Floating Wind Turbine Dynamics Identification

Floa ing Wind Tu bine Dynamics Iden i ica ion Iden i icaci´on de la din´amica de una u bina e´olica lo an e Juan Tecedo Roa Bachelo Deg ee in So wa e Enginee ing FACULTY OF COMPUTER SCIENCE Supe ised by Ma ilde San os Pe˜nas Ca los Luis Se ano Ba e o MADRID, 2021-2022 Floa ing Wind Tu bine Dynamics Iden i ica ion Iden i icaci´on de la din´amica de una u bina e´olica lo an e Bachello in So wa e Enginee ing Final P ojec Juan Tecedo Roa Supe ised by Ma ilde San os Pe˜nas Ca los Luis Se ano Ba e o Depa men o Compu e A chi ec u e and Au oma ion Facul y o Compu e Science Uni e sidad Complu ense de Mad id MADRID, 2021-2022 Acknowledgemen s To my supe iso s, Ma ilde San os Pe˜nas and Ca los Luis Se ano Ba e o om he Complu ense Uni e si y o Mad id, o hei guiding h oughou all he wo k, and o En ique Sie a Ga c´ıa om he Uni e si y o Bu gos o his sugges ions and guiding ea ly in he p ojec . Wi hou hei inpu and knowledge his wo k would ha e been a ca wi hou wheels. To he wo ke s a Biblio eca Complu ense o hei s yle guide in he o mal aspec s o w i ing a inal bachelo p ojec [14]. To he de elope s, c ea o s and main aine s o he ee-open-sou ce so wa e ha enabled he c ea ion o his wo k as well as he companies ha acili a e educa ional pe sonnel he pa ial o comple e use o hei licensed ools. i Abs ac Clima e change is one o he bigges and mos wo ying p oblems in he cu en wo ld. Renewable ene gy sou ces a e one o he main ools ha will allow he humani y o igh agains i . Mo e p ecisely, loa ing wind u bines o e unp eceden ed amoun s o gene a ed powe compa ed o hei onsho e o o sho e (bo om- ixed) coun e pa s. The echnology is howe e , a an ea ly s age o de elopmen , wi h a lo o imp o emen s o be made and coun less ields o s udy. This p ojec aims o s udy he beha io o he scale model o a loa ing wind u bine by elabo a ing se e al s a is ical models ha can p edic some o i s mos impo an s a is ical me ics. These s a is ical models a e dependen on he wind speed and he blade pi ch angle o he wind u bine. Addi ionally, a pe iodici y analysis o he wind u bine is also made in o de o de e mine i he e a e equencies associa ed wi h i a di e en wind speeds and pi ch angles. In his wo k, a da a p ep ocessing phase is ca ied ou wi h he aid o s a is ics and g aphical ep esen a ions. Then, wo s udies a e made: a pe iodici y analysis by se e al Fou ie T ans o ms, and mul iple eg ession supe ised models. The supe ised models used we e: Linea Reg ession, Polynomial Reg ession, Ridge Reg esso , Hube Reg esso , Gaussian Reg esso and a Neu al Ne wo k (MLP Reg esso ). Mos o he supe ised models we e e y success ul and could be used o c ea e a i ual model o a wind u bine. The pe iodici y analysis was also success ul and was consis en wi h he physical analysis o he wind u bine. Keywo ds — Wind u bine, loa ing, dynamics, iden i ica ion, eg ession, neu al ne - wo ks, s a is ical analysis, da a p ep ocessing, da a ep esen a ion, Fas Fou ie T ans- o ms. iii Resumen El cambio clim´a ico es uno de los p oblemas m´as impo an es y p eocupan es ac ualmen e en el mundo. Las ene g´ıas eno ables son una de las he amien as que enemos disponibles pa a pode lucha con a ´el. En conc e o, las u binas e´olicas lo an es pueden o ece una p opo ci´on de ene g´ıa el´ec ica gene ada sin p eceden es, especialmen e si las compa amos con las u binas emplazadas en ie a o en el ma a baja p o undidad ce ca de la cos a. Sin emba go, la ecnolog´ıa de las u binas e´olicas lo an es es ´a en sus comienzos, con incon ables mejo as po implemen a y di e sos campos de es udio. Es e p oyec o es udia el compo amien o de un modelo a escala de una u bina lo an e, median e la elabo aci´on de a ios modelos es ad´ıs icos que puedan p edeci las m´e icas es ad´ıs icas m´as ele an es de la u bina. Es os modelos es ad´ıs icos dependen de la elocidad del ien o y del ´angulo de a aque de las palas de la u bina. Adicional- men e, se ha ealizado ambi´en un an´alisis de la pe iodicidad de la u bina de ien o pa a de e mina qu´e ecuencias es ´an asociadas con ella y a qu´e elocidades de ien o y a qu´e ´angulos de a aque. En es e abajo se ealiza una p ime a ase de p ep ocesado de da os, con ayuda de he- amien as es ad´ıs icas y ep esen aciones g ´a icas. Pos e io men e, se ealizan dos es u- dios: un an´alisis de la pe iodicidad median e a ias T ans o madas R´apidas de Fou ie , y m´ul iples modelos supe isados de eg esi´on. Los modelos supe isados ue on los siguien es: Reg esi´on lineal, Reg esi´on Polin´omica, Reg esi´on de Ridge, Reg esi´on de Hube , Reg esi´on Gaussiana y una ed neu onal ( eg eso MLP). La mayo ´ıa de los modelos supe isados ob u ie on esul ados muy sa is ac o ios y pod ´ıan se usados pa a c ea un modelo i ual de la u bina de ien o. El an´alisis de pe iodicidad ue ambi´en exi oso y consis en e con el an´alisis ´ısico de la u bina de ien o. Palab as cla e — Tu bina e´olica, lo an e, din´amica, iden i icaci´on, eg esi´on, edes neu onales, an´alisis es ad´ıs ico, p ep ocesamien o de da os, ep esen aci´on de da os, FFT. LIST OF FIGURES xii 6.12 P edic ions o he Hube model o he mean o he leng h o he accele a ion ec o ...................................... 83 6.13 P edic ions o he MLPR model o he s anda d de ia ion o he leng h o heaccele a ion ec o ............................. 84 Lis o Tables 1.1 Wo kplan able. ............................... 9 1.2 O icialdeadlines................................ 9 1.3 Mainp ojec asks............................... 10 1.4 Py hon packages used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1 Equi alence be ween ol ages and wind speeds. . . . . . . . . . . . . . . 19 3.2 Sample o da a a 130V (8.5ms-1) and 1◦................... 23 3.3 Sample o da a a 200V (13.8ms-1) and 1◦. ................. 24 3.4 Da a a a wind speed o 8.5ms−1and 1◦pi ch angle. . . . . . . . . . . . . 24 3.5 Da a a a wind speed o 13.8ms−1and 1◦pi ch angle. . . . . . . . . . . . 24 3.6 G ouped da a se columns. . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.1 Da ase columns................................ 40 4.2 Da a se columns wi h leng h ec o s. . . . . . . . . . . . . . . . . . . . . 41 4.3 S a is ical unc ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.4 G ouped da a se columns. . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.5 G ouped da a se columns. . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.1 Da a s a is ics a a wind speed o 8.5ms−1and 30◦pi ch angle. . . . . . . 54 5.2 Da a s a is ics a a wind speed o 8.5ms−1and 30◦pi ch angle. . . . . . . 65 6.1 Da ase columns................................ 67 6.2 Model sco es b eakdown. Run 1. . . . . . . . . . . . . . . . . . . . . . . 71 6.3 Model sco es b eakdown. Run 2. . . . . . . . . . . . . . . . . . . . . . . 71 6.4 Model sco es b eakdown. Run 3. . . . . . . . . . . . . . . . . . . . . . . 71 6.5 Model sco es b eakdown. Run 4. . . . . . . . . . . . . . . . . . . . . . . 72 6.6 P edic ions o inpu s ou side he expe imen , only models wi h sco es o e 0.90. ...................................... 86 xiii Chap e 1 In oduc ion 1.1 Clima e change Clima e change is one o he mos impo an and wo ying p oblems in he cu en wo ld. Labeled as ” he bigges heal h h ea acing humani y” by he Wo ld Heal h O ganiza ion [33], wi h expec a ions o 250 000 addi ional dea hs pe yea be ween 2030 and 2050 due o clima e change alone. The causes o his dea h oll a e om: malnu i ion, mala ia, dia hea and hea s ess. Di ec damage cos s o heal h a e es ima ed o each be ween USD 2-4 billion pe yea by 2030 [32]. The main con ibu o o clima e change is ai pollu ion, and mo e conc e ely CO2 emissions. The e o e, we mus push o educing ai pollu ion d as ically. 