Mardfekri Et Al-2015-Wind Energy

RESEARCH ARTICLE

Multi-hazard reliability assessment of offshore windturbinesMaryam Mardfekri1 and Paolo Gardoni2

1 Zachry Department of Civil Engineering, Texas A&M University, College Station, Texas 77843-3136, USA2 Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 3118 Newmark Civil EngineeringLaboratory, Urbana, Illinois 61801, USA

ABSTRACT

A probabilistic framework is developed to assess the structural performance of offshore wind turbines undermultiple hazards. Amulti-hazard fragility surface of a given wind turbine support structure and the seismic and wind hazards at a specic sitelocation are incorporated into the probabilistic framework to assess the structural damage due to multiple hazards. A databaseof virtual experiments is generated using detailed three-dimensional nite element analyses of a set of typical wind turbinesystems subject to extreme wind speeds and earthquake ground motions. The generated data are used to develop probabilisticmodels to predict the shear and moment demands on support structures. A Bayesian approach is used to assess the modelparameters incorporating the information from virtual experiment data. The developed demandmodels are then used to estimatethe fragility of the support structure of a given wind turbine. As an example of the proposed framework, the annual probabilitiesof the occurrence of different structural damage levels are calculated for two identical wind turbines, one located in the Gulf ofMexico of the Texas Coast (prone to hurricanes) and one off the California Coast (a high seismic region). Copyright 2014John Wiley & Sons, Ltd.

KEYWORDS

multi-hazard; reliability; probabilistic models; fragility; offshore wind turbines

Correspondence

P. Gardoni, Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 3118 Newmark CivilEngineering Laboratory, 205 N. Mathews Ave. Urbana, Illinois 61801, USA.E-mail: [email protected]

Received 16 November 2012; Revised 1 May 2014; Accepted 6 May 2014

1. INTRODUCTION

According to the annual reports by the Global Wind Energy Council,1 the global cumulative-installed wind capacity hasbeen doubling every 3 years, and it is projected to continue to grow at a similar rate. Offshore wind turbines installedextensively around the world are subject to different hazards (e.g. earthquake, hurricane and typhoon) raising concernsabout the reliability of the wind turbine support structure. For instance, Japan is the worlds 13th largest producer of windpower according to the World Wind Energy Association,2 despite having a considerably high occurrence rate of earth-quakes and typhoons. Likewise, according to the National Renewable Energy Laboratory (NREL3), California, a highlyseismic region, is the third largest wind power producer in the nation. Furthermore, the wind industry is recently consider-ing installing offshore wind farms in the south coast of the USA, and in particular in the Gulf of Mexico, because of thesuperior wind resources available in this region.4 However, a considerably high hurricane occurrence rate in the Gulf ofMexico raises a new concern about the safety of wind turbine support structures subject to hurricane. To investigate thereliability of a wind turbine support structure, all possible hazards that can occur during the wind turbines life have tobe considered. To this end, a probabilistic framework is needed to evaluate the safety of the support structure under multiplehazards and predict its expected structural damage. The results can assist the wind industry decision-makers choosingoptimum design and location for future wind energy projects. In addition, the assessment of the expected structural damagecan be used for an optimal design of wind turbines to maximize the power production and minimize manufacturing, oper-ation and maintenance cost. Current standards for the design of wind turbines structural components (IEC 61400-15 and

WIND ENERGY

Wind Energ. 2015; 18:14331450

Published online 30 May 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/we.1768

Copyright 2014 John Wiley & Sons, Ltd. 1433

DNV-OS-J1016) are based on the partial safety factor methods, in which the target safety level is obtained by applyingsafety factors to load and resistance terms in the design equation. The structural reliability analysis presented in this papercan be valuable to calibrate the load and material factors to be used in the partial safety factor method and in particular toupdate current standards such as IEC 61400-1 and IEC 61400-3. This paper develops estimates of the annual probabilitiesof being in different structural damage states for offshore wind turbines installed in water depths less than 30m consideringmultiple hazards including seismic and hurricane.

Several probabilistic studies have been conducted on wind turbines. Walford7 and Tavner et al.8 investigated the reliabil-ity of operation and power production systems on the basis of historical data of failures and their associated costs. Walford7

also discussed the means for reducing operation and maintenance costs. Madsen et al.,9 Agarwal and Manuel,10 andManuel et al.11 employed probabilistic frameworks to predict the extreme and fatigue loads for the design of onshoreand offshore wind turbines on the basis of the dynamic response of the support structures. However, while the aeroelasticinteraction is successfully considered in the analyses, only limited work is available that incorporates the foundationstiffness in the dynamic response of offshore wind turbines, including Andersen et al.12

Offshore wind turbine support structures installed in water depths less than 30m are typically supported by monopilefoundations. Bush and Manuel13 investigated the effect that the use of alternative models for monopile foundation of shal-low-water offshore wind turbines has on the design extreme loads. Their results showed the importance of incorporatingfoundation stiffness in the simulations. The stiffness of monopile foundation in their study was obtained using py curves.However, using a three-dimensional (3D) non-linear nite element (FE) model, Mardfekri et al.14 showed that, dependingon the pile diameter and soil type, using common simple models such as py curves and particularly modeling the pileusing 1D beamcolumn elements may result in inaccurate responses. This is true in particular for the pile sizes typicalof foundations of offshore wind turbines.

Aeroelastic simulators such as FAST,15 ADAMS16 and GH Bladed17 successfully include the aeroelastic interactions inthe analysis of dynamic response of the support structure. However, an important limitation of these simulators is that theyare not capable of continuous modeling of the non-linear foundation behavior and the dynamic soil-structure interaction. Adetailed non-linear FE analysis of the support structure and the foundation can be carried out to account for the non-linearfoundation behavior and the dynamic soil-structure interaction. However, a detailed non-linear FE analysis can be quiteexpensive and time consuming both in developing and running it. In addition, assessing the reliability of a wind turbine requiresaccounting for the uncertainties inherent in the structural material, soil and geometrical properties. To account for such uncer-tainties, a high number of FE analyses would need to be carried out making this approach too time consuming.

Mardfekri and Gardoni18 conducted detailed 3D non-linear FE analyses of the support structures of a suite of typicaloffshore wind turbines supported by monopile foundations. The FE models included a continuous modeling of the pileand the surrounding soil. As a result, the FE models successfully incorporated the dynamic soil-structure interaction intothe response of the support structure. In addition, Mardfekri and Gardoni18 used the data generated from the FE analysisto calibrate simplied probabilistic models for the deformation, shear force and bending moment demands on the supportstructure under day-to-day loading in operating conditions (i.e. day-to-day wind, wave and current loads.) The developedprobabilistic models were then used to assess the reliability of support structures in a more efcient way than by simulatingthe structural responses using the FE models. However, Mardfekri and Gardoni18 only considered operating wind turbinesunder day-to-day environmental loads and not extreme loading from earthquakes and hurricanes.

