Comparative study of four methods for estimating Weibull parameters for Halabja, Iraq

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1 Iteratoal Joural of Physcal Sceces ol. 8(5), pp. 86-9, 9 February, 3 Avalable ole at DOI:.5897/IJPS.697 ISSN Academc Jourals Full Legth Research Paper Comparatve study of four methods for estmatg Webull parameters for Halabja, Iraq Salahadd A. Ahmed Departmet of Physcs, Faculty of Scece ad Scece Educato, School of Scece, Uversty of Sulama, Iraq. E-mal: salahaddahmed@gmal.com. Accepted 8 February, 3 The Webull dstrbuto s the stadard fucto used by the wd eergy commuty to model the wd speed frequecy dstrbuto. I ths study, four methods are preseted for estmatg Webull parameters (Shape ad Scale), amely, Maxmum lelhood method (MLM), Ra regresso method (RRM), Mea-stadard devato method (MSD), ad Power desty method (PDM). To compare the methods, a perod of 4 years ( - 4) of mothly tme seres data of Halabja cty was cosdered. Two dstct aalytcal methods are studed to determe the parameter estmato accuracy of these methods; coeffcet of determato ad root mea square error (RMSE) are used as measuremet tools. The Ra regresso ad MSDs are recommeded to estmate the shape parameter; also the Ra regresso s recommeded for use wth our tme seres wd data to estmate the scale parameter. Key words: Webull dstrbuto, parameter estmato, eergy patter factor, accuracy. INTRODUCTION Webull has bee recogzed as a approprate model relablty studes ad lfe testg problems such as tme to falure or lfe legth of a compoet or product. Over the years, estmato of the shape ad scale parameters for a Webull dstrbuto fucto has bee approached through Maxmum lelhood method (MLM), lear method, ad several versos of regresso aalyss. I recet years, Webull dstrbuto has bee oe of the most commoly used, accepted, recommeded dstrbuto to determe wd eergy potetal ad t s also used as a referece dstrbuto for commercal wd eergy softwares such as Wd Atlas Aalyss ad Applcato Program (WAsP). The two-parameter Webull dstrbuto fucto s commoly used to ft the wd speed frequecy dstrbuto. The preferred method of estmatg the Webull parameters was a graphcal way usg the cumulatve wd speed dstrbuto, plottg t o specal Webull graph paper. Estmato of the two-parameter Webull dstrbuto occurs may real-lfe problems. The Webull dstrbuto s a mportat model especally for relablty ad mataablty aalyss. Webull dstrbuto ca be used to model the wd speed dstrbuto at a partcular ste ad hece, t ca help wd resource assessmet of a ste. By calculatg the two parameters (shape ad scale) for Webull dstrbuto the wd speed frequecy curve for a ste ca be made (Prasad et al., 9) ad the ey to perform wd turbe ad wd farm eergy calculato. Several methods have bee proposed to estmate Webull parameters (Mars, 5; Rder, 96; Kao, 959; Pag et al., ; Padey et al., ; Seguro ad Lambert, ; Steves ad Smulders, 979; Bhattacharya ad Bhattacharjee, ). I lterature about wd eergy, these methods are compared several tmes ad dfferet ways (Adag ad Al, 9; Slva et al., 4; Ylmaz et al., 5; Gupta, 986; Rahma et al., 994; Le, 8; Katar ad Seoglu, 7), however, results ad coclusos of the prevous studes are dfferet. Several of ft tests are used lterature. A method for estmatg parameters of mxed dstrbutos usg sample momets has bee outled by Paul (96) who cosdered compoud Posso, bomal, ad a specal case of the mxed Webull dstrbuto. A graphcal method for estmatg the mxed Webull parameters lfe testg of electro tubes s proposed by Joh (959). For these reasos, accordg to the results of the studes, t mght be cocluded that sutablty of the method may

