Safety-factor Calibration for Wind Turbine Extreme Loads

Transcription

1 WIND ENERGY Wind Energ. 2008; 11: Published online in Wiley Interscience ( Research Article Safety-factor Calibration for Wind Turbine Extreme Loads Patrick Moriarty*, National Renewable Energy Laboratory, Golden, CO, USA Key words: loads extrapolation; safety factor; extreme events Proper prediction of long-term extreme values for operating wind turbine loads and deflections is a critical component of wind turbine design. Direct observations or simulations of long-term extremes are not yet available; therefore, these predictions rely on some combination of large numbers of simulations and extrapolation. Extrapolation methods themselves can have significant uncertainty, and they also require that the wind turbine designer have a greater level of statistical expertise factors that make the methods less attractive for industrial application. As an alternative to extrapolation, safety factors can be calibrated using techniques that allow designers to use smaller data sets. To calculate such factors, a series of simulations was used to extrapolate 50 year extreme values for a 5 MW wind turbine. Two methods are proposed for calculating such safety factors: one based on the mean and standard deviation of extreme values, and one based on the median of extreme values. Through a process of random sampling without replacement, the safety factor based on the median of extreme values was found to be less variable and also more independent of the number of simulations. The safety factors required were as large as 1.7, or were only 1.25 if rotor thrust loads were considered the dominant design drivers. Copyright 2008 John Wiley & Sons, Ltd. Received 24 May 2008; Revised 2 October 2008; Accepted 4 October 2008 Introduction Extrapolation methods now are formally included in wind turbine design standards, such as the International Electrotechnical Commission (IEC) Ed. 3, 1 which has focused a great amount of attention on the strengths and weaknesses of these techniques. Many different authors 2 8 have shown that the variability of long-term extrapolated loads is highly dependent on the number of simulations needed, the probability distribution used for fitting, the data used for extrapolation, the load or deflection that is extrapolated, the wind turbine response and control system, the external wind environment and even the designer. Maturation of these techniques continues, including reducing the uncertainties of extrapolated loads through confidence interval estimation. 9 More simple methods, however, such as load multipliers or safety factors, 3 might provide a conservative way of estimating long-term loads that bypasses the complexity and uncertainty of current loads extrapolation techniques. This paper explores the use of these simpler techniques, which are not proposed as a complete replacement to loads extrapolation techniques, but rather provide an easily implemented and conservative estimate for long-term loads. These techniques could enable designers to quickly estimate long-term loads during the initial phases of the design cycle, and to perform more accurate and detailed analysis using extrapolation techniques for the final design. The study concentrates on estimating the 50 year extreme values required by the IEC * Correspondence to: P. Moriarty, National Renewable Energy Laboratory, 1617 Cole Blvd., MS 3811, Golden, CO 80401, USA. patrick_moriarty@nrel.gov Copyright 2008 John Wiley & Sons, Ltd.