To address the concern related to the installation of wind farms in moderate and high seismic regions, a number ofresearchers conducted studies on the seismic response of wind turbines. Early publications on the analysis of dynamic responseof wind turbines during earthquake19,20 were based on the simplied models that lumped the nacelle and rotor as a point mass atthe top of the tower. As a result, the aeroelastic interaction was not accounted for. More recently, Witcher21 and Prowell et al.22

developed more rened models that considered the aeroelastic interaction. Specically, Witcher21 studied the seismic responseof support structures for both operating and parked wind turbines. The results showed the importance of accounting foraeroelastic interaction for operating wind turbines. Prowell et al.22 calibrated the aeroelastic interaction modeled in FAST usingexperimental data from a shake-table test of a small onshore 65 kW wind turbine.23 Yet both studies fail to incorporate thedynamic soil-structure interaction. To address this limitation, Mardfekri and Gardoni24 used the detailed 3D non-linear FEmodels developed in Mardfekri and Gardoni18 to conduct time history analyses of offshore wind turbines subject to seismicloading in addition to day-to-day operational loading accounting for the dynamic soil-structure interaction. Using the generateddata, Mardfekri and Gardoni24 also developed probabilistic models for the seismic demands on the support structure. Thedeveloped probabilistic models were then used to assess the reliability of support structures conditioning on spectral accelera-tion and the mean wind speed acting on the structure. However, there is still a need for a probabilistic framework to assess themulti-hazard reliability of wind turbine support structures and predict their annual failure probabilities.

This paper addresses this need by proposing a probabilistic framework to assess the multi-hazard reliability of offshorewind turbines. As a rst step, we develop novel probabilistic models for the shear and moment demands on the supportstructures subject to extreme wind loads in addition to seismic loads. The models are developed by updating availableprobabilistic seismic demand models24 using additional virtual experiment data generated for support structures subjectto extreme wind loads like those experienced during hurricanes. The virtual experimental data are obtained by developing

Multi-hazard reliability assessment of offshore wind turbines M. Mardfekri and P. Gardoni

1434 Wind Energ. 2015; 18:14331450 2014 John Wiley & Sons, Ltd.DOI: 10.1002/we

detailed 3D non-linear FE models of wind turbines accounting for the dynamic soil-structure interaction. The probabilisticdemand models are calibrated using a Bayesian approach. The proposed probabilistic models provide unbiased predictionsfor the shear and moment demands on the support structures, accounting for the inherent uncertainties, including thestatistical uncertainty (associated to the nite sample size) and the modeling errors (associated with the selection of thevariables in the models and the model forms.) The probabilistic models are then used to develop the fragility curves of windturbines for given intensity measures of the seismic and wind loading, namely, the spectral acceleration Sa and the meanwind speed Ws. The fragility curves and site-specic hazard functions are then used to assess the annual probabilities ofstructural damage states. The developed methodology proposes three structural damage state classications for windturbines. As an illustration, fragility curves and the annual probabilities of being in specied damage states are estimatedfor two identical 5 MW offshore wind turbines one located in the Gulf of Mexico of the Texas Coast (prone to hurricanes)and one off the California Coast (a high seismic region).

The next section introduces the probabilistic framework to assess the multi-hazard reliability of wind turbine supportstructures. The third section discusses the proposed probabilistic demand models. In the fourth section, we present the for-mulation of fragility estimates for offshore wind turbine support structures along with the assessment of importance andsensitivity measures. Finally, fragility estimates, and importance and sensitivity measures are developed for the exampleswind turbines along with their annual probabilities of three structural damage states.

2. MULTI-HAZARD ASSESSMENT

According to the total probability rule,25 the probability of failure to meet a specied performance level for a component orsystem, Pf, can be written as

Pf wF w f w dw (1)where w = vector of loading variables, f(w) = the joint probability density function (PDF) of occurrence of w andF(w) = conditional probability of failure (or fragility) to meet a specied performance level given the occurrence of w.

The focus of this paper is on the two most signicant hazards for offshore wind turbines support structures: seismic andwind. Given the intensity measures Sa= spectral acceleration at the natural period of the wind turbine, and Ws=mean windspeed, equation (1) can be written as

Pf wF Sa;Ws f Sa;Ws dSadWs (2)where F(Sa,Ws) = probability of failure conditioned on Sa and Ws, and f(Sa,Ws) = joint PDF of Sa and Ws. Given that theoccurrence, or non-occurrence, of earthquake does not affect the probability of occurrence of any particular level of windspeed and vice versa, Sa and Ws, can be assumed to be statistically independent. Therefore, equation (2) can be written as

Pf wF Sa;Ws f Sa f Ws dSadWs (3)where f(Sa) and f(Ws) =marginal PDF of Sa and Ws, respectively.

2.1. Seismic contribution to probability of failure

To quantify the probability of future seismic activity at a particular location, we use the seismic hazard function, Q(Sa),dened as the expected annual frequency of experiencing a spectral acceleration equal to Sa or greater. Assuming the arrivalof earthquakes at a site is a Poisson process,26 f(Sa) can be expressed in terms of Q(Sa) as

f Sa exp Q Sa dQ Sa dSa

(4)

The United States Geological Survey (USGS) provides annualized seismic hazard exceedance curves, containingdiscrete values of Q(Sa) for locations throughout the USA, on the basis of the available information about past earthquakes,deformation of the earth crust, geologic site conditions and seismic attenuation relationships.26

2.2. Wind contribution to probability of failure

To develop the annual PDF for wind hazard, we combine the PDF for day-to-day wind speed with the one for extreme windspeed during hurricanes. Morgan et al.27 investigated annual probability distributions for offshore wind speeds on the basis

Multi-hazard reliability assessment of offshore wind turbinesM. Mardfekri and P. Gardoni

1435Wind Energ. 2015; 18:14331450 2014 John Wiley & Sons, Ltd.DOI: 10.1002/we

statistical analysis of day-to-day 10 min average wind speed data. Wind speed data were recorded at 178 ocean buoy stationsaround North America, by the National Data Buoy Center.28 On the basis of Morgan et al.,27 we use the bimodal Weibullmixture distribution (BIW) to model the day-to-day wind speed. The BIW is a combination of two Weibull distributions andhas two different modes. Using the BIW, the conditional PDF of Ws given that there is no hurricane f WsjH

is expressed as

f Ws H b1Wsb11

a1b1exp Ws

a1

b1" # 1 b2Ws

b21

a2b2exp Ws

a2

b2" #(5)

where shape b and scale a parameters have subscripts corresponding to the two different modes, =mixing parameter and Hindicates the event of non-occurrence of a hurricane.

