TIME-ADAPTIVE KERNEL DENSITY FORECAST: A NEW METHOD FOR WIND POWER UNCERTAINTY MODELING

Transcription

1 TIME-ADAPTIVE EREL DESITY FORECAST: A EW METHOD FOR WID POWER UCERTAITY MODELIG R.J. Bessa, J. Sumal, V. Mranda, A. Boerud 2, J. Wang 2, E. Consannescu 3 Insuo de Engenhara de Ssemas e Compuadores do Poro (IESC Poro), Faculdade de Engenhara, Unversdade do Poro Poro, Porugal rbessa@nescporo.p, jean.sumal@nescporo.p, vmranda@nescporo.p 2,3 Argonne aonal Laboraory, 2 Decson and Informaon Scences Dv., 3 Mahemacs and Compuer Scence Dv. Argonne, IL, USA aboerud@anl.gov, janhu.wang@anl.gov, emconsa@mcs.anl.gov Absrac Ths paper repors new conrbuons o he advancemen of wnd power uncerany forecasng beyond he curren sae-of-he-ar. A new ernel densy forecas (DF) mehod appled o he wnd power problem s descrbed. The mehod s based on he adaraya-wason esmaor, and a me-adapve verson of he algorhm s also proposed. Resuls are presened for dfferen casesudes and compared wh lnear and splnes quanle regresson. eywords: Wnd power forecasng, uncerany, ernel densy esmaon, me-adapve. ITRODUCTIO A wnd power forecaser sees he perfec wnd power forecas, bu s common sense ha hs represens a uopan mage. everheless, successful effors have been made o decrease he wnd power forecas error []. Furhermore, wh he growng peneraon of wnd power and he economc mporance of forecasng errors [2], s becomng ncreasngly mporan o also forecas he uncerany assocaed wh wnd power generaon predcon. An exensve sae-of-he-ar repor on algorhms for wnd power uncerany forecasng can be found n [3]. The mos popular sascal algorhms are: splnes quanle regresson [4], whch consss of a lnear quanle regresson wh bass funcons formulaed as cubc B-splnes; adaped resamplng [5], whch s a process for generang alernave scenaros of power producon, and hs way, s possble o change he weghs obaned by a fuzzy nference sysem. Physcal approaches for uncerany forecasng can be found n [6] and [7]. The nformaon provded by probablsc forecasng algorhms creaes addonal value (e.g. economc) n several decson-mang problems. Boerud e al. [8] presened several bddng sraeges for wnd power n he elecrcy mare, such as expeced uly maxmzaon and rade-off beween expeced value and rs. Maos and Bessa [9] presened a decson mang approach for seng he operang reserve requremens usng non-paramerc probablsc forecass (a se of quanles) as npu. Wang e al. [] descrbed a sochasc un commmen ha uses forecased scenaros of wnd power generaon as npu. Usaola [] presened a probablsc power flow ha aes no accoun correlaon beween wnd farms and uses bea dsrbuons for modelng he wnd power uncerany. Wnd power uncerany can ae he form of probablsc forecass [4]-[4] or scenaros for shor-erm wnd power generaon [2]. Probablsc forecasng consss of expressng he wnd power generaon or forecas error n probablsc erms, such as: paramerc represenaon (e.g. Gaussan dsrbuon); momens of he dsrbuons (e.g. sandard devaon, sewness); a se of quanles; probably densy funcon (pdf). ormally, he uncerany represenaon s deermned by he algorhm used, e.g. f quanle regresson s used, he uncerany s represened by a se of quanles. Models raned n an offlne mode (e.g. [4][4]) are unable o cope wh (non-saonary) changes n he underlyng dsrbuons of he npu varables. The rend n he sae-of-he-ar s o develop algorhms capable of adapng o changes n daa [3]. Hence, an algorhm for wnd power uncerany forecasng should deally have as requses: ) a hgh flexbly o represen wnd power uncerany; ) meadapve characerscs. In hs paper we propose a novel ernel Densy Forecas (DF) algorhm ha addresses hese wo requses. The oupu s a pdf of he forecased wnd power, and snce hs represenaon s generc can be ransformed o several uncerany forms, such as quanles, sandard devaon, or sewness. A me-adapve verson of he algorhm s descrbed, whch means ha he model s capable of learnng from recen nformaon whle dscounng older nformaon. Mehods based on ernel Densy Esmaon (DE) are no new n he sae-of-he-ar, one example can be found n [4]. The auhors presen an adapaon of he classc adaraya-wason ernel densy esmaor, where all he ernel funcons are bwegh funcons. The reflecon mehod was used for bounded varables. Our approach dffers n wo mporan ways: ) our mehod s based on selecng adequae ernels for modelng he dfferen varables ypes, 2) our mehod s me-adapve. 7 h Power Sysems Compuaon Conference Socholm Sweden - Augus 22-26, 2

2 The paper s organzed as follows: secon 2 descrbes he DF mehodology; n secon 3 resuls are presened for dfferen case-sudes and compared wh lnear and splnes quanle regresson; secon 4 presens he conclusons. 2 EREL DESITY FORECAST METHODLOGY 2. ernel Densy Esmaon DE consss of a non-paramerc esmaor of a probably densy funcon (pdf) [5]. Gven ndependen and dencally dsrbued daa (..d.),, n drawn from an unnown densy funcon f, he unvarae DE s gven by: x ( x) = () h = h where s he number of samples, s a ernel funcon and h he bandwdh parameer. Ths equaon, places a ernel around each sample. Gven..d. mulvarae daa d,, nd from d dfferen varables drawn from an unnown mulvarae densy funcon f, he mulvarae DE s gven by he produc ernel esmaor [6]: d x j j ( x ) =,..., xd (2) j = j= h hj where j s he ernel funcon for varable j wh bandwdh h j. 2.2 adaraya-wason Esmaor Condonal densy esmaon consss of esmang he densy of a random varable Y, nowng ha he explanaory random varable s equal o x. In oher words, consss of esmang he densy of Y condoned o =x, f(y =x). The condonal densy can be formulaed as follows: f ( x y) Y, y = x = (3) f x where f Y (x,y) s he mulvarae densy funcon of and Y (jon pdf) and f (x) s he margnal densy of. I s also possble o have nonparamerc condonal densy esmaon usng Eq. and 2. The classc approach s he modfed adaraya-wason ernel smooher [7]: havng ( y = x) = ( y Y ) w ( x) (4) w ( x) = = = hx hy ( x ) hx ( x ) (5) where he bandwdh h y conrols he smoohness of each condonal densy n he y drecon, whle h x conrols he smoohness beween condonal denses n he x drecon. 2.3 Formulaon for he Wnd Power Problem The wnd power densy forecas problem can be formulaed as: forecas he wnd power pdf a me sep for each loo-ahead me sep + of a gven mehorzon (e.g. up o 72 hours ahead) nowng a se of explanaory varables (numercal weaher predcon (WP) varables, wnd power measured values, hour of he day, ec.). Translang hs senence o an equaon, we have: f P ( p x ), +, + P p+ = x+ = (6) f x + where p + s he wnd power forecased for loo-ahead me +, x + are he explanaory varables forecased for loo-ahead me sep + and avalable/launched a me sep. Eq. 6 can be solved usng Eq. 4 and 5, where he varable Y s he wnd power, and he explanaory varables are for nsance: WP varables (wnd speed, wnd