2 602 P. Moriarty standards using the response from simulations of a single 5 MW wind turbine. The variability of the calculated load or deflection when using a small number of simulations (6 20) is an important consideration for calibration of the proposed safety factors below. 50 Year Extreme Value The first step in determining the safety factor for the 50 year extreme value is to determine the value used for calibration. Modern wind turbines have not yet operated for 50 years or longer, therefore, operational data cannot be used for the purposes of calibration, and simulation results must be relied upon for this purpose. To date, fully stochastic time simulations also have been limited in their length because of the amount of time required to simulate the operating response of a wind turbine. One of the longest sets of simulations at this time is examined in the companion paper by Moriarty found in this issue of Wind Energy. 10 The data sets in this paper consist of 5 years worth of simulation for a variable-speed, variable-pitch 5 MW onshore wind turbine. Details of the simulations, including turbine specifications and data verification techniques, can be found in Moriarty s study. 10 Altogether, the data set was comprised of 243, min simulations within the standard operating wind speed range of the turbine. As described in the Moriarty paper, the simulations contained time series for 24 different channels of operating loads and deflections, which can be broken into rotor loads, blade deflections and other loads. They are: Blade root flap and edge bending moments; Blade root in-plane and out-of-plane bending moments; Blade tip in-plane and out-of-plane displacements; Rotor torque; Low-speed shaft bending moments; and Tower base bending moments. For extrapolation purposes, a single value was extracted from each 10 min simulation for each channel: the maximum value in 10 min. These values were aggregated into empirical distributions that were fit with a three-parameter Weibull distribution using the method of moments. Moriarty 7 demonstrated that the threeparameter Weibull distribution is widely applicable to wind turbine loads and produces reasonable fits to empirical data under most operating conditions. The number of points used in fitting the empirical distribution has a significant effect on the shape of the fitted distribution and also the extrapolated load. A first attempt at estimating the 50 year extremes was done by fitting a probability distribution to all 5 years worth of operational data. When all of the data points (243,758) are used to determine the extrapolated load, the results are often overly conservative relative to the value that would likely be chosen by an experienced designer. Figure 1 shows the empirical and fitted distributions for the flap bending moment at the root of blade 1. In this instance, the 50 year extreme from the fitted distribution is approximately knm. Based on the shape of the distribution, however, a designer probably would select something closer to knm (15% less), which would result in a less costly design than the extrapolated load from the three-parameter Weibull. The reason for this overly conservative estimate of the extrapolated load is the large number of points below the change in slope of the distribution at approximately knm. A full 70% of the data in this empirical distribution lies below the slope change and could have a disproportionate influence the shape of the final distribution of extreme loads that lie at lower probabilities. The rather uniform distribution of maxima (see Figure 2) below the slope change results in a distribution that is not in the typical shape of a Weibull distribution. This study mainly focused on extrapolating values of the extremes below the change in slope in the exceedence plot, and concentrated on that population to get better fits to the lowest probability data in the tail, although it was also important to balance the need to use only representative points in the tail of the data, as well as retain enough statistical information to reliably fit that tail. To get better fits, we first propose examina-

3 Safety Factor for Extreme Loads Probability of Exceedence in 10 minutes year year Flap Bending Moment (knm) x 10 4 Figure 1. Empirical and fi tted exceedence distributions for fl ap bending moment of blade Number of Extremes Flap Bending Moment (knm) x 10 4 Figure 2. Histogram for fl ap bending moment of blade 1 tion of only the largest 1000 values. As shown in Figure 3, this results in a distribution with a shape that is much closer to that of a Weibull distribution. As expected, the fitted distribution to these largest values produces an extrapolated 50 year load (see Figure 4) that appears to be more realistic than that shown in Figure 1. Using the largest 1000 values to obtain a fitted distribution appears to be a reasonable method for most of the 24 channels examined in this study. For a couple of channels, however, such as low-speed shaft bending moment, this produces overly conservative results for the 50 year extrapolated load. The reasons for this conservatism again are related to slope changes in load distribution as shown in Figure 5. As with most loads, the low-speed shaft bending moment* has a distinctive change in slope that occurs near 9000 knm. By including values below this slope change when calculating the fitted distribution, the resulting distribution predicts a load close to 12,000 knm, which visually appears to be too great. Instead of using the * Low-speed shaft bending moments are calculated in a rotating coordinate system, where the axis of bending is perpendicular to the pitch axis of blade 1.

4 604 P. Moriarty Number of Extremes Flap Bending Moment (knm) x 10 4 Figure 3. Histogram of largest 1000 values for fl ap bending moment of blade Probability of Exceedence in 10 minutes year 50 year Flap Bending Moment (knm) x 10 4 Figure 4. Exceedence plot of fl ap bending moment from blade 1 and fi tted distribution using the largest 1000 values largest 1000 points, however, if the largest 100 points (Figure 5) are used, then the fitted distribution predicts a 50 year load closer to 11,000 knm, which seems more reasonable, but also possibly is conservative. The problem with using so few points when fitting the distribution is that there is significant variability in the largest and smallest probability loads. This means that if there were another 5 years of simulation available, then the tail of the data particularly the largest 10 or so points could appear at slightly different probability levels and therefore produce a much different 50 year extreme. The largest value in Figure 5, for example, actually could be a lower probability event than is shown, but because there is only one realization, its variability and the effect of that variability on the extrapolated load cannot be quantified. To strike a balance between the need to use only data points below slope changes in the data and also including enough data points in the distribution to produce a stable shape, a fourth approach is available. This approach involves examining each year of the 5 years worth of simulation separately and fitting appropriate

