In the present paper methods for statistical load extrapolation of wind turbine response are studied using a stationary Gaussian process model which has approximately the same spectral properties as the response for the flap bending moment of a wind turbine blade. For a Gaussian process an approximate analytical solution for the distribution of the peaks is given by Rice. In the present paper three different methods for statistical load extrapolation are compared with the analytical solution for one mean wind speed. The methods considered are global maxima, block maxima and the peak over threshold method with two different threshold values. The comparisons show that the goodness of fit for the local distribution has a significant influence on the results, but the peak over threshold method with a threshold value on the mean plus 1.4 standard deviations generally gives the best results. By considering Gaussian processes for twelve mean wind speeds the ‘fitting before aggregation’ and ‘aggregation before fitting’ approaches are studied. The results show that the ‘fitting before aggregation’ approach gives the best results.
I. Introduction Statistical load extrapolation for wind turbines during operation is required according to the wind turbine standard IEC 61400-1 3.edition 20051 in order to determine the characteristic load during operation. The characteristic load is determined as the load which on average occurs once every 50 years – corresponding to a return period of 50 years. The structural loads on onshore wind turbines during operation are dependent on (among other things):
• Mean wind speed • Turbulence intensity • Type and settings of the control system
The effect from these parameters must be taken into account when the characteristic load is determined for the
individual wind turbine components. Several methods for determining the long-term distribution for the load by statistical load extrapolation have
been proposed in recent years. Common for the methods is that they use a limited number of 10 minutes simulations of the structural response for the statistical load extrapolation, but the methods do in general deviate in the way they extract the peaks from the time series. Different methods for loads extrapolation have been considered in [2-5].
*PhD-student, Department of Civil Engineering, Sohngaardsholmsvej 57, AIAA member. †Professor, Department of Civil Engineering, Sohngaardsholmsvej 57, Wind Energy Division, Frederiksborgvej 399. ‡ Senior Specialist, R&D Department, Dr. Langemaijerweg 1a.
48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition4 - 7 January 2010, Orlando, Florida
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However, it is generally difficult to verify which of these methods is the best one since the correct long-term distribution for the wind-turbine response is unknown. Often different methods are compared by performing many 10 min. simulations of the wind turbine during operation and then comparing the characteristic loads obtained by the different methods. A database with time series of wind turbine response was published by Moriarty (NREL) [6].
In the present paper some of the methods for statistical load extrapolation are studied for the special case where the wind turbine response at the individual mean wind speeds is modelled by a stationary Gaussian process. The main reason for this is that for stationary Gaussian processes there exists an approximate analytical solution for the distribution of the extremes given by Rice7 which can be used as reference for comparing the different methods The ability to correctly predict the extremes of a Gaussian process may be regarded as a minimum requirement for a good extrapolation method.
Realizations of the stationary Gaussian process are simulated such that they have approximately the same spectral properties as the flap bending moment in a ‘real’ wind turbine blade. In general, the flap bending moment response and other responses in the wind turbine do not follow a Gaussian process partly due to the influence of the control system. However, also in this case the methods used for statistical load extrapolation should be capable of determining the long-term distribution with a satisfactory accuracy.
Besides, the statistical uncertainty in the long-term distribution due to a limited number of time series used in the extrapolation procedure is studied using the bootstrapping technique. Further, two procedures for aggregation of the distribution at the individual mean wind speeds “fitting before aggregation” and “aggregation before fitting” are investigated.
The methods for statistical load extrapolation investigated are global maxima, block maxima and the peak over threshold method. The stationary Gaussian response is simulated for 12 different mean wind speeds between cut-in at 3 m/s and cut-out at 25 m/s.
II. Gaussian Blade Response In order to determine the spectral properties for the flap bending moment of the blade some of the simulations
from the database [6] are used. The simulations are based on a 5 MW reference onshore wind turbine with a rotor diameter of 126 m and a hub height of 90 m. The wind turbine is pitch controlled with variable speed and has a rated wind speed of 11.5 m/s. The maximum rotor speed is 12.1 rpm. The natural frequencies for the blade are shown in table 1 together with the rotational frequencies at the nominal wind speed8.
