wind energy pro.odt

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Capitolul 2 - Caracteristicile vantului

WIND

ENERGY

SYSTEMS

TABLE OFCONTENTS1

Introduction..............................................................................1-11.1 Historical Uses of Wind...................................................................1-21.2 History of Wind Electric Generation...............................................1-31.3 Horizontal Axis Wind Turbine Research..........................................1-5

1.4 Darrieus Wind Turbines.................................................................1-131.5 Innovative Wind Turbines..............................................................1-161.6 California Wind farms....................................................................1-212 Wind Characteristics................................................................2-12.1 Meteorology of Wind........................................................................2-12.2 World Distribution of Wind.............................................................2-72.3 Wind Speed Distribution in the United States................................2-82.4 Atmospheric Stability.....................................................................2-142.5 Wind Speed Variation With Height...............................................2-232.6 Wind Speed Statistics....................................................................2-262.7 Weibull Statistics...........................................................................2-302.8 Determining the Weibull Parameters.............................................2-382.9 Rayleigh and Normal Distributions................................................2-442.10 Distribution of Extreme Winds....................................................2-552.11 Problems.......................................................................................2-623 Wind Measurements.................................................................3-1


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3.1 Eolian Features................................................................................3-13.2 Biological Indicators........................................................................3-23.3 Rotational Anemometers.................................................................3-53.4 Other Anemometers.......................................................................3-143.5 Wind Direction..............................................................................3-173.6 Wind Measurements with Balloons...............................................3-283.7 Problems3-344 Wind Turbine Power, Energy, and Torque...............................4-14.1 Power Output from an Ideal Turbine..............................................4-14.2 Aerodynamics..................................................................................4-44.3 Power Output from Practical Turbines...........................................4-74.4 Transmission and Generator Efficiencies........................................4-134.5 Energy Production and Capacity Factor........................................4-214.6 Torque at Constant Speeds............................................................4-29

4.7 Drive Train Oscillations.................................................................4-334.8 Starting a Darrieus Turbine...........................................................4-394.9 Turbine Shaft Power and Torque at Variable Speeds....................4-434.10 Problems........................................................................................4-495 Wind Turbine on the Electrical Network.................................5-15.1 Methods of Generating Synchronous Power....................................5-15.2 AC Circuits........................................................................................5-45.3 The Synchronous Generator.............................................................5-145.4 Per Unit Calculations........................................................................5-225.5 The Induction Machine.....................................................................5-275.6 Motor Starting...................................................................................5-375.7 Capacity Credit.................................................................................5-405.8 Features of the Electrical Network....................................................5-485.9 Problems...........................................................................................5-586 Asynchronous Electrical Generators........................................6-16.1 Asynchronous Systems....................................................................6-26.2 DC Shunt Generator with Battery Load............................................6-56.3 Permanent Magnet Generators........................................................6-116.4 AC Generators.................................................................................6-18

6.5 Self-Excitation of the Induction Generator.......................................6-206.6 Single-Phase Operation of the Induction Generator.......................6-32

6.7 Field Modulated Generator............................................................6-376.8 Roesel Generator...........................................................................6-396.9 Problems........................................................................................6-427 Asynchronous Loads.................................................................7-1


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7.1 Piston Water Pumps.......................................................................7-27.2 Centrifugal Pumps..........................................................................7-107.3 Paddle Wheel Water Heaters.........................................................7-217.4 Batteries.........................................................................................7-237.5 Hydrogen Economy........................................................................7-337.6 Electrolysis Cells............................................................................7-407.7 Problems........................................................................................7-478 Economics of Wind Systems.....................................................8-18.1 Capital Costs...................................................................................8-18.2 Economic Concepts.........................................................................8-98.3 Revenue Requirements...................................................................8-158.4 Value of Wind Generated Electricity................................................8-208.5 Hidden Costs in Industrialized Nations............................................8-238.6 Economic Factors in Developing Countries.....................................8-24

8.7 Problems.........................................................................................8-269 Wind Power Plants..................................................................9-19.1 Turbine Placement..........................................................................9-19.2 Site Preparation..............................................................................9-29.3 Electrical Network...........................................................................9-49.4 Selection of Sizes, Low Voltage Equipment...................................9-79.5 Selection of Sizes, Distribution Voltage Equipment........................9-149.6 Voltage Drop...................................................................................9-229.7 Losses.............................................................................................9-249.8 Protective Relays...........................................................................9-28

9.9 Wind farm Costs..............................................................................9-309.10 Problems.......................................................................................9-34

Appendix A: Conversion Factors...........................................................9-36Appendix B: Answers To Selected Problems........................................9-37Appendix C: Wire Sizes.........................................................................9-39Appendix D: Streams and Waterways.................................................9-45


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INTRODUCERE

Vantul (energia eoliana) este o surs de energie liber, curat, i inepuizabil. Ea a servitomenirea de-a lungul vremii ajutand la propulsia navelor ,turbinelor eoliene cu ajutorul carorase macinau cereale si pompatul apei. Interesul fa de energia eolian a fost unulextraordinarinsa cu toate acestea, atunci cnd produsele petroliere ieftine i din abunden au devenit disponibile, dup al doilea rzboi mondial, costurile ridicate i incertitudinea financiaraau plasat energia eolian ntr-un dezavantaj economic. Apoi, n 1973, na iunile arabe pus unembargo asupra petrolului, sugerand ca zilele petrolului ieftin i din bel ug au fost numarate.

Oamenii au nceput s realizeze c livrrile de petrol din lume nu ar dura la nesfr it i c proviziile rmase ar trebui s fie conservat pentru industria petrochimic.De exemplu utilizareadrept combustibil pentru cazanele de incalzit apa in locuinta a trebuit sa fie stopata,iar altesurse de energie n afar de petrol i gaze naturale au trebuit s fie dezvoltate.Cele dou surse de energie n afar de petrol care s-au presupus a fi n msur s furnizezenevoileenergetice pe termen ale Statelor Unite sunt energia crbunelui i energia nuclear.Energia nuclear are mai multe avantaje fa de crbune, datorita faptului ca in urmaobtineriiniciun fel de dioxid de carbon sau dioxid de sulf sunt produse, opera iunile miniere fiindde dimensiuni mai mici, neavand nici o alt utilizare major n afar de furnizarea de cldur.Din cauza problemelor de crbune, energia eolian i alte forme de energie ca de exemplu ceasolar sunt puternic ncurajate. Energia eolian ar putea deveni o surs major de energie, n

ciuda costurilor pu in mai mari dect cea obtinuta cu crbune sau cea nuclear, Acest lucruinsa nu inseamna ca energia eolian va fi ntotdeauna mai scump dect crbunele sau energienuclear, deoarece s-au facutprogrese considerabile pentru ca energia eolian sa fie mai pu incostisitoare. Dar chiar si fara un avantaj de cost clar, energia eolian poate deveni cu adevratun tronson important in furnizarea de energie la nivel mondial


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1.2 UTILIZREA VANTULUI DE-A LUNGUL ISTORIEI

Vntul a fost folosit pentru propulsia navelor cu vele pentru multe secole.Lumea Noua a fostexplorata de catre oameni cu ajutorul acestor corabii.Este foarte adevarat ca vantul a fostsingura energie folosita in transportul maritim pana in secolul al 18-lea cand Watt a inventatmotorul pe aburi. Eroul din Alexandria, care a trit n secolul III .Hr., a descris un simplu

dispozitiv cu patru vele, care a fost folosit pentru a impinge aer prin tevile unei orgi .Persaniiau folosit turbine eoliene pe scar larg ctre mijlocul secolului al 7-lea acestia construindun fel de ma in cu axa vertical, cu un numr de 4 pnze montate radial. Acestea au fostrealizate din materiale locale datorita costurilor de productie foarte mici,iar dimensiunea lor afost, probabil, determinat de materialele disponibile. S-a ajuns la concluzia ca pentru a fieficienti trebuiau sa construiasca un numar mai mare de turbine eoliene de dimensiuni maimici fiind mai putin costisitoare in comparatie c cele mariCea mai veche turbin eolian englez inregistrata este datat la 1191.iar prima moarapentru macinarea cerealelor fost construit n Olanda n 1439. Au fost o serie de evolu iitehnologicede-a lungul secolelor, si prin anii 1600 cel mai frecvent tip de moara construita era moaraturn.