5 Safety Factor for Extreme Loads Probability of Exceedence in 10 minutes year year Low Speed Shaft Bending Moment (knm) Figure 5. Exceedence plot of low-speed shaft bending moment and fi tted distribution using the largest 100 values 10 0 Probability of Exceedence in 10 minutes year 50 year Low Speed Shaft Bending Moment (knm) Figure 6. Exceedence plots of low-speed shaft bending moment and fi tted distributions using the largest 20 values of each of 5 years separately distributions to each year. Figure 6 shows the results from this exercise, again examining the low-speed shaft bending moment that has the greatest variability in the distribution tail. Through a process of trial and error, we decided to use the largest 20 points in each of these probability distributions to fit the shape of the tail appropriately. The resulting 50 year loads range from 10,000 to nearly 12,000 knm a substantial variation. To eliminate this variation, the mean of these five values was calculated and used as the single extrapolated value. Compared with the other extrapolation results using the methods described for Figures 1, 4 and 5, the resulting mean extrapolated the 50 year load of 10,700 knm seemed the most reasonable. Using this method for all 24 channels also produced reasonable results based on intuitive examination of the 5 year empirical distributions for these loads and deflections. Therefore, this method was employed to predict the 50 year extrapolated values to use with the safety factor for bypassing extrapolation. More scientifically rigorous methods for

6 606 P. Moriarty choosing 50 year loads may have been used, but we felt that this intuitive approach reflected the methods used by current designers, who often rely on visual inspection and experience as a final check for realistic results. Future work may study the use of statistical tools to quantify the optimal number of data points to use for fitting a distribution, although the accuracy of these methods may not be known until 50 years of simulation or more are available for comparison. Calibration Data Because of time constraints, wind turbine designers typically are unable to run nearly 250,000 simulations to properly estimate the 50 year extreme values. A number less than 20 over a range of wind speeds seems to be more appropriate for the current generation of computational methods and hardware. The goal of this study was to calculate a multiplication or safety factor that designers can apply to smaller simulation sets to estimate long-term loads without doing a full extrapolation. To achieve this, a second data set was produced by Moriarty 10 using the same turbine model and fewer simulations. The set of simulations was intended to replicate a typical set used for determining design loads in the certification process. Instead of randomly sampling over the distribution of annual mean wind speed, 1200 simulations were performed at discrete wind speeds between 3 and 25 m s 1, with an increment of 2 m s 1. To mimic what a typical designer would use for loads estimation, the 1200 simulations at each wind speed were randomly sampled without replacement, and then grouped in sets of simulations having between 6 and 20 simulations per set. A total of 60 sets of simulations per wind speed were assembled, such that all 1200 simulations were sampled when there were 20 simulations per set, but only 360 simulations were used when there were six simulations per set. Again, the largest value in 10 min for each of the channels in each simulation was used. Once the sets of randomly sampled simulations were created, a distribution of maximum values for each load and each wind speed was produced from each of the 60 sets. Figure 7 shows the median and the 95th percentile of the maximum values of low-speed shaft bending moment using 20 simulations per set. For each wind speed, there are 60 values of median and 95th percentile maximum load forming a distribution of Number of Samples = 20 Low Speed Shaft Bending Moment (knm) Mean Wind Speed (m/s) Figure 7. Median (asterisks) and 95th percentile (circles) of the maximum load from 60 sets of 20 simulations per wind speed of low-speed shaft bending moment

