Recipes for Correcting the Impact of Effective Mesoscale Resolution on the Estimation of Extreme Winds

Transcription

1 MARCH 2012 L A R S É N ET AL. 521 Recipes for Correcting the Impact of Effective Mesoscale Resolution on the Estimation of Extreme Winds XIAOLI GUO LARSÉN, SØREN OTT, JAKE BADGER, ANDREA N. HAHMANN, AND JAKOB MANN Wind Energy Division, Risø National Laboratory for Sustainable Energy, Technical University of Denmark, Roskilde, Denmark (Manuscript received 27 April 2011, in final form 8 September 2011) ABSTRACT Extreme winds derived from simulations using mesoscale models are underestimated because of the effective spatial and temporal resolutions. This is reflected in the spectral domain as an energy deficit in the mesoscale range. The energy deficit implies smaller spectral moments and thus underestimation in the extreme winds. The authors have developed two approaches for correcting the smoothing effect resulting from the mesoscale model resolution that impacts the extreme wind estimation by taking into account the difference between the modeled and measured spectra in the high-frequency range. Both approaches give estimates of the smoothing effect that are in good agreement with measurements from several sites in Denmark and Germany. 1. Introduction Recently mesoscale modeling has been shown to have great potential in providing mean wind statistics for the dramatically growing wind energy sector. At the same time, the results directly from mesoscale modeling have also been used to estimate the extreme wind (Pryor et al. 2012; Hofherr and Kunz 2010; Kunz et al. 2010). The extreme wind is a parameter that has to be calculated for each wind turbine site, thus ensuring that the wind does not exceed the turbine s design specification. It is well known that for winds simulated with mesoscale models, the variations in the mesoscale range are smeared because of spatial and temporal averaging effects (e.g., Skamarock 2004, 2011; Frehlich and Sharman 2008). This is reflected in the spectral domain as the lower spectral energy level in comparison with measurements in the mesoscale range, corresponding from a few kilometers to tens of kilometers, or equivalently, from a few minutes to a few hours. This issue is illustrated in Fig. 1a, where for frequency f greater than 2day 21, the spectrum of the observed wind speed S(f) follows f 25/3, while the spectrum of the simulated wind speed has a steeper slope, with f 23. This decay of f 23 Corresponding author address: Xiaoli Guo Larsén, Wind Energy Division, Risø National Laboratory for Sustainable Energy, 4000, Roskilde, Denmark. xgal@risoe.dtu.dk indicates energy removal by the model s dissipation mechanisms, related to the filtering and damping schemes of the model (Skamarock 2004). In the literature there is a series of studies concerning the mesoscale spectral form based on measurements, mesoscale modeling, and theoretical arguments (Brown and Robinson 1979; Gage and Nastrom 1986; Högström et al. 1999; Lindborg 1999; Tung and Orlando 2003; Lindborg et al. 2010; Larsén et al. 2011). The spectra were studied mostly in the wavenumber (k) domain instead of the frequency domain. Measurements suggest that, for the range from a few kilometers to several hundreds of kilometers, S(k)followsk 25/3 but changes to approximately follow k 23 at an even larger scale up to a few thousands of kilometers. This spectral behavior is also supported by many theoretical arguments, even though it is not thoroughly clarified (e.g., Lindborg 1999; Lindborg et al. 2010; Tung and Orlando 2003). For wind energy applications, it is more relevant to focus on the f domain. Considering that the scales are consistent in space and time, it is expected that S(f) followsf 25/3 in high frequency and through a transition range changes to approximately follow f 23 for lower frequencies, according to the scale analysis through Taylor s transformation: 2pf 5 U 0 k,withu 0 as the background mean wind speed (Brown and Robinson 1979; Gage and Nastrom 1986; Högström et al. 1999). This spectrum description is supported by our measurements from several sites and Fig. 1a is for the Horns Rev site (west DOI: /JAMC-D Ó 2012 American Meteorological Society

2 522 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 51 FIG. 1. Spectra of wind speed at 10 m from WRF 15-km simulation and observation at Horns Rev during : (a) S( f), (b) fs(f), and (c) f 3 S(f) vs frequency f. The thin line in (a) shows the spectral form with a slope of 2 5 /3 in higher frequency, transforming to a slope of 23 in lower frequency. The thick lines in (a) show the reference slopes 2 5 /3 and 23. of Jutland, Denmark), where the thin solid line reads S(f) 5 a 1 f 25/3 1 a 2 f 23, in analog to Eq. (71) in Lindborg (1999); here the simple coefficients a 1 5 a are used, with the intention not to model the spectrum but to show the transition of the slope from 2 5 /3 to 23. It could be observed from Fig. 1a that the slope transition occurs at f ; 1 day 21 ; the physical meaning of this is discussed in section 5a. Wind variations in the mesoscale range are important for the estimation of extreme winds. The same spectra as in Fig. 1a are presented as fs(f) versus f as well as f 3 S(f) versus f for the entire range in Figs. 1b,c, respectively. Overall they show good agreement between the observed and Weather Research and Forecasting model (WRF)-simulated wind variation for f, 2 day 21 but clear deviation in higher frequencies. In this paper, results from simulations from three mesoscale models are used to examine the spectral behavior of the wind time series in the mesoscale range. The mesoscale range is referring to approximately 1, f, 72 day 21, with 1 day 21 being approximately where the slope transition occurs, and 72 day 21 being the Nyquist frequency of the time series block averaged over 10 min. The three models are a high-resolution limited area model (HIRHAM5; Christensen et al. 2006), regional climate model (REMO; de), and WRF (Skamarock et al. 2007). These are three well-used models for wind studies in northern