drecon, pressure), wnd power pon forecas, measured wnd power. Fg. depcs he jon pdf compued usng Eq. 2 for daa from a real wnd farm. Ths jon pdf represens he probably densy assocaed o each jon realzaon of forecased wnd speed and realzed wnd power. Fg. 2 s a dscree represenaon of he nformaon conaned n Eq. 6. I allows seeng he changes n he wnd power densy funcon condoned o dfferen values of forecased wnd (.e. he explanaory varable). Jon Densy Funcon Wnd Power (p.u.) Fgure : Jon probably densy funcon of forecased wnd speed and measured wnd power..8.6 Wnd Power (p.u.).4 Fgure 2: Condonal DE for forecased wnd speed and wnd power generaon Wnd Speed (m/s) Wnd Speed (m/s) 7 h Power Sysems Compuaon Conference Socholm Sweden - Augus 22-26, 2

3 2.4 ernel Funcon Choce The choce of he ernel funcon for he wnd power forecasng problem s a crcal ssue. The choce depends on he ype of varable. We have n he wnd power problem four dfferen varable ypes: wnd power bounded beween and (e.g. raed power); wnd speed bounded beween and +Inf; crcular varables le he wnd drecon; varables such as emperaure, beween -Inf and + Inf. For hese four ypes, dfferen ernels should be consdered. For varables wh range [,] we use he followng bea ernel (used for wnd power n Fg. ) [8]: ( x) = = + ( ) + x h x h (7) / /, where p,q s he densy funcon of a Bea(p,q) random varable defned by: p q ( u; p, q) = u ( u), u [, ] (8) B p, q wh B(.) denong he bea funcon, p and q are he wo posve shape parameers, and h beng he bandwdh parameer. For he varables wh suppor [,+Inf) he gamma ernel (used for wnd speed n Fg. ) [9] was used: ( x) = = + x h h (9) /, where p,q s he densy funcon of a Gamma(p,q) random varable defned by: p exp ( u / q) u; p, q = u, u [, + [ () p Γ p q wh Γ(.) denong he gamma funcon, p as he shape parameer, q as he scale parameer, and h s he bandwdh parameer of p,q. For varables wh unbounded suppor, he naural choces are he Gaussan ernel or he bwegh ernel. For crcular varables he approach s o use crcular dsrbuons such as he von Mses dsrbuon [2]: κ cos( θ µ ) g ( θ; µ, κ ) = e () 2π I κ where I s he modfed Bessel funcon of he frs nd and order and defned by: 2π κ cos( θ ) I ( κ ) = e dθ (2) 2π The parameer µ s he dreconal cener of he dsrbuon, κ s he concenraon parameer and θ belongs o any nerval of lengh 2π. The concenraon parameer can be used o conrol he degree of smoohng n crcular DE, and s analogous o he bandwdh parameer bu larger values lead o less smoohng. oe ha he negrals compued from he bea and gamma ernels may lead o dsrbuons ha do no have an negral (area of he dsrbuon) equal o one. Hence, we use he dea of a modfed bea ernel esmaor [2]: ˆ ˆ ' f ( x) f x = (3) x dx Snce hs s only a change of scale, he normalzaon s employed over he condonal funcon of Eq Tme-adapve Algorhm A recursve formula descrbed n he leraure [22] can be used for he DE esmaor: n x = + n x n x (4) n n h h The exenson o he mulvarae case (Eq. 2) s sraghforward. Eq. 4 allows updang he densy funcon when new samples are avalable whou he need o enrely recompue he whole densy funcon. However, as he number of ncreases, he rao (n-)/n approaches one (and /n approaches zero), and hen he new samples become redundan. Moreover, f here s a change n he general srucure of he daa (non-saonary daa), hs recursve esmaon s ncapable of auomacally dscard older daa. In