Short-Term Electrical Load Forecasting Using a Fuzzy ARTMAP Neural Network

Transcription

1 Short-Term Electricl Lod Forecsting Using Fuzzy ARTMAP Neurl Network Stefn E. Skrmn, Michel Georgiopoulos, nd Avelino J. Gonzlez Deprtment of Electricl nd Computer Engineering, College of Engineering, University of Centrl Florid, Orlndo, FL, ABSTRACT Accurte electricl lod forecsting is necessry prt of resource mngement for power generting compnies. The better the hourly lod forecst, the more closely the power generting ssets of the compny cn be configured to minimize the cost. Automtion of this process is profitble gol nd neurl networks hve shown promising results in chieving this gol. The most often used neurl network to solve the electric lod forecsting problem is the bck-propgtion neurl network rchitecture. Although the performnce of the bck-propgtion neurl network rchitecture hs been encourging, it is worth noting tht it suffers from the slow convergence problem nd the difficulty of interpreting the nswers tht the rchitecture provides. A neurl network rchitecture tht does not suffer from the bove mentioned drwbcks is the Fuzzy ARTMAP neurl network, developed by Crpenter, Grossberg, nd their collegues t Boston University. In this work we pplied the Fuzzy ARTMAP neurl network to the electric lod forecsting problem. We performed numerous experiments to forecst the electricl lod. The experiments showed tht the Fuzzy ARTMAP rchitecture yields s ccurte electricl lod forecsts s bckpropgtion neurl network with trining time smll frction of the trining time required by the bckpropgtion neurl network. 1. INTRODUCTION AND BACKGROUND Short-term lod forecsting is n essentil function in plnning the opertion of n electricl power system. The better the forecst, the more closely the power genertion ssets of the compny cn be configured to minimize the cost. If the forecst lod is higher thn the ctul lod, power generting units will be needlessly ctivted. If, on the other hnd, the forecst is too low, the deficit hs to be mde up through the opertion of "pekers," smll stndby power units tht cn be brought online reltively quickly. The drwbck with pekers is tht they re expensive to run compred to lrge generting units. However, lrge generting units tke more thn dy to go online from cold stte. Therefore, they cnnot be used to respond to sudden lod chnges in rel time. Florid Power Corportion (FPC), the power compny whose lod forecsting requirements nd dt were used s the bsis for this study, nd other power generting utilities generlly employ humn to perform the electricl lod forecsting function. The forecster typiclly predicts the lod t lest once every work dy. Since the forecster typiclly does not work during the weekends, the forecst for Sturdy, Sundy, nd Mondy re mde on Fridys. Forecsters typiclly receive five-dy wether forecst every dy. A wether forecst becomes more inccurte the further in dvnce it is mde becuse of the difficulty in mking such predictions. The lod forecster fces the sme problem, but the results become even worse becuse of the inccurcy of the wether forecsts tht hve to be used to mke the lod forecst. The forecster typiclly hs t his/her disposl severl yers of wether nd lod dt vilble in printout form. The old vlues re used to find dys of the sme type (e.g., weekdys, weekends, holidys) nd with similr wether conditions s the dy whose lod is to be forecst. The historicl lod profiles, the forecst wether prmeters, nd the type of dy re then used in the determintion of the forecst, bsed on their reltive importnce. The min objective of short-term electricl lod forecsting is to predict the hourly lods one dy or even up to one week hed of time. This is necessry for the opertionl plnning of power system. Since the electricl lod demnd is function of wether vribles nd humn socil ctivities, time series models [3,4], regression models [5,6], nd expert systems [7] hve been proposed to solve this problem. Time series models employ historicl lod dt for extrpoltion to obtin future hourly lods. These models ssume tht the lod trend is sttionry, nd