Colonistii olandezi au adus acest tip de turbine eoliene n America pe la mijlocul anilor 1700insa numarul de turbine eoliene construite atunci era infim fata de cel al turbinelor dejaexistente in Europa.

Apoi,la jumatatea anilor 1800 au fost dezvoltate turbine eoliene de dimensiuni mai micipentru pomparea apei.Vestul statelor unite a fost impanzit de terenuri de p unat care insa nu aveau apa lasuprafata dar puteau fi irigate cu ajutorul pomparii apei folosind acest tip de turbina!Se estimeaza ca 6,5 milioane de turbine s-au construit n Statele Unite ntre 1880 i 1930 de ctre o varietate de companii. Multe dintre acestea nc func ioneaz n mod satisfctor.


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ajutand la furnizarea de ap pentru animale, si pasune. aceste masini au jucat un rolimportant n dezvoltarea partii de vest a americii

1.3 Istoria generarii energiei electrice cu ajutorul vantului

1. Danemarca a fost prima ar care a utilizat vntul pentru generarea de energieelectric.Danezii au folosit o turbina de 23 m diametru n 1890 pentru a generaelectricitate. Pn n 1910, mai multe sute de unit ile cu capacit i de 5 pn la 25 kW au fost n func iune n Danemarca.

Prin anul 1925, au aparut in comert pe piata americana turbine eoliene ce foloseau elicecu dou i trei pale.Brandurile cele mai frecvente au fost Wincharger (200 - 1200W) i Jacobs (1,5 la 3 kW).Costurile mai mici insa de energie electric produs decentrale au dus la declinul rapid a ceea ce insemnat energia eoliana la vremea aceea.Dup 1940, costul energiei electrice generate au continuat sa scada coborand sub 3centi pe kWh la nceputul anilor 1970. Cea mai mare Turbina eolian construita naintede sfr itul anilor 1970 a fost o masina de 1250 kW, construita, in Vermont, n 1941.

ntre 1941 i 1945 ma in Smith-Putnam acumulat aproximativ 1100 de ore de functionare Mai multe ar fi fost acumulat, cu excep ia preblemei de a ob ine piese pentru repara ii critice n timpul rzboiului. Rezultatele tehnice ale turbinei eoliene Smith-Putnam l-a determinat pe inginerul Percy H. Thomas, un inginer ce facea parte dinComisia Federal de Administrare a Energiei, s- i petreac aproximativ 10 ani ntr-oanaliz detaliat a modului de generare a energiei eoliene electrice. El a proiectat douturbine de dimensiuni mari considerand ca acestea sunt potrivite proiectului sau, una de6500 kW, iar cellalta de 7500 kW.nl imea turnului turbinei de 6500 kW a fost 145 m cu dou rotoare fiecare de 61 de metrii in diametru

3 Cercetarea turbinelor cu axa orizontala in statele unite

Programul Federal de Energie Eolian si-a avut nceputul n 1972 cnd un grup de cercetatoriin domeniul energiei solare ,apartinand de National Science Foundation (NSF) i National

Aeronautics and Space Administration (NASA) a recomandat ca energia eolian s fiedezvoltata pentru a extinde op iunile natiunii in legatura cu surse noi de energie .

Sa decis foarte devreme n program ca MOD-0-ar fi evaluat la 100 kW si aveau un rotor cudou lame de 38 m in diametru. Aceast ma in a va incorpora multe progrese naerodinamica, materiale, interpretare a datelor etc. Aceste tipuri de turbina cu 2 lame au fostconstruite n dimensiuni mai mari i au fost operate mai multe ore dect orice alt tip, prinurmare, a avut cea mai mare probabilitate de a lucra destul de bine de la nceput.

O diagrama a turbinei i con inutul nacelei (structura sau partea de sus a turnului, care

con ine cutia de viteze, generatorul, i controale)Aceasta structura era asezata pe un suport de otel cu 4 picioare avand aproximativ 30 de metrii inaltime fiint pozitionat in amonte de vant!MOD-0 a fost operat pentru perioade scurte de timp, pentru a evalua unele dintre efectele defunc ionare in amonte pe direc ia vntului privind incarcatura structurale i a cerin elor de control ale ma inii.


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Lamele din lemn cntresc 1360 kg fiecare, cu 320 kg mai mult decat lamele dinaluminiu, dar durata de via anticipat a fost mai mare dect a omologilor dinaluminiu. Un MOD-0A este prezentat n Fig. 2.

Figura 2: MOD-0A in zona Kahuku Point, Oahu, Hawaii.

In urma MOD-0 i MOD-0 au fost proiectate fost o serie de alte maini, MOD-1, MOD-2, etc ,iar parametrii de proiectare pentru mai multe dintre aceste maini suntprezentati n tabelul 1. 1.MOD-1 a fost construit ca o main de 2000-kW, cu undiametru al rotorului de 61 metri. Acesta este reprezentata n Fig. 3.


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Figura 3: MOD-1 Situata la Boone, North Carolina.

A fost construita la Knob Howard, n apropiere de Boone, Carolina de Nord lasfritul anului 1978. Se poate observa din tabelul 1.1, ca viteza apreciata avntului pentru MOD-1 a fost de 14,6 m / s la nlimea butucului, cu mult mai mare

dect celelalte. Acest lucru a permis MOD-1 s aib o putere nominal de 10 ori maimare MOD-0 cu o zona de contact de doar 2.65 ori mai mare

Boeing Compania de inginerie si constructii din Seattle, Washington, a fabricat celedou lame de oel. Aparatul urmtor n serie, MOD-2, a reprezentat un efort de aconstrui o main cu adevrat competitiva nu numai financiar ci si in materie defiabilitate incluznd toate informaiile obinute de la mainile anterioare. Rotorulavut dou lame, a fost de 91,5 m n diametru, i a fost pozitionat in direcia opusvntului Viteza rotorului a fost controlat la o constant 17.5 rot / min. Putereanominal a fost de 2500 kW (2,5 MW), la o vitez a vntului de 12,4 m / s ,msuratla nlimea butucului de 61 m. n scopul de a simplifica configuratia i pentru aatinge o greutate mai mic i diminua costurile, acestui tip de turbina i s-aimplementat un nou tip de lamele ce puteau fi controlate mult mai usor.

Aceast caracteristic de construcie se poate vedea n Fig. 4.


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Figura 4: MOD-2 situata in Goodnoe Hills aproape de zona Goldendale, Washington

Cele mai multe turbine eoliene concepute pentru producia de energie electric auconstat ntr-o elice lam cu dou sau trei lamele ce se rotesc n jurul unei axeorizontale. Aceste lame tind s fie scumpe, precum i turbina trebuie s fieorientat n vnt, o alt sarcin costisitoare pentru masinile eoliene mai mari.

Fig. 5 prezinta un Darrieus 17 de metri, construit la Sandia.Diametrul paleteloreste aceeai ca i nlime, 17 m. Lamele din aluminiu extrudat au fost fcute dectre Alcoa (compania de aluminiu ale Americii, Alcoa Centru, Pennsylvania).

Aceast main a fost evaluata la 60 kW la un vant constant de 12,5 m / s . Maimulte modele de aceast tip de main au fost construite n jurul anilor 1980.Darrieus are mai multe caracteristici atractive. Una dintre ele este faptul c

maina se rotete n jurul unei axe verticale, prin urmare, nu are nevoie s fiepozitionata catre vnt. Aceast form se numete troposkein, de la termenul grecpentru rasucirea sforii. Deoarece lama funcioneaz n tensiune aproape pura estesuficienta o singura lama usoara si ieftina. Un alt avantaj este faptul cgeneratoruli controalele sunt toate situate n apropiere de nivelul solului, prin urmare, sunt maiuor de a modifica i menine in functiune.Eficiena este aproape la fel de bun cacea a turbinei cu elice cu axa orizontala astfel nct Darrieus se intrepta cu pasi

repezi spre obtinerea unei masini de energie eoliana cu randament mare si costuriscazute!


