Comparison of Estimators of Extreme Value Distributions for Wind Data Analysis

Transcription

1 Bofrg Iteratoal Joural of Data g, Vol., o. 3, September 0 6 Comparso of Estmators of Extreme Value Dstrbutos for d Data Aalyss. Vvekaada Abstract--- Estmato of extreme wd speed potetal at a rego s of mportace whle desgg tall structures such as coolg towers, stacks, trasmsso le towers, etc. Assessmet of wd speed ca expedetly be carred out by probablstc modellg of hstorc wd speed data usg extreme value dstrbuto, or by usg stadard procedures avalable uder BIS code of practces for buldg ad structures. Ths paper llustrates the use of extreme value dstrbutos such as umbel, rechet ad ebull for modellg wd speed data recorded at Kayakumar. ethod of Least Squares (LS), maxmum lkelhood method ad order statstcs approach are used for determato of parameters of the dstrbutos. Kolmogorov-Smrov test s used for checkg the adequacy of fttg the method/ dstrbuto to the recorded data. D-dex s used for selecto of sutable method/ dstrbuto for estmato of desg wd speed. The study shows that the umbel dstrbuto usg LS s better suted for estmato of desg wd speed for Kayakumar. A comparatve study of wd speed estmates obtaed usg LS of umbel ad wth BIS code of practces s carred out; ad results preseted. Keywords--- D-dex, rechet, umbel, Kolmogorov- Smrov, Least Squares, Order Statstcs, ebull I. ITRODUCTIO ID egeerg aalyses the effect of wd the atural ad bult evromet ad the damage due to the wd. or a structural egeer, the terest s strog wds whch may cause dscomfort. th the structures becomg tall ad sleder the effect of wd o these structures s becomg crtcal. The dstrbuto of wd speed s also mportat determg servceablty of buldgs []. The basc wd speed s arrved by cosderg 3-secods (sec) average wd speed at a heght of 0m; ad the estmated value of 000-year (yr) retur perod s used to arrve at a desg load, whch the structure must wthstad durg ts lfetme [-3]. or arrvg at these values, wd records over a perod of tme are observed ad statstcal aalyss s carred out. I statstcal theory, geeralzed extreme value dstrbuto s detfed as a famly of cotuous probablty dstrbutos that clude umbel, rechet ad ebull. As defed by the extreme value theorem statstcs, the asymptotc dstrbutos of the extremes ted to coverge o certa. Vvekaada, Assstat Research Offcer, Cetral ater ad Power Research Stato, Pue 404. E-mal: aadaa@redffmal.com lmtg forms for large sample; specfcally to the double expoetal form, or to two dfferet sgle expoetal forms. Sce the extreme values of a radom varable are varably assocated wth the tals of ts probablty desty fucto, the covergece of the dstrbuto fucto of ts extreme value to a partcular lmtg form wll deped o the behavour at tal ed of the tal dstrbuto the drecto of the extreme [4-6]. I the preset study, -parameter umbel, rechet ad ebull dstrbutos are cosdered for estmato of desg wd speed for Kayakumar rego. Aalytcal procedures lke ethod of Least Squares (LS), axmum Lkelhood ethod (L) ad Leble techque based o Order Statstcs Approach (OSA) are appled for determato of parameters of the dstrbutos. Qualtatve assessmet s made from the probablty plot of the recorded ad estmated extreme wd speed. oodess-of-t (o) test volvg Kolmogorov-Smrov (KS) s appled for checkg the adequacy of the method/ dstrbuto to the recorded wd speed data. I addto to o test, dagostc test volvg D-dex s appled upper tal rego for selecto of sutable probablty