Probabilistic Design of Wind Turbines

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1 Downloaded from orbt.dtu.dk on: Oct 7, 28 Probablstc Desgn of Wnd Turbnes Sørensen, John Dalsgaard; Toft, H.S. Publshed n: Energes Lnk to artcle, DOI:.339/en3224 Publcaton date: 2 Document Verson Publsher's PDF, also known as Verson of record Lnk back to DTU Orbt Ctaton (APA): Sørensen, J. D., & Toft, H. S. (2). Probablstc Desgn of Wnd Turbnes. Energes, 3(2), DOI:.339/en3224 General rghts Copyrght and moral rghts for the publcatons made accessble n the publc portal are retaned by the authors and/or other copyrght owners and t s a condton of accessng publcatons that users recognse and abde by the legal requrements assocated wth these rghts. Users may download and prnt one copy of any publcaton from the publc portal for the purpose of prvate study or research. You may not further dstrbute the materal or use t for any proft-makng actvty or commercal gan You may freely dstrbute the URL dentfyng the publcaton n the publc portal If you beleve that ths document breaches copyrght please contact us provdng detals, and we wll remove access to the work mmedately and nvestgate your clam.

2 Energes 2, 3, ; do:.339/en3224 Artcle OPEN ACCESS energes ISSN Probablstc Desgn of Wnd Turbnes John D. Sørensen, * and Henrk S. Toft 2 2 Aalborg Unversty and Rsø-DTU, Sohngaardsholmsvej 57, DK-9 Aalborg, Denmark Aalborg Unversty, Sohngaardsholmsvej 57, DK-9 Aalborg, Denmark; E-Mal: hst@cvl.aau.dk * Author to whom correspondence should be addressed; E-Mal: jds@cvl.aau.dk; Tel.: ; Fax: Receved: 4 January 2 / Accepted: 29 January 2 / Publshed: 23 February 2 Abstract: Probablstc desgn of wnd turbnes requres defnton of the structural elements to be ncluded n the probablstc bass: e.g., blades, tower, foundaton; dentfcaton of mportant falure modes; careful stochastc modelng of the uncertan parameters; recommendatons for target relablty levels and recommendaton for consderaton of system aspects. The uncertantes are characterzed as aleatorc (physcal uncertanty) or epstemc (statstcal, measurement and model uncertantes). Methods for uncertanty modelng consstent wth methods for estmatng the relablty are descrbed. It s descrbed how uncertantes n wnd turbne desgn related to computatonal models, statstcal data from test specmens, results from a few full-scale tests and from prototype wnd turbnes can be accounted for usng the Maxmum Lkelhood Method and a Bayesan approach. Assessment of the optmal relablty level by cost-beneft optmzaton s llustrated by an offshore wnd turbne example. Uncertanty modelng s llustrated by an example where physcal, statstcal and model uncertantes are estmated. Keywords: wnd turbne; relablty; stochastc model; relablty level. Introducton Hgh relablty and cost reductons are substantal requrements n order that offshore and land-based wnd turbnes can become compettve compared to other energy supply methods. In tradtonal determnstc, code-based desgn, the structural costs are among other thngs determned by

3 Energes 2, the value of the safety factors, whch reflects the uncertanty related to the desgn parameters. Improved desgn wth a consstent relablty level for all components can be obtaned by use of probablstc desgn methods, where explct account of uncertantes connected to loads, strengths and calculaton methods s made. In probablstc desgn the sngle components are desgned to a level of safety, whch accounts for an optmal balance between falure consequences, materal consumpton and the probablty of falure. Furthermore, usng a probablstc desgn bass t s possble to desgn wnd turbnes such that ste-specfc nformaton on clmate parameters can be used. As bass for the desgn of wnd turbnes tests wth mportant components and materal strength parameters are often performed. There s no ratonal desgn method where the test results and the assocated uncertanty can be appled n the desgn process. However, usng a probablstc desgn bass t s possble explctly to take nto account the nformaton from tests. In probablstc desgn asssted by testng the uncertanty related to the tests s accounted for, and the test results are combned wth the pror nformaton n the probablstc model, for example usng Bayesan statstcs. Durng the last 5 years desgn analyss usng probablstc methods has been used n other ndustral areas, e.g., offshore nstallatons, large brdges and tunnels. In developng methods for probablstc desgn of wnd turbnes experence from those applcatons can be used. It s noted that wnd turbnes are seres produced whereas most cvl engneerng structures are one-of-a-knd. The seres producton and followng Type Approval allows for a more refned relablty assessment. However, n several aspects wnd turbnes are more complcated than the above mentoned structures, especally because wnd turbnes can be consdered as a machne where the control system nfluences the magntude of the loads. On the other hand wnd turbnes are produced n large numbers, allowng for ratonal updatng of the uncertantes. Probablstc desgn of structural wnd turbne components can be used for drect desgn of components, thereby ensurng a more unform and economc desgn than obtaned by tradtonal desgn usng standards such as the IEC 64 seres. Formulaton of the probablstc bass ncludes the followng aspects descrbed n the paper: () defnton of the structural elements to be ncluded n the probablstc bass: e.g., blades, tower, substructure and foundaton; (2) dentfcaton of mportant falure modes and stochastc models for the uncertan parameters; (3) recommendaton of methods for estmaton of the relablty; (4) recommendatons for target relablty levels for the dfferent groups of elements; (5) recommendaton for consderaton of system aspects and damage tolerant desgn. An mportant aspect n obtanng wnd turbne systems wth hgh relablty and avalablty s to account for system relablty effects and to secure a system that s robust to unexpected ncdents and errors. The applcaton of a general framework for structural, rsk-based robustness/damage-tolerant assessment to wnd turbne systems s descrbed. 2. Relablty Modelng of Wnd Turbne The relablty modelng n ths secton consders one wnd turbne modeled as a system of components. The model can be extended to nclude more wnd turbnes n a larger system, e.g., a wnd farm. The components are dvded n two groups: Electrcal and mechancal components where the relablty s estmated usng classcal relablty models,.e. the man descrptor s the falure rate and the MTBF (Mean Tme between Falure).