order o overcome hese problems, he DE esmaor wh exponenal smoohng [22] can be used: ( ) x λ = + n x λ n x (5) h h where λ s called forgeng facor and conrols how qucly or slowly he exponenal smoohng adaps o he new daa (exponenal forgeng). A value of λ close o one means ha he exponenal smoohng pus more wegh on he hsorcal daa and lle wegh on he mos recen values, whle when λ s closer o zero means he oppose suaon; λ can be represened n erms of n, and we have: λ=n/(n+). The adaraya-wason esmaor descrbed n secon 2.2 can be convered o a me-adapve esmaor usng Eq. 5. The esmaor becomes: ( y = x) = λ + ( ) x, y λ h h x y (6) λ x hx x ( x) + ( λ) hx y Y hy where f (y =x) means he nowledge of he model a me nsan, whch s updaed usng recen values of Y and. The me-adapve wnd power forecas problem consss of he followng man seps:. f ˆ ( p+ = x+ ): DF model wh nowledge a me sep ; 2. Oban new values of measured wnd power generaon and correspondng WP daa for he same perod. Ths recen daa s used o updae 7 h Power Sysems Compuaon Conference Socholm Sweden - Augus 22-26, 2

4 he nowledge of he model (usng Eq. 6), and he model n () becomes f ˆ ( p+ = x+ ); hs process s repeaed when new values are avalable. 3 CASE STUDIES 3. Descrpon Two dfferen ses of daa are used as case sudes. The frs daase consss of day-ahead wnd power forecased and realzed values for 5 hypohecal wnd power ses n he sae of Illnos, obaned from REL s Easern Wnd Inegraon and Transmsson Sudy [23]. We used he wnd power daa for he perod Jan-Aug o ran he uncerany forecas models. The monhs beween Sepember and December are used as a es daase. The second daase s from a large wnd farm locaed n fla erran n he U.S. Mdwes. The complee daase (SCADA and WP) correspond o he perod beween January s 29 and February 2h 2. The WP daa was generaed wh he Weaher Research and Forecasng (WRF) model [24] by Argonne aonal Laboraory and consss of several weaher varables (e.g. wnd speed, drecon, emperaure) for one reference pon nsde he wnd farm. The emporal horzon of he WP predcons was as follows: wnd power s forecased a 6 AM for he emporal horzon of +6 up o +48 hours. The emporal resoluon of he forecass s one hour. The ranng daase was seleced o have 7% of he all avalable daa (3% of for esng): ranng se from January 29 o 2 ovember 29 (269 pons), and he esng se from 22 ovember 29 o 2 February 2 (523 pons). 3.2 Evaluaon Framewor The resuls obaned wh he adaraya-wason esmaor were compared wh he lnear quanle regresson model and he splnes quanle regresson [4]. A framewor o evaluae wnd power probablsc forecass dealed n [25] was followed n hs paper. Three mercs were used for evaluaon: calbraon, sharpness and sll score. Calbraon s a measure of he agreemen beween nomnal proporons (forecased probables) and he ones compued from he evaluaon sample. In oher words, for a quanle he emprcal proporon should equal he nomnal, e.g. an 85% quanle should conan 85% of he observed values lower or equal o s value. In order o evaluae quanle forecass, s necessary o defne he ndcaor varable. An ndcaor varable for a quanle forecas wh nomnal proporon s: q ˆ + f p + qˆ + ξ = (7) oherwse The ndcaor varable refers o he acual oucome of p + a me +. Furhermore, hese ndcaors are defned as follows: n n {, = } =, = # ξ ξ (8), = { = } = n, = ξ,, # (9) ha s, as sums of hs and msses, respecvely, for a gven