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Figure 5: Sandia Laboratories 17-m Darrieus, rated at 60 kW in a 12.5-m/s wind. (Courtesy ofAluminum Company of America.)

Prima turbina Darrieus a fost construita in 1977 de ctre Dominion FabricatorsAluminium, Limited din Ontario, Canada. . Cnd viteza de rotaie aparatul a ajuns la60 rot / min, cu mult peste viteza nominal de 41 rot / min, un urub s-a rupt i apermis unei lame sa zboare si sa taie unul din firele exterioare. Aparatul sa prabusit,apoi la sol.Accidente ca acestea nu sunt mai puin frecvente n zonele de test pt noiletehnologii, dar ele sunt cu siguran frustrante pentru persoanele implicate. Se parec diferitele probleme sunt rezolvabile, dar irurile de accidente cu siguran au

ncetinit desfurarea constructiilor de turbine Darrieus n raport cu cea a turbinelorcu axa orizontal.

5 Turbine eoliene inovatoare

Un alt tip de turbina la care se lucra intens in aceea perioada era turbina Savonius proiectata inFinlanda de S.J Savonius, o turbina cu axa verticala ce nu necesita sa fie orientata spre directiavantului. Entuziastii pro energie alternativa de la aceea vreme obijnuiau sa construiasca acesteturbine din butoaie vechi din tabla in care fusese depozitat petrol! Acestea erau taiate pelungime iar apoi cele 2 jumatati sudate cap la cap in compensare una fata de cealalta pentru aprinde curentii de aer.


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O astfel de turbina se regtaseste in Fig. 7.

Figura 7: Turbina cu o putere de 5 kW in raport cu un vant de 12-m/s

Turnul Savonius a fost de 11 m nlime i 6 m lime. Fiecare rotor avea 3 mnlime respectiv1,75 m n diametru. Rotoarele au fost conectate impreuna si a

condus generator cu magnet permanent 5 kW, care avea trei faze. La vitezavntului nominal de 12 m / s, viteza rotorului a fost de 103 rot / min, iar vitezageneratorului a fost de 1800 rot / min, iar frecvena de 60 Hz. Tensiunea de ieire ifrecvena au variat in functie de viteza vntului i de sarcin, ceea ce nsemna caceast turbina , nu a putut fi legata direct n paralel cu reeaua de electricitate!Principalele avantaje ale Savonius sunt un cuplu foarte mare de pornire i simpla

construcie. Dezavantajele sunt greutatea materialelor i dificultatea de proiectare arotorului pentru a rezista la viteze mari ale vantului. O alt main de axa vertical,care are oameni interesai de mai muli ani este rotorul Madaras. Acest sistem a fostinventat de catre Julius D. Madaras, care a efectuat teste considerabile pe ideea lui

ntre 1929 i 1934. Sistemul iniial propus de Madaras a constat in niste cilindrii de27 de metri inaltime si 6,8m in diametru montati vertical i rotite de motoareelectrice pentru a converti energia eolian in efect MagnusToate turbinele eoliene discutate pn acum insa aveau o problema cu costurile de

productie. Dei aceste maini funcioneaz satisfctor, costurile de capital suntmari. Cele Darrieus poate deveni mai rentabile dect turbina cu elice cu dou pale,dar nu este posibil sa se produca electricitate cu adevarat ieftina. Turnul turbineiare paletele verticale, care conduc vntului ntr-un traseu circular in interiorul


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turnului. Efectul produs este acela de tornada unde interiorul spiralei are presiunejoasa ,iar viteza vortexului este mare. Diferena de presiune dintre nucleu vortex iaerul exterior este apoi utilizata pentru roti, turbina de mare viteza de la bazaturnului. Aceasta masina mai degraba se bazeaza pe extragerea energiei datoritadiferentelor de presiune si potentialului energetic din aer si nu se bazeaza in

principiu pe utilizarea de energie cinetic a aerului n micare.Multe alte maini eoliene au fost inventate de-a lungul ultimele cteva sute de ani.

Turbina cu elice i Darrieus au prut a fi de ncredere, masini ieftine si competitive,care pot oferi o cantitate semnificativ de energie electric. Costul in comparatie cuproductia este relativ mic astfel incat ne putem atepta s vedem o implementarepe scara larg a acestor masini in urmatoarele cateva decenii.

Caracteristici ale vantului

Atmosfera pmntului poate fi considerata ca un motor termic gigantic. Extrage energie de laun rezervor (soare) i livreaz cldur la un alt rezervor, la o temperatur mai joas (spa iu). n urma acestui proces, se produc schimbari la nivelul gazelor din atmosfera precum si la nivelulcelor aproape de sol. Vor fi regiuni n care presiunea aerului este temporar mai mare sau mai

mic dect in mod normal. Aceast diferen de presiune de aer provoaca gazele atmosfericesau de vnt s curg de la regiunea de presiune mai mare la o presiune mai mica . Acesteregiuni sunt de obicei de sute de kilometri n diametru.

Radia ia solar, evaporarea apei, stratul de nori, i de rugozitatea suprafetei joaca roluri importante n stabilirea condi iilor de atmosfera.Studiul de interac iunile dintre aceste efecte este un subiect complex numit meteorologie, care este acoperit de mai multe manuale excelente.Prin urmare, doar o scurt introducere la acea parte a meteorologiei privind fluxul de vnt va fiprezentata in cele ce urmeaza.

1 Meteorologia Vantului

Ideea de baz a mi crii aerului este o diferen de presiune de aer ntre dou regiuni. Aceast presiune de aer este descris de mai multe legi fizice. Una dintre acestea este legea luiBoyle, care prevede c produsul dintre presiune i volumul unui gaz la o temperatur constanttrebuie s fie o constant, sau:

p1V1 = p2V2 (1)


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O alt lege este legea lui Charles ", care spune c, pentru o presiune constant, volumul unuigaz variaza direct cu temperatura absolut s fie o constant, sau:

Daca in urma masuratorilor s-ar creea un grafic al volumelor in comparatie cu temperatura s-arobserva ca starea de zero absolut este considerata a fi -273.15oC sau 0 Kelvin.

Legile i Charles Boyle pot fi combinate n legea gazului ideal :

pV = nRT (3)

In conditii standard de 0 grade celsius la o presiune de o atmosfera un kilomol de gaz ocupa22.414 m3 iar constanta universala a gazului este 8314.5 J/(kmolK) unde J reprezinta un joulesau un newton metru de energie,adica presiunea unei atmosfere la 0 Grade Celsius sedetermina cu relatia:

Densitatea a unui gaz este masa m a unui kilomol impartita la volumul V a acelui kilomol

Volumul unui kmol variaza in functie de presiune si temperatura asa cum am specificat mai susla ecuatia nr 3. iar daca o introducem in ecuatia nr 5 vom observa ca densitatea este calculataastfel:

unde p este in kPa si Tin grade kelvin. Aceast expresie ne arata densitatea aerului uscat n condiii standardde 1.293 kg/m3.

Unitatea comun de presiune folosita n trecut pentru activitatea de meteorologiea fost barul (1 bar =100 kPa) i milibari (100 Pa). n aceast notaie o atmosfer

standard a fost mentionata ca 1,01325 bar sau 1013.25 milibari.