4 Energes 2, Further, the bath-tub model s often used to descrbe the typcal tme dependent behavor of the hazard rate. The relablty s often modeled by a Webull dstrbuton, see e.g. [,2]. Usng e.g. FMEA (Falure Mode and Effect Analyss) or FTA (Falure Tree Analyss), system models can be establshed and the systems relablty can be estmated, see [2]. Relablty modelng for electrcal and mechancal components s not consdered further n ths paper. Structural members such as tower, man frame, blades and foundaton where a lmt state equaton can be formulated defnng falure or unacceptable behavor. Falure of the foundaton could be overturnng. Falure of a blade could be large deflectons wth nonlnear effects and delamnatons. The parameters n the lmt state equaton are modeled by stochastc varables and the relablty s estmated usng Structural Relablty Methods, e.g., FORM/SORM methods, see [3-5]. Relablty analyss of structural components and systems are consdered n the followng. Further, an mportant part of a wnd turbne s the control system whch regulates the energy output and lmts the loads on the wnd turbne components. Falure of the control system can be very crtcal for both the electrcal/mechancal and the structural components snce the loads on these can ncrease dramatcally e.g., loss of torque due to falure n control system may cause problems n blades or tower-nacelle moton whch agan may mply large edgewse vbratons n the blades. Therefore the relablty of the control system should be ncluded n a relablty assessment of the whole wnd turbne system. Relablty modelng of the control system s not consdered further n ths paper. In [6] relablty modelng of wnd turbnes related to unavalablty due to large wnd nduced acceleratons usng a fraglty curve approach s consdered. 3. Modelng of Uncertantes Parameters subject to uncertanty are assumed to be modeled by stochastc varables and/or stochastc processes/stochastc felds. Uncertantes modeled by stochastc varables = (,..., n ) are dvded n the followng groups:. Physcal uncertanty also denoted nherent uncertanty s related to the natural randomness of a quantty, for example the annual maxmum mean wnd speed or the uncertanty n the yeld stress due to producton varablty. 2. Measurement uncertanty s related to mperfect measurements of for example a geometrcal quantty. 3. Statstcal uncertanty s due to lmted sample szes of observed quanttes. Data of observatons are n many cases scarce and lmted. Therefore, the parameters of the consdered random varables cannot be determned exactly. They are uncertan themselves and may therefore also be modeled as random varables. Are addtonal observatons provded then the statstcal uncertanty may be reduced. 4. Model uncertanty s the uncertanty related to mperfect knowledge or dealzatons of the mathematcal models used or uncertanty related to the choce of probablty dstrbuton types for the stochastc varables.