dstrbuto for estmato of desg wd speed for Kayakumar rego. The 000-yr retur perod value of estmated 3-sec average extreme wd speed gve by umbel, rechet ad ebull dstrbutos are compared wth the correspodg values of IS 875 procedure to arrve at a desg parameter for Kayakumar rego. The methodology adopted estmato of desg wd speed usg extreme value dstrbutos ad IS 875 procedures, o ad dagostc tests are brefly descrbed the esug sectos. II. ETHODOLOY The Cumulatve Dstrbuto ucto (CD) of umbel ad rechet dstrbutos are gve by: () e e ( λ ), >0 ad - < < (for umbel) () e, >0 ad - < < (for rechet) where ad are the locato ad scale parameters of umbel dstrbuto. The estmates of extreme wd speed usg umbel ( ) for dfferet retur perods (T) are computed from + Y ad Y T -l(-l(-(/t)). T Smlarly, ad λ are the scale ad shape parameters of rechet dstrbuto. Based o extreme value theory, rechet dstrbuto ca be trasformed to umbel dstrbuto through logarthmc trasformato. Uder ths trasformato, the estmates of extreme wd speed usg rechet ( ) for DOI: /BIJD.503 ISS Bofrg

2 Bofrg Iteratoal Joural of Data g, Vol., o. 3, September 0 7 dfferet retur perods are computed from Exp( ), Exp( ) ad λ /. The CD of ebull dstrbuto s gve by: λ () e, >0 ad - < < where ad λ are the scale ad shape parameters of ebull dstrbuto. The estmates of extreme wd speed ( ) usg ebull dstrbuto for dfferet retur perods / λ are computed from [ Log( ( T) )]. A. Estmato of Dstrbuto Parameters OSA s appled for estmatg the parameters of umbel ad rechet dstrbutos of the seres cosstg of the largest values. I addto, LS s appled for determg the parameters of umbel dstrbuto. The procedure adopted determg the parameters of the dstrbutos usg LS ad OSA s as follows: ethod of Least Squares: The recorded extreme wd speed data are arraged a creasg order of magtude by assgg rak of,, 3,...,. The o-exceedace probablty for each data pot for rak ` s assged by usg the followg relato: P + Usg LS, the parameters of umbel dstrbuto ca be obtaed from the followg equatos: ( l( l( P ))) l( l( P )) ( l( P )) l + Here l (-l (P )) defes the cumulatve probablty of oexceedace for each. The Stadard Error (SE) o the estmated extreme wd speed s computed by: ( Y. ) 0. 5 SE T + Y T Order Statstcs Approach: The approach s based o the assumpto that the set of extreme values costtutes a statstcally depedet seres of observatos. The OSA parameters of umbel dstrbuto are gve by: r + r ; r + r where r ad r are proportoalty factors, whch ca be obtaed from the selected values of k, ad usg the relatos r k / ad r /. Here, s the sample sze cotas basc data that are dvded to k sub groups of elemets each leavg remaders [7]. I OSA, ad are the dstrbuto parameters of the groups ad ad are the parameters of the remaders, f ay. These ca be computed from the followg equatos: ( k) S ( k) S k ad ad where S j, j,,3,..,. The values of the weghts ad are gve Table. The stadard error o the estmated wd speed s computed by: SE [Var( / )], Var ( ) r + r where, k r ad r ad are defed k. by the geeral form as ( ) AY + BY + C. The values of A, B, ad C are gve Table. / Table : eghts ad for Computato of OSA Parameters Table : Varace Determators for A B C The procedures adopted determg the L parameters of ebull dstrbuto ad stadard error o the estmated wd speed for dfferet retur perods are detaled the text book o lood requecy Aalyss by Rao ad Hamed [8]. The lower ad upper cofdece lmts (LCL ad UCL) for the expected Extreme d Speed (ES) estmates ca by obtaed from LCLES-.96SE ad UCLES+.96SE. ISS Bofrg

