A bi-level mode decomposition framework for multi-step wind power forecasting using deep neural network

Ene gy Con e sion and Managemen : X 23 (2024) 100650 A ailable online 22 June 2024 2590-1745/© 2024 The Au ho (s). Published by Else ie L d. This is an open access a icle unde he CC BY-NC license (h p://c ea i ecommons.o g/licenses/by- nc/4.0/). A bi-le el mode decomposi ion amewo k o mul i-s ep wind powe o ecas ing using deep neu al ne wo k ★ Jingxuan Wu a , * , Shu ing Li b , ** , Juan C. Vasquez a , Josep M. Gue e o a , b , c a Aalbo g Uni e si y, Pon oppidans aede 111, Aalbo g 9220, Denma k b Ba celonaTech (UPC), Ba celona Eas School o Enginee ing (EEBE), Ba celona 08019, Spain c Ca alan Ins i u ion o Resea ch and Ad anced S udies (ICREA), Pg. Lluís Companys 23, Ba celona 08010, Spain ARTICLE INFO Keywo ds: BiLSTM Deep lea ning Mode decomposi ion Wind powe p edic ion ABSTRACT The p opo ion o wind ene gy in global ene gy s uc u e is g owing apidly, p omo ing he de elopmen o wind powe o ecas ing (WPF) echnologies o sol e he unce ain y and in e mi ence o wind powe gene a ion. Howe e , he nonlinea and s ochas ic ea u es o wind powe ime se ies es ain he accu acy o mul i-s ep p edic ion pe o mance. A mul i-s ep WPF (MS-WPF) app oach based on a ime se ies bi-le el empi ical mode decomposi ion (BLEMD) me hod and BiLSTM neu al ne wo k is p oposed in his pape o imp o e he WPF accu acy o egional wind powe gene a o s. Since he unce ain y is always gene a ed h ough coupled ac o s om bo h wind and wea he - o-powe con e sion, he linea i y ea u e is i s in oduced as an aspec apa om he equency in he p oposed app oach o decompose he wind powe ime sequence da a. The p oposed BLEMD in oduces Pea son p oduc -momen co ela ion coe icien o e alua e he linea i y o ime se ies and a linea i y- based decomposi ion algo i hm is designed acco dingly. To u he enhance he p ecision and elease compu- a ion bu dens, a DL-based p edic ion s a egy, including a BiLSTM ne wo k, a CNN-BiLSTM ne wo k, and a mean weigh es ima ion me hod a e implemen ed o p edic he componen s sepa a ely. The p oposed me hod only elies on local da a, g ea ly educing he da a acquisi ion and compu a ion cos . The p ecision o he p o- posed MS-WPF is e i ied by a 2.5 kW wind u bine wi h ho izons om 5 s o 30 s, a 1.5 MW wind u bine wi h ho izons om 10 min o 1 h, and a 51 MW wind a m wi h ho izons om 1 h o 6 h. The compa a i e expe - imen al esul s wi h o he cu ing-edge me hods indica ed ha he p oposed MS-WPF has supe io p edic ion accu acy and s able pe o mance o mul i-s ep p edic ion. 1. In oduc ion Wind ene gy has eme ged as a highly sough -a e o m o enewable ene gy due o he inc easing awa eness o en i onmen al conse a ion. The de elopmen o wind powe gene a ion has become one o he mos widely discussed opics a ound he wo ld. Besides he supe io i ies o non-pollu ion and enewabili y, wind powe gene a ion p omo ing is g ea ly es ic ed by he unce ain y and in e mi ence o wind powe . This p oblem becomes much mo e c i ical in some pa icula small-scale applica ions like egional Mic og id applica ions. Di e en om la ge- scale wind a m scena ios [1], he wind in Mic og ids becomes a mo e complex na u al phenomenon. No only wind oscilla ion, including speed, di ec ion, and gus bu also he mechanical and elec ical pa o he wind u bines (WTs), o ins ance, deg ada ion and ic ion o he gene a o , ene gy con e sion, and ansmission losses, migh aise he powe oscilla ion in Mic og id sys ems [2]. In his sense, wind powe o ecas ing (WPF) becomes an impo an pa o ene gy managemen s a egies o Mic og ids [3,4]. The dynamic cha ac e is ics o wind powe and wind speed ac as chao ic ime se ies wi h high s ochas ic ea u es, wind p edic ion is consequen ly ega ded as a complex eg ession ask. Exis ing WPF esea ch can be classi ied in o model- based and da a-d i en me hods. Model-based me hods include nume ical wea he p edic ion (NWP)- based physical me hods and pe sis ence models (PM). NWP can gi e egional long- e m wea he o ecas s (wind speed, cloud co e , em- pe a u e, ai p essu e, e c.) acco ding o a la ge amoun o wea he da a ★ This wo k was suppo ed in pa by he China Schola ship Council (CSC). * Co esponding au ho . ** P incipal co esponding au ho . E-mail add esses: [email p o ec ed] (J. Wu), [email p o ec ed] (S. Li). Con en s lis s a ailable a ScienceDi ec Ene gy Con e sion and Managemen : X jou nal homepage: www.sciencedi ec .com/jou nal/ene gy-con e sion-and-managemen -x h ps://doi.o g/10.1016/j.ecmx.2024.100650 Ene gy Con e sion and Managemen : X 23 (2024) 100650 2 [5]. I s pe o mance on WPF is highly es ic ed by he addi ional un- ce ain y caused by he wea he - o-powe con e sion. Besides, he high compu a ion bu den o his me hod also es ains i s applica ion on ul a-sho - e m [6]. The pe sis ence model, on he con a y, is a s aigh o wa d me hod. The