2 regrd ny bnorml dt s bd dt. Regression models nlyze the reltionship between lod, wether, nd socil ctivities. The disdvntge with regression models is tht they require complex modeling techniques nd hevy computtions to produce ccurte results. Expert systems use n expert forecster s experience nd heuristics to cpture the reltionship between socil ctivities, wether, nd future lods. However, to cpture the expert s experience nd knowledge, trnsforming the knowledge into IF-THEN rules tht cn be used in computer progrm is not n esy tsk. The problem with the described models is tht they lck the desired ccurcy nd they hve difficulty producing ccurte forecsts in the cse of rpid wether chnges [8]. Other reported problems [9] re tht model developed for one utility cnnot esily be modified for use t nother utility nd tht instlltion of the model t new site is often complex nd time-consuming procedure. During the lst severl yers, rtificil neurl networks (ANN`s) hve been pplied to the Short-Term Lod Forecsting (STLF) problem with considerble success. A problem reported by Gerber [2] is tht very long trining time is required for trining of the most commonly used ANN rchitecture (the Bck-Prop. NN). The excessive Bck-Prop trining time prevented Gerber [2] from doing extensive experimenttion with the vilble dt. The power generting compnies need system tht cn mke forecst up to seven dys hed of time. However, this would require even longer trining time thn wht is needed to trin system to mke n hourly forecst 24 hours hed of time. The hypothesis of this pper ws to pply the Fuzzy ARTMAP neurl network to the lod forecsting problem. The short trining time of the Fuzzy ARTMAP NN llowed us to conduct more experiments in order to identify n optimum input prmeter set, nd led eventully to more ccurte forecsts. Another problem with the Bck- Prop. NN is the difficulty to interpret its nswers. The opertion of the Fuzzy ARTMAP rchitecture is better understood nd the nswers derived by the rchitecture cn be logiclly explined. Therefore, it should be possible to determine why certin dys re more difficult to forecst thn others, nd hence wht to do to improve the forecst ccurcy for those dys. In section 2, the Fuzzy ARTMAP NN lgorithm is described. Section 3 describes the experiments nd presents the results chieved in this reserch. Section 4 summrizes nd describes the conclusions drwn from this work. 2. FUZZY ARTMAP NEURAL NETWORK The Fuzzy ARTMAP neurl network is n rchitecture tht cn lern rbitrry mppings from nlog or digitl inputs of ny dimensionlity to nlog or digitl outputs of ny dimensionlity. The rchitecture pplies incrementl supervised lerning of recognition ctegories nd multidimensionl mps in response to rbitrry sequences of nlog or binry input vectors, which my represent fuzzy or crisp set of fetures. [10]. The Fuzzy ARTMAP NN consists of two Fuzzy ART modules designted s Fuzzy ART nd Fuzzy ART b, s well s n inter Fuzzy ART module, s shown in Figure 1. Inputs re presented t the Fuzzy ART module, while the outputs re presented t the Fuzzy ART b module. The Inter-Fuzzy ART module includes MAP field, whose purpose is to determine whether the mpping between the presented input nd output is the desired one. The F 0 lyer receives the input vector nd produces the complement encoded input vector I. The complement encoded vector I is creted s: I = (, c ) = ( 1,, M, 1 c,, M c ) (1) i c = 1 - i, 1 i M, where M is the dimensionlity of the input vector (2) Hence, the F 0 lyer cts s preprocessor to the Fuzzy ART rchitecture. The vector I is subsequently pplied t the F 1 lyer. In most cses the complement encoding is not necessry for the output vector O. F 2 nd F 2 b re the lyers where compressed representtions of the input ptterns nd the output ptterns re estblished respectively.