3 Bofrg Iteratoal Joural of Data g, Vol., o. 3, September 0 8 B. oodess-of-t Test The KS statstcs s defed by: KS ax( e ( ) D ( )) Here e ( ) ( 0.35) D ( ) () s the emprcal CD of ad s the computed CD of [9]. If the computed values of KS statstcs of the method/ dstrbuto are less tha that of crtcal value at the desred sgfcace level the the selected method/ dstrbuto s accepted to be adequate tha ay other method/ dstrbuto for modellg wd speed data. C. Dagostc Test The D-dex s gve by: 6 D-dex ( / ) () where ad are the th hghest recorded ad estmated wd speed values by umbel (usg LS ad OSA), rechet (usg OSA) ad ebull (usg L) dstrbutos, ad s the average wd speed of the recorded data. The method/ dstrbuto havg the least value for D-dex s cosdered as the best method/ dstrbuto for estmato of desg wd speed [0]. D. Cocept of IS 875 Procedure ollowg IS 875 procedure, the basc wd speed ( b ) for a rego s obtaed ad subsequetly modfed to accout for dfferet effects ad get desg wd speed ( z ) at heght z( meter) for the chose class of structure []. The relatoshp betwee b ad z ca be expressed by: z bkk k (3) 3 where k s the probablty-factor/ rsk-coeffcet, k s the terra ad heght factor ad k 3 s the topography factor. Value of k for dfferet classes ad mea probable desg lfe of structures ca be computed from Eq. (4). k YR A B l l( PYR ) (4) A + 4B Here YR s the mea probable desg lfe (yr) of the structure, P YR the rsk level YR cosecutve years, A ad B approprate coeffcets for the basc wd speed zoe. III. RESULTS AD DISCUSSIOS A. Estmato of Desg d Speed usg Probablty Dstrbutos By applyg the procedures, as descrbed above, a computer program was developed ad used to estmate the 3-sec average extreme wd speed for Kayakumar rego. Hourly wd speed data for the perod was used to estmate the 3-sec average extreme wd speeds for dfferet retur perods adoptg umbel (usg LS ad OSA), rechet (usg OSA) ad ebull (usg L) dstrbutos []. By usg the moogram o ormalzed wd speed, the recorded 3-sec average extreme wd speed s obtaed by multplyg the factor of.5 wth hourly data. Table 3 gves the estmates of 3-sec average extreme wd speeds for dfferet retur perods gve by umbel, rechet ad ebull dstrbutos for Kayakumar rego. rom Table 3, t may be oted that the 3-sec average extreme wd speed gve by umbel dstrbuto usg LS, s cosstetly hgher for retur perods above -yr whe compared wth the correspodg values gve by umbel ad rechet dstrbutos usg OSA, ad ebull usg L. Also, from Table 3, t may be oted that the stadard error o the estmated extreme wd speed for dfferet retur perods by ebull (usg L) ad umbel (usg OSA) dstrbutos are cosderably mmum whe compared wth the correspodg values of umbel (usg LS) ad rechet (usg OSA) for the rego uder study. Table 3: 3-sec Average Extreme d Speed (ES) Estmates wth Stadard Error (SE) for Dfferet Retur Perods usg umbel, rechet ad ebull Dstrbutos Retur Estmated 3-sec average extreme wd speed (km/ hr) usg perod umbel rechet ebull (yr) LS OSA (OSA) (L) ES SE ES SE ES SE ES SE gure shows the plot of recorded ad estmated 3-sec average extreme wd speeds for dfferet retur perods gve by umbel, rechet ad ebull dstrbutos. gure : Plot of Recorded ad Estmated 3-sec Average Extreme d Speeds for Dfferet Retur Perods usg umbel, rechet ad ebull Dstrbutos for Kayakumar rom gure, t ca be see that the ftted curves usg umbel ad ebull dstrbutos show covergece whle rechet shows dvergg tred o estmato of 3-sec average extreme wd speeds for the rego uder study. B. Aalyss Based o o Test KS statstcs for umbel (usg LS ad OSA), rechet (usg OSA) ad ebull (usg L) dstrbutos usg the seres of 3-sec average extreme wd speed were computed from Eq. (), ad gve Table 4. ISS Bofrg