p edic ion alue equals he la es obse - a ion alue. The pe o mance o he pe sis ence model deg ades as he ime ho izon ex ends. Hence, i p o ides highe accu acy in he ul a- sho - e m o ecas and he pe o mance is limi ed in medium- e m and long- e m p edic ion asks [7]. Da a-d i en me hods, using lagged obse a ion as inpu , ge inc easing a en ion due o he popula i y o supe iso y con ol and da a acquisi ion (SCADA) sys ems. Da a-d i en me hods can be di ided in o con en ional s a is ical and machine lea ning-based ypes. The au o eg essi e mo ing a e age (ARMA) model is o g ea impo ance in s a is ical me hods [8], and he NWP model is u he deployed based on he ARMA model o p o ide inpu da a [9]. The ec o au o eg essi e (VAR) model wi h he Leas Absolu e Sh inkage and Selec ion Ope a o is also e i ied as an e ec i e way o enhance he accu acy [10]. In- o ma ion usion is ano he solu ion o enhance p edic ion pe o mance by me ging mul i-sou ces da a [11]. Fo wind powe in e al p edic ion, uzzy models [12], Ma ko chain [13], and Lo enz dis u bance sequence [14] can be e ec i e solu ions. Howe e , he pe o mance o hese s a is ical me hods is highly dependen on he accu acy o s a is ical models and he non-linea i y o he inpu ime se ies will dec ease he p edic ion accu acy [15]. The machine-lea ning-based me hod is de eloped o sol e he nonlinea ime-se ies p oblem. Thanks o i s da a mining and ea u e ex ac ion abili ies, i ou pe o ms s a is ical me hods in mos cases. Since adi ional machine lea ning-based p e- dic ion, including suppo ec o machine, decision ee, o neu al ne wo k [16], equi es lo s o expe ience and skill o de elop shallow models, deep lea ning (DL) [17] shows i s ad an ages in au oma ion and high gene aliza ion e iciency [18]. Plen y o esea ch is o ganized wi h a i icial neu al ne wo k (ANN) [19], con olu ional neu al ne wo k (CNN) [20], au oencode [21], ecu en neu al ne wo k (RNN) [22], e c. In DL app oaches, RNN is heo e ically designed o ime se ies eg ession asks due o i s sequence da a memo y. Long sho - e m memo y (LSTM), bidi ec ional LSTM (BiLSTM) and ga ed ecu en uni s (GRU) [23,24], as a ian s o RNN, inhe i he ad an age and sol e he exploding o g adien anishing p oblem by in oducing an ex a o ge ga e o he memo y cell, making hem he mos e icien neu al ne wo k a chi ec u es o ime-sequence da a o ecas ing [25]. How- e e , he signi ican cha ac e is ic o wind, unce ain y, is s ill oo s o- chas ic o be i ed. In his sense, decomposi ion me hods a e deployed o spli ime- se ies signals in o wo o mo e sub-signals and p edic hem sepa- a ely. Empi ical mode decomposi ion (EMD), a ia ional mode decomposi ion (VMD), wa ele decomposi ion (WD), and Fou ie decomposi ion (FD) a e he mos common me hods o wind da a decomposi ion. EMD spli s he ime-se ies signal acco ding o he da a ea u e [26]. A complemen a y ensemble EMD is p oposed o decom- pose he wind powe da a in o ou in insic mode unc ions (IMFs) [27] and build he long- e m p edic ion wi h he sigma poin Kalman il e and sho - e m p edic ion wi h ein o cemen lea ning. EMD can also be used o wind speed p edic ion [28]. A s uc u ing elemen is in oduced o ex ac ea u es om he o iginal wind speed signal. Howe e , he main sho coming o EMD is he lack o ma hema ic heo y, which makes his me hod highly dependen on expe ience. VMD decomposes he o iginal signal in o se e al subse ies wi h limi ed bandwid h in he spec al domain [29], while WD and FD do i acco ding o he equency ea u e o he o iginal signal [30]. Based on he assump ion ha he wind ime se ies con ain di e en equencies, hese me hods will lead o an o e all imp o emen in o ecas ing. While conside ing he wind is a non-s a iona y andom p ocess, decomposi ion acco ding o equency and bandwid h migh ace he challenge o de ining a undamen al wa e o wa ele basis. To sol e his p oblem, his pape p oposes a mul i-s ep BiLSTM- based WPF (MS-WPF) me hod, including a Bi-le el EMD (BLEMD) wi h ma hema ical analysis and coope a ed DL-based p edic ion app oaches. The main con ibu ions o his app oach can be summa ized as ollows. 1. A BLEMD conside ing bo h linea i y and equency ea u es o ime se ies is p oposed o assign he eg ession algo i hm and compu a- ion esou ce. The design o he BLEMD is suppo ed by igo ous ma hema ical analysis, and a no el linea i y e alua ion me hod based on Pea son p oduc -momen co ela ion coe icien (PCCs) is p oposed o ex ac he linea end o he ime se ies. 2. A MS-WPF s uc u e is p oposed o handle he eg ession ask o ime se ies componen s. The MS-WPF, including a BiLSTM ne wo k, a CNN-BiLSTM ne wo k, and a weigh mean es ima ion (WME), p o- ides high-p ecision o ecas ing esul s and educes he compu a ion bu den. 3. The p oposed MS-WPF is alida ed wi h ealis ic da a om dis inc a eas. The mul i-s ep p edic ion pe o mance is e i ied by compa ing i wi h cu ing-edge app oaches. The esul s indica e ha he p oposed app oach can achie e a high p edic ion accu acy. The es o he a icle is o ganized as ollows. The amewo k o he mul i-le el WPF me hod is p esen ed in Sec ion 2. The p oposed BLEMD algo i hm is discussed in Sec ion 3. The s uc u e and algo i hm o he BiLSTM ne wo k, CNN-BiLSTM ne wo k, and he p oposed WME a e p esen ed in Sec ion 4. In Sec ion 5, he expe imen al esul s wi h h ee dis inc scena ios a e analyzed, and he compa ison be ween he p o- posed app oach and exis ing me hods is s udied. A conclusion is d awn in Sec ion 6. 