# Probabilistic Design of Wind Turbines

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## Abstract

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## 1. Introduction

## 2. Reliability Modeling of Wind Turbine

- Electrical and mechanical components where the reliability is estimated using classical reliability models, i.e. the main descriptor is the failure rate and the MTBF (Mean Time between Failure). Further, the bath-tub model is often used to describe the typical time dependent behavior of the hazard rate. The reliability is often modeled by a Weibull distribution, see e.g. [1,2]. Using e.g. FMEA (Failure Mode and Effect Analysis) or FTA (Failure Tree Analysis), system models can be established and the systems reliability can be estimated, see [2]. Reliability modeling for electrical and mechanical components is not considered further in this paper.
- Structural members such as tower, main frame, blades and foundation where a limit state equation can be formulated defining failure or unacceptable behavior. Failure of the foundation could be overturning. Failure of a blade could be large deflections with nonlinear effects and delaminations. The parameters in the limit state equation are modeled by stochastic variables and the reliability is estimated using Structural Reliability Methods, e.g., FORM/SORM methods, see [3,4,5]. Reliability analysis of structural components and systems are considered in the following.

## 3. Modeling of Uncertainties

- Physical uncertainty also denoted inherent uncertainty is related to the natural randomness of a quantity, for example the annual maximum mean wind speed or the uncertainty in the yield stress due to production variability.
- Measurement uncertainty is related to imperfect measurements of for example a geometrical quantity.
- Statistical uncertainty is due to limited sample sizes of observed quantities. Data of observations are in many cases scarce and limited. Therefore, the parameters of the considered random variables cannot be determined exactly. They are uncertain themselves and may therefore also be modeled as random variables. Are additional observations provided then the statistical uncertainty may be reduced.
- Model uncertainty is the uncertainty related to imperfect knowledge or idealizations of the mathematical models used or uncertainty related to the choice of probability distribution types for the stochastic variables.

_{i}denotes the statistical parameters. Dependency between the stochastic variables can be modeled by joint distribution functions or correlation coefficients. A number of methods can be used to estimate the statistical parameters α

_{i}in distribution functions, e.g., the Maximum Likelihood method, the Moment method, the Least Square method or Bayesian statistics.

**H**is the Hessian matrix with second order derivatives of the log-Likelihood function. The statistical uncertainty can easily be included in a reliability analysis using FORM (First Order Reliability Method), see below.

**X**with realizations denoted

**x**. Further, the model is assumed to be a function of a number of regression parameters denoted ${R}_{1},...,{R}_{m}$. The regression parameters are determined by statistical methods, and are therefore subject to statistical uncertainty. The model is not perfect; therefore model uncertainty has in general also to be introduced. This is often done by a multiplicative stochastic variable R

_{0}. The model can thus be written:

_{0}is modeled by a LogNormal distributed stochastic variable with mean ${\mu}_{0}$ and standard deviation ${\sigma}_{0}$. The statistical parameters ${R}_{1},...,{R}_{m},{\mu}_{0},{\sigma}_{0}$ can be determined by the Maximum Likelihood Method using the Likelihood function:

_{0}) and statistical uncertainties (${R}_{1},...,{R}_{m},{\mu}_{0},{\sigma}_{0}$) can be modeled and estimated.

**X**) is available. This model will typically be a function of a number of uncertain parameters

**X**, e.g., strength and stiffness parameters and parameters describing the load. Further, the model will be subject to model uncertainty R

_{0}. The statistical parameters describing the physical uncertainties

**X**and the model uncertainty R

_{0}are in many cases determined by experiments and measurements, and therefore subject to statistical uncertainty. In Section 10 an example is presented where the different types of uncertainties are modeled. Another example is estimation of the long term energy production of a wind turbine/wind farm using e.g. WASP [10]. Examples of uncertainties to be taken into account are long-term site air density, turbulence intensity and long term wind speed (combined physical and model/prediction uncertainties); topography over the site and surrounding area (model uncertainty); power curve (model uncertainty); losses due to wakes (model uncertainty). Further statistical uncertainties will be associated with estimation of the statistical parameters using availble data. Similar uncertainties should be modelled when estimating the extreme loads, see example in Section 9.