2 Ahmed 87 vary wth the sample data sze, sample data dstrbuto, sample data format ad goodess of ft test (Adag ad Al, 9). The preset wor s based o the tme seres wd data collected over a perod of 4 years ( - 4) (hourly). The locato cocered ths study amed Halabja s stuated east Sulama/North Iraq 35 ' 7" North lattude, 45 58' 4" East logtude ad t s at a elevato of 69 m above sea level. There s o obstacle aroud wd speed measurg locato, the wd data recorded from a mechacal cup type aemometer at heght of m above the groud level. I preset study, four methods for estmatg the parameters of the Webull wd speed dstrbuto are preseted [MLM, Ra regresso method (RRM), Meastadard devato method (MSD), ad the Power desty method (PDM)] by Adag ' ad Al (9). The am of ths wor was to select a method that gves more accurate estmato for the Webull parameters at ths locato order to reduce ucertates related to the wd eergy output calculato from ay Wd Eergy Coverso Systems (WECS). WEIBULL DISTRIBUTION The Webull dstrbuto s characterzed by two parameters, oe s the scale parameter c (m/s) ad the other s the shape parameter (dmesoless). I Webull dstrbuto, the varatos wd speed are characterzed by two fuctos whch are the probablty desty fucto (PDF) ad the cumulatve dstrbuto fucto (CDF). The PDF, v,,c dcates the fracto of tme (or probablty) for whch the wd s at a gve speed. It s gve by Bhattacharya ad Bhattacharjee () ad Wesser (3). f v,, c c e () c c () Where ν >, ad, c > The CDF of the speed gves the fracto of the tme (or probablty) that the wd speed s equal or lower tha, thus, the cumulatve dstrbuto v,,c s the tegral of the PDF, gve by Tae v c x, d x dx (6) c (6) Equato 5 ca be smplfed as: x c e x dx (7) Ths s the form of the stadard gamma fucto, whch s gve by x e x dx (8) From Equatos 7 ad 8, let be expressed as: (7) (8) the average speed ca c (9) (9) The stadard devato of wd speed s gve by v v f ( v ) dv () v f ( v ) dv v vf ( v ) dv v or Usg v f v dv v v v ( ). () v v v f ( v ) dv v dv c x dv c c c c () () F,, f d e c () c The average wd speed ca be expressed as: () Equatg to x c x e dx () () f ( ) d (3) c ( ) e d (4) c c (3) (4) Ad puttg, the the followg equato ca be obtaed. Hece, get the stadard devato c c (3) (3) Ths ca be rearraged as: or v v c e d (5) c (5) c (4) (4)

3 88 It. J. Phys. Sc. METHODS FOR ESTIMATING WEIBULL PARAMETERS Maxmum lelhood method (MLM) Maxmum lelhood techque, wth may requred features s the most wdely used techque amog parameter estmato techques. The MLM method has may large sample propertes that mae t attractve for use; t s asymptotcally cosstet, whch meas that as the sample sze gets larger, the estmate coverges to the true values. Let ν,ν,ν 3..ν be a radom sample sze draw from a PDF ƒ(ν,θ) where θ s a uow parameter. The lelhood fucto of ths radom sample s the jot desty of the radom varables ad s a fucto of the uow parameter. Thus, (Ylmaz et al., 5; Nlse, ), L f ( v, ) (5) (5) v The maxmum lelhood estmator of say s the value of that maxmzes L or, equvalet, the logarthm of L. Ofte but ot always, the MLM of s a soluto of d log L. d Now, we apply the MLM to estmate the Webull parameters, ad c. Cosder the Webull PDF gve Equato, the lelhood fucto wll be (Ylmaz et al., 5; Nlse, ): c L(,,...,, c) e (6) c c (6) O tae the logarthms of Equato 6, dfferetatg wth respect to ad c tur, ad equatg to zero, oe ca obta the estmatg equatos l L l l (7) c (7) l L (8) c c (8) I elmatg c betwee Equatos 7 ad 8 ad smplfyg, oe ca get l l (9) (9) Ths may be solved to get the estmate of. Ths ca be accomplshed by the use of stadard teratve procedures (that s, Newto-Raphso method), whch ca be wrtte the form f ( x ) x x () ' f ( x ) () Where f l ( ) l () () ' f ( ) (l ) l l l () () The shape parameter ca be estmated usg Equatos ad wth Equato as: l l (3) (3) Oce s determed, c ca be estmated usg Equato 8 as follows: c (4) Ra regresso method (RRM) (4) The secod estmato techque, we shall dscuss s ow as the least squares method. Ths s, essece, a more formalzed method of the maual probablty plottg techque, that t provdes a mathematcal method for fttg a le to plotted falure data pots. It s so commoly appled egeerg ad mathematcs problems that are ofte ot thought of as a estmato problem. Wth the help of ths method the parameters are estmated wth regresso le equato by cumulatve desty fucto. From Equato, the cumulatve desty fucto of Webull dstrbuto fucto wth two parameters ca be wrtte as (Justus et al., 978): c F( ) e (5) Ths fucto ca be arraged as: c { F( )} e (6) If we tae the atural logarthm of Equato 6 (5) (6) l{ F ( )} (7) c (7) Ad the retae the atural logarthm of Equato 7, we get the followg equato: l[ l{ F( )}] lc l (8) (8) Equato 8 represets a drect relatoshp betwee (lν ) ad l{ F ( )} whch should be mmzed F E F l[ l( ( )] l[ l( ( ( ))] (9) (9) Ad Parameters of Webull dstrbuto wth two parameters are

