Optimized Calculation of Hourly Price Forward Curve (HPFC)

Transcription

1 World Academy of Scece, Egeerg ad Techology Ieraoal Joural of Elecrcal ad Comuer Egeerg Vol:, No:9, 008 Omzed Calculao of Hourly Prce Forward Curve (HPFC) Ahmed Abdolkhalg Ieraoal Scece Idex, Elecrcal ad Comuer Egeerg Vol:, No:9, 008 wase.org/publcao/3000 Absrac Ths aer exames may mahemacal mehods for moldg he hourly rce forward curve (HPFC); he model wll be cosruced by umerous regresso mehods, lke olyomal regresso, radal basc fuco eural eworks & a furrer seres. Examao he models goodess of f wll be doe by meas of sascal & grahcal ools. The crera for choosg he model wll deed o mmze he Roo Mea Squared Error (RMSE), usg he correlao aalyss aroach for he regresso aalyss he omal model wll be dsc, whch are robus agas model mssecfcao. Learg & suervso echque emloyed o deerme he form of he omal arameers corresodg o each measure of overall loss. By usg all he umercal mehods ha meoed revously; he exlc exressos for he omal model derved ad he omal desgs wll be mlemeed. Keywords Forward curve, furrer seres, regresso, radal basc fuco eural eworks. I. INTRODUCTION HIS aer o llusrae a model for he valuao ay T eergy marke, he quaave aalyss wll be deely emhaszed o creaes a model ha reflec he marke behavor, hs model wll be he hourly rce forward curve (HPFC), whch s he goal of hs aer. The model rovded wll be cosruced bascally from he so rces of ay arbrary eergy marke; he omum model wll be soughed ou for he valuao he markes. Secal aeo wll ad o Albera elecrcy rces. Albera's elecrcy marke srucure wll revewed followed wh a descro of he fudameal ecoomc drvers of he so rces jus o udersad he behavor of he rces hs marke, he, he quaave aalyss s aled o creae & mleme he model (HPFC), bu before ha, a h of he forward rce curve s gve. II. HOURLY PRICE FORWARD CURVE A. Defo of he "Hourly Forward Prce Curve? A Hourly Prce Forward Curve ca be defed as (HPFC) Mauscr receved Seember 9, 008.Ths work was suored by he Dearme of Elecrcal Egeerg, Omar Al Mukhar Uversy, Albda, Lbya. Ahmed Abdolkhalg s a Asssa Lecurer he Dearme of Elecrcal ad Elecroc Egeerg, The Uversy of Omar Al Mukhar, Lbya (hoe: ; e-mal:almsray@homal.com). as a ls of rces as of oday for he delvery of elecrcy for examle a a seres of dffere os of me he fuure as show Fg.. These rces may be average rces for each hour erod for a secfc over a gve erod (e.g. day / week ec) as show Fg., rereses he erm-srucure of forward rces hourly resoluo. The secfc curve requred deeds o he urose of he curve. For examle, he lookg o rce a swa based o he average of rces over a week hs s he forward curve would requred []. Prce Mw/h Tme Hours Fg. A vsoary lo of Hourly Prce Forward Curve B. Dsco bewee a Forward Curve ad a Forecas The forward rce curve s used o mark rades o marke. I lqud ad dee markes such as eres rae, foreg currecy, ad wdely raded hyscal commodes, forward curves are easly avalable ad derved. Bu youg elecrcy markes, s less clear wha s mea by a forward curve []. Ofe hs s derved from a schedule of so rce forecass, so ha he curve s used o redc he lkely so rce of elecrcy several years he fuure. There are some radoal models [3,5,7,8] for racoer o choose for modelg so rces curve. Fscher Black ad Myro Scholes ublshed 973 her semal aer o oos rcg. The Black-Scholes oo rcg model whch was based o he exeso of Browa moo corbued o he exlosve growh radg of dervaves. The orgal Black-Scholes model allowed oly for rcg oos o a o-dvded ayg sock [6]. Exesos o he Black-Scholes model, such as he Garma-Kohl Hage ad Black (976) model, allowed for rcg comrehesve commody oos, ad oos o fuures resecvely. Regardless of he srog assumos ha uderlay hs model, he smlcy of he Black's formula drecly made a acceed choce amog racoers. Uforuaely, Black's model makes several resrcve assumos. I arcular, assumes ha he evoluo of fuures rces ca Ieraoal Scholarly ad Scefc Research & Iovao (9) scholar.wase.org/ /3000

