Discovering Patterns in Electricity Price Using Clustering Techniques

Disco e ing Pa e ns in Elec ici y P ice Using Clus e ing Techniques F. Ma ínez Ál a ez1, A. T oncoso2, J. C. Riquelme1, J. M. Riquelme3 1 Depa amen o de Lenguajes y Sis emas In o má icos Escuela Técnica Supe io de Ingenie ía In o má ica. Uni e sidad de Se illa Phone: +0034 954 552775, e-mail: [email p o ec ed], [email p o ec ed] 2 Á ea de Lenguajes y Sis emas In o má icos Escuela Poli écnica Supe io . Uni e sidad Pablo de Ola ide Phone: +0034 954 977522, e-mail: [email p o ec ed] 3 Depa amen o de Ingenie ía Eléc ica Escuela Supe io de Ingenie os. Uni e sidad de Se illa Phone: +0034 954 481274 e-mail: [email p o ec ed] Abs ac . Clus e ing is a p ocess o g ouping simila elemen s ga he ed o occu ed closely oge he . This pape p esen s wo clus e ing echniques, K-means and Fuzzy C- means, o he analysis o he elec ici y p ices ime se ies. Bo h algo i hms a e ocused on ex ac ing use ul in o ma ion om he da a wi h he aim o model he ime se ies beha iou and ind pa e ns o imp o e he p ice o ecas ing. The main objec i e, hus, is o ind a ep esen a ion ha p ese es he o iginal in o ma ion and desc ibes he shape o he ime se ies da a as accu a ely as possible. This esea ch demons a es ha he applica ion o clus e ing echniques is e ec i e in o de o dis inguish se e al kinds o days. To be p ecise, wo majo g oups can be dis inguished hanks o he clus e ing: he i s one ha includes he wo king days and he second one ha includes weekends and es i i ies. Equally ema kable is he simila i y shown among days belonging o a same season. Key wo ds Clus e ing, p ice o ecas ing, ime se ies model. 1. In oduc ion I is impo an o ob ain an app oach o op imize he bidding s a egies ca ied ou by elec ici y-p oduce companies [1]. Consequen ly, he de elopmen o o ecas ing echniques is becoming inc easingly ele an in he cu en hec ic Spanish elec ici y-ma ke de egula ion. This wo k is ocused on ex ac ing meaning ul in o ma ion o he p ices ime se ies by using clus e ing echniques. Clus e ing is he basis o many classi ica ion and sys em modelling algo i hms. The main a ge o clus e ing is o gene a e g oupings o da a om a la ge da ase wi h he in en ion o p oducing an accu a e ep esen a ion o he beha iou o a sys em. Thus, he esea ch is based on he applica ion o wo well-known clus e ing me hods, K-means and uzzy clus e ing [2], o inding hose g oups o p ices which show a simila beha iou unde some pa icula condi ions such as wo king / non-wo king days o seasons. La e , his in o ma ion can be used o p edic ing how he p ices will p og ess h oughou he nex day. O he esea che s ha e de eloped echniques o o ecas he p ices ime se ies. Recen ly, A. J. Conejo e al. [3] p oposed a o ecas ing model using he wa ele ans o m and ARIMA models. Equally, R. C. Ga cía e al. [4] p esen ed a o ecas ing echnique based on a GARCH model. In [5] a me hod combining A i icial Neu al Ne wo ks wi h uzzy logic is p oposed. In [6] an adap i e non-pa ame ic eg ession app oach is applied o o ecas he hou ly On a io ene gy p ice. In [7] a simple model based on he Weigh ed Nea es Neighbou s me hodology is p esen ed and i s pe o mance is compa ed wi h o he s ecen ly published echniques. Howe e , i can be s a ed ha he o ecas ing echniques o he nex -day elec ici y p ices published in he cu en li e a u e do no use p e ious clus e ing echniques. Consequen ly, i is necessa y o disco e pa e ns in he elec ici y p ices ime se ies o imp o e he p edic ion models. The inal goal is o p o ide se e al p edic ions o he p ice e olu ion cu e o he subsequen day. This in o ma ion would be used in op imiza ion models in o de o help o ma ke agen s o gene a e hei op imal bidding s a egies. Thus, he i s objec i e is o de e mine a p e ious clus e ing o e eal p ices popula ion cu es in o de o ob ain g oups o days acco ding o he p ice o he elec ici y hou by hou . F om his di ision, he connec ion be ween belonging o a ce ain clus e and he model o p edic ion o each clus e could be ound. The e o e, i would be chosen he clus e o which he cu en day i belong in o de o p edic he p ices cu e o he ollowing day. Finally, he p edic ion would be gene a ed by means o he co esponding model o his clus e . The no el and main con ibu ion o he pape is o apply clus e ing echniques o he elec ici y p ices ime se ies o disco e simila pa e ns. The pa e ns p o ide use ul h ps://doi.o g/10.24084/ epqj05.245 174 RE&PQJ, Vol. 1, No.5, Ma ch 2007 in o ma ion o imp o e he o ecas ing echniques. The ime se ies is he a ia ion o he p ice o he elec ici y h oughou he day. The mo e days conside ed in he da ase , he mo e p ecise will be he p edic ion. The es o he pape is o ganized as ollows. Sec ion 2 de ails he wo clus e ing echniques applied o ind pa e ns in ime se ies. As selec ing he numbe o clus e s esul s a key p ocess, Sec ion 3 explains he mo i a ion o choosing he numbe o clus e s in bo h algo i hms. Sec ion 4 p esen s all he esul s ob ained, as well as i compa es bo h echniques. A desc ip ion o he da ase used is also shown in his sec ion. Finally, Sec ion 5 expounds he conclusions achie ed and he u u e wo k. 2. Me hodology Clus e ing is a p ocess o g ouping an unlabeled se o examples in o a numbe o clus e s such ha a simila pa e n is associa ed o e e y clus e , ha is o say, clus e ing ope a es on a se o examples ha mus be pa i ioned acco ding o some no ion o simila i y. Clus e analysis echniques ha e been classi ied in o wo majo me hods: 1) C isp clus e ing (o ha d clus e ing) in which he bounda y be ween clus e s is ully de ined. 2) Fuzzy clus e ing in which he bounda y be ween clus e s can no be clea ly de ined (such is he case o many eal cases). Bo h app oaches p esen a la ge se o algo i hms, mos o hem designed o speci ic p oblems. In his pape wo di e en clus e ing-based echniques ha e been used in o de o iden i y pa e ns o beha iou in he p ices cu es: K-means, ep esen ing he c isp clus e ing and he Fuzzy C-means (FCM) ep esen ing he uzzy clus e ing. A. k-means algo i hm K-means is a as me hod o pe o m clus e ing. The basic in ui ion behind K-means is he con inuous eassignmen o objec s in o di e en clus e s so ha he wi hin-clus e dis ance is minimized. I uses an i e a i e algo i hm di ided in wo phases o minimize he sum o poin - o-cen oid dis ances, o e all k clus e s. In he i s phase, each i e a ion consis s o eassigning poin s o hei nea es clus e cen oid and hen i ecalcula es he clus e cen oids. In he second