3 Inter-Fuzzy ART module F b Fuzzy ART module W j Fuzzy ARTb module Rho F 2 F 2 b w j W j Reset w k b W k b Reset F 1 F 1 b I = (, c ) F 0 Rho_ O Rho_b Figure 1 - Block digrm of the Fuzzy ARTMAP rchitecture. Index i is used to designte the nodes in the F 1 lyer, index j is used to designte nodes in the F 2 lyer, index k is used to designte nodes in the Fuzzy MAP lyer (F b ) nd the F 2 b lyer, index l is used to designte nodes in the F 1 b lyer. Hence W j = (W 1j,, W (2M)j ) is the vector of bottom-up weights emnting from node i in the F 1 lyer to node j in the F 2 lyer. The vector of top-down weights emnting from node j to node i is denoted s w j = (w j1,, w j(2m) ). Also, W k b = (W 1k b,, W (Mb)k b ) is the vector of bottom-up weights converging to node k in F 2 b. Furthermore, w k b = (w k1 b,, w k (M b ) b ) is the vector of top-down weights converging to node l from node k in F 2 b. Finlly, w j = (w j1, w jnb ) is the vector of weights emnting from node j in F 2 nd converging to the nodes of the MAP lyer F b. A vector whose components re the top-down weights emnting from single node in F 2 /F 2 b is clled templte of the Fuzzy ART /b module. An uncommitted templte is defined to be the vector of top-down weights ssocited with node in F 2 /F 2 b which hs not yet been chosen to represent n input/output pttern. Ech of the components of n uncommitted templte is equl to one. The vigilnce prmeters, ρ /b (designted s Rho_/b in Figure 1) constrints the mximum size of the compressed representtions of the input/output ptterns. The rnge of the vigilnce prmeter is the intervl [0,1]. Smll vlues of the vigilnce prmeter results in corse clustering of the ptterns, while lrge vlues led to fine clustering. The choice prmeters β /b ffect the choice of node. The rnge of the choice prmeter is the intervl (0, ). Relisticlly, the vlue should not exceed vlue tht is proportionl to the numbers M nd M b respectively.

4 3. RESULTS 3.1 Network prmeter evlution The Fuzzy ARTMAP network performnce depends on the vlues of the prmeters β /b nd ρ /b. In order to identify n optiml set of network prmeter vlues for the problem t hnd, we conducted number of experiments. The network prmeter evlution ws performed using one dy hed, one hour forecst, predicting Wednesdys lod t 7:00 m. The trining exmples used for prmeter evlution consist of ll norml Wednesdys (Wednesdys tht re not holidys) from 1993 nd 1994, up to 10/12/93 nd 10/12/94 respectively. Wednesdys fter 10/12 re not used for trining due to the fct tht 1995 dt (used for testing) includes dt only up to 10/12/95. The test exmples used consist of ll norml Wednesdys from 1995, up to 10/12/95. The trining exmples correspond to 94 dys nd the test exmples correspond to 40 dys. A bsic trining set (the sme trining set s used by Gerber [2]) ws used for the network prmeter evlution. This trining set included 27 tempertures nd 6 lods. The 27 tempertures were spred over three dy period; the dy to forecst, the current dy (the dy before the dy to forecst), nd the previous dy (the dy two dys before the dy to forecst). For ech dy, the hour to forecst nd the two preceding hours were used s inputs. The temperture dt is vilble for Orlndo, St. Petersburg, nd Tllhssee. This yields 27 temperture inputs (3 cities * 3 hours* 3 dys = 27). The six lods used for the input were spred over two dy period, the current dy, nd the previous dy. For ech dy the hour to forecst nd the two preceding hours were used for the lod inputs. This yields six lod inputs (2 dys*3 hours = 6). For ech experiment, the ρ ws chosen s 0, 0.3, 0.6, or 0.9, while the ρ b ws chosen s 0.95, 0.99, 0.995, 1.0, nd β = β b used vlues of 0.01, 1, 5, 10. Tble 1 shows the network prmeter evlution results by choosing specific ρ vlue nd obtining the network performnce for ll possible combintions of the other network prmeters. The MAPE (Men Absolute Percentge Error) vlue is clculted s the verge APE (Absolute Percentge Error) vlue for ll test Wednesdys, where APE = Actul lod Pr edicted lod Actul lod 100 (3) The best result chieved ws MAPE = 7.4% for ρ = 0.0, β = β b = 10, nd ρ b = These prmeter vlues creted firly smll network with 17 nodes in F 2 nd 7 nodes in F 2 b. The corresponding result obtined by Gerber [2] ws MAPE = 6.8%, using the sme trining exmples nd the sme test set. However, the trining time required to trin the Fuzzy ARTMAP neurl network ws very short ( few minutes) compred to hours of trining for the Bck-Propgtion neurl network used by Gerber. The order of trining pttern presenttion to Fuzzy ARTMAP ffects its performnce. Therefore by trining 10 networks with the sme trining set, but with different orders of pttern presenttion nd tking the predicted lod to be the verge of the outputs from the 10 networks, we expected performnce improvement. Tble 2 show the results for the ten network pproch. The best result ws MAPE = 7.3%. This result is only 0.1% points better thn the one network pproch. The minor performnce improvement, the incresed trining time (fctor 10), nd incresed weight storge (fctor 10) of the ten network pproch llows us to conclude tht for this experiment the ten network pproch is inferior to the one network pproch.