## 4. Modeling of Structural Failure Modes and Reliability

- Local or global buckling failure of tower
- Fatigue failure of blade or details in substructure
- Foundation failure by sliding

**x**denote realizations of the stochastic variables

**X**which includes physical, model, statistical and measurement uncertainties. Realizations of

**x**where $g\left(x\right)<0$ denote failure states.

## 5. Recommendation of Target/Minimum Reliability Level

**z**(e.g., cross-sectional geometrical parameters) are determined from the optimization problem:

**z**and other practical, geometrical limits can be added. A similar optimization problem can be formulated if a systems reliability constraint is introduced.

- Cost benefit analysis, see below. The wind turbine design (including decisions on strategy for operation and maintenance) is optimized such that a minimum of all costs minus benefits is obtained. The corresponding reliability levels for different components can be used to assess ${\beta}_{i}^{\mathrm{min}}$.
- The Life Quality Index (LQI) concept can be used to assess the minimum acceptable reliability level in case wind turbine failure implies risk of loss of human lives, see below.

_{0}is the reference initial cost of corresponding to a reference design

**z**

_{0}. The optimal design

**z*** is determined by the solution to (8). The corresponding probability of failure, ${P}_{F}\left({z}^{*}\right)$ can be considered the optimal probability of failure related to the failure event and the actual cost-benefit ratios used. The failure rate λ and probability of failure can be estimated for the considered failure event, if a limit state equation, $g\left({X}_{1},...,{X}_{n},z\right)$, and a stochastic model for the stochastic variables, $\left({X}_{1},...,{X}_{n}\right)$, are established. If more than one failure event is critical, then a series-parallel system model of the relevant failure modes can be used, see below.

## 6. System Aspects

_{i}limit state equations ${g}_{i,j}\left(x\right)=0$, j = 1, 2,…, m

_{i}, i = 1, 2, …, m, the probability of failure of the whole system is given by:

_{dir}) or indirect (C

_{ind}). Direct consequences are considered to result from damage states of individual component(s). Indirect consequences are incurred due to loss of system functionality or failure and can be attributed to lack of robustness [9,16].

_{dir,ij}is the consequence (cost) of damage (local failure) D

_{j}due to exposure EX

_{i}, C

_{ind,ijis}the consequence (cost) of comprehensive damages (indirect) S

_{k}given local damage D

_{j}due to exposure EX

_{i}, P(EX

_{i}) is the probability of exposure EX

_{i}, P(D

_{j}|EX

_{i}) is the probability of damage D

_{j}given exposure EX

_{i}and P(S

_{k}|...) is the probability of comprehensive damages S

_{k}given local damage D

_{j}due to exposure EX

_{i}. The first term $\sum _{i}P\left({D}_{j}|E{X}_{i}\right)P\left(E{X}_{i}\right)$ express the probability of a local damage D

_{j}considering all exposures. The second term $\sum _{i}{\displaystyle \sum _{j}P\left({S}_{k}|{D}_{j}\cap E{X}_{i}\right)P\left({D}_{j}|E{X}_{i}\right)P\left(E{X}_{i}\right)}$ express the probability of comprehensive damage S

_{k}considering all exposures and local damages.

- Reducing one or more of the probabilities of exposures P(EX
_{i})—prevention of exposure or event control - Reducing one or more of the probabilities of damages P(D
_{j}|EX_{i})—related to element/component behaviour - Reducing one or more of the probabilities P(collapse|D
_{j}∩ EX_{i})

_{dir,ij}and comprehensive (indirect) consequences, C

_{ind,ij}are important. It is noted that increasing the robustness at the design stage will in many cases only increase the cost of the structure marginally—the key point is often to use a reasonable combination of a suitable structural system and materials with a ductile behaviour, if possible. In other cases increased robustness will influence the cost of the structural system.

## 7. Bayesian Statistical Methods

- Observation of events described by one or more stochastic variables. The observation can be modeled by an event margin. Updated/conditional probabilities of failure can then be estimated.
- Test samples/measurements of one or more stochastic variables, X. Updating can in this case be performed using Bayesian statistics.

- Inspection or monitoring events such as inspection of cracks. The event margin can include measurement uncertainty and the reliability of the inspection method.
- Proof loading where a well defined load is applied to the wind turbine and the level of damage is observed.
- No-failure events where the observation that the wind turbine/component considered is well-functioning after some time in use.