4 Bofrg Iteratoal Joural of Data g, Vol., o. 3, September 0 9 Table 4: Computed Values of KS Statstcs Usg Extreme Value Dstrbutos Extreme wd Computed values of KS statstcs usg data seres umbel rechet ebull LS OSA (OSA) (L) 3-sec average rom Table 4, t may be oted that the computed value of KS statstcs gve by umbel, rechet ad ebull dstrbutos are less tha the crtcal value (KS 39, ) fve percet level, ad at ths level, all three dstrbutos are foud to sutable for modellg wd speed data for Kayakumar. C. Aalyss Based o Dagostc Test D-dex values for umbel, rechet ad ebull dstrbutos usg the seres of 3-sec average extreme wd speed were computed from Eq. (), ad gve Table 5. Table 5: Computed Values of D-dex for Extreme Value Dstrbutos Extreme wd Computed values of D-dex for data seres umbel (LS) umbel (OSA) rechet (OSA) ebull (L) 3-sec average rom Table 5, t may be oted that the value of D-dex gve by umbel dstrbuto usg LS s mmum, whe compared wth the correspodg value gve by umbel ad rechet usg OSA, ad ebull usg L. Based o o ad dagostc tests, t s suggested that umbel dstrbuto usg LS could be used for estmato of desg wd speed for Kayakumar rego. D. Estmato of Desg d Speed usg IS 875 Procedure Based o wd speed map gve IS 875, the basc wd speed for Kayakumar rego s foud to be 39 m/ s; ad the coeffcets of A ad B correspodg to the basc wd speed s 84 ad 4 respectvely. Sce Kayakumar s cosdered to be a Terra Category.e. exposed ope terra wth few or o obstructos ad whch the average heght of ay object surroudg the structure s less tha.5m, the values of k ad k 3 are cosdered as.05 ad.00 respectvely. The values of k for dfferet retur perods such as 50-yr, 00-yr, 00-yr, 500-yr ad 000-yr are computed from Eq. (4). By usg the values of b, k, k ad k 3, the values of desg wd speed for dfferet retur perods, at a stadard heght of 0m, are computed from Eq. (3). Table 6 gves a comparso of 3- sec average extreme wd speed estmates obtaed from umbel dstrbuto usg LS ad IS 875 procedures. Table 6: Comparso of Estmated 3-sec Average Extreme d Speed by umbel Dstrbuto Usg LS ad IS 875 Procedures Retur perod (yr) IS 875 (at stadard heght 0m) Estmated 3-sec average extreme wd speed (m/ s) by umbel dstrbuto usg LS d speed Stadard Error rom Table 6, t may be oted that the values of extreme wd speed for 50-yr, 00-yr, 00-yr, 500-yr ad 000-yr gve by LS of umbel dstrbuto are comparatvely less tha the correspodg values obtaed from IS 875 procedure. rom the results of the data aalyss, t may be oted that the percetages of varato o estmated 3-sec average extreme wd speed usg LS of umbel dstrbuto, wth referece to desg wd speed obtaed from IS 875 procedures correspodg to 50-yr, 00-yr, 00- yr, 500-yr ad 000-yr are about %, 9%, 8%, 6% ad 5% respectvely. The study recommeded that the 3-sec average extreme wd speed wth stadard error of 50. m/ s (80.4 km/ hr) for 000-yr retur perod may be cosdered for desg purposes Kayakumar rego. gure shows the plot of recorded ad estmated 3-sec average extreme wd speed by umbel dstrbuto usg LS together wth cofdece lmts at 95% level for Kayakumar rego. gure : Plot of Recorded ad Estmated 3-sec Average