Europe. The models use a variety of dynamical cores and physical parameterization schemes, with different largescale forcing, and horizontal and vertical resolutions. Despite all of these differences, the wind speed time series from the three models show consistent spectral behavior in the mesoscale range, similar to the case showninfig.1. It is the intention of this paper to develop approaches for estimating the smoothing effects of mesoscale modeling on extreme wind estimation, and thus to bridge the gap between the current models effective resolution and a time scale of approximately 10 min (or equivalently a few kilometers), which is approximately the upper limit of the mesoscale range and also the reference scale as used in the European standard (Eurocode 1995). The two approaches developed here utilize the annual maximum method (AMM) (Gumbel 1958; Larsén and Mann 2009) to estimate the extreme winds. An often-used definition for the extreme wind, as used in the European Load Code (Eurocode 1995), is the 50-yr wind (i.e., the 10-min averaged wind, which on average occurs with a return period of 50 yr). Note that the extreme wind here is different from the gust factor that has time scales ranging from several minutes to fractions of a second (e.g., Kristensen et al. 1991). In applying AMM, we sort the set of annual maximum winds from an n-year record (U max 5 U max,i,withi 5 1,..., n) in ascending order and use the Gumbel (type I) extreme wind distribution to fit this set of sorted U max (Gumbel 1958). Together with the relation between the cumulative probability, which is a double

3 MARCH 2012 L A R S É N ET AL. 523 exponential and the reoccurrence interval T r, here 50 yr, we obtain the T r -year wind speed as U Tr 5 a lnt r 1 b, (1) where a and b are coefficients calculated from the probability-weighted moment procedure (Abild 1994; Hosking 1985) and functions of U max. The relative underestimation in the dataset of U max should, in principle, be the same as that of U Tr. The impact of the effective resolution on U max is estimated in terms of the peak factor k p, defined as k p 5 U max 2 U, (2) s with U max 5 (1/n)å n i51 U max,i as the mean of the annual wind maxima and U and s as the mean and standard deviation of the wind speed. The three meteorological models are briefly described in section 2 and examples of the spectra from the modeled winds are presented in section 3. The measurements used for data validation are described in section 4. The two approaches we developed for correcting the impact of model effective resolution are introduced in section 5. The results and their validation against data follow in section 6. A discussion is given in section 7 and brief conclusions in section 8. A list of symbols is provided in Table 1 for readability. 2. The mesoscale atmospheric models HIRHAM5 combines the dynamical core of the numerical weather prediction model HIRLAM, version 7, (e.g., schemes for diffusion, damping, and advection filter) and the physical parameterization schemes of the global climate model ECHAM5 (e.g., microphysics, radiation, surface thermal exchange, boundary layer parameterization, and cloud physics parameterization). Details are available in, for example, Christensen et al. (2006) and Roeckner et al. (2006). We use data from two HIRHAM5 runs, one driven by global data from ECHAM5 and another driven by data from the 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re- Analysis (ERA-40). The resolution of the ECHAM5 data is T63 and that of the ERA-40 data is T159 for spectral fields and an N80 reduced Gaussian grid for the surface fields. The global data are downscaled directly to 25 km without nesting. No spectral nudging is used. There are 19 vertical levels. The simulation was done by the Danish Meteorological Institute. REMO is a three-dimensional hydrostatic atmospheric model. It was developed from the Europa-Modell of the TABLE 1. List of symbols. Variables Definition f Cyclic frequency (day 21 ) f c Start frequency of the spectral correction approach (day 21 ) F Once per year exceedance k Wavenumber k p Peak factor, in definition and in recipe I k p,1h Peak factor for the 1-h time series k p,corr Peak factor for modified spectrum with a tail slope of 2 5 /3 from f c to f 5 72 day 21 k pt Peak factor in recipe II k pt,ta Peak factor in recipe II, with averaging time T a k pt,10min Peak factor in recipe II, with averaging time 10 min m 0 Zero-order moment of the wind speed spectrum m 2 Second-order moment of the wind speed spectrum P Probability of wind speed S Power spectrum for wind speed SE kp Smoothing effect as in k p, in recipe I SE kpt Smoothing effect as in k pt, in recipe II SE Umax Smoothing effect as in U max T Integral time scale T a Averaging time T r Return period in years for extreme wind T 0 Basis period of 1 yr U Wind speed U Mean wind speed u Time series of wind speed minus its mean, U 2 U _u The time derivatives of u U max The set of annual wind maxima U max The mean value of the annual wind max, 10 min U max,1h The mean value of the annual wind max of the 1-h simulated time series U max,corr Equivalent 10-min annual max U 1 k p,corr s corr, recipe I 1 U max,corr,ts U max,1h, recipe I 1 2 SE Umax u max Annual wind max from u, thus u max 5 U max 2 U U Tr Extreme wind with T r -year return period a, b Coefficients in the Gumbel distribution for U Tr v 2pf, angular frequency u Filter as used in the calculation of the moment l Rate of occurrence s Std dev of u s Umax Std dev of U max from 10-min measured time series s Umax,1h Std dev of U max from 1-h simulated time series s corr (m 0 ) 1/2 from the modified spectrum r Autocorrelation coef of the wind speed time series r Ta Correlation coef of two blocks of wind speed time series German Weather Service. The physical parameterization package of ECHAM (version 4) is implemented. The standard setup is used, which does not employ the spectral nudging technique. More details about the model setup can be found in Jacob and Podzun (1997),