2 World Academy of Scece, Egeerg ad Techology Ieraoal Joural of Elecrcal ad Comuer Egeerg Vol:, No:9, 008 Ieraoal Scece Idex, Elecrcal ad Comuer Egeerg Vol:, No:9, 008 wase.org/publcao/3000 be modeled usg Geomerc Browa Moo (GBM). The use of Black's model may be arorae cases where hs assumo cao be made. The Albera Elecrcy marke s a clear examle of a marke where he assumo of Geomerc Browa Moo o model so res curve erms of hourly rces s ureasoable. The occurrece of la ouages a radom ervals eds o geerae dscouous skes so rces. Such skes are cosse wh a couous me radom walk, ad mus be exlcly accoued for a model. III. ANALYSIS OF ALBERTA ELECTRICITY MARKET PRICES DRIVERS Today Albera Power Pool Marke orgazes ad oeraes he hyscal delvery of elecrcy he rovce (Albera, Caada) ad he ere elecrcy urchased he rovce [9] mus ass hrough hs marke. All he geeraors offer o he ool o suly a secfc amou of ower for a arcular durao a a secfc rce. A he same me, cosumers u forward bds o buy elecrcy a or below a exacg rce. The ower ool afer ha uses hese bds ad offers o creae a mer order. Ths mer order allows us o be dsached from lowes rce o hghes rce o serve load. Usg he mer order, he sysem margal rce ( SMP ) s deermed, ad he average of hs rce for each hour becomes he offcal ool rce. The rce for a gve hour may rage from $0 o $000, he curre rce ca Albera. A. Albera Power Pool Prce Equlbrao If he demad Albera Pool as show Fg., s remedously sesve eve o large chages rce. Ths s because, he average cosumer of elecrcy, he household, dvduals gve o hough o he curre ool rce whe hey ur o her alaces or cook der. The resul ha ca be assumed a ay o me, demad s relavely elasc, assumg he shae of a vercal le. Demad s served mosly by coal-fred, gas-fred, ad hydro geerao facles. The order whch hese us are dsached deeds o her varable cos of roduco ad bddg behavor, whle suly s currely a kked curve, becomg very see a hgh quaes. From a urely hyscal o of vew, suly ad demad mus always be equlbrum. Ths s because elecrcy cao easly be sored for laer cosumo. Ths characersc of elecrcy markes makes cosderably dffere from oher commody markes lke aural gas ad crude ol, boh of whch ca be sored ad resold laer whe rces are hgher, he rmary cosequece of o beg able o sore elecrcy he Albera elecrcy marke s exreme rce volaly whe suly s cosraed. Fg. Pool Prce &Pool Demad Forecas vs. Acual B. The chagg Albera Elecrcy Prces (Volaly) Fg. 3 llusraes a examle of hourly rces, whch are measured dollars er megawa hour ($/MWH, Caada dollars beg he moeary u here ad he remader of hs hess), durg he erod Ja, 000 o Oc 3, 00. [6], a hgh chagg he rces durg he seasoally see. Here, elecrcy rces exhbs he mos comlcaed cyclcal aers of all eergy commodes, here are eaks reacg o heag ad coolg eeds. Fg. 3 A lo of hourly Albera elecrcy rces from Ja 000 o Oc 3, 00 The skess of he daase could be worh clearg umeral values as follow: whle he mea of he daase s oly $56.47 ($/Mwh), he daase s characerzed by occasoal excursos o he $ ad $ level. These erods of hgh rces are usually followed by a reur o a rce of uder $ IV. MATHEMATICAL METHODS FOR MODELING HPFC Modelg he HPFC ay Eergy Marke s ecessary for kowg he rces behavor he fuure. Regresso Techque wll be used for hs o cosruc he omum model. There are may Regresso mehods for modelg ad each mehod ca roduce redco wh dffere accuracy. I he followg secos, hese mehods wll be dscussed: A. Polyomal Regresso The geeral equao of he Polyomal Regresso [] has he followg form: Y ( ) 3... = () Where a ooal cosa s erm ad 0 hrough are coeffces of creasg owers of. The order of he Ieraoal Scholarly ad Scefc Research & Iovao (9) scholar.wase.org/ /3000

