Day-ahead Electricity Price Forecasting Using PSO -Based LLWNN Model

Transcription

1 Inernaonal Journal of Energy Engneerng (IJEE) Aug. 3, Vol. 3 Iss. 4, PP Day-ahead Elecrcy Prce Forecasng Usng PSO -Based LLWNN Model Prasana kuar Pany, Sak Prasad Ghoshal Deparen of Elecrcal Engneerng, DRIEMS,Cuack, Odsha, Inda Deparen of Elecrcal Engneerng, Naonal Insue of Technology, Durgapur, Wes Bengal, Inda Prasanpany@gal.co, spghoshalndgp@gal.co Absrac - Prce forecasng has becoe an poran acvy for arke parcpans n elecrc power ndusry for developng her bddng sraeges. The work presened n hs paper akes use of parcle swar opzaon based local lnear wavele neural neworks (LLWNN) o fnd he Marke Clearng Prce (MCP) for a gven perod, wh a ceran confdence level. The resuls of he new ehod show sgnfcan proveen n he prce forecasng process. Keywords- Elecrcy Prce, Forecasng, Wavele Neural Nework (WNN), Local Lnear Wavele Neural Nework (LLWNN), Parcle Swar Opzaon (PSO), Marke Clearng Prce (MCP), Weekly Mean Absolue Percenage Error (WMAPE) I. INTRODUCTION The elecrc power ndusry n any counres all around he world s evolvng no an era of arke econoy wh deregulaon and free copeon. The undersandng of elecrc power supply as a publc servce s beng replaced by he noon ha a copeve arke s a ore approprae echans o supply energy o consuers wh hgh relably and low cos. A key eleen of he elecrcy secor resrucurng s he esablshen of a arke-drven prce for elecrcy. The prcng syse of elecrcy plays an poran role n a copeve arke. In he power arke, he elecrcy prce depends on he evoluon of balance beween he deand for elecrcy and he avalable supply. A he sae e, any oher arke facors also nfluence he elecrcy prce, such as econoc growh, weaher, he power-plan x, he prces of fuels and he sraegc behavor of large players (usually on he generaon sde). An acve, fully copeve and lqud spo arke for wholesale elecrcy wll ranslae he physcal rsk of nadequae capacy no a fnancal rsk of hgh prces and place hgher requreens on prce forecasng. Producers and consuers rely on prce forecasng nforaon o propose her correspondng bddng sraeges. If a producer has an accurae forecas of he prces, can develop a bddng sraegy o axze s prof. On he oher hand, a consuer can ake a plan o nze hs own elecrcy cos f an accurae prce forecas s avalable. Due o he coplcaed bddng sraeges lnked wh he gang by arke parcpans and specal elecrc prce characerscs [], such as hgh frequency, nonsaonary behavor, ulple seasonaly, calendar effec, hgh volaly, hgh percenage of unusual prces, hard nonlnear behavor, ec. and led nforaon o he arke parcpans, an accurae elecrcy prce forecasng s a challengng ask. In he pas few years dfferen echnques have been proposed o forecas elecrcy prce. Saonary e seres and non-saonary e seres odels, neural nework and s exended odels [- 7],suppor vecor achne(svm) [8-9], and an npu/oupu hdden Markov odel(iohmm) [], ec. have been appled for elecrcy prce forecasng. Auo regressve negraed ovng average (ARIMA) [], dynac regresson (DR) and ransfer funcon (TF) [], and generalzed auo regressve condonal heeroscedascy (GARCH) [3] are he os wdely used e seres odels. Alhough, e seres echnques are well esablshed o have good perforance, however, due o he use of lnear odelng os of he have dffcules n predcng he hard nonlnear behavors and rapd changes of he prce sgnals. As elecrcy prce s a non-lnear funcon of s npu feaures, he behavor of elecrcy prce sgnal can no be copleely capured by he e seres echnques. On he oher hand, arfcal nellgence (AI) echnques have been exensvely used by any researchers for he elecrcy prce forecasng. A wavele neural nework (WNN)--, frs proposed by Zhang e al. [4] as an alernave o he classcal feed- forward neural nework (FFNN) for approxang arbrary nonlnear funcons, nspred by boh he FFNN and wavele heory has been eerged as a powerful new ype of ANN. A shorcong of WNN s ha for hgher densonal probles any hdden layer uns are needed. Curse- of densonaly s anly an unsolved proble n WNN heory whch brngs soe dffcules n applyng he WNN o hgh densonal probles. In order o ake advanage of he local capacy of he wavele bass funcons whle no havng oo any hdden uns, an alernave ype of wavele neural nework known as local lnear wavele neural nework (LLWNN) has been proposed [5]. The Parcle Swar Opzaon (PSO), a populaon based opzaon ehod frs proposed by Kennedy and Eberhar [6] s nroduced for ranng he local lnear wavele neural nework. To he bes of he auhors knowledge, a PSO based