4 524 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 51 TABLE 2. Description of the model simulations used in this study. Models Spatial resolution Period of integration Large-scale forcing HIRHAM5 ECHAM5 25 km ECHAM5 HIRHAM5 ERA km ERA-40 REMO 50 km ERA-15 and ECMWF analysis REMO 10 km ERA-15 and ECMWF analysis WRF 45 km NCEP reanalysis II WRF 15 km NCEP reanalysis II Jacob et al. (2007), and Larsén et al. (2010). The lateral boundary conditions are provided by ERA-15 data for the period and by ECMWF analysis data for the period ; both are of a horizontal resolution of about 120 km (T106). REMO was first nested with a horizontal resolution of 50 km using a one-way nesting technique and again nested with a resolution of 10 km using a double nesting strategy. There are 27 vertical levels. The simulation was done by the Max Planck Institute for Meteorology in Hamburg, Germany. The WRF (version 3.1.1) setup uses standard physical parameterizations including the Yonsei University planetary boundary layer scheme [see details in Peña and Hahmann (2011)]. The initial and lateral boundary conditions are provided by National Centers for Environmental Prediction U.S. Department of Energy (NCEP DOE) reanalysis data II from 1999 to 2009, with a horizontal resolution of The model was integrated within two domains at the horizontal resolutions of 45 and 15 km. Grid nudging above the planetary boundary layer is done in the outer domain only. There are 45 vertical levels. The simulation was done by the authors at the Risø National Laboratory for Sustainable Energy at the Technical University of Denmark (Risø DTU). For each run, the model resolution, period of integration, and large-scale forcing are listed in Table 2. The time step of the simulations is a few minutes. For all simulations, hourly output is used. section 4). Their spectral behavior in the mesoscale range is similar to that from Horns Rev. To focus on the mesoscale variability, the group of spectra in Fig. 2 is shown for the range f. 1 day 21. The four simulations with spatial resolution ranging from 10 to 25 km (HIRHAM5 ECHAM5, HIRHAM5 ERA- 40, REMO 10 km, and WRF 15 km) provide similar spectral form for 2, f, 12 day 21, with S(f) approximately following f 23, while REMO 50-km and WRF 3. Wind speed spectra from the three models Similar to Fig. 1, Fig. 2 demonstrates the mesoscale spectral behavior of wind speed at 10 m from the HIRHAM5 simulations forced by ECHAM5 (HIRHAM5 ECHAM5) and by ERA-40 (HIRHAM5 ERA-40), the REMO simulations at 10- and 50-km resolutions, respectively, and the WRF simulations at 15- and 45-km resolutions. All time series were extracted from the grid point closest to the Horns Rev site. Horns Rev is an offshore site in the North Sea ( In addition to Horns Rev, measurements from five other sites are used (Table 3; FIG. 2. Spectra of wind speed at 10 m at Horns Rev from measurement and the six model simulations, S( f ) vs frequency f (see data details in Table 2). Thick short straight lines show the three reference slopes of 2 5 /3, 23, and 24.

5 MARCH 2012 L A R S É N ET AL. 525 TABLE 3. Details of measurements and the peak factors calculated using Eq. (2). Stations Data period Location Measurement height k p 6 s(k p ) Horns Rev N, E 62 m Sprogø N, E 11 m Tystofte N, E 10 m Kegnæs N, E 10 m Jylex N, E 10 m FINO N, E 50 m km simulations give a more significant smoothing effect, with S(f) approximately following f 24. Note that when making the spectra, the time series from the entire period for each dataset is used. Thus the data from different models do not have the same period and length. We examined the spectrum for each time series using different periods and lengths and found that this only affects the low-frequency range, while there is almost no difference for f day Measurements Wind measurements of 10-min averages from six sites are analyzed. The Horns Rev, Tystofte, Sprogø, Kegnæs, and Jylex sites are in Denmark and FINO is in Germany, offshore in the North Sea (see Fig. 3). For each site, the coordinates and data period are listed in Table 3. The recording systems at these sites have been kept consistent during the period. Regarding the yearly strongest winds, it is crucial for the calculation of extreme wind that these data are recorded; the instruments can sometimes be destroyed by the storms. Measurements from all levels available are therefore examined together to reduce the chance of missing the highest wind speed data. The spectra were calculated with measurements at one level as given in Table 3. All time series have been despiked. When making the power spectrum, those years where data coverage is less than 90% are disregarded; otherwise, missing data are filled with linear interpolation. approach, we use the statistical model derived in Larsén and Mann (2006, hereinafter LM2006) for correcting the impact of the temporal resolution. In this second method we approximate the combined spatial and temporal resolution effects in the modeled time series into a temporal effect. a. Recipe I: Spectral correction approach Assume that the once-per-year exceedance F follows a Poisson process F 5 exp(2lt 0 ), where T 0 is the period (here 1 yr) and l is the rate of occurrence, calculated with l 5 ð 0 P(u, _u) ud_u _ 5 s _u pffiffiffiffiffiffi P(u), (3) 2p where P(u, _u) is the conditional probability of u, P(u) is the probability of u, _u is the time derivatives of u, and it is assumed that u and _u are independent. Here u is the 5. Two recipes to correct the impact of the effective model resolution Two independent recipes are provided for estimating the impact on the annual wind maximum of the energy deficit in the simulated wind spectra in the mesoscale range. In the first recipe, called spectral correction approach, we parameterize different spectral tails through the spectral moments that describe the rate of occurrence of the annual maximum wind. In the second recipe, here called the effective temporal averaging FIG. 3. The map of Denmark and the stations marked with their initials (H for Horns Rev, S for Sprogø, T for Tystofte, K for Kegnæs, J for Jylex, and F for FINO 1).