Wang29 characterized the hurricane event on the basis of the statistical analysis of 4776 hurricanes simulated to occur in10,000 years with landfall position assumed to occur with equal probability along the length of the Texas coastline.Specically, using the existing hurricane tracking models, event-based simulation techniques and information extracted fromhistorical hurricane data, Wang29 developed a database of synthetic hurricanes simulated to occur during a 10,000 year period.He then found the lognormal distributions that t the marginal cumulative distribution functions of the maximum wind speedand the radius of maximum wind speed. On the basis of Wang,29 we use the lognormal distribution to model the extreme windspeed associated to hurricanes. The PDF of Ws given the occurrence of a hurricane f(Ws|H) is written as

f Ws Hj 1Ws

2

p exp 12

lnWs

2" #(6)

where the location parameter and scale parameter are the mean and standard deviation of the natural logarithm of Ws,respectively. The PDF of Ws can now be written as using the total probability rule

25 as

f Ws f Ws Hj P H f Ws H 1 P H (7)

where P(H) = annual probability of occurrence of a hurricane. With the assumption of arrival of hurricane being a Poissonprocess, P(H) = 1 exp[T], in which = annual occurrence rate of hurricane and T=1 year.

3. PROBABILISTIC DEMAND MODELS

In this paper, a model is an analytical expression or procedure that relates the demand on a structural component to theproperties x of the considered system (such as material properties, structural dimensions and boundary conditions) listedin Table I and the loading variables w listed in Table II. To facilitate the use of the proposed probabilistic models in prac-tice, Gardoni et al.30,31 suggested developing a demand model starting from a commonly accepted deterministic model orprocedure and adding correction terms and a model error to, respectively, correct for the inherent bias and capture theuncertainty in the developed model. Accordingly, we write the general form of the proposed demand models as

Dk x;w;k d^ k x;w k x;w; k kk k v;m (8)

where Dk(x,w,k) = kth probabilistic demand model, in which k= v or m to indicate the shear and moment demand models,respectively, d^ k x;w selected deterministic demand model, k(x,w, k) = correction term for the bias inherent in thedeterministic model, in which k = vector of unknown model parameters, kk =model error, in which k = random variablewith zero mean and unit variance, and k = unknown standard deviation of the model error, nally k = (k,k) = vector ofall the unknown parameters. We also dene the vector of all the unknown parameters = (v,m, ), where = the corre-lation between v and m. On the basis of a preliminary study of the data, we employ a variance stabilizing logarithmic trans-formation of the quantity of interest to satisfy the homoskedasticity assumption (k is constant), the normality assumption(k follows the normal distribution) and the additive form used in equation (8).

3.1. Deterministic model

Being a popular wind turbine simulator in the wind industry around the world makes FAST a proper candidate fordeterministic predictions of the demands on the wind turbine support structures. FAST employs a combined modal andmulti-body dynamics formulation to simulate the aerodynamics and structural response of wind turbines.15 For givenvalues of the mean wind speed and turbulence intensity, a time history of wind speed is generated by TurbSim and used



as an input for the dynamic analysis in FAST. TurbSim uses a statistical model to numerically simulate time series of three-com-ponent wind speed vectors.32 TurbSim supports the Kaimal spectrum33 to simulate the turbulence, while using the standard IECcategories of turbulence characteristics to estimate turbulence. More details on the turbulence modeling can be found inMardfekri and Gardoni.18 In addition, FAST supports the JONSWAP/PiersonMoskowitz spectrum34 to model linear irregularsea states (stochastic waves), which represents the superposition of a large number of periodic and parallel wave components. Asea state is specied by a wave frequency spectrum with a given signicant wave height and wave peak period, which areconsidered stationary within periods of typically 3 h duration, according to DNV-OS-J101.6 It then uses the Morisons equationto determine the hydrodynamic forces on the tower. Current loading is also incorporated in the Morisons equation. Additionaldetails on the modeling of wave and current loading can be found in Mardfekri and Gardoni.18

In this paper, we use FAST to predict deterministic seismic demands on the support structure of wind turbines. Thecurrently in-practice version of FAST does not include the seismic module. However, FAST allows forces and momentsto be applied at the tower base platform. The platform has six degrees of freedom, and it provides the possibility to modelthe earthquake ground motions in a time marching simulation.

In a practice consistent with those conducted for calculation of wind and wave loads, for given intensity and durationparameters and frequency content of the ground motion, Mardfekri and Gardoni24 generated synthetic ground motionsusing a stochastic model proposed by Rezaeian and Der Kiureghian.35 Generated synthetic ground motion time historiesare then applied as a time history of force F(t) to the platform. Using an articially large mass for the support platform,the force F t Mx t produces the desirable acceleration x t at the base of the turbine support structure, where M isthe total mass of the support platform and the wind turbine.

3.2. Model correction

The correction term k(x,w, k) is added to incorporate the missing terms in the deterministic model into the developeddemand models and to correct for the potential bias in d^ k x;w . It is written as

Table I. Geometrical and mechanical parameters x used in experimental design.

Property Symbol Ranges Unit

Rotor diameter RD 40126 mTower height HH 4090 mTower top diameter dt 1.94.0 mTower diameter to wall thickness ratio t 100200 Water depth HWr 2030 mSteel type ST S235, S275, S355 Material damping ratio 0.05 Support structure vibration period First mode Tn 0.911.9 s

Second mode Ts 0.53.6 sPile diameter dp 3.06.0 mPile penetration Hp 1050 mPile diameter to wall thickness ratio p 50100 Soil modulus of elasticity Esoil 13200 MPaFriction between pile and soil frp s 0.20.3

Soil type Clay SandSoil cohesion Csoil 10200 080 kPaSoil friction angle soil 1025 3545

Table II. Loading parameters w used in experimental design.