The normalized power spectrum for the flap bending moment of the blade at 15 m/s is calculated by FFT-
analysis and is shown in figure 1. The power spectrum contains a large amount of energy at the smallest frequencies which corresponds to background turbulence. For the frequency corresponding to 1P (0.20 Hz) a dominant peak is observed and additional smaller peaks are observed at 2P and 3P. A small peak is also observed for the 1st natural frequency in the edgewise direction.
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100
101
Frequency [Hz]
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ized
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Figure 1. Normalized power spectrum for blade flap bending moment at 15m/s.
In order to investigate whether the response of the blade is Gaussian the skewness and kurtosis are calculated
based on 200 time series at each mean wind speed, see figure 2. It is seen from the figure that the flap bending moment of the blade does not follow a Gaussian distribution since the skewness differs from 0 and the kurtosis differs from 3, which are the skewness and kurtosis for a Gaussian process. The skewness and kurtosis vary significantly over the different mean wind speeds which may be due to the nonlinear behaviour of the control system.
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tosi
s [-
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Figure 2. Skewness and kurtosis for blade flap bending moment.
In the present paper it is assumed that the response for the flap bending moment can be approximated by a
stationary Gaussian process with a rational spectrum from which realizations can be simulated. The process is assumed to have a double-sided spectral density function S(ω), where ω is the rotational frequency (in rad/s). One way of simulating realizations of a Gaussian process is the autoregressive moving average (ARMA) models. These models are formulated in discrete time and treated by Box, Jenkins and Reinsel9. Closely related to ARMA models are methods formulated in continuous time with realizations generated at equidistant time steps as used in the present paper, see Franklin10,11.
In order to compare the spectrum for the stationary Gaussian response (rational spectrum) with the spectrum for the blade response given in figure 1 (basic spectrum) several spectral quantities are calculated. The jth order spectral moment λj is defined as:
( )0
2 jj XS dλ ω ω ω
∞
= ∫ (1)
The zero up-crossing rate ν0 and the peak rate νm are defined as:
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20
0
12
λν
π λ= (2)
4
2
12m
λν
π λ= (3)
The irregularity factor α and the spectral width parameters ε and δ are defined as:
0
m
να
ν= (4)
2
22
0 4
1 1λ
ε αλ λ
= − = − (5)
2
1
0 2
1λ
δλ λ
= − (6)
The intervals for the spectral width parameters are 0 ≤ ε ≤ 1 and 0 ≤ δ ≤ 1 where low values correspond to a narrow-banded process and high values correspond to a broad-banded process. In table 2 the spectral quantities for the basic spectrum and the rational spectrum are compared. The basic spectrum is normalized before the fitting procedure to have variance (0th order moment) equal to one.
Table 2: Comparison of basic spectrum and rational spectrum for blade flap bending moment at 15 m/s. Parameter Basic spectrum Rational spectrum Error [%] λ0 - 0th order moment (variance) 1.00 0.97 -2.8 λ1 - 1st order moment 0.72 0.70 -3.4 λ2 - 2nd order moment 1.62 1.56 -3.8 λ4 - 4th order moment 29.25 27.96 -4.4 ν0 - zero crossing rate 0.20 0.20 -0.5 νm - peak rate 0.68 0.67 -0.3 α - irregularity factor 0.30 0.30 -0.2 ε - spectral width parameter 0.95 0.95 0.0 δ - spectral width parameter 0.82 0.83 0.1
The differences between the spectral quantities for the basic spectrum and the rational spectrum are in general
below 5%. The response for the blade can according to the spectral width parameters ε and δ be characterized as broad-banded. In figure 3 the basic spectrum and the rational spectrum are compared. From the figure it is seen that the rational spectrum fits the general behaviour of the basic spectrum well. However, small fluctuations in the basic spectrum are as expected not captured by the rational spectrum.
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Figure 3. Normalized power spectrum for blade flap bending moment at 15m/s.