2. F amewo k o he p oposed MS-WPF Due o he da a mining abili y, DL me hods always pe o m well wi hin long- e m o sho - e m ime se ies p edic ion. BiLSTM p o ides sa is ying accu acy due o i s memo y cell and bidi ec ional aining o Fig. 1. S uc u e o he p oposed MS-WPF. J. Wu e al. Ene gy Con e sion and Managemen : X 23 (2024) 100650 3 imp o e sequence lea ning. Howe e , o sho - e m WPF, he in luence o unce ain y migh lead o he p edic ion e o being ampli ied. The MS-WPF app oach is p oposed o enhance he p edic ion accu acy. The main s uc u e is shown in Fig. 1. The o line pa upda es he eg ession model o online p edic ion wi h his o ical da a and he online pa conduc s he mul i-s ep p edic ion in eal ime. Real- ime (RT) da a is s o ed in a da abase o deep neu al ne wo k aining. In he o line aining pa , he wind powe ime se ies will be no malized and decomposed by he BLEMD. The wind powe da a will be spli in o h ee pa allel ime se ies acco ding o equency ea u e and linea i y. Componen 1 (C1) desc ibes he end o wind powe . Componen 2 (C2) and Componen 3 (C3) ollow no mal dis ibu ion a ound ze o wi h egula ed equency. The o line aining p ocess implemen s a BiLSTM ne wo k and a CNN-BiLSTM ne wo k o C1 and C2 eg ession model aining. Meanwhile, he dis ibu ion o C3 is analyzed o p o ide he es ima ion pa ame e s o WME. The Mul i-s ep WPF DL Ne wo k ma ix is he amewo k o he eg ession model, which consis s o mul iple laye s de ined by he look-ahead s eps. The DL ne wo k ma ix will be gene a ed by he o line aining p ocess and each laye includes wo DL ne wo ks and a WME algo i hm o he h ee componen s. The mul i- le el eg ession model will be upda ed in a long- e m ho izon. In he sho - e m ho izon, RT da a would be sampled and s o ed in a da a bu e and hen decomposed in o h ee elemen s as in he o line aining p ocess. The mul i-le el eg ession model ou pu s he p edic ed wind powe using he componen s and wind speed da a, inally, he esul s o he componen s p edic ion will be combined in o he WPF ou pu . 3. P oposed BLEMD algo i hm The unce ain y o wind powe and wind speed signi ican ly inhibi s he WPF accu acy. In he ime se ies o wind powe sampling da a, he unce ain y ac s like a andom e o , and he ampli ude inc eases as he ime ho izon dec eases. This Sec ion will in oduce he BLEMD algo- i hm o wind powe da a and analyze he equency and linea i y ea u es o componen s. 3.1. F equency and linea i y-based BLEMD BiLSTM ne wo k ou pe o ms o he DL me hods on ime sequence eg ession and p edic ion, bu s ochas ic componen s migh inc ease he non-linea i y and lead o he pe o mance deg ada ion o BiLSTM. To es ain he in luence o s ochas ic componen s and alloca e he compu a ion esou ce app op ia ely, he equency and linea i y-based BLEMD is implemen ed o di ide he ime se ies in o h ee compo- nen s, he p ocess is shown in Fig. 2. To ex ac he unp edic able unce ain y om he wind powe da a, a lowpass il e is deployed o spli he high- equency componen s. The i e a ion p ocess o he il e can be desc ibed as {LFS(n) = TS ω cP(n) + (1−TS ω c)LFS(n−1); C3(n) = P(n) − LFS(n). (1) in which low- equency sequence (LFS) is he componen ex ac ed by he il e , he P is he wind powe ime se ies, Ts ep esen s he sample ime, and ω c means passband equency. The di e ence be ween he aw signal and he ou pu LFS is he C3. The ω c mus be se as high as possible o limi he ampli ude o C3 since i s mean and s anda d de ia ion a e in e sely p opo ional o ω c. He e he ω c is de ined acco ding o he look-ahead s eps and i should be less han hal o he sample a e. A e he equency decomposi ion, he ime se ies is di ided in o wo sub-signals wi h speci ic equency cha ac e is ics, and he esidual will be u he decomposed acco ding o he linea i y ea u es. A Gaussian-weigh ed mo ing mean (GWMM)-based da a smoo he [31] is deployed he e o ex ac he end o wind powe ime se ies, which is C1, and he C2 is he emainde . The window mus be igh -aligned as ⎧ ⎪ ⎨ ⎪ ⎩ C1(n)=Σn i=n−m+1LFS(i)wi Σn i=n−m+1wi ; C2(n) = LFS(n) − C1(n). (2) whe e C1(n)dona es he ou pu alue o GWMM, also he C1 ime se ies, a n h da a poin , he LFS is he low- equency sequence de ined in (1). The wi is he weigh calcula ed by he a e age alue and s anda d de- ia ion o he sliding window, and m is he leng h o he window. The window leng h is he c i ical pa