phase, poin s a e indi idually eassigned i doing so educe he sum o dis ances; clus e cen oids a e ecompu ed a e each eassignmen . Each i e a ion consis s o one pass h ough all he poin s. Bo h phases a e summa ized in Table I, which desc ibes he k-means in e ms o i s basic s eps. Table I. – Ou line o he k-means algo i hm STEP DESCRIPTION 1 Decide a alue o k 2 Ini ialize he k clus e cen es 3 Assign an example o he nea es clus e cen e 4 Re-calcula e he k clus e cen es assuming ha he membe ships ound in s ep 3 a e co ec 5 Exi i no example changes o clus e in he las i e a ion. O he wise go o s ep 3. B. Fuzzy C-means algo i hm The Fuzzy C-means clus e ing, whe e C is he numbe o clus e s o classi y, he da a is a echnique whe ein each da a belongs o a clus e o some deg ee speci ied by a membe ship g ade. I p o ides a me hod ha shows how o g oup da a poin s ha popula e some mul idimensional space in o a speci ic numbe o di e en clus e s. The FCM algo i hm ocuses on minimizing he alue o an objec i e unc ion which calcula es he weigh ed wi hin-g oup sum o squa ed e o s. To measu e he quali y o he pa i ioning i compa es he dis ance om an example o he cu en candida e clus e cen e wi h he dis ance o o he candida e clus e cen es. Table II shows he summa ized s eps ollowed in he algo i hm. Table II. – Ou line o he Fuzzy C-means algo i hm STEP DESCRIPTION 1 Decide a alue o C 2 Ini ialize he clus e cen e ma ix, W( =0) 3 Ini ialize he membe ship ma ix, U( =0) 4 Inc ease by one and compu e W( ) 5 Compu e U( ) 6 I (U( ) – U( -1)) is lowe han a gi en e o s op. O he wise go o s ep 4. While hese wo algo i hms a e ypically used in he li e a u e ela i e o clus e ing app oaches in ime se ies, hey p esen a well-known sho coming: he numbe o clus e s mus be speci ied in ad ance. The choice o his pa ame e will be jus i ied in he subsequen sec ion. 3. Selec ion o he numbe o clus e s The numbe o clus e s selec ed is one o he mos c i ical decisions in clus e ing echniques. The ac o choosing a la ge numbe o clus e s does no necessa ily imply ha e a be e quali y o in o ma ion. On he con a y, esul s could be unclea and could muddle he pa e n ecogni ion up. This limi a ion can be mi iga ed by es ing all alues o K o C clus e s wi hin a la ge ange. Fu he s a is ical es can, hen, be used o de e mine which alue o K o C i s be e . Sec ions 3.A and 3.B show a me hodical way o selec he op imal numbe o clus e s o bo h echniques. A. Numbe o clus e s in K-means The silhoue e unc ion in Ma lab p o ides a measu e o he clus e s sepa a ion. I s alue a ies be ween –1 and h ps://doi.o g/10.24084/ epqj05.245 175 RE&PQJ, Vol. 1, No.5, Ma ch 2007 +1, whe e +1 deno es clea clus e sepa a ion and –1 ma ks poin s wi h ques ionable clus e assignmen . A success ul clus e ing has a mean silhoue e alue highe han 0,6 o all clus e s. Howe e , in eal ime se ies i is almos impossible o each his alue and no ha ing nega i e alues in he igu e is usually enough o decide how many clus e s ha e o be chosen. Figu es 1.a, 1.b and 1.c show he plo ed silhoue e unc ion o 4, 5 and 6 clus e s espec i ely o he p ices o he elec ici y o he yea 2005. The me ic used was squa ed Euclidean dis ance since cosine me ics ga e wo se esul s. Fo u he analysis, 4 clus