1.1.1 Main pollu e s In o de o educe ai pollu ion we mus i s iden i y which elemen s con ibu e he mos o CO2emissions. In he ollowing g aph we analyze he main pollu e s: 1 CHAPTER 1. INTRODUCTION 2 Figu e 1.1: Global His o ial CO2Emissions by Sec o [11]. We can see ha h ee qua e s o he CO2emissions a e due o ene gy gene a ion. Elec ici y gene a ion is, he e o e, he i s pollu e , and i is whe e mos o he wo k mus be done. On he elec ici y gene a ion mix, he main non- enewable sou ces ep esen he ol- lowing pe cen ages: oil (31.2%), coal (27.2%) and gas (24.7%). The es o he gene a ion is made by enewables (5.7%), hyd oelec ic (6.9%) and o he low-ca bon gene a o s such as nuclea (4.3%) [9]. The oom o g ow h in enewable sou ces is he e o e la ge. I mus eplace he main pollu e s such as coal, oil and gas. The ollowing g aph illus a es he accele a ing g ow h in wind as a enewable ene gy sou ce in he ecen yea s: CHAPTER 1. INTRODUCTION 3 Figu e 1.2: Renewable Ene gy Gene a ion [24] [9]. In conclusion, wind as a enewable ene gy sou ce is undamen al o educe he global CO2emissions. I is also cu en ly in a pe iod o apid adop ion. Wind echnology is, howe e , s ill e ol ing echnologically and he e a e s ill p oblems o be sol ed and op imiza ions o be made. 1.2 How a u bine gene a o wo ks Wind powe gene a ion is made by he use o gene a o s inside he wind u bines. Tu - bines ex ac he kine ic ene gy o a luid (wind) ha is passing h ough i s blades (in his case i is wind) and con e s i in o a o a o y mo ion. This mechanical mo ion is ed in o an elec ical gene a o , ha con e s he o a o y mechanical ene gy in o elec ical ene gy (no mally al e na ing ol age (AC)). 1.2.1 Pa s o a wind u bine To u he explain how a u bine wo ks we p esen a diag am: CHAPTER 1. INTRODUCTION 4 Figu e 1.3: Pa s o a wind u bine as seen om he back and o he le side. [31]. F om igu e 1.3 we mus i s ocus on he h ee main componen s: he owe ( he col- umn ha suppo s he nacelle), he nacelle (s eamlined body ha houses he mechanical and elec ical componen s) and he blades. Fu he decomposi ion o componen s can be made. We highligh he gene a o and he sha s ha connec he o o o he gene a o . Finally, no e he pi ch a ows, indi- ca ing ha he angle a which he u bine blades encoun e he wind can be changed. 1.2.2 Physics o powe gene a ion The amoun o ene gy gene a ed by a wind u bine depends on se e al ac o s [20]. Fi s , we ake in o accoun he kine ic ene gy o he wind: Ek=1 2m 2(1.1) Equa ion 1.1 is he s anda d kine ic ene gy equa ion. Knowing ha ρ(densi y) is exp essed in [kgm−3], eloci y in [ms−1], ime in [s] and Aa ea in [m2] we can subs i u e m[kg] in he p e ious equa ion: m= ( kg m3)(m s)(s)(m2) = ρ A =kg (1.2) E=1 2m 2=1 2(A ρ) 2=1 2A ρ 2=1 2A ρ 3(1.3) Powe is exp essed in [W] (wa s) which is he same as [E −1] (Joules pe second). F om equa ion 1.3: P=E =1 2Aρ 3(1.4) CHAPTER 1. INTRODUCTION 5 F om equa ion 1.4 we conclude ha wind powe is p opo ionally dependen on he a ea swep by he blades o he u bine, densi y o he ai and wind speed (las one is cubed, he e o e much mo e impo an ). Howe e , i mus be no ed ha equa ion 1.4 is a simpli ica ion o ideal condi ions, whe e a u bine can ex ac all he ene gy om he wind. In eal li e, powe gene a ed will be much lowe [20]. None heless, equa ion 1.4 is s ill use ul as i in oduces us o he concep s and a iables in ol ed. In conclusion, in o de o maximize he powe ou pu om a u bine we ha e o ake in o accoun he p e ious a iables. F om equa ion 1.4 he A e m, he a ea, can be changed easonably, building la ge u bines wi h longe blades, ha sweep a bigge a ea. The second e m, p, he ai densi y, can ha dly be changed, as i depends in se e al ac o s such as empe a u e, humidi y o p essu e. The las and mos impo an e m is cubed, we he e o e ha e o ca e ully conside whe e o build he wind u bines (each geog aphical loca ion will ha e an a e age wind speed). The u bine should be buil in a eas wi h high wind speeds. 1.2.3 Blade pi ch angle In he p e ious sec ion we concluded ha wind speed was he de e minan ac o in how much powe was gene a ed. The e is ano he ac o o conside , and ha is he blade pi ch angle. Pi ch con olled u bines allow he con olle o change he angle a which he blade con ac s he wind, ins ead o ha ing a ixed, de e mined angle. A 0◦pi ch is conside ed o ha e he blade is pa allel o he wind, wi h no o a o y mo ion induced. Posi i e angles inc ease his pi ch om he 0◦ e e ence, and nega i e alues indica e he same mo ion bu in he opposi e di ec ion, esul ing in a con a y o a ion compa ed o he posi i e alues. The ange o o a ion o he blades in a wind u bine is usually [0, 90] deg ees ( om pa allel o he wind o pe pendicula o i ). Pi ch con ol can he e o e be used o con ol he powe gene a ed by he u bine and o p o ec i by keeping i s o o speed in a con ollable limi as well as op imizing he ene gy gene a ion [28]. 1.3 Types o wind u bines The e a e h ee ypes o u bines depending on hei physical loca ion: onsho e (placed inland), o sho e (ins alled in a pla o m wi h a solid ounda ion ha is ixed o he sea bed) and loa ing (a loa ing pla o m ha is p e en ed om d i ing by he use o cables and ancho ing de ices). Figu e 1.4 illus a es his: CHAPTER 1. INTRODUCTION 6 Figu e 1.4: Th ee ypes o u bine, wi h he ypes labeled. Adap ed om [50]. This hi d ype o u bine is specially impo an a dep hs beyond 60m, whe e he easibili y o a pla o m ha uns down un il he sea bed is no longe economically iable o easible [19]. These u bines a e no mally placed much a he om he coas han hei o sho e coun e pa s. Howe e , his geog aphical loca ion poses se e al p oblems ha inc ease cos and he di icul y o deploymen . Some o hese p oblems a e: •Ha she wea he compa ed o inland o o sho e coun e pa s (wi h some ema ks in he ad an ages sec ion). •Uns able, ma e ial a igue-p one beha io due o cons an oscilla ion and ib a- ions. •G ea ly inc eased main enance cos s due o he di icul y o eaching he pla o m by he main enance pe sonnel and i s non-s a ic beha io . •Powe lines ha e o be esis an o he unde wa e en i onmen , esul ing in mo e expenses. Wha a e hen he mo i a ions o using o sho e echnology? •Highe wind speeds compa ed o onsho e ins alla ions, leading o inc eased powe gene a ion. CHAPTER 1. INTRODUCTION 7 Figu e 1.5: Global Wind Speed in Janua y and July yea 2001 [30]. F om igu e 1.5 no e ha he ligh shaded a eas a e loca ed in he sea and in some si ua ions close o he coas , bu e y a ely inland. As a gene al ule, a e age wind speed inc eases wi h dis ance om he coas [23]. •La ge a eas wi hou obs uc ions o es ic ions on whe e o build he u bine, allowing la ge p ojec s. This p ope y is known as e ch [23]. •Mo e s able and p edic able wind, wi h educed u bulence and wind-shea [19] [8]. 1.4 Goals and speci ic objec i es 1.4.1 Main objec i e The main objec i e o his wo k is o c ea e a i ual model o a loa ing wind u bine om he expe imen al da a. The model should be able o ep oduce he beha io o he u bine, ou pu ing a iables wi h alues ha a e iden ical o he eal model. Addi ionally, i will ex apola e p edic ions o alues ha a e no p esen in he da a, ha can be la e on e i ied expe imen ally. This i ual model would allow us o, up o a ce ain deg ee, know how a u bine will beha e a di e en wind condi ions o blade pi ch angle con igu a ions, wi hou ac ually needing o measu e, build o be physically nea a eal wind u bine. Mo e p ecisely, he i ual models s udied in his wo k a e dependen on he wind speed and on he pi ch angle o he blades o he u bine. Gi en hese wo a iables, he models ou pu a me ic o he wind u bine such as median accele a ion. CHAPTER 2. STATE OF THE ART 14 Figu e 2.1: A e age a ed ou pu o a wind u bine o selec ed yea s [21]. F om igu e 2.1 we can clea ly see he jump in a ed ou pu om he yea s 1990 o 2000, mul iplying he ou pu by 33 imes. La e on, be ween he yea s 2000 and 2010, he ou pu mul iplied by 1.2 imes and be ween he yea s 2010 and 2016 by 1.42 imes, which is s ill a e y good inc ease. The o al inc ease in powe om 1990 o 2016 is 56.96 imes (2848kW / 50kW). The end in a e age a ed ou pu om he yea 2000 onwa d looks as e han linea bu i will e y likely slow down due o manu ac u ing limi a ions, anspo logis ics (a e manu ac u e) and p ac ical cons ain s. We can also conclude ha mode n u bines a e mo e e icien and la ge . These imp o emen s in powe a e no ob iously ee, as