Risk Assessment and Sensitivity Analysis for Offshore Wind Turbines

Alexandros A. Taflanidis Department of Civil Engineering and Geological Sciences, University of Notre Dame

Notre Dame, IN, U.S.A

Eva Loukogeorgaki and Demos C. Angelides Department of Civil Engineering, Aristotle University of Thessaloniki

Thessaloniki, Greece

ABSTRACT

A comprehensive risk assessment framework is discussed in this paper for the support structure and the tower of an offshore wind turbine under extreme wind and wave conditions. The framework is founded on a probabilistic characterization of the uncertainty in the models for the excitation, the turbine and its performance. A comprehensive computational model is used for describing the dynamic behavior of the turbine and stochastic simulation is proposed for evaluating the associated stochastic integral quantifying risk. For improvement of the computational efficiency, a surrogate modeling approach is introduced based on moving least squares response surface approximations.

KEY WORDS: Offshore wind turbines; risk assessment; sensitivity analysis; stochastic simulation; probabilistic uncertainty.

INTRODUCTION

Offshore Wind Turbines (OWTs) (Fig. 1) represent nowadays an attractive alternative solution to the onshore wind turbines, offering multiple benefits and addressing effectively the well- known obstacles and problems associated with the latter ones (Henderson et al. 2003; Breton and Moe 2009). However, their design and operation are characterized by high complexity and uncertainty due to extensive variability of components, intense interaction among components and assemblies, multiple uncertain loading sources acting on the OWTs parts, and different OWTs operating/loading conditions. For an efficient design such uncertainties need to be explicitly addressed, indicating the necessity for a risk-informed approach. Under this consideration, Cheng at al. (2003) presented a reliability-based approach for determining the extreme response distribution of OWTs. Thns et al. (2008) and Thns et al. (2010) performed a reliability analysis for the support structure of a fixed bottom OWT, considering the ultimate, the fatigue and the serviceability limit states. The analysis was performed utilizing stochastic finite elements in conjunction with an adaptive response surface algorithm and an importance sampling

Monte-Carlo algorithm. The dynamic response analysis of the support structure of an OWT under wave and seismic loading including uncertainty was performed by Kawano et al. (2010). The dynamic response was obtained using the substructure method for a two-dimensional model of the OWTs support structure with pile-soil foundation. The Monte Carlo Simulation method was applied to evaluate the maximum response characteristics of the OWTs support structure, in conjunction with the second moment approach.

Fig.1: Offshore wind turbine.

In the present paper, a comprehensive risk assessment framework is presented for the support structure and the tower of an OWT under extreme wind and wave conditions. The structural model is based on the Finite Element Method (FEM). The Morison equation is applied in order to calculate the hydrodynamic forces on the support structure of the OWT, with water particle kinematics evaluated using a higher order wave theory. Characterization of the uncertainty in the parameters of these structural and excitation models, through appropriate probabilistic descriptions, leads then to quantification of the overall risk. This risk is

Tower

Support Structure

(including

foundation)

479

Proceedings of the Twenty-first (2011) International Offshore and Polar Engineering ConferenceMaui, Hawaii, USA, June 19-24, 2011Copyright 2011 by the International Society of Offshore and Polar Engineers (ISOPE)ISBN 978-1-880653-96-8 (Set); ISSN 1098-6189 (Set); www.isope.org

ultimately expressed as a stochastic integral of the response over the uncertain parameter space. To efficiently address the complexity of the adopted models evaluation of this integral through stochastic simulation is suggested and for improved computational efficiency a surrogate modeling approach is discussed. In particular a moving least squares response surface approximation is considered for this task. The framework for risk assessment is also extended to an efficient sensitivity analysis. Such analysis aims to identify which are the critical risk factors contributing most to the overall risk and is efficiently performed here with minimal additional computational effort over the risk assessment task. This is established through an information entropy definition of the sensitivity, and efficient calculation based on advanced stochastic sampling techniques.

RISK QUANTIFICATION

The risk estimation approach adopted here is an extension of the systematic framework proposed by Taflanidis and Beck (2009a). Quantification of this risk requires adoption of appropriate models for (i) the wind turbine, (ii) the excitation (wind and wave), and (iii) the system performance (Fig. 2). The combination of the first two models provides the structural response z. The performance evaluation model assesses, then, the favorability of this response based on the chosen criteria of the systems stakeholders and provides vector y composed of the desired performance quantities.