Property Symbol Ranges Unit

Mean wind speed Ws 3.075 ms1

Turbulence intensity ITw 00.16 Signicant wave height Hs 1.010 mWave peak period Tp 3:6

Hs

p 5:0 Hsp sRated wind speed Ws rated 10.311.7 ms

1

Earthquake moment magnitude Meq 5.87.0a

Distance between earthquake record and rupture zone Req 1.060.0a km

aThese ranges are not used in the experimental design. Earthquake records are selected using the bin approach within these ranges.



k x;w; k XNp1

kphkp x;w (9)

where k = [kp], hk1(x,w),, hkN(x,w) normalized explanatory functions that might be signicant in correcting d^ k x;w and N= the number of unknown model parameters. In order to facilitate the use in practice of the probabilistic model,we identify the most parsimonious model for k(x,w, k) by removing the unimportant explanatory function, which isidentied by examining the posterior statistics of k.

The same candidate explanatory functions as those of Mardfekri and Gardoni24 considered in their study are selected inthis paper. Table III presents the candidate explanatory functions. Specically, hk1(x,w) = 1 captures the potential bias inthe model that is independent of x and w, hk2 x;w d^ k x;w captures any possible under-estimation or over-estimationof the deterministic models, functions hk3 hk6 characterize the possible inuence of wind and wave parameters, hk7 hk13incorporates the possible inuence of the characteristic of the ground motion, hk14 brings in the size-related characteristicsof the wind turbine and nally functions hk15 hk18 incorporate the effects of the foundation stiffness in the model.

The proposed probabilistic models are calibrated using a Bayesian updating rule36 on the basis of experimental obser-vations. However in this study, because of the lack of data from eld observations or laboratory experiments, virtual dataare generated by conducting dynamic analyses of detailed 3D non-linear FE models of the support structures.

3.3. Virtual data

A set of representative congurations are selected to generate virtual data. We select the representative congurations usinga space lling experimental design technique to ensure that the congurations have a good coverage of the design space.Tables I and II present, respectively, the ranges of the variables considered to characterize the wind turbine congurationsand loading parameters. The proposed ranges for the rst and second modes of vibration are estimated on the basis of modalanalyses on the support structures of wind turbines with rated powers of 500 kW to 5MW. The upper limit of the range forthe mean wind speed Ws reaches 75m s

1 to incorporate the extreme wind velocities during hurricanes.The FE models are developed in ABAQUS37 to simulate the dynamic response of the support structure of a broad range

of offshore wind turbines, subject to different load cases including seismic excitations in addition to day-to-day environ-mental loads on operating and parked wind turbines, and extreme wind velocities due to hurricanes on parked windturbines. It is noteworthy that no specic design load cases are considered in the analyses. However, as mentioned earlierand presented in Table II, ranges for loading parameters are considered wide enough to develop a comprehensive demandmodel for the structural behavior of wind turbines.

Witcher21 conducted time domain simulations of a 2 MW wind turbine with 80m diameter rotor mounted on a 60 mhigh tubular steel tower in different load cases including continuous operation throughout earthquakes, emergencyshutdown initiated during an earthquake and parked throughout earthquakes. The results showed a signicant difference

Table III. Explanatory functions for demand models.

Explanatory function Formula Parameter

hk1 1 k= v or mhk2 d^ k d^ k =deterministic shear or moment demandhk3 ln(Ws Tn/HH) Ws=mean wind speed; HH=Hub height

Tn=natural period of the support structurehk4 ln(ITw) ITw=wind turbulence intensityhk5 ln(Hs/HH) Hs=signicant wave heighthk6 ln(Tp/Tn) Tp=wave peak periodhk7 ln(Sa/g) Sa=spectral acceleration; g=ground accelerationhk8 ln(Sd/HH) Sd=spectral displacementhk9 ln(PGA/g) PGA=peak ground accelerationhk10 ln(PGV Tn/HH) PGV=peak ground velocityhk11 ln(PGD/HH) PGD=peak ground displacementhk12 ln[2PGV/(PGA Tn)]hk13 ln[2PGD/(PGV Tn)]hk14 ln(RD/HH) RD= rotor diameterhk15 ln(Cs/Csmax) Cs=soil shear wave velocity; Csmax = 194.594ms

1

hk16 ln(Csoil/Esoil) Csoil=soil cohesion; Esoil=soil modulus of elasticityhk17 ln[tan(soil)] soil=soil friction anglehk18 ln(kt/kf) kt= tower stiffness; kf= foundation stiffness



in the responses of operating and parked wind turbines. He concluded that this difference is due to the absence ofaerodynamic damping in the parked condition. Comparing the peak loads resulted from time domain analyses withthose obtained using frequency domain procedure, Witcher21 showed that the aerodynamic damping experienced byan operational wind turbine is close to the typical 5% value used for the design spectra in building codes. Prowellet al.23 estimated the structural damping of a 65 kW wind turbine in idling (parked) condition through a full-scaletest on a wind turbine mounted on the Network for Earthquake Engineering Simulation (NEES) shake table at theUniversity of California, San Diego. They suggested a value of 0.6% for the structural damping of a parked windturbine. In another study on the seismic response of wind turbines, Ishihara and Sarwar38 suggested a structuraldamping of 0.5% for parked wind turbines with rated powers of 400 kW and 2MW. In this paper, we accountfor the aerodynamic damping of an operating wind turbine by considering a 5% structural damping for the steeltower. The structural damping for parked wind turbines is considered to be 0.5%.

Foundation non-linearities are considered explicitly in dening non-linear behavior of the soil and soil-structureinteraction. We use the MohrCoulomb plasticity model to dene the non-linear behavior of the soil. The Rayleighdamping formulation is used to include frequency-dependent soil damping in the structural response. Soil-pileinteraction is modeled using contact pair, a formulation used in ABAQUS to dene the non-linear contact propertiesof two bodies. Forces at the top of the tower due to the wind and the rotation of the rotor in case of operating windturbine are obtained using FAST. The resulted time histories are then used in the FE model of the support structureas an external loading in addition to wave, current and earthquake. Ground motion records are selected from the PacicEarthquake Engineering Research Center Next Generation Attenuation (NGA) database.39 We select the ground motionrecords on the basis of the bin approach, proposed by Shome and Cornell.40 We use ve bins on the basis of the momentmagnitude (Meq) and the closest distance between the record location and the rupture zone (Req) to capture all possiblecharacteristics of the earthquake. More details on the analytical modeling to generate virtual experiment data arepresented in Mardfekri and Gardoni.24

3.3.1. Equality and lower bound dataThe accuracy of the results of FE analyses is sensitive to how the solution method handles large displacements and

second-order effects. To include the data from the analyses that lead to large deformations without letting inaccurate valueswrongfully inuence the model parameters, the data from virtual experiments are divided into equality and lower bounddata.30,41 A threshold of 5% is considered for the drift ratio, such that if the maximum drift ratio during one time historyanalysis is less than 5%, then the shear and moment data are considered to be accurate and taken as equality data. If ananalysis produces a drift ratio that exceeds 5%, then we consider the maximum shear and moment that occurred prior toreaching the 5% drift ratio as lower bound data for the shear and moment, respectively.