ame e o he GWMM, i a ec s he linea i y o C1 and he co ela ion be ween C1 and C2. The p edic ion o C1 and C2 will bene i om a alid choice o window leng h m. 3.2. F equency and linea i y analysis The main pa ame e s o he p oposed BLEMD a e he ω c and he m, which a e ela ed o he equency and linea i y sepa a ely. The ω c will comp ess he equency o C1 and C2 in a limi ed egion, which will g ea ly bene i he pe o mance o he deep lea ning-based p edic ion me hod. Howe e , he ampli ude o C3 g ows as he ω c dec eases, so he ω c is se as ω c=1 2 α Ts (3) ω c is ela ed o he maximum obse a ion ime Ts in he aining p ocess and he look-ahead s eps α . The α mus be smalle han he look- ahead s ep o enhance he pe iodici y o C1 and C2 wi h a cycle ime equal o he look-ahead ho izon. Hence, he p edic ion pe o mance o C1 and C2 can be imp o ed. Besides, he ampli ude o C3 inc eases as he α dec eases, which will gene a e inc easing andom e o s in he p edic ion o C3. In his sense, he α should be as la ge as possible. By join ly conside ing he e ec s on he wo aspec s, he α should be equal o he look-ahead s ep. Apa om he equency, linea i y is also an essen ial ea u e a ec ing he p ecision o DL-based p edic ion me hods. Howe e , he e a e s ill no widely accep ed linea i y e alua ion me hods, so a PCC- based linea i y analysis is designed o guide he linea i y-based decomposi ion. I is de ined as he PCCs be ween he aw signal and Fig. 2. wind powe da a BLEMD p ocess. J. Wu e al. Ene gy Con e sion and Managemen : X 23 (2024) 100650 4 i s 2-o de polynomial eg ession i ing cu e in a sliding window. The ma hema ical p ocedu e o he p oposed linea i y e alua ion includes wo s eps. Fi s ly, a sliding window is ac i a ed o he local i ing. A 2- o de polynomial i ing cu e in each window is gene a ed h ough o dina y leas squa es. As he window mo es, a new disc e e signal wi h he same end as he aw signal is p oduced as he linea i y e e ence. The second s ep is calcula ing he PCCs be ween he aw da a and he e e ence, which can be deno ed as C1= [C1(1),C1(2),…,C1(n)]; (4) and C1 e =[C1 e (1),C1 e (2),…,C1 e (n)]; (5) The linea i y l can be calcula ed by l=Σn i=1(C1(i)−C1)(C1 e (i)−C1 e )  Σn i=1(C1(i) − C1)2Σn i=1(C1 e (i)−C1 e )2 √(6) whe e l anges in (0, 1], and he close l app oaches 1, he highe he aw da a’s linea i y. C2 is de ined as he emainde o he LFS a e ex ac ing C1. The co ela ion be ween hem can be e alua ed by hei co ela ion coe i- cien 1−2, 1−2= Σn i=1(C2(i)−C2)(C1(i)−C1)  Σn i=1(C2(i) − C2)2Σn i=1(C1(i) − C1)2 √ (7) The ange o 1−2 is (0, 1], and he highe he 1−2, he close he co ela ion be ween he wo elemen s. The BLEMD could be designed based on he linea i y e alua ion l o C1 and co ela ion coe icien 1−2 be ween C1 and C2. I is easie o he DL me hods o ain an accu a e eg ession model wi h a linea and highly co esponding inpu , which means highe l and 1−2. The sliding window leng h o he smoo he is an essen ial ac o o hem. The l is ising while he 1−2 is alling wi h he sliding window leng h inc easing. To balance he l and 1−2, a p edic abili y coe icien ε is de ined as, ε =lΣC1+ 1−2Σ|C2| ΣC1+ 1−2Σ|C2|(8) he ε is a ma hema ical unc ion o he GWMM window leng h m, he op imal window leng h m is he index o he maximum p edic abili y coe icien . m∈ [2,3,4,…]: ε (m)⩾ ε (x), o ∀x∈ [2,3,4,…](9) The p edic abili y o he ain da a will be analyzed and he op i- mized window leng h will be igu ed ou o he BLEMD in each aining p ocess. 4. P oposed DL-based p edic ion and e o compensa ion s a egy The h ee componen s o wind powe will be p edic ed sepa a ely. The eg ession model o C1 will be gene a ed by a BiLSTM ne wo k, and he inpu in ol es wind speed. The o ecas ing o C2 is ealized ia a CNN-BiLSTM ne wo k o achie e be e accu acy, he inpu in ol es wind speed and C1. The C3 is p edic ed using a WME me hod o sa e compu a ion esou ces. 4.1. BiLSTM-based sequence o sequence DL ne wo k The BiLSTM is a u he s uc u e based on LSTM, which is an enhanced RNN wi h a memo y cell and a o ge ga e. The i e a ion p ocess o LSTM, which is al eady well-known, can be summa ized as ou s eps. Fi s ly, he o ge ga e o k h cycle is igu ed by, (k)= σ (W ⋅[h(k−1),x (k)]+b )(10) In (10), he x is he inpu ime sequence, he h is he hidden laye s a us o LSTM, σ is he passing a e wi hin [0, 1], hen W will decide which pa o he memo y and inpu sequence should be o go en. Secondly, he memo y ga e is calcula ed, i (k) = σ (Wi⋅[h(k−1),x (k)] + bi)(11) whe e i is he memo y ga e. The  C is he empo a y cell s a us, and he cell s a us C could be upda ed in he hi d s ep as,  C (k)= anh(Wc⋅[h(k−1),x (k)]+bc)(12) and C (k)= (k)⋅C (k−1)+i (k)⋅ C (k)(13) A e he cell s a us is upda ed, he ou pu ga e o is calcula ed, and hen he hidden laye is upda ed as, o (k) = σ (Wo⋅[h(k−1),x (k)] + bo)(14) and h(k) = o (k)⋅ anh(C (k)) (15) The his o ical inpu and cell s a us will be