5 Tble 1 - MAPE using 1 neurl network to forecst the lod for Wednesdys t 7:00 m. Rho_b Rho_b Tble 1.1,MAPE, rho_ = 0.0 Tble 1.2, MAPE, rho_ = 0.3 Rho_b Rho_b Tble 1.3, MAPE, rho_ = 0.6 Tble 1.4, MAPE, rho_ = 0.9 Tble 2 - MAPE using 10 neurl network to forecst the lod for Wednesdys t 7:00 m. Rho_b Rho_b Tble 2.1, MAPE, rho_ = 0.0 Tble 2.2, MAPE, rho_=0.3 Rho_b Rho_b Tble 2.3, MAPE, rho_=0.6 Tble 2.4, MAPE, rho_ = hour forecst for Wednesdys The forecst ws mde one dy hed of time for ll 24 hours of Wednesdys in 1995, up to 10/12/95, for the sme resons mentioned in section 3.1. Every hour is predicted with one nd ten seprte networks, using the prmeter vlues tht yielded the best performnce during the network prmeter evlution phse (ρ = 0.0, ρ b = 0.95, β = β b = 10). The bsic trining set using 27 tempertures nd 6 lods ws used for trining nd testing. The network ws trined using dt from 1993 nd 1994 (Wednesdys only, 94 dys) nd tested using dt from 1995 (Wednesdys only, 40 dys). Tbles 3 nd 4 show the performnce when one nd ten networks were used to forecst ech hour, respectively. MAPE denotes the verge performnce for ll test dys. MAX is the mximum error for ll test dys. MIN is the minimum error for ll testing dys. For this experiment there is significnt improvement by using 10 networks, the MAPE decreses by 1.2% points, however, the MAPE vlue is still high.

6 Tble 3 - MAPE, Mx MAPE, Min MAPE, nd stndrd devition of MAPE for one dy hed, 24 hour forecst (Wednesdys) using 1 NN. MAPE = 7.7% MAX = 25.4% MIN = 3.0% Std dev = 4.5% MAPE (Dys bove 10% excluded (7 dys)) = 6.0% Tble 4 - MAPE, Mx MAPE, Min MAPE, nd stndrd devition of MAPE for one dy hed, 24 hour forecst (Wednesdys) using 10 NNs. MAPE = 6.5% MAX = 25.5% MIN = 1.9% Std dev = 4.5% MAPE (Dys bove 10% excluded (7 dys)) = 5.2% 3.3 Grouping of dys According to Hsu nd Yng [11], severl dys of the week hve similr lod profiles nd cn therefore be lumped into the sme group. According to Hsu nd Yng, Mondys nd dys fter holidys hve similr lod profile, Tuesdys through Fridys hve similr lod profile, Sturdys nd dys before holidys hve similr lod profile, nd finlly Sundys nd holidys hve similr lod profile. One possible method to decrese the reltively high MAPE chieved so fr, is to increse the number of trining exmples. Therefore, the method of grouping dys together ws tried. The trining/testing set described in previous sections were used for this group of experiments. The network prmeter set derived in section 3.1 ws used. Two sets of experiments were mde: 1. Identicl to n experiment evluted by Gerber [2], where the forecst ws mde one dy hed of time, predicting the lod t 7:00 m. 2. Lod forecst one dy hed of time,predicting ll 24 hours, where ech hour is predicted with one nd ten networks, respectively. Tble 5 shows the performnce for the one dy hed forecst, predicting the lod t 7:00 m. The MAPE for the forecst of Wednesdys t 7:00 m used in the grouping of Tuesdys trough Fridys is equl to 5.2%, the corresponding MAPE when Wednesdys lone were used for trining is 7.4% (Tble 1). The grouping of dys significntly improves the performnce, the MAPE decresed by 2.2% points nd the stndrd devition decresed by 3.1% points. The corresponding MAPE reported by Gerber [2] is 4.4%, 0.8% points better thn the result chieved here. The reson tht the Bck-Propgtion neurl network yields better results thn the