Presiunea atmosferic a fost de asemenea data de nlimea mercurului ntr-un tubvidat. Aceast nlime este 29.92 inci sau 760 milimetri coloana de mercur pentruo atmosfer standard. Aceste numere pot fi utile n folosirea instrumentelormeteorologice.De remarcat aici faptul c mai multe definiii ale condiiilor standard


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sunt n uz incepand cu temperatura ca de exemplu: chimistul folosete 0oC castandard de temperatur n timp ce inginerii au folosit adesea 68oF (20 C) sau77oF (25 C), n calitate de temperatur standard. Pentru ca o regiune de naltpresiune sa se menina n timp ce aerul de la nivelul solului se misca, trebuie sexiste aer care il inlocuieste intrand in aceiasi regiune, n acelai timp. Singura surs

pentru acest aer este deasupra regiunii de nalt presiune. aerul va curge n jos ninteriorul regiunii de nalt presiune (i in sus in interiorul unei regiuni de joaspresiune) pentru a menine presiunea. Acest aer descendent va fi nclzit adiabatic(adic fr transfer de cldur sau de mas cu mprejurimile sale) i va tinde sdevin uscat. n interiorul regiunii de joas presiune, aerul este rcit adiabatic ncretere, ceea ce poate duce la nori i precipitaii.O linie trasat prin punctele de presiune egal pe o hart meteo se numete o

izobar. Aceste presiuni sunt trasate la o altitudine comuna, cum ar fi nivelul mrii.Pentru a facilita reprezentarea, intervalele dintre isobare sunt, de obicei, 300, 400,sau 500 Pa Astfel, isobare succesive ar fi trasate prin punctele cu rezultate de 100.0100.4, 100.8 kPa, etc O astfel de hart este prezentat n Fig. 1. n descriereadireciei vntului, ne referim ntotdeauna la direcia de origine a vntului. .Diferentaorizontala de presiune prescrie determina viteza si directia vantului. Deoarecedirecia forelor este orientata de la presiuni inalte la presiuni joase siperpendiculare pe isobare, tendina iniial a vntului este sa se miste in paralel cugradientul de presiune orizontal i sa se miste perpendicular pe isobara. Cu toateacestea, de ndat miscarea vantului este stabilit, o for deflectoare esteprodusa,forta care modific direcia de micare. Aceast for se numete forCoriolis.Efectul de baz este prezentat n Fig. 2. Cele dou linii curbe sunt linii de

latitudine constant, cu punctul B situat direct la sud de punctul A. Un pachet de aer

(reprezentat ca un proiectil de tun) se deplaseaz spre sud, la punctul A. Dac neputem imagina parcela noastr de aer ca avand 0 frictiune cu aerul, atunci aceleimase de aer va rmne constant . Direcie se va schimba insa datorita miscariipamantului sub masa de aer.


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.

Figure 1: Weather map showing isobars

Figure 2: Coriolis force

2 Impartirea vantului in zone la nivel global

Calmele ecuatoriale sunt datorate unei centuri de joas presiune, care nconjoarPmntul, n zona ecuatorial, ca urmare a supranclzirii medie a pmntului naceast regiune . Aerul cald se ridica aici, ntr-un flux puternic de convectie. Aversele dup-amiaza trziu sunt comune datorita racirii adiabatice rezultat, care este cea mai pronun at latemperatura diurna cea mai ridicata. Aceste du uri pstreaz umiditatea foarte mare, fr aoferi suprafeteiun timp de uscare/racire foarte mare.Atmosfera tinde s fie opresiva, calda, ilipicioasa, cu vnturi calme,iar apele marilor si oceanelor sunt calme aparand ca u suprafata


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alunecoasa si sticloasa .Cu exceptia unor cazuri rare care schimba tiparele meteorologice inaceasta regiune zona ecuatoriala nu este preferata pentru obtinerea energiei eoliene.

Figure 5: Ideal terrestrial pressure and wind systems

4 Stabilitatea atmosferica

Aa cum am menionat, cele mai multe msurtori pentru determinarea vitezeivantului sunt fcute la aproximativ 10 m deasupra solului. Turbinele eoliene mici

sunt montate de obicei de la 20 la 30 m deasupra nivelului solului, n timp ce vrfulelicei poate ajunge la o nlime mai mare de 100 m pe turbine de mari dimensiuni,astfel nct o estimare a variaiei vitezei vntului cu nlimea estenecesar.Formula matematic a acestei reformri va fi luata n considerare nseciunea urmtoare, dar mai nti trebuie s examinm o proprietate a atmosfereicare controleaz aceast variaie pe vertical a vitezei vntului. Aceast proprietatese numete stabilitate atmosferic, care se refer la cantitatea de mixtiune natmosfer. Vom ncepe aceast discuie prin examinarea variaia de presiune cuinaltimea in atmosfera inferioar.O parcela de aer are masa si este atrasa de pmnt de fora de gravitaie. Pentru ca

parcela s nu cad la suprafaa pmntului, trebuie s existe o for egal i opuscare sa o impinga departe de pmnt. Aceast for provine din scderea presiuniiaerului cu creterea nlimii.Cu cat este mai mare densitatea aerului,cu atat mairapid trebuie s scad sub presiunea ascendent pentru a mentine aerul la oinaltime constant mpotriva forei de gravitaie. Prin urmare, presiunea scaderapid, cu nlimea la altitudini joase, unde densitatea este mare, i ncet laaltitudini mari, unde densitatea este sczut. Aceast condiie se numete echilibruhidrostatic sau balans hidrostatic


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Se presupune ca n Fig. 9 presiunea nu varieza cu temperatura locala. sepresupune c aceasta coloana de aer direct deasupra punctului de interes isimenine aceeai mas de-a lungul ciclului de temperaturi

Figure 9:Variaia de presiune cu altitudinea in Atmosfera Standard a SUA

Temperatura ambienta trebuie folosita in aceasta ecuatie

Exemplu

O turbina eoliana este evaluata la 100 kW la o viteza a vantului de 10 m / s n aer n condi iistandard. n cazul n care puterea este direct proportionala cu densitatea aerului, care ar fiputerea produsa de turbina la o viteza a vantului de 10 m/s la 2000 m altitudine, unde seinregistreaza o temperatura de 20C?

In figura 9 citim o presiune medie de 79.4 kPa. ,iar dSensitatea la 20oC = 293 K astfel:

In aceste conditii puterea obtinuta este raportul dintre aceasta densitate si densitatea inconditiile standard inmultita cu puterea turbinei in conditii standard.


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Puterea turbinei a scazut de la 100 kW la 73 kW avand aceiasi viteza a vantului, doar pentrusimplul fapt ca densitatea aerului este mai mica.

Dup ce ne vom indeparta de nivelul solului insa, in primele primele cteva mii de metri deatmosfer, vom gsi o scdere de temperatur mult mai previzibila in raport cu altitudinea. Vomutiliza cuvantul altitudine pentru a ne referii la nl imea unui obiect deasupra nivelului solului, i cuvantul cota spentru a exprima nl imea deasupra nivelului mrii. Cu aceste defini ii, un pilot care zboar n mun i este mult mai preocupat de altitudine lui dect cota lui.

Aceast varia ie de temperatur cu altitudinea este foarte importanta pentru caracterul vnturilor incepand cu primii 200 m deasupra suprafe ei Pmntului, a a c trebuie examinat n detaliu. Observm c ecua ia. 3 (legea gazului ideal sau ecua ia de stare) poate fi satisfcut de presiune i temperatur n scdere cu altitudinea n timp ce volumul unui kmol (volumul specific) este n cre tere. Presiunea i temperatura sunt u or de msurat n timp ce volumul specific nu este. Prin urmare,se va utiliza prima lege a termodinamicii (care prevede c energia esteconservat) i vom elimina termenul de volumul din ecua ia. 3. Atunci cnd se face acest lucru pentru un proces adiabatic, rezultatul este

n cazul n care T1 i P1 sunt temperatura i presiunea la starea 1,iar To i po reprezinta temperatura i presiunea la starea 0, R este constanta universal a gazelor, iar cp este clduraconstant-presiunea specific a aerului. Derivarea acestei ecua ii poate fi gsita n cr ile cele mai introductive de termodinamica. Valoarea medie pentru raportul R / CP n atmosfera

inferioar este 0.286.EQ. 9, uneori numit ecua ia lui Poisson, se refer schimbri de temperatur adiabatic

experimentata de ctre o parcel de aer in curs de deplasare pe vertical a cmpului depresiune, prin care se mi c. Dac am cunoa te condi iile ini iale To i po, putem calcula T1 temperatura la orice presiune p1, atta timp ct procesul este adiabatic i implic doar gazeideale. Gazele ideale pot sa nu contina nici un material lichid sau solid i tot ar putea ficonsiderate gaze ideale. Aerulse comport ca un gaz ideal, atta timp ct vaporii de ap din aenu sunt saturati. Atunci cnd apare satura ia, apa ncepe s condenseze, i n condensare renun la cldura latent de vaporizare. Aceast intrare de energie termic ncalcconstrngerea adiabatic, n timp ce prezen a apei lichide scoate din ecuatie aerul ca fiind un

gaz ideal.