3 World Academy of Scece, Egeerg ad Techology Ieraoal Joural of Elecrcal ad Comuer Egeerg Vol:, No:9, 008 Ieraoal Scece Idex, Elecrcal ad Comuer Egeerg Vol:, No:9, 008 wase.org/publcao/3000 olyomal wshed whch f he daa mus be secfed. A lear equao he form Y ( ) = 0 + x () A quadrac olyomal equao he form Y ( ) = (3) A cubc olyomal equao he form 3 Y ( ) = (4) Hgher ( h h 4 or 5 ) order olyomals are useful for aems o descrbe daa os as fully as ossble, bu he erms geerally cao be meagfully erreed ay hyscal sese. Hgher order erms ca lead o odd ad ureasoable resuls, esecally beyod he rage of he values []. Geeral olyomal model (Polyomal Models Oe Varable) s j Y ( ) = + (5) 0 j = Assumed ha s ossble ha he daa ca be modeled by a quadrac olyomal fuco gve equao (4); = \ y ad he ukow coeffces, ad 0 ca be comued by dog a leas squares f, whch mmzes he sum of he squares of he devaos of he daa from he model. If here are sx equaos hree ukows, he here are sx daa recorded assumed. y y y 3 = y 4 y 5 y The soluo s foud wh he backslash oeraor ( j y= X = \Y ). = X? (6) Alg hs o he daa of Albera Marke Prces, he smulao of he hourly forward rce curve by meas of ler olyomal regresso fouded as show Fg. 4. B. Radal Bass Fucos Neural Neworks Le's suose ha he aroxmao of a real valued fuco y (x) by Y (x) s gve by he se of values Y = ( y,..., y ) a he dsc os d X = ( x,... x ) R. Ths could be doe by alyg he secod roosed mehod whch s "RBF" ) Wha s a RBF? Le y(x) o be a Radal Bass Fuco he form: Y ( x) = ( x) = = k j= w G( x x a γ ( x) j j ) + ( x) Where, (x) s a olyomal of degree a mos k, s a real-valued wegh,. deoes he Eucldea orm, G s a basc Gaussa fuco, G: R + R x x, ad s smly a dsace -- how far x x s from he o.(x) s a bass he sace of olyomals of degree m, ad m deeds o G. Noe ha he mos oular Gaussa RBF does o eed (x),.e., (x) = 0. Thus, he aroxmag fuco s a weghed sum of RBF's G ( x x ). ) Comuao of he Coeffces (Weghs w ) The coeffces are foud by solvg he lear sysem: T ( G + λ I) w + Γ a = d (8) Γw = 0 Where I s he dey marx, ad G s a so-called desg marx ad d = y = [ y y... y ] T.Whe, here s o olyomal erm he soluo follows he from ( G + λ I) w = d Look a hs smle examle for formg a Gaussa G marx for clarfyg ad avodg ambgues. w (7) Fg. 4 Smulao of HPFC by usg quadrac olyomal regresso Ieraoal Scholarly ad Scefc Research & Iovao (9) scholar.wase.org/ /3000

4 World Academy of Scece, Egeerg ad Techology Ieraoal Joural of Elecrcal ad Comuer Egeerg Vol:, No:9, 008 3) Neural Neworks Ierreao of he RBF Aroxmao Scheme A src erolag regularzao ework [4] for a oe dmesoal u x. The rag daa se comrses 5 examles as show Fg. 5. Ceers corresod o he us, ad all varaces are equal. Bas show s o madaory ad does o follow from he equaos below: So, sead of: y( x) = w G( x ) () = x subsue by a aroxmao : Y( x) = wg( x ) () = c Where << P; he vecors c are called ceers ( 0% N). Ths ca be rereseed aga grahcally afer he APPROXIMATION as follows: Ieraoal Scece Idex, Elecrcal ad Comuer Egeerg Vol:, No:9, 008 wase.org/publcao/3000 Fg. 5 Ierolag regularzao ework for a oe dmesoal u x Thus, our model wll be erms of weghed ad ceered Gaussa he form y( x) = Here x exress he me. = w G( x x Ths ca be rereseed grahcally as follows: The soluo s gve by y = G w w = G y (0) ya = Gw = y Ths wll be comuaoally exesve for P >>, roblem! ) (9) Alg hs o he daa of Albera Marke Prces wh sgma equals 900 ad umber of euros equals 9, he smulao of he hourly forward rce curve by meas of RBF s show he Fg. 6 below. Fg. 6 Smulao of HPFC by he Radal Bass Fucos Neural Neworks From Fg. 6, seems ha usg Radal Bass Fucos Neural Neworks o smulae he HPFC s much beer ha he oher model smulaed by quadrac olyomal regresso. Laer, he valdao of all of hese models ad he model by meas of Fourer seres whch wll roduce he ex seco wll be checked. C. A Fourer seres (Trgoomerc Polyomals) Fourer seres s a mahemacal ool used for aalyzg a arbrary fuco by decomosg o a weghed sum of much smler susodal comoe fucos somemes referred o as ormal Fourer modes. The Fourer seres exaso of s: a0 Y ( ) = + [ a cos( w) + b s( w) ] (3) = Ieraoal Scholarly ad Scefc Research & Iovao (9) scholar.wase.org/ /3000