Fuzzy ARTMAP lgorithm is most likely due to the fct tht the Bck-Prop. NN provides s nswers to test inputs interpoltions of the nswers to trining inputs. The Fuzzy ARTMAP, fter trining, provides s nswers to test inputs only the nswers provided to trining inputs in the trining phse. Tbles 6 nd 7 show the result for the 24 hour forecst. The MAPE for the 24 hour forecst of Wednesdys used in the grouping of Tuesdys trough Fridys is equl to 6.3% (Tble 6), the corresponding MAPE when Wednesdys lone were used for trining is 7.7% (Tble 3). The result for the Wednesdys used in the grouping of dys is 1.4% points better thn when only those Wednesdys were used for trining. The result for the 10 network pproch is MAPE = 5.8%, 0.5% points better thn the one network pproch. Once gin, it is demonstrted tht the performnce when severl networks re used to forecst ech hour is better thn the performnce when single network is used to forecst ech hour.

7 The grouping of dys hd positive effect on the verge MAPE for Wednesdys. Since the results for the other groups of dys (see [1]) re close to the result for the Tuesdys-Fridys group, one cn conclude tht n incresed number of trining exmples hs positive effect on the performnce of the network. Tble 5 - MAPE, Mx MAPE, Min MAPE, nd Stndrd devition of MAPE for one dy hed forecst (forecst for Tuesdys-Fridys t 7:00m). MAPE = 5.4% MAX = 28.8% MIN = 0.05% Std dev = 4.9% MAPE (Dys bove 10% excluded) = 3.9% MAPE (Wednesdy's only) = 5.2% Tble 6 - MAPE, Mx MAPE, Min MAPE, nd Stndrd devition of MAPE for one dy hed 24 hour forecst using 1NN(forecst for Tuesdys-Fridys). MAPE = 6.3% MAX = 20.1% MIN = 2.9 Std dev = 2.8% MAPE (Wednesdy's only) = 6.3% Tble 7 - MAPE, Mx MAPE, Min MAPE, nd Stndrd devition of MAPE for one dy hed 24 hour forecst using 10 NNs (forecst for Tuesdys-Fridys). MAPE = 5.8% MAX = 20.1% MIN = 2.3 Std dev = 2.8% MAPE (Wednesdy's only) = 6.4% 3.4 Sesonl influence The lod profile for dys belonging to certin group chnges s the seson chnges. To tke the sesonl effect into considertion, 24 hour forecst ws produced using networks trined with dt from June, July, August of 1993, nd 1994 nd tested with dt from June, July, nd August of A 24 hour forecst ws lso performed using networks tht re trined with dt from Jnury, nd Februry of 1993, nd 1994 nd tested with dt from Jnury, nd Februry of The bsic trining set used included 27 tempertures nd 6 lods. The sme network prmeter set tht yielded the best results during the prmeter evlution section ws used. Both trining nd testing were performed on the Tuesdys to Fridys group (100 trining exmples nd 43 testing exmples for the June-August group, 64 trining exmples nd 28 testing exmples for the Jnury-Februry group). As shown in Tbles 8 nd 9 (the left columns show the performnce with sesonl influence nd the right columns shows the performnce without sesonl influence). There is smll improvement by using sesonl dt for trining, since the verge MAPE decresed by 0.3% points nd 0.1% points respectively. An importnt dvntge is tht the mximum error for most dys is lower when the sesonl effect is tken into considertion. For certin dys the mximum error decresed significntly (e.g. from 21.6% to 8.3% for 6/20/95) for the June-August period. The Jnury-Februry period shows similr behvior, for certin dys the mximum error decresed significntly (e.g. from 28% to 10% for 2/7/95).

