STATISTICAL ANALYSIS OF WIND SPEED DATA

Transcription

1 Esşehr Osmagaz Üerstes Müh.Mm.Fa.Dergs C. XVIII, S.2, 2005 Eg.&Arh.Fa. Esşehr Osmagaz Uersty, Vol. XVIII, No: 2, 2005 STATISTICAL ANALYSIS OF WIND SPEED DATA Veysel YILMAZ, Haydar ARAS 2, H.Eray ÇELİK 3 ABSTRACT : Wd speed s the most mportat parameter the desg ad study of wd eergy oerso dees. The eergy whh s obtaed from wd s dretly proportoal wth the ub power of the wd speed. As the wd speed reases, the ost of the wd eergy s redued. I may studes lterature, t s assumed that the probablty dstrbuto related to wd speeds a be desrbed by Webull dstrbuto, ad t s aepted so wthout ay statstal examato. I ths study, the theoretal dstrbutos of wd potetals ft to Webull dstrbuto for fe dfferet topograph stuatos from Turey Wd Atlas are estgated ad reported. KEYWORDS : Wd Eergy; Webull Dstrbuto; Statstal Tests. RÜZGAR HIZ VERİLERİNİN İSTATİSTİKSEL ANALİZİ ÖZET : Rüzgar eerjs döüşüm sstemlerde rüzgar hızı e öeml parametrelerde brdr. Rüzgarda elde edle eerj, rüzgar hızıı üpüyle doğruda oratılıdır. Rüzgar hızı arttıça rüzgar eerjs malyetler azalmatadır. Lteratürde pe ço çalışmada rüzgar hızı olasılı dağılımları hç br statst test yapılmasızı Webull dağılımı olara taımlamatadır. Bu çalışmada Türye Rüzgar atlasıda erle beş farlı stasyoda rüzgar hız erler teor dağılımıı Webull a uyup uymadığı araştırılmıştır. ANAHTAR KELİMELER : Rüzgar Eerjs; Webull Dağılımı; İstatstsel Testler.,3 Esşehr Osmagaz Üerstes,Fe Edebyat Faültes,İstatst Bölümü,26480 ESKİŞEHİR 2 Esşehr Osmagaz Üerstes, Mühedsl Mmarlı Faültes, Maa Mühedslğ Bölümü, Batı Meşel, ESKİŞEHİR

2 I.INTRODUCTION As the world populato grows rapdly, the eergy demad reases proportoally. I spte of reasg eergy demad, the lmted reseres of fossl fuel, good qualty eergy, ad produte usage of the produed eergy beomes a mportat matter all outres espeally deelopg outres suh as Turey []. Turey has a osderably hgh leel of reewable eergy resoures that a be utlzed to satsfy a porto of the total eergy demad [4]. I ths study, statstal estmato tehques whh are used estmato of Webull dstrbuto parameters are quoted, the the wd speed dstrbuto of fe dfferet regos ftted to Webull dstrbuto was examed usg Ch-Square ad Kolmogoro-Smro tests. Geographal oordates for fe dfferet regos are ge Table. The data for ths study was obtaed from Turey wd atlas publshed by the Geeral Dretorate of Tursh State Meteorologal Sere ad the Geeral Dretorate of Eletral Power Resoures Surey Admstrato. Table : Coordate data for the fe stato Stato Stato Coordates Heght of the sea Aemograph heght (Degree-Mute-Seod leel Afyo ı 59 ıı N ı 38 ıı E 034m 3m Va ı 4 ıı N ı 42 ıı E 66m 0m Sop ı 5 ıı N ı 8 ıı E 32m 0m Bozaada ı 00 ıı N ı 25 ıı E 28m 0m Slfe ı 58 ıı N ı 9 ıı E 5m 0m Webull dstrbuto wth two parameters s usually the used for probablty dstrbuto of wd speeds. It s geerally aepted that measured wd data a be best haraterzed by Webull dstrbuto [3, 6-, 3]. But most studes fttg of data set to Webull dstrbuto was ot examed. To determe f the frequey seres ft to a theoretal dstrbuto or whh dstrbuto fts better, a oly be deded by statstal hypothess tests.