e s ha e been chosen due o ha only one clus e has nega i e alues and i s g aphical ep esen a ion p o ides sa is ac o y esul s. B. Numbe o clus e s in Fuzzy C-means The FCM clus e ing algo i hm is sensi i e o he si ua ion o he ini ializa ion and easy o all in o a local minimum o a saddle poin when i e a ing. To sol e his p oblem se e al o he echniques ha e been de eloped ha a e based on global op imiza ion me hods [8]. Howe e , in many p ac ical applica ions he clus e ing me hod ha is used is FCM wi h mul iple es a s o escaping om he sensibili y o ini ial alue. The subclus unc ion in Ma lab inds clus e cen es and i is commonly used in o de o ob ain he op imum numbe o clus e s in i e a i e op imiza ion-based clus e ing me hods such as FCM. This unc ion es ima es he clus e cen es in a se o da a by using he sub ac i e clus e ing me hod. I assumes ha each da a poin is a po en ial clus e cen e and calcula es a measu e o he likelihood ha each da a poin would de ine he clus e cen e, based on he densi y o su ounding da a poin s. A e he execu ion o his algo i hm, i was ound ha 6 is he op imum numbe o clus e s. Fig. 1.a. Silhoue e alues wi h 4 clus e s. Fig. 1.b. Silhoue e alues wi h 5 clus e s. Fig. 1.c. Silhoue e alues wi h 6 clus e s. 4. Resul s A. Da ase desc ip ion The da a sou ce is he p ices o he elec ici y ma ke o mainland Spain o he yea 2005 (OMEL) [9]. Be o e ope a ing wi h he elec ici y p ices, da a no maliza ion was ca ied ou wi h he aim o a oiding he e ec s o he g ow h o he in a-annual p ices. The no maliza ion was pe o med by di iding he hou ly p ices by he a e age p ice o he whole day. B. K-means Figu e 2 shows he yea 2005 classi ied in o 4 clus e s, as jus i ied in sec ion 3.B, ia he k-means algo i hm. Wi h jus a quick look, i can be clea ly di e en ia ed wo kinds o clus e s: clus e s 1 and 2 g oup all he wo king days and clus e s 3 and 4 he weekends. Ne e heless, he e a e some days ha ha e an appa en ly disco dan beha iou . Table IV shows he pe cen age o days classi ied in o he 4 clus e s. h ps://doi.o g/10.24084/ epqj05.245 176 RE&PQJ, Vol. 1, No.5, Ma ch 2007 Table IV. – G ade o membe ship o days o clus e s Clus e 1 Clus e 2 Clus e 3 Clus e 4 Monday 36,54% 51,92% 3,85% 7,69% Tuesday 31,48% 57,41% 3,70% 7,41% Wednesday 30,77% 63,46% 3,85% 1,92% Thu sday 32,69% 59,62% 5,77% 1,92% F iday 28,85% 59,62% 3,85% 7,69% Sa u day 11,32% 0,00% 39,62% 49,06% Sunday 0,00% 0,00% 44,23% 55,77% The e a e 22 wo king days ha ha e been g ouped in clus e s 3 o 4. A me iculous analysis e eals ha mos o hese days we e holiday. A de ailed lis o his ac is summa ised in Table V. Table V. – W ong classi ica ion o wo king days Nº OF DAY DATE FESTIVITY 6 06-01 Epiphany 70 11-03 None 75 16-03 None 77 18-03 F iday p e-Eas e 82 23-03 83 24-03 84 25-03 Eas e 87 28-03 Monday pos -Eas e 98 08-04 None 122 02-05 Wo king es i i y 123 03-05 Mad id es i i y 125 05-05 Long weekend 01-05 126 06-05 Long weekend 01-05 227 15-08 Assump ion o Ma y 231 19-08 None 235 23-08 None 285 12-10 Columbus Day 304 31-10 Long weekend 01-11 305 01-11 All Sain s 340 06-12 Spanish Cons i u ion Day 342 08-12 Immacula e Concep ion 360 26-12 Monday a e Ch is mas One commen has o be done abou he i s week o May. The eal holiday