Extreme d Speed by umbel Usg LS wth Cofdece Lmts at 95% Level for Kayakumar IV. COCLUSIOS The paper preseted a computer aded procedure for estmato of desg wd speed for Kayakumar rego adoptg umbel (usg LS ad OSA), rechet (usg OSA) ad ebull (usg L) dstrbutos. IS 875 procedure was also used to determe the desg wd speed usg basc wd speed for the rego ad the results were compared wth the correspodg values obtaed from extreme value dstrbutos. o ad dagostc tests results showed that the umbel dstrbuto usg LS s foud to be approprate for estmato of desg wd speed for dfferet retur perods. The results also showed that the percetages of varato o ISS Bofrg

5 Bofrg Iteratoal Joural of Data g, Vol., o. 3, September 0 0 the estmated values of 3-sec average extreme wd speed wth referece to desg wd speed correspodg to 50-yr, 00- yr, 00-yr, 500-yr ad 000-yr are about %, 9%, 8%, 6% ad 5% respectvely. The study showed that the upper lmt of 3-sec average extreme wd speed of 50. m/ s (80.4 km/ hr) for 000-yr retur perod gve by umbel usg LS may be adopted for desg purposes for Kayakumar rego. ACKOLEDETS The author s thakful to Ida eteorologcal Departmet, Pue, for makg avalable the wd speed data. The author s grateful to the Drector, Cetral ater ad Power Research Stato, Pue, for provdg the research facltes to carry out the study. REERECES [] Iteratoal Atomc Eergy Agecy (IAEA, 003), eteorologcal evets ste evaluato for uclear Power Plats IAEA Safety ude, o. S--3.4, Iteratoal Atomc Eergy Agecy, Vea. [] S. Bvoa, R. Burlo ad C. Leoe, Hourly wd speed aalyss Scly, Reewable Eergy, Vol. 8, o. 9, Pp , 003. [3] A.. Celk, O the dstrbutoal parameters used assessmet of the sutablty of wd speed probablty desty fuctos, Eergy Coverso ad aagemet,vol.45, o. -, Pp , 004. [4]. Deaves ad I.. Les, O the fttg of low mea wd speed data to the ebull dstrbuto, Joural of d Egeerg ad Idustral Aerodyamcs, Vol. 66, o. 3, Pp 69-78, 997. [5].J. Cook, Dscusso o oder estmato of the parameters of the ebull wd speed dstrbuto for wd eergy aalyss, Joural of d Egeerg ad Idustral Aerodyamcs, Vol. 89, o. 0, Pp , 00. [6] J. Zhag ad Y. u, Lkelhood rato tests for ormalty, Computatoal Statstcs ad Data Aalyss, Vol. 49, o. 3, Pp 709-7, 005. [7] Atomc Eergy Regulatory Board (AERB, 008), Extreme values of meteorologcal parameters, AERB Safety ude AERB/ / S/ S-3. [8] R. Rao ad H.K. Hamed, lood requecy Aalyss, /s CRC Publcatos, ew York, Lodo, 000 [9] R.B. D Agosto ad.a. Stephes, oodess of t Techques, arcel Dekker Ic, ew York, 986. [0] Uted States ater Resources Coucl (USRC), udeles for Determg lood low requecy, Bullet o. 7B, Pp 5-9, 98. [] IS: 875 (Part 3), Code of practce for desg loads (other tha earthquake) for buldg ad structures, Bureau of Ida Stadards, ew Kayakumar, 003. [] Ida eteorologcal Departmet (ID), Report o avalablty of meteorologcal data, 00.. Vvekaada (Jue 969) post graduated mathematcs from adura Kamaraj Uversty 99. He obtaed post graduate degree hydrology from Uversty of Roorkee 000 ad master of phlosophy degree mathematcs from Bharathar Uversty 006. rom ay 993 to arch 006, he worked as Research Assstat ad aalyzed studes. rom Aprl 006 to tll date, he s workg Assstat Research Offcer where carryg out hydrologcal studes usg statstcal ad determstc approaches, predcto of hydrometeorologcal varables usg soft computg techques ad optmzato of hydrometrc etwork usg spatal regresso approach ad formato theory for varous water resources projects. ISS Bofrg