8 Using sesonl dt for trining hs the dvntge tht for certin dys it decresed the mximum APE. This implies tht the sesonl trining is worthwhile procedure, despite the fct tht the verge MAPE does not decrese significntly. Tble 8 - MAPE, Mx MAPE, Min MAPE, nd stndrd devition of MAPE for one dy hed, 24 hour forecst for Tuesdys-Fridys, with nd without sesonl influence (June, July, August). MAPE MAX MIN Std dev Tble 9 - MAPE, Mx MAPE, Min MAPE, nd stndrd devition of MAPE for one dy hed, 24 hour forecst for Tuesdys-Fridys, with nd without sesonl influence (Jnury, Februry). MAPE MAX MIN Std dev Dily forecst using 1 neurl network In previous sections, one or ten seprte neurl networks hve been used to predict the lod for every single hour of the dy. In this section, one network is used to predict the lod for ll 24 hours of the next dy. The one dy hed, 24 hour forecst ws performed using networks trined with dt from June, July, nd August of 1993, nd 1994 nd tested with dt from June, July, nd August of 1995 (100 trining exmples nd 43 testing exmples). Also, one dy hed, 24 hour forecst using networks trined with dt from Jnury, nd Februry of 1993, nd 1994, nd tested with dt from Jnury, nd Februry of 1995 ws performed (64 trining exmples nd 28 testing exmples). The trining/testing uses yesterdys ctul tempertures nd lods, plus the forecst tempertures for the dy to forecst. This result in 168 input prmeters (24 tempertures for 3 cities for 2 dys = 144 inputs + 24 lods = 168). Both trining nd testing ws performed on the Tuesdys to Fridys group. The left column of Tble 10 shows the performnce for the June-August group. The results re resonble with MAPE = 6.3%. The Jnury-Februry group yield slightly worse results with MAPE = 7.6% (left column of Tble 11). In both cses, the shpe of the lod curves ws correctly predicted for ll the test dtes. However, both predictions underestimte, in most instnces, the ctul lod. Therefore, n effort to correct the underestimtion ws undertken. The lod consumption hs incresed from yer to yer over the time period covered in this reserch. Since the forecst vlue in lmost every test cse ws found to be lower thn the ctul vlue, the simple pproch to increse the forecst vlue by certin percentge ws evluted in this section. Experiments hve showed tht by incresing the forecst lod by 4% yielded the gretest performnce improvement. The verge performnce improved by 1.2% points for the June-August group (compre the left column of Tble 10 to the right column of Tble 10) nd by 1.1% points for the Jnury-Februry group (compre the left column of Tble 11 to the right column of Tble 11). As shown by Tble 10 the MAPE decreses but the mximum nd the minimum APE vlue increses for the June-August group. The reson for this is becuse the forecst lod for some dys is lredy higher thn the ctul lod before the increment of the forecst is pplied. However, this type of dys re uncommon compred to the number of dys tht hve forecst vlue tht is lower thn the ctul vlue. On the other hnd, for the Jnury-Februry group (Tble 11) the MAPE, the Mx, nd the Min decreses when the forecst vlue is incresed. The reson for this is due to the fct tht the forecst vlue is more often lower thn the ctul vlue for the Jnury-Februry group thn for the June-August group.