4 Ahmed 89 estmated by mmzg wth Equato 9. The two parameters c ad are tersectg by the followg equatos: l l[ l{ F ( )}] l l[ l{ F ( )}] (3) c l { l } l l[ l{ F( )}] exp{ } (3) (3) (3) From Equatos 3 ad 3, ad c ca be estmated, respectvely. Mea-stadard devato method (MSD) The Webull factors ad c ca also be estmated from the mea ad stadard devato σ of wd data, cosder the expresso for average ad stadard devato gve Equatos 9 ad 4, from these, oe has (Fug et al., 7; Wesser ad Foxo, 3): Ad hece the cubc mea wd speed s gve as: cu c (38) (38) 3 3 To determe the eergy patter factor (E pf ) oe ca wrte Equatos 37 ad 38 as: E pf 3 3 cu 3 (39) 3 (39) Equato 39 s ow as eergy patter factor (Epf) method. Webull parameters ca be estmated wth solvg eergy patter factor Equato 39 umercally or approxmately by power desty techque usg the smple formula as follows: 3.69 (4) ( E ) pf (4) v (3) Where Ad (3) (33) (34) s the umber of wd observato. Oce v ad (33) (34) calculated for a gve data set, the ca be determed by solvg Equato 3 umercally, oce s determed, c s gve by c (35) (35) I a smpler approach, a acceptable approxmato for s (Ahlaque et al., 6):.86 v (36) (36) Power desty method (PDM) Ths s a ew method suggested by Adag ad Al (9). It s used to estmate the two-webull parameters, depeds o the eergy patter factor method; t s related to the averaged data of wd speed. Ths method has smpler formulato, easer mplemetato ad also requres less computato. Accordg to the Webull probablty dstrbuto, the mea wd speed of Equato 35 ca be wrtte as (Slva et al., 4; Paula et al., ): c (37) (37) are Oce s determed, c ca be estmated usg Equato 37. COMPARISON AND ACCURACY OF THE METHODS Four methods for estmatg the parameters of the Webull wd speed dstrbuto for wd eergy aalyss for Halabja cty are preseted. The applcato of each method s demostrated usg a sample wd speed data set, ad a comparso of the accuracy of each method s also performed wth the actual tme seres data for the our case study (Halabja cty). I order to compare the methods, mothly mea wd data used for Halabja rego s obtaed from meteorologcal automatc stato whch covers the perod of 4 years ( - 4). Two tests were employed to determe the accuracy of the four methods gve ths artcle, frst s the coeffcet of determato R of Equato 4 used to how well the regresso model descrbes the data, ad secod s root mea square error (RMSE) of Equato 4. N X x R N (4) X X N RMSE X x (4) N (4) (4) Where N s the total umber of tervals, X the frequeces of observed wd speed data, x the frequeces dstrbuto value estmated wth Webull dstrbuto, X the mea of X values. RESULTS AND DISCUSSION Oce coeffcet of determato ad RMSEs are computed the dfferece methods ca be compared accuracy as show Tables ad. Webull parameters have bee estmated mothly accordg to the four methods wth the actual tme seres data for all the years ( - 4). Fgure shows the hstogram of the actual frequecy dstrbuto of dural wd speed for all these years wth the Webull fucto for fttg a wd data probablty dstrbuto. Fgures ad 3 show the

5 9 It. J. Phys. Sc. Table. Mothly estmated Webull parameters wth actual data. Moth Mea (m/s) Actual Data ML method RR method MSD method PD method K C (m/s) K C (m/s) K C (m/s) K C (m/s) K C (m/s) Ja Feb Mar Aprl May Ju July Aug Sep Oct Nov Dec Mea Table. Statstcal aalyss for all the methods wth actual data. Method arace Stadard devato Coeffcet of varato Coeffcet of determato Root mea square error K C (m/s) K C (m/s) Actual data ML RR MSD PD estmated parameters c ad, respectvely versus the moths of years, the smlarty ca be see amog the methods wth the true data almost for all the moths for parameter c, whle for parameter, the dvergece of the methods wth the actual data obtaed due to the dfferece the estmated values, these also have appear the RMSE results as show Table. The ra regresso ad mea stadard devato methods gve satsfactory results for the shape parameter estmato, whle ra regresso method gve satsfactory result for the scale parameter estmato. Graphcally, Fgure 4 shows the aual mea wd data of the probablty desty fucto for Webull dstrbutos usg the estmated Webull parameters by all methods have bee compared wth the aual mea wd true tme seres data, as a result the power desty method s the most ftted method to estmate the Webull parameters our study case. We ca also see that all methods are smlar eough to show that each method would be suffcet for determg our parameter estmates. Cocluso Accordg to the results, t mght be cocluded that sutablty of these methods may vary wth the sample such as data sze, sample data dstrbuto (moths), sample data format, ad of ft tests. Whe wd data s avalable tme seres format, accordg to the R ad RMSE tests both the RRM ad MSD, respectvely are the recommeded methods for estmatg the shape parameter, whle for the scale parameter ad for the both tests, the RRM s recommeded method to estmate. Graphcally, the curves of the methods show that the best way to estmate the two Webull parameters s the PDM. Ths fact s also bee supported by meas of the RMSE ad R statstcal tests (Table ). From ths comparatve study, t s observed that the values of RMSE ad R have magtudes that are almost smlar for all the methods. ACKNOWLEDGEMENT The author would le to tha Dr. Samra Mhamad