2 Inernaonal Journal of Energy Engneerng (IJEE) Aug. 3, Vol. 3 Iss. 4, PP Local Lnear Wavele Neural Nework (LLWNN) has no ye been esed for elecrcy prce forecasng. In hs paper an LLWNN odel whch soohly bases funcon of hdden layer neurons accordng o ranng daa se aps he npu-oupu space by adapng he shape of wavele s exaned for elecrcy prce predcon of he Onaro elecrcy arke. The proposed odel does no requre exernal decoposer/coposer. So rsk of losng hgh frequency coponens of elecrcy prce sgnal s avered. I s found ha predcon of elecrcy prce based on LLWNN odel gves beer perforance because of s favorable propery of odelng he non-saonary hgh frequency sgnals such as elecrcy prce. The res of he paper s organzed as follows: Secon II descrbes an characerscs of he elecrcy prce seres. Elecrcy prce forecasng usng LLWNN odel s descrbed n secon III. Tranng of LLWNN odel by PSO algorh s descrbed n secon IV. Secon V descrbes he sascal easures used o evaluae he forecasng perforance. Secon VI presens resuls and dscussons on elecrcy prce forecas of Onaro elecrcy arke. Fnally, secon VII provdes concludng rearks. II. PRICE-DATA ANALYSIS To develop an approprae odel for prce forecasng, we exane he an characerscs of he hourly prce seres n hs secon. To llusrae he forecasng procedure he elecrcy prces for he Onaro power arke fro s June 4 o 6 h Dec., 4 s used for predcon. An analyss repored n [8], [9], [] was o fnd ou whose paraeers could be used o successfully predc he average Marke Clearng Prce (MCP). Accordng o he daa saples for each hour of he day and each day of he onh, s clear ha he prce dynacs have ulple seasonal paerns, correspondng o a daly and weekly perodcy, respecvely, and are also nfluenced by a calendar effec,.e. weekends and holdays. These properes are jus he sae as hose of load. However, n conras o he load-e seres, here are several parcular properes of prce. The hourly prce curve s vared and flucuaes wh a hgh frequency, and here s also a hgh percenage of an abrup change or spkes n he prce curve (anly n perods of hgh deand). The prce presens hgh volaly and non-consan ean. The abrup changes and volaly of prce can be refleced as a swch n he prce seres dynacs owng o he dscree behavors n copeors sraeges. In oher words, here exs dfferen reges n he prce e seres, whch generally gve rse o pece-wse-saonary dynacs. Based on such analyss, we use a hybrd odel o classfy he non-saonary prce-daa se o several pecewse saonary daa subses, on whch hghly accurae learnng and predcon can be expeced, copared wh he convenonal approaches. If prce a hour h (Ph) s o be forecased, he prce nforaon of prevous hours up o hours.e. p h-, p h-,... p h- should be aken as a par of he npu of shor-er prce forecasng (STPF) odel. The auo co-relaon funcon (ACF) can be used o denfy he degree of assocaon beween daa n he prce seres separaed by dfferen e lags.e. prevous prce. Oher knd of sensvy analyss can also be very helpful n deernng he varable whch has sgnfcan nfluence on he syse prce. In order o denfy he load nfluence on prce, load a hour o be predced a dfferen lagged hours ( d h-, d h-,... d h- ) s also ncluded as an exogenous varable n he npu se of he forecasng odels. The hsorcal hourly daa of 7 days pror o he day whose prce o be predced have been consdered o buld he forecasng odel. Hence he oal daa pons are equal o 7 x 4 = 68. Snce he proposed odel uses prce daa 7 hours ago o predc he prce ph, 68-7=6 npu vecors are used o develop he forecas odel. III. ELECTRICITY PRICE FORECASTING USING LLWNN The LLWNN odel for he hourly Onaro energy prce s developed o forecas for hree e perods. The frs perod coprses wo consequen weeks fro Aprl 6 o May 9, 4, whch are referred as Week-, and Week- respecvely n hs paper, he Onaro arke presened s lowes sprng deand durng hs perod. The second perod conans suer peak deand weeks fro July 6 o Augus 8, 4, whch are referred o as Week-3 o Week-4, respecvely. The las perod ncludes wo hgh deands wner weeks n 4, sarng on Deceber 3 and endng on Deceber 6, and hese weeks are referred as Week-5 and Week-6, respecvely. One hour ahead prce forecasng usng seven hours before prce daa, wenyfour hours ahead forecasng usng seven days before prce daa have been used n he proposed odel. Afer he one sep ahead ranng, he nex hour predcon s evaluaed. Mulple seps ahead are reached va recurson.e. by feedng npu varables wh odel s oupus. The nex hour forecass are perfored for every hour of he day. The odel s reraned a he end of each day o ncorporae he os recen nforaon. The concaenaon of 7 days ranng wndows, for a parcular day, s shfed one day-ahead and forecass for he nex 4 hours are copued. Accordng o wavele ransforaon heory, wavele n he followng for s a faly of funcons generaed fro one sngle funcon ψ(x) by he operaon of dlaon and ranslaon. Ψ(x) whch s localzed n boh e space and he frequency space s called a oher wavele. - -