7 Safety Factor for Extreme Loads 607 loads as a function of wind speed. The quantiles for all distributions in this study were estimated using rank ordering of the data with linear interpolation. For quantiles outside the data range, either the minimum or maximum value is chosen. From this point, several different values can be used for safety-factor calibration, depending on the preferred method. One method could use the maximum value among wind speeds, but this value was found to vary greatly with the number of simulations per set and also depending upon how the simulations were grouped between sets. Values that have less variation would be the mean of the maximum values along with the standard deviation to retain some information about the variability of the load. The median values (asterisks in Figure 7) were found to be the most stable values to use. They were even more stable than the mean values because the median value is less influenced by the maximum values in each set. When selecting values for safety-factor calibration, the wind speeds were searched to identify the largest mean or median of maxima value, and the values from this speed were retained. If the mean of maxima value was used, then the standard deviation of maxima value from the same wind speed was also retained. All other wind speeds were ignored. Using data from only the wind speed with the largest mean or median of maxima is reasonable, because these wind speeds will contribute the greatest number of maxima to the tail of the load distribution, even if the largest single maxima may occur at a different speed. Figure 8 shows the median of maxima values from each of the 60 sets as a function of the number of simulations per set. Also shown are the median (circles), 5th and 95th percentile (error bars) of all 60 median values, and a horizontal line representing the 50 year load (estimated using the extrapolation technique described in the previous section). Safety Factor Based on Mean and Standard Deviation One method for calibrating a safety factor for the 50 year extrapolated extreme is to use the mean and standard deviation of the largest values, similar to those shown in Figure 7. Using the median and 95th percentile (or other quantile) shown in Figure 7 is acceptable, but using the mean and standard deviation of the maximum values is a simpler approach that produces similar results. By using a safety factor that incorporates the variation of extremes, this method should reflect the uncertainty of the prediction based on the number of Low Speed Shaft Bending Moment (knm) Number of Simulations Figure 8. Median values (asterisks) of the maximum low-speed shaft bending moment for 60 sets of simulations with a varying number of simulations per set; the median (circles), 5th and 95th percentile of those median values (error bars) are also shown, along with the 50 year extrapolated load (horizontal line)

8 608 P. Moriarty simulations used to predict the load. To implement such a safety factor, a designer would use the following formula. L50 = µ N + SFσ N (1) where L 50 represents the 50 year value, m N is the largest mean of maxima value over all wind speeds for N simulations, SF represents the required safety factor, and s N is the related standard deviation of maxima from N simulations at the same wind speed. To calibrate the safety factor value, we used the simulation sets with 1200 simulations per wind speed and different sets of data grouped according to the previous section. For each of the 60 sets of simulations, with the number of simulations ranging between 6 and 20 per wind speed, the mean and standard deviation of the largest extreme values among the N simulations are calculated at each wind speed. The wind speed with the largest mean load, then, is identified as the design-driving case, and its standard deviation also is retained. This process is repeated 60 times, and the design-driving mean and standard deviation are recorded for each set. Depending on the type of load, the dominant wind speed can change with each set of N simulations. For this study, we calculated a separate safety factor for each of the 60 sets and N simulations using equation (1). These safety factors were then assembled into a distribution of values dependent on the number of simulations. From this distribution, the 95th percentile safety factor was chosen as a conservative estimate for all loads and deflections. Figure 9 shows the safety factor calculated in this manner for all 24 loads and deflections, where each line represents a different channel. In this figure, the largest values are from the tower fore aft bending moment. The rest of the lines are not identified because there was large variation in the calculated safety factor depending upon how the 60 sets of simulations were grouped. As a general trend, the safety factor was found to be larger for a lower number of simulations and steadily decreased as the number of simulations approached 20. Note that the values for the safety factor are much larger than multipliers calculated by Freudeneich and Argyriadis 3 because they considered only one set of simulations and not the upper quantile of a distribution of the safety factors. The large variability of this method with the actual groupings of data sets made this method less attractive than the alternative method proposed for calculating a safety factor, discussed below. 22 Muliplication factor of Std. Dev. from Mean value Number of Simulations Figure 9. Safety factor (95th percentile level) as a function of the number of simulations for all 24 of loads and defl ections