he heigh o he u bine and he adius o i s o o s is also inc easing. None heless, he mos ecen , la ge u bines a e mo e e icien . This g aphic u he e i ies he equa ion 1.4 ( om he in oduc ion), whe e one o he de e mining ac o s in he powe ou pu o a u bine is he a ea and hus by inc easing a ea swep by he blades, we inc ease he powe ou pu ed. CHAPTER 2. STATE OF THE ART 15 2.3 Types o pla o ms o loa ing wind u bines One o he mos in luen ial componen s on he u bine dynamics is he pla o m ha suppo s i . Di e en designs ade o s abili y o cos o ease o manu ac u e. The ollowing subsec ion gi es an o e iew o he main ypes. The e a e a leas ou ypes o pla o ms ha can suppo a u bine [23]: Figu e 2.2: Fou ypes o loa ing wind u bines. Adap ed om [23]. •Ba ge This design maximizes he su ace a ea in con ac wi h he wa e . The pla o m is wide han i s heigh , which inc eases s abili y. This ype esembles he design philosophy o a ship and esul s in a low amoun o d a 1. •Semi-Subme sible This model minimizes he su ace a ea ha ouches he wa e by maximizing olume. Due o manu ac u ing limi a ions, a sphe e ha would be he ideal shape, is no easible. The pla o m is composed hen by a se o cylinde s o simple shapes, placed a ce ain dis ance apa . The sepa a ion be ween hese buoyan elemen s p o ides s abili y. •Spa This concep is designed a ound densi ies. A la ge mass is placed a he deepes pa , while a hollow s uc u e comple es he pla o m up o he op. This densi y dis ibu ion makes he model loa s aigh . This design has he mos d a compa ed wi h he es . Buoyancy is p o ided by he ligh es ma e ials while he dense mass a he bo om gi es s abili y and p e en s o a o y mo ions. •Tensioned Legs Pla o m (TLP) This s uc u e is he mos complex o he ou and he newes de elopmen . The pla o m has excess buoyancy and has o be kep unde he wa e line by a se o cables ha a e connec ed o a se o weigh s a he seabed and he a ms o he pla o m a he o he end. The dis ibu ion and sepa a ion o he ex emes o hese a ms s abilize he pla o m [10]. This design seeks o minimize manu ac u ing cos s. 1The d a o a ship is he e ical dis ance be ween he wa e line and he deepes poin o he hull o he ship ( he keel). CHAPTER 2. STATE OF THE ART 16 2.4 Types o loa ing wind u bines ancho s Ano he componen ha in luences he s abili y o he u bine and i s beha io is he ancho ing sys em. The e a e se e al s udies being made on he ypes o ancho s o he loa ing u bines, as well as hei placemen [29] o a e en a sha ed ancho concep in o de o educe cos [16]. Depending on he seabed, he e a e a leas ou ypes o ancho ing sys ems. The ollowing sec ion o e s an o e iew o he s a e o he a sys ems cu en ly [23]: •D agging ancho s This ype is simila o a boa ancho , suppo ing ension in only one di ec ion. •Suc ion bucke s Only app op ia e o sandy seabeds. The coun e o ce is enabled by suc ioning o ces ha keep he bucke in place. •D illed piles Same p inciple han in ixed ounda ions. La ge me al cilinde s a e o ced in o he seabed o d illed in he case o ha de bo oms suck as ocks. •G a i y ancho s A simple, massi e and hea y mass (such as a conc e e block) is deposi ed on he seabed. This ype o ancho has a e y la ge oo p in bu he lowes complexi y in e ms o ins alla ion p ocedu es. 2.5 Fields o s udy 2.5.1 O e lap be ween loa ing wind u bines and o sho e u - bines As we explained in he in oduc ion, loa ing u bines allow he placemen o gene a o s in places whe e cons uc ing a ounda ion is no longe economically easible. In he p ocess, loa ing wind u bines addi ionally enable s onge winds hanks o he dis ance om he coas . Howe e , hanks o ecen de elopmen s, loa ing wind u bines can be placed in shallow wa e s. This is specially use ul in shallow seabeds ha allow he placemen o a ixed- ounda ion u bine bu p ohibi i because o he ype o seabed [23]. This ad ancemen blu s he line be ween he usual dep hs we e each u bine is no mally placed, bu allows mo e op ions and lexibili y. 2.5.2 O e lap wi h he oil indus y Be o e loa ing wind u bines designs we e adop ed, he oil indus y had al eady de el- oped loa ing pla o ms ha enabled hem o ex ac oil a deepe seabeds. No all he knowledge and designs can be ans e ed. This is due o he ac ha he economical easibili y app oach is e y di e en o a single oil pla o m han o dozens o u bines in a wind a m. None heless, when s udying loa ing wind u bines, p io knowledge om he oil in- dus y can be ex ac ed, specially in ancho ing and pla o m s udies [23] [10]. The oil CHAPTER 2. STATE OF THE ART 17 indus y can he e o e also be conside ed in some aspec s s a e o he a in ela ion o pla o m and ancho ing echnology. 2.6 Cu en loa ing wind u bines s udy ields Se e al s udies ha e been made abou he in e ac ions o he u bine wi h he medium. These a e he main ields o s udy: •Pla o m design As we lea ned in p e ious sec ions, se e al ypes o pla o m a e a ailable and a e being s udied and ha e a signi ican impac on he u bine s abili y and hus, a ec cos and e iciency. I is also impe a i e in hese designs ha he u bine does no lip o all, i.e: i is sa e and s able enough. •Moo ing design (ancho ing also in s udy) This ield sol es p oblems ela ed o ixing he u bine ancho s o he seabed. As we lea ned p e iously, he chosen design is hea ily dependen on he seabed ype. •Tu bine con ol, s a egies planning In o de o achie e maximum powe gen- e a ion and sa e ope a ion, some pa ame e s o he u bine can be weaked, such as he pi ch angle. S udies in his a ea op imize hese pa ame e s. •Ae odynamic modeling This a ea s udies he ae odynamic p ope ies o he u bine, wi h a special ocus on he shape o he blades. An e icien blade design, ha ex ac s he mos powe is key o he wind gene a ion. •Hyd odynamic modeling This a ea s udies he in luence o he hyd odynamic ac o s in he loa ing u bines, mainly he wa e . •Wind and wa e modeling, o ecas ing This ield ocuses on he p edic ion o he wind and wa es. Wi h he in o ma ion and models ex ac ed om he s udies we can ind he ideal geog aphical spo s o he u bines. Chap e 3 Ma e ials and Me hods 3.1 In oduc ion o he expe imen al se up The expe imen al da a was measu ed using a scale model o he wind u bine ha will be desc ibed sho ly. The u bine was subjec ed o di e en wind speeds a a ious blade pi ch angles. The wind speed was con olled by egula ing he ol age in a wind gene a o and he blade pi ch was changed by adjus ing i s own mechanism. The dis ance om he wind gene a o o he u bine was no changed du ing expe imen s in o de o ha e consis en measu emen s. The expe imen al da a was s o ed wi h he wind speed in ela ion o ol age. As ol age alone is no e en a measu e o powe (wa s could be much mo e in o ma i e o example) a ans o ma ion o a mo e use ul me ic is needed. Thank ully, he da a included an equi alence. The ollowing able shows he equi alence be ween ol ages and wind speeds 1: Table 3.1: Equi alence be ween ol ages and wind speeds. Vol age [V] Wind speed [ms-1] 130 8,5 140 9,3 150 10,1 160 11 170 11,6 180 12,4 190 13,1 200 13,8 The alues om able 3.1 will be used la e on o con e he ol ages o he gene a o in o wind speed a he u bine. 1In e es ingly enough, he ela ion be ween ol age and wind speed is almos linea : wind = 0.07571× ol age −1.34285. In any case, i is mo e p ecise o use wind speed. 