3 Inernaonal Journal of Energy Engneerng (IJEE) Aug. 3, Vol. 3 Iss. 4, PP y= -/ n { y = a yç : a, bîr, Îz} x= ( x, x,... x ) a = ( a, a,... a b = ( b, b... b ) n n n æ x-bö ç a è ø ) () The paraeers a and b are he scale and ranslaon paraeers, respecvely. Accordng o he prevous researches, he wo paraeers can eher be predeerned based on wavele ransforaon heory or be deerned by a ranng algorh. In he sandard for of wavele neural nework, he oupu of a WNN s gven by -/ æ x-b ö f ( x) = wy ( x) = w a y ç a bî R Îz} a :,, () = = è ø The above wavele neural nework s a knd of bass funcon neural nework n he sense of ha he waveles consss of he bass funcon. An nrnsc feaure of he bass funcon neworks s he localzed acvaon of he hdden layer uns, so ha he connecon weghs assocaed wh he uns can be vewed as locally accurae pecewse consan odels whose valdy for a gven npu s ndcaed by he acvaon funcons. Copared o he ullayer percepron neural nework, hs local capacy provdes soe advanages such as he learnng effcency and he srucure ransparency. However, he proble of bass funcon neworks s also led by. Due o he crudeness of he local approxaon, a large nuber of bass funcon uns have o be eployed o approxae a gven syse. A shorcong of he wavele neural nework s ha for hgher densonal probles any hdden layer uns are needed. In order o ake advanage of he local capacy of he wavele bass funcons whle no havng oo any hdden uns, LLWNN has been used as an alernave neural nework. The dfference beween a local lnear wavele neural nework (LLWNN) and a convenonal wavele neural nework (WNN) s ha he connecon weghs beween he hdden layer and oupu layer of convenonal WNN are replaced by a local lnear odel. The oupu of LLWNN s gven by Y= = ( w + w x wn xn) y ( x) (3) Where, nsead of he sragh forward wegh w (pecewse consan odel), a lnear odel v = w + w x w x n ns nroduced. The acves of he lnear odels ψ (x) (=,, ,n), hus v (=,, n) are deerned by he assocaed locally acve wavele funcons v s only locally sgnfcan. The archecure of he proposed odel s shown n Fg.-. Y Fg. General srucure of a local lnear wavele neural nework. - -