condi ionally conside ed du ing he i e a ion. Howe e , he e is s ill one sho coming: he e ec o he nex cycle is no conside ed. As a esul , BiLSTM is de eloped o sol e his p oblem. The p oposed sequence- o-sequence BiLSTM ne wo k is es ablished based on he LSTM, as shown in Fig. 3. The inpu sequence o he C1 ne wo k includes wo dimensions which a e wind Fig. 3. A chi ec u e o he p oposed BiLSTM aining ne wo k. J. Wu e al. Ene gy Con e sion and Managemen : X 23 (2024) 100650 5 speed and i s his o ical sequence, while he inpu sequence o he C2 ne wo k con ains one mo e dimension: C1. The BiLSTM laye has a o wa d LSTM and a backwa d one, he hidden laye o he wo LSTM ma ked as h and hb will be combined o achie e he imp o ed hidden laye sequence h . Then he d op laye will dele e some cells andomly o p e en o e i ing. The ully connec ed laye is employed o e ch he ea u e and a e he ac i a ion o he anh Laye , ano he is deployed o achie e he mapping be ween he hidden laye and esponses. 4.2. CNN-BiLSTM-based DL ne wo k The C2 sequence is p edic ed h ough a CNN-BiLSTM ne wo k since i con ains mo e mu a ions han C1. The C2 p edic ion p ocess in ol es C1, C2, and wind speed sequences, a con olu ion il e will scan he inpu ma ix and de ec he lead and lag o di e en sequences. The esul is ansmi ed o he BiLSTM ne wo k o ime sequence da a mining. The C2 sequence has limi ed equency and ollows no mal dis ibu ion, an essen ial challenge o i s p edic ion ask is a oiding o e - i ing. Apa om employing d op laye s in he DL ne wo k s uc u es, he lea ning a e is se o dec ease piecewise and a alida ion p ocess is es ablished o p e en o e - i ing p oblems. 4.3. High- equency componen es ima ion me hods Since he C3 is s ochas ic and unp edic able, an es ima ion me hod is designed ins ead o compensa e o he andom e o in he WPF. Based on he wo ea u es o he C3: 1) The equency o he C3 has limi ed abo e he pass-band equency; 2) The dis ibu ion o C3 ollows he no mal dis ibu ion a ound ze o; he WME is designed. The p ocedu e has h ee s eps. S ep 1: e ch a da a window o his o ical da a wi h he leng h o nWME and s o e i in ma ix X (So by ime in descending o de ), he es ima- ion elemen s ma ix  E will be calcula ed by, { E= [e(1),e(2),…,e(nWME −1)]; e(k)=(k+1) μ −Σk i=1X(i). (16) whe e μ ep esen s he long ho izon mean. S ep 2: The ma ix E is he absolu e es ima ion e o o each elemen in he p e ious cycle, and he weigh ma ix W is de ined by he accu acy o each elemen in he las i e a ion, {E= [e(1),e(2),…,e(nWME −1)]; e(k)= |X(1)−(k+1) μ +Σk+1 i=2X(i)|. (17) ⎧ ⎪ ⎨ ⎪ ⎩ W= [w(1),w(2),…,w(nWME −1)]; w(k)=ΣE−E(k) (nWME −2)ΣE. (18) S ep 3: calcula e he WME ou pu as, WME =WET(19) Bo h he long ho izon and sho ho izon mean a e calcula ed in  E, and he weigh ma ix W will adjus he dynamic esponse o WME acco dingly. The accu acy o he p oposed WME elies on he dis ibu- ion o C3. 5. Case s udy Compu a ional expe imen s a e implemen ed o alida e he pe - o mance o he p oposed MS-WPF on di e en look-ahead s eps and ho izons agains exis ing me hods. Th ee da ase s wi h di e gen sce- na ios, lis ed in Table 1, a e u ilized o es he ad ance o he p oposed me hod. Da a om scena io 1 comes om he eal- ime da a acquisi ion sys em o he IoT labo a o y, Aalbo g Uni e si y, Denma k, in which a 2.5 kW WT is es ablished and analyzed in his scena io. Scena ios 2 and 3 a e ex ac ed om a public da ase om KDD Cup 2022 [9], he da a comes om he SCADA sys em o a wind a m in China. Scena io 2 is he ou pu powe o one single WT. Scena io 3 uses he o al wind a m ou pu powe and a e age wind speed da a o 34 WTs. The ho izons o he scena ios a e dis inc , om ul a-sho - e m (5s) o long- e m (1 h). The pe sis ence model is deployed as one o he baselines o p edic ion me hodology due o i s high pe o mance in sho - e m and ul a-sho - e m p edic ion. The o he baselines include ARIMA [28,32], GRU [24], CNN-BiLSTM [20,23] and WD-BiLSTM [30]. Fo all he DL-based me hods, 80% o da a is alloca ed o aining, 1% o alida ion du - ing he aining and 19% o es ing. Mean absolu e pe cen e o (MAPE) and nominal oo mean squa ed e o (nRMSE) a e he wo c i ical pa ame e s o WPF accu acy e alua ion. MAPE =1 NΣN i=1 yi−yi yi(20) in which yi is he ac ual wind powe and yi is he wind powe p edic ion. The nRMSE is de ined by, nRMSE = ΣN i=1(yi−yi)2 ΣN i=1(yi−y)2 √(21) y is he mean o ac ual alue. MAPE is a e y in ui i e in e p e a ion o ela i e e o , and nRMSE is mo e sensi i e o ou lie s. Addi ionally, he coe icien o de e mina ion, R2, is deployed o e alua e he eg ession pe o mance, which is de ined as, R2=1−ΣN i=1(yi−yi)2 ΣN i=1(yi−y)2(22) when he R2 app oaches 1, he p edic ion is posi i ely co ela ed o he o iginal signal. I he R2 app oaches −1, he p edic ion is nega i ely co ela ed o he o iginal signal. R2=0 indica es ha y and y a e no ela ed. Table 1 Con igu a ion o he expe imen al da ase s. Pa ame e s Scena io 1 Scena io 2 Scena io 3 Ho izon 6*5 s 6*10 min 6*1 h Nominal powe 2.5 kW 1.5 MW 1.5 MW*34 Loca ion Denma k China China Fig. 4. The wind powe ime se ies and ou pu componen s o BLEMD in 5-s, 10-min, and 1-h scena ios. J. Wu e al. Ene gy Con e sion and Managemen : X 23 (2024) 100650 6 5.1. 