3.4. Bayesian model updating

A Bayesian formulation is used to calibrate the proposed probabilistic models with the virtual data. In a Bayesian approach,the unknown parameters are estimated using the following updating rule:36

f L p (10)where p() = the prior distribution of that reects the state of knowledge about available before generating the virtualdata, L() = the likelihood function that represents the objective information on contained in the virtual experiment data, = a normalizing factor and f() = the posterior distribution of that represents the updated state of knowledge about .The posterior distribution f() incorporates both the information about included in p() and L(). Application of theupdating rule in equation (10) can be repeated to update our present state of knowledge as new information on becomesavailable. Because of lack of prior information on the unknown parameters, we use a non-informative prior that reects thatno knowledge about is available a priori. Gardoni et al.30 showed that a non-informative prior for the parameters canbe written as

p 1 1

3=2Y2i1

1i

(11)

3.4.1. Likelihood functionThe likelihood is a function that is proportional to the conditional probability of observing the results from the virtual

experiments for given values of the model parameters. Under the assumption of statistically independent observations,following Gardoni et al.,30 L() is written as



where rki k Dki d^ k xi;wi k xi;wi; k and Dki = ith observed value for the kth demand for given xi and wi.However, the events in the aforementioned expression are dependent because of the correlation between ki for differentdemands k. To consider this dependence, the probability term for each observation in equation (12) can be computed usingthe multi-normal probability density and cumulative distribution functions for both shear and moment demand observa-tions. Table IV lists the formulation for probability terms proposed by Gardoni et al.30 for bivariate probabilistic models.The likelihood function is formulated on the basis of the type and form of the available data. In this paper, we divide thevirtual experiments into equality and lower bound data.

3.5. Model selection

To develop parsimonious probabilistic demand models (i.e. with only the explanatory functions that are strictly needed), amodel selection process is used to identify the important explanatory functions among the candidates presented in Table III.For the model selection, we use a stepwise deletion procedure developed by Gardoni et al.30 for the case in which data arein the form not only of equality data but also of lower (or upper) bound data. Starting with a comprehensive candidate formof k(x,w, k), unnecessary terms are deleted in a stepwise manner on the basis of the posterior statistics of model param-eters. At each step, the term hkj whose coefcient kj has the largest posterior coefcient of variation (COV) is deleted.Model reduction is continued until an unacceptable increase is seen in the value of k.

Figure 1 summarizes the stepwise deletion process for shear demand model. At each step, the solid dots show the pos-terior COVs of the model parameters vi, and the open circle shows the posterior mean of v. It is seen that after 15 steps,further model reduction deteriorates the accuracy of the model (i.e. v increases signicantly.) Stopping at this step, themodel is left with four terms.

Table IV. Probability terms for likelihood function with equality and lower bound data.30

Moment model

Shear model Equality datum Lower bound

Equality datum 1vjm

rvi vjmvjm

h i1m

rmi m

h i rvi vjmvmh i

1m

rmi m

h iLower bound rmi mjvmjv

h i1v rvi v

h i

rmi

rvi vjvjm

1m

m

d

k|l= kl (k/l)rli, k lk12kl

p , k, l= v,m and v|= vm(v/m).

Figure 1. Stepwise deletion process for shear demand model, where () indicates term to be removed.

L Y

Observation i

P Equalitydata k

kki rki k Lower bound

data k

kki > rki k 8

Upon carrying out the model selection process, nal probabilistic seismic shear demand model is written as

Dv x;w;v d^ v x;w v1 v2d^ v x;w v9 ln PGAg

v18 ln KtKf

vv (13)

where d^ v the natural logarithm of the deterministic shear demand at the base of the tower normalized by themean value of theyield shear force, dened as V^ y f^ y A 3=4 R2 r2

= R2 Rr r2 , in which f^ y expected yield stress of steel, and A, R

and r= tower base cross section area, outer and inner diameter, respectively; PGA=peak ground acceleration, g=gravitationalacceleration, Kt= tower stiffness and Kf= foundation stiffness.

Likewise, the model selection process for the moment model is carried out. Figure 2 shows the results of thestepwise deletion process. As in Figure 1, solid dots show the posterior COVs of the model parameters mi, and opencircles show the posterior mean of m at each step. Figure 2 shows that after 13 steps of model reduction, the largest COV(for parameter m15) is close in magnitude to m, and further reduction deteriorates the quality of the model (m increases).Stopping at this step, the moment demand model is left with six correction terms and is written as

Dm x;w;m d^m x;w m1 m2d^m x;w m11 ln PGDHH

m13 ln 2 PGDPGV Tn

m15 ln CsCsmax

m18 ln KtKf

mm

(14)

where d^m the natural logarithm of the deterministic moment demand at the tower base normalized by M^ y f^ yS, in whichS= elastic section modulus at the tower base; PGD=peak ground displacement,HH=hub height, PGV=peak ground velocity,Tn=natural period of the support structure,Cs= soil shear wave velocity andCsmax =maximum shear wave velocity consideredin the analyses (194.6m s1.)

3.6. Proposed probabilistic models

Table V gives the developed posterior statistics of the parameters v= (v1, v2, v9, v18,v). Figure 3 shows a comparisonbetween measured and predicted shear demands on the support structure on the basis of the deterministic (left) and prob-abilistic (right) models. For the probabilistic model, the median predictions are shown. Solid dots and open triangles indi-cate equality and lower bound data, respectively. The dashed lines in the Figure 3(b) delimit the region within one standarddeviation of the model.

Figure 2. Stepwise deletion process for moment demand model, where () indicates term to be removed.

Table V. Posterior statistics of the parameters in the shear demand model.

Standard deviation

Correlation coefcient

Parameter Mean v1 v2 v9 v18 v

v1 3.050 0.480 1v2 0.737 0.067 0.74 1v9 0.259 0.044 0.15 0.41 1v18 0.228 0.046 0.74 0.15 0.12 1v 0.508 0.033 0.07 0.11 0.04 0.01 1



The deterministic model on the left is strongly biased on the non-conservative side, because almost all equality data andmost of the lower bound data lie below the 1:1 line. For a perfect model, all the equality data should be lined up along the1:1 line, and all the lower bound data should lie above the 1:1 line. The proposed probabilistic demand model providesunbiased estimates as shown in Figure 3(b).