5 Variatia vitezei vantului in raport cu inaltimea

Dup cum am vzut, o cunoatere de vitezei vntului la nlimi de 20 pn la 120m deasupra solului este foarte de dorita n orice decizie cu privire la locatie si tipulde turbine eoliene ce urmeaza a fi instalate. De multe ori, aceste date nu sunt


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disponibile, iar unele estimri trebuie s se fac de la viteze ale vntului, msuratla aproximativ 10 metri. Acest lucru necesit o ecuaie care prezice viteza vntuluila o nlime n in corelatie cu viteza msurat la o inaltime mai mica. Aceste ecuaiisunt dezvoltate n textele de mecanica fluidelor.Aceste derivatii insa sunt dincolo dedomeniul de aplicare al acestui curs, astfel vom folosi doar rezultatele. O form

posibil pentru variaie a vitezei vntului u (z) cu nlimea z este :

Aici UF este viteza de frecare, K este constanta lui von K'arm'an (normal, se presupune a fi0,4), ZO este rugozitatea lungimii suprafe ei , iar L este un factor de scara numit Moninlungimea Obukov [17, 13]. Func ia (z / L) este determinat de radia ia solar net la sit.

Aceast ecua ie este valabil pentru termen scurt ca de exemplu vitezele medii ale vntului pedurata unui minut, dar nu la mediile lunare sau anuale.

Rugozitatea suprafe ei zo va influentata att de mrimea rugozitatilor cat i de spa ierea elementelor de rugozitate, cum ar fi iarba, culturi, cldiri, etc Valorile tipice ale zo sunt deaproximativ 0,01 cm pentru suprafe e de ap sau zpad, 1 cm pentru gazon, 25 cm pentruiarb nalt sau culturi, i 1 la 4 m de pdure i ora . n practic, zo nu poate fi determinat cu precizie din aspectul unui sit, dar este determinat de msurtori ale vntului. Acela i lucru este valabil pentru UF viteza de frecare, care este o func ie de frecare de suprafa i densitatea aerului, i (z / l). Parametrii sunt gsiti prin msurarea vntului la trei nl imi, si scrierea ecuatiilor pentru fiecare inaltime iar mai apoi rezolvand cele 3 necunoscute. Aceasta nu este oecua ie liniar, astfel o analiz neliniar trebuie s fie utilizata. Rezultatele trebuie s fieclasificate n func ie de direc ia vntului i de perioada din an, deoarece zo variaz n func ie de rugozitatea suprafe ei in amonte pe direc ia vntului i starea de vegeta ie. Rezultatele trebuie s fie, de asemenea, clasificate n func ie de cantitatea paturii de radiatii astfel nct formaadecvat func ional a (z / L) poate fi folosita.

Acest lucru este destul de satisfacator pentru studii detaliate cu privire la anumitesit-uri importante, dar este prea dificil de folosit pentru studii de inginerie generale.Acest lucru a condus muli oameni s caute expresii simple, care vor aducerezultate satisfctoare, chiar dac acestea nu sunt, teoretic exacte. Cele maifrecvente dintre aceste expresii simple este legea de putere, exprimat astfel

n aceast ecuaie, z1 este luat de obicei ca nlimea de msurare, aproximativ10 m, iar Z2 este nlimea la care viteza vntului este dorita a fiestimata.Parametrul este determinat empiric.Ecuaia se poate folosii pentruobservatii ale vitezei vantului in intervalul de la 10 la probabil, 100 sau 150 m, ncazul nu exist limite sau impedimente in miscarea vantului


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Exponentul variaz n funcie de nlime, de momentul zilei, de anotimpulanului, natura terenului viteza vantului si temperatura . O serie de modele au fostpropuse pentru variaia cu aceste variabile [. Vom utiliza parcela liniarlogaritmic n Fig. 14. Aceast cifr arat o parcela pentru zi si alta pentru noapte,fiecare variind cu viteza vntului n conformitate cu ecuaia

Coeficien ii a i b pot fi determinate printr-un program de regresie liniar. Valorile tipice ale unei b i sunt 0.11 i 0.061 n timpul zilei i 0,38 i 0.209 pe timp de noapte.

Figura 14: Varia ia vantului profil exponent cu viteza vntului de referin u (z1).

Valoarea medie a a fost determinat prin msurtori multe in ntreaga lume si s-a constatat ca aceasta este aproximativ o eptime. (Ecuaia. 13, legea celei de-a a7-a parti). Aceast valoare medie ar trebui s fie utilizate numai n cazul n care

datele anumit site nu sunt disponibile, din cauza gamei largi de valori pe care poate asuma.

6 Statistici ale vitezei vantului


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The speed of the wind is continuously changing, making it desirable to describe the wind bystatistical methods. We shall pause here to examine a few of the basic concepts of probabilityand statistics, leaving a more detailed treatment to the many books written on the subject.

One statistical quantity which we have mentioned earlier is the average or arithmetic mean. Ifwe have a set of numbers ui, such as a set of measured wind speeds, the mean of the set isdefined as

The sample size or the number of measured values is n.

Another quantity seen occasionally in the literature is the median. If n is odd, the median is themiddle number after all the numbers have been arranged in order of size. As many numbers lie

below the median as above it. If n is even the median is halfway between the two middlenumbers when we rank the numbers.

In addition to the mean, we are interested in the variability of the set of numbers. We want tofind the discrepancy or deviation of each number from the mean and then find some sort ofaverage of these deviations. The mean of the deviations ui - u is zero, which does not tell usmuch. We therefore square each deviation to get all positive quantities. The variance 2 of thedata is then defined as

The term n - 1 is used rather than n for theoretical reasons we shall not discuss here.

The standard deviation is then defined as the square root of the variance.

Example

Five measured wind speeds are 2,4,7,8, and 9 m/s. Find the mean, the variance, and thestandard deviation.


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Wind speeds are normally measured in integer values, so that each integer value is observedmany times during a year of observations. The numbers of observations of a specific windspeed ui will be defined as mi. The mean is then

where w is the number of different values of wind speed observed and n is still the total numberof observations.

It can be shown that the variance is given by

The two terms inside the brackets are nearly equal to each other so full precision needs to bemaintained during the computation. This is not difficult with most of the hand calculators that areavailable.

Example

A wind data acquisition system located in the tradewinds on the northeast coast of Puerto Ricomeasures 6 m/s 19 times, 7 m/s 54 times, and 8 m/s 42 times during a given period. Find themean, variance, and standard deviation.


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We shall now define the probability p of the discrete wind speed ui being observed as

By this definition, the probability of an 8 m/s wind speed being observed in the previousexample would be 42/115 = 0.365. With this definition, the sum of all probabilities will be unity.

Note that we are using the same symbol p for both pressure and probability. Hopefully, the

context will be clear enough that this will not be too confusing.We shall also define a cumulative distribution function F(ui) as the probability that a measured

wind speed will be less than or equal to ui.

The cumulative distribution function has the properties

Example

A set of measured wind speeds is given in Table 2.2. Find p(u i) and F(ui) for each speed. Thetotal number of observations is n = 211.


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The values of p(ui) and F(ui) are computed from Eqs. 20 and 22 and tabulated in the table.

We also occasionally need a probability that the wind speed is between certain values orabove a certain value. We shall name this probability P(ua u ub) where ub may be a verylarge number. It is defined as

For example, the probability P(5 u ) that the wind speed is 5 m/s or greater in the previousexample is 0.242 + 0.128 = 0.370.