6-s ep p edic ion esul s o he p oposed MS-WPF The pe o mance o he MS-WPF is e i ied in h ee di e en sce- na ios wi h di e en scales and ho izons. An abla ion expe imen is conduc ed o alida e he e ec i eness o he p oposed BLEMD. A con- en ional EMD using he de end unc ion, which also decomposes he o iginal signal in o h ee componen s, is deployed o coope a e wi h he same DL amewo k. The wind powe ime se ies aw da a should be p ep ocessed h ough he BLEMD. The ou pu is h ee componen s wi h di e en linea i y and complemen a y equency ea u es. The ou pu o BLEMD o MS-WPF o he h ee scena ios is p esen ed in Fig. 4. The pa ame e s o he BLEMD a e op imized by he me hodology p oposed in Sec ion 3. The sliding window leng h is 39 o scena io 1, 5 o scena io 2, and 35 o scena io 3, as Fig. 4 shows, C1 has he same end as wind powe , C2 and C3 oscilla e a ound 0. The h ee componen s a e ansmi ed o he pa allel DL ne wo ks o ain he eg ession models, he hype pa ame e s a e p esen ed in Table 2. The 1-s ep p edic ion esul s o he MS-WPF and he EMD- BiLSTM o he h ee componen s a e p esen ed in Fig. 5. The p edic- ion esul s p esen ed by he do s a e compa ed wi h he ac ual wind powe alue in he same igu e, and he dis ance be ween he do s and he e e ence line ep esen s he p edic ion e o . I is e iden ha he BiLSTM and he CNN-BiLSTM ne wo k ha e sa is ying accu acy wi h he p edic ion ask o C1 and C2. In Fig. 5(g–i), he p oposed WME me hod also shows p omising pe o mance. Table 2 Hype pa ame e s o he p oposed MS-WPF Laye s BiLSTM CNN-BiLSTM Con olu ion Laye NA C[4,1] – F[256] BiLSTM B[128] B[256] D op Laye 0.1 0.1 Fully connec Laye 1 F[256] F[512] Lea n a e/D op ac o 0.005/0.5 0.005/0.5 Fig. 5. 1-s ep componen s p edic ion esul s o 5-s scena io, 10-min scena io and 1-h scena io. J. Wu e al. Ene gy Con e sion and Managemen : X 23 (2024) 100650 7 The dis ibu ion o black do s and blue do s indica es he esul s o he MS-WPF and he EMD-BiLSTM. In bo h decomposi ion me hods, he gene al end o he wind powe ime se ies is ex ac ed as C1. The p oposed BiLSTM and CNN-BiLSTM can e ec i ely es ima e he C1 and C2. Howe e , con en ional EMD canno ensu e he equency and linea i y ea u es o he C1 and C2, so he accu acy is es ained. The p oposed BLEMD decomposes he wind powe ime se ies in o h ee componen s wi h speci ic cha ac e is ics: C1 (high linea i y, low e- quency), C2 (low linea i y, low equency), and C3 (low linea i y, high equency). The accu acy o he wind powe end is g ea ly enhanced i s by p edic ing he C1. The p edic ion accu acy o C2 bene i s om he egula ed equency o C2. The ampli ude o he C3 is es ained by he BLEMD and will ha e a limi ed e ec on o e all pe o mance. The 6-s ep p edic ion esul s o he p oposed MS-WPF e alua ed h ough MAPE, nRMSE, and R2 a e shown in Fig. 6. The p edic ion p ecision deg ades as he look-ahead s ep g ows as expec ed, and he MAPE is es ained by he p oposed me hodology. The MAPE o 1-s ep p edic ion is below 10 %, which g ea ly bene i s om he high p eci- sion o C1 and C2 p edic ion, as shown in Fig. 5. The in luence o C3 p edic ion e o is es ained by he low ampli ude o C3. As he look- ahead s ep g ows, he ω c o equency decomposi ion mus inc ease o achie e be e C1 and C2 p edic ion esul s. Hence he ampli ude o Fig. 6. S a is ical p edic ion esul e alua ion o a) 5-s scena io, b) 10-min scena io, and c) 1-h scena io. Fig. 7. Mul i-s ep p edic ion esul s o he p oposed MS-WPF and exis ing app oaches. J. Wu e al. Ene gy Con e sion and Managemen : X 23 (2024) 100650 8 C3 is ising, which leads o he educ ion o o ecas ing accu acy. The R2 indica es a good eg ession pe o mance. The p edic ion esul s ack he ealis ic wind powe well. The p edic ion accu acy o he 10-min scena io is he bes among he h ee cases. Scena io 1 ep esen s he ul a-sho - e m p edic ion which in ol es d ama ic wind dynamics. The dynamics ga he in he C3 ime se ies and di ec ly a ec he p ecision o he MS-WPF. Scena io 3 is a long- e m p edic ion case ha indica es a long- e m ho izon. The 1- s ep p edic ion in scena io 3 eaches an accu a e le el (MAPE = 0.056,nRMSE =0.059,R2=0,996). Howe e , he p ecision deg ades exponen ially as he ho izon g ows due o he s ochas ic o wind. In summa y, he p oposed MS-WPF p o ides high-pe o mance p edic ion o wind powe in di e en scena ios and ho izons. The 1- Table 3 Hype pa ame e s o he benchma ks Benchma ks Hype pa ame e s PM NA GRU G[1024] – D op[0.1] – F[1024] Lea n a e/D op ac o – [0.001/0.2] CNN-BiLSTM C[4,32] – B[512] – D op[0.1] – F[512] Lea n a e/D op ac o – [0.001/0.5] WD-BiLSTM WD[‘db3’, 