19 CHAPTER 3. MATERIALS AND METHODS 20 3.2 Expe imen se up Ins ead o ha ing he u bine loa ing on a pla o m, a se o sp ings was placed a ound he squa e pe ime e o he pla o m o suppo i and imi a e a u bine loa ing on he wa e . This con igu a ion should p o ide a simila beha io o a eal loa ing pla o m. The u bine was placed inside a wind unnel ha had he se o winds om able 3.1 calib a ed. The ollowing igu e illus a es he wind u bine se up ha was inside he wind unnel: Figu e 3.1: F on iew o he wind u bine wi h labeled axes. All axes a e pe pendicula . The Z axis is pe pendicula o he g ound plane, he Y axis is pa allel o he axis o he u bine, poin ing owa ds he back, and he X axis is poin ing igh , pe pendicula o he Y and Z axes. No e some o he sp ings isible a he igh o he pho og aph. On igu e 3.1, a se o sp ings is isible a he igh o he pho og aph, wi h blue ci cles along he con ac poin s on he pla o m. On he le , nex o he u bine, he A duino mic o-con olle (a blue PCB2wi h a black squa e chip on i and a USB ype B po ) can be seen. The A duino is esponsible o egis e ing he measu emen s om he IMU 3and sending hem o a compu e . Fu he o he le and o he back, wi h a g een cable connec ing o i , almos a he edge o he pla o m, a second blue PCB can pa ially be seen. This blue PCB is he IMU. The IMU is esponsible o measu ing accele a ion, angula eloci y and magne ic lux densi y. No e ha he choice o all he axes ep esen a ions and con igu a ions is he same h oughou all he wo k. This con igu a ion is also he exac same han he one gi en by 2P in ed Ci cui Boa d. 3Ine ial Measu emen Uni , a ype o senso . Explained la e mo e in dep h. CHAPTER 3. MATERIALS AND METHODS 21 he senso documen a ion and by he expe imen se up. I.e.: The axes a e labeled in such a way ha hey a e in he same con igu a ion ha as senso uses o i s measu emen s. A simpli ied diag am o he u bine con igu a ion is shown in igu es 3.2 and 3.3: Figu e 3.2: F on iew o he u bine wi h labeled axes, simpli ied. No e he sp ings on he bo om o he pla o m. Figu e 3.3: Top iew o he u bine wi h labeled axes, simpli ied. CHAPTER 3. MATERIALS AND METHODS 22 The IMU is he senso esponsible o he measu emen s. I will la e be de ailed. Fo now, jus know ha i measu es me ics in ela ions o a di ec ion, and he e o e we need axes. Two diag ams o IMU axes and i s pola i ies a e also shown, o illus a ion pu poses and axis labeling: Figu e 3.4: IMU wi h i s axes. Figu e 3.5: Pola i y o o a ion and o ien a ion o he IMU axes. Adap ed om [25]. CHAPTER 3. MATERIALS AND METHODS 23 Again, he u bine axes, diag am axes and IMU axes a e all he same h oughou he wo k. This on pu pose o a oid con usion. Wind u bine componen s •Mic ocon olle : A duino Mega. •Ine ial Measu emen Uni : MPU6050. •Elec ic mo o : B ushless mo o 22M-1000 GPMG4500. •Wind u bine blade dimensions: 10×2 cen ime e s. •Pi ch angle ange o he u bine blades: -30 o 30 deg ees. •Pla o m dimensions: 16×24 cen ime e s. 3.3 Ma e ials: Da a om expe imen s A o al o 48 simula ions we e ca ied ou , each o which las ing 10 seconds. The 48 expe imen s co espond o he ollowing combina ion o ol ages: 130, 140, 150, 160, 170, 180, 190 and 200 [V] and he ollowing blade pi ch angles: -30, -20, -10, 10, 20 and 30 [deg ees]. Each expe imen las ed 10 seconds. Ins an aneous measu emen s we e aken a a equency o 10Hz o e e y 100ms, o a o al o 99 measu emen s. Each o hese mea- su emen s consis ed o a measu emen o he accele a ion [g]4, angula eloci y [◦s−1] and magne ic lux densi y [µT]5 o each axis (x, y, z). In o al, 8 ol ages (o wind speeds) we e used, 6 pi ch angles and 99 measu emen s we e made, o aling 99 ∗8∗6 = 4752 ows o da a. This, in combina ion wi h he h ee measu emen s made (accele a ion, angula eloci y and magne ic lux densi y), o e e y axis, equa es o 4752 ∗3∗3 = 42768 samples. The ollowing ables illus a e how he da a looks like be o e any p ep ocessing, a e i is loaded. Acc x is he accele a ion o he x axis, Gy o y is he angula eloci y o he y axis and Mag z is he magne ic lux densi y o he z axis, inally Time (0.1s) is he imes amp a which he ow was s o ed (in en hs o a second). Two ables a e shown, bo h ha e he same pi ch angle bu hey ha e di e en wind speeds. Table 3.2: Sample o da a a 130V (8.5ms-1) and 1◦. Time [0.1s] Acc x [g] Acc y [g] Acc z [g] Gy o x [◦s−1] Gy o y [◦s−1] Gy o z [◦s−1] Mag x [µT] Mag y [µT] Mag z [µT ] 1 -0,001062012 0,000982666 0,978607178 -0,002361081 -0,010571808 -0,010266725 -26,67796875 -6,1425 -9,7734375 2 -0,010552979 -0,00043335 1,0246521 0,041690331 0,010730982 -0,008038288 -26,16773438 -5,338125 -9,7734375 . . .. . .. . .. . .. . .. . .. . .. . .. . .. . . 99 -0,00824585 -0,018615723 1,048773193 0,004443608 -0,017681582 -0,026210657 -26,31351563 -5,630625 -9,7734375 41g= 9.8ms−2 5The da a shee om [25] did no speci y uni s o he magne ome e . Howe e , as he Ea h’s magne ic ield anges om 25 o 65 µT and he da a om he expe imen was in his ange, we will conside i o be in µT. CHAPTER 3. MATERIALS AND METHODS 30 Figu e 3.6: Neu al ne wo k example. Illus a ion made hanks o NN-SVG [3]. No e he 2 inpu nodes, he wo hidden laye s wi h 4 and 8 nodes and he ou pu laye wi h h ee nodes. The numbe o nodes in each laye is ex emely impo an as i de e mines he di- mensions o he ma ices ha will be used inside he neu al ne wo k in o de o make he calcula ions. I was chosen because i is e y di e en om he es o he models and could ha e be e esul s. 3.4.7 E alua ion o model pe o mance No mally, 80% o he da a se is used o aining he models ( ain se ). The 20% o he da a se emaining is he es se and i is piped in o he sklea n.pipeline.Pipeline.sco e unc ion [44] in o de o ind ou he pe o mance o he model. This unc ion e u ns a loa wi h he sco e o he model. No e ha depending on he model ( o example R2 sco e) i can be nega i e, i.e: sco e ∈(−∞,1]. I is no possible o say wha a good sco e is in gene al. Fo example: A neu al ne wo k is ge ing a sco e o 95% when ying o p edic i a ain is la e. This sco e looks g ea . Howe e , he neu al ne wo k is always p edic ing ”no” and he e o e guessing co ec mos o he ime jus by saying always ”no”, because he ains a e a ely la e (i is also p obably igno ing he inpu da a). The e a e se e al echniques and di e en sco ing me ics in o de o a oid hese pi alls. Fo his wo k, howe e we will s a conside ing sco es a and abo e 50 (≥50%) and conside sco es excellen when o e o a 90 (≥90%). In o de o a oid he p e iously men ioned pi all ( he one wi h he ain delay es i- ma ion) and he di icul y when picking a good sco e, we will also use dis ances and e o . Fo example, i he p edic ed alue is 10 and he eal alue is 10.8 we could calcula e he e o using he ollowing o mula: CHAPTER 3. MATERIALS AND METHODS 31 E=| −p| |p|×100 = |10.8−10| |10|×100 = 8% (3.8) Whe e is he eal alue and pis he p edic ion. Again, we canno ix an o e all accep able o good e o a his poin in he wo k. Howe e , we will conside a ≈5% e o o be accep able and ≤1% o be excellen . Models wi h e o s ≫5% will be conside ed useless. 3.4.8 O e i ing in Machine Lea ning The in oduc ion o non-linea , mo e complex equa ions has i s disad an ages. The main one is ha we can make a model ”memo ize” he da a. Figu e 3.7: O e i ed da a [18]. In igu e 3.7 we can see wo lines: in black, a s aigh line ha could ep esen a adi ional linea eg ession and in blue a cu ed line ha ep esen s a polynomial o 10 deg ees ha has been made o i pe ec ly he da a. The i s disad an age is he complexi y in oduced by a 10 deg ee polynomial, leading o inc eased un ime, due o ha ing o calcula e a mo e pa ame e s compa ed o he linea model. Howe e , he mos wo ying p oblem is he ollowing: The blue line has an almos pe ec i , a mo e p ecise han he black line. Howe e , his model will all apa once new poin s a e in oduced, i.e.: i will ail o gene alize, i will p obably be much mo e imp ecise han he linea coun e pa once enough new poin s a e in oduced. The e a e se e al ways o managing his p oblem, one being examining he models manually o penalizing each ime he model inc eases complexi y. Ne e heless, in his wo k he main way used was di iding he da a in o wo se s. We will use app oxima ely 80% o he da a o ain ou models, and he 20% le will be used o sco e how good o how bad he models a e. The da a o his se s will also be shu led in o de o a oid epea edly c ea ing a biased subse . 3.4.9 Hype pa ame e uning Hype pa ame e uning is he selec ion o he op imal pa ame e s o he models. One example could be choosing he bes deg ee o he polynomial in a polynomial eg esso CHAPTER 3. MATERIALS AND METHODS 32 model. In o de o pick he bes pa ame e s, an asso men o pa ame e combina ions was manually made and piped in o sk-lea n’s G idSea chCV [42] [34]. The G idSea chCV model selec o has a me hod ha e u ns he bes es ima o . A e some manual checks ( o make su e ha he pa ame e s make sense), one o he bes es ima o s was chosen o each model. 