A joint probability density function of wind speed and direction for wind energy analysis

A very flexible joint probability density function of wind speed and direction is presented in this paper for use in wind energy analysis. A method that enables angular–linear distributions to be obtained with specified marginal distributions has been used for this purpose. For the marginal distribution of wind speed we use a singly truncated from below Normal–Weibull mixture distribution. The marginal distribution of wind direction comprises a finite mixture of von Mises distributions. The proposed model is applied in this paper to wind direction and wind speed hourly data recorded at several weather stations located in the Canary Islands (Spain). The suitability of the distributions is judged from the coefficient of determination R².

The conclusions reached are that the joint distribution proposed in this paper: (a) can represent unimodal, bimodal and bitangential wind speed frequency distributions, (b) takes into account the frequency of null winds, (c) represents the wind direction regimes in zones with several modes or prevailing wind directions, (d) takes into account the correlation between wind speeds and its directions. It can therefore be used in several tasks involved in the evaluation process of the wind resources available at a potential site. We also conclude that, in the case of the Canary Islands, the proposed model provides better fits in all the cases analysed than those obtained with the models used in the specialised literature on wind energy.     

An M‐Rice wind speed frequency distribution

Several known statistical distributions can describe wind speed data, the most commonly used being the Weibull family. In this paper, a new law, called ‘M‐Rice’, is proposed for modeling wind speed frequency distributions. Inspired by recent empirical findings that suggest the existence of some cascading process in the mesoscale range, we consider that wind speed can be described by a seasonal AutoRegressive Moving Average (ARMA) model where the noise term is ‘multifractal’, i.e. associated with a random cascade. This leads to the distribution of wind speeds according to the M‐Rice probability distribution function, i.e. a Rice distribution multiplicatively convolved with a normal law. A comparison based on the estimation of the mean wind speed and power density values as well as on the different goodness‐of‐fit tests (the Kolmogorov–Smirnov test, the Kuiper test and the quantile–quantile plot) was made between this new distribution and the Weibull distribution for 35 data sets of wind speed from the Netherlands and Corsica (France) sites. Accordingly, the M‐Rice and Weibull distributions provided comparable performances; however, the quantile–quantile plots suggest that the M‐Rice distribution provides a better fit of extreme wind speed data. Beyond these good results, our approach allows one to interpret the observed values of Weibull parameters. Copyright © 2011 John Wiley & Sons, Ltd.     

Determination of Weibull parameters for wind energy analysis of İzmir,Turkey

In this study, the two Weibull parameters of the wind speed distribution function, the shape parameter k (dimensionless) and the scale parameter c (ms^?1), were computed from the wind speed data for ?zmir. Wind data, consisting of hourly wind speed records over a 5‐year period, 1995–1999, were measured in the Solar/Wind‐Meteorological Station of the Solar Energy Institute at Ege University. Based on the experimental data, it was found that the numerical values of both Weibull parameters (k and c) for ?zmir vary over a wide range. The yearly values of k range from 1.378 to 1.634 with a mean value of 1.552, while those of c are in the range of 2.956–3.444 with a mean value of 3.222. The average seasonal Weibull distributions for ?zmir are also given. The wind speed distributions are represented by Weibull distribution and also by Rayleigh distribution, with a special case of the Weibull distribution for k=2. As a result, the Weibull distribution is found to be suitable to represent the actual probability of wind speed data for ?zmir (at annual average wind speeds up to 3 ms^?1). Copyright © 2002 John Wiley & Sons, Ltd.     