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For the flap bending moment of the blade 1200 10 min. time series with Gaussian response are generated at 12 different mean wind speeds between 3 and 25 m/s. The time series are generated with a time step on 0.05s corresponding to a frequency on 20 Hz.
III. Statistical Load Extrapolation
A. Analytical Solution for a Stationary Gaussian Process For a stationary Gaussian response (mean zero and standard deviation one) the individual peaks ξ follows the
following distribution function, according to Rice12,13:
( )2
2 2exp
21 1mF ξ ξ αξξ α
α α
⎛ ⎞ ⎛ ⎞⎛ ⎞= Φ − − Φ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎝ ⎠⎝ ⎠ ⎝ ⎠
(7)
where Φ is the standard Normal distribution function and α is the regularity factor defined in equation (4). If α is equal to zero the individual peaks are Gaussian distributed, and if α is equal to one the individual peaks are Rayleigh distributed. The short-term distribution for the largest peak within the time interval T can be obtained by assuming that the individual peaks are independent: ( ) ( ) mN
T mF Fξ ξ= (8) where Nm is the number of maxima within the time interval T. Nm can be obtained by time series analysis or from the peak rate νm given in equation (3): m mN Tν= (9)
Several asymptotic distributions for FT(ξ) have been proposed, e.g. by Davenport7,14. The asymptotic distribution by Davenport is based on the assumption that the number of zero-crossings is large which is fulfilled for long time intervals T. In the present case a time interval of 10min. is used which can be characterized as short and the approximation by Davenport therefore overestimates the load with 10-15% dependent on the return period. In the present paper different methods for statistical load extrapolation are compared to the approximate analytical solution given by Rice7.
B. Statistical Load Extrapolation The long-term distribution of the extremes can be determined by two different approaches ‘fitting before
aggregation’ and ‘aggregation before fitting’3. In the following the two approaches are elaborated. In the first approach, ‘fitting before aggregation’, a local distribution is fitted to the peaks for each mean wind
speed and the long-term distribution is obtained by weighting the short-term distributions according to the mean wind speed. The local distribution denotes the distribution for the peaks within a time series whereas the short-term distribution denotes the distribution for the largest peak within a time series. In the present paper the peaks are extracted from the time series in the following three ways:
• Global Maxima • Block Maxima • Peak over Threshold
For the global maxima method only the largest peak within each time series is extracted. In block maxima the
time series are divided into six blocks with a length on 100s and the largest peak from each block is extracted. The peak over threshold method extracts peaks above a certain threshold. Two different thresholds have been used corresponding to the mean value plus 1.4 and 2.0 standard deviations. In order to secure that the individual peaks are independent a time separation between the peaks of at least 10s is applied.
For the extracted peaks a local distribution function is fitted by the Maximum Likelihood Method. Several different local distributions have been used for real wind turbine response15. However, the 3-parameter Weibull distribution is often preferred. In the present paper the Weibull, Normal, Rayleigh and Gumbel distributions are compared but mainly the Weibull distribution is considered because it is often used for real wind turbine response. Based on the local distribution the short-term distribution for the maximum response, l within the time interval [0,T] is obtained from: ( ) ( ) ( ),| , | , n U T
short term localF l T U F l T U− = (10) where n(U,T) is the expected number of independent peaks at the mean wind speed U within the time interval [0,T]. The long-term distribution for the maximum response within the time interval [0,T] can be determined from the
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short-term distribution by integrating over the mean wind speeds given by the density function fU(U). According to IEC 61400-11 the mean wind speed is modelled by a Rayleigh distribution and wind class II is assumed (reference wind speed 42.5m/s).
( ) ( ) ( )| | ,out
in
U
long term short term UU
F l T F l T U f U dU− −= ∫ (11)
In equation (11) it is assumed that fU(U) is truncated to the interval [Uin,Uout]. The characteristic value of the response Lc with return period Tr (years) can then be determined from the long-term distribution by assuming that the individual time series are independent.