o he Wo king Day is he 1s May and o he Mad id Fes i i y he 2nd May. Howe e , 1s May 2005 was Sunday and bo h es i i ies we e pos poned one day. Wi h e e ence o weekends, he e a e six Sa u days ha ha e been g ouped ha ha e been g ouped as i hey we e wo king days, conc e ely, in clus e 1. A de ailed lis o hese Sa u days is shown in Table VI. Table VI. – Sa u days classi ied in w ong clus e s NUMBER OF DAY DATE 169 18 h June 176 25 h June 183 2nd July 197 16 h July 204 23 d July 211 30 h July Fig. 2. Days classi ied in o 4 clus e s ia K-means. h ps://doi.o g/10.24084/ epqj05.245 177 RE&PQJ, Vol. 1, No.5, Ma ch 2007 Fig. 3. Cha ac e is ic cu es o clus e s ob ained by K-means algo i hm in yea 2005. No e ha almos all six Sa u days a e consecu i e and belong o summe , excep o he 9 h July ha has been classi ied in o clus e 4. The whole yea is di ided in o 261 wo king days and 104 weekends o es i i ies. In Table V, i e days we e imp ope ly classi ied (11 h Ma ch, 16 h Ma ch, 8 h Ap il, 19 h Augus and 23 d Augus ). Hence, he a e age e o in wo king days is 1,92% (5 days ou o 261). Wi h ega d o weekends and es i i ies, he e a e 6 Sa u days which ha e been imp ope ly g ouped (18 h June, 25 h June, 2nd July, 16 h July, 23 d July and 30 h July). Gi en ha , he a e age e o o weekends and es i i ies is 5,77% (6 days o ou 104), he o al e o is 3,01% (11 days ou o 365). The ollowing ask consis s in explaining when a wo king day belongs o clus e 1 o o clus e 2 as well as when es i i ies belong o clus e 3 o o clus e 4: he e a e h ee zones clea ly di e en ia ed in Figu e 2 o bo h wo king days and es i i ies. F om he 1s Janua y un il he 18 h May (day numbe 144), mos o he wo king days belong o clus e 2. F om his day un il he 20 h Sep embe (day numbe 263) hey belong o clus e 1. Finally, om he 21s Sep embe (day numbe 264) un il he yea ends he wo king days belong again o clus e 2. In es i i ies he e is a simila si ua ion. F om he 1s Janua y un il he 27 h Ma ch (day numbe 86) mos o he es i i ies and weekends belong o clus e 3. F om his weekend un il 30 h Oc obe (day numbe 303) hey belong o clus e 4. Finally, om his weekend un il he yea ends he es i i ies and weekend belong o clus e 3. Consequen ly, a seasonal beha iou can be obse ed in he ene gy p ices ime se ies. The cha ac e is ic cu es o each clus e a e depic ed by Figu e 3. Especially ema kable is ha cu es associa ed o clus e s 3 and 4 (weekends and es i i ies) ha e s a ing and ending p ices highe han he ones associa ed o he wo king days (clus e s 1 and 2). The i s ones show hei highe alues in he la e a e noon. I is due o people consuming mo e elec ici y all he nigh long du ing weekends. On he o he hand, he second ones ha e hei peak p ices a midday when indus ies, comme ce and en e p ises a e ully unc ioning. C. FCM. Figu e 4 p esen s he six pa e ns ound by he FCM algo i hm o he ene gy p ices o he yea 2005. I can be no ed ha hese pa e ns a e no e y di e en o he pa e ns ob ained by using he K-means app oach. Fo he ep esen a ion o hese cu es he ollowing me hodology has been used. Fi s , he clus e wi h he maximum g ade o membe ship was assigned o e e y day. Then, he ep esen a ion was pe o med like in K-means algo i hm as i has depic ed in Figu e 4. h