5 World Academy of Scece, Egeerg ad Techology Ieraoal Joural of Elecrcal ad Comuer Egeerg Vol:, No:9, 008 Ieraoal Scece Idex, Elecrcal ad Comuer Egeerg Vol:, No:9, 008 wase.org/publcao/3000 Where, for ay o-egave eger ; π w = T Alg hs o he daa of Albera Marke Prces, he smulao of he hourly forward rce curve by meas of A Fourer seres s show Fg. 7. Fg. 7 Smulao of HPFC by he Fourer seres V. EVALUATING THE GOODNESS OF FIT OF THE MODELS Oce he daa rces model mlemeed, he evaluao o es he goodess of f should be doe. Ths wll be doe by a vsual examao measure of he curve dslayed & by umercal measures; hese wll be vald for all. Meas, he goodess of f measures for boh lear ad olear aramerc fs should rovde: Resduals Goodess of f sascs These measures could be groued o wo yes: grahcal ad umercal. The resduals are grahcal measures, whle he goodess of f sascs s umercal measures. Geerally seakg, grahcal measures are more beefcal ha umercal measures because hey allow us o vew he ere daa se a oce, ad hey ca easly dslay a wde rage of relaoshs bewee he model ad he daa. The umercal measures are more arrowly focused o a arcular asec of he daa ad ofe ry o comress ha formao o a sgle umber. Also, s ossble ha oe of he fs by he model cosruced ca be cosdered he bes oe. I hs case, mgh be ha eeded o selec a dffere model. Coversely, s also ossble ha all he goodess of f measures dcae ha a arcular f s he bes oe. However, he crera ha wll follow are ha; exame boh he goodess of f sascs ad he grahcal measures o check he mmum error should be doe. A. Some Defos of Goodess of F Sascs The sum of squares due o error (SSE) r-square Adjused r-square Roo mea squared error (RMSE) ) Sum of Squares Due o Error: hs sasc he oal varao of he resose values from he f o he resose values s measured. I s also called he summed square of resduals ad s usually labeled as SSE. SSE = w ( y Y ) (4) = Where w are he weghs. You ca aroxmae he weghs usg a equao such as = w ( y y ) (5) = A value of SSE closer o 0 dcaes a beer f. ) r-square: hs sasc measures how successful he f s clearg u he varao of he daa. By oher words, r- square s he square of he correlao bewee he resose values ad he redced resose values (so &forward rces). I s also called he square of he mulle correlao coeffces ad he coeffce of mulle deermaos. So, r-square ca be defed as he rao of he sum of squares of he regresso (SSR) ad he oal sum of squares (SST). SSR s defed as: SSR = = w ( ) Y y (6) SST s also called he sum of squares abou he mea, 's defe as SST = w ( y y) (7) = r-square s exressed as: SSR r square = (8) SST 3) Degrees of Freedom Adjused r-square: Ths sasc uses he r-square sasc defed above, ad adjuss based o he resdual degrees of freedom. The resdual degrees of freedom s defed as he umber of resose values mus he umber of fed coeffces m esmaed from he resose. SSE( ) adjused r square = (9) SST ( v) Where; v = -m The adjused r-square sasc ca ake o ay value less ha or equal o, a value closer o dcag a beer f. 4) Roo Mea Squared Error: Ths sasc s also kow as he f sadard error ad he sadard error of he regresso RMSE = s = MSE (0) Where MSE s he mea square error or he resdual mea square SSE MSE = () v A RMSE value closer o 0 dcaes a beer f. B. The Crera for he Valdao of he Models There are may mahemacal ools as meoed before for he model valdao, bu he rmary ool for mos modelg alcaos s grahcal resdual aalyss. Dffere yes of Ieraoal Scholarly ad Scefc Research & Iovao (9) scholar.wase.org/ /3000