5 Energes 2, The above types of uncertanty are usually treated by the relablty methods whch wll be descrbed below. Another type of uncertanty whch s not covered by these methods s gross errors or human errors. These types of errors can be defned as devaton of an event or process from acceptable engneerng practce and s generally handled by qualty control measures. Realzatons of uncertan parameters = (,..., n ), such as wnd and wave clmate, strengths, degradaton parameters, model uncertantes wll take place durng the lfetme. The uncertantes can be dvded n aleatory and epstemc uncertantes. Aleatory uncertanty s nherent varaton assocated wth the physcal system or the envronment (physcal uncertanty) t can be characterzed as rreducble uncertanty or random uncertanty. Epstemc uncertanty s uncertanty due to lack of knowledge of the system or the envronment t can be characterzed as subjectve uncertanty, whch can be reduced by better models, more data, etc. It s noted that some aleatory uncertantes change to epstemc uncertantes when the system s realzed. Model, measurement and statstcal uncertantes can be characterzed as epstemc uncertantes. Whereas epstemc uncertanty can be reduced by mproved models and/or addtonal observatons, the aleatory uncertanty remans unchanged. It only changes f the quantty of nterest s modfed tself. In many problems, natural fluctuaton (physcal uncertanty) and nsuffcent nformaton (model uncertanty) are the most mportant sources of uncertanty. The reference perod for the use of the stochastc model s also very mportant when modelng stochastc varables and processes. It s often assumed that ergodc stochastc processes may be used. However, the nfluence of long-term effects (e.g., clmate change) may also need to be consdered. Some uncertantes may for short reference perods appear reasonable but when predctve models are extrapolated for long reference perods then uncertantes can easly propagate and ncrease to unrealstc levels. Each of the stochastc varables, =,2,..., n s assumed to be modeled by a dstrbuton functon F ( x ; α ) where α denotes the statstcal parameters. Dependency between the stochastc varables can be modeled by jont dstrbuton functons or correlaton coeffcents. A number of methods can be used to estmate the statstcal parameters α n dstrbuton functons, e.g., the Maxmum Lkelhood method, the Moment method, the Least Square method or Bayesan statstcs. The Maxmum Lkelhood method gves a consstent estmate of the statstcal uncertantes. In Bayesan statstcs t s possble to take consstently nto account subjectve (pror) nformaton through a pror dstrbuton. In the Maxmum Lkelhood method the densty and dstrbuton functons for a stochastc varable are denoted: f ( xα,..., α m ) and F ( xα,..., α ) where m α,...,α m are statstcal parameters. N observatons are assumed to be avalable: xˆ,..., xˆ N. The statstcal parameters are determned usng the Maxmum-Lkelhood method by maxmzng the LogLkelhood functon usng a standard nonlnear optmzer, e.g., the NLPQL algorthm, see [7]. In general the parameters α,...,α m are determned usng a lmted number data and are therefore subject to statstcal uncertanty. Snce the parameters are estmated by the Maxmum Lkelhood technque they become asymptotcally (number of data should be larger than 25 3) Normally dstrbuted stochastc varables wth expected values equal to the Maxmum Lkelhood estmators and covarance matrx equal to, see e.g. [8]:

6 Energes 2, α ρ αα 2 α α ρ 2 αα m α αm 2 = [ ] = ρα α 2 α α 2 α ρ 2 α2α m α 2 αm C α,..., α H m () 2 ρα α m α α ρ m α2α m α 2 α m αm where H s the Hessan matrx wth second order dervatves of the log-lkelhood functon. The statstcal uncertanty can easly be ncluded n a relablty analyss usng FORM (Frst Order Relablty Method), see below. Model uncertanty, see [5,9], can be assessed f a mathematcal model h s ntroduced to descrbe/approxmate a physcal phenomenon (e.g., the load bearng capacty of a wnd turbne component). The mathematcal model s assumed to be a functon of a number of physcal uncertantes (e.g., strength parameters) modeled by stochastc varables wth realzatons denoted x. Further, the model s assumed to be a functon of a number of regresson parameters denoted R,..., Rm. The regresson parameters are determned by statstcal methods, and are therefore subject to statstcal uncertanty. The model s not perfect; therefore model uncertanty has n general also to be ntroduced. Ths s often done by a multplcatve stochastc varable R. The model can thus be wrtten: ( ) R h( R ) f,,..., R m (2) It s assumed that N data sets are avalable from measurements or tests: x 2 x 2 f 2... N f x N N It s assumed that the model uncertanty R s modeled by a LogNormal dstrbuted stochastc varable wth mean μ and standard devaton. The statstcal parameters R,..., R m, μ, can be determned by the Maxmum Lkelhood Method usng the Lkelhood functon: where ( r μ ) N ( x, R,..., R ) L( R,..., Rm, μ, ) = fln R (ln f ln h m μ, ) (4) = fln R, s the densty functon for ln R wth mean μ and standard devaton. The optmal parameters R,..., R m, μ, are determned as soluton to the optmzaton problem max ln L( R,..., Rm, μ, ). The statstcal uncertanty assocated wth a lmted number of data s R,..., Rm, μ, modelled by treatng the parameters R,..., R m, μ, as stochastc varables wth standard devatons and correlaton coeffcents determned from () f the number of data sets s larger than Ths llustrates how model (R ) and statstcal uncertantes ( R,...,, μ, ) can be modeled and estmated. The model descrbed by (2) has many applcatons wthn turbne desgn. One example s desgn of structural detals where an ncomplete/approxmate computatonal model h() s avalable. Ths model wll typcally be a functon of a number of uncertan parameters, e.g., strength and stffness parameters and parameters descrbng the load. Further, the model wll be subject to model uncertanty R. The statstcal parameters descrbng the physcal uncertantes and the model uncertanty R are n many cases determned by experments and measurements, and therefore subject to statstcal R m (3)