It is convenient for a number of theoretical reasons to model the wind speed frequency curveby a continuous mathematical function rather than a table of discrete values. When we do this,

the probability values p(ui) become a density function f(u). The density function f(u) representsthe probability that the wind speed is in a 1 m/s interval centered on u. The discrete probabilitiesp(ui) have the same meaning if they were computed from data collected at 1 m/s intervals. Thearea under the density function is unity, which is shown by the integral equivalent of Eq. 21.

The cumulative distribution function F(u) is given by


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Both of the above integrations start at zero because the wind speed cannot be negative. Whenthe wind speed is considered as a continuous random variable, the cumulative distributionfunction has the properties F(0) = 0 and F(~) = 1. The quantity F(0) will not necessarily be zeroin the discrete case because there will normally be some zero wind speeds measured which areincluded in F(0). In the continuous case, however, F(0) is the integral of Eq. 26 with integration

limits both starting and ending at zero. Since f(u) is a well behaved function at u = 0, theintegration has to yield a result of zero. This is a minor technical point which should not causeany difficulties later.

We will sometimes need the inverse of Eq. 26 for computational purposes. This is given by

The general relationship between f(u) and F(u) is sketched in Fig. 15. F(u) starts at 0, changesmost rapidly at the peak of f(u), and approaches 1 asymptotically.

The mean value of the density function f(u) is given by

The variance is given by

These equations are used to compute theoretical values of mean and variance for a widevariety of statistical functions that are used in various applications.

7 WEIBULL STATISTICS

There are several density functions which can be used to describe the wind speed frequencycurve. The two most common are the Weibull and the Rayleigh functions. For the statisticallyinclined reader, the Weibull is a special case of the Pearson Type III or generalized gammadistribution, while the Rayleigh [or chi with two degrees of freedom(chi-2)] distribution is asubset of the Weibull. The Weibull is a two parameter distribution while the Rayleigh has onlyone parameter. This makes the Weibull somewhat more versatile and the Rayleigh somewhatsimpler to use. We shall present the Weibull distribution first.


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Figure 15: General relationship between (a) a density function f(u) and (b) a distribution

function F(u).

The wind speed u is distributed as the Weibull distribution if its probability density

function is

We are using the expression exp(x) to represent ex.


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This is a two parameter distribution where c and k are the scale parameter and the shape

parameter, respectively. Curves of f(u) are given in Fig. 16, for the scale parameter c = 1. It

can be seen that the Weibull density function gets relatively more narrow and more peaked

as k gets larger. The peak also moves in the direction of higher wind speeds as k increases.

A comparison of Figs. 16 and 7 shows that the Weibull has generally the right shape to fit windspeed frequency curves, at least for these two locations.

If c is different from unity, the values of the vertical axis have to divided by c, as seen by Eq.30. Since one of the properties of a probability density function is that the area under the curvehas to be unity, then as the curve is compressed vertically, it has to expand horizontally. For c =10, the peak value of the k = 2.0 curve is only 0.0858 but this occurs at a speed u of 7 ratherthan 0.7. If the new curve were graphed with vertical increments of 1/10 those of Fig. 16 andhorizontal increments of 10 times, the new curve would have the same appearance as the one

in Fig. 16. Therefore this figure may be used for any value of c with the appropriate scaling.A possible problem in fitting data is that the Weibull density function is defined for all values of

u for 0 u whereas the actual number of observations will be zero above some maximumwind speed. Fitting a nonzero function to zero data can be difficult. Again, this is not normally aserious problem because f(u) goes to zero for all practical purposes for u/c greater than 2 or 3,depending on the value of k. Both ends of the curve have to receive special attention becauseof these possible problems, as will be seen later.

The mean wind speed computed from Eq. 28 is

If we make the change of variable

then the mean wind speed can be written.


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Figure 16: Weibull density function f(u) for scale parameter c = 1.

This is a complicated expression which may not look very familiar to us. However, it is basicallyin the form of another mathematical function, the gamma function. Tables of values exist for thegamma function and it is also implemented in the large mathematical software packages justlike trigonometric and exponential functions. The gamma function , (y), is usually written in theform

Equations 33 and 34 have the same integrand if y = 1 + 1/k. Therefore the mean wind speed is

Published tables that are available for the gamma function (y) are only given for 1 y 2.


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If an argument y lies outside this range, the recursive relation

must be used. If y is an integer,

The factorial y! is implemented on the more powerful hand calculators. The argument y

is not restricted to an integer, so the quantity computed is actually (y + 1). This may be

the most convenient way of calculating the gamma function in many situations.

Normally, the wind data collected at a site will be used to directly calculate the mean

speed Pu. We then want to find c and k from the data. A good estimate for c can be obtained

quickly from Eq. 35 by considering the function c/Pu as a function of k which is given in Fig.17.

For values of k below unity, the ratio c/Pu decreases rapidly. For k above 1.5 and less than 3

or 4, however, the ratio c/Pu is essentially a constant, with a value of about 1.12. This means

that the scale parameter is directly proportional to the mean wind speed for this range of k.

Most good wind regimes will have the shape parameterkin this range, so this estimate

ofcin terms of u will have wide application.

It can be shown by substitution that the Weibull distribution function F(u) which satisfies

Eq. 27, and also meets the other requirements of a distribution function, i.e. F(0) = 0and

F() = 1, is

The variance of the Weibull density function can be shown to be


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The probability of the wind speed u being equal to or greater than u a is

Figure 17: Weibull scale parameter c divided by mean wind speed u versus Weibull shapeparameter k

The probability of the wind speed being within a 1 m/s interval centered on the wind speed ua is


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Example

The Weibull parameters at a given site are c = 6 m/s and k = 1.8. Estimate the number ofhours per year that the wind speed will be between 6.5 and 7.5 m/s. Estimate the number ofhours per year that the wind speed is greater than or equal to 15 m/s.

From Eq. 42, the probability that the wind is between 6.5 and 7.5 m/s is just f(7), which can beevaluated from Eq. 30 as

This means that the wind speed will be in this interval 9.07 % of the time, so the number ofhours per year with wind speeds in this interval would be

(0.0907) (8760) = 794 h/year

From Eq. 41, the probability that the wind speed is greater than or equal to 15 m/s is

which represents

(0.0055) (8760) = 48 h/year

If a particular wind turbine had to be shut down in wind speeds above 15 m/s, about 2 days peryear of operating time would be lost.

We shall see in Chapter 4 that the power in the wind passing through an area A perpendicularto the wind is given by


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The average power in the wind is then

We may think of this value as the true or actual power in the wind if the probabilities p(ui) aredetermined from the actual wind speed data.

If we model the actual wind data by a probability density function f(u), then the average powerin the wind is

It can be shown[13] that when f(u) is the Weibull density function, the average power is

If the Weibull density function fits the actual wind data exactly, then the power in the windpredicted by Eq. 46 will be the same as that predicted by Eq. 44. The greater the differencebetween the values obtained from these two equations, the poorer is the fit of the Weibull

density function to the actual data.The function u3f(u) starts at zero at u = 0, reaches a peak value at some wind speed ume, and

finally returns to zero at large values of u. The yearly energy production at wind speed ui is thepower in the wind times the fraction of time that power is observed times the number of hours inthe year. The wind speed ume is the speed which produces more energy (the product of powerand time) than any other wind speed. Therefore, the maximum energy obtained from any onewind speed is

The turbine should be designed so this wind speed with maximum energy content is includedin its best operating wind speed range. Some applications will even require the turbine to bedesigned with a rated wind speed equal to this maximum energy wind speed. We therefore wantto find the wind speed ume. This can be found by multiplying Eq. eq:2.30 by u3, setting thederivative equal to zero, and solving for u. After a moderate amount of algebra the result can beshown to be


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We see that ume is greater than c so it will therefore be greater than the mean speed .If the

mean speed is 6 m/s, then ume will typically be about 8 or 9 m/s.

We see that a number of interesting results can be obtained by modeling the wind speedhistogram by a Weibull density function. Other results applicable to the power output of windturbines are developed in Chapter 4.