9] B[128, 128] – D op [0.3, 0.7] Lea n a e/D op ac o – [0.01/0.5] ARIMA P[4] – D[0] – Q[5] Table 4 Mul i-s ep p edic ion esul s e alua ion o p oposed MS-WPF and exis ing app oaches. S eps S ep-1 S ep-2 S ep-3 S ep-4 S ep-5 S ep-6 Case 1 (5s) MAPE MS-WPF 0.067 0.126 0.187 0.232 0.256 0.332 PM 0.281 0.590 0.708 0.781 0.875 0.961 GRU 0.108 0.185 0.376 0.578 0.683 0.744 CNN-BiLSTM 0.152 0.300 0.547 0.734 0.790 0.801 WD-BiLSTM 0.307 0.425 0.547 0.638 0.685 0.740 ARIMA 0.234 0.430 0.712 0.928 1.072 1.169 nRMSE MS-WPF 0.110 0.215 0.312 0.361 0.407 0.451 PM 0.546 0.941 1.039 1.113 1.177 1.236 GRU 0.191 0.288 0.524 0.755 0.843 0.865 CNN-BiLSTM 0.240 0.465 0.766 0.905 0.929 0.936 WD-BiLSTM 0.488 0.630 0.719 0.781 0.819 0.848 ARIMA 0.468 0.905 1.361 1.625 1.802 1.884 R2 MS-WPF 0.987 0.953 0.902 0.869 0.834 0.797 PM 0.702 0.114 −0.079 −0.238 −0.385 −0.528 GRU 0.963 0.917 0.725 0.430 0.289 0.252 CNN-BiLSTM 0.942 0.783 0.413 0.181 0.137 0.123 WD-BiLSTM 0.762 0.602 0.482 0.389 0.329 0.280 ARIMA 0.780 0.180 −0.851 −1.639 −2.245 −2.548 Case 2 (10 min) MAPE MS-WPF 0.067 0.179 0.196 0.223 0.235 0.263 PM 0.291 0.462 0.592 0.664 0.716 0.798 GRU 0.150 0.216 0.344 0.487 0.587 0.642 CNN-BiLSTM 0.149 0.244 0.416 0.570 0.653 0.702 WD-BiLSTM 0.332 0.381 0.450 0.490 0.548 0.589 ARIMA 0.140 0.355 0.504 0.583 0.693 0.823 nRMSE MS-WPF 0.058 0.150 0.154 0.195 0.215 0.237 PM 0.272 0.445 0.561 0.644 0.711 0.767 GRU 0.108 0.163 0.283 0.401 0.468 0.501 CNN-BiLSTM 0.121 0.205 0.355 0.466 0.511 0.541 WD-BiLSTM 0.242 0.309 0.365 0.412 0.458 0.498 ARIMA 0.148 0.352 0.494 0.603 0.697 0.793 R2 MS-WPF 0.996 0.977 0.976 0.962 0.953 0.943 PM 0.925 0.801 0.684 0.584 0.493 0.411 GRU 0.988 0.973 0.919 0.839 0.780 0.749 CNN-BiLSTM 0.985 0.958 0.873 0.782 0.738 0.707 WD-BiLSTM 0.941 0.904 0.867 0.830 0.790 0.752 ARIMA 0.978 0.876 0.755 0.636 0.514 0.371 Case 3 (1 h) MAPE MS-WPF 0.056 0.120 0.163 0.204 0.276 0.437 PM 0.261 0.578 0.828 0.977 1.026 1.022 GRU 0.105 0.181 0.325 0.464 0.564 0.611 CNN-BiLSTM 0.188 0.374 0.544 0.645 0.758 0.852 WD-BiLSTM 0.232 0.330 0.416 0.501 0.574 0.637 ARIMA 0.341 0.676 0.812 0.707 0.670 0.796 nRMSE MS-WPF 0.059 0.116 0.158 0.185 0.249 0.378 PM 0.301 0.595 0.782 0.885 0.942 0.975 GRU 0.097 0.188 0.350 0.504 0.597 0.649 CNN-BiLSTM 0.184 0.353 0.493 0.565 0.619 0.676 WD-BiLSTM 0.238 0.348 0.438 0.516 0.587 0.646 ARIMA 0.300 0.602 0.708 0.636 0.613 0.674 R2 MS-WPF 0.996 0.986 0.975 0.965 0.937 0.856 PM 0.909 0.645 0.388 0.215 0.111 0.049 GRU 0.991 0.964 0.877 0.745 0.643 0.578 CNN-BiLSTM 0.966 0.875 0.756 0.679 0.616 0.541 WD-BiLSTM 0.942 0.878 0.808 0.733 0.655 0.582 ARIMA 0.909 0.637 0.498 0.595 0.623 0.544 J. Wu e al. Ene gy Con e sion and Managemen : X 23 (2024) 100650 9 s ep p edic ion e o is supp essed wi hin 7%, and he accu acy deg a- da ion as he ho izon g ows is e ec i ely es ained by he p oposed MS- WPF app oach. The p oposed BLEMD, which decomposes he complex wind powe ime se ies acco ding o linea i y and equency, has a signi ican e ec on enhancing he accu acy o he MS-WPF. 5.2. Compa ison be ween MS-WPF and exis ing app oaches The supe io i y o he p oposed MS-WPF me hod is alida ed h ough compa ison wi h cu ing-edge app oaches in h ee scena ios. The p edic ion esul s o he p oposed MS-WPF and exis ing app oaches wi h di e en p edic ion s eps a e shown in Fig. 7. The hype pa ame e s o he DL-based p edic ion me hod in he p oposed MS-WPF a e p e- sen ed in Table 2 and hype pa ame e s o exis ing app oaches a e lis ed in Table 3. The 1-s ep ahead p edic ion esul s o wo scena ios a e shown in Fig. 7(a), (d), (g), while he 3-s ep ahead esul s a e shown in (b), (e), (h) and he 6-s ep ahead in (c), ( ), (i). The blue cu e ep esen ing he e- sul s o he p oposed me hod always acks he ac ual alue well, he acking o wind powe end is p ecise, and he esponse o wind powe co ne s is as . The accu acy o he p oposed MS-WPF deg ades in 3-s ep and 6-s ep p edic ions due o he dec eased sensi i i y o he wind powe dynamics. Howe e , he o e all end o he wind powe change can s ill be e ec i ely acked. The PM esul desc ibes he cha ac e is ics o he wind powe da ase , which is nonlinea , s ochas ic, and ha d o o ecas . Mos o he me hods ha e p omising esul s on 1-s ep p edic ion, bu he p ecision deg ades d ama ically. Fo 6-s ep p edic ion, mos o he exis ing app oaches al eady lose he end o wind powe da a. The p oposed MS-WPF ou pe o ms all he exis ing app oaches in e ec i ely cap u ing he end o wind powe . Hence, he o e all accu acy o MS- WPF o mul i-s ep p edic ion is he highes among he selec ed me hods. F om he igu es, i can be ound ha DL-based me hods handle nonlinea p edic ion asks be e han con en ional ma hema - ical algo i hms. GRU ne wo k can e icien ly p edic wind powe e en wi hou any decomposi ion app oaches. I compa ing he 1-s ep p edic ion in he h ee scena ios, i can be concluded ha he p edic ion