3.4.10 Pipelining All o he a o emen ioned subsec ions de ail a complex wo k low. Fo una ely, sk-lea n o e s us a Pipe ha can be se up o a g oup o models o only one model [35]. The conc e e pipeline usage will is de ailed in he Supe ised Models sec ion, bu a gene al pipe example is o e ed (adap ed om [35]): 1. Scale S anda dScale () 2. Classi ie o classi ie s PCA(), Logis icReg ession() A e he pipe se up, only a call o i and sco e o he necessa y me hods is needed ins ead o se e al indi idual calls and pa ame e passings o each componen . 3.4.11 Pe iodici y S udy Ano he a ibu e s udied in his wo k is he pe iodici y o he u bine. The objec i e o his pa is o ind ou i he u bine is ib a ing in some p edic able way. And i i is ib a ing, how he equency migh change o di e en wind speeds o pi ch angles. The p oblem is he e o e o ex ac equencies om a da a se ha consis s o mea- su emen s made a a ce ain equency. This is enabled by Fou ie analysis. Fou ie analysis Fou ie analysis s udies he decomposi ion o a unc ion in o he sum o se e al, simple igonome ic unc ions. This p ocess i sel is called a Fou ie T ans o m. One example o an applica ion o a Fou ie T ans o m is o de ec and emo e high equencies om a eco ding ha may be i ele an o dis ac ing. The Fou ie T ans o m will be able o, om he signal wi h all he mixed sounds, de ec he peak in he high equency, allowing o i o be emo ed. In o de o use Fou ie Analysis, he da a mus be equally spaced. Some app oaches exis o unequally spaced da a bu i is ou side o he scope o his wo k. Fas Fou ie T ans o m A Fas Fou ie T ans o m is an algo i hm ha compu es he Disc e e Fou ie T ans o m o a sequence [1]. The ”Fas ” keywo d in FFT is a a ian o he algo i hm ha educes he un ime complexi y om O(N2) in a Disc e e Fou ie T ans o m (DFT) o O(Nlog N) [1]. CHAPTER 3. MATERIALS AND METHODS 33 Fas Fou ie T ans o m ou pu ep esen a ion The ou pu om he FFT is ul ima ely a ep esen a ion in he equency domain. Figu e 3.8 shows an example: Figu e 3.8: FFT o a Cosine Summa ion Func ion esona ing a 10, 20, 30, 40, and 50 Hz. [2]. No e om he bo om g aph o igu e 3.8 he peaks a x= 10, x= 20, x= 30, x= 40 and x= 50; he FFT has success ully de ec ed he indi idual equencies ha whe e mixed in he op g aph. An FFT analysis will be pe o med on all he me ics in he da a la e on, in o de o ind ou i he u bine is ib a ing in a p edic able manne . 3.4.12 Da a o ganiza ion The da a was dis ibu ed in se e al excel iles and inside hem, in di e en shee s. The i s s ep in he p og am was o load he da a using Pandas’s ead excel unc ion. This unc ion e u ns a Da aF ame which is simila o an SQL able. In o de o o ganize he 48 ables (8 wind speeds and 6 angles), each Da aF ame was placed inside a Py hon dic iona y ha con ained ano he dic iona y. The s uc u e o he nes ed dic iona ies was he ollowing: da a = (key =angle : alue =dic 1) (3.9) dic 1 = (key =wind speed : alue =Da aF ame) (3.10) CHAPTER 3. MATERIALS AND METHODS 34 Da aF ame om equa ion 3.10 would con ain he da a om he u bine a an angle o angle and a wind speed o wind speed. This ype o da a s uc u e allowed O(1) access ime (because i is a Py hon dic io- na y) o he Da aF ames while i also p ese ed an use - iendly access me hod. I.e: I is e y easy o eques he da a om he u bine o a gi en angle and wind speed. Fu he down he wo k, he supe ised models equi ed ma ices o da a wi hou a b eakdown by angle and wind speed. As he da a s uc u e desc ibed in he p e ious pa ag aphs did no p o e o be ap- p op ia e, he new da a s uc u e ha was adop ed combined all he p ep ocessed da a in a single Da aF ame. The Da aF ame will be de ailed in he ollowing chap e s. In any case o he pu poses o his sec ion he ollowing able is shown de ailing he columns: Table 3.6: G ouped da a se columns. angle windspeed median Acc x mean Acc x a Acc x p p Acc x amax Acc x amin Acc x s d Acc x . . . min abs len Magn ... ... ... ... ... ... ... ... ... ... ... The da a om able 3.6 can be easily accessed (again in O(1) access ime): columns can be selec ed in o de o use hem as inpu o ou pu s o he supe ised models and his columns can also be subdi ided in o de o di ide he da a se in o es and ain se s. We will also highligh ha able 3.6 has 48 ows ( ecall om he p e ious pa ag aphs 8 wind speeds and 6 angles) and 122 columns ( he column coun will be explained la e on). Chap e 4 S a is ical Analysis 4.1 Da a isualiza ion The i s s ep made in he wo k a e loading he da a was o isualize he da a. All he plo s we e gene a ed in Py hon using Ma plo lib. In any s a is ical s udy i is always ecommended o isualize he da a as e y di e en da a se s can ha e he same s a is ical me ics. The ollowing igu es show a g aph o wo 10 second expe imen s a he same angle o 10◦. One g aph is o a wind speed o 8.5ms-1 and he nex o 13.8ms-1: Figu e 4.1: 2D plo s o he da a co esponding o a wind speed o 8.5ms-1 and a pi ch angle o 10◦. 35 CHAPTER 4. STATISTICAL ANALYSIS 36 Figu e 4.2: 2D plo s o he da a co esponding o a wind speed o 13.8ms-1 and a pi ch angle o 10◦. F om igu e 4.1 we can see some ex eme alues ha migh sugges he p esence o ou lie s, specially in he magne ome e (”Magn”) column. The mos ob ious one being in he ”Magn y” subplo . In igu e 4.2 we poin ou again ex eme alues, he mos ema kable one in he ”Magn z” subplo . No much mo e in o ma ion om hese g aphs can be ex ac ed isually. We can howe e see ha in g aph 4.2 compa ed o he g aph 4.1 he ange o alues o e e y subplo excep some (like in Magn x o Magn z) has inc eased g ea ly. This sugges s ha wind speed is a ec ing he a iables and is co ela ed wi h hem. La e on in he wo k, i was concluded ha he ou lie emo al was no necessa y, and ha in ac i migh emo e use ul in o ma ion and he e o e i may be ha m ul. This conclusion was eached a e he supe ised models pe o med e y good wi hou he possible ou lie s emo al, and a e he no mali y analysis o he da a, discussed la e on he wo k. F om hese g aphs no much in o ma ion can be ex ac ed. We can howe e conclude ha he e a e no missing o e oneous alues in he da a se . The exis ence o absence o ou lie s is no clea bu i is unimpo an . CHAPTER 4. STATISTICAL ANALYSIS 37 4.2 Dis ibu ion s udy 4.2.1 His og ams In o de o be able o use pa ame ic me hods we i s ha e o obse e he dis ibu ion o he da a. Pa ame ic s a is ics assume ha he da a is modeled by a p obabili y dis ibu ion. I he da a is no mally dis ibu ed, some models will wo k much be e han o he s. Fi s , a se o his og ams was ob ained. The he e a e ou his og ams, allowing he compa ison o wo di e en angles, and o he same angle, wo wind speeds: Figu e 4.3: His og am o he da a co esponding o a wind speed o 8.5ms-1 and a pi ch angle o 10◦. CHAPTER 4. STATISTICAL ANALYSIS 38 Figu e 4.4: His og am o he da a co esponding o a wind speed o 8.5ms-1 and a pi ch angle o 10◦. Figu e 4.5: His og am o he da a co esponding o a wind speed o 13.8ms-1 and a pi ch angle o 30◦. CHAPTER 4. STATISTICAL ANALYSIS 39 Figu e 4.6: His og am o he da a co esponding o a wind speed o 13.8ms-1 and a pi ch angle o 30◦. F om igu es 4.3, 4.4, 4.5 and 4.6 i can be seen ha mos o he g aphs a e compa able o a no mal dis ibu ion. Examples o a no mal dis ibu ion shape a e: om igu e 4.3 (Acc x, Acc y), igu e 4.4 (Gy o x, Gy o y), igu e 4.5 (Gy o x, Acc y) and om igu e 4.6 (Gy o x, Gy o y). Examples o a dubious (o no ) no mal dis ibu ion shape a e: om igu e 4.3 (Magn y, Magn z), igu e 4.4 (Acc x, Acc y), igu e 4.5 (Gy o z, Magn y, Magn z) and om igu e 4.6 (Acc y, Acc z). 