6 World Academy of Scece, Egeerg ad Techology Ieraoal Joural of Elecrcal ad Comuer Egeerg Vol:, No:9, 008 Ieraoal Scece Idex, Elecrcal ad Comuer Egeerg Vol:, No:9, 008 wase.org/publcao/3000 los of he resduals from a fed model are rovded he ex secos o gve formao o he adequacy of dffere asecs of he model, bu, as meoed before, he crera ha wll followed s ha, boh he goodess of f sascs ad he grahcal measures should be examed o sure beer f. The defo of he resduals from a fed model s he dffereces bewee he resoses observed a each recorded values of he exlaaory varables ad he corresodg redco of he resose comued usg he regresso model. Mahemacally, he defo of he resdual for he h observao he daa se s wre ( e = y Y ). C. Comarso he Calculaos of he Dffere Models I he ex secos, a frs Comarso bewee he resuls of he dffere models wll be creaed. The sar wll be wh he aalyss of he resduals curve resul, ad he go o aalyze all he resul of sascs measures. The dffere yes of los of he resduals from fed models are as show Fgs. 8-. ) The Polyomal Regresso Models Fg. 8 Resuls of he model by usg ler olyomal regresso Fg. 9 Resuls of he model by usg quadrac olyomal regresso Fg. Resuls of he model by usg 4h degree olyomal regresso The quadrac model aears o f he daa, bu, he resduals of all he models aear o be radomly dsrbued aroud zero. Therefore, a grahcal evaluao of he fs does o reveal ay obvous dffereces bewee he all equaos. The umercal f resuls for all he models fed by he olyomal regresso are show below he able: TABLE I RESULTS OF FITS OF ALL THE POLYNOMIAL REGRESSION MODELS Name of Regresso Tae # Coeffce HPFC_LPR Ler.04e +004 HPFC_QPR Quadra e c +003 HPFC_CPR cubc 4.44e +004 HPFC_4h 4h 5.457e degree PR +004 HPFC_5h 5h e degree PR +004 HPFC_6h 6h e degree PR +004 SSE RMSE r- square The goodess of f sascs s show. The sascs reveal a subsaal dfferece bewee he equaos, he secod h model he able ( degree olyomal regresso) reveals he bes resuls (mmum RMSE ha all he ohers models), bu, he r-square close o oe ha he frs, hrd & forh. (Bu s o beer he ffh &sxh). ) The Radal Bass Fucos Neural Neworks Models Here, he resduals of he model for wo arameers sgma ad he umber of he euros are examed, he followg lo rereses hs for sgma=900 &NN=0, The; he wo arameers are creased o sgma=600 & NN=0, ad observe he resul as show Fgs., 3. Fg. 0 Resuls of he model by usg cubc olyomal regresso Fg. Resuls of he model by usg RBF wh σ=900 & NN=9 Ieraoal Scholarly ad Scefc Research & Iovao (9) scholar.wase.org/ /3000

7 World Academy of Scece, Egeerg ad Techology Ieraoal Joural of Elecrcal ad Comuer Egeerg Vol:, No:9, 008 Ieraoal Scece Idex, Elecrcal ad Comuer Egeerg Vol:, No:9, 008 wase.org/publcao/3000 Fg. 3 Resuls of he model by usg RBF wh σ=600 & NN=0 I he frs lo, he models aear o f he daa oor. Lookg o he resduals, 's aear o be almos radomly dsrbued aroud zero ha he oher model (σ = 600& NN=0), whch seems more f good for he daa, sce he resduals are more radomly dsrbued aroud zero. The umercal f resuls for he models fed by he RBF are (RMSE= ) for he secod model, hs meas, he goodess of f sascs s, he sascs reveal a dfferece bewee hs model ad he ere las model. The las oe model (RBF regresso model) reveals he bes resuls (mmum RMSE ha all he ohers models ad beer R-square close o oe ha oher. 3) A Fourer Seres (Trgoomerc Polyomals) Fally, comg o he Fourer seres mehod, here, he examao of he mehod wll be doe uder crease he umber of he harmocs ad lookg o he bes me erod by chagg. The resduals wll be checked, he aga he umercal f resuls for he models fed by hs mehod s checked oo. The followg lo; Fg. 4 rereses hs for umber of harmoes= &me erod equals o 6000, The he wo arameers mus be creased o me erod =0000 & NH=5, ad he resul s observed Fg. 5. Fg. 4 Resuls of he model by usg a Fourer seres (NH= &T=6000) VI. CORRELATION ANALYSIS Correlao aalyss s a sascal echque ha evaluaes he relaosh bewee wo varables;.e., how closely hey mach each oher erms of her dvdual mahemacal chage. The queso addressed s: f oe varable ( x ) moves or a chage a cera dreco does he secod varable ( y ) also move or chage a smlar or comlemeary dreco? A. The Coeffce of Correlao The coeffce of correlao