4 Inernaonal Journal of Energy Engneerng (IJEE) Aug. 3, Vol. 3 Iss. 4, PP Here = n s he order of he dynacal syse whch s predeerned consan. The oher wavele s y (x) = - x -x s e (4) æ x-cö -ç è s ø y ( x ) = e (5) Where x = p + p pn IV. LEARNING ALGORITHM The usually used learnng algorh for LLWNN s graden decen ehod o ge all he unknown paraeers of nework.e. ranslaon and dlaon coeffcens, weghs whch are randoly nalzed a begnnng snce he funcon copued by he LLWNN odel s dfferenable wh respec o all enoned unknown paraeers. Bu s dsadvanages are slow convergence speed and easy say a local nu. Hence he proposed odel s raned by he PSO algorh. Parcle swar opzaon s bascally developed hrough sulaon of brd flockng n wo-denson space. The poson of each agen s represened by XY axs poson and also he velocy s expressed by vx and vy. Modfcaon of he agen poson s realzed by he poson and he velocy nforaon. Brd flockng opzes a ceran objecve funcon. Each agen knows s bes value so far (pbes) and s XY poson. Moreover, each agen knows he bes value so far n he group (gbes) aong pbes. Manly each agen res o odfy s poson usng he followng nforaon. (a) The dsance beween he curren poson and pbes. (b) The dsance beween he curren poson and gbes. Velocy of each agen can be odfed by he followng equaon: v v ĺƽǵ ƅ ŢŖȖ Ϝ ĺƽǵ ƅ ŖȖ Ϝ (6) where, v k s he velocy of agen a eraon k, w s called nera facor, c and c are known as acceleraon coeffcens, s k s he curren poson of agen a eraon k, pbes s he prevous bes of agen and gbes s he global bes parcle of he group. Usng he above equaon, a ceran velocy, whch gradually ges close o pbes and gbes, can be calculaed. The curren poson (searchng pon n he soluon space) can be odfed by he followng equaon: k+ k k+ S = S + V (7) The frs er of (6) s he prevous velocy of he agen. The second and hrd ers are used o change he velocy of he agen. The nera wegh w s nroduced o prove PSO perforance. Suable selecon of nera wegh w provdes a balance beween global and local exploraon and exploaon. The general flow char of PSO for opzng a local lnear wavele neural nework can be descrbed as follows: Sep. Generaon of nal condon of each agen Inal searchng pons ( S ) and velocy ( V ) of each agen are usually generaed randoly whn he allowable range. Noe ha he denson of search space consss of all he paraeers used n he local lnear wavele neural nework as shown n Equaons () and (3). The curren searchng pon s se o pbes for each agen. The bes-evaluaed value of pbes s se o gbes and he agen nuber wh he bes value s sored. Sep. Evaluaon of searchng pons of each agen. The objecve funcon value s calculaed for each agen. If he value s beer han he curren pbes of he agen, he pbes value s replaced by he curren value. If he bes value of pbes s beer han he curren gbes, gbes s replaced by he bes value and he agen nuber wh he bes value s sored. Sep.3 Modfcaon of each searchng pon The curren searchng pon of each agen s changed usng (6) and (7). Sep.4 Checkng he ex condon. - -

5 Inernaonal Journal of Energy Engneerng (IJEE) Aug. 3, Vol. 3 Iss. 4, PP The curren eraon nuber reaches he predeerned axu eraon nuber, hen exs oherwse goes o Sep. V. ACCURACY MEASURES Several s easures defned n [7] have been used o evaluae he perforance of LLWNN based forecasng odel. Mean absolue percenage (MAPE) s used o assess predcon accuracy of he developed odels n he paper. The absolue (AE) s defned as AE Pa, - Pf, = (8) P a, The daly ean absolue (DMAE) can becoe copued as follows: The daly ean absolue percenage The weekly ean absolue And The weekly ean absolue percenage DMAE = (DMAPE) = (WMAE) = (WMAPE) = 4 4 = AE (9) 4 = 68 = 68 = VI. RESULTS & ANALYSIS AE () AE () AE () The effecveness of he LLWNN odel s deonsraed on SMP predcon n Onaro elecrcy arke for he year 4. The forecased prce obaned wh proposed odel durng sprng es weeks (Week-, Week-) are shown n Fg. and Fg.4 along wh acual prce and he correspondng s shown n Fg. 3. and Fg.5. forecased prce es prce Fg.. Dynac syse oupu and odel oupu for Week- daa se Fg. 3. ly for Week- daa se - 3 -