7 Energes 2, uncertanty. In Secton an example s presented where the dfferent types of uncertantes are modeled. Another example s estmaton of the long term energy producton of a wnd turbne/wnd farm usng e.g. WASP []. Examples of uncertantes to be taken nto account are long-term ste ar densty, turbulence ntensty and long term wnd speed (combned physcal and model/predcton uncertantes); topography over the ste and surroundng area (model uncertanty); power curve (model uncertanty); losses due to wakes (model uncertanty). Further statstcal uncertantes wll be assocated wth estmaton of the statstcal parameters usng avalble data. Smlar uncertantes should be modelled when estmatng the extreme loads, see example n Secton 9. It s mportant to note that the model uncertanty s assocated wth a mathematcal model of the consdered problem. The mathematcal model can be more general than the model n (2), e.g., the model output could be vector valued and more general models for the model uncertanty can be ntroduced. 4. Modelng of Structural Falure Modes and Relablty Examples of ultmate lmt state (ULS) modelng structural falure modes n wnd turbne desgn are: Local or global bucklng falure of tower Fatgue falure of blade or detals n substructure Foundaton falure by sldng For each of the falure modes t s assumed that a lmt state equaton can be formulated: g ( x) = (5) where x denote realzatons of the stochastc varables whch ncludes physcal, model, statstcal and measurement uncertantes. Realzatons of x where g ( x) < denote falure states. The probablty of falure for a falure mode descrbed by a lmt state equaton g ( x) = s gven by P F = P( g( ) ) and can be estmated by smulaton methods (crude Monte Carlo, Importance samplng, drectonal samplng, etc.) or by FORM/SORM methods where a relablty ndex β s determned and: ( g( ) ) Φ( β ) where Φ () s the standard Normal dstrbuton functon, see e.g. [4]. 5. Recommendaton of Target/Mnmum Relablty Level P F = P (6) In probablstc desgn the wnd turbne desgn parameters z (e.g., cross-sectonal geometrcal parameters) are determned from the optmzaton problem: mn z s.t. W ( z) mn ( ) β, M β z =,..., where W ( z) s an objectve functon to be mnmzed, e.g., the weght or cost of the wnd turbne/component, β ( z) s the relablty ndex for falure mode no, (7) mn β s the mnmum acceptable relablty ndex for falure mode and M s the number of falure modes to be checked. In addton to the constrants n (7) also smple bounds on the desgn parameters z and other practcal,

8 Energes 2, geometrcal lmts can be added. A smlar optmzaton problem can be formulated f a systems relablty constrant s ntroduced. The mnmum, target relablty level, mn β can be assessed based on, see e.g. [9,]: Cost beneft analyss, see below. The wnd turbne desgn (ncludng decsons on strategy for operaton and mantenance) s optmzed such that a mnmum of all costs mnus benefts s mn obtaned. The correspondng relablty levels for dfferent components can be used to assess β. The Lfe Qualty Index (LQI) concept can be used to assess the mnmum acceptable relablty level n case wnd turbne falure mples rsk of loss of human lves, see below. Further, experence from well-functonng wnd turbnes and statstcal analyss of reported falures should be used to assess and verfy the requred relablty level. Assessment of the relablty analyss usng cost-beneft analyses can be made n dfferent ways. Here t s assumed for smplcty that one wnd turbne s consdered, and that the wnd turbne s systematcally rebult n case of falure. The man desgn varables are denoted z = ( z,..., z N ), e.g., dameter and thckness of tower and man dmensons of blades. The ntal (fabrcaton) costs are denoted C I ( z), the drect falure costs are C F, the benefts per year (ncome from producton of electrcty) are b, and the real rate of nterest s r. Falure events are modeled by a Posson process wth falure rate λ. The probablty of falure s P F ( z), and the falure rate s then λ P F ( z). The optmal desgn can be determned from the followng optmzaton problem [2] based on maxmzng the total dscounted expected benefts mnus costs (cost beneft analyss): ( z) CI ( z) C F λpf ( z) + C C r + λp ( z) b CI max W ( z) = (8) z r C C F where C s the reference ntal cost of correspondng to a reference desgn z. The optmal desgn z* * s determned by the soluton to (8). The correspondng probablty of falure, P F ( z ) can be consdered the optmal probablty of falure related to the falure event and the actual cost-beneft ratos used. The falure rate λ and probablty of falure can be estmated for the consdered falure event, f a lmt state equaton, g (,..., n,z), and a stochastc model for the stochastc varables, (,..., n ), are establshed. If more than one falure event s crtcal, then a seres-parallel system model of the relevant falure modes can be used, see below. The lfe qualty ndex (LQI) can be used quantfy what s necessary and what s affordable for a socety to nvest nto rsk reducton. The lfe qualty ndex s a functon of the gross domestc product, the lfe expectancy at brth and the fracton of tme necessary to rase the gross natonal product by work. If margnal changes are consdered an acceptablty crteron from a socetal pont of vew can be formulated. If wnd turbne falure mples a rsk of human lves then LQI prncple mples a mnmum acceptable relablty level whch can be obtaned from the acceptablty crtera, see [9,]: d d CI ( z) GΔ k N PE λ PF ( z) (9) dz dz where G Δ s the socetal value of a statstcal lfe (related to the gross natonal product per capta), k s the probablty of beng klled n or by the faclty n case of falure and N PE s the number of people

