ) - Q m a z (2 - 1) ~ Q m z n ( 2 ) = Q m z n ( 2 ) - Qmzn(i - 1) The errors in the weather parameters and load-correc- tion values are fuzzified using six fuzzy sets such as SP (small positive), MP (medium positive), LP (large posi- tive), SN (small negative), NM (medium negative) and LN (large negative). Figs. 6 and 7 show the fuzzy sets for temperature and humidity errors. These sets are obtained using the n-function given in eqn. 2. Both positive and negative fuzzy sets are symmetrical about the origin. -0.06 -0.03 0 0.03 0.06 error (normalised) Fig. 6 ~ small -0- medium -x- large Fuzzy membership grade for temperature errors -0.12 -0.06 -0.04 0 0.04 0.06 0.12 error (normalised) Fig. 7 ~ small -0- medium -x- large Fuzzy membership grade for humidity errors The load correction output sets have six members and use a linear fuzzification principle for obtaining membership grades as shown in Figs. 8 and 9. Nonlinear membership grades can also be used for obtaining load corrections for the final forecast. The sets for load corrections are classified as SPC (small positive correction), MPC (medium positive correc- tion), LPC (large positive correction) etc. The corre- sponding negative fuzzy sets are SNC, MNC, and LNC, respectively. 1 . O R t I f -1.5 -1.2 -0.6 0 0.6 1.2 1.5 correction (normalised) Fig. 8 ~ small -0- medium -x- large Membership grades for load correction (linear) 0 0.4 0.8 1.2 1.6 2.0 correction (normalised) Membership grades for load correction (linear) Fig. 9 The membership values of the load correction APc output is given by b m a z where C,,, is the slope of the load error correction and eLc is the maximum load error correction for the corre- sponding linguistic set (for which the membership value becomes unity). The value of C,,, for different output linguistic sets SPC, MPC and LPC is C L a x , C$,, CL,, respectively, and these values are obtained by observing the load prediction errors over a two-week period prior to forecasting. The fuzzy rule base is formed by trial and error to reduce the load correction to a very small value during training. However, a fuzzy basis function approach [19] can be used to select the appropriate rules automati- cally out of a large number of possible combinations. Two sample rules for an all-temperature model for average load forecast will be Rule 1: I F AOmaX(i) is LN and AO,,(i) is SP, THEN AP,(i) is MPC Rule 2: I F AOmaX(i) is MP and AOmin(i) is LP, THEN AP,(i) is LPC In a similar way the two-sample rule for the peak- load forecast are Rule 1: IF AOmaX(i) is SP THEN AP,(i) is SPC Rule 2: I F AO,,,(i) is LN THEN AP,(i) is MNC. IEE Proc-Gener. Transm. Distrib., Vol. 143, No. 1, January 1996 109 However, if a load forecasting model with both temper- ature and humidity parameters is used, the rules are of the form Rule 1: I F AQ(i) is SP and H ( i ) is MP THEN AP,(i) is MPC Rule 2: I F AQ(i) is SN and AH(i) is LN THEN AP,(i) is LNC. The total number of production rules in the fuzzy knowledge base using two variables and six categories of sets is 36. Because of partial matching of the fuzzy rules and the fact that preconditions do overlap, more than one fuzzy control rule can fire at a time. For the fuzzy rules used for load forecasting, the truth values of the pre- conditions are (considering a temperature-humidity model) i = 1, 2, ..., k and k = number of rules fired for a given value of AO(i), and AH(i) belonging to fuzzy classifiers SP, MP, LP etc. and A denotes a conjunction operator (usually a minimum operator). The output of rule (i) is calculated by applying the matching strength of its pre- condition on its conclusion as j = 1, 2, ..., 6 and cALax is the value of C,,, for the out- put set Ay If two rules have the same consequent out- put set, the Lukasiewickz OR rule is used as ai = A [ P { A @ ( i ) ) , Pu(AH(411 (9) A P c ( i ) = a,c2a, (10) a; = min[I,p{AO(i)} + p { A H ( i ) } ] (11) A centroid defuzzification technique is used to yield the load correction as A P c ( i ) = aiAPc/ a, = a:C2u,l ai (12) A P c ( i ) = a:c2,z/ a2 (13) However, by taking the load correction as CAAax, for which the output membership function is unity, the value of AP,(i) is obtained as The final value of forecast load is thus obtained by summing the ANN output, and the output from the fuzzy expert system is The block diagram for the integrated fuzzy neural net- work and fuzzy expert