Similarly, Table VI lists the posterior statistics of the parameters m= (m1, m2, m11, m13, m15, m18,m) for themoment demand model. Figure 4 shows plots of predicted versus measured moment demands on the basis of the deterministic(left) and probabilistic (right) models. The same comments as those already made in relation to Figure 3 apply. We note that

Figure 3. Measured versus predicted shear demands based on (a) deterministic and (b) probabilistic models.

Table VI. Posterior statistics of the parameters in the moment demand model.

StandardDeviation

Correlation coefcient

Parameter Mean m1 m2 m11 m13 m15 m18 m

m1 0.568 0.694 1m2 0.584 0.071 0.16 1m11 0.132 0.047 0.40 0.48 1m13 0.134 0.088 0.54 0.49 0.68 1m15 0.387 0.192 0.62 0.40 0.11 0.05 1m18 0.226 0.069 0.88 0.33 0.01 0.15 0.69 1m 0.517 0.034 0.01 0.09 0.02 0.01 0.05 0.03 1

Figure 4. Measured versus predicted moment demands based on (a) deterministic and (b) probabilistic models.



whereas the deterministic model is strongly biased on the non-conservative side, the proposed probabilistic model correctsthe bias. In addition to the values of v and m, it is found that has a mean and standard deviation equal to 0.719 and0.041, respectively.

4. FRAGILITY AND MEASURES OF IMPORTANCE AND SENSITIVITY

With the developed demand models, we can estimate the fragility of wind turbine support structures considering differentfailure modes. We dene the fragility as the conditional probability that the shear or moment demands exceeds speciedcapacity levels for given value of the vector w. According to the conventional notation in structural reliability theory,42

a limit state function gkj(x,w,k) is dened such that the event {gkj(x,w,k) 0} denotes the exceedance of the jth capac-ity level of kth limit state. Using the probabilistic demand model described earlier, gkj(x,w,k) is formulated as

gkj x;w;k Ckj x Dk x;w;k (15)in which Ckj(x) represents the capacity corresponding to Dk(x,w,k). In this paper, we consider yield and ultimate capacitylevels identied as j= y and u, respectively.

Table VII presents proposed damage states and the corresponding performance levels. The shear capacity is dened as theallowable yield/ultimate shear force in the hollow cross section of the steel tubular tower Vj= fj A(3/4)(R

2 + r2)/(R2 +Rr+ r2),normalized using V^ y, in which fj= fy or fu, for yield and ultimate stresses. Therefore, by simplifying the expression, the shear

capacity can be written as Cvj f j=f^ y . Similarly, the yield-bending moment capacity is dened as Cmy My=M^y , which isthe resultant of the allowable yield-bending moment My= fyS, normalized using M^ y. Finally, the ultimate moment capacity is

dened asMu=M^y, in whichMu=maximum moment in the moment-curvature diagram, constructed for a tubular cross sectionof tower, on the basis of the stressstrain curve for structural steel of grade S235, to calculate the ultimate bending momentcapacity.

The fragility is then formulated as

Fj w; P k

gkj x;w;k 0

w; (16)

where P[B|w] denotes the conditional probability of event B for the given values of w. The uncertainty in the event forgiven w arises from the inherent randomness in the structural and material properties, the inexact nature of the limit statemodel and the uncertainty inherent in the model parameters.

4.1. Predictive estimates of fragility

Following Gardoni et al.,30 a predictive estimate of the fragility is formulated as

eFj w Fj w; f d (17)which incorporates the uncertainties in the model parameters by considering as random variables and taking theexpected value of Fj(w,) over the posterior distribution of . Following Ditlevsen and Madsen,42 the correspondingreliability index can be dened as ej eFj .4.2. Sensitivity measures

A sensitivity analysis can be carried out to identify which parameter(s) inuences the most the reliability of wind turbinesupport structures. Sensitivity measures can provide insight into the behavior of support structures and are useful for

Table VII. Proposed damage states.

Damage state Description Performance level

No signicant damage (ND) No structural damage Tower base shear or momentexceeds yield limit

Permanently out-of-service (PO) Support structure yields. Permanentexcessive deformations

Tower base shear or momentexceeds ultimate limitComplete (C) Support structure is unable to carry additional loads



optimal design and resource allocation. Hohenbichler and Rackwitz43 dene the sensitivity of a reliability index as thegradient of with respect to the parameters g in the limit state function

g 1Gk kgg z*;g

(18)

where G(u) =g(z(u)) = limit state function expressed in terms of the standard normal variables.42 Onceg is known, the gra-dient of the rst-order reliability approximation of the failure probability is obtained using the chain rule of differentiation as

g p1 g (19)

where () = standard normal PDF.

4.3. Importance measures

Among several random variables in the limit state function, some have larger effect on the variance of the limit statefunction and thus are more important. Der Kiureghian and Ke44 proposed the measure of importance dened as

T TJu*;z*SD

TJu*;z*SD

(20)where = unit vector at the design point directed toward the failure set, z= vector of random variables, Ju *,z * = Jacobian ofthe probability transformation from the original space z to the standard normal space u with respect to the coordinates of thedesign point z* and nally, SD= standard deviation matrix of equivalent normal variables z dened by the linearizedinverse transformation z= z * + Jz *,u *(u u *) at the design point. The elements of SD are the square roots of thecorresponding diagonal elements of the covariance matrix = z * + Jz *,u *J

Tz *,u * of z.

5. ILLUSTRATION

As an illustration of the proposed probabilistic models and framework, we assess the reliability of a typical 5 MW windturbine with the same characteristics as those of the NREL offshore 5 MW baseline wind turbine, introduced by Jonkmanet al.45 The considered wind turbine is installed in a 20 m water depth and supported by a monopile foundation that istypical for this water depth. Table VIII lists all the relevant properties.

5.1. Predictive fragility

Monte Carlo simulations are used to estimate the predictive fragility for the example wind turbine support structure, wherethe parameters in the demand models are considered to be random variables with a Nataf distribution46 such thatv andmare jointly normal, v and m are lognormal and follows a beta distribution. Statistical properties of the model parameters

Table VIII. Properties of the NREL offshore 5-MW baseline wind turbine.