s a measure of he sregh of he lear relaosh bewee wo varables ad. I s comued [0] (for a samle of measuremes o x ad y ) as follows: SS xy () r = SS SS SS SS xy yy = = = = ( y y), xx yy ( x x)( y y), SSxx = ( x x) (3) x = = x, = y = y = B. The Coeffce of Deermao Aoher way o measure he corbuo of x redcg Y s o cosder how much he errors of redco of Y ca be reduced by usg he formao rovded by x. The samle coeffce of deermao s develoed from he relaosh bewee wo kds of varao: he varao of he Y values a daa se aroud: The fed regresso le. Ther ow mea. The erm varao boh cases s used s usual sascal sese o mea he sum of a grou of squared devaos. The frs varao s he varao of Y values aroud he regresso le,.e., aroud her redced values. Ths varao s he sum of squares for error ( SSE ) of he regresso model SSE = ( y Y ) (4) = The secod varao s he varao of y values aroud her ow mea SS = y y) (5) I s easy o verfy ha yy = r SS SSE SSE = = SS SS yy (6) yy Where; r s he coeffce of correlao, defed he equao (). yy Fg. 5 Resuls of he model by usg a Fourer seres (NH=5 &T=0000) Ieraoal Scholarly ad Scefc Research & Iovao (9) scholar.wase.org/ /3000

8 World Academy of Scece, Egeerg ad Techology Ieraoal Joural of Elecrcal ad Comuer Egeerg Vol:, No:9, 008 Ieraoal Scece Idex, Elecrcal ad Comuer Egeerg Vol:, No:9, 008 wase.org/publcao/3000 VII. PRACTICAL ANALYSIS OF THE COEFFICIENT OF DETERMINATION, r Now, by alyg equaos (-6) gve revous secos o all he models cosruced ad by alyg o Albera Marke rces, he resuls of each oe as follows: A. Correlao Aalyss of he resul of Polyomal Regresso Model: Defed equao () ha he geeral Polyomal Regresso equao has he followg form: 3 Y ( ) = (7) The correlao aalyss of he ler olyomal, quadrac olyomal, cubc olyomal, 4 h order olyomal ad he 5 h order olyomal wll be calculaed. 5 h order wll be cosdered eough o gve ermssble error). ) Resuls of he Model by Usg Ler Polyomal Regresso The ler olyomal s he form Y ( ) = 0 +. Coeffces: = , = Goodess of f: SSE:.04e+004 r-square: RMSE: ) Resuls of he Model by Usg Quadrac Polyomal Quadrac olyomal s he form Y ( ) = Coeffces: = , = , = Goodess of f: SSE: e+003 r-square: RMSE: 50.3 Regresso summary Quadrac erm clearly very mora Farly mressve r-square Decrease RMSE o ) Resuls of he Model by Usg Cubc Polyomal Cubc olyomal s he form 3 Y ( ) = Coeffces: = , = , = , = Goodess of f: SSE:.44e+004 r-square: RMSE: Regresso summary Cubc erm clearly adds o f. Hgh correlao bewee esmaes for lear & cubc Imressve crease RMSE o 80.7% 4) Resuls of he Model by usg 4h Degree Polyomal 4h degree olyomal s he form 3 4 Y ( ) = Coeffces: =.0e-005, =.0e-005, =.0e-005 * , =.0e-005 * , =.0e-005 Goodess of f: SSE:.457e+004 r-square: 0.8 RMSE: Regresso summary Addg 4h ower does mrove f a all. Imressve crease RMSE o 80.7% 5) Resuls of he Model by usg 5h Degree Polyomal 5h degree olyomal s he form Y ( ) = Coeffces: =.0e-009 * 0, =.0e-009 *0, =.0e-009 *0 =.0e-009 * =.0e-009 * =.0e-009 * Goodess of f: SSE:.5048e+004 r-square: 0.70 RMSE: Regresso summary Addg 5h ower does mrove f a all. Imressve crease RMSE o 89.% B. Correlao Aalyss of he RBF Neural Neworks Models The geeral form of our hourly rce forward curve (HPFC) modeled by RBF for umber of euros NN wll be as he ex form: ) ) (8) Y( ) = w.ex( (( c ) / σ + w.ex( (( c ) / σ w.ex( (( c ) / σ ) NN NN ) Resuls of he model by usg RBF wh σ=60 & NN=0 The model wll be he form Y( ) = w.ex( (( c ) / σ ) + w.ex( (( c ) / σ ) + (9)... + w.ex( (( c ) / σ ) 0 0 Goodess of f: SSE: r-square: RMSE: Regresso summary Imressve crease r-square 64 % ha 6h degree olyomal. Resduals sll show eed for more Neural. Ieraoal Scholarly ad Scefc Research & Iovao (9) scholar.wase.org/ /3000

