Steven R. Winterstein

Predicting Design Wind Turbine Loads from Limited Data: Comparing Random Process and Random Peak Models

This paper considers two distinct topics that arise in reliability-based wind turbine design. First, it illustrates how general probability models can be used to predict long-term design loads from a set of limited-duration, short-term load histories. Second, it considers in detail the precise choice of probability model to be adopted, for both flap and edge bending loads in both parked and operating turbine conditions. In particular, a 3-moment random peak model and a 3- or 4-moment random process model are applied and compared. For a parked turbine, all models are found to be virtually unbiased and to notably reduce uncertainty in estimating extreme loads (e.g., by roughly 50 percent). For an operating turbine, however, only the random peak model is found to retain these beneficial features. This suggests the advantage of the random peak model, which appears to capture the rotating blade behavior sufficiently well to accurately predict extremes.

PARAMETRIC MODELS FOR ESTIMATING WIND TURBINE FATIGUE LOADS FOR DESIGN

International standards for wind turbine certification depend on finding long-term fatigue load distributions that are conservative with respect to the state of knowledge for a given system. Statistical models of loads for fatigue application are described and demonstrated using flap and edge blade-bending data from a commercial turbine in complex terrain. Distributions of rainflow-counted range data for each ten-minute segment are characterized by parameters related to their first three statistical moments (mean, coefficient of variation, and skewness). Quadratic Weibull distribution functions based on these three moments are shown to match the measured load distributions if the non-damaging low-amplitude ranges are first eliminated. The moments are mapped to the wind conditions with a two-dimensional regression over ten-minute average wind speed and turbulence intensity. With this mapping, the short-term distribution of ranges is known for any combination of average wind speed and turbulence intensity. The longterm distribution of ranges is determined by integrating over the annual distribution of input conditions. First, we study long-term loads derived by integration over wind speed distribution alone, using standard-specified turbulence levels. Next, we perform this integration over both wind speed and turbulence distribution for the example site. Results are compared between standarddriven and site-driven load estimates. Finally, using statistics based on the regression of the statistical moments over the input conditions, the uncertainty (due to the limited data set) in the long-term load distribution is represented by 95% confidence bounds on predicted loads.

Application of Measured Loads to Wind Turbine Fatigue and Reliability Analysis

Cyclic loadings produce progressive damage that can ultimately result in wind turbine structural failure. There are many issues that must be dealt with in turning load measurements into estimates of component fatigue life. This paper deals with how the measured loads can be analyzed and processed to meet the needs of both fatigue life calculations and reliability estimates. It is recommended that moments of the distribution of rainflow-range load amplitudes be calculated and used to characterize the fatigue loading. These moments reflect successively more detailed physical characteristics of the loading (mean, spread, tail behavior). Moments can be calculated from data samples and functional forms can be fitted to wind conditions, such as wind speed and turbulence intensity, with standard regression techniques. Distributions of load amplitudes that accurately reflect the damaging potential of the loadings can be estimated from the moments at any wind condition of interest. Fatigue life can then be calculated from the estimated load distributions, and the overall, long-term, or design spectrum can be generated for any particular wind-speed distribution. Characterizing the uncertainty in the distribution of cyclic loads is facilitated by using a small set of descriptive statistics for which uncertainties can be estimated. The effects of loading parameter uncertainty can then be transferred to the fatigue life estimate and compared with other uncertainties, such as material durability.

Extremes of Nonlinear Vibration: Models Based on Moments, L-Moments, and Maximum Entropy

Nonlinear effects beset virtually all aspects of offshore structural loading and response. These nonlinearities cause non-Gaussian statistical effects, which are often most consequential in the extreme events—e.g., 100- to 10,000-year conditions—that govern structural reliability. Thus there is engineering interest in forming accurate non-Gaussian models of time-varying loads and responses, and calibrating them from the limited data at hand. We compare here a variety of non-Gaussian models. We first survey moment-based models; in particular, the 4-moment “Hermite” model, a cubic transformation often used in wind and wave applications. We then derive an “L-Hermite” model, an alternative cubic transformation calibrated by the response “L-moments” rather than its ordinary statistical moments. These L-moments have recently found increasing use, in part because they show less sensitivity to distribution tails than ordinary moments. We find here, however, that these L-moments may not convey sufficient information to accurately estimate extreme response statistics. Finally, we show that 4-moment maximum entropy models, also applied in the literature, may be inappropriate to model broader-than-Gaussian cases (e.g., responses to wind and wave loads).Copyright