9 Energes 2, exposed to the falure. Based on the acceptablty crtera a relablty constrant to the optmzaton problem n (8) can be formulated. Offshore wnd turbnes are characterzed by a very low rsk of human njury n case of falure when compared to onshore wnd turbnes, and to cvl engneerng structures n general. The acceptablty crtera n (9) s then not relevant and the mnmum relablty level for structural desgn can be assessed on the bass of relablty-based cost optmzaton n (8) consderng the whole lfecycle of the turbnes. It s noted that dfferent falure modes/components can have dfferent target/mnmum relablty levels snce the margnal cost of safety measures (margnal costs of ncreasng the relablty at the desgn stage) and the consequences of falure can be dfferent, see [3]. 6. System Aspects Generally a structural relablty model for the whole wnd turbne system wll consst of m falure modes each modelng a sequence of element falures. Ths can be modeled by a seres system of parallel systems. If the falure events/elements n falure mode no are descrbed by m lmt state equatons g ( x), j =, 2,, m, =, 2,, m, the probablty of falure of the whole system s gven by:, j = P = P { g ( ) } F, sys, j =, m j =, m () Methods to estmate the system probablty of falure are descrbed n e.g. [4,5]. Another aspect of system relablty s desgn prncples related to damage tolerant desgn and robustness. The damage tolerant desgn (fal safe) desgn phlosophy requres that the structure s able to wthstand damage due to e.g. fatgue, corroson and accdental damage at probable locatons. Further, t s assumed that a mantenance program s mplemented that wll result n detecton and repar of the damage before such damage degrades structural strength below an acceptable lmt, see e.g. [4]. Structural robustness can be defned as a structure shall be desgned and executed n such a way that t wll not be damaged by events such as exploson, mpact, and the consequences of human errors, to an extent dsproportonate to the orgnal cause, see [5]. Assessment of robustness starts by consderaton and modellng of exposures (E) that can cause damage to the components of the wnd turbne. The term exposures refers to extreme values of desgn loads, accdental loads and deteroraton processes but also ncludes human errors n the desgn, executon and use of the structure. The term damage refers to reduced performance or falure of ndvdual components of the system. After the exposure event occurs, the components of the structural system ether reman n an undamaged state ( D ) as before or change to a damage state (D). Each damage state can then ether lead to the falure of the structure (F) or no falure ( F ). Consequences are assocated wth each of the possble damage and falure scenaros, and are classfed as ether drect (C dr ) or ndrect (C nd ). Drect consequences are consdered to result from damage states of ndvdual component(s). Indrect consequences are ncurred due to loss of system functonalty or falure and can be attrbuted to lack of robustness [9,6].