9 Tble 10 - MAPE, Mx MAPE, Min MAPE, nd stndrd devition of MAPE for one dy hed, dily forecst, using 1NN without nd with incremented prediction (June-August). MAPE MAX MIN Std dev Tble 11 - MAPE, Mx MAPE, Min MAPE, nd stndrd devition of MAPE for one dy hed, dily forecst, using 1NN without nd with incremented prediction (Jnury-Februry). MAPE MAX MIN Std dev Input prmeter set evlution The input prmeter set used in previous sections hs been bsic trining set including few tempertures nd lods s inputs. The reson the bsic input prmeter set hs been used ws to fcilitte fir comprisons with the results obtined by Gerber [2]. However, since different neurl network rchitecture is used in this reserch, different input prmeter set might improve the performnce. The input prmeter sets evluted in this section ws: Bsic: 27 tempertures (tempertures for the hour to forecst plus the two preceding hours for yesterdy, tody nd the dy to forecst for Orlndo, St. Petersburg, nd Tllhssee) nd 6 lods (electricl lod for the hour to forecst plus the two preceding hours for yesterdy nd tody), re used in this dt set. 6h: 54 tempertures (tempertures for the hour to forecst plus the five preceding hours for yesterdy, tody nd the dy to forecst for Orlndo, St. Petersburg, nd Tllhssee) nd 12 lods (electricl lod for the hour to forecst plus the five preceding hours for yesterdy nd tody), re used in this dt set. 12h: 36 tempertures (tempertures for the hour to forecst plus the eleven preceding hours) nd 11 lods (electricl lod for the eleven hours preceding the hour to forecst), re used in this dt set. Averge: 9 tempertures (tempertures for the hour to forecst plus the two preceding hours for yesterdy, tody nd the dy to forecst; the temperture is set to be the verge temperture of Orlndo, St. Petersburg, nd Tllhssee) nd 6 lods (electricl lod for the hour to forecst plus the two preceding hours for yesterdy nd tody), re used in this dt set. The trining/testing exmples used were from the Tuesdy-Thursdy group (100 trining exmples nd 43 testing exmples for the June-August group, nd 64 trining exmples nd 28 testing exmples for the Jnury-Februry group). Tble 12 shows the results for ρ b = The top hlf of Tble 12 shows the results for the Jnury-Februry group nd the bottom hlf shows the results for the June-August group. Similrly, Tble 13 shows the results for ρ b = The bsic input prmeter set yields n APE of 5.1% with stndrd devition of 3.4% for ρ b = for the June- August group. Both the 6h nd the 12h input prmeters set yield better results. This mkes sense, since more informtion is used in the input pttern. In prticulr the 12h input prmeters set, yields the most promising results. This lso mkes sense since dt from the hour to forecst plus the eleven preceding hours re used for the input, for both lod nd temperture. The gretest improvement is noticed for the Jnury-Februry group where the 12h input prmeters set is 4.2% points better thn the 6h input prmeter set nd 4.5% points better thn the bsic input prmeter set (Tble 12). The comprison between the 12h input prmeter set nd the other input sets is not totlly fir becuse during this test ctul vlues were used for both tempertures nd lods in the 12h input

10 prmeters set. However, during rel execution of the system this would not be possible if the forecst is to be mde t lest one dy hed of time. Therefore, the 12h input prmeters set should be evluted using forecst vlues. The use of verge temperture shows smll performnce improvement compred to the bsic input prmeters set for the June-August group. For the Jnury-Februry group, however, the result for the input prmeter set with verge temperture is the worst with APE = 13.2% nd stndrd devition = 10.9%. The reson for this behvior is due to the fct tht there re lrge temperture differences between the three cities during the winter months. On the other hnd, during the summer months, the tempertures in the three cities re not significntly different from ech other, nd hence the verge temperture ide works. When the verge temperture ide is used to forecst the lod for the winter months, lot of temperture informtion is lost nd therefore, worse forecst is produced. Tble 12 - MAPE, Mx MAPE, Min