horzon over realzaons. The emprcal proporons are compued wh he Eq. 7 and 8 as follows: n, ˆ = (2) n + n,, The dfference beween emprcal and nomnal proporons s consdered he bas of he probablsc forecasng mehod. Sharpness s he endency of probably forecass owards dscree forecass. Quanles are gahered by pars n order o oban nervals wh dfferen nomnal coverage raes. Le δ = ˆ q be he sze of / 2 / 2 he + q + ˆ+ nerval forecas wh nomnal coverage rae - esmaed a me for lead me +. In hs paper, sharpness s measured by he mean sze of he dsance beween quanles: = δ, δ = (2) We also calculaed a sll score from Eq. 22 whch gves nformaon abou a model s performance (e.g. calbraon, sharpness, ec.) n a sngle measure for a se of m quanles: S c m (, p ) = ( )( p qˆ ) + + ξ + + = (22) where p + s he realzed wnd power, s he quanle proporon, q + s he forecased quanle and ξ s he ndcaor varable of Eq. 7. The hgher he scorng rule, he beer: he maxmum value s for perfec probablsc forecass. For reasons of comparson, he probablsc forecas s represened hrough a se of quanles rangng from 5% o 95% wh a 5% ncremen. 3.3 Evaluaon Resuls: REL s EWITS Sudy 3.3. Offlne Resuls The ernel funcon used n he adaraya-wason (W) esmaors was Chen s bea ernel for boh realzed and forecased wnd power (.e. he explanaory varable). The ernel sze was. for boh varables (deermned expermenally by ral-error). Fg. 3 shows he average calbraon for he whole me horzon (24 hours) for probablsc forecass obaned wh he lnear quanle regresson (Lnear QR), splnes quanle regresson (Splnes QR) and he W esmaor. oe ha wha s depced n he dagram s he dfference beween forecased and emprcal quanle proporons. 7 h Power Sysems Compuaon Conference Socholm Sweden - Augus 22-26, 2

5 For he quanles above 55% he W esmaor presen a lower devaon han he QR mehods. For quanles below he medan he splnes QR s compeve wh he W, and for some quanles acheves he lowes devaon. On average, he mehods overforecas he quanles snce he forecased quanle proporons are greaer han he emprcal ones. The ess performed wh dfferen bandwdhs showed ha by changng he ernel bandwdh he model changes from underesmaon o overesmaon and vce-versa. Devaon [%] W Lnear QR Splnes QR omnal proporon rae [%] Fgure 3: Calbraon dagram for he offlne es wh REL daa. Inervals mean lengh [% of raed power] W Lnear QR Splnes QR omnal coverage rae [%] Fgure 4: Sharpness dagram for he offlne es wh REL daa. Fg. 4 depcs a sharpness dagram where he x-axs s he nomnal coverage of he forecas nerval (-) and he y-axs s he average sze of he nervals. In hs case wha s desred s o have nervals wh smaller sze for all coverage raes. In erms of sharpness he forecased quanles presened relavely narrow ampludes for all mehods, alhough splnes QR has he lowes sharpness. There s a rade-off beween relably and sharpness, meanng ha mprovng he relably wll generally degrade he sharpness and vce-versa [4] Tme-adapve Algorhm: Proof of Concep The am of hs secon s o demonsrae he valdy of he me-adapve concep presened n secon 2.5. However, n order o nroduce a change n he daa srucure, we dsconneced wo ses (one of 2.6 MW and anoher of 66. MW, ou of a 5.9 GW oal) durng Jan-Sep and conneced hem afer Oc. Ths change was creaed arfcally; however, reproduces a suaon ha could acually happen. For nsance: a sysem operaor s recevng forecass from 3 wnd farms (hese forecass