Analysis of two-component mixture Weibull statistics for estimation of wind speed distributions

The typical two-parameter Weibull is a flexible distribution that is useful for describing unimodal frequency distributions of wind speeds at many sites. A two-component mixture Weibull distribution (WW-probability distribution function (pdf)) is even more useful because it is additionally able to represent heterogenous wind regimes in which there is evidence of bimodality or bitangentiality or, simply, unimodality.An analysis is made in this paper of three of the most frequently used methods in the estimation of the five parameters of the WW-pdf and the numerical methods employed are described. Hourly mean wind speed data recorded at four weather stations located in the island of Gran Canaria (Spain) are used to analyse the estimation methods. Prior identification of the sample components of the mixture is not required.The suitability of the distributions is judged from the various tests-of-fit commonly used in the specialised literature on wind energy. A comparison is also made of the ability to describe the experimental wind power density distribution. The general conclusion is that if the sample data are independent then maximum likelihood (ML) estimators should be used due to their large sampling efficiency. However, they require elaborate calculation techniques. The least-square (LS) method provides a robust and computationally efficient alternative to the techniques currently in use. The method of moments has the disadvantage that it does not always supply a feasible result and lacks the desirable optimality properties of ML and LS estimators.     

Estimating wind speed probability distribution based on measured data at Burla in Odisha,India

In this work, the wind speed probability distribution is estimated for Burla location in the state of Odisha in the east coast of India. For this purpose, 10 min averaged wind speed data collected over one year period at Burla is utilized. Specifically, Weibull, Gamma, Lognormal, Inverse Gaussian distributions; mixture distributions such as Weibull-Weibull, Gamma-Weibull, Normal-Weibull, and Normal-Normal are examined to evaluate their suitability to represent the measured wind speed. The non-parametric kernel density method is also used to represent the measured wind speed wherever the parametric distributions are not suitable. Chi-square test and Kolmogorov-Smirnov goodness-of-fit tests are used to evaluate the suitability of each of the above distributions.     

随机风速下风力发电机传动系统动态外载荷计算

通过建立实际风场的随机风速模型、风轮气动性能模型、传动链模型、发电机模型,获得了风力发电机整机的动力学分析模型。根据变速风力发电机的控制策略,计算了随机风速和发电机负载条件下传动系统的动态外载荷,并比较了采用刚性、柔性传动链模型对传动系统动态外载荷的影响,为进一步研究风力发电机齿轮传动系统的动态特性及可靠性奠定了基础。     

Data efficient measure-correlate-predict approaches to wind resource assessment for small-scale wind energy

The feasibility of predicting the long-term wind resource at 22 UK sites using a measure-correlate-predict (MCP) approach based on just three months onsite wind speed measurements has been investigated. Three regression based techniques were compared in terms of their ability to predict the wind resource at a target site based on measurements at a nearby reference site. The accuracy of the predicted parameters of mean wind speed, mean wind power density, standard deviation of wind speeds and the Weibull shape factor was assessed, and their associated error distributions were investigated, using long-term measurements recorded over a period of 10 years. For each site, 120 wind resource predictions covering the entire data period were obtained using a sliding window approach to account for inter-annual and seasonal variations. Both the magnitude and sign of the prediction errors were found to be strongly dependent on the season used for onsite measurements. Averaged across 22 sites and all seasons, the best performing MCP approach resulted in mean absolute and percentage errors in the mean wind speed of 0.21 ms⁻¹ and 4.8% respectively, and in the mean wind power density of 11 Wm⁻² and 14%. The average errors were reduced to 3.6% in the mean wind speed and 10% in the mean wind power density when using the optimum season for onsite wind measurements. These values were shown to be a large improvement on the predictions obtained using an established semi-empirical model based on boundary layer scaling. The results indicate that the MCP approaches applied to very short onsite measurement periods have the potential to be a valuable addition to the wind resource assessment toolkit for small-scale wind developers.     