Rating 5MWRotor diameter 126mHub height 90mCut-in, rated, cut-out wind speeds 3ms1, 11.4ms1, 25ms1

Rotor mass 110 000 kgNacelle mass 240 000 kgTower mass 347 460 kgNatural period of the tower 2.5 sTower top diameter and wall thickness 3.87m, 0.019mTower base diameter and wall thickness 6.00m, 0.027mWater depth 20mTurbulence intensity 0.10Signicant wave height 1.0mWave peak period 4.87 s



are given in Tables V and VI. The random variables that dene the capacity models are described in Table IX. A COV of10% is assumed for yield and ultimate strength of steel.47 The statistics ofMu is obtained using moment-curvature diagramsconstructed for the tubular cross section of the tower base, considering the stressstrain curve of structural steel of GradeS235. A COV of 30% is selected for soil strength properties on the basis of data collected from literature.48

Figure 5(a) presents the predictive fragility estimates plotted as a function of spectral acceleration Sa at the natural periodof the support structure (Tn= 2.5 s) within its linear elastic range, for both the yield and ultimate limit states. The signicantwave height is set to Hs= 1m. The dotted, solid and dashed lines in the gure show the fragilities for cut-in, rated and cut-out wind speeds, respectively, where cut-in and cut-out wind speeds are the lower and upper limits of the range of windspeeds in which a turbine is operating and producing power. The rated wind speed is the wind speed at which a controlsystem is activated to keep the power-generated constant by changing the blade pitch angle. The pitch control system limitsthe aerodynamic forces on the blades and consequently the operational loadings on the support structure of the windturbine. As shown in the gure, the fragility at the rated wind speed is higher than the fragilities at the other two windspeeds because of the higher wind speed than the cut-in wind speed and higher operational loading than at the cut-out windspeed. However, the contribution of the wind loading in the operational range of wind turbines is not signicant comparedwith the seismic excitation even for small earthquakes. In addition, the fragility in shear failure mode is found to benegligible compared with that in the bending failure mode, as expected for slender elements like wind turbines towers.

Predictive fragility estimates due to yield and ultimate limit states are also plotted as a function of the mean wind speed, forHs=1m and in absence of earthquakes, Sa=0 (Figure 5(b).) The gure shows how the fragility rapidly increases after the cut-out wind speed due to the lack of aerodynamic damping for parked (idle) wind turbine in the presence of high wind speeds.

5.2. Sensitivity measures

We dene g= [E(kt/kf),E(Cs),E(Mu),E(fy),PGD,PGV], where E() = the expected value of the variable. Table X lists thesensitivity measures for the moment failure mode for both yield and ultimate capacity levels. Results show that increasingthe mean of the tower to foundation stiffness ratio kt/kf is the most effective way of increasing the bending moment reliabil-ity (reducing the probability of failure.) Also, the shear wave velocity Cs happens to be the second most important param-eter, whose increment (increasing the soil stiffness) will increase the reliability of the support structure.

Figure 5. Fragility estimates for a typical 5 MW offshore wind turbine as a function of (a) spectral acceleration and (b) mean wind speed.

Table IX. Distribution, mean and COV for the random variables in the capacities in the limit statefunction.

Random variables Distribution Mean COV (%)

fy Lognormal 300.0 10fu Lognormal 410.0 10Mu Lognormal 390.6 10.13Cs Lognormal 109.2 30kt/kf Lognormal 0.0020 30



5.3. Importance measures

Table XI shows the importance measures for the moment failure mode, for yield and ultimate capacity performance levels,where z= (xp,m), in which xp= (kt/kf,Cs,Mu, fy, fu, m) are the random variables in the limit state function, in addition tothe model parametersm considered in a Bayesian approach as random variables. We see that in addition to the model errorm, some of the model parameters m1, m18 and m13 are also important random variables that affect the variance of thelimit state function. On the other hand, there are random variables in the limit state function, which are not important,and we can ignore their uncertainty in developing fragility estimates without a signicant loss of accuracy. Therefore,we partition z in a vector of constant parameters zc= (kt/kf,Cs,Mu, fy, fu,m, m2, m11, m15), which includes the point estimatesof unimportant random variables equal to their mean values, and a vector of random variables zp = (m, m1, m18, m13),so that z can be written as z = (zc, zp). By reducing the number of random variables in the limit state function, we makethe computation of fragilities faster without signicant loss of accuracy. This observation can be helpful to expedite thecomputation of the reliability of similar wind turbines.

5.4. Structural damage states

Once the fragilities and the annual PDFs for seismic and wind hazards are available, we can estimate the annual probabil-ities of being in any specied damage state for a wind turbine support structure at any particular locations using the totalprobability rule (equation (3)).

In this paper, we select two locations: site I in the Gulf of Mexico, about 70Km east of Galveston, Texas, with latitude of2925N and longitude of 9403W, which is prone to hurricanes; and site II in the west coast, about 90Km west of SantaBabara, California; with latitude of 3416N and longitude of 12042W, which is in a high seismic region.

We use USGS seismic hazard exceedance curves for the Gulf of Mexico and the west coast of the USA. Figure 6 showsa comparison of USGS seismic hazard curves at the two sites of interest. The gure clearly shows that the annual proba-bility of occurrence of an earthquake at site II is signicantly larger than at site I.

Table X. Sensitivity measures for the moment failure mode for both yield and ultimate performancelevels.

xc

Parameter, xc Symbol Yield Ultimate

Mean of tower to foundation stiffness ratio E(kt/kf) 220.1 124.0Mean of shear wave velocity of soil E(Cs) 1.333 1.542Mean of ultimate bending moment capacity E(Mu) 0.000 0.000Mean of yield stress of steel E(fy) 0.000 0.000Peak ground displacement PGD 0.001 0.003Peak ground velocity PGV 0.294 0.062

Table XI. Importance measures for the moment failure mode for both yield and ultimate performancelevels.