are summed up and esmaes for he uncerany assocaed o he oal wnd power generaon are produced); hen, n Ocober wo new wnd farms are conneced o he grd. In hs case, he nowledge from pas observaons s no longer vald. By usng a me-adapve model he sysem operaor s able o adap o he new suaon whou requrng an offlne ranng of he model. Fg. 5 depcs he calbraon dagram obaned wh he offlne and he me-adapve W esmaors wh hree dfferen values of λ. Due o he ncrease n he wnd power generaon wh he connecon of wo wnd farms, s expeced ha he offlne model gves an underesmaon of he quanles for values below he 5% quanle and an overesmaon of he quanles for greaer values. As an example, he 95% quanle means ha he probably of havng a wnd generaon above s value s only 5%; however, he emprcal analyss wh he offlne approach shows ha hs probably s 3%. Wh he me-adapve verson, under and overesmaons are parly correced. The calbraon obaned wh λ equal o.999 and.995 s much beer han he offlne approach. For nsance, for he quanle 95% he emprcal proporons obaned wh he me-adapve approach s 92.3% wh λ=.999 and 9.2% wh λ= Evaluaon Resuls: Mdwes Wnd Farm 3.4. Offlne Resuls The followng ernel funcons were used: Chen s bea ernel wh h=.8 for he wnd power generaon; Chen s gamma ernel wh h=.5 for he wnd speed forecas; von Mses dsrbuon wh κ= 2.5 for he forecased wnd drecon; Chen s bea ernel wh h=. for he loo-ahead me sep. The ernel bandwdh values were deermned expermenally (ral-error) and usng as sarng pon he values suggesed by he R pacage hdrcde [26]. Fg. 6 depcs he calbraon obaned wh W and splnes QR (wh 6 degrees of freedom). The bes calbraon performance s from he W esmaor. As prevously menoned, due o he rade-off beween calbraon and sharpness, s expeced from he splnes QR a beer sharpness performance, as depced n Fg. 7. Fg. 8 depcs he sll score compued for each looahead me sep for boh W and QR esmaors. The W esmaor has almos he same performance as QR 7 h Power Sysems Compuaon Conference Socholm Sweden - Augus 22-26, 2

6 n erms of sll score. QR s beer han DF for some loo-ahead seps, bu s worse n ohers. Devaon [%] Offlne Lambda=.999 Lambda=.995 Lambda= omnal proporon rae [%] Fgure 5: Calbraon dagram for he REL daase wh concep change. Devaon [%] -5 5 W SplnesQR omnal proporon rae [%] Fgure 6: Calbraon dagram for he Mdwes wnd farm wh offlne W and QR esmaors. Inervals mean lengh [% of raed power] W SplnesQR omnal coverage rae [%] Fgure 7: Sharpness dagram for he Mdwes wnd farm wh offlne W and QR esmaors. oe ha he sll score does no nform on he conrbuons from calbraon or sharpness. Hence, calbraon should be assessed (as a prmary requremen), and hen he nformaon provded by sll score allows dervng conclusons abou he remanng mercs [25]. Sll Score W SplnesQR Loo-ahead Tme [hours] Fgure 8: Sll score dagram for he Mdwes wnd farm wh offlne W and QR esmaors Tme-adapve Resuls The me-adapve W verson was compared wh he offlne verson for dfferen values of he forgeng facor (λ). For a beer undersandng, λ was represened by he correspondng n value. So, hree values for λ were consdered: (corresponds o n=2738 pons),.999 (corresponds o n= pons) and.995 (corresponds o n=2 pons). The same ernel and bandwdhs as n he offlne verson was used. Fg. 9 depcs he calbraon resuls. The meadapve verson wh n=2738 and n= acheved he bes performance, whle a small number of pons