10 Energes 2, The basc framework for rsk analyss s based on the followng equaton where rsk contrbutons from local damages (drect consequences) and comprehensve damages (follow-up/ndrect consequences) are added, see [6]: R = j C P ( D E ) P( E ) + C P( S D E ) P( D E ) P( E ) dr, j j nd, jk k j j () k j where C dr,j s the consequence (cost) of damage (local falure) D j due to exposure E, C nd,j s the consequence (cost) of comprehensve damages (ndrect) S k gven local damage D j due to exposure E, P(E ) s the probablty of exposure E, P(D j E ) s the probablty of damage D j gven exposure E and P(S k...) s the probablty of comprehensve damages S k gven local damage D j due to exposure E. The frst term P ( D E ) P( E ) express the probablty of a local damage D j consderng all j exposures. The second term ( S D E ) P( D E ) P( E ) P k j j express the probablty of comprehensve j damage S k consderng all exposures and local damages. The optmal desgn (decson) s the one mnmzng the sum of costs of mtgatng measures and the total rsk R. A detaled descrpton of the theoretcal bass for rsk analyss can be found n [9]. It s noted that an mportant step n the rsk analyss s to defne the system and the system boundares. The total probablty of comprehensve damages/collapse assocated to () s: P( collapse ) = P( collapse D j E ) P( D j E ) P( E ) j where P( ) damage collapse D j E s the probablty of collapse (comprehensve damage) gven local D j due to exposure (2) E. Note that compared to () only one comprehensve damage state (collapse) s ncluded n (2). From Equaton (2) t s obvous that the probablty of collapse can be reduced by: Reducng one or more of the probabltes of exposures P(E ) preventon of exposure or event control Reducng one or more of the probabltes of damages P(D j E ) related to element/component behavour Reducng one or more of the probabltes P collapse D j E ) ( If the consequences are ncluded n a rsk analyss then also reducton of drect (local) consequences, C dr,j and comprehensve (ndrect) consequences, C nd,j are mportant. It s noted that ncreasng the robustness at the desgn stage wll n many cases only ncrease the cost of the structure margnally the key pont s often to use a reasonable combnaton of a sutable structural system and materals wth a ductle behavour, f possble. In other cases ncreased robustness wll nfluence the cost of the structural system. For wnd turbnes examples of exposures are extreme wnd condtons (e.g., hurrcanes), human errors n desgn, fabrcaton and operaton; examples of local damages are defect(s) n a blade, falure of a welded detal. Robustness can n general be ncreased by ncreased redundancy through mechancally load sharng and statstcal parallel system effects, ductlty of falure modes and reducng the probabltes of the

11 Energes 2, 3 25 exposures by protectng the wnd turbne to (unforeseen) ncdents and securng a good qualty control n all phases. 7. Bayesan Statstcal Methods When new nformaton from tests and observatons become avalable they can be used to update the stochastc models and the estmates of the relablty (probablty of falure). The new nformaton can conssts of:. Observaton of events descrbed by one or more stochastc varables. The observaton can be modeled by an event margn. Updated/condtonal probabltes of falure can then be estmated. 2. Test samples/measurements of one or more stochastc varables,. Updatng can n ths case be performed usng Bayesan statstcs. In order to model the observed events an event functon H = H () s ntroduced. The event functon H corresponds to the lmt state functon. The actual observatons are consdered as realzatons (samples) of the stochastc varable H. Ths type of nformaton, for example, can be: Inspecton or montorng events such as nspecton of cracks. The event margn can nclude measurement uncertanty and the relablty of the nspecton method. Proof loadng where a well defned load s appled to the wnd turbne and the level of damage s observed. No-falure events where the observaton that the wnd turbne/component consdered s well-functonng after some tme n use. These observatons can be modeled by nequalty events { H } or equalty events { H = }. If nequalty events are used the updated probablty of falure s estmated by: P( g( ) H( ) ) P U F = P( g( ) H( ) ) = P( H( ) ) (3) For equalty events the updated probablty of falure can be estmated as descrbed n [7,8]. When samples x ˆ = ( xˆ, xˆ 2,..., xˆ N ) of a stochastc varable wth statstcal parameters α are avalable Bayesan statstcal technques can be used to update a pror stochastc model f Α( α), see e.g. [8,9]. The posteror, updated stochastc model s denoted f Α( α xˆ ). The predctve, updated stochastc model for s: f ( x xˆ) = f ( x α) f Α( α xˆ dα (4) ) By use of Bayesan technques, both the physcal uncertanty related to the consdered varable as well as the statstcal uncertanty related to the model parameters can be quantfed and engneerng judgment can be ncorporated. Bayesan statstcal methods can n a smlar way be used for uncertanty quanttes n regresson models used e.g. to model spatal varablty n e.g. sol strength or blade strength parameters, see [2].