6 526 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 51 wind speed minus its mean (U 2 U). Later we use u max as the annual wind maximum from the wind time series minus its mean, thus u max 5 U max 2 U. With a large threshold, such a distribution of the exceedance is valid for a Gaussian process for which P(u) 5 p 1 ffiffiffiffiffiffi 2p s u exp 2 u2 2s 2. (4) u Note that, in reality, the wind speed distribution follows a Weibull rather than a Gaussian distribution, which means that the assumption of u and _u being independent could be questionable. However, we proceed with this simple approach to test its sensitivity. Substituting Eq. (4) into Eq. (3) gives l 5 1 2p s _u exp 2 u2 s u 2s 2 u Equation (5) can be rewritten as l 5 1 sffiffiffiffiffiffi m 2 2p m 0. (5) exp 2 u2, (6) 2m 0 with the spectral moments, m 0 and m 2, defined as m j 5 2 ð 0 u 2 (v)v j S(v) dv, (7) where S(v) is the power spectrum of the Gaussian process u(t), v 5 2pf, and u is a filter due to the temporal resolution given by u 5 [sin(vt a /2)]/(vT a /2), with T a the averaging time. With this filter, the integration of Eq. (7) is done from v 5 0tov 5 2p[1/(2T a )] (i.e., the upper limit of the integration becomes the Nyquist frequency). Thus, the integration does not diverge when the tails have slopes equal to or greater than 22. For the maximum wind that occurs once a year, lt 0 5 1; using this in Eq. (6) gives vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p u max 5 ffiffiffiffiffiffi m 0 2ln 1 sffiffiffiffiffiffi! u t m 2 T 2p 0. (8) Together with Eq. (2), the peak factor k p can be expressed as a function of the spectral moments: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k p 5 2ln 1 sffiffiffiffiffiffi! u t m 2 T 2p 0. (9) Thus, the peak factors with different spectral tails can be estimated. The wind variation in the high-frequency m 0 m 0 range does not affect U, and brings only small pffiffiffiffiffiffiffiffiffiffiffiffiffi differences to s, but it contributes significantly to m 2 /m 0, and hence k p and U max. In Figs. 1b,c, the integrated areas below the curves correspond to m 0 and m 2, respectively, where it can clearly be seen that the difference is greatest in magnitude in m 2 (Fig. 1c). The core of the approach is to replace the spectrum of the modeled winds in the mesoscale range with a spectral slope of 2 5 /3, and extend it to the frequency of 72 day 21, which is the Nyquist frequency of 10-min time series. This approach should increase the extremes from the mesoscale models just enough to match the measurements. In principle, the spectral correction should start where the modeled wind spectrum starts to deviate from the expected spectrum. This corrected, or modified, spectrum and the original modeled hourly spectrum provide two different sets of m 0 and m 2,through Eq. (9), and the corresponding peak factors are denoted as k p,corr and k p,1h, respectively. The smoothing effect can then be estimated as SE kp k p,1h /k p,corr. In the conversion to U max using Eq. (2), we obtain the mean annual wind maximum from the modified spectrum U max,corr 5 U 1 k p,corr s corr,wheres corr is calculated from m 0 of the modified spectrum. At the same time, the underestimation of U max can be obtained through SE Umax (U 1 k p,1h s)/ (U 1 k p,corr s corr ). The annual wind maxima from the simulation, U max,1h, can thus be corrected to U max,corr,ts 5 U max,1h /(12SE Umax ). The values of U max,corr,ts and U max,corr should be comparable. The spectra of the six simulated winds are plotted together with that from measurements in Fig. 4. It should be noted that there is only a negligible difference in the power spectrum made from measurements at different heights from 10 to 62 m for the mesoscale ranges. The starting frequency of the correction f c can now be identified. It was found that f c 5 2day 21 suits all cases except for WRF 45 km (Fig. 4f) for which f c 5 1day 21 is a better choice. Results on the correction of the smoothing effect are given in section 6a, together with a sensitivity test on f c. Measurements, even just for several months, can provide enough information for identifying f c. However, in the wind energy business, sometimes there are no data available at all. For this situation, there are a couple of parameters that can be considered to help determine f c. One parameter is the Nyquist frequency of the largescale forcing, which for the models analyzed here, is 2day 21. This frequency can be used as f c if the global model data fully resolved the variability up to their recording frequency. The spectrum of the wind speed at 10 m from the NCEP reanalysis data (with a horizontal resolution of about 250 km) was calculated and

7 MARCH 2012 L A R S É N ET AL. 527 FIG. 4. In connection with recipe I, the spectral correction approach modification of the spectrum of the hourly simulated wind speed (thick dashed line) by replacing the tail for f. f c with a spectrum of tail slope of 2 5 /3 is shown as the solid lines with f c 5 1and2 day 21. The spectrum from measurements is plotted in dots: (a) HIRHAM5 ECHAM5, (b) HIRHAM5 ERA-40, (c) REMO 10 km, (d) REMO 50 km, (e) WRF 15 km, and (f) WRF 45 km. it was found that there is an energy drop with a slope of 23 atf 1day 21 (not shown), implying that the spectrum with f. 1day 21 is not accountable. Comparing this spectrum with those from mesoscale modeling as in Fig. 4, it can be concluded that the mesoscale modeling has added values in the mesoscale variability, with some models adding more than others. Also, this comparison suggests that, from an engineering loading point of view, a conservative choice of f c of 1 day 21 is preferable compared to the Nyquist frequency of the large-scale forcing. The second parameter to be considered when choosing f c is related to