**α**are available Bayesian statistical techniques can be used to update a prior stochastic model ${f}_{{\rm A}}^{\prime}\left(\mathsf{\alpha}\right)$, see e.g. [8,19]. The posterior, updated stochastic model is denoted ${f}_{{\rm A}}^{\u2033}\left(\mathsf{\alpha}|\widehat{x}\right)$. The predictive, updated stochastic model for X is:

## 8. Framework for Integrated Uncertainty Modeling in Wind Turbine Design

- Coupon tests with basic material and measurements of climatic parameters performed at an early stage of the design process can be used to update the statistical description of the physical variables
**X**using Bayesian methods, see Section 7. - Tests and measurements of response parameters from prototype and 0-series wind turbine(s) and wind turbine parts/components (e.g., blades or drive train) can be used to update the model uncertainties associated with the mathematical models for the wind turbine behavior and failure modes. Bayesian methods can be used to update both physical and model uncertainties, see Section 7. When updating the model uncertainties it is assumed that the physical uncertainties are measured (or are known) such that the methods described in Section 3 can be used. The test results are often of the ‘event’ type, e.g., no failure of a wind turbine blade. It is noted that usually only a very limited number of prototypes or wind turbines parts (e.g., blades) are tested in full-scale implying a significant statistical uncertainty.
- When the wind turbine is in series production and many wind turbines are in operation then continuous condition monitoring of various parameters can be used to update physical and model uncertainties, and to decrease the statistical uncertainties. This information can be used to update/modify the design of new wind turbines of the same type, as information (prior knowledge) to development of new wind turbines, and as decision basis for possible life time extension (especially relevant for offshore wind turbines). Again the Bayesian methods described in Section 7 can be used to handle the information in a rational way.

## 9. Example–Optimal Reliability Level

_{0}is the reference cost corresponding to the reference radius R

_{0}= 8.5 m and area A

_{0}= 3/26 m

^{2}. The failure costs are assumed to be C

_{F}/C

_{0}= 1/36. The benefits per year are b/C

_{0}= 1/8 and the real rate of interest is assumed to be r = 0.05.

**Table 1.**Stochastic variables for local buckling failure mode. Variables denoted X model model-uncertainties. LN: Lognormal, G: Gumbel. COV: coefficient of variation.

Variable | Distribution type | Expected value | COV | |
---|---|---|---|---|

P | Annual maximummean wind pressure | G | 538 kPa | 0.23 |

I | Turbulence intensity | LN | 0.05 | 0.05 |

${C}_{T}A$ | Thrust coeff. x rotor disk area | 340 m^{2} | ||

${k}_{p}$ | Peak factor | 3.3 | ||

${X}_{\mathrm{exp}}$ | Exposure (terrain) | LN | 1 | 0.20 |

${X}_{st}$ | Climate statistics | LN | 1 | 0.10 |

${X}_{dyn}$ | Structural dynamics | LN | 1 | 0.10 |

${X}_{aero}$ | Shape factor/model scale | G | 1 | 0.10 |

${X}_{str}$ | Stress evaluation | LN | 1 | 0.03 |

${F}_{y}$ | Yield stress, structural steel | LN | 240 MPa | 0.05 |

E | Young’s modulus | LN | 2.1 × 10^{5} MPa | 0.02 |

${X}_{y,ss}$ | Yield stress, structural steel | LN | 1 | 0.05 |

${X}_{E,ss}$ | Young’s modulus | LN | 1 | 0.02 |

${X}_{cr}$ | Critical load capacity | LN | 1 | 0.10 |

^{−4}–10

^{−3}, corresponding to reliability indices in the interval 3.0–3.5. This reliability level is significantly lower than for civil engineering structures in general, but is of the same level as can be estimated from reported structural failures of wind turbines, see e.g. [22] where failure rates for blades are described. Further, this reliability level also corresponds to the reliability used to calibrate partial safety factors in the IEC 61400-1 [23], and IEC 61400-3 [24], standards, see [25,26] where the stochastic model in Table 1 has been used.

## 10. Example–Statistical Modeling Using Test Results

- Physical uncertainty on the SN-curves.
- Statistical uncertainty on the SN-curves due to a limited number of tests.
- Model uncertainty related to Miners rule.