4.2.2 Examples o no mal dis ibu ions Figu e 4.7 shows se e al no mal dis ibu ions wi h di e en means and a iances (and he e o e shapes): CHAPTER 4. STATISTICAL ANALYSIS 46 Figu e 4.13: Median alue o he modulus o he magne ome e ec o in he z axis. I is clea ha igu es 4.12 and 4.13 can be app oxima ed again by a plane. One impo an di e ence is ha his ime he co ela ion is in e ed (in e se co ela ion); he highe he wind speed, he lowe he me ic measu ed. Finally, o he g aphs did no look p edic able ( hey seem andom): CHAPTER 4. STATISTICAL ANALYSIS 47 Figu e 4.14: Median absolu e alue o he measu emen s o he magne ome e in he y axis. Figu e 4.15: Median angula eloci y measu emen s in he z axis. CHAPTER 4. STATISTICAL ANALYSIS 48 Wi h hese g aphs we can ge a g aphical ep esen a ion o how he me ics change wi h di e en wind speeds and angles. We can also see how as hey seem o scale, which in mos o he examples looked linea (and by ex ension can be app oxima ed by a plane). In con as , o he g aphs look andom and he me ics associa ed wi h hem will be hus e y ha d o p edic . 4.6 Co ela ion ma ix Recalling he g ouped da a able: Table 4.5: G ouped da a se columns. angle windspeed median Acc x mean Acc x a Acc x p p Acc x amax Acc x amin Acc x s d Acc x . . . min abs len Magn ... ... ... ... ... ... ... ... ... ... ... Using he da a om able 4.5 we c ea e a co ela ion ma ix (disca ding angle and wind speed). Ma ix om igu e 4.16 is e y dense because o he 120 columns om he da a se . None heless, we will y o commen he ma ix: CHAPTER 4. STATISTICAL ANALYSIS 49 Figu e 4.16: A e age modulus o he magne ome e ec o . CHAPTER 4. STATISTICAL ANALYSIS 50 Figu e 4.17: A e age modulus o he magne ome e ec o . CHAPTER 4. STATISTICAL ANALYSIS 51 F om igu e 4.17 we highligh wo a eas: 1. Red: We can see se e al in e se co ela ions: s d Gy o y - median abs Magn z, median abs Gy o y - median abs Magn z, e c. This is expec ed as he changes in angle will also a ec he magne ome e componen s. 2. Pu ple: We can see some di ec co ela ions: median abs len Acc - median Acc z, max abs len Acc - amax Acc z, e c. This is expec ed, as he modulus o he ec o and he median should be co ela ed, specially wi h he z axis ha is he one wi h he la ge alues o he h ee. O e all, he ma ix is no e y in o ma i e as he e is a lo o edundancy on i . Fo example: maximum and minimum will be co ela ed wi h he ange, as i is a linea combina ion o he wo. Ano he example: All he me ics o he accele a ion in an axis will be co ela ed wi h he o he me ics o he accele a ion in he same axis. The e we e howe e , mo e in e es ing pa s o he igu e, as he ones desc ibed in he ed a ea, bu o e all, no e y in o ma i e. Chap e 5 Pe iodici y analysis A pe iodici y analysis was pe o med. The aim o his analysis was o y o ind i he u bine ib a es a a ce ain equency. This analysis canno be easily done i we we e o emo e he possible ou lie s, because hese alues a e e ased om he da a se , esul ing in gaps in he measu emen s. The goals om his sec ion a e: ind ou i any o he me ics o he u bine p esen a pe iodic beha io , i hey do, ob ain he equency and igu e ou how he equency changes wi h di e en wind speeds and angles. Addi ionally, in e p e he indings in o de o know i hey make sense. In o de o ex ac he equency om he da a, a Fas Fou ie T ans o m (FFT) was used. The Py hon lib a y used was Scipy 1.8.0 wi h he scipy. pack module and he and eq unc ions. 5.1 Tes se up The ollowing lis desc ibes he con igu a ion o he pe iodici y analysis: •The expe imen las s 10 seconds. •The numbe o samples is 99 o each me ic. •Each measu emen (sample) is spaced by 100ms gaps ( he sampling a e is 10Hz). •Be o e he analysis he FFT lib a y will be es ed wi h a es unc ion o con i m co ec usage. •The analysis will be made o each column o he da a se o a gi en angle and wind speed. Fo example: a e age Acc z o a wind speed o 8.5 and an angle o 1◦. •The mi o ed ou pu om he algo i hm, on he nega i e x ange, is igno ed and no shown. 5.2 Tes ing he lib a y Fi s , a es on he unc ion was made o make su e ha he lib a ies we e wo king and ha hey we e being used co ec ly: 53 CHAPTER 5. PERIODICITY ANALYSIS 54 Figu e 5.1: FFT Tes wi h a sin unc ion a 2Hz, wi h a clea peak a x= 2. The g aph om igu e 5.1 clea ly ep esen s a peak a 2Hz. The FFT has success ully de ec ed he only equency a 2Hz. The same wo k low will be used in he eal analysis o make su e ha he da a is co ec ly passed o he unc ion. 5.3 Plo ing he da a se Fas Fou ie T ans o ms 5.3.1 DC Componen Mos o he g aphs showed a peak a x= 0, which is a p obable side-e ec o a DC componen . I.e.: The FFT is de ec ing he median o he da a and plo ing i a x= 0. I we ake a look a able 5.1, he mean o he len Magn column ( he mean o he modulus o he Magn ec o ) is 28.56. I we mul iply his alue by 2 we ob ain 57.12 which is oughly equal o he alue a x= 0 on igu e 5.2: Table 5.1: Da a s a is ics a a wind speed o 8.5ms−1and 30◦pi ch angle. Me ic Time (0.1s) Acc x Acc y Acc z Gy o x Gy o y Gy o z Magn x Magn y Magn z len Acc len Gy o len Magn coun 99.000000 99.000000 99.000000 99.000000 99.000000 99.000000 99.000000 99.000000 99.000000 99.000000 99.000000 99.000000 99.000000 mean 50.000000 0.001525 -0.021226 1.033122 -0.028354 0.001044 -0.000453 -26.313516 -5.573011 -9.607244 1.033699 0.089341 28.564459 s d 28.722813 0.011927 0.024573 0.060953 0.081943 0.025032 0.041457 0.289135 0.286139 0.298388 0.060954 0.042546 0.288391 min 1.000000 -0.028040 -0.079266 0.801373 -0.187839 -0.061255 -0.100001 -26.969531 -6.288750 -10.476562 0.801555 0.009376 27.976938 25% 25.500000 -0.005765 -0.040277 1.014185 -0.094118 -0.016368 -0.029460 -26.532187 -5.776875 -9.843750 1.014505 0.058281 28.377556 50% 50.000000 0.002338 -0.020795 1.031024 -0.031516 0.003900 -0.001884 -26.313516 -5.557500 -9.562500 1.031553 0.084312 28.540237 75% 74.500000 0.009515 -0.004721 1.046768 0.034030 0.017615 0.027431 -26.167734 -5.411250 -9.421875 1.047564 0.118908 28.766974 max 99.000000 0.032532 0.034198 1.258600 0.151109 0.050737 0.093170 -25.584609 -4.826250 -8.648438 1.259381 0.191606 29.395060 CHAPTER 5. PERIODICITY ANALYSIS 55 Figu e 5.2: FFT wi h wind speed o 8.5 and an angle o 30◦, accele a ion o y. This obse a ion: y0≈2×median(column), holds o mos i no all he g aphs. Please no e ha in igu e 5.2 he e is in o ma ion on he y axis in he = (0,5] ange. Howe e , i is shadowed by he alue a x= 0 (y0), because i is much, much, la ge han he alues o he ange (ya, a ∈ ). Ma plo lib adap s he axes o he ange o he da a. The alue y0≈57 (a x= 0, x /∈ ) is he maximum alue in he da a and in he y axis ange, while he es o he da a is a maximum y≈0.15. This will be mo e clea in g aph 5.3. In o de o a oid he dis o ion on he y axis, we can se he y alue a x= 0 o 0. We will no des oy any in o ma ion as we al eady know ha i is app oxima ely equal o wo imes he median o he da a: CHAPTER 5. PERIODICITY ANALYSIS 62 Figu e 5.11: FFT wi h wind speed o 13.8 and an angle o 30◦, accele a ion o z. DC componen emo ed. F om igu es 5.9, 5.10 and 5.11 he e is no isible pa e n no displacemen o he peaks om he di e en wind speeds. I is possible ha he wind u bine is ib a ing a a equency along he accele a ion in he z axis ha is much highe han he sample a e and he e o e canno be de ec ed. This is also suppo ed by he magni ude o he DC componen compa ed o he peaks in he FFT (≈0.05 ampli ude e sus ≈2.1). As he accele a ion is he de i a i e o he eloci y, we expec he changes o his me ic o be much as e han on he eloci y. We will now analyze he angula eloci y, hope ully being able o de ec a pa e n and clea e equencies hanks o expec ed slowe changes. Angula eloci y on he X axis (changes in pi ch) The g aphs om he angula eloci y on he x axis (gy o x) we e much mo e clea e . In ac , i was no necessa y o emo e he DC componen , indica ing ha he peaks a e p opo ional o i and ha he e is a high p obabili y ha he equencies ha e been de ec ed: CHAPTER 5. PERIODICITY ANALYSIS 63 Figu e 5.12: FFT wi h wind speed o 8.5 and an angle o 30◦, angula eloci y on x. DC componen no emo ed. Figu e 5.13: FFT wi h wind speed o 10.1 and an angle o 30◦, angula eloci y on x. DC componen no emo ed. CHAPTER 5. PERIODICITY ANALYSIS 64 Figu e 5.14: FFT wi h wind speed o 11.6 and an angle o 30◦, angula eloci y on x. DC componen no emo ed. Figu e 5.15: FFT wi h wind speed o 13.8 and an angle o 30◦, angula eloci y on x. DC componen no emo ed. F om he p og ession o igu es 5.12, 5.13, 5.14 and 5.15 wo hings can be obse ed: The peak p esen a x≈4.35Hz in 5.12 clea ly mo