8 DETERMINING THE WEIBULL PARAMETERS

There are several methods available for determining the Weibull parameters c and k [13]. If the

mean and variance of the wind speed are known, then Eqs. 35 and 40 can be solved for c and kdirectly. At first glance, this would seem impossible because k is buried in the argument of agamma function. However, Justus[13] has determined that an acceptable approximation for kfrom Eq. 40 is

This is a reasonably good approximation over the range 1 ~k~ 10. Once k has beendetermined, Eq. 35 can be solved for c.

The variance of a histogram of wind speeds is not difficult to find from Eq. 19, so this methodyields the parameters c and k rather easily. The method can even be used when the variance isnot known, by simply estimating k. Justus examined the wind speed distributions at 140 sitesacross the continental United States measured at heights near 10 m, and found that k appearsto be proportional to the square root of the mean wind speed.

The proportionality constant d1 is a site specific constant with an average value of 0.94

when the mean wind speed is given in meters per second. The constant d1 is between 0.73

and 1.05 for 80 % of the sites. The average value of d1 is normally adequate for wind power


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calculations, but if more accuracy is desired, several months of wind data can be collected and

analyzed in more detail to compute c and k. These values of k can be plotted versus

on log-log paper, a line drawn through the points, and d1 determined from the slope of the line.

Another method of determining c and k which lends itself to computer analysis, is the least

squares approximation to a straight line. That is, we perform the necessary mathematical errorbetween the linearized ideal curve and the actual data points of p(ui). The process is somewhatof an art and there may be more than one procedure which will yield a satisfactory result.Whether the result is satisfactory or not has to be judged by the agreement between the Weibullcurve and the raw data, particularly as it is used in wind power computations.

The first step of linearization is to integrate Eq. 27. This yields the distribution function F(u)which is given by Eq. 39. As can be seen in Fig. 15, F(u) is more nearly describable by astraight line than f(u), but is still quite nonlinear. We note that F(u) contains an exponential and

that, in general, exponentials are linearized by taking the logarithm. In this case, because theexponent is itself raised to a power, we must take logarithms twice.

This is in the form of an equation of a straight line

where x and y are variables, a is the slope, and b is the intercept of the line on the y axis. Inparticular,

Data will be expressed in the form of pairs of values of ui and F(ui). For each wind speed ui

there is a corresponding value of the cumulative distribution function F(ui). When given valuesfor u = ui and F(u) = F(ui) we can find values for x = xi and y = yi in Eqs. 55. Being actual data,these pairs of values do not fall exactly on a straight line, of course. The idea is to determine thevalues of a and b in Eq. 53 such that a straight line drawn through these points has the bestpossible fit. It can be shown that the proper values for a and b are


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In these equations x and y are the mean values of xi and yi, and w is the total number of pairsof values available. The final results for the Weibull parameters are

One of the implied assumptions of the above process is that each pair of data points is equallylikely to occur and therefore would have the same weight in determining the equation of the line.For typical wind data, this means that one reading per year at 20 m/s has the same weight as100 readings per year at 5 m/s. To remedy this situation and assure that we have the bestpossible fit through the range of most common wind speeds, it is possible to redefine a weightedcoefficient a in place of Eq. 55 as

Example

The actual wind data for Kansas City and Dodge City for the year 1970 are given in Table 2.3.The wind speed ui is given in knots. Calm includes 0 and 1 knot because 2 knots are required tospin the anemometer enough to give a non zero reading. The parameter xi is ln(ui) as given in

Eq. 55. The number mi is the number of readings taken during that year at each wind speed.The total number of readings n at each site was 2912 because readings were taken every threehours. The function p(ui) is the measured probability of each wind speed at each site as givenby Eq. 20. Compute the Weibull parameters c and k using the linearization method.


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We first use Eq. 22 to compute F(u i) for each ui and Eqs 55 to compute yi for each F(ui). Theseare listed in Table 2.3 as well as the original data.

We are now ready to use Eqs. 58 and 56 to find a and b. First, however, we plot the point pairs(xi, yi) for each ui as shown in Fig. 18 for both sites. The readings for calm (0 and 1 knot) areassumed to be at 1 knot so that xi = ln u i is zero rather than negative infinity.

Placing a straight edge along the sets of points shows the points to be in reasonable alignmentexcept for calm and 2 knots for Kansas City, and calm for Dodge City. As mentioned earlier, thegoal is to describe the data mathematically over the most common wind speeds. The Weibull

function is zero for wind speed u equal to zero (if k > 1) so the Weibull cannot describe calms.Therefore, it is desirable to ignore calms and perhaps 2 knots in order to get the best fit over thewind speeds of greater interest.


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Figure 18: y versus x for Kansas City and Dodge City, 1970

9 RAYLEIGH AND NORMAL DISTRIBUTIONS

We now return to a discussion of the other popular probability density function in wind powerstudies, the Rayleigh or chi-2. The Rayleigh probability density function is given by

The Rayleigh cumulative distribution function is


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The probability that the wind speed u is greater than or equal to u a is just

The variance of this density function is

Wind power density as computed from Eq. 44 for standard conditions is Pw/A = 396 W/m2. The

power density computed from the Weibull model, Eq. 46, is 467 W/m2, while the power densitycomputed from the Rayleigh density function by a process similar to Eq. 44 is 565 W/m2. TheWeibull is 18 % high while the Rayleigh is 43 % high.

Although the actual wind speed distribution can be described by either a Weibull or a Rayleighdensity function, there are other quantities which are better described by a normal distribution.The distribution of monthly or yearly mean speeds is likely to be normally distributed around along-term mean wind speed, for example. The normal curve is certainly the best known andmost widely used distribution for a continuous random variable, so we shall mention a few of itsproperties.

The density function f(u) of a normal distribution is

Where is the mean and is the standard deviation. In this expression, the variable u isallowed to vary from - to +. It is physically impossible for a wind speed to be negative, ofcourse, so we cannot forget the reality of the observed quantity and follow the mathematicalmodel past this point. This will not usually present any difficulty in examining mean wind speeds.

The cumulative distribution function F(u) is given by


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This integral does not have a simple closed form solution, so tables of values are determinedfrom approximate integration methods. The variable in this table is usually defined as

Thus q is the number of standard deviations that u is away from . A brief version of this tableis shown in Table 2.4. We see from the table, for example, that F(u) for a wind speed onestandard deviation below the mean is 0.159. This means there is a 15.9 % probability that themean speed for any period of interest will be more than one standard deviation below the longterm mean. Since the normal density function is symmetrical, there is also a 15.9 % probabilitythat the mean speed for some period will be more than one standard deviation above the longterm mean.

It is also of interest to know the probability of a given data point being within a certain distanceof the mean. A few of these probabilities are shown in Table 2.5. For example, the fraction0.6827 of all measured values fall within one standard deviation of the mean if they are normallydistributed. Also, 95 % of all measured values fall within 1.96 standard deviations of the mean.

We can define the em 90 percent confidence interval as the range within 1.645 standard

deviations of the mean. If the mean is 10 m/s and the standard deviation happens to be 1 m/s,then the 90 % confidence interval will be between 8.355 and 11.645 m/s. That is, 90 % ofindividual values would be expected to lie in this interval.


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Example

The monthly mean wind speeds at Dodge City for 1958 were 11.78, 13.66, 11.16, 12.94,12.10, 13.47, 12.56, 10.86, 13.77, 11.76, 12.44, and 12.55 knots. Find the yearly mean

(assuming all months have the same number of days), the standard deviation, and the windspeeds one and two standard deviations from the mean. What monthly mean will be exceeded95 % of the time? What is the 90 % confidence interval?

By a hand held calculator, we find

= 12.42

= 0.94

The wind speeds one standard deviation from the mean are 11.48 and 13.36 knots, while thespeeds two standard deviations from the mean are 10.54 and 14.30 knots. From Table 2.4 wesee that F(u) = 0.05 (indicating 95 % of the values are larger) for q = -1.645. From Eq. 65 wefind

u = 12.42 + (-1.645) (0.94) = 10.87 knots

Based on this one years data we can say that the monthly mean wind speed at Dodge Cityshould exceed 10.87 knots (5.59 m/s) for 95 % of all months.