9 Safety Factor for Extreme Loads 609 Safety Factor Based on Median Value The next method for calculating a safety factor for the 50 year extremes is to use the median of the median values shown in Figure 8. The median is thought to be a more stable value than the mean of maxima because it is less sensitive to outliers than is the mean value. As noted above, 1200 simulations per wind speed were used and these were randomly grouped into 60 sets of simulations, with the number of simulations ranging between 6 and 20 per wind speed. For each set of simulations, the median of the largest extreme values is calculated as shown in Figure 7. Again, the largest median value over all wind speeds is found and retained for the safety-factor calculation. The resulting set of 60 median values is plotted in Figure 8 as a function of the number of simulations per wind speed. This distribution of median values also has its own median (shown in the figure as an open circle) and the other quantiles. For this method of safety-factor calibration, a designer would use the following formula to calculate the extreme value. L 50 = SFq 50, N (2) where L 50 represents the 50 year value, q 50,N is the largest median value over all wind speeds for N simulations, and SF represents the required safety factor. For the purposes of this study and to maintain a level of conservatism, the 5th percentile median value from Figure 8 was used to calibrate the safety factor. This suggests at least for this data set and wind turbine that the safety factor will be a conservative 95% of the time. This value represents the lower error bar in Figure 8, and the safety factor is the ratio of this value to that of the 50 year extrapolated value (horizontal line). The quantitative difference between using the 95th percentile value and the median of the median value to calculate safety factor in this case is only about 5%; however, the more conservative approach was followed. The results from the safety-factor calculation for each of the different loads and deflections from the simulations available are shown in Figures The calculated safety factors for each of these loads range between 1.1 and 1.7, and are greatly dependent upon the loading type. Comparing these with the values provided by Freudeneich and Argyriadis 3 produces slightly different answers, because this study considered a random distribution of simulation sets and not just one sample. The observation that each blade has a slightly different safety factor in Figures deserves some additional discussion. The variation is highest among in-plane quantities (about 6%), which, as discussed by 1.9 Ratio of 50 year Extrapolated to 5%le of Max Simulated In Plane Deflections Out of Plane Deflections Number of Simulations Figure 10. Safety factor as a function of the number of simulations for blade-tip defl ections

10 610 P. Moriarty 1.36 Ratio of 50 year Extrapolated to 5%le of Max Simulated Root In Plane Bending Moments Root Out of Plane Bending Moments Number of Simulations Figure 11. Safety factor as a function of the number of simulations for blade root in-plane and out-of-plane bending moments Ratio of 50 year Extrapolated to 5%le of Max Simulated Root Edge Bending Moments Root Flap Bending Moments Number of Simulations Figure 12. Safety factor as a function of the number of simulations for blade root fl ap- and edge-bending moments Moriarty, 10 also have the greatest amount of variability among loads. This variability causes variability in the 50 year load prediction as shown in Figure 6. Therefore, while the median value used among blades is fairly constant, the 50 year extrapolated load varies (about 6%), creating a noticeable difference in safety factor between blades. The difference between blades may be reduced by using more points in the 50 year extrapolation procedure, but that may also produce a more conservative safety factor following the logic in the previous section. Since the difference is not great, we chose the largest value among blades to err on the side of conservatism. Future work could focus on reducing this variability and examining the resulting impact on safety factor.