es igh and ends up a x≈4.8Hz in CHAPTER 5. PERIODICITY ANALYSIS 65 5.15, his sugges s an inc ease in equency wi h an inc ease in wind speed. Addi ionally, he peak also inc eases in heigh , om J( he a)≈0.052 o J( he a)≈0.077 in he las igu e. This obse a ion s ongly sugges s ha he wind u bine is ib a ing as e a highe wind speeds, in he pi ch axis as i was expec ed. Res o he me ics and axes The es o he FFTs did no show clea pa e ns no dis inc equency de ec ions. The magne ome e me ic was no conside ed use ul o his analysis, as i s main use would be o ge he o ien a ion o he u bine in ela ion wi h he ea h’s magne ic ield. 5.3.3 Pe iodici y analysis esul s In his chap e we success ully con i med he expec ed beha io o he wind u bine. Addi ionally, we managed o de ec a di ec co ela ion be ween he wind speed and he equency a which he u bine was ib a ing. This inding was e y clea on he angula eloci y on he x axis, bu no e y conclusi e on he displacemen s on he z axis. One las obse a ion o he magne ome e is made: Table 5.2: Da a s a is ics a a wind speed o 8.5ms−1and 30◦pi ch angle. Me ic Magn x [µT] Magn y [µT] Magn z [µT] min -26.969531 -6.288750 -10.476562 max -25.584609 -4.826250 -8.648438 ange 1.384922 1.4625 1.828124 F om able 5.2 we can obse e ha he la ges ange is egis e ed a he magne ome e z axis, ollowed by y and hen x. This g ea e change in he z and y componen s o he magne ome e ec o u he con i ms ha he main o a ion is along he x axis ( a ia ion in pi ch) and no he o he axes (i he wind u bine o a es a ound an axis A, he magne ome e alues measu ed a ha axis will be a ec ed he leas , because he ec o o he magne ic ield will emain a he same angle wi h he A axis, while he o he axes will no main ain he angle). Chap e 6 Supe ised Models 6.1 In oduc ion The main goal o his sec ion is o p edic , gi en a blade pi ch angle αand a wind speed υ, a s a is ical me ic. This would allow us o ha e a gene al idea o he beha io o he wind u bine a di e en wind speeds and angles wi hou ac ually ha ing o pe o m he expe imen . Fo example, knowing he maximum accele a ion ha he wind u bine will be exposed o a a ce ain wind speed will allow us o plan how s ong should he s uc u e ha suppo s i be, in o de o i o no b eak and o no was e unnecessa y ma e ial o a oid ha de o cos lie suppo s uc u es. This is a eg ession p oblem: Gi en a se o eal alues, p edic a new eal alue based on p e ious examples. Mo e conc e ely his is a supe ised machine lea ning p oblem. Se e al supe ised linea lea ning models om sciki lea n we e used in his sec ion as hey a e al eady p epa ed o use he numpy da a o ma s. 6.2 Da a p epa a ion Table 6.1: Da a se columns. angle windspeed median Acc x mean Acc x a Acc x . . . median Magn z mean Magn z . . . a e age abs len Magn 1 8.5 0.00187 0.00153 0.00035 . . . -9.56250 -9.60014 . . . 9.60014 ... ... ... ... ... ... ... ... ... ... Recalling he columns om ea lie in able 6.1 he inpu ec o Xwill be he columns angle and wind speed and he a ge ec o Ywill be any o he o he columns. In o de o ha e be e esul s, lowe complexi y and mo e in e p e able models, he decision was made o ha e one model pe p edic ed me ic, ins ead o a highly complex model ha p edic s all he ou pu columns a he same ime. Each model is ained o p edic one ou pu me ic. In he case o he Neu al Ne wo k, his esul s in wo inpu nodes and one ou pu node o each neu al ne wo k. The hidden laye s can also be adjus ed in sciki lea n, 67 CHAPTER 6. SUPERVISED MODELS 68 along wi h o he pa ame e s such as he algo i hm used, α(L2 penal y o egula iza ion e m), e c. Fo each o he lea ne s he ollowing s eps we e aken: 1. C ea e he Xand Yinpu and ou pu ec o s. 2. Shu le ( andomize) he o de o he ows o he da a se in o de o a oid epea ed biases. 3. Spli he da a se by ows in o wo se s. The aining se and he es se , in 80% and 20% p opo ion, espec i ely. 4. Scale he da a. 5. T ain he model using he aining se . 6. E alua e he model using he es se . 6.3 Models used A o al o 6 models we e used in his sec ion: Linea eg ession [39], Linea wi h poly- nomial ea u es [39][45], Ridge [40], Hube [38], Gaussian [37] and a MLP Reg esso [43] [36]. All o he models used a S anda dScale [46] be o e i ing he da a. The pipelines o all he models we e: Scale →Es ima o . Excep he polynomial model ha was: Scale →PolynomialFea u es →Es ima o [45]. Wi h es ima o being he model (linea , gaussian, e c). 6.4 Scale selec ion The S anda dScale was chosen because i was concluded in S a is ical Analysis chap e , Dis ibu ion s udy sec ion, ha he da a was no mally dis ibu ed and ha pa ame ic s a is ical me hods could be used. Sk-lea n men ions he ollowing abou he S anda dScale :”[...] migh beha e badly i he indi idual ea u es do no look like s anda d no mally dis ibu ed da a [...] [46]. We he e o e conclude ha he S anda dScale is he app op ia e scale in his wo k, because o i s compa ibili y wi h no mally dis ibu ed da a. 6.5 Hype pa ame e s uning The hype pa ame e uning was pe o med semi-au oma ically using sk-lea n’s G id- Sea chCV [42]. CHAPTER 6. SUPERVISED MODELS 69 The de aul pa ame e s used in G idSea chCV we e he ollowing: sco ing =None, n jobs =None, e i =T ue,c =None (5- old c oss alida ion) and e o sco e = np.nan. Fo example, in he case o polynomial eg ession, G idSea chCV allows us o speci y an a ay o deg ees o es wi h. A e he es ing is done, G idSea chCV has a me hod ha e u ns he bes es ima o . Addi ionally, G idSea chCV does c oss- alida ion [34]. By semi-au oma ically we mean ha a numbe o pa ame e s we e ed in o he G id- Sea chCV model selec o . These pa ame e s we e selec ed acco ding o he size o he da a and he complexi y o i , which was hough o no be ex emely high in his case 1. Fo example, in he case o he deg ee o he polynomial model, he alues ed we e in he ange [2,8], and he bes esul s we e ob ained wi h alues unde 4. Ano he example is he neu al ne wo k, he MLPReg esso . I did no make any sense o use a model wi h a lo o hidden laye s and nodes, because he inpu s we e wo and he ou pu is only one. 6.6 Models pa ame e s Some models did no allow o did no ha e pa ame e s o une. O he models, howe e , equi ed adjus men in o de o ha e accep able pe o mance. I no hing is speci ied, he de aul alues om sk-lea n we e used. Fo de ailed docu- men a ion on he de aul s, check he o icial documen a ion o he models in he p e ious e e ences (we e we lis he models used). The ollowing lis de ails he chosen model pa ame e s: •Linea No changes, de aul s used: i in e cep =T ue,no malize =False and posi i e =False. •Polynomial Deg ees 2 o 4 we e ound o be he bes pe o me s. The de aul pa ame e s we e: deg ee = 2, in e ac ion only =False,include bias =T ue and o de =C. Deg ee 2 was chosen in he end o i s good pe o mance and lowe chance o o e i ing. Same pa ame e s o he model as in Linea . •Ridge No changes, de aul s used. Regula iza ion s eng h α= 1.0, i in e cep = T ue,no malize =False,max i e =None, ol = 10−3(p ecision o he solu ion) and sol e =au o. •Hube No pa ame e s speci ied, de aul s used. Numbe o samples ha should be classi ied as ou lie s ε= 1.35, max i e = 100, egula iza ion pa ame e α= 0.0001, wa m s a =False, i in e cep =T ue and ol = 10−5. •Gaussian No uning, de aul s used. ke nel =None (Cons an Ke nel(1.0, con- s an alue bounds=” ixed” * RBF(1.0, leng h scale bounds=” ixed”)), alue added o he diagonal o he ke nel ma ix du ing i ing α= 10−10,op imize = min l b gs b, n es a s op imize = 0, no malize y =False, and andom s a e =None. 1Recall om he Ma e ials and Me hods chap e ha he numbe o samples was 42768. Addi ionally, he models s udied ha e wo inpu s and one ou pu . CHAPTER 6. SUPERVISED MODELS 70 •MLPReg esso Changes: hidden laye sizes = (2,4,2). The es , we e de- aul pa ame e s: ac i a ion = elu,sol e =lb gs, L2 penal y ( egula iza ion e m) α= 0.0001, ba ch size =au o =min(200, n samples), max i e = 1000, andom =None, ol = 10−4,wa m s a =False and max un = 15000. The de aul hidden laye size was (100,), conside ed o be excessi e. The sol e pa- ame e lb gs was i al o he model o pe o m accep ably. Acco ding o sk-lea n: ”Fo small da a se s, lb gs can con e ge as e and pe o m be e ” [43]. This was ound o be he case. The ollowing igu e illus a es he men ioned (2, 4, 2) MLPReg esso : Figu e 6.1: Rep esen a ion o he (2,4,2) Neu al Ne wo k used. This ne wo k has 3 hidden laye s. Illus a ion made hanks o NN-SVG [3]. No e om igu e 6.1 he 2 nodes on he inpu laye (wind, angle), he (2, 4, 2) node s uc u e in he hidden laye s (3 hidden laye s, wi h 2, 4 and 2 nodes espec i ely) and he single node on he ou pu laye . 