ps://doi.o g/10.24084/ epqj05.245 178 RE&PQJ, Vol. 1, No.5, Ma ch 2007 Focusing on Figu e 4, i can be clea ly di e en ia ed wo kinds o clus e s: clus e s 2, 3, 4 and 5 g oup all he wo king days while clus e s 1 and 6 he weekends. Ne e heless, he e a e some days ha ha e an appa en ly disco dan beha iou . Table VII shows he pe cen age o days classi ied in o he 6 clus e s. Table VII. – G ade o membe ship o days o clus e s CLUSTER 1 CLUSTER 2 CLUSTER 3 Monday 7,69% 15,38% 32,69% Tuesday 0,00% 23,08% 28,85% Wednesday 0,00% 28,85% 26,92% Thu sday 3,85% 25,00% 26,92% F iday 5,77% 25,00% 26,92% Sa u day 66,04% 3,77% 5,66% Sunday 53,85% 0,00% 0,00% CLUSTER 4 CLUSTER 5 CLUSTER 6 Monday 38,46% 1,92% 3,85% Tuesday 44,23% 0,00% 3,85% Wednesday 40,38% 0,00% 3,85% Thu sday 40,38% 0,00% 3,85% F iday 36,54% 1,92% 3,85% Sa u day 3,77% 0,00% 20,75% Sunday 0,00% 0,00% 46,15% The e a e 19 wo king days ha ha e been g ouped in clus e s 1 o 6. Table VIII summa izes he es i i ies ound in hese days. Table VIII. – W ong classi ica ion o wo king days Nº OF DAY DATE FESTIVITY 6 06-01 Epiphany 70 11-03 None 75 16-03 None 77 18-03 F iday p e-Eas e 82 23-03 Eas e 83 24-03 Eas e 84 25-03 Eas e 87 28-03 Monday pos -Eas e 98 08-04 None 122 02-05 Wo king es i i y 125 05-05 Long weekend 01-05 126 06-05 Long weekend 01-05 136 16-05 None 227 15-08 Assump ion o Ma y 304 31-10 Long weekend 01-11 305 01-11 All Sain s 340 06-12 Spanish Cons i u ion Day 342 08-12 Immacula e Concep ion 360 26-12 Monday a e Ch is mas Wi h e e ence o weekends, he e a e six Sa u days ha ha e been g ouped as i hey we e wo king days, conc e ely, in clus e 3. I is shown in Table IX. Table IX. – Sa u days classi ied in w ong clus e s NUMBER OF DAY DATE 176 25 h June 183 2nd July 190 9 h July 204 23 d July 211 30 h July 330 26 h No embe Fig. 4. Days classi ied in o 6 clus e s ia FCM. h ps://doi.o g/10.24084/ epqj05.245 179 RE&PQJ, Vol. 1, No.5, Ma ch 2007 Fig. 5. Cha ac e is ic cu es o clus e s ob ained by K-means algo i hm in yea 2005. No e ha almos all six Sa u days a e consecu i e and belong o summe , excep o he 16 h July ha has been classi ied in o clus e 1. The dis ance om 16 h July o clus e 1 is 0,7910 ( he clus e o which i belongs) while o clus e 2 is 0,7970 ( he clus e o which i should belong, assuming ha all he Sa u days in summe beha e as i hey we e a wo king day). The whole yea is di ided in o 261 wo king days and 104 weekends o es i i ies. In able VIII, ou days we e imp ope ly classi ied (11 h Ma ch, 16 h Ma ch, 8 h Ap il, 16 h May). Hence, he a e age e o in wo king days is 1,53% (4 days ou o 261). Wi h ega d o weekends and es i i ies, he e a e six Sa u days which ha e been imp ope ly g ouped (25 h June, 2nd July, 9 h July, 23 d July, 30 h July and 26 h No embe ). The e is also a es i i y, Columbus Day, which has been g ouped in clus e 2. Gi en ha , he a e age e o o weekends and es i i ies is 6,73% (7 days o ou 104), he o al e o is 3,01% (11 days ou o 365). In con as o wha i happened wi h K-means clus e ing, i is no ob ious o de e mine clea pe iods o he yea o days o belong o a speci ic clus e . The cha ac e is ic cu es o each clus e a e depic ed by Figu e 5. Especially ema kable is ha cu es associa ed o clus e s 1 and 6 (weekends and es i i ies) ha e s a ing and ending p ices highe han he ones associa ed o he wo king days (clus e s 2, 3, 4 and 5). The i s ones show hei highe alues in he la e a e noon. I is due o people consuming mo e elec ici y all he nigh long du ing weekends. On he o he hand, he second ones ha e hei peak p ices a midday when indus ies, comme ce and en e p ises a e ully unc ioning. 