The response vector for a specific excitation scenario and system configuration may be accurately estimated by numerical simulation, once an appropriate high fidelity wind turbine model is established. This task will be discussed in more detail in following sections. Since the high-fidelity model requires extensive computational effort for each analysis, a surrogate model will be also developed for simplification of the risk evaluation. This surrogate model is based on information provided by a number of pre-computed evaluations of the computationally intensive high-fidelity model, and ultimately establishes an efficient approximation for the response z and the associated performance vector y with dimension ny. This approximation is denoted herein. The relationship between each component of the actual performance vector yi and corresponding component that is provided through the surrogate model, i, is

i i iy y = + (1)

where i is the model prediction error that is established through the introduced approximations to the system performance.

Excitation model with

parameters qq=Q(q)

Performance Evaluation model with parameters p

y=J(z;p)

Response z

Risk occurrence measure h()for specific excitation & turbine configuration

Uncertainty in={q, s, p }: p()

+Risk

Excitation q

Wind Turbine

model with parameters s

x=F(q;s)1z=G(x;s)21 State evolution equation

2 Output (response) equation

Fig. 2: Risk quantification concept

The characteristics of these models used to calculate the wind turbine performance are not known with absolute certainty. Uncertainties may pertain to: (i) the properties of the wind turbine, for example, related to damping characteristics of the structure or to soil properties for the foundation, (ii) the variability of future excitation events, i.e., the wind speed or the significant wave height, or (iii) parameters related to the performance of the system, for example, thresholds defining fragility of system components. A probability logic approach provides a rational and consistent framework for quantifying all these uncertainties and explicitly incorporating them into the system description. In this approach, probability can be interpreted as a means of describing the incomplete or missing information (Jaynes 2003) about the system under consideration and its environment, representing future excitation hazards, through the entire life-cycle (Taflanidis and Beck 2009a).

To formalize these ideas let n , denote the augmented n-dimensional vector of model parameters where represents the space of possible model parameter values. Vector is composed of all the uncertain model parameters for the individual structural system, s, excitation, q, and performance evaluation, p, models as illustrated in Fig. 2. Let the performance of the augmented model, for specific , be characterized by the performance measure ( ) : nh + , which represents the utility-loss from a decision-theoretic point of view (i.e., corresponds to a risk-occurrence measure) and ultimately is estimated based on the predicted performance vector y. The assumption that risk corresponds to a strictly positive quantity is used herein. For addressing the uncertainty in a PDF (probability density function) p(), is assigned to it, quantifying the relative likelihood of different model parameter values. This PDF incorporates the available knowledge about the wind turbine system and its environment into the respective models. In this stochastic setting, risk, H, is finally described by the following stochastic integral that corresponds to the expected value of h() over the probability models:

( ) ( ) ( )

H E h h p d= = (2)

where E[.] denotes expectation over the uncertain parameters. Different selections for the risk-occurrence measure h() lead to different quantifications for risk H. For example, if h()=Cin()+Clif(), where Cin() corresponds to the initial cost and Clif() to the additional cost over the lifetime due to repairs or downtime, then risk corresponds to life cycle cost (Taflanidis and Beck 2009a); if h()=IF(), where IF() is the indicator function for some event F (one if F has occurred and zero if not), then risk corresponds to the probability of unacceptable performance (Taflanidis and Beck 2009b).

In this setting it is evident that risk, as described by the stochastic integral (Eq. 2), is directly related to the uncertainty in the model parameter vector . This leads to a direct characterization of the risk factors contributing to this risk; they correspond to the uncertain parameters in the augmented mathematical model (Fig. 2) that describes the response and performance of the real system for anticipated future excitations.

For evaluation of Eq. 2 a stochastic-simulation approach will be discussed next. Along with the risk quantification described above (and demonstrated in Fig. 2) this establishes a versatile, end-to-end simulation based framework for detailed description of wind turbine risk. This framework puts no restrictions in the complexity of the models used allowing for incorporation of all important sources of nonlinearities as well as consideration of complex probability models for description of the uncertainties in the turbine response and its environment.