6 Inernaonal Journal of Energy Engneerng (IJEE) Aug. 3, Vol. 3 Iss. 4, PP forecased prce es prce Fg. 4. Dynac syse oupu and odel oupu for Week- daa se. Fg. 5. ly for Week- daa se. forecased prce erro.. es prce Fg. 6. Dynac syse oupu and odel oupu for Week-3 daa se. Fg. 7. ly for Week-3 daa se forecased prce es prce forecased prce Fg. 8. Dynac syse oupu and odel oupu for Week-4 daa se Fg. 9. ly for Week-4 daa se es prce Fg.. Dynac syse oupu and odel oupu for Week-5 daa se. Fg.. ly for Week-5 daa se

7 Inernaonal Journal of Energy Engneerng (IJEE) Aug. 3, Vol. 3 Iss. 4, PP forecased prce es prce Fg.. Dynac syse oupu and odel oupu for Week-6 daa se. Fg. 3. ly for Week -6 daa se The forecased prce obaned wh proposed odel durng suer es weeks (Week-3, Week-4) are shown n Fg.6 and Fg.8 along wh acual prce and he correspondng s shown n Fg. 7 and Fg.9. The forecased prces obaned wh proposed odel durng wner es weeks (Week-5, Week-6) are shown n Fg. and Fg. along wh acual prce and he correspondng s shown n Fg. and Fg.3. I can be seen fro fgures ha he predcaed elecrcy prce of he es weeks are que close o he acual one. The weekly MAPEs of he generaed forecass, usng he odels developed n hs paper for he sx weeks under sudy, are presened n Table I. For coparson purposes, he weekly MAPEs of he generaed forecas, usng heursc ehod (PM),ndependen elecrcy syse operaor (IESO) odel(pm), ulple lnear regresson (MLR) odel (PM3), neural nework (NN) odel (PM4), wavele NN odel (PM5) [8] are also presened n hs able. Accuracy of LLWNN odel s beer han he oher odels n Week-, Week-3, Week-5 and Week-6. Overall, accuracy of LLWNN odel s beer han he oher odels The bes resuls were acheved for Week-3, whch was one of he hgh deand weeks of 4 suer. Despe he hgh deand, prces on all seven days were n he expeced range durng hs week. TABLE I WMAPE IN ONTARIO MARKET FOR SIX WEEKS Tes perod Week no. PM PM PM 3 P M 4 PM 5 LLWNN Apr. 6 o May,4 Week May 3-9,4 Week July 6 o Aug.,4 Week Aug. -8,4 Week Dec. 3-9,4 Week Dec. -6,4 Week Average The hghes forecas s occurred durng Week-4. In hs week, he prces are unusually volale for he frs wo days of he week and unusually seady for he res. Local Lnear Wavele Neural Nework raned by PSO algorh has been convergen a eraon 38 wh average weekly ean absolue percenage (WMAPE) of for es daa se. We beleve ha hese resuls are reasonably accurae for a sudy spannng one whole year. Very less ranng e shows he hgher convergence rae of LLWNN odel o predc he wnd power generaon wh hgher accuracy. A LLWNN perfors beer han all consdered ehods, because boh sooh global and sharp local varaons of elecrcy prce sgnal can be effecvely represened by he wavele bass acvaon funcon for hdden layer neurons whou any exernal decoposer / coposer and also no havng oo any hdden uns. Consderng all hese pons, he perforance of he proposed odel s sasfacory. VII. CONCLUSIONS In hs paper, energy prce forecasng by usng a local lnear wavele neural nework (LLWNN) odel s used. The characersc of he nework s ha he sragh forward wegh s replaced by a local lnear odel and hereby needs only saller waveles for a gven proble han he coon wavele neural neworks. I s also observed ha accuracy of LLWNN odel s beer han oher odels wh hgh converges and ou -perfored n he forecasng of he elecrcy prce because of s favorable propery for odelng he non-saonary and hgh frequency sgnal such as elecrcy prce. ACKNOWLEDGMENT The auhors would lke o hank he expers who have conrbued owards developen of he eplae

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