12 Energes 2, Framework for Integrated Uncertanty Modelng n Wnd Turbne Desgn In desgn of wnd turbnes nformaton on uncertantes are obtaned n all phases of the desgn process and should be used n combnaton wth the mathematcal models of the falure modes to mprove the relablty of the desgn and possbly decrease costs. In Sectons 3 and 4 a mathematcal model and correspondng lmt state equaton for the falure modes are ntroduced. The followng nformaton sources can be ntegrated: Coupon tests wth basc materal and measurements of clmatc parameters performed at an early stage of the desgn process can be used to update the statstcal descrpton of the physcal varables usng Bayesan methods, see Secton 7. Tests and measurements of response parameters from prototype and -seres wnd turbne(s) and wnd turbne parts/components (e.g., blades or drve tran) can be used to update the model uncertantes assocated wth the mathematcal models for the wnd turbne behavor and falure modes. Bayesan methods can be used to update both physcal and model uncertantes, see Secton 7. When updatng the model uncertantes t s assumed that the physcal uncertantes are measured (or are known) such that the methods descrbed n Secton 3 can be used. The test results are often of the event type, e.g., no falure of a wnd turbne blade. It s noted that usually only a very lmted number of prototypes or wnd turbnes parts (e.g., blades) are tested n full-scale mplyng a sgnfcant statstcal uncertanty. When the wnd turbne s n seres producton and many wnd turbnes are n operaton then contnuous condton montorng of varous parameters can be used to update physcal and model uncertantes, and to decrease the statstcal uncertantes. Ths nformaton can be used to update/modfy the desgn of new wnd turbnes of the same type, as nformaton (pror knowledge) to development of new wnd turbnes, and as decson bass for possble lfe tme extenson (especally relevant for offshore wnd turbnes). Agan the Bayesan methods descrbed n Secton 7 can be used to handle the nformaton n a ratonal way. 9. Example Optmal Relablty Level Ths example s based on a smplfed model for local bucklng falure of an offshore wnd turbne support structures n shallow waters. The lmt state functons and the economy model for the followng example are descrbed more detaled n [2]. As an example consder the falure mode local bucklng of tower. The lmt state equaton s wrtten: where h s the rotor heght. The resstance s: and the load effect s: g = M cr Qh (5) 3 3 ( D ( D 2t) ) y ss cr Fy D F = (6) y, ss y M cr.84, 6 t E, sse

13 Energes 2, Q ( + 2k picamp dyn ) aero exp st str = P CT A (7) where D and t are dameter and thckness of the tower. The other parameters are descrbed n Table where varables denoted wth some subscrpt are model uncertantes. The followng desgn varables are used: radus of foundaton, R; dameter of tower, D; thckness of tower, t. The representatve cost model conssts of ntal costs, falure costs, and benefts. The ntal costs are modeled by contrbutons from foundaton, turbne and others: CI R Dt = Cfoundaton + Cturbne + Cothers = + + C 6 R 6 A 3 (8) where C s the reference cost correspondng to the reference radus R = 8.5 m and area A = 3/26 m 2. The falure costs are assumed to be C F /C = /36. The benefts per year are b/c = /8 and the real rate of nterest s assumed to be r =.5. Table. Stochastc varables for local bucklng falure mode. Varables denoted model model-uncertantes. LN: Lognormal, G: Gumbel. COV: coeffcent of varaton. Varable Dstrbuton type Expected value COV P Annual maxmum mean wnd pressure G 538 kpa.23 I Turbulence ntensty LN.5.5 C T A Thrust coeff. x rotor dsk area 34 m 2 k p Peak factor 3.3 exp Exposure (terran) LN.2 st Clmate statstcs LN. dyn Structural dynamcs LN. aero Shape factor/model scale G. str Stress evaluaton LN.3 F y Yeld stress, structural steel LN 24 MPa.5 E Young s modulus LN 2. 5 MPa.2 Yeld stress, y, ss structural steel LN.5 E, ss Young s modulus LN.2 Crtcal load capacty LN. cr Usng ths smple, but representatve cost model the optmum desgn s determned based on (8). The results show that the correspondng optmal relablty level for offshore wnd turbnes related to structural falure corresponds to annual probabltes of falure equal to 2 4 3, correspondng to relablty ndces n the nterval Ths relablty level s sgnfcantly lower than for cvl engneerng structures n general, but s of the same level as can be estmated from reported structural falures of wnd turbnes, see e.g. [22] where falure rates for blades are descrbed. Further, ths relablty level also corresponds to the relablty used to calbrate partal safety factors n the IEC 64- [23], and IEC 64-3 [24], standards, see [25,26] where the stochastc model n Table has been used.