n he sldng wndow leads o a worse performance comparng o he offlne resuls. The verson wh hgher λ does no have a sgnfcan mpac on he sharpness, as depced n Fg.. Devaon [%] Offlne Tme-adapve (n=2738) Tme-adapve (n=) Tme-adapve (n=2) omnal proporon rae [%] Fgure 9: Calbraon dagram for he Mdwes wnd farm wh offlne and me-adapve W. Fg. depcs he sll score for boh versons. The bes performance was obaned wh 2738 and pons. The dfference beween offlne and me adapve versons s only noceable n he frs 28 loo-ahead me seps. 7 h Power Sysems Compuaon Conference Socholm Sweden - Augus 22-26, 2

7 Inervals mean lengh [% of raed power] Offlne Tme-adapve (n=2738) Tme-adapve (n=) Tme-adapve (n=2) omnal coverage rae [%] Fgure : Sharpness dagram for he Mdwes wnd farm wh offlne and me-adapve W. Sll Score Offlne Tme-adapve (n=2738) Tme-adapve (n=) Tme-adapve (n=2) Loo-ahead Tme [hours] Fgure : Sll score dagram for he Mdwes wnd farm wh offlne and me-adapve W. 4 COCLUSIOS Ths paper presens a new approach o esmae he uncerany n shor erm wnd power forecass, based on ernel densy esmaon, ncludng a new me-adapve model. Our sudes demonsraed ha ernel densy forecass wh he W esmaor have a endency o presen beer performance n erms of calbraon, whle he QR mehods have a endency o presen a beer sharpness performance. The sll score of boh mehods s raher smlar. We mus underlne ha he calbraon merc s he prmary requremen for wnd power probablsc forecasng. The new me-adapve verson mproves he bas of he probablsc forecass (calbraon), whle only slghly changng he sharpness; mproves he sll score when compared wh he offlne approach. From a qualave perspecve, densy esmaon models offer an mporan advanage over oher models. The oupu s a complee descrpon of he forecased pdf, somehng ha s of mporance for several decson problems n he power sysem doman. Moreover, he full pdf s very valuable for forecasng mulmodal dsrbuons (.e. compue he modes nsead of jus compung he expeced value). Hence, he wor repored n hs paper s seen as a vald conrbuon o he wnd power forecasng acvy. ACOWLEDGEMETS The submed manuscrp has been creaed by UChcago Argonne, LLC, Operaor of Argonne aonal Laboraory ( Argonne ). Argonne, a U.S. Deparmen of Energy Offce of Scence laboraory, s operaed under Conrac o. DE AC2-6CH357. The U.S. Governmen reans for self, and ohers acng on s behalf, a pad-up non-exclusve, rrevocable worldwde lcense n sad arcle o reproduce, prepare dervave wors, dsrbue copes o he publc, and perform publcly and dsplay publcly, by or on behalf of he Governmen. The auhors acnowledge Horzon Wnd Energy for provdng some of he wnd farm daa used n he analyss. The auhor Rcardo J. Bessa acnowledges Fundação para a Cênca e a Tecnologa (FCT) for PhD Scholarshp SFRH/BD/33738/29. REFERECES [] R.J. Bessa, V. Mranda and J. Gama, Enropy and Correnropy Agans Mnmum Square Error n Offlne and Onlne Three-day Ahead Wnd Power Forecasng, IEEE Trans. on Power Sys., vol. 24, no. 4, pp , ov. 29. [2] R.J. Bessa, V. Mranda, A. Boerud and J. Wang, Good or Bad Wnd Power Forecass: A Relave Concep, Wnd Energy, 2, In Press. DOI:.2/we.444 [3] C. Monero, R. Bessa, V. Mranda, A. Boerud, J. Wang and G. Conzelmann, Wnd Power Forecasng: Sae-of-he-Ar 29, Repor AL/DIS--, Argonne aonal Laboraory, ov. 29. [4] H.A. elsen, H. Madsen and T. S. elsen, Usng Quanle Regresson o Exend an Exsng Wnd Power