5. Conclusions I has been p o en ha u ilising clus e ing echniques in he p ices ime se ies is as powe ul as use ul. Two algo i hms ha e been used in o de o classi y he elec ici y p ice cu es o he Spanish Ma ke : K-means and Fuzzy C-means. The clus e analysis ca ied ou ia bo h K-means and Fuzzy C-means algo i hms yielded e y ele an in o ma ion: wo king days ha e beha iou diame ically opposi e o weekend and es i i ies. The a e age e o commi ed in hei classi ica ion was 3,01%. Only 11 days o he yea 2005 we e imp ope ly g ouped which means a g ea deg ee o accu acy o bo h echniques wi h ei he 4 (K-means) o 6 (Fuzzy C-means) clus e s. The esul s ob ained may be ex emely p o i able. Fu u e wo ks will be di ec ed in he p edic ion o day-ahead p ices once known he p e ious clus e ing. In sho , N clus e s will be ob ained and di e en models will be applied o e e y clus e o imp o e he quali y o he ma ke p ice o ecas ing. Acknowledgemen s The au ho s would like o acknowledge he inancial suppo om he Spanish Minis y o Science and Technology, p ojec s TIN2004-00159 and ENE-2004- 03342/CON, and om he Jun a de Andalucía, p ojec P05-TIC-00531. h ps://doi.o g/10.24084/ epqj05.245 180 RE&PQJ, Vol. 1, No.5, Ma ch 2007 Re e ences [1] M. A. Plazas, A. J. Conejo and F. J. P ie o, "Mul ima ke Op imal Bidding o a Powe P oduce ", IEEE T ansac ions on Powe Sys ems, ol. 20, no. 4, No embe 2005. [2] Rui Xu and Donald C. Wunsch II, “Su ey o Clus e ing Algo i hms”, IEEE T ansac ions on Neu al Ne wo ks, ol. 16, no. 3, May 2005. [3] A. J. Conejo, M. A. Plazas, R. Espínola and A. B. Molina, “Day-Ahead Elec ici y P ice Fo ecas ing Using he Wa ele T ans o m and ARIMA Models”, IEEE T ansac ions on Powe Sys ems, ol. 20, no. 1, May 2005. [4] R. C. Ga cía, J. Con e as, M. an Akke en and J. B. C. Ga cía, “A GARCH Fo ecas ing Model o P edic Day-Ahead Elec ici y P ices”, IEEE T ansac ions on Powe Sys ems, ol. 20, no. 1, May 2005. [5] N. Amjady, “Day-Ahead P ice Fo ecas ing o Elec ici y Ma ke s by a New Fuzzy Neu al Ne wo k”, IEEE T ansac ions on Powe Sys ems, ol. 21, no. 2, May 2006. [6] H. Ze ipou , K. Bha acha ya and C. A. Cañiza es, “Fo ecas ing he Hou ly On a io Ene gy P ice by Mul i a ia e Adap i e Reg ession Splines”, IEEE Powe Enginee ing Socie y Gene al Mee ing, 2006. [7] A. T oncoso, J. M. Riquelme, A. Gómez Expósi o, J. L. Ma ínez Ramos and J. C. Riquelme, “Elec ici y Ma ke P ice Fo ecas ing Based on Weigh ed Nea es Neighbo s Techniques”, IEEE T ansac ions on Powe Sys ems, in p ess, 2006. [8] W. Wang, Y. Zhang, Y. Li and Xiaona Zhang, “The Global Fuzzy C-Means Clus e ing Algo i hm”, P oceedings o he 6 h Wo ld Cong ess on In elligen Con ol and Au oma ion, June 2006. [9] OMEL. Ma ke Ope a o o he Elec ici y Ma ke o Mainland Spain. [Online] A ailable: h p://www.omel.es. h ps://doi.o g/10.24084/ epqj05.245 181 RE&PQJ, Vol. 1, No.5, Ma ch 2007