Application of measured loads to wind turbine fatigue and reliability analysis

Cyclic loadings produce progressive damage that can ultimately result in wind turbine structural failure. There are many issues that must be dealt with in turning load measurements into estimates of component fatigue life. This paper deals with how the measured loads can be analyzed and processed to meet the needs of both fatigue life calculations and reliability estimates. It is recommended that moments of the distribution of rainflow-range load amplitudes be calculated and used to characterize the fatigue loading. These moments reflect successively more detailed physical characteristics of the loading (mean, spread, tail behavior). Moments can be calculated from data samples and functional forms can be fitted to wind conditions, such as wind speed and turbulence intensity, with standard recession techniques. Distributions of load amplitudes that accurately reflect the damaging potential of the loadings can be estimated from the moments at any, wind condition of interest. Fatigue life can then be calculated from the estimated load distributions, and the overall, long-term, or design spectrum can be generated for any particular wind-speed distribution. Characterizing the uncertainty in the distribution of cyclic loads is facilitated by using a small set of descriptive statistics for which uncertainties can be estimated. The effects of loading parameter uncertainty can then be transferred to the fatigue life estimate and compared with other uncertainties, such as material durability.

RELIABILITY OF FLOATING STRUCTURES: EXTREME RESPONSE AND LOAD FACTOR DESIGN

The reliability of a floating offshore platform against extreme offsets is studied. Methods are illustrated to perform more refined analysis of extreme loads and load effects. These include the effect of nonlinear diffraction under random wave excitation. They permit inclusion of randomness in significant wave height H{sub s}, in peak spectral period T{sub p} given H{sub s}, and also in the extreme load effect given both H{sub s} and T{sub p}. Numerical effects are demonstrated by applying these methods to a specific floating structure: a deep-draft spar buoy. Design of the spar has been considered in two deep-water sites, one in the Gulf of Mexico and another in the Northern North Sea. Appropriate joint contours of significant wave height and peak period are developed, and used to develop load and resistance factors for each of the two sites.

Analytical Predictions of the Air Gap Response of Floating Structures

Two separate studies are presented here that deal with analytical predictions of the air gap for floating structures. (1) To obtain an understanding of the importance of firstand second-order incident and diffracted wave effects as well as to determine the influence of the structure’s motions on the instantaneous air gap, statistics of the air gap response are studied under various modeling assumptions. For these detailed studies, a single field point is studied here – one at the geometric center (in plan) of the Troll semi-submersible. (2) A comparison of the air gap at different locations is studied by examining response statistics at different field points for the semi-submersible. These include locations close to columns of the four-columned semi-submersible. Analytical predictions, including firstand second-order diffracted wave effects, are compared with wave tank measurements at several locations. In particular, the gross root-mean-square response and the 3-hour extreme response are compared. BACKGROUND The air gap response, and potential deck impact, of ocean structures under random waves is generally of considerable interest. While air gap modeling is of interest both for fixed and floating structures, it is particularly complicated in the case of floaters because of their large volume, and the resulting effects of wave diffraction and radiation. These give rise to two distinct effects: (1) global forces and resulting motions are significantly affected by diffraction effects; and (2) local wave elevation modeling can also be considerably influenced by diffraction, particularly at locations above a pontoon and/or near a major column. Both effects are important in air gap prediction: we need to know how high the wave rise (step 2), and how low the deck translates vertically (due to net heave and pitch) at a given point to meet the waves. Moreover, effects (1) and (2) are correlated in time, as they result from the same underlying incident wave excitation process. We focus here on analytical diffraction models of air gap response, and its resulting stochastic nature and numerical predictions under random wave excitation. Attention is focused on a semi-submersible platform, for which both slow-drift motions (heave/pitch) and diffraction effects are potentially significant. This air gap response presents several new and interesting challenges. It is the first response limit state where we need to simultaneously include both second-order sumand difference-frequency effects (on the wave surface), and second-order differencefrequency effects (on slow drift motions and generally, on the wave surface as well). The sumand differencefrequency waves and the difference-frequency heave and pitch motions can both influence the air gap. The air gap response is further complicated because the heave, pitch, and roll motions of the floating structure are generally coupled. Moreover, the motions and the net wave elevation, both of which affect the air gap, are correlated in time as they result from the same underlying incident wave excitation process. Note that air gap modeling has been the subject of previous work within the Reliability of Marine Structures Program at Stanford University. For example, Winterstein and Sweetman (1999) apply a fractile-based approach to develop a scaling factor between the statistics of the incident waves and those of the associated air gap demand. Results are shown here from frequency-domain analyses which permit careful study and isolation of various effects: e.g., wave forces on a fixed (locked-down) structure, the effect of structural motions on air gap response, and finally, the effect of different local wave elevation models. For reference, a complete second-order diffraction model is formulated and studied. Compared with this complete model, various simplified models are imposed and evaluated: (a) second-order wave elevation effects are neglected completely, or (b) these second-order effects are approximated by analytical Stokes theory, which retains second-order effects on the incident wave but neglects second-order diffraction. Use of (a) and (b) would significantly simplify the analysis, avoiding the costly step of second-order diffraction. The local wave modeling in step (b) is found to be quite important in predicting the air gap response of two semi-submersibles studied here. THEORY Volterra Series and Transfer Functions In modeling nonlinear systems, such as floating structures, it is common to employ Volterra series that permit one to describe the response (output) of such systems. The nonlinear system is defined in terms of firstand second-order transfer functions. For floating structures, these transfer functions are obtained from firstand second-order wave diffraction analysis programs (e.g., WAMIT, 1995). In order to study the response of a floating structure to random seas, we start by defining an irregular sea surface elevation, η(t), as a sum of sinusoidal components at N distinct frequencies: ) exp( ; ) ( 2 where ; ) exp( Re ) cos( ) (