system is shown in Fig. 10. p(t-1)- w - 1 ) - defuzzification e o ) -- I Pf(2) = P(2) + APc(i) (14) fuzzyfication i - ANN defuzzification \ - - M t ) -+- fuzzy rule base and fuzzy inference Fig. 10 Integrated fuzzy neural network-fuzzy expert system 5. I Adaptive load correction For small load correction eqns. 10 and 11 are adequate to produce accurate forecasts (usually for a lead time from 1 - 6h). However, as the lead time increases to 110 24, 48, 72 or 168h, the membership function for the load-error correction is adapted as 1 p{apc(z)~ = c,,, f ac,,, apc(~) (15) the value of AC,, is obtained as a function of load- correction errors from the training cases using different lead times for predictions. Fig. 11 shows the linear adaptive correction ACmax as a function of the normal- ised error. The nonlinear adaptive version of the load- error correction is shown in Fig. 12. For adaptive cor- rections, eqns. 12 and 13 are rewritten as and Some of the results are given below for the various models considered in this paper. 80 I increment / decrement \ I -801 I I I I I I I I 0 0.002 0.004 0.006 0.008 ,010 ,012 014 maximum error (normalised) Fig. 11 60 I Linear adaptive load correction decrement -20 -40 increment I 1 I I I 1 I I 0001 0002 0 003 0 004 0005 maximum error (normalised) Fig. 12 Nonlinear adaptive load correction 6 Forecasting results 6. I Average load forecasting To evaluate the fuzzy ANN and fuzzy expert system approach, load forecasting is performed on the load data collected at the Virginia Polytechnic Institute and State University. The fuzzy neural network (FNN) is compared with the combined FNN and fuzzy expert system model using ordinary backpropagation algo- rithm for obtaining one-day-ahead average load fore- casting during a 14 day period in the month of May. The input layer of the FNN comprises 15 neurons for five input features and the output layer has three neu- rons. The number of neurons in the hidden layer is fixed as 17 for this particular forecast, to obtain the best results. If the day-of-the-week indicator is used, one more neuron is added to the input layer. Back- propagated errors are assigned appropriate weightage for weight updating depending on the membership val- ues at corresponding outputs. The learning rate r\ and IEE Pvoc -Genev Tvansm Distrib Vol 143, No I , Januavy 1996 momentum coefficient a are gradually decreased to prevent oscillations as the neural network converges to a minimum-error solution in a maximal number of sweeps through the training set. Note that the parameters q and a traverse a range of values in the course of computations and one may choose 0.6 < a < 1.0, and 0.0001 < q < 0.1. The values of p and y are chosen for best performance as 0 < p < 0.02 0.2 < y < 0.6 For the present study the initial learning rate q and momentum a are and The percentage error PE is evaluated as 7 = 0.01, a = 0.8 /!? = -0.015, 7 = 0.4 forecast load - actual load PE = x 100 forecast load and percentage absolute error PAE = lPEl The average load profile for the 14 day period in May is shown in Fig. 13. Fig. 14 presents the PAE for 24h period ahead forecast. The maximum PAE for a 24h-ahead forecast during a 14 day period in May is 1.75 using the FNN and fuzzy expert system in com- parison with 3.40 for the FNN only. Weekdays, week- ends and Sundays are all included in this model. 28 3 2 7 - - 2 2 6 - E- 25 24 29 t - - - t 30 t "I 6 W Q 4 4 2 0 1 6 11 16 21 26 3' 2311 I I 1 I I I I 8 I I I . ( 1 1 6 11 days Fig. 13 Average loudprojile forecast for the month of May c I davs Fig. 16 month of Mav Percentage absolute error in peak load profile forecast for the ~ ~ N N bnly -+- FNN with fuzzy corrections 1 6 11 d a y s Fig. 14 month of May ~ FNN only -e- FNN with fuzzy corrections I E E Proc-Gener Transm. Distrib., Vol. 143, No. 1. January 1996 Percentage absolute error in average load profile forecast for the 6.2 Peak-load forecast Figs. 15-18 show the actual peak loads and PAEs for a 31 day period during May and December. The maxi- mum temperature errors are used for fuzzy corrections. The combined model produces a maximum PAE of about 1.65 for a 24h-ahead peak-load forecast during the month of December. During May, the maximum PAE is 7.5 for a 24h-ahead peak-load forecast using the FNN model. However, using fuzzy correction, the maximum PAE is reduced to 5.5. Both the forecasts shown in these Figures include weekdays, and week- ends etc. Special holidays like Christmas etc. are included in obtaining the peak-load forecast for the month of December. These errors can be reduced fur- ther by using adaptive load-correction schemes. 