3 II. METHODS FOR ESTIMATING THE PARAMETERS OF THE WEIBULL DISTRUBITION The wd speed probablty desty futo a be alulated as f ( exp ( where f( s the probablty of obserg wd speed, s the Webull sale parameter ad s the Webull shape parameter. Basally, the sale parameter,, dates how wdy a wd loato uder osderato s, whereas the shape parameter,, dates how peaed the wd dstrbuto s [3, 0, ]. Statstally the parameters of Webull dstrbuto a be dered by usg arous estmato tehques. Amog those, some of the most wdely used tehques are; Maxmum Lelhood Estmato (MLE, Method of Momets (MOM ad, Least-Squares Method (LMS related wth graphal tehque [4]. II.. The Maxmum Lelhood Method Maxmum Lelhood Tehque, wth may requred features s the most wdely used tehque amog parameter estmato tehques. The MLE method has may large sample propertes that mae t attrate for use. It s asymptotally osstet, whh meas that as the sample sze gets larger, the estmates oerge to the true alues. It s asymptotally effet, whh meas that for large samples, t produes the most prese estmates. It s also asymptotally ubased, whh meas that for large samples, oe expets to get the true alue o aerage. The estmates themseles are ormally dstrbuted f the sample s large eough. These are all exellet large sample propertes. Lelhood futo for Webull dstrbuto wth two parameters s as follows [0, 2], L(, f ( ; θ (2 ad usg Eq ( Eq (2, we get

4 L exp(, ( (3 If we tae the atural logarthm of the lelhood futo ge by Eq (3 ( (, ( + I I I IL (4 or + I I I IL (, ( (5 s obtaed. The tag the derate of the lelhood futo (Eq.(5 wth respet to ad, ad set them to zero. 0, ( ( + + IL (6 ad 0 (l l l, ( l + L β (7 The shape fator ad the sale fator are estmated usg the followg two equatos: l l l (8 ad

5 ( ( l l (9 where s the wd speed tme step ad the umber of ozero wd speed data pots. Eq. (9 must be soled usg terate proedure, after whh Eq. (0 a be soled expltly. Care must be tae to apply Eq. (9 oly to the ozero wd speed data pots. (0 II.2. Least Squares Method Ths method s also alled Graphal Method. Wth the help of ths method the parameters are estmated wth the regresso le equato by usg umulate desty futo. The umulate desty futo of Webull dstrbuto wth two parameters a be wrtte as, F ( - exp (- - ( Ths futo a be arraged as, {- F ( } - exp ( - (2 If we tae the atural logarthm of Eq.(2, -l {- F ( } ( - (3 ad the retae the atural logarthm of Eq. (3, we get the followg equato:

6 l[ -l {- F( } ] -l + l (4 I Eq. (4 represets a dret relatoshp betwee l ad l [-l {- F ( }] whh should be mmzed. 2 {l[ l( F( ] l[ l( E( F( ]} (5 Parameters of Webull dstrbuto wth two parameters are estmated by mmzg wth Eq. (5. Parameters ad tersets by usg Eq. (6 ad (7 as follows, l l[ l{ F( }] l l[ l{ F( }] (6 2 l } 2 l { l l[ l{ F( }] exp{ } (7 II.3. Method of Momets Method of Momet s oe of the oldest of the estmato methods. If there s a j parameter to estmate, whe j uerse momet oeted wth these parameters s equated to orrelate sample momets, qualty j, whh otas these parameters, s aheed. Estmato alues of ths equato, obtaed o ths j umber, are foud by solg the uow parameters. I the estmato of the Webull Dstrbuto parameters wth the method of Momets the frst ad seod momets of dstrbuto aroud zero are used. The j th momet of the two-parametered Webull Dstrbuto aroud zero s as j j j E( V Γ( + (8 where Γ s the gamma futo.

7 Whe equalty umber (8 s used, the frst two momets aroud zero s as follows. E ( V Γ( + ( ad E ( V Γ( + (20 Oly oe futo whh depeds o the parameters of j s obtaed by the dso of the square of E(V to E(V 2, V Γ( + (2 ad 2 { } 2 V { Γ( + } 2 2 V Γ( + Parameter s estmated by usg Eq. (22 wth stadard terate tehque [6]. The parameter s estmated by plag Eq. (23. V (23 Γ( + (22 III. RESULTS & DISCUSSION I ths study to eable the probablty dstrbuto whh estmates the potetals of the wd speed were determed for Afyo, Va, Sop, Bozaada, Slfe. Wth ths purpose, wd speeds of fe dfferet regos Turey Wd Atlas hae bee reorded for ad ts approprateess to Webull dstrbuto has bee studed wth the help of statstal hypothess test. p-alues are ofte used hypothess tests where you ether aept or rejet a ull hypothess. The p-alue represets the probablty of mag a Type error, or rejetg the ull hypothess whe t s true. A utoff alue ofte used s 0.05 that s, rejet the ull hypothess whe the p-alue s less tha I may areas of researh, the p-leel of 0.05 s ustomarly treated as a "border-le aeptable" error leel. Aordg to results of statstal test, for the obtaed data from