ask can be di ided in o wo sub- objec i es: 1) ollow he end, and 2) de ec he u ning poin . RNN- based DL app oaches can ollow he end e y well while con en- ional ma hema ical me hods and CNN-based DL algo i hms a e sensi- i e o u ning poin s. The p oposed MS-WPF can e ec i ely conside wo sub-objec i es join ly since he BLEMD has decoupled he end and oscilla ion componen s. The end is p edic ed by a BiLSTM ne wo k, he u ning poin is de ec ed by a CNN ne wo k, and he d ama ic oscilla ion is compensa ed by a ma hema ical app oach. In his way, he in e ac ion o componen s wi h di e en linea i y and equency in he DL ne wo k aining p ocess is a oided. The objec i es o he DL ne wo ks a e simpli ied and conc e ized. The MAPE, nRMSE, and R2 o he mul i-s ep p edic ion me hods wi h h ee scena ios a e p esen ed in Table 4. Gene ally, DL-based me hods ha e highe accu acy and eg ession pe o mance compa ed o con- en ional ma hema ical app oaches. The p oposed MS-WPF shows he highes accu acy, and o he DL-based app oaches also show good esul s wi h one o wo-s ep-ahead p edic ion. Howe e , wi h mo e look-ahead s eps, mos exis ing me hods will mee a p ecision deg ada ion due o he o e - i ing p oblem. The s ochas ic and nonlinea componen s in he ime se ies will a ec he p edic ion o linea componen s, leading o he algo i hm losing he o e all end o wind powe , u he dec easing he o e all WPF accu acy. This p oblem is signi ican ly supp essed in he p oposed MS-WPF, he in e ac ion o he linea and nonlinea compo- nen s in he p edic ion algo i hm is decoupled by he linea i y- equency BLEMD. The accu a e p edic ion o linea componen s (C1) ensu es he s able pe o mance o mul i-s ep o ecas ing. 6. Conclusion In his pape , an MS-WPF is p oposed o enhance he p edic ion ac- cu acy o wind powe p edic ion and imp o e he pe o mance as he look-ahead s ep g ows. The pe o mance o he p oposed me hod is e i ied h ough h ee c oss-expe imen s. The ad ancemen o he p o- posed MS-WPF is p o ed by compa ing i o i e con en ional me hods. The main wo ks o he pape can be summa ized in he ollowing aspec s. 1. An imp o ed BLEMD wi h igo ous ma hema ical analysis is designed o wind powe ime se ies decomposi ion. Apa om equency ea u es, a linea i y ea u e e alua ed by PCCs is p oposed o decompose he ime se ies legi ima ely. The p oposed BLEMD e ec i ely decouples he nonlinea and linea componen s in he wind powe ime se ies and es ains hei in e ac ions, u he enhancing he accu acy and s abili y o mul i-s ep o ecas ing. 2. A mul i-le el p edic ion s uc u e, including a BiLSTM-based neu al ne wo k, a CNN-BiLSTM-based neu al ne wo k, and a WME me hod, is es ablished o sol e he eg ession ask o s ochas ic ime se ies while e icien ly alloca ing he compu a ion esou ce. The C1, high linea i y and low- equency componen , is p edic ed wi h less compu a ion esou ce; he C2, low linea i y and low- equency componen , is sol ed by a mo e complex neu al ne wo k o ach- ie e be e esul s; he C3, low linea i y and high- equency componen , is compensa ed h ough he p oposed WME o elease compu a ion bu den. 3. The p oposed MS-WPF me hod ou pe o ms he exis ing app oaches in his pape . The p ecision eaches 93% and emains a ound 70 % a e 6-s ep-ahead o ecas ing in all cases. The p oposed BLEMD and DL-based mul i-s ep p edic ion me hod p o ide an essen ial e e - ence o o ecas ing asks o ime se ies wi h a high p opo ion o unce ain y. CRediT au ho ship con ibu ion s a emen Jingxuan Wu: Concep ualiza ion, Fo mal analysis, Da a cu a ion, So wa e, In es iga ion, Valida ion, W i ing – o iginal d a . Shu ing Li: Me hodology, Visualiza ion, W i ing – e iew & edi ing. Juan C. Vas- quez: Resou ces, W i ing – e iew & edi ing. Josep M. Gue e o: P ojec adminis a ion, Supe ision. Decla a ion o compe ing in e es The au ho s decla e ha hey ha e no known compe ing inancial in e es s o pe sonal ela ionships ha could ha e appea ed o in luence he wo k epo ed in his pape . Da a a ailabili y Da a will be made a ailable on eques . Re e ences [1] Li Menglin, Yang Ming, Yu Yixiao, Lee Wei Jen. A wind speed co ec ion me hod based on modi ied hidden ma ko model o enhancing wind powe o ecas . IEEE T ans Ind Appl 2022;58:656–66. [2] Wang Jianzhou, Qian Yuansheng, Zhang Linyue, Wang Kang, Zhang Haipeng. A no el wind powe o ecas ing sys em in eg a ing ime se ies e ining, nonlinea mul i-objec i e op imized deep lea ning and linea e o co ec ion. Ene gy Con e s. Manage. 2024;299:117818. [3] Khodaya Mahdi, Wang Jianhui. Spa io- empo al g aph deep neu al ne wo k o sho - e m wind speed o ecas ing. IEEE T ans Sus ain Ene gy 2019;10:670–81. [4] Wu Jingxuan, Li Shu ing, Fu Aihui, C e ko ic Miloˇ s, Palensky Pe e , Vasquez Juan C, Gue e o Josep M. Hie a chical online ene gy managemen o esiden ial mic og ids wi h hyb id hyd ogen-elec ici y s o age sys em. Appl Ene gy 2024; 363:123020. J. Wu e al.