480

RISK ASSESSMENT AND SENSITIVITY ANALYSIS

Assessment

Since the models adopted for the wind turbine risk characterization can be complex, i.e. include a large number of model parameters and various sources of nonlinearities, the expected value (Eq. 2) cannot be calculated, or even accurately approximated, analytically. An efficient alternative approach is to estimate this integral by stochastic simulation (Robert and Casella 2004). In this case, using a finite number, N, of samples of simulated from some importance sampling density q(), an estimate for Eq. 2 is given by the stochastic analysis:

1 1 / ( ) ( ) / ( ) N j j jjH N h p q== (3)

where vector j denotes the sample of the uncertain parameters used in the jth

simulation and {j} corresponds to the entire sample set. As N, then H but even for finite, large enough N, Eq. 3 gives a good approximation for Eq. 2. The quality of this approximation is assessed through its coefficient of variation, . An estimate for may be obtained through the information already available for the risk assessment (Eq. 3) through the following expression (Robert and Casella 2004):

21

2

1 ( ) ( ) / ( )1 1

N j j ji

h p qN

N H

=

(4)

Thus the stochastic analysis provides not only with an estimate for the risk integral, but simultaneously with a measure for the accuracy of that estimate. The importance sampling density q() may be used to improve this accuracy and, ultimately, the efficiency of the estimation of Eq. 3. This is established by focusing the computational effort on regions of the space that contribute more to the integrand of the stochastic integral in Eq. 2 (Taflanidis and Beck 2008). The simplest selection is to use q()=p(), then the evaluation in Eq. 3 corresponds to direct Monte Carlo analysis. For problems with large number of model parameters choosing efficient importance sampling densities for all components of is challenging and can lead to convergence problems for the estimator in Eq. 3; thus it is preferable to formulate importance sampling densities only for the important components of , i.e. the ones that have biggest influence on the system performance, and use q(.)=p(.) for the rest (Taflanidis and Beck 2008).

In the proposed computational framework, the risk occurrence measure in Eq. 3 is approximated for each sample through the developed surrogate model. Thus estimation 3 may be efficiently performed, even for a large selection for N, once this model has been established. This also indicates that the surrogate model should be formulated so that it provides good accuracy for components of the model parameter vector and ranges within their region of possible values that have higher contributions to the integrand of Eq. 2.

Sensitivity Analysis

This simulation-based framework for risk assessment may be extended to additionally investigate the sensitivity of the total risk with respect to each of the uncertain model parameters. Such a sensitivity analysis aims to identify which of the risk factors has higher overall importance when calculating the total risk. This innovative sensitivity analysis is based on the ideas initially proposed by Taflanidis (2009). Foundation of this methodology is the definition of an auxiliary density function that is proportional to the integrand of the risk integral

( ) ( )( ) ( ) ( )( ) ( )

h p h ph p d

pi =

(5)

where denotes proportionality. The sensitivity analysis is established by comparing this auxiliary distribution pi() and the prior probability model p(); based on the definition of pi() in Eq. 5 such a comparison provides directly information for h(). Bigger discrepancies between distributions pi() and p() indicate greater importance of in affecting the system performance, since they ultimately correspond to higher values for h(). More importantly, though, this idea can be implemented to each specific model parameter i (or even to groups of them), by looking at the marginal distribution pi(i)

( ) ( ) ( ) ( ) ( | )i i i i i id p h p dpi pi = x x x (6)

where xi corresponds to the rest of the components of , excluding i. Comparison between this marginal distribution pi(i) and the prior distribution p(i) expresses the probabilistic sensitivity of the risk with respect to i. Uncertainty in all other model parameters and stochastic excitation is explicitly considered by appropriate integration of the joint probability distribution pi() to calculate the marginal probability distribution pi(i). A quantitative metric to characterize this sensitivity is the relative information entropy, which is a measure of the difference between probability distributions pi(i) and p(i)

( ) ( )( ) || ( ) ( ) log ( )i

i i i ii

D p dppi

pi pi

=

(7)

The importance of each parameter (or sets of them) is then directly investigated by comparison of the relative information entropy value for each of them; larger values for D(pi(i)||p(i)) indicate bigger importance.