Air Gap Response of Floating Structures: Statistical Predictions Versus Observed Behavior

Summary and Conclusions • A new, fractile-based approach has been proposed to definean amplification factor,b p , on extreme wave crest levels due todiffraction. This is calculated as the rms-ratio between input andoutput peaks in all wave cycles ~Eq. ~8!!. The scaling factor re-sulting from this new approach is found to describe the outputfractile results quite well, even at extreme levels within the 18 hof seastate test results, e.g. Figs. 5–7.• Figure 8 shows that the largest observed amplification,roughly 1.4, is found at location 1 ~in front of the up-wave col-umn; see Fig. 1!. Other near-column locations ~4, 5, 6, and 9!show amplification factors of roughly 1.2. As might be expected,somewhat greater amplifications generally occur for the smallerT P case ~with shorter wavelengths, hence larger relative effect ofthe structure!. The largest amplifications, however, at the near-column locations are relatively constant for the two T P cases con-sidered.• An analogous amplification factor, b

Relating Turbulence to Wind Turbine Blade Loads: Parametric Study with Multiple Regression Analysis

Different wind parameters are studied to find a set that is most useful in estimating fatigue loads on wind turbine blades. The histograms of rainflow counted stress ranges are summarized through their first three statistical moments and regression analysis is used to estimate these moments in various wind conditions. A systematic method of comparing the ability of different wind parameters to estimate the moments is described and results are shown for flapwise loads on three HAWTs. In the case of two of these turbines, the stress ranges are shown to be highly correlated with a turbulence measure obtained by removing a portion of the low-frequency content of the wind.

Stochastic Fatigue Damage Accumulation Due to Nonlinear Ship Loads

Efficient methods are described here to predict the stochastic accumulation of fatigue damage due to nonlinear ship loads that are produced in random seas. The stochastic analysis method, which may be applied both to overload and fatigue limit states, is based on a relatively new concept: the “nonlinear transfer function” (NTF) method. The basic goal of this method is to require the use of a generally expensive, nonlinear time-domain ship load analysis for only a limited set of idealized, regular waves. This establishes the socalled nonlinear transfer function; i.e., the generally nonlinear transformation from wave amplitude and period to the load amplitude measure of interest (e.g., total load range for rainflow-counting, tensile portion for crack propagation, etc.). Stochastic process theory is used (1) to identify a minimal set of regular waves (i.e., heights and periods) to be applied, (2) to assign an appropriate set of “side–waves” to be spatially distributed along the ship, and (3) to determine how these results should be weighted in predicting statistics of the loads produced in random seas. The result is compared here with full nonlinear analysis of a specific ship, over long simulations of an irregular sea. A ship with relatively flared cross-section is chosen, which