the spectral slope transition at f ; 1 day 21 (Fig. 1a). Our interpretation of this frequency of spectral transition is that it represents a scale of the local weather systems. The integral time scale T is such a parameter, where T 5 Ð 0 r(t) dt, with r being the autocorrelation coefficient; T is an average memory time scale, related to the geographical weather characteristics (Kristensen et al. 2002; Pope 2000). Thus we propose f c 1/T. In the area of the current study, namely, midlatitudes around Denmark and the North Sea, T was calculated from measurements to be about 0.8 day (LM2006). In LM2006, by fitting the spectrum from time series to a power spectrum model S(f ) 5 f(2t)/[11 (2pTf ) 2 ]gs 2, T and s were obtained for several sites. This spectrum model has a tail slope of 22 inthemesoscale range, but this is not important for the estimation of T. The fitting, and hence T and s, are mostly dependent on the spectral form in the energy-containing range, which, for data length on the order of a year, is frequencies a lot lower than 1 day 21. Both the data from

8 528 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 51 TABLE 4. For the Horns Rev site, mean annual mean wind speed (U) and standard deviation (s), mean annual wind maximum (U max ) and standard deviation of the maxima (s Umax ) directly from the time series, measurements (10 min), and modeling (1 h), all at 10 m. The integral time scales T are also given. HIRHAM5 REMO WRF Variables Obs ECHAM5 ERA km 50 km 15 km 45 km U (m s 21 ) s (m s 21 ) U max (m s 21 ) 27.2 s Umax (m s 21 ) 5.1 U max,1h (m s 21 ) s Umax,1h (m s 21 ) T (day) the global climate and mesoscale models could be used for this purpose. For the Horns Rev case, the T values are obtained from the fitting to the wind spectra for measurements and mesoscale modeled winds and they are listed in Table 4. The values range between about 0.8 and 1 day. The fitting of the NCEP reanalysis data gives T day, which is also satisfactory. Of course, it is preconditioned that the long-term wind climate within one model grid box is representative for the site of interest. According to Fig. 4, these are very good estimates of f c, supported by physical meaning. Using f c 1/T in the absence of measurements is recommended. b. Recipe II: Effective temporal averaging approach Using a longer averaging time (T a ) gives smaller extreme winds. For point measurements, LM2006 derived an approach to take into account the temporal resolution effect on the annual maximum wind by assuming the time series to be a Gaussian process. The peak factor in connection with the temporal averaging is denoted as k pt here [as F in LM2006 s Eqs. (10) and (15)] and is a function of the correlation coefficient between the two blocks of time series r Ta : vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 v N 1 u Ta u1 2 r Ta k pt 5 t (1 1 r Ta )ln@ t A, (10) p 1 1 r Ta where N Ta is the number of values in a year at a certain T a [if the number of 10-min values in a year is N, N Ta 5 N/(T a /10), if T a is in minutes]. LM2006 showed that r Ta 5 a(1 2 a m ) 2 =[m(1 2 a 2 ) 2 2a(1 2 a m )], where m is the number of consecutive blocks in a year at T a and a 5 exp(2dt/t), Dt being the time between two values and T again being the integral time scale as introduced in section 5a. Both N Ta and r Ta [LM2006 s Eq. (14)] decrease with increasing T a, and so does k pt. Thus, the underestimation of k pt and hence U max due to temporal resolution can be estimated. Corresponding to an averaging time ranging from 10 min to 6 h, the spectral energy becomes smaller and smaller in the mesoscale range, as shown in Fig. 5. Here the data (thin lines) are 10-min winds measured at Horns Rev, converted from 62 to 10 m using the logarithmic wind law and the Charnock formulation for roughness length. The spectrum at 62 m is very similar. For f. 2day 21 the spectrum from the 10-min time series approximately follows f 25/3, but at longer running-averaging time, it drops with a steeper slope with f. We denote the smoothing effect due to temporal averaging as SE kpt and calculate it with SE kpt k pt,ta /k pt,10min,withk pt,ta as the peak factor at an averaging time of T a. The theoretical relationship between SE kpt and T a from LM2006 is shown in Fig. 6. The values of the wind speed from the mesoscale models here are from a single time step (a few minutes) but sampled every 1 h, and at the same time, it is spatially inherently smoothed over the model gridbox size. The spectra from the simulations are plotted in Fig. 5. Taking HIRHAM5 ECHAM5 as an example (the large circles in Fig. 5), the combined impact of spatial smoothing of 25 km and a sampling interval of 1 h is comparable to an effective averaging time of about 2 h, resulting in the spectrum following approximately the one with T a 2 h. Thus, if we approximated the spatial averaging and the disjunctive sampling effect to an effective averaging time of about 2 h, we get SE kpt 5 12:3% (Fig. 6). 6. Results and validation The peak factors were first calculated directly from the 10-min measured time series at six sites using Eq. (2) for each year, and then averaged; thus we obtain a mean value and a standard deviation (Table 3). All in all, the mean value over all sites is At the offshore site Horns Rev, the measurements give k p The mean annual wind maxima were calculated directly from the time series of the 10-min measurements,