An analytical expression, though, is not readily available for the marginal distribution pi(i) since evaluation of Eq. 6 is challenging. An alternative stochastic-sampling approach is discussed next, based on generation of a set samples, {k}, from the joint distribution pi(). Such samples may be obtained by any appropriate stochastic sampling algorithm, for example by the accept-reject method (Robert and Casella 2004). Furthermore this task may be seemingly integrated within the stochastic analysis (Eq. 3): each of the samples j from Eq. 3 can be used as a candidate sample in the context of the Accept-Reject algorithm. Projection, now, of the samples from pi() to the space of each of the model parameters provides samples for the marginal distributions pi(i) for each of them separately. Thus using the same sample set {k} this approach provides simultaneously information for all model parameters. For scalar quantities, as in this case, the relative entropy (Eq. 7) may be efficiently calculated by establishing an analytically approximation for pi(i), based on the available samples, through Kernel density estimation (Beirlant et al. 1997). This estimate will not necessarily have high accuracy, but it can still provide an adequate approximation for computing the information entropy integral. A Gaussian Kernel density estimator may be used for this purpose. Using the n available samples for i, with ik denoting the kth sample and si the standard deviation for these samples, the approximation for pi(i) would be

2

2( )

2 1/5

1

1( ) ; 1.062

ki i

lin

i li sikli

e nn

pi pi

=

= = (8)

For establishing better consistency in the relative information entropy

481

calculation p(i) may also be approximated by Eq. 8 based on samples, even when an analytical expression is available for it. This way, any type of error introduced by the Kernel density estimation is similar for both of the densities compared. The approximation for the relative information entropy is finally (Beirlant et al. 1997):

( ),

,

( )( ) || ( ) ( ) log ( )i u

i l

bi

i i i iib

D p dppi

pi pi

(9)

where the last scalar integral can be numerically evaluated, for example using the trapezoidal rule, and [bi,l, bi,u] is the region for which samples for pi(i) and q(i) are available.

This approach ultimately leads to an efficient sampling-based approach for calculating the relative information entropy for different parameters, which can be performed concurrently with the risk assessment, exploiting the readily available system model evaluations to minimize computational burden. Comparing the value for this entropy between the various model parameters leads then to a direct identification of the importance of each of them in affecting risk. Parameters with higher value for the relative entropy will have greater importance.

HIGH FIDELITY OFFSHORE WIND TURBINE MODEL

The OWT is modeled using a comprehensive structural numerical model (MICROSAS) appropriate for the integrated modeling, analysis and design of any kind of offshore structures. The model is based on the Finite Element Method (FEM) utilizing the direct stiffness method for the structural analysis and furthermore has the capability of performing stress analysis according to American Petroleum Institute (2000). In this numerical model, the structural system of the OWT consists of the tower, the support structure (including piles foundation) and the Rotor Nacelle Assembly (RNA). The tower and the support structure are modeled as a frame composed of tubular members connected with joints. The tubular members are further discritized in segments of specific geometric and material characteristics. The RNA is modeled using joints that correspond to the rotor and the nacelle centers of mass and are rigidly connected to the tower top joint. Fig. 3 includes the modeled structural system of an OWT with a tripod support structure, which corresponds to a prototype of the MULTIBRID M5000 OWT [Bicker (Offshore Wind Technology GmBH), personal communication].

Fig.3: Modeled structural system of the MULTIBRID M5000 OWT.

With regard to the foundation, the nonlinear soil-pile interaction is taken into account considering nonlinear horizontal and vertical springs distributed along the length of each pile (Fig. 4). The nonlinear characteristics of these springs are presented by P-Y and T-Z curves, respectively, P-Y curves describe the nonlinear relation between the lateral resistance of the soil per unit length of the pile, P (KN/m), and, the lateral deflection of the pile, Y (m). T-Z curves describe the nonlinear relation between the axial skin friction per unit area of the pile surface, T (KN/m2), and the relative axial pile-soil displacement, Z (m), necessary to mobilize this skin friction. The P-Y curves, which are input in the numerical model, correspond to N different elevations below the mudline. This enables the inclusion in the numerical analysis of a soil consisting of layers with different soil characteristics. As far as the T-Z curves, these are calculated internally in the numerical model, considering as input the corresponding soils skin friction curves.

For the OWT loading we will be focusing in this study on extreme environmental conditions, under which the turbine is in standstill situation (non-operational). It should be noted, though, that the generalized framework for risk quantification and assessment can be extended to any operational situation. Loads due to dead weight of the whole structure (including the weight of the RNA), buoyancy, marine growth, waves, wind and inertia are taken into account and calculated in detail. For a specific wave excitation, described by the significant wave height Hs and up-crossing period Tz, the Morison equation is applied in order to calculate the hydrodynamic forces on the support structure of the OWT, with water particle kinematics evaluated using linear or high-order wave theory. In Morison equation the drag CD and inertia CM coefficients present inputs to the numerical model that should be appropriately estimated. On the other hand, for a specific wind excitation, described by a reference wind velocity Vhub , selected here at the hub height, and the chosen wind velocity profile (following a specified power low). The inertia loads are calculated based on an eigenvalue analysis, where the foundation piles are replaced with equivalent linear and rotational springs, after an appropriate linearization of the piles behavior. All the above loads are then combined using appropriately defined partial safety factors, in order to form the loading combination for the structural and stress analysis. Based on the results of this analysis, the quantities that describe the performance of the OWT, i.e. joints displacements, member forces, stresses ratios, etc, are, finally, obtained.