6.3 Hourly load forecasts The data from a utility in Virginia is used to produce 24h- and 48h-ahead hourly forecasts. For this utility, both temperature and humidity records are available during all seasons of the year. However, the forecasting errors for 21 and 22 January over a 24h period are shown in Figs. 19-22. The maximum PAEs for 24h- and 48h-ahead forecasts are 2.8 and 3.5 without fuzzy corrections and 0.69 and 1.72 with fuzzy corrections, respectively. The corresponding load profiles are also shown in these Figures. 111 40c 20 1 6 11 16 21 26 31 days Peak load profile forecast for the month of December Fig. 17 4 3 W 2 2 1 6 11 16 21 26 31 0 1 days Fig. 18 month of December ____ FNN only -+- FNN with fuzzy corrections Percentage absolute error in peak load profile jorecasl f o r the 9500 - 9000 - 8500 - 8000- 7500- 7000 - 6500 - *- 0 - 6000 5500 i 1 1 1 1 1 1 * 1 1 I I 1 6 11 16 21 time,h \ Fig. 19 records for utility data 24h ahead hourly load forecasts with temperature and humidity 6.4 Hourly load forecasts using adaptive correction Figs. 23 and 24 show the 24h ahead forecast errors for a summer day (27 May) using an all-temperature model and the data from the experimental setup at the Virginia Polytechnic. Both linear and nonlinear adap- tive fuzzy corrections are used to provide a more accu- rate forecast. From the Figure it is found that the percentage absolute-load-prediction error over the entire 24h period comes down significantly using both adaptive-fuzzy-correction formulations. However, the difference between the two versions is not very signifi- 112 cant, and thus the PAE with the nonlinear version is shown in the Figure. The nonlinear version is pre- ferred, as it is expected to produce significant accuracy for 1 to 2h ahead load forecast. The hourly-load-fore- cast accuracy is significant in this case as the University load profile (shown in the Figure) does not show signif- icant changes during the 24h period. ‘I I -3 1 6 11 16 21 time,h Fig.20 records for utility data ~ without fuzzy corfection -+- with fuzzy correction 24h ahead hourly load forecasts with temperature and humidity 9500 t 6500 1 6 11 16 21 time,h Fig.21 recordr f o r utility data 48h ahead hourly load forecast with temperature and humidity ’I I time, h Fig.22 records for utility data ~ without fuzzy correction -+- with fuzzy correction 48h ahead hourly load forecasts with temperature and humidity IEE Proc.-Gener. Transm. Distrib., Vol. 143, No. I , Januavy 1996 Figs. 25 and 26 show the 24h-ahead forecast errors for 15 February (winter day) using the nonlinear ver- sion of the fuzzy adaptive correction scheme along with FNN and FNN with fuzzy corrections. Significant accuracy in the hourly forecast is also obtained in this case using adaptive corrections. The above two days are nonspecial days and are chosen to illustrate the accuracy of the adaptive correction scheme. 22.55 22 t 21. 21 3 20. 2- 20 d 9 19. 19 18. 18 17.5 1 6 11 16 21 I ’ ’ I I ’ ’ I ‘ I ’ I I I ‘ ‘ I I ‘ I ‘ I time, h Fig. 23 mer day) Hourly loadjorecast with adaptive corrections f o r 27 May (sum- 0.5l””v”ttJis I 0 1 6 11 16 21 time, h Fig.24 Houvly load forecast with adaptive corrections for 27 M a y (sum- mer day) ~ without fuzzy correction -*- with fuzzy correction --A- with nonlinear adaptive correction 7 Discussion The proposed fuzzy-neural-networklfuzzy-expert-sys- tem approach is found to be very powerful and robust for short-term load predictions. Although the results for two seasons of the year are presented in this paper for validating the effectiveness of this approach, extensive tests have been conducted for other seasons, Sundays, holidays and special days of the year. From the results presented in this paper, it is observed that significant accuracy can be achieved for 24h ahead hourly load forecasts and the PES could be even less than 1%. Adaptive fuzzy load correction schemes enhance the accuracy of the predictions in most cases. However, if the lead time increases to 48h, the percentage error will be around 2%. The accuracy of load predictions using both adaptive corrections will be the highest with loads which do not show large IEE Proc.