Random Variable Symbol

i

Yield Ultimate

Model parameter for hm1 m1 0.674 0.673Model error/m m 0.501 0.500Model parameter for hm18 m18 0.419 0.421Model parameter for hm13 m13 0.275 0.275Model parameter for hm15 m15 0.104 0.116Shear wave velocity of soil Cs 0.115 0.110Ultimate bending moment capacity Mu 0.000 0.098Model parameter for hm11 m11 0.071 0.071Tower to foundation stiffness ratio kt/kf 0.066 0.064Model parameter for hm2 m2 0.014 0.014Standard deviation of moment model error m 0.029 0.001Yield stress of steel fy 0.097 0.000



For the day-to-day wind, we use the BIW as presented in equation (5), with the distribution parameters estimated byMorgan et al.27 for sample ocean buoy stations at both sites I and II. Table XII lists the BIW distribution parameters forthe two locations. Depending on the type of buoy, the wind speed recorded by National Data Buoy Center28 is measuredat either 5 or 10m above sea level.27 However, we are interested in wind speed data at the height of turbine hub. To obtainthe wind speed at the turbine hub height, we use an empirical approximation of wind speed prole Ws(h) written as

49

Ws h Vr hHr

(21)

In which, Vr =wind velocity at a reference height, where a common choice for the reference height is Hr= 10m. TheGuidelines for Design of Wind Turbines (DNV/Riso)49 suggests a value of = 0.12 for offshore winds.

The PDFs of the wind speed at both sites I and II are then modied to represent the probability density of wind speeds atthe wind turbine hub height. As mentioned earlier, Wang29 found that the lognormal distribution provides the best t tomeasured data for the annual PDF of the gradient wind speeds during hurricane along the Texas coastline. The gradientlevel is generally taken as between 500 and 2000m. Lee and Rosowsky50 summarized the gradient-to-surface wind speedconversion factor for 10 min sustained wind speeds for different locations. They suggested a value of 0.65 for offshoresites. The surface wind speed is the value of wind velocity at 10m height above the ground or sea level. However, the windspeed at a wind turbine hub height is of interest in this study. We convert the gradient wind speed to hub height wind speedby rst bringing it down to surface by applying the gradient-to-surface conversion factor and then taking it up to hub heightusing equation (21). Finally for site I, we use the lognormal distribution function presented in equation (6) for the hubheight wind speed during the hurricane with location and scale parameters of = 3.348 and = 0.34, respectively.

The occurrence of hurricane is modeled as a Poisson process with an annual occurrence rate of hurricane to be = 0.1689for site I and = 0 for site II, on the basis of the Historical Hurricane Tracks database at the National Oceanic andAtmospheric Administration.51 Wind hazard curves are then developed using equation (7). Figure 7 shows the wind hazardcurves for the two particular locations of interest. It can be seen that site II has generally higher day-to-day wind speeds andis not expected to experience hurricanes.

Table XIII lists the annual probabilities of being in the damage states specied in Table VII for the NREL offshore 5MW wind turbine subject to different hazards. Table XIII shows that even though the occurrence rate of hurricane issignicantly larger at site I than site II, the wind hazard alone happens to result in the same failure probabilities for the

Figure 6. Annual probability density function for spectral acceleration at site I (dotted line) and site II (solid line).

Table XII. BIW distribution parameters.

ParameterValue

Site Ia Site IIb

1 4.455 1.6941 5.809 5.5152 2.067 4.4052 6.368 9.848 0.141 0.467aBuoy 42035.29bBuoy 46063.29



two sites of interest. Figure 7(b) can explain the reason for the relatively high probabilities of exceeding the performancelevels at site II due to wind hazard. The PDF of day-to-day wind speed for site II has a considerably higher density aroundthe rated wind velocity than site I, which has most of its PDF density at the low wind speeds (where the fragility is small).For the case of seismic hazard in the presence of day-to-day wind for operating wind turbine, the wind turbine installed atsite II has considerably larger probabilities of exceeding the performance levels than the one installed at site I, as expectedon the basis of the seismic hazard curves. Finally with the consideration of multiple hazards (wind and seismic), it can beseen that site II results in an overall higher failure probability for the wind turbine considered in this study.

It is noteworthy to mention that the obtained probabilities of complete damage (damage state C) are generally higher thanthe nominal annual failure probability of 104, taken by DNV-OS-J1016 as target safety level for structural design of supportstructures and foundations for wind turbines to the normal safety class. Given the fact that the DNV-OS-J101 guideline6 onlyconsiders wave loads and normal and extreme wind conditionsexcluding conditions experienced in tropical storms6inaddition to permanent loads, the higher values show the importance of considering hurricane and seismic loading.

6. CONCLUSIONS

A probabilistic framework is proposed to evaluate the multi-hazard structural reliability of offshore wind turbines. Probabilisticmodels were developed for shear andmoment demands on the support structure of wind turbines using the information obtainedfrom detailed 3D non-linear FE models. The FE models incorporated the aeroelastic interaction as well as the inuence of soil-structure interaction in the dynamic response of the support structures. The developed demand models are used to assess thefragilities of an example offshore wind turbine subject to day-to-day and extreme wind speeds in addition to seismic loading.No specic design load cases are considered in the simulations. However, one could use the methodology described in thispaper for specic design load cases to assess the corresponding probability of failure. Incorporating the hazard informationfor two particular locations in the USA (one in the Gulf of Mexico and one off the California Coast), the annual probabilitiesof specied structural damage states are evaluated for a typical 5 MW offshore wind turbine subject to day-to-day and extremewind loads during hurricane in the presence of the seismic risk. The fact that obtained values for annual probability of failureconsidering seismic and tropical storm hazards are higher than target safety level recommended by guidelines for industry prac-tice, excluding those hazards, shows the importance of considering seismic and hurricane loading for design purposes, althoughwith the wind turbines operating at their maximum rate of power production at the rated wind speed, theWest Coast could be an

Figure 7. Annual probability density function for wind speeds at (a) site I and (b) site II.

Table XIII. Annual probabilities of specied damage states.

Damage statePerformance

level

Annual probabilities of being in specied damage states

Wind hazard Seismic hazard Multi-hazard

Site I Site II Site I Site II Site I Site II

NDYield limitexceedance

0.9869 0.9861 0.9902 0.9771 0.9857 0.9733

PO 0.0116 0.0123 0.0088 0.0196 0.0126 0.0230Ultimate limitexceedance

C 0.0015 0.0016 0.0010 0.0033 0.0017 0.0037

ND, no signicant damage; PO, permanently out-of-service; C, complete.



ideal location for an offshore wind farm on the power production side. However, on the basis of the resulted annual probabilitiesof failure, there is a higher failure risk for wind turbines installed in the West Coast of the USA, because of high seismicity.Given that choosing a location for a wind farm is always a trade-off between multiple considerations, including possible loss,power production potential and construction cost, this paper proposes a valuable framework to estimate the failure probability tobe used in estimation of possible losses.

ACKNOWLEDGEMENT

The authors acknowledge the Texas A&M Supercomputing Facility (http://sc.tamu.edu/) for providing computingresources useful in conducting the research reported in this paper.

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