6 Percet of observato Ahmed % 4848% 44% Percet of observato 3434% 77% % 44% 7 7% % Wd vvelocty (m/sec) (m/s) Fgure. Show the hstogram of the tme seres dstrbuto of the actual wd data. Fgure 4. The Webull probablty desty fucto usg all methods wth actual wd data. (Departmet of Statstcs) for provdg her formato to ths paper. REFERENCES Fgure. Estmated Webull parameter C (m/s) versus the moths. Fgure 3. Estmated Webull parameter K versus the moths. Adag SA, Al D (9). A ew method to estmate Webull parameters for wd eergy applcatos. Eergy Covers. Maag. 5: Ahlaque AM, Froz A, Wasm AM (6). Assessmet of wd power potetal for coastal areas of Pasta. Tur. J. Phys. 3:7-35. Bhattacharya P, Bhattacharjee R (). A study o Webull dstrbuto for estmatg the parameters. J. Appl. Quat. Methods 5():34-4. Fug CH, Ym HL, Lau KH, Kot SC (7). W eegy potetal Guagdog provdece. J. Geophys. Res. :-. Gupta BK (986). Webull parameters for aual ad mothly wd speed dstrbutos for fve locatos Ida. Sol. Eergy 37(6): Justus CG, Hargraves W, Amr M, Dese R (978). Methods for estmatg wd speed frequecy dstrbuto. J. Appl. Meteorol. 7: Katar YM, Seoglu B (7). Estmatg the locato ad scale parameters of the Webull dstrbuto: a applcato from egeerg. Proceedgs of the 6th IASTED Iteratoal Coferece, Appled Smulato ad Modellg. Palma de Mallorca, Spa. Kao JH (959). A graphcal estmato of mxed Webull parameters lfe testg of electro tubes. Techometrcs : Le Y (8). Evaluato of three methods for estmatg the Webull dstrbuto parameters of Chese pe (Pus tabulaeforms). J. For. Sc. 54(): Mars NB (5). Estmato of Webull parameters from commo percetles. J. Appl. Stat. 3():7-4. Nlse MA (). Parameter estmato for the two-parameter Webull dstrbuto. Msc. Thess. Departmet of Statstcs, Brgham Youg Uversty. Padey BN, Ndh D, Pulastya B (). Comparso betwee Bayesa ad maxmum lelhood estmato of scale parameter Webull dstrbuto wth ow shape uder lex loss fucto. J. Sc. Res. 55:63-7. Pag WK, Joatha JF, Marv DT (). Estmato of wd speed dstrbuto usg marov cha Mote Carlo techques. J. Appl. Meteorol. 4: Paula AC, Rcardo CS, Carla FD, Mara E (). Comparso of seve umercal methods for determg Webull parameters for wd eergy geerato I the ortheast rego of Brazl. Appl. Eergy 89: Prasad RD, Basal RC, Sauturaga M (9). Wd Modelg based o wd put data codtos usg Webull dstrbuto. Iter. J. Glob.

7 9 It. J. Phys. Sc. Eergy 3(3):7-4. Rahma S, Halawa TO, Husa T (994). Webull parameters for wd speed dstrbuto Saud Araba. Sol. Eergy 53(6): Rder PR (96). Estmatg the parameters of mxed Posso, Bomal ad Webull dstrbuto by the method of momets. Bull. Isttute It. Destatstque :38. Seguro J, Lambert TW (). Moder estmato of parameters of the Webull wd speed dstrbuto for wd eergy aalyss. J. Wd Eg. Id. Aerodyamc 85: Slva G, Alexadre P, Dael F, Everaldo F (4). O the accuracy of the Webull parameters estmators. Europea Wd Eergy Coferece ad Exhbto (EWEC). Lodo UK. Steves MJ, Smulders PT (979). The estmato of parameters of the Webull wd speed dstrbuto for wd eergy utlzato purpose. Wd Eg. 3():3-45. Wesser D, Foxo TJ (3). Implcatos of seasoal ad dural varatos of wd velocty for power output estmato of a turbe: A case study of greada. It. J. Eergy Res. : Wesser DA (3). Wd eergy aalyss of Greada, a estmato usg Webull desty fucto. Reew. Eergy 8:83-8. Ylmaz, Aras H, Cel HE (5). Statstcal aalyss of wd speed data. Eg. & Arch. Fac. Essehr Osmagaz Uversty. 98():-.

Comparing Different Estimators of three Parameters for Transmuted Weibull Distribution

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