8 statos Afyo ad Sop are approprate theoretal models for Webull dstrbuto (Null hypothess aept beause p>α ad D< Dα ; α0.05. But for the other statos, t the test yelded ot approprate (Null hypothess rejet beause p<α ad D > Dα ; α0.05. These results exposes that ot all wd speed data fts Webull dstrbuto. Beause of ths, t s ery mportat to exame the wd speed dstrbuto wth statstal tests before mag estmets to a rego just loog at the wd speed potetals. Otherwse suh a estmet would ot be a feasble oe. If the dstrbuto of wd speed was ot determed orretly, obtaed eergy amout wll ot be as expeted. Thus, t s ery mportat to determe the probablty dstrbuto of wd speed potetal orretly. Parameter estmatos were doe by usg data set related to frequey dstrbuto ad umulate frequey dstrbuto whh s ge Table 2 [5]. Table 2: Obsered wd speed data frequey dstrbuto ( O Speed Iteral(m/ s Afyo ( Va ( Statos Sop ( Bozaada ( Slfe ( < > The results of the aalyss for parameter estmato alues related dstrbuto are ge Table 3. Whe Table 3 examed, parameter alues obtaed by three

9 aalytal methods (Ra Regresso o X [RRX], Ra Regresso o Y [RRY], MOM ad MLE a be see. Table 3: Shape ad sale parameter estmato alues of Webull dstrbutos aordg to RRX, RRY, MLE ad MOM methods. Stato Parame ter Estmat o RRX Method RRY Method MLE Method MOM Method Afyo Va Sop Bozaa da Slfe Parameter estmatos were serted equato (4 by usg RRY. Estmated alues a be see Table 4. LSM, or least sum of squares, regresso requres that a straght le to be ftted to a set of data pots, suh that the sum of the squares the dstae of the pots to the ftted le s mmzed. Table 4: Estmated wd speed data frequey dstrbuto ( E Speed Iteral Statos (m/s Afyo Va Sop Bozaada Slfe < >

10 Ths mmzato a be performed ether the ertal or horzotal dreto. If the regresso s o the x-axs, the the le s ftted so that the horzotal deatos from the pots to the le are mmzed (RRX. If the regresso s o the y-axs, the the le s ftted so that the ertal deatos from the pots to the le are mmzed (RRY. The ra regresso estmato method s qute good for futos that a be learzed. As most of the dstrbutos used lfe data aalyss are apable of beg learzed. Further, ths tehque prodes a good measure of the goodess-of-ft of the hose dstrbuto. Therefore, ths study RRY method was used. For data sets otag large quattes of suspeded data pots, MLE may be the preferable form of aalyss. Obsered ad expeted wd speed alues of fe dfferet regos were ge Fg. a,b,,d,e. Whether the measured wd speed for these fe statos ftted well wth the Webull dstrbuto or ot, a be doe oly by usg statstal tests. Null hypothess, prog that statstally there s o sesble dfferee betwee obsered frequees ad expeted frequees, tested by Ch-Square ad Kolmogoro - Smro tests. Formulas of testg statsts used ths test are ge equatos 24 ad 25, 2 χ r ( O E 2 E (24 where O s the obsered frequey for eah lass, E s the frequey of eah lass estmated aordg to theoretal dstrbuto ad, r s the umber of the lass. Degree of freedom (df of Ch-Square df s, r--m, where m s the umber of parameters of oered theoretal dstrbuto. F S D max ( V ( V (25 x T Cumulate frequey of the dstrbuto obsered by Kolmogoro-Smro test based o the maxmum absolute dfferee betwee (F x (V ad umulate frequey of the estmated dstrbuto (S T (V. Maxmum absolute alue of D amog alulated D alues, are ompared wth table alues ( Dα at meag leel α whh were prepared for Kolmogoro-Smro test. If D> Dα, the hypothess sayg that data omes from estmated dstrbuto s rejeted.