11 Safety Factor for Extreme Loads Ratio of 50 year Extrapolated to 5%le of Max Simulated Side to Side Tower Bending Moment Low Speed Shaft Bending Moments Torsional Tower Bending Moment Fore Aft Tower Bending Moment Rotor Torque Number of Simulations Figure 13. Safety factor as a function of the number of simulations for non-rotor loads Of greatest interest in all these figures is that the safety factor is relatively insensitive to the number of simulations performed, making it a much more stable value to use than that based on the mean and standard deviation (discussed above). Also, each time the calculation was performed (i.e. different groupings of simulations), the calculated safety factors changed very little, indicating further stability. Figure 10 shows the safety factor required for deflections for each of the three blades on the turbine. Note that the in-plane deflections require a much greater safety factor near 1.7 than a safety factor for the out-of-plane deflections, which is closer to This indicates that the 50 year extrapolated values for in-plane deflections are much more sensitive to rare events that are not captured using a smaller number of simulations. This is discussed by Moriarty, 10 who states that in-plane loads and deflections tend to be dominated by loading situations at less frequently occurring wind speeds near cut-out. The same behavior manifests when comparing the in-plane and edge root bending moments of the blades shown in Figures 11 and 12. Although in both of these instances, the safety factors are neither as great as for the in-plane deflections, nor are the differences between either the out-of-plane or the flap bending moments as great. Also, the safety factors for the other rotor quantities may be more important because in-plane deflections rarely drive the design. Figure 13 shows the non-rotor loads, including shaft bending moments, tower bending moments and rotor torque. For this set of loads, the low-speed shaft bending moments have the highest required safety factors (nearly 1.7). Again, these are also loads dominated by less frequently occurring wind speeds near cut-out, as described in Moriarty. 10 Examining these figures as a whole, it can be concluded that a safety factor of 1.7 is required to cover all of the loads and deflections of this turbine design. The out-of-plane rotor loads and deflections are often considered the most critical, however, in which case, a safety factor of 1.25 might be sufficient. This factor increases to 1.35 if the tower bending moment is included, and remains at 1.7 if the low-speed shaft bending is critically important to the design. While these safety factors may be useful for this particular 5 MW turbine, the applicability of these values to other turbines should be the study of future work. By performing the same analysis over a range of turbine designs with, in particular, different control system algorithms, we will be able to discern what appropriate levels for these factors should be for a variety of turbine designs.

12 612 P. Moriarty Conclusions A safety factor applied to the largest median of maximum loads and deflections from a small number of simulations may allow designers to bypass the complexity of extrapolation methods. For the loads and deflections on a 5 MW baseline turbine, a safety factor of 1.7 applied to the median value was necessary to conservatively estimate each of the 50 year loads and deflections. If only out-of-plane rotor loads are considered to be design drivers, then a factor of 1.25 can be used. These safety factors should be conservative 95% of the time. It was discovered that loads and deflections that are dominated by high wind speeds, often in-plane, had the highest required safety factors, most likely because small numbers of simulations are less likely to produce low-probability events in these wind speeds. Note that the results here are for a single turbine design and should be reproduced with other designs before being considered widely applicable. Future work should focus on extending these results to a wide range of turbine and control systems designs. A more detailed quantification of why in-plane loads have higher safety factors is also recommended. Acknowledgements Thanks very much to Paul Veers (Sandia National Laboratories) and other members of the IEC Loads Extrapolation Evaluation Exercise committee for the many useful discussions that guided this work. References 1. International Electrotechnical Commission. IEC/TC88, ed. 3, Wind Turbines Part 1: Design Requirements. IHS: Geneva, Freudenreich K, Argyriadis K. The load level of modern wind turbines according to IEC , the science of making torque from wind. Journal of Physics: Conference Series 2007; 75: Freudenreich K, Argyriadis K. Wind turbine load level based on extrapolation and simplified methods. Wind Energy 2008; 11(6): DOI: /we Agarwal P, Manuel L. Simulation of the offshore wind turbine response for extreme limit states. Proceedings of OMAE 2007, 26th International Conference on Offshore Mechanics and Arctic Engineering, June 2007, San Diego, CA, USA. 5. Ragan P, Manuel L. Statistical extrapolation methods for estimating wind turbine extreme loads. Proceedings of ASME Wind Energy Symposium. AIAA: Reno, NV, Genz R, Nielsen KB, Madsen PH. An investigation of load extrapolation according to IEC ed.3. Proceedings of the European Wind Energy Association Conference, Athens, Greece, 27 February 2 March Moriarty PJ, Holley WE, Butterfield CP. Extrapolation of Extreme and Fatigue Loads Using Probabilistic Methods, NREL/TP National Renewable Energy Laboratory: Golden, CO, Cheng PW. A reliability based design methodology for extreme response of offshore wind turbines. PhD dissertation. Delft University of Technology, Fogle J, Agarwal P, Manuel L. Towards an improved understanding of statistical extrapolation for wind turbine extreme loads. Wind Energy 2008; 11(6): DOI: /we Moriarty P. Database for validation of design load extrapolation techniques. Wind Energy 2008; 11(6): DOI: /we.305.