( )| 160 24 365long term c
r
TF L TT− = −
⋅ ⋅ ⋅ (12)
In the second approach ‘aggregation before fitting’ the number of time series at each mean wind speed is weighted according to the wind distribution. For each time series the peaks are extracted and the long-term distribution is fitted to the peaks. The peaks can be extracted for the time series in the same way as in the ‘fitting before aggregation’ approach. However, in the present paper only the global maxima and the peak over threshold methods are used. In order to improve the fit of the long-term distribution to the tail only the largest 10% of the extracted peaks are used in the fitting procedure.
In the ‘aggregation before fitting’ approach the highest wind speeds are only represented by a very limited number of time series. For load cases where the highest loads occur around the nominal wind speed, this will not necessarily be a problem. However, for load cases where the highest loads occur close to the cut-off wind speed the limited number of simulations will lead to large statistical uncertainty on the long-term distribution.
IV. Statistical Load Extrapolation of Gaussian Response at 15m/s The methods for statistical load extrapolation are first compared with the analytical solution for the Gaussian
response for one mean wind speed at 15m/s. In the comparison the characteristic load is calculated based on simN time series which are selected randomly. In order to determine the statistical uncertainty in the characteristic loads the bootstrapping technique16 is applied by recalculating the characteristic load 200 times. The long-term distributions are compared using characteristic loads corresponding to 1, 5, 50 and 100 year return periods. In order to compare the different methods for statistical load extrapolations the calculated characteristic loads have been normalized with the approximate analytical solution in equation (8) for the given return period.
A. Local Distribution Function First the local distribution function at 15 m/s is considered by comparing the Weibull, Normal, Rayleigh and
Gumbel distributions. Since the Rayleigh distribution does not contain a location parameter the peaks for this distribution are shifted. In figure 4 an exceedence plot is shown for each of the four distributions and compared with the analytical solution obtained from Rice. The peaks have been extracted by the method of global maxima and 100 time series are used.
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0.7 0.8 0.9 110
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roba
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100
Normalized Load [-]
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roba
bilit
y [-
]
Gumbel
Global MaximaRiceGumbel
Figure 4. Comparison of local distributions with Rice, 100 time series, peaks extracted by global maxima.
Operation at 15 m/s. From figure 4 it is seen that the largest peaks fits the analytical solution by Rice well. The smaller peaks follow
the analytical solution less well which is due to the way the peaks are extracted. The analytical solution is the distribution for the highest peaks which not necessarily will occur in separated time series as assumed by using global maxima. From figure 4 it is seen that the Weibull, Normal and Rayleigh distributions tend to underestimate the characteristic load for small exceedence probabilities whereas the Gumbel distribution overestimates the characteristic load. The Weibull distribution contains for a specific choice of the parameters the Rayleigh distribution and the simulated blade response has an irregularity factor α equal to 0.30. Figure 4 shows that the Rayleigh and Weibull distributions in the present case result in the best fits to the smallest exceedence probabilities. The global extremes for a Gaussian process will in general converge to a Gumbel distribution if the number of zero-crossings in each time-series is large. This assumption is not fulfilled for the 10 min. series used in the present study for which reason the Gumbel distribution overestimate the local distribution.
In the following only the Weibull distribution is considered, because it is often used for real wind turbine response.
B. Comparison of Load Extrapolation Methods for Weibull Distribution In the following the different methods for statistical load extrapolation using a Weibull distribution for the fit are
compared with the analytical solution obtained from Rice, see figure 5. The load extrapolation methods are compared for different number of simulations, Nsim equal to 25, 50, 75 and 100. The result shown in the figure corresponds to a return period on 50 years and the statistical uncertainty obtained by bootstrapping technique is indicated by the error-bar which symbolizes one standard deviation.