11 Frequey Wd speed (m/s O(Afyo E(Afyo Frequey Wd speed (m/s O (Va E (Va Frequey Wd speed O (Sop E (Sop Frequey O (Bozaada E (Bozaada Wd speed (m/s Frequey O (Slfe E (Slfe W d speed (m/s Fgure. Obsered ad estmated wd speeds for fe statos ( O : Obsered frequey E : Estmated frequey The results of Ch-Square ad Kolmogoro-Smro tests are ge Table 5. Table 5: Test results of Ch-Square ad Kolmogoro-Smro Stato Ch-Square alue ad ts meagfuless (α0.05 Afyo 0.64 ; df 06 ; p Null hypothess aept (p>α Va ; df 2 ; p Null hypothess rejet (p<α Sop ; df 06 ; p Null hypothess aept (p>α Boza ; df 3 ; p ada Null hypothess rejet (p<α Slfe ; df 05 ; p Null hypothess rejet (p<α D e Dα 0.04 omparso(α <0.04 Null hypothess aept 0.04>0.04 Null hypothess rejet 0.04<0.04 Null hypothess aept 0.095>0.04 Null hypothess rejet 0.04>0.04 Null hypothess rejet

12 IV.CONCLUSIONS Nowadays, wd eergy s the most rapdly deelopg tehology ad eergy soure, ad t s reusable. Due to ts leaess ad reusablty, there hae bee rapd deelopmets made o trasferrg the wd eergy systems to eletr eergy systems. Wd eergy s a alterate lea eergy soure omparg to all other fossl orgated eergy soures whh pollute the lower parts of the atmosphere [2]. As problems le eromet polluto ad supplyg eergy eeds gets bgger, t s mportat to mae use of a polluto free eergy suh as wd. Coertg the wd eergy to more wdely used eletral eergy a be doe oly wth the wd turbes. The most mportat parameter of the power whh was obtaed from wd turbes s the wd speed. Due to ths fat, t s ery mportat to determe the probablty dstrbuto of wd speed alues of a area orretly whh wd eergy potetal wll be utlzed. The wd potetal of a rego depeds o arous wd measuremets of a loato for years. To determe the wd eergy potetal of a rego or feld a be determed by orretly estmatg the probablty dstrbuto. REFERENCES [] Aras H, Codto ad deelopmet of the ogeerato faltes based o autoproduto estmet model Turey. Reewable ad Sustaable Eergy Reews 2003; 7(6: [2] Aras H, Wd eergy status ad ts assessmet Turey. Reewable Eergy 2003; 28(4: [3] Boa S, Burlo R, Leoe C. Hourly wd speed aalyss Sly. Reewable Eergy 2003 ;28: [4] Che Zhem. Statstal feree about the shape parameter of the Webull Dstrbuto. Statsts ad Probablty Letters. 997; 36: [5] Düdar C, Cabaz M, Agü N, Ural G. Turey wd atlas. Publshed by the Geeral Dretorate of Tursh State Meteorologal Sere ad the Geeral Dretorate of Eletral Power Resoures Surey Admstrato, Aara, Turey, [ Tursh]

13 [6] Dorlo S.S. A. Estmatg wd speed dstrbuto. Eergy Coerso & Maagemet 2002; 43: [7] Gupta B.K. Webull parameters for aual ad mothly wd speed dstrbutos for fe loatos Ida. Solar Eergy 986; 37(6: [8] AL-Hasa M, Ngmatull R.R. Idefato of the geeralzed Webull dstrbuto wd speed data by the egeoordates method. Reewable Eergy 2003; 28: [9] Rehma S, HalawaT.O, Husa T. Webull parameters for wd speed dstrbuto Saud Araba. Solar Eergy 994;53(6: [0] Seguro J.V, Lambert T.W. Moder estmato of the parameters of the Webull wd speed dstrbuto for wd eergy aalyss. Joural of Wd Egeerg ad Idustral Aerodyams 2000; 85: [] Wesser D. A Wd eergy aalyss of Greada: a estmato usg the Webull desty futo. Reewable Eergy 2003; 28: [2] Xe M, Tag Y, Goh T.N. A modfed Webull exteso wth bathtub-shaped falure rate futo. Relablty Egeerg ad System Safety 2002;76: [3] Çel A.N. A statstal aalyss of wd power desty based o the Webull ad Raylegh models at the souther rego of Turey. Reewable Eergy 2004; 29 (4: [4] Hepbasl A, Ozgeer O. A reew o the deelopmet of wd eergy Turey. Reewable & Sustaable Eergy Reews 2004; 8(3: NOMENCLATURE F ( : Webull Cumulate Dstrbuto Futo D : Kolmogoro Smro test statsts χ 2 : Pearso Ch-Square test statsts E : Estmated frequey O : Obsered frequey : Webull Shape Parameter L (, : Lelhood futo : Wd speed (m/s

14 : Webull Sale Parameter (m/s : The Total umber of samples p : The p-alue represets the probablty of mag a Type error, or rejetg the ull hypothess whe t s true. f ( : Probablty desty futo m : Number of parameters V : Mea of wd speed Dα : Kolmogoro Smro theoretal test statsts α : Error leel -α : Cofdee leel r : Number of the Class