-Gener. Trunsm. Distrib., Vol. 143, No. I , January 1996 hourly variations. However, significant accuracy can still be achieved with highly stochastic load variations, if a fuzzy basis function approach is used to arrive at adaptive corrections instead of the trial-and-error method presented in this paper. 4017----- 35 21 151 I I I I I I I ’ ‘ I I ’ ’ I ’ ’ ’ 1 6 11 16 time,h Fig. 25 (winter day) Hourly load forecast with adaptive corrections for 15 February 0.5 1 1 0 1 6 11 16 21 tirne,h Fig. 26 (winter day) ~ without fuzzy correction -*-- with fuzzy correction -A- with nonlinear adaptive correction Hourly load forecast with aduptive corrections f o r 15 February The results for average load forecast are very encour- aging and the maximum absolute percentage error is around 1.8. From the results for peak-load forecast presented in this paper, it is observed that except for 14 and 15 May, the PAE is less than 2 for most of the days. The mean absolute error is evaluated for this month and is found to be 2.824 with the FNN model only and 1.150 using fuzzy corrections. The large errors for the above two days are probably due to other factors which have not been used in the training patterns. Although the results for 168 h-ahead load forecast using the above models have not been reported in this paper, the computations reveal that the mean absolute percentage error is around 2 and the maximum PE is 4.28 using the FNN model. With fuzzy corrections, the maximum PE is reduced to 2.16. This is quite comparable with the results for the ANN-based load forecasting technique presented by Karady et al. [12, 131. Although the studies reported here have utilised a few simple examples and models, they are extremely 113 valuable in identifying a promising hybrid forecasting methodology which can be investigated for larger number of inputs, more weather parameters, special load patterns, seasonal load changes, peak and total load forecasts one week ahead etc. 8 Conclusion A new hybrid model integrating an artificial neural net- work and a fuzzy expert system is developed for 24 h ahead average and peak load forecasts and tested with historical load data. The hybrid model uses a fuzzy neural network for obtaining the initial forecast from the fuzzified input data and a fuzzy expert system gen- erating the load correction to produce the final fore- cast. The simulation results of the proposed method using historical data show that the forecasting errors are less than 2% for both 24h ahead average and peak load predictions. The selection of training pattern pre- sented in this paper does not classify the patterns to weekday and weekend day and thus the hybrid model is a promising approach for short-term load forecast for all days of the year. The paper also presents an adaptive fuzzy correction scheme to minimise the fore- cast errors more precisely and 24h ahead hourly fore- casts using this scheme yield significant accuracy. The fuzzy neural network and fuzzy expert system approach has also been applied to one week ahead load forecast, and the results are found to be very promising. 9 Acknowledgment The authors acknowledge funds from the US National Science Foundation (NSF grants INT-9209 103 and INT-9 1 17624) for undertaking this research. 10 References 1 RAHMAN. S.. and BHATNAGAR. R.: ‘An exuert svstem based algorithm for ’ short-term load forecast’, I E g E T;ans.. 1988, PWRS-3, ( 2 ) , pp. 392-399 2 PARK. J.. PARK. Y.. and LEE. 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Control, 1965, 8, pp. 318- 358 18 PAL, S.K., and MAJUMDAR, D.D.: ‘Fuzzy mathematical approach to pattern recognition’ (Halstead Press, New York, 1986) 19 WANG, L.-X., and MENDEL, J.M.: ‘Fuzzy basis functions, Universal approximation, and orthogonal least-square learning’, I E E E Trans., 1992, “-3, (5), pp. 807-814 9 11 Appendix Weight updating using error backpropagation for fuzzy neural network The least-mean-square error in output vectors is mini- mised by using a gradient-descent algorithm by starting with any set of weights and repeatedly updating each weight by an amount where q = learning rate a, p = momentum coefficients n = iteration number E = error cost function Further, the learning rate q is adaptively varied as (19) where y determines the tuning of learning rate q . 114 IEE Pvoc.-Genev. Tvansm. Distvib., Vol. 143, N o . 1, January 1996