9 World Academy of Scece, Egeerg ad Techology Ieraoal Joural of Elecrcal ad Comuer Egeerg Vol:, No:9, 008 C. Correlao Aalyss of a Fourer Seres Models The geeral form of our hourly rce forward curve (HPFC) modeled by Fourer seres wll be as he ex form: Y ( ) = a0 + a.cos( w) + b s( w) + a cos( w) + (30) b s( w)... + an cos( N w) + bn s( N w) ) Resuls of he model by usg a Fourer seres (NH= &T=6000) Ieraoal Scece Idex, Elecrcal ad Comuer Egeerg Vol:, No:9, 008 wase.org/publcao/3000 Y ( ) = a0 + a.cos( w) + b s( w) + (3) a cos( w) + b s( w) Coeffces: = = =.5590 = = Goodess of f: SSE: 9.685e+003 r-square: RMSE: VIII. OPTIMIZATION (MINIMIZING THE LOSS) Ay egeer s eresed deermg omal segs of he model Facors (arameers); ha s, o deerme for each arameer a level of ermssble error ha omzes he model resose error. The loss or he error (RMSE) rovdes a objecve measure of redcve error for a secfc choce of model arameers. Thus; he goal s o fd he values of he model arameers ha mmze he error. The roblem of fdg mmum error ca be solved by a erave umercal echque called grade desce. I works as follows:. Choose some (radom) al values for he model arameers.. Calculae he error fuco wh resec o each model arameer. 3. The model arameers chage so ha a shor dsace he dreco of he greaes rae of decrease of he error s recorded. 4. Ses ad 3 wll reeaed ul error ges close o zero. A. Omzao for HPFC Modeled by usg a Fourer Seres To omze he model by chagg he me erod T ul geg he smalles error. The me wll chage from zero o double of he me erod ha rces recorded. Ths chage mus occur ul guaraeed ha here s mmum error. The mmum error was recorded a T=5300 as show Fg. 6, resuls show ha, here s decrease he error o mmum value of a umber of harmocs equals 0 ad crease he correlao coeffces, meas beer curve smulao Fg. 7. Fg. 6 Plo of he error RMSE versus me erod Resuls afer omzao: RMSE = , r_square= Fg. 7 Plo of he HPFC afer omzg he model by usg (NH=0 &T=5300) B. Omzao for HPFC modeled by RBF Neural Neworks Models I he omzao of he models cosruced by RBF regresso, models f o daa colleced usg resose surface desgs s used. How he omal rego o ru a rocess does s deermed? The aswer s he omzao wll be doe a erave rocess aga. So, he omzao wll be doe here by chagg boh Sgma & umber of Neuros ul he arge ouu s h, whch s mmze rocess ouu error, afer reeag he erao umber of mes ul he goal s h whch s fdg values of he arameers o mmze he error of he rocess. The error s foud mmum by usg Ma lab code whe sgma=00 & NN=9, he umber of erave was 0*0 =400.The followg lo (Fg. 8) show he chage he error wh chagg he arameers sgma &NN. Fg. 8 Plo of he error RMSE versus NN & sgma Ieraoal Scholarly ad Scefc Research & Iovao (9) scholar.wase.org/ /3000

10 World Academy of Scece, Egeerg ad Techology Ieraoal Joural of Elecrcal ad Comuer Egeerg Vol:, No:9, 008 Ieraoal Scece Idex, Elecrcal ad Comuer Egeerg Vol:, No:9, 008 wase.org/publcao/3000 Now; hs value of he arameer T; whch gve almos he omal model corresodg o he varous choces of overall measure of loss, he oher arameers "Sgma & umber of Neuros" mus be chaged aga o deerme he coeffces "Weghs" whch gves he omum model. The lo of he HPFC wh arameers sgma=80 & NN=39 s show (fg.9).eve hough, he error could be aga mmzed f he umber of he euros s creased, bu hs really comuaoally exesve ( for umber of euros equal o 60, ake a me of sx hours o our comuer! ).So, f he me eeded o make beer omzao s val, he a sohscaed mache o erform hs ask s eeded. Fg. 9 Plo of he HPFC afer omzg he model Here, modelg he HPFC by RBF regresso wll coss of wey weghg coeffces, hese coeffces & he omum corresodg arameers wll form he equao for redco he forward rces he fuure. *Resuls afer omzao: r_square = 0.45, RMSE = The resuls show ha, here s decrease he error o RMSE= a arameers of sgma=00 ad NN =9,also he correlao coeffce come closer o oe he before, meas beer curve smulao. IX. IMPLEMENTATION I hs seco, he omal model derved he las secos s mlemeed. The mlemeao modeled by RBF Neural Neworks Models wll aly o EEX marke for daa readg of more ha fve years. However, hs