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Figure 5. Comparison of global maxima, block maxima and peak over threshold, Weibull distribution, 50
year return period. Operation at 15 m/s (1=Rice solution). From figure 5 it is seen that both the method of global maxima and block maxima underestimate the characteristic load with 6.7% and 9.3%, respectively. The poor estimation of the characteristic load is primary due to the poor fit of the Weibull distribution, see figure 4 (Weibull) and figure 6. For the method of global maxima which only extracts one peak from each time series, the statistical uncertainty is between 5% and 15% whereas it for the method of block maxima which extracts six peaks for each time series is between 2% and 5%. The peak over threshold method has been used with two different threshold values corresponding to the mean value plus 1.4 and 2.0 standard deviations. For a threshold value of 1.4 the characteristic load is overestimated by 0.5% and for a threshold value on 2.0 the characteristic load is underestimated by 6.1%. The difference between the characteristic loads shows that the threshold value has a significant influence. The good performance of the peak over threshold method (1.4) is primarily due to the good fit of the Weibull distribution to the extracted peaks, see figure 6. By comparison of figure 4 (Weibull) and figure 6 it is seen that the general behaviour of the extracted peaks changes slightly dependent on which method is used for extracting the peaks.
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Normalized Load [-]
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100Peak over Threshold 2.0
Normalized Load [-]
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eden
ce P
roba
bilit
y [-
]
PoTRiceWeibull
Figure 6. Comparison of local Weibull distribution with Rice, 100 time series. Operation at 15 m/s.
Besides predicting the characteristic load with a return period of 50 years accurately, the methods for statistical load extrapolation should be robust and give a good approximation of the short-term distribution for all return periods. In figure 7 the characteristic loads obtained for four different return periods are shown using the method of global maxima, block maxima and the peak over threshold method together with the Weibull distribution. It is expected that the deviation between the characteristic load obtained by statistical load extrapolation and the analytical solution will increase with the return period, since the local distribution does not fit the tail of the distribution perfectly. For the method of global maxima and block maxima this behaviour is seen, whereas the characteristic load obtained by the peak over threshold method only shows a small variation with increasing return period.
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Figure 7. Comparison for different return periods, Weibull distribution, 100 time series. Operation at 15 m/s
(1 = Rice solution).
From the comparison of the different load extrapolation methods it is seen that the ability to predict the correct characteristic load is highly dependent on how well the local distribution function fits the extracted peaks. By comparison of the Weibull fits in figure 4 (Weibull) and figure 6 it is seen that the Weibull distribution fits the peaks extracted by the peak over threshold method (1.4) well. The statistical uncertainty on the characteristic load is dependent on how many peaks are extracted from the time series. However, all methods are based on the assumption that the individual peaks are independent which approximately has been obtained by applying a time separation of 10s for the peak over threshold method. For the characteristic loads shown in figure 7 the statistical uncertainty varies between 3 and 5% for the method based on global maxima and between 1 and 3% for the peak over threshold method.
V. Statistical Load Extrapolation of Gaussian Response for all Wind Speeds In this section the methods for statistical load extrapolation are compared with the analytical solution using
simulated Gaussian response from 12 different mean wind speeds between cut-in 3 m/s and cut-out 25 m/s. For each
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mean wind speed 1200 – 10 min time series are simulated with the same mean and standard deviation as for the real wind turbine response. The statistical load extrapolation is performed for the Weibull distribution and the peaks have been extracted by the method of global maxima and the peak over threshold method (1.4).
The long-term distribution for the analytical solution is determined by equation (11) where the short-term distribution is determined by equation (7) and (8). Thereby, the long-term distribution for the analytical solution is based on the ‘fitting before aggregation’ approach. Because a Gaussian process is used for generating the response at all twelve mean wind speeds the long-term distribution is dominated by the mean wind speed where the largest loads are observed. In the present case 13 m/s is dominating the long-term distribution.
For the ‘fitting before aggregation’ approach Nsim corresponds to the number of time series at each mean wind speed and for the ‘aggregation before fitting’ approach Nsim corresponds to the total number of simulations. The values of Nsim are chosen such that the total number of simulations for all wind speeds is the same for the two approaches. In order to obtain a good fit for the largest peaks in the ‘aggregation before fitting’ approach only the largest 10% of the peaks are used for determining the long-term distribution.