6.7 Resul s This sec ion will de ail he o e all sco es ob ained. 6.7.1 R2sco e All he sco es we e calcula ed using he R2sco e (also known as he coe icien o de e - mina ion) is calcula ed using he ollowing o mula [13] [41]: R2= 1 −SS es SS o (6.1) whe e SS es ( he esidual sum o squa es) is: SS es =X i (yi− i)2=X i e2 i(6.2) SS o ( he o al sum o squa es) is: SS o =X i (yi−¯y)2(6.3) CHAPTER 6. SUPERVISED MODELS 71 and ¯y( he mean o he obse ed da a) is: ¯y=1 n n X i=1 yi(6.4) Acco ding o sklea n R2∈(−∞,1], whe e 1 is he bes sco e possible [41]. Sco es o e .75 a e conside ed good and sco es o e .90 e y good. Finally, sco es unde 0.50 will be conside ed e y bad. The R2sco e is in e sely co ela ed wi h he dis ance om he p edic ion o he eal alue, i.e: he close he p edic ion is o he eal alue ( he smalle he dis ance), he highe he R2sco e is. I his dis ance is equal o 0, he R2sco e will be equal o 1. 6.7.2 O e all sco es The ini ializa ion o he model pa ame e s is andom, he e o e, some a iance be ween uns is expec ed. The ollowing ables show he numbe o sco es o e a ce ain h eshold o ou di e en uns: Table 6.2: Model sco es b eakdown. Run 1. Model Sco es o e 0.0 Sco es o e 0.25 Sco es o e 0.5 Sco es o e 0.75 Sco es o e 0.9 Sco es o e 0.95 Linea 82 50 23 4 1 0 Polynomial 66 54 30 8 2 0 Ridge 76 47 22 10 2 1 Hube 78 50 22 8 4 0 Gaussian 10 8 3 0 0 0 Mlp 42 27 14 4 0 0 Table 6.3: Model sco es b eakdown. Run 2. Model Sco es o e 0.0 Sco es o e 0.25 Sco es o e 0.5 Sco es o e 0.75 Sco es o e 0.9 Sco es o e 0.95 Linea 79 52 22 7 3 0 Polynomial 77 66 36 8 1 1 Ridge 75 49 29 5 2 0 Hube 75 49 26 8 2 0 Gaussian 7 5 2 1 0 0 Mlp 31 24 14 1 0 0 Table 6.4: Model sco es b eakdown. Run 3. Model Sco es o e 0.0 Sco es o e 0.25 Sco es o e 0.5 Sco es o e 0.75 Sco es o e 0.9 Sco es o e 0.95 Linea 73 52 29 10 1 0 Polynomial 77 57 36 13 3 1 Ridge 80 58 27 6 2 0 Hube 64 41 15 7 2 1 Gaussian 7 6 3 1 0 0 Mlp 45 24 12 4 1 0 CHAPTER 6. SUPERVISED MODELS 78 Figu e 6.7: P edic ions o he MLPR model o he ange o he gy oscope in he z axis. CHAPTER 6. SUPERVISED MODELS 79 Figu e 6.8: P edic ions o he Polynomial model o a iance o he gy oscope in he y axis. Figu es 6.7 and 6.8 show mo e e y good sco es and igu e 6.7 in oduces he p edic- ions o he MLPR model. CHAPTER 6. SUPERVISED MODELS 80 Accele a ion The accele a ion ec o is p obably he mos impo an me ic o p edic oge he wi h he gy oscope alues. The ollowing sec ion de ails se e al o he bes pe o me models: Figu e 6.9: P edic ions o he Linea model o he mean o he accele a ion in he z axis. CHAPTER 6. SUPERVISED MODELS 81 Figu e 6.10: P edic ions o he Polynomial model o he mean o he accele a ion in he z axis. CHAPTER 6. SUPERVISED MODELS 82 Figu e 6.11: P edic ions o he Ridge model o he mean o he accele a ion in he z axis. CHAPTER 6. SUPERVISED MODELS 83 Figu e 6.12: P edic ions o he Hube model o he mean o he leng h o he accele a ion ec o . CHAPTER 6. SUPERVISED MODELS 84 Figu e 6.13: P edic ions o he MLPR model o he s anda d de ia ion o he leng h o he accele a ion ec o . As we can see om he p e ious igu es, he models managed o ha e e y good sco es, wi h some e en b eaking o e 0.95 like he Polynomial model o he mean o he accele a ion in he z axis on igu e 6.10. CHAPTER 6. SUPERVISED MODELS 85 As an impo an ema k o his sec ion, he Gaussian model did no manage o ha e an sco e o e 0.75. This is u he suppo ed by he ables 6.2, 6.3, 6.4 and 6.5, whe e he Gaussian model consis en ly showed he wo se sco es. A e age e o a e Using equa ion 3.8 and he p edic ions om he p e iously men ioned models, he a e age e o a e was calcula ed o be ≈1%. This e o a e was conside ed o be e y low and he p edic ions a e he e o e also conside ed o be e y p ecise. I a i ual model o he u bine we e o be made, he ecommended models o build i would be: Linea , Polynomial, Ridge and Hube . MLPR model could also be conside ed in some ins ances and he Gaussian model is s ongly discou aged in his case. No all he g aphs a e shown, as he e a e 107 (107 models wi h sco es o e 0.75), bu his o e iew should gi e a good idea o he pe o mance o he models. In conclusion, wi h hese esul s, we can say ha a i ual model o he g ouped me ics o he wind u bine can be done, ei he by combining he bes models o ce ain me ics o jus using one o he bes pe o me s o all he me ics. Addi ionaly, hese models can be used o p edic alues ou side he aining da a se (new pi ch angle con igu a ions and wind condi ions ha we e no p esen in he expe imen ). This should wo k ai ly well wi h in e media e alues like 15◦, bu no wi h alues nea 0◦( his is an special case we e he wind u bine blades will no be o a ing). One las ema k mus be made, and ha is ha he alues ying o be p edic ed should no be e y a om he expe imen al anges, o example: 20ms−1( he expe i- men had a maximum o 13.8ms−1), as hese alues a e e y a om he measu emen s and he models will p obably ail. Some p edic ions a e shown in he ollowing subsec ion. 6.7.4 P edic ions o inpu s ou side he expe imen Using he p e ious models, a able was de eloped ha p esen s he p edic ions o wind speeds ou side he expe imen , bu only o models ha had a sco e o e 0.90: CHAPTER 6. SUPERVISED MODELS 86 Table 6.6: P edic ions o inpu s ou side he expe imen , only models wi h sco es o e 0.90. Model Me ic Angle [◦] Wind speed [ms−1] P edic ion linea mean Acc z 10 15.0 1.104912 linea mean Acc z 10 18.0 1.142317 linea mean Acc z 10 20.0 1.167254 linea mean Acc z 30 15.0 1.103160 linea mean Acc z 30 18.0 1.140565 linea mean Acc z 30 20.0 1.165502 idge mean Acc z 10 15.0 1.104175 idge mean Acc z 10 18.0 1.140750 idge mean Acc z 10 20.0 1.165133 idge mean Acc z 30 15.0 1.103168 idge mean Acc z 30 18.0 1.139742 idge mean Acc z 30 20.0 1.164125 linea a Gy o y 10 15.0 0.004726 linea a Gy o y 10 18.0 0.006617 linea a Gy o y 10 20.0 0.007877 linea a Gy o y 30 15.0 0.005020 linea a Gy o y 30 18.0 0.006911 linea a Gy o y 30 20.0 0.008171 linea mean len Acc 10 15.0 1.111576 linea mean len Acc 10 18.0 1.149678 linea mean len Acc 10 20.0 1.175079 linea mean len Acc 30 15.0 1.110320 linea mean len Acc 30 18.0 1.148421 linea mean len Acc 30 20.0 1.173822 polynomial mean len Acc 10 15.0 1.116658 polynomial mean len Acc 10 18.0 1.161558 polynomial mean len Acc 10 20.0 1.193637 polynomial mean len Acc 30 15.0 1.109985 polynomial mean len Acc 30 18.0 1.155171 polynomial mean len Acc 30 20.0 1.187439 I we compa e he ou pu s o mean len Acc z wi h he g aphs om 6.12 we can see an inc ease in he p edic ion me ic (mo e accele a ion, in he linea model, om a ound 1.25ga 13.8ms−1 o 1.11ga 15.0ms−1and up o 1.17ga 20.0ms−1). Las ly, an obse a ion is made: The change in angle om 10◦ o 30◦does no change he accele a ion me ics o he same wind speed d as ically. I does howe e , a ec he a iance o he gy oscope along he y axis. This sugges s ha wind speed is much mo e in luen ial in he accele a ion ec o han he angle, a leas o hese wo angles. CHAPTER 6. SUPERVISED MODELS 87 As he e is no eal expe imen al da a o compa e his ou pu s wi h, no u he com- men a y will be done. 6.8 Conclusions In his sec ion we showed a ep esen a ion o he p edic ions o he bes models. All he models had good esul s, bu he e we e some ha we e limi ed o only some me ics. Addi ionally, some p edic ions we e made o alues ou side he expe imen da a se and an expec ed inc ease in some me ics was obse ed. Bibliog aphy [1] Fas ou ie ans o m. h ps://en.wikipedia.o g/wiki/Fas _Fou ie _ ans o m, 2022. [2] AkanoToE. 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