9 MARCH 2012 L A R S É N ET AL. 529 FIG. 6. Smoothing effect SE kpt k pt,ta /k pt,10min (%) varying with averaging time T a with k pt calculated using Eq. (10). FIG. 5. Spectra S(f) as a function of frequency f, from measurements and simulations. The lines are from measurements with averaging time T a from 10 min to 6 h. All values are derived from wind speed at 10 m for the site Horns Rev. as well as those of the hourly output from simulations. The results are given in Table 4, together with the standard deviation of the annual wind maxima. From the 10-min wind measurements at Horns Rev, the mean value of the annual maximum winds at 10 m U max is 27.2 m s 21 with a standard deviation of 5.1 m s 21. Here, the wind measurements at 62 m from 1999 to 2005 from Horns Rev were used to demonstrate the spectral correction. The data coverage of 2005 is 77% and from 2000 to 2004 it varies between about 95% and 100%. Data from 2006 were disregarded because the data coverage is only 47%. To make comparisons with the simulated winds at 10 m, the annual maximum winds are extrapolated from 62 to 10 m using the logarithmic wind law and the Charnock formulation for the water surface roughness length. The wind profiles during many storms were examined and they showed to be logarithmic throughout the measuring levels from 15 to 62 m. This suggests that neutral stability is a reasonable description for such strong wind conditions. The Horns Rev data since 2003 are affected by the wakes of the wind farm. This wind farm has a length scale of about 6 km, at a mean wind speed of 8 m s 21 ; the wind variability of time scale on the order of 10 min is affected (Frandsen et al. 2007). At the extreme wind strength of about 27 m s 21, this effect corresponds to a smaller time scale of about 4 min, which is beyond our studied frequency range. At the same time, we estimated the mean annual maximum wind before 2003 (free of wake effects) and obtained the corresponding U max as 27.4 m s 21,thoughwithalargers Umax as 6.7 m s 21.Itseemsthatthe10-minmeanextreme wind estimation is not sensitive to the possible wake effects from the wind farm, probably because the turbines are generally shut down at this wind strength. a. From recipe I: The spectral correction approach The following parameters were calculated for the offshore site Horns Rev: d d d d d the peak factors using Eq. (9) based on the spectra of hourly simulated wind time series (in Table 5 as k p,1h ), the peak factors using Eq. (9) based on the modified spectra as shown in Fig. 4 (in Table 5 as k p,corr ), smoothing effects on k p and U max, calculated with SE kp k p,1h /k p,corr and SE Umax (U 1 k p,1h s)/ (U 1 k p,corr s corr ) (in Table 5 as SE kp and SE Umax ), the mean annual wind maximum with the modified spectra as indicated in Fig. 4 (in Table 5 as U max,corr ), and equivalent 10-min mean value of the annual wind maxima calculated as U max,1h /(1 2 SE Umax )(intable 5asU max,corr,ts ). As shown together in Table 4, the corresponding values of the mean annual wind maximum from the hourly modeled time series are underestimated, being in the range of 21.6 and 23.8 m s 21, as compared with 27.2 m s 21 from measurements. The peak factors of the simulated hourly time series calculated from the moments m 0 and m 2 of the discrete power spectra of all models for Horns Rev, k p,1h, are also underestimated and they range between 4.07 and 4.18 (Table 5). We calculate m 0 and m 2 from the actual spectrum of the measured 10-min time series at Horns Rev and obtain k p It is very close to the value 4.97 as calculated from the measured time series using Eq. (2) (Table 3). By applying the modified spectrum shown in Fig. 4, the peak factors are obtained from the six simulations at the grid point closest to Horns Rev and

10 530 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 51 TABLE 5. For the Horns Rev site, peak factors k p,corr and k p,1h, smoothing effect of the peak factor and of the mean annual wind maximum SE kp and SE Umax, and the corrected mean annual wind maxima U max,corr and U max,corr,ts through the spectral correction approach. In this approach, f c 5 2 day 21 is used. HIRHAM5 REMO WRF Variables ECHAM5 ERA km 50 km 15 km 45 km* k p,1h k p,corr SE kp 15.5% 15.6% 15.9% 16.5% 15.5% 18.1% SE Umax 10.7% 10.8% 11.0% 11.6% 10.7% 12.6% U max,corr (m s 21 ) U max,corr,ts (m s 21 ) * For WRF 45 km, f c 5 1 day 21 is used (see Fig. 4). they are given in Table 5 as k p,corr. According to Fig. 4, we use f c 5 2 day 21 for all, except for WRF 45 km where f c 5 1 day 21 is used. The corrected peak factors for the simulations range between 4.91 and 4.97, in good agreement with the peak factor from the measurements, and the values of U max,corr are greatly improved and they agree well with measurements, being in the range of m s 21. This suggests the success of this approach at Horns Rev in correcting the effective mesoscale resolution effect, in spite of the uncertainties introduced by the differences in U and s among different models and measurements, as well as the different periods that are used (Table 4). It is a pity that we cannot use an overlapping period long enough for all six different model simulations and measurements. Some simulations are not overlapping with any other simulations or with the measurements. However, the yearly mean as well as maximum wind speeds from all sites is examined and no trend of statistical significance is observed. This implies that our assumption of a rather stationary wind climate, both in the mean and strong winds, in the past several decades is reasonable and does not necessarily bring bias in the results. The smoothing effect SE kp for all models is estimated using SE kp k p,1h /k p,corr to be 15.5% 18.1% for different simulations. Accordingly, the underestimation in U max,se Umax, is estimated and it is slightly below 11% for simulations with resolutions coarser than 25 km and more than 11% for 45- and 50-km resolutions. We use SE Umax, to correct the mean annual wind maximum from the 1-h time series from simulations and obtain greatly improved results (see U max,corr,ts in Table 5). The sensitivity of U max,corr to f c was