This defines a computationally expensive high fidelity numerical model for prediction of the dynamic response of the OWT. An approximate response-surface-based surrogate model is further developed, based on information obtained by a small number of high fidelity runs, for efficient approximation of Eq. 3.

Fig. 4: Modeling of nonlinear soil-pile interaction.

482

MOVING LEAST SQUARES (MLS) RESPONSE SURFACES

Response surface approaches (Myers and Montgomery 2002) aim to approximate a complex process, requiring large computational cost for its evaluation, by a simpler mathematical model. This is established by expressing the function corresponding to the initial process fj() : n , where =[1n] n denotes the vector of free variables, through a number of NB prescribed basis functions bi() : n . The approximation is expressed as a linear combination of the bi() by introduction of coefficients ai{}; i=1,,NB. The latter can be constant or depend on the location for which the interpolation is established. The approximation is

1 ( ) ( ) { } ( ) { }NB Tj i iif b a== = b a (10)

where b() is the vector of basis functions and a{} is the vector containing the coefficients for the basis functions. Different classes of basis functions have been suggested and used in the literature. A common choice is a full second order approximation:

1

1

2 21 1 1 2

1 11 12

( ) { } { }( 3) 2

{ } ; 2

( ) 1 ... ...

( ) { } { } ... { } { } { } ... { }y

n

j o i ii

n n

ik i ki k j

n n

o n n n

f a an n

a NB

a a a a a a

=

=

= +

+ ++ =

=

=

x

b

a

(11)

The coefficients a{} are calculated by initially evaluating fj() in a set of NS>NB support points {; I=1,,NS}, and then by minimizing the mean squared error over these points between fj() and the approximation established through Eq. 10. In the Moving Least Squares (MLS) approach the coefficients are dependent on , and are selected by minimizing a weighted sum of squared error, with weights that are a function of as well

2

1{ } { } ( ) ( )

{ } { } { }

NSR j I j II

T

j j

J w f f=

=

=

Ba F W Ba F (12)

where the following quantities are introduced

( ) ( )

1 1

1

( ) ... ( ) ; ( ) ... ( ){ } ( ; ) ... ( ; )

TTNS j j j NS

NS

f fdiag w d w d

= =

=

B b b F

W (13)

and w{d(;I)} is a variable weight function with a compact support that depends on some measure of the distance between the interpolation point and each of the supporting points. A typical selection for this distance is a weighted quadratic vector norm

( )

( ) ( )

2 2,1

1

( ; )

; ( ... )

n

I I i i I ii

T

I I n

d v

diag v v

== =

= =

v

v v v

(14)

with vi representing the relative weight for each component of xi. Since v is diagonal, the transformation v corresponds to scaling of each component xi by vi. The introduction of the dependence on the distance weights w{d} aims at reducing the approximation error at each point by performing a weighted local averaging of the information obtained by

the support points that are closer to it. Without these weights, the coefficient vector, a, would be constant over the whole domain for , which means that a global approximation would be established (global least squares). The efficiency i.e, fit to fj() of global approximations depends significantly on the selection of the basis functions, which should be chosen to resemble fj() as closely as possible. Such a selection is not always straightforward. The MLS circumvents such problems by establishing a local approximation for a{} around each point in the interpolation domain. This leads to a smaller dependence of the fit on the type of basis functions used (Breitkopf et al. 2005). On the other hand the efficiency of the MLS interpolation depends on the weighing function chosen. This function should prioritize support points that are close to the interpolation point, and should vanish after an influence radius D. An appropriate support size D should be selected at any point so that a sufficient number of neighboring supporting points are included to avoid singularity in the solution for a{}. This means that D should include at least NB points. Many types of weighting functions have been suggested in the literature. One of the most common is the exponential type of function

2 2 21 1

{ } [ ] / [1 ] if 0 else

k k kdcD c cw d e e e d D

=