MAPE, nd stndrd devition of MAPE for four prmeter sets, ρ b = Bsic 6h 12h Averge MAPE MAX MIN Std dev MAPE MAX MIN Std dev Tble 13 - MAPE, Mx MAPE, Min MAPE, nd stndrd devition of MAPE for four prmeter sets, ρ b = Bsic 6h 12h Averge MAPE MAX MIN Std dev MAPE MAX MIN Std dev REVIEW AND CONCLUSIONS In n erlier work, Gerber [2] used the Bck-Prop. NN to forecst the electricl lod for the Florid Power dt. Gerber reported excessive trining time to trin the Bck-Prop. NN. Therefore, we pplied the Fuzzy ARTMAP NN to the electricl lod forecsting problem nd conducted identicl experiments s the ones crried out by Gerber. The forecsting results chieved with the Fuzzy ARTMAP NN were slightly worse thn wht Gerber chieved with the Bck-Prop. NN. The reson for this is tht the Bck-Prop. NN hs better interpolting bility thn Fuzzy ARTMAP. However, the trining time required by the Fuzzy ARTMAP NN is only smll frction of wht the Bck-Prop. NN requires to solve the sme problem. Experiments hve showed tht by using 10 networks to forecst ech hour nd use the verge output from the 10 networks s the forecst vlue, the network s performnce is significntly improved compred to single network s performnce. Especilly the fluctutions in the forecst lod profile experienced with the one network pproch is significntly decresed. The grouping of dys hd positive effect on the network s

11 performnce. By grouping dys with similr lod profiles together, nd thereby incresing the number of trining exmples, the performnce improved by s much s 2.2 % points. The seson hd n effect on the extremely high APE vlues tht occurred for certin dys. When the seson ws tken into considertion, the APE vlue for some hours decresed with s much s 18% points. Since historicl dt re used for trining nd the fct tht the lod demnd increses over the covered time period, it is expected tht the forecst lod should be lower thn the ctul lod. This observtion is especilly true when one network is used to forecst ll 24 hours of the next dy. The input prmeter set hs significnt effect on the network s performnce. Experiments showed 6.1% points difference between the best nd the worst input prmeter set. More reserch hs to be done to find Fuzzy ARTMAP s optimum input prmeter set for the short term electricl lod forecsting problem. ACKNOWLEDGEMENTS The uthors re grteful to the Florid Power Corportion s Energy Control Center, St. Petersburg, Florid for the dt provided for this reserch. REFERENCES [1] Stefn E. Skrmn, Short-Term Electricl Lod Forecsting Using Fuzzy ARTMAP Neurl Network, Mster s thesis, University of Centrl Florid, Orlndo, [2] Willim J. Gerber, Prmetric nlysis of electricl lod forecsting using rtificil neurl networks, Mster s Thesis, University of Centrl Florid, Nov [3] W. Christinse, Short-term lod forecsting using generl exponentil smoothing, IEEE Trnsctions on Power Apprtus nd Systems. vol. PAS-90, April, 1971, pp [4] D. W. Bunn nd E. D. Frmer, Comprtive models for electricl lod forecsting. New York: John Wiley & Son, [5] C. Asbury, Wether lod model for electricl demnd energy forecsting, IEEE Trnsctions on Power Apprtus nd Systems, vol. PAS-94, no. 4, 1975, pp [6] A. D. Pplexopoulos nd T. C. Hesterberg, A regression bsed pproch to short-term system lod IEEE Trnsctions on Power Systems, vol. 5, Nov. 1990, pp [7] S. Rhmn nd R. Bhtngr, An expert system bsed pproch for short-term lod forecst, IEEE Trnsctions on Power Systems, vol. 3, No. 2, My 1988, pp [8] A. Khtonzd, R-C. Hwng, A. Abye, nd D. Mrtukulm, An dptive modulr rtificil neurl network hourly lod forecster nd its implementtion t electric utilities, IEEE-PES Winter Meeting [9] Shin-Tzo Chen, Dvid C. Yu, nd A. R. Moghddmjo, Whether sensitive short-term lod forecsting using non-fully connected rtificil neurl networks, IEEE Trnsctions on Power Systems, vol. 7, no. 3, Aug [10] G. A Crpenter, S. Grossberg, N. Mrkuzon, H. Reynolds, nd D. B Rosen, Fuzzy ARTMAP: A neurl network rchitecture for incrementl supervised lerning of nlog multidimensionl mps, IEEE Trnsctions on Neurl Networks, vol. 3, no. 5, Sept. 1992, pp [11] Y-Y. Hsu, nd C-C. Yng, Electricl lod forecsting, Aln F. Murry, Applictions of Neurl Networks, Kluwer Acdemic Publishers, 1995, pp