examined using f c equal to 1 and 2 day 21, respectively. Thus we expect the largest difference in U max,corr for the WRF 45-km data in Table 5; for these data, using f c 5 2day 21 gives a 1.5 m s 21 smaller U max,corr than using f c 5 1day 21.Fortherestof the modeled data in Table 5, using f c 5 2day 21 gives a m s 21 smaller U max,corr than using f c 5 1 day 21. b. From recipe II: The effective temporal averaging approach The smoothing effect SE kpt for T a 5 2hwascalculated from the observed 10-min wind speed from several sites using Eq. (2) and it is 10% at Horns Rev, 9% at Sprogø, 12% at Tystofte, 11% at Kegnæs, 11% at Jylex, and 14% at FINO. When compared with the value 12.3% obtained in section 5b, it seems that through Eq. (10), reasonable estimation of the effective resolutioneffectcanbeobtained,whichisalsotheconclusion from LM2006. The drawback of this method is that sometimes it is difficult to obtain a good fit such as the HIRHAM data in Fig. 5. Even for the HIRHAM data, the circles and the dots are actually located in between the curves for T a h for the most part; to be more accurate, SE kpt is somewhere between 12.5% and 15% (SE kpt for T a 5 3h). For the REMO simulated winds, the spectrum corresponding to 10-km resolution is distributed across T a ; 2 5 h and that for 50 km across T a ; 3 5 h, thus corresponding to SE kpt in an even broader range of about 12.5% 17% and 15% 17%, respectively. For the WRF simulated winds, the spectrum is distributed across T a ; 2 3 h for the 15-km-resolution data and across T a ; 4 5 h for the 45-km-resolution data, corresponding to SE kpt in the range of 12.5% 15% and 16.5% 17%, respectively. This suggests that the characteristics of the spectra from the REMO simulated winds result in larger uncertainty than the HIRHAM5 and WRF simulated winds when using this approach. However, these ranges agree well with SE kp in section 6a using the spectral correction approach. 7. Discussion The smoothing effect is a common issue in the mesoscale modeled winds (Skamarock 2004, 2011; Frehlich and Sharman 2008). It is also demonstrated here by wind

11 MARCH 2012 L A R S É N ET AL. 531 simulations from three well-used mesoscale models, reflected as the spectral energy deficit in the mesoscale range. Accurate reproduction of the energy in this range is shown by the statistical analysis to be essential for extreme wind estimation. It is the target of this paper to derive approaches that can take this smoothing effect into account, and thus correct the extreme winds from a resolution of tens of kilometers to a scale of a few kilometers, or equivalently, from a time scale of hours to 10 min. The smoothing effects of the spatial and temporal resolutions are related and they are embedded together in the output of mesoscale simulations. The spectral energy deficit in the mesoscale range is related to many factors in the mesoscale modeling, including the temporal and spatial resolutions of the largescale forcing, the grid nudging technique, the filtering and damping schemes of the model, etc. However, when applying the approaches proposed, it is the eventual spectral behavior, not these causes, that matters. In other words, the approaches take into account the synthesized smoothing effects of the mesoscale modeling. Two simple approaches are provided. The first approach describes the occurrence rate in terms of the spectral moments. The moments, especially higher-order moments, directly account for the spectral tail form that reflects the smoothing effect to different degrees. The spectral energy deficit and the temporal resolution are easily taken into account in this approach. The procedure for the correction is straightforward and is given in section 5a. The exact frequency where the mesoscale energy starts to deviate from the measurements reflects the effective resolution of the model. This frequency f c can be determined from measurements. Even just for several months or one year, the spectrum for f. 1day 21 can be made available; f c can then easily be found with the aid of a plot like Fig. 1. In the absence of measurements, we proposed a conservative estimate for f c by relating this frequency to a time scale representing the local weather, the integral time scale T: f c 1/T. The estimates of T using both the data from mesoscale models and global models are satisfactory. Another way to conservatively estimate f c was also discussed, that is, by assuming no added values of the wind variability from the mesoscale modeling to the global climate data and choosing f c based on the spectral behavior of the global climate data, which for the studied case here is 1 day 21. For our test case, the conservative estimates were shown to bring a larger value in U max,corr on the order of about 1 m s 21, which is within the standard deviation of the series for annual wind maximum and therefore indicates a rather low extra cost. It is anticipated that if a model can resolve finer-scale variability, f c is higher (Skamarock 2004) and accordingly it becomes less important which exact value is chosen for f c in the range from 1/T to f c. If the model can resolve the variability up to the limit of mesoscale (Larsén et al. 2011), then the spectral correction is not needed and applying it will not make a difference. The second approach, the effective temporal averaging approach, is based on the method derived in LM2006 for correcting the temporal resolution for point measurements and it is used here to resolve the effective averaging time that combines the spatial and temporal smoothing. With measurements at hand, one can also plot the spectra at different T a, similar to Fig. 5, and fit the modeled spectrum to measured spectra of different averaging times. Without measurements, T can be estimated from modeled