The 90 % confidence interval is given by the interval 12.42 1.645(0.94) or between 10.87and 13.97 knots. We would expect from this analysis that 9 out of 10 monthly means would bein this interval. In examining the original data set, we find that only one month out of 12 isoutside the interval, and it is just barely outside. This type of result is rather typical with suchsmall data sets. If we considered a much larger data set such as a 40 year period with 480monthly means, then we could expect approximately 48 months to actually fall outside this 90 %confidence interval.

The monthly means were distributed around the long term measured monthly mean m


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with an average standard deviation of 0.098 m where m is the mean wind speed for a givenmonth of the year, e.g. all the April average wind speeds are averaged over the entire periodof observation to get a long term average for that month. For a normal distribution the 90 %confidence interval would be, using Table 2.5, m /1.645(0.098) m or the interval between0.84 m and 1.16m. We can therefore say that we have 90 % confidence that a measuredmonthly mean speed will fall in the interval 0.84 m to 1.16 m. If we say that each measuredmonthly mean lies in this interval, we will be correct 90 times out of 100, and wrong 10 times.

The above argument applies on the average. That is, it is valid for sites with an averagestandard deviation. We cannot be as confident of sites with more variable winds and hencehigher standard deviations. Since we do not know the standard deviation of the wind speed atthe candidate site, we have to allow for the possibility of it being a larger number. According toJustus [14], 90 % of all observed monthly standard deviations were less than 0. 145um. This isthe 90 percentile level. The appropriate interval is now um 1.645(0.145)um or the intervalbetween 0.76um and 1.24um. That is, any single monthly mean speed at this highly variablesite will fall within the interval 0.76um and 1.24um with 90 % confidence. The converse is also

true. That is, if we designate the mean for one month as um1, the unknown long term monthlymean um will fall within the interval 0.76um1 and 1.24um1 with 90 % confidence. We shalltry to clarify this statement with Fig. 22.

Figure 22: Confidence intervals for two measured monthly means, m1 and m2 .

Suppose that we measure the average monthly wind speed for April during two successive

years and call these values m1 and m2. These are shown in Fig. 22 a long with the intervals


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0.76m1to 1.24m1, 0.76 m2 to 1.24 m2, and 0.76m to 1.24m.The confidence interval centeredon um contains 90 % of measured means for individual months. The mean speed is m1 belowm but its confidence interval includes m.The mean speed m2 is outside the confidence intervalform and the reverse is also true. If we say that the true mean m is between 0.76 m2 and 1.24m2, we will be wrong. But since only 10 % of the individual monthly means are outside theconfidence interval form, we will be wrong only 10 % of the time. Therefore we can say with 90% confidence that mi lies in the range of 0.76mito 1.24miwhere mi is some measuredmonthly mean.

Example

The monthly mean speeds for April in Dodge City for the years 1948 through 1973 are: 15.95,12.88, 13.15, 14.10, 13.31, 15.14, 14.82, 15.58, 13.95, 14.76, 12.94, 15.03, 16.57, 13.88, 10.49,11.30, 13.80, 10.99, 12.47, 14.40, 13.18, 12.04, 12.52, 12.05, 12.62, and 13.15 knots. Find the

mean m and the normalized standard deviation m/m. Find the 90 % confidence intervals form, using the 50 percentile standard deviation (m/m= 0.098), the 90 percentile standarddeviation (m/m= 0.145), and the actual m/m. How many years fall outside each of theseconfidence intervals? Also find the confidence intervals for the best and worst months using

m/m = 0.145.

From a hand held calculator, the mean speed m= 13.50 knots and the standard deviation m =1.52 knots are calculated. The normalized standard deviation is then

This is above the average, indicating that Dodge City has rather variable winds.

The 90 % confidence interval using the 50 percentile standard deviation is between 0.84m and1.16m, or between 11.34 and 15.66 knots. Of the actual data, 2 months have means above thisinterval, with 21 or 81 % of the monthly means inside the interval.

The 90 % confidence interval using the nationwide 90 percentile standard deviation is between

0.76m and 1.24m, or between 10.26 and 16.74 knots. All the monthly means fall within thisinterval.

The 90 % confidence interval using the actual standard deviation of the site is between the limitsm 1.645(0.113) mor between 10.99 and 16.01 knots. Of the actual data, 24 months or 92 %of the data points fall within this interval.

If only the best month was measured, the 90 % confidence interval using the 90 percentilestandard deviation would be 16.57 (0.24)16.57 or between 12.59 and 20.55 knots. Thecorresponding interval for the worst month would be 10.49 (0.24)10.49 or between 7.97 and


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13.01 knots. The latter case is the only monthly confidence interval which does not contain thelong term mean of 13.5 knots. This shows that we can make confident estimates of the possiblerange of wind speeds from one months data without undue concern about the possibility of thatone month being an extreme value.

The wind speed at a given site will vary seasonally because of differences in large scaleweather patterns and also because of the local terrain at the site. Winds from one direction mayexperience an increase in speed because of the shape of the hills nearby, and may experiencea decrease from another direction. It is therefore advisable to make measurements of speed

over a full yearly cycle. We then get a yearly mean speed y. Each yearly mean speed y willbe approximately normally distributed around the long term mean speed . Justus found thatthe 90 % confidence interval for the yearly means was between 0.9 and 1.1 for the median(50 percentile) standard deviation and between 0.85 and 1.15 for the 90 percentile standarddeviation. As expected, these are narrower confidence intervals than were observed for thesingle monthly mean.

Example

The yearly mean speed at Dodge City was 11.44 knots at 7 m above the ground level in 1973.

Assume a 50 percentile standard deviation and determine the 90 % confidence interval for thetrue long term mean speed.

The confidence interval extends from 11.44/1.1 = 10.40 to 11.44/0.9 = 12.71 knots. We cantherefore say that the long term mean speed lies between 10.40 and 12.71 knots with 90 %confidence. Our estimate of the long term mean speed by one years data can conceivably beimproved by comparing our one year mean speed to that of a nearby National Weather Service

station. If the yearly mean speed at the NWS station was higher than the long term mean speedthere, the measured mean speed at the candidate site can be adjusted upward by the samefactor. This assumes that all the winds within a geographical region of similar topography and adiameter up to a few hundred kilometers will have similar year to year variations. If the long term

mean speed at the NWS station is , the mean for one year is a, and the mean for the sameyear at the candidate site is b, then the corrected or estimated long term mean speed cat thecandidate site is

Note that we do not know how to assign a confidence interval to this estimate. It is a single

number whose accuracy depends on both the accuracy of and the correlation between a andb. The accuracy of should be reasonably good after 30 or more years of measurements, as iscommon in many NWS stations. However, the assumed correlation between a and b may notbe very good. Justu found that the correlation was poor enough that the estimate of long term


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means was not improved by using data from nearby stations. Equally good results would beobtained by applying the 90 % confidence interval approach as compared to using Eq. 66. Thereason for this phenomenon was not determined. One possibility is that the type of anemometerused by the National Weather Service can easily get dirty and yield results that are low by 10 to20 % until the next maintenance period. One NWS station may have a few months of low

readings one year while another NWS station may have a few months of low readings the nextyear, due to the measuring equipment rather than the wind. This would make a correction likeEq. 66 very difficult to use. If this is the problem, it can be reduced by using an average ofseveral NWS stations and, of course, by more frequent maintenance. Other studies arenecessary to clarify this situation.

The actual correlation between two sites can be defined in terms of a correlation coefficient r,where

In this expression,a is the long term mean speed at site a, bis the long term mean speed atsite b,bi is the observed monthly or yearly mean at site b, and w is the number of months oryears being examined. We assume that the wind speed at site b is linearly related to the speed

at site a. We can then plot each mean speed biversus the corresponding ai. We then find thebest straight line through this cluster of points by a least squares or linear regression process.

The correlation coefficient then describes how closely our data fits this straight line. Its valuecan range from r = +1 to r = -1. At r = +1, the data falls exactly onto a straight line with positiveslope, while at r = -1, the data falls exactly onto a straight line with negative slope. At r = 0, thedata cannot be approximated at all by a straight line.

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