Forecasng Sysem wh Probablsc forecass, Wnd Energy, vol. 9, no., pp. 95-8, 26. [5] P. Pnson and G. arnoas, Condonal Predcon Inervals of Wnd Power Generaon, IEEE Trans. on Power Sys., vol. 25, no. 4, pp , ov. 2. [6].J. Culer, H.R. Ouhred, I.F. MacGll, M.J. ay and J.D. eper, Characerzng Fuure Large, Rapd Changes n Aggregaed Wnd Power Usng umercal Weaher Predcon Spaal Felds, Wnd Energy, vol. 2, pp , 29. [7] E. Consannescu, V. Zavala, M. Rocln, S. Lee, and M. Anescu, A Compuaonal Framewor for Uncerany Quanfcaon and Sochasc Opmzaon n Un Commmen wh Wnd Power Genera- 7 h Power Sysems Compuaon Conference Socholm Sweden - Augus 22-26, 2

8 Powered by TCPDF ( on, IEEE Trans. on Power Sys., vol. 26, no., pp , 2. [8] A. Boerud, J. Wang, R.J. Bessa, H. eo and V. Mranda, Rs Managemen and Opmal Bddng for a Wnd Power Producer, IEEE PES General Meeng, Mnneapols, USA, Jul. 2. [9] M.A. Maos and R.J. Bessa, Seng he Operang Reserve Usng Probablsc Wnd Power Forecass, IEEE Trans. on Power Sys., vol. 26, no. 2, pp , May 2.. []J. Wang, A. Boerud, R. Bessa, H. eo, L. Carvalho, D. Isscaba, J. Sumal, V. Mranda, Represenng Wnd Power Forecasng Uncerany n Un Commmen, Appled Energy, 2, In Press: DOI:.6/j.apenergy []J. Usaola, Probablsc Load Flow wh Correlaed Wnd Power Injecons, Elec. Power Sys. Res., vol. 8, no. 5, pp , May 2. [2]P. Pnson, G. Papaefhymou, B. locl, H.Aa. elsen and H. Madsen, From Probablsc Forecass o Sascal Scenaros of Shor-erm Wnd Power Producon, Wnd Energy, vol. 2, no., pp. 5-62, 29. [3]J.. Møller, H.A. elsen and H. Madsen, Tme- Adapve Quanle Regresson, Comp. Sa. & Daa Anal., vol. 52, no. 3, pp , Jan. 28. [4]J. Juban,. Seber and G. arnoas, Probablsc Shor-erm Wnd Power Forecasng for he Opmal Managemen of Wnd Generaon IEEE Power Tech Conference, Swzerland, July 27. [5]M. Rosenbla, Remars on Some onparamerc Esmaes of a Densy Funcon, The Annals of Mah. Sa., vol. 27, no. 3, pp , 956. [6]M.P. Wand and M.C. Jones, Mulvarae Plug-n Bandwdh Selecon, Comp. Sa., vol. 9, pp. 97-6, 994. [7]R.J. Hyndman, D.M. Bashanny and G.. Grunwald, Esmang and Vsualzng Condonal Denses, Journal of Comp. and Graph. Sa., vol. 5, no. 4, pp , Dec [8]S.. Chen, Bea ernel Esmaors for Densy Funcons, Comp. Sa. & Daa Anal., vol. 3, no. 2, pp. 3-45, 999. [9]S.. Chen, Probably Densy Funcon Esmaon usng Gamma ernels, Annals of he Ins. of Sa. Mah., vol. 52, no. 3, pp , Sep. 2. [2]. V. Marda and P. E. Jupp, Dreconal Sascs, Wley, ov ISB: [2]C. Goureroux and A. Monfor, (on) Conssency of he Bea ernel Esmaor for Recovery Rae Dsrbuon, Worng Paper º26-3, Isu aonal de la Sasque e des Eudes Economques, Dec. 26. [22]E.J. Wegman and D.J. Marchee, On Some Technques for Sreamng Daa: a Case Sudy of Inerne Pace Headers, Journal of Comp. and Graph. Sa., vol. 2, no. 4, pp , 23. [23]Easern Wnd Inegraon and Transmsson Sudy (EWITS), aonal Renewable Energy Laboraory (REL).Avalable: hp:// ml [24]W. Samaroc, e al., A Descrpon of he Advanced Research WRF Verson 2, CAR/T 468+STR Techncal noe, Jun. 25. [25]P. Pnson, H.A. elsen, J.. Moller, H. Madsen and G. arnoas, onparamerc Probablsc Forecass of Wnd Power: Requred Properes and Evaluaon, Wnd Energy, vol., no. 6, pp , ov. 27. [26]R.J. Hyndman, J. Enbec and M. Wand, Hghes Densy Regons and Condonal Densy Esmaon. Pacage hdrcde, Techncal Repor, ov h Power Sysems Compuaon Conference Socholm Sweden - Augus 22-26, 2