_{F}is the number of cycles to failure, $\Delta \sigma $ is the stress range and ε models the lack of fit and is assumed normal distributed with mean value zero and standard deviation ${\sigma}_{\epsilon}$. The constants K and m are material dependent parameters. If N constant amplitude tests and N

_{0}run-outs are available then m is obtained using the least squares method. The parameters log K and ${\sigma}_{\epsilon}$ can be estimated using the Maximum Likelihood Method, see Section 3. The statistical uncertainty represented by the standard deviations ${\sigma}_{\mathrm{log}K}$ and ${\sigma}_{{\sigma}_{\epsilon}}$ and the correlation coefficient ${\rho}_{\mathrm{log}K,{\sigma}_{\epsilon}}$ is obtained using (1).

R-ratio | Number of tests | Number of run-outs | m | $\mathbf{log}\mathit{K}$ | ${\mathit{\sigma}}_{\mathit{\epsilon}}$ | ${\mathit{\sigma}}_{\mathbf{log}\mathit{K}}$ | ${\mathit{\sigma}}_{{\mathit{\sigma}}_{\mathit{\epsilon}}}$ |
---|---|---|---|---|---|---|---|

0.5 | 15 | 0 | 10.541 | 27.768 | 0.358 | 0.092 | 0.065 |

0.1 | 45 | 2 | 9.508 | 27.191 | 0.259 | 0.039 | 0.027 |

–0.4 | 28 | 0 | 7.582 | 23.398 | 0.435 | 0.082 | 0.058 |

–1.0 | 84 | 3 | 6.719 | 21.359 | 0.878 | 0.094 | 0.068 |

–2.5 | 10 | 2 | 11.983 | 35.231 | 0.633 | 0.197 | 0.143 |

10.0 | 34 | 0 | 22.211 | 58.664 | 0.644 | 0.110 | 0.078 |

2.0 | 6 | 3 | 29.686 | 73.780 | 0.354 | 0.143 | 0.103 |

_{C}is the number of stress cycles with stress ranges $\Delta {\sigma}_{i},i=1,2,...,{N}_{C}$. Fatigue failure occurs when the accumulated fatigue damage exceeds 1.

_{Δ}, standard deviation, σ

_{Δ}and coefficient of variation, COV

_{Δ}. The results show that except for the Wisper spectrum the estimated mean accumulated damage at failure is significantly below one and that the coefficients of variations are quite high. It is noted that the uncertainty for fatigue damage accumulation often is modeled by a Lognormal distribution in order to avoid negative values of Miners rule.

Spectrum | Number of tests | Mean ${\mathit{\mu}}_{\mathbf{\Delta}}$ | Std. dev. ${\mathit{\sigma}}_{\mathbf{\Delta}}$ | $\mathit{C}\mathit{O}{\mathit{V}}_{\mathbf{\Delta}}$ |
---|---|---|---|---|

Wisper | 10 | 0.90 | 0.54 | 0.60 |

Wisperx | 13 | 0.28 | 0.20 | 0.72 |

Reverse Wisper | 2 | 0.20 | ||

Reverse Wisperx | 10 | 0.32 | 0.16 | 0.50 |

All | 35 | 0.46 | 0.42 | 0.91 |

- Physical uncertainty is modeled by ε (Normal distributed)
- Statistical uncertainty is modeled by $\mathrm{log}K$ and ${\sigma}_{\epsilon}$ (Normal distributed)
- Model uncertainty is modeled by Δ (LogNormal distributed)

## 11. Conclusions

- Identification and selection of structural elements to be included in the probabilistic basis: e.g., blades, tower, substructure and foundation;
- Identification and modeling by limit states of important failure modes;
- Stochastic models for the uncertain parameters;
- Recommendation of methods for estimation of the reliability;
- Recommendations for target reliability levels for the different groups of elements;
- Recommendation for considerations of system aspects and damage tolerant design.

## Acknowledgements

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**MDPI and ACS Style**

Sørensen, J.D.; Toft, H.S.
Probabilistic Design of Wind Turbines. *Energies* **2010**, *3*, 241-257.
https://doi.org/10.3390/en3020241

**AMA Style**

Sørensen JD, Toft HS.
Probabilistic Design of Wind Turbines. *Energies*. 2010; 3(2):241-257.
https://doi.org/10.3390/en3020241

**Chicago/Turabian Style**

Sørensen, John D., and Henrik S. Toft.
2010. "Probabilistic Design of Wind Turbines" *Energies* 3, no. 2: 241-257.
https://doi.org/10.3390/en3020241