A. Comparison of Load Extrapolation Methods for Weibull Distribution In the following, the two approaches ‘fitting before aggregation’ and ‘aggregation before fitting’ are compared
with the analytical solution using a Weibull distribution for the local peaks. The peaks have been extracted by the method of global maxima and the peak over threshold method (1.4). The load extrapolation methods are compared for a return period of 50 years and the results are shown in figure 8.
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Figure 8. Comparison of ‘fitting before aggregation’ and ‘aggregation before fitting’, fit: Weibull
distribution, 50 year return period. Operation at all wind speeds (1 = Rice solution).
From figure 8 it is seen that the characteristic loads determined by the ‘fitting before aggregation’ approach show the same behaviour as when only one wind speed is considered. The methods of global maxima underestimate the characteristic load with 5% using 100 time series at each mean wind speed. For the peak over threshold method the characteristic load is underestimated with 1% using the same number of time series. For the ‘aggregation before fitting’ approaches the characteristic load is overestimated with 7% and 8% for the method of global maxima and the peak over threshold method, respectively, using 1200 time series. The statistical uncertainty using the
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‘aggregation before fitting’ approach is significantly higher than for the ‘fitting before aggregation’ approach. The higher statistical uncertainty is primary due to that only the largest 10% of the peaks are used for fitting the long-term distribution.
The characteristic loads obtained for different return periods using the ‘fitting before aggregation’ and ‘aggregation before fitting’ approach for the methods of global maxima and the peak over threshold method are shown in figure 9. For the ‘fitting before aggregation’ approach the peak over threshold method is within 2% of the characteristic load for all return periods whereas the method of global maxima is within 5%. The methods based on the ‘aggregation before fitting’ approaches estimates the characteristic load within 9%.
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Figure 9. Comparison for different return periods, fit: Weibull distribution, Nsim=100 for ‘fitting before
aggregation’ and Nsim=1200 for ‘aggregation before fitting’. Operation at all wind speeds (1 = Rice solution).
VI. Conclusion In this paper three different methods for statistical load extrapolation are studied for a stationary Gaussian
process with approximately the same spectral properties as the flap bending moment for a representative wind turbine blade. The methods considered are the method of global maxima, block maxima and the peak over threshold method with two different threshold values. First, the different methods are compared for one mean wind speed using a Weibull distribution for the local peaks. The results show that the goodness of fit for the local distribution function is very important for obtaining the ‘correct’ characteristic load which is determined analytically for the Gaussian distributed response. However, the different methods for extracting the peaks lead to different ‘optimal’ (best fit) distributions for the local peaks. Therefore the distribution function should be chosen individually for each mean wind speed dependent on which methods is used for extracting the peaks. For the stationary Gaussian response the Weibull distribution fits the peaks extracted by the peak over threshold method (1.4) well and good results are obtained.
The statistical uncertainty decreases as expected with the number of peaks extracted from the time series and is also dependent on the type of local distribution. For 25 time series at each mean wind speed and using a Weibull distribution for the local maxima the statistical uncertainty on the characteristic load (50 year return period) is 15% if the method of global maxima is used and only 5% if the block maxima or peak over threshold methods are used.
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By approximating a stationary Gaussian process for 12 different mean wind speeds between cut-in 3 m/s and cut-out 25 m/s the ‘fitting before aggregation’ and ‘aggregation before fitting’ approaches are compared. The results show that the ‘fitting before aggregation’ approach gives the best results if again the peak over threshold method is used.
The results in the present paper are based on simulated Gaussian response. Even though this response has approximately the same spectral properties as a real wind turbine’s response, not all details in the response are captured, such as the influence of the control system at high loads. The influence of the control system on the response and on the statistical load extrapolation methods should be studied further. However, based on the present study it can be concluded that the fit of the local distribution function to the extracted peaks is the most important aspect in order to determine the characteristic load with high accuracy. The best results for the Gaussian response where obtained by extracting the peaks with the peak over threshold method using a threshold value on the mean plus 1.4 standard deviations together with a Weibull distribution.
Acknowledgments The work presented in this paper is part of the project ‘Probabilistic design of wind turbines’ supported by the
Danish Research Agency, grant no. 2104-05-0075. The financial support is greatly appreciated.
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