data as discussed before, and the underestimation of k p can be calculated. Its drawback is that the fit is sometimes not particularly good, leading to considerable uncertainty. For the offshore site, we used SE kp from the spectral correction approach to obtain U max and obtained a very good agreement with measurements. This owes partly to the simplicity of the offshore site Horns Rev where surface complexity is not an issue, whereas this might well be an issue at the land sites. For this reason, we did not make such a conversion from k p to U max for the land sites; we do not intend to go into discussions about the models performance induced by factors other than resolution (e.g., the issue of the quality of mesoscale modeling itself). This approach can in principle be applied to the global reanalysis data as well if U and s from the global reanalysis data were reliable. But there is very little chance that the wind field averaged over a grid box of 100 km or coarser can represent the wind conditions for a specific site. That the surface condition at the measuring point is not the same as that averaged over the model grid box is one of the reasons for the discrepancies between the modeled and measured U and s. The chance of this kind of inconsistency is greater when the model resolution is coarser. In comparison with the NCEP reanalysis winds, the mesoscale modeling not only provides a better mean and standard deviation of the wind speed but also adds significant value to the mesoscale wind variability. Thus it is necessary to make the corrections from mesoscale modeled results. Even though the mesoscale modeling is limited in resolving the wind variabilities at fine scales, by improving the simulation we can get improved wind statistics in terms of U and s. This will eventually improve the application of the approaches. The good prediction of the annual wind maxima suggests that, despite the assumption of a Gaussian process, the smoothing effect on the extreme winds is well modeled, at least at the sites in this study. It remains to be seen if the assumption is valid for sites in other locations (e.g., the tropics).

12 532 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 51 We discussed the extreme winds in terms of the parameters k p and the mean annual maximum winds U max, but not the T r -year return wind U Tr. Theoretically, the effective resolution effect on U Tr shouldbethesameas that on U max. Another reason for not using U Tr is that there are other factors that may affect the estimation of U Tr (e.g., the data length and period). The measurements are not always long enough and the simulation periods from different models are hardly overlapping. But this will not affect the analysis of the effective resolution effect. 8. Conclusions Simulations from three mesoscale models that are well used for wind energy studies in northern Europe are analyzed. They cover a range of model setups, physics, and dynamical schemes as well as large-scale forcing, thus providing an ensemble of mesoscale modeled wind time series and demonstrating that mesoscale smoothing affecting the extreme wind estimation is a common issue. There is a spectral energy deficit of the mesoscale modeled winds in the mesoscale range, here ;2 day 21, f, 72 day 21, in comparison with measurements. The spectra from measurements in this range show a slope of approximately 2 5 /3 while those from mesoscale modeling show steeper slopes of 23 or more. This energy deficit has serious implications for extreme wind estimation. The first recipe, the spectral correction approach, is robust in handling the combined impact of the effective spatial and temporal resolutions. It corrects the spectrum of the modeled winds to match that of measured 10-min time series. Thus, spectral moments can be calculated from the modified spectra and then be used for calculating the peak factor. Together with the mean and the standard deviation, the first recipe predicts the mean annual wind maximum. The estimation is in good agreement with measurements. The second recipe, the effective temporal averaging approach, combines the spatial and temporal resolutions into an effective temporal resolution. It is simple but often the estimation of the smoothing effect is less certain. The result is also supported by measurements. Measurements from a short period of a half-year or one year would be helpful in determining the start frequency of the energy deficit f c for recipe I and the integral time scale T for recipe II. Acknowledgments. This work is supported by Danish Grant , the Nordic project Climate and Energy System ( ), and EU SafeWind project (213740). We thank Morten Nielsen, Mark Kelly, and Niels-Erik Clausen from Risø DTU and Martin Drew from DMI for discussions and help. Data from Horns Rev are provided by DONG Energy, data from FINO are provided by Deutsches Windenergie Institut (German Wind Energy Institute) through EU-NORSEWIND, the REMO data are provided by the Max Planck Institute for Meteorology, and HIRHAM5 data are provided by the Danish Meteorological Institute. REFERENCES Abild, J., 1994: Application of the wind atlas method to extremes of wind climatology. Tech. Rep. Risoe-R-722(EN), Risø National Laboratory, Roskilde, Denmark, 174 pp. Brown, P., and G. Robinson, 1979: The variance spectrum of tropospheric winds over eastern Europe. J. Atmos. Sci., 36, Christensen, O., M. Drews, J. Christensen, K. Dethloff, K. Ketelsen, I. Hebestadt, and A. Rinke, cited 2006: The HIRHAM regional climate model version 5(b). [Available online at dmi.dk/dmi/index/viden/dmi-publikationer/tekniskerapporter. htm.] Eurocode, 1995: Eurocode 1, Basis of design and actions on structure Parts 2 4: Actions on structure Wind actions. 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