14 Energes 2, Example Statstcal Modelng Usng Test Results Ths example on fatgue models for composte materals n wnd turbne blades llustrates how physcal, statstcal and model uncertantes can be obtaned on bass of test results usng the models and technques descrbes n Sectons 2 and 3, see also [27]. The Mner rule for lnear damage accumulaton s recommended n [23] for modelng fatgue n composte materals n wnd turbne blades even though the model s subject to sgnfcant uncertanty. The uncertantes n the damage accumulaton based on Mners rule can be dvded n three parts: Physcal uncertanty on the SN-curves. Statstcal uncertanty on the SN-curves due to a lmted number of tests. Model uncertanty related to Mners rule. The physcal uncertanty on the SN-curves s due to the natural nherent uncertanty n the materal whch cannot be reduced. The statstcal uncertanty can be reduced by performng addtonal fatgue tests and the model uncertanty can n prncple be reduced by adoptng a better model. Constant ampltude and varable ampltude fatgue tests are avalable n the OptDAT database [28] for geometry R4 MD (MultDrectonal lamnate). Ths geometry has been selected because many fatgue tests are performed wth ths geometry. For composte materals the mean stress can have a sgnfcant nfluence on the fatgue propertes. Ths s taken nto account by estmaton of SN-curves for dfferent R-ratos and arrangng these n a constant lfe dagram. The R-rato s defned by: mn R = (8) where mn and max are mnmum and maxmum stresses n a stress cycle respectvely. A lnear SN-curve model s used: log N F = log K m log Δ + ε (9) where N F s the number of cycles to falure, Δ s the stress range and ε models the lack of ft and s assumed normal dstrbuted wth mean value zero and standard devaton ε. The constants K and m are materal dependent parameters. If N constant ampltude tests and N run-outs are avalable then m s obtaned usng the least squares method. The parameters log K and ε can be estmated usng the Maxmum Lkelhood Method, see Secton 3. The statstcal uncertanty represented by the standard devatons log K and and the correlaton coeffcent ρ ε log K, s obtaned usng (). ε Table 2 shows the estmated parameters n the SN-curves for dfferent R-ratos wth run-outs taken nto account. The results show that log K and ε can be assumed uncorrelated. It s noted that ε represents the physcal uncertanty and that log K and represents the statstcal uncertanty. ε Fgure shows the constant lfe dagram based on the estmated SN-curves. max

15 Energes 2, R-rato Table 2. SN-curves for dfferent R-ratos for geometry R4 MD. Number of tests Number of run-outs m log K ε log K ε Varable ampltude fatgue tests are also performed wth geometry R4 MD. The load spectrum used s the Wsper and Wsperx spectra developed for representng the flap bendng moment of a wnd turbne blade. In order to calculate the accumulated damage D Mners rule for lnear damage accumulaton s used: N C D = = N F ( Δ ) where N C s the number of stress cycles wth stress ranges Δ, =,2,..., N when the accumulated fatgue damage exceeds. Fgure. Constant lfe dagram for geometry R4 MD. C (2). Fatgue falure occurs amp [MPa] 3 2 N= 3 N= 4 N= 5 N= 6 N= 7 N= 8 N= 9 R=2. R=. R=-2.5 R=-. R=-.4 R=. R= mean [MPa] A lmt state functon ncludng model uncertanty can be formulated by: N C g = Δ = N F ( Δ ) (2) where Δ s a stochastc varable modelng the model uncertanty. Table 3 shows the varable fatgue tests performed and lsted together wth the estmated mean μ Δ, standard devaton, Δ and coeffcent

16 Energes 2, of varaton, COV Δ. The results show that except for the Wsper spectrum the estmated mean accumulated damage at falure s sgnfcantly below one and that the coeffcents of varatons are qute hgh. It s noted that the uncertanty for fatgue damage accumulaton often s modeled by a Lognormal dstrbuton n order to avod negatve values of Mners rule. Table 3. Mean and standard devatons for estmated damage at falure for varable ampltude tests. Spectrum Number Mean Std. dev. of tests μ Δ Δ COV Δ Wsper Wsperx Reverse Wsper 2.2 Reverse Wsperx All In summary ths example llustrated how the followng types of uncertanty can be modeled: Physcal uncertanty s modeled by ε Statstcal uncertanty s modeled by log K and ε Model uncertanty s modeled by Δ (Normal dstrbuted) (Normal dstrbuted) (LogNormal dstrbuted). Conclusons A probablstc bass s descrbed for relablty-based desgn of wnd turbnes. Probablstc methods can be used as decson tool for desgn of structural wnd turbne components, thereby ensurng a more unform and economc desgn than obtaned by tradtonal desgn usng standards such as the IEC 64 seres. The followng aspects are descrbed: Identfcaton and selecton of structural elements to be ncluded n the probablstc bass: e.g., blades, tower, substructure and foundaton; Identfcaton and modelng by lmt states of mportant falure modes; Stochastc models for the uncertan parameters; Recommendaton of methods for estmaton of the relablty; Recommendatons for target relablty levels for the dfferent groups of elements; Recommendaton for consderatons of system aspects and damage tolerant desgn. An mportant aspect n obtanng wnd turbne systems wth hgh relablty and avalablty s to account for system relablty effects and to secure a system that s robust to unexpected ncdents and errors. The applcaton of a general framework for structural, rsk-based robustness assessment to wnd turbne systems s presented. It s descrbed how uncertantes n wnd turbne desgn related to ntegrated desgn usng computatonal models, statstcal data from small (coupon) test specmens, results from a few full-scale tests and from prototype wnd turbnes can be accounted for usng the Maxmum Lkelhood Method

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