mlemeao s uvely ca be aled o ay oher Marke Daa, bu, keeg md ha for he markes ha have hgh chagg rces bewee he seasos of he year, he regresso for erod ha gve arameers whch draw curve should be ru o deely exress he rces whou care abou he me eeded for modelg, ha s, he erod of he erave should be exaded as ossble o ge ermssble error. The followg lo shows he behavor of he rces of EEX marke he erod of (from Jue 00 ul Jue 006): Fg. 0 Plo of EEX Marke Pres The rces wh average equal 3.98 & sadard devao equal 5.57, whch are meas he res are chagg bewee 4.54 & 4.4, whch mles low volaly ha Albera Marke Prces. So, he regresso could be reeaed wh less umber of he erave for geg he ermssble error.aleravly, f he me s o val, hgh umber of erao ca be ru, bu, he error resulg from he model wll o be sgfcaly dffer ha f low umber s ru! Fg. Plo of he omzed HPFC afer omzg of EEX Marke As s clear from he lo, he curve f wlls he rces alog he fve years erod wh smalles ermssble error. The followg are he resuls of he model: Name of Regresso Fally, he omzed HPFC for EEX Marke uder hese crera could be wre e as follows: Y( ) = w ex Ta e... # Coeffce + w SSE RMSE r- square HPFC_RBF RBF e ex ( 0.5*( c ) / sgma) ( 0.5*( c ) / sgma) ( 0.5*( c9 ) / sgma) w9 ex + Ieraoal Scholarly ad Scefc Research & Iovao (9) scholar.wase.org/ /3000

11 World Academy of Scece, Egeerg ad Techology Ieraoal Joural of Elecrcal ad Comuer Egeerg Vol:, No:9, 008 Ieraoal Scece Idex, Elecrcal ad Comuer Egeerg Vol:, No:9, 008 wase.org/publcao/3000 w= w= w3= 0.09 w4= w5= w6= w7= w8= w9= w0= -0.0 w= w= w3= w4= w5= w6= w7= w8= w9= w0= HPFC for wo years he fuure w= w= w3= w4= w5= 0.07 w6= w7= w8= 0.89 w9= w30= Fg. Omzed HPFC of EEX Marke wh me of wo years he fuure Fg. 3 Omzed HPFC of EEX Marke for me of wo years he fuure X. CONCLUSION I hs aer, he omzed mehod for modelg he Hourly Prce Forward Curve (HPFC) usg he regresso echque s mlemeed. Ths mlemeao s aroached usg hree mehods (olyomal regresso, RBF eural ework regresso & usg a Fourer seres (Trgoomerc Polyomals Regresso). The comarsos amog hese hree mehods usg he RMSE (Roo Mea Square Error) as overall measure of error have made. I comarg hese hree mehods, he maxmum loss values are observed close o ffy before omzao. For he mehod oe, he degree of he olyomal coag of he loss values, he s o wde ha s, he varao of loss s hgh. For all he mehods, s observed ha he loss s o moooc for Y () as a fuco of he degree for mehod oe ad o moooc for he oher Y () whch s defed he oher mehods. From a comuaoal o of vew, Mehod s smle. Mehod s more comlcaed comared o he mehod 3. Mehod s comuaoally harder ha he oher wo mehods because we aly he omzao echque o solve for N aurals, (where, N >>) wo arameers. ACKNOWLEDGMENT Ths work was suored by he De. of Elecrcal Egeerg, Omar Al Mukhar Uversy, Albda, Lbya. REFERENCES [] Eergy Ausrala Py Ld, Brsbae, Ausrala, ( ) [] Plovc Dragaa.997. Valug ad Maagg Eergy Dervaves. McGraw-Hll. [3] Alexader Eydelad &Krzyszof Wolye.003.Eergy ad Power Rsk Maageme. Joh Wley &Sos, Ic. [4] Kecma Vojslav.00.Learg ad Sof Comug. A Bradford Book, he MIT Press. Cambrdge, Massachuses. [5] Sadegh ad Ware, Mea Reverg Models for Eergy Oo Prcg, Workg Paer, Uversy of Calgary, 00. [6] Paagos A. Dafas, Esmag he arameers of a mea-reverg Markov- swchg jum-dffuso model for crude ol so rces Workg Paer, Uversy of Calgary, 004. [7] Alvaro Carea ad Marcelo G. Fgueroa, Prcg Elecrcy Markes: a mea reverg jum dffuso model wh seasoaly, Workg Paer, Uversy of Lodo, 005. [8] Le Xog, Sochasc Models for Elecrcy Prces, Workg Paer, Uversy of Calgary, 004. [9] Parck MacDoald Parck, Coge Clams he Albera Elecrcy Marke. Workg Paer, Uversy of Calgary.003. [0] The Iformao Techology Laboraory (ITL) a he Naoal Isue of Sadards ad Techology (NIST). (h:// [] h:// [] h://e.wkeda.org/wk/nolear_regresso Ieraoal Scholarly ad Scefc Research & Iovao (9) scholar.wase.org/ /3000