Estimation of wind energy potential using different probability density functions

In addition to the probability density function (pdf) derived with maximum entropy principle (MEP), several kinds of mixture probability functions have already been applied to estimate wind energy potential in scientific literature, such as the bimodal Weibull function (WW) and truncated Normal Weibull function (NW). In this paper, two other mixture functions are proposed for the first time to wind energy field, i.e. the mixture Gamma–Weibull function (GW) and mixture truncated normal function (NN). These five functions will be reviewed and compared together with conventional Weibull function. Wind speed data measured from 2006 to 2008 at three wind farms experiencing different climatic environments in Taiwan are selected as sample data to test their performance. Judgment criteria include four kinds of statistical errors, i.e. the max error in Kolmogorov–Smirnov test, root mean square error, Chi-square error and relative error of wind potential energy. The results show that all the mixture functions and the maximum entropy function describe wind characterizations better than the conventional Weibull function if wind regime presents two humps on it, irrespective of wind speed and power density. For wind speed distributions, the proposed GW pdf describes best according to the Kolmogorov–Smirnov test followed by the NW and WW pdfs, while the NN pdf performs worst. As for wind power density, the MEP and GW pdfs perform best followed by the WW and NW pdfs. The GW pdf could be a useful alternative to the conventional Weibull function in estimating wind energy potential.     

Wind energy estimation at Quetta

Hourly wind data at Quetta airport (Samungli) for the years 1984–1985, were obtained using a standard anemometer height of 10 m, with a view to work out the feasibility of wind energy utilization. Quetta (lat. 30°11 'N long. 66°57'E) is elevated at 1799 m above sea level. In this paper, we analyse the wind energy data by using the Weibull distribution. Scaling and shaping parameters are determined by using least-squares approximation to a straight line. Actual wind data, weighted Weibull density function and weighted Rayleigh probability density function for regular and continuous periods of 4 weeks up to a year (wind characteristics are being recorded and summarized as diurnal and monthly wind velocity distributions and wind power density roses) are plotted which shows that the Weibull distribution is generally of the right shape to fit low-averaged wind speed frequency curves. However, density function of a normal distribution is also determined. Deviations in wind speed distributions at very low-averaged wind speeds and at comparatively large-averaged wind speeds are found.     

Probabilistic maximum‐value wind prediction for offshore environments

High wind speeds can pose a great risk to structures and operations conducted in offshore environments. When forecasting wind speeds, most models focus on the average wind speeds over a given period, but this value alone represents only a small part of the true wind conditions. We present statistical models to predict the full distribution of the maximum‐value wind speeds in a 3 h interval. We take a detailed look at the performance of linear models, generalized additive models and multivariate adaptive regression splines models using meteorological covariates such as gust speed, wind speed, convective available potential energy, Charnock, mean sea‐level pressure and temperature, as given by the European Center for Medium‐Range Weather Forecasts forecasts. The models are trained to predict the mean value of maximum wind speed, and the residuals from training the models are used to develop the full probabilistic distribution of maximum wind speed. Knowledge of the maximum wind speed for an offshore location within a given period can inform decision‐making regarding turbine operations, planned maintenance operations and power grid scheduling in order to improve safety and reliability, and probabilistic forecasts result in greater value to the end‐user. The models outperform traditional baseline forecast methods and achieve low predictive errors on the order of 1–2 m s^?1. We show the results of their predictive accuracy for different lead times and different training methodologies. Copyright © 2014 John Wiley & Sons, Ltd.     

Influence of atmospheric stability on wind turbine loads

Simulations of wind turbine loads for the NREL 5 MW reference wind turbine under diabatic conditions are performed. The diabatic conditions are incorporated in the input wind field in the form of wind profile and turbulence. The simulations are carried out for mean wind speeds between 3 and 16 m s ^{? 1} at the turbine hub height. The loads are quantified as the cumulative sum of the damage equivalent load for different wind speeds that are weighted according to the wind speed and stability distribution. Four sites with a different wind speed and stability distribution are used for comparison. The turbulence and wind profile from only one site is used in the load calculations, which are then weighted according to wind speed and stability distributions at different sites. It is observed that atmospheric stability influences the tower and rotor loads. The difference in the calculated tower loads using diabatic wind conditions and those obtained assuming neutral conditions only is up to 17%, whereas the difference for the rotor loads is up to 13%. The blade loads are hardly influenced by atmospheric stability, where the difference between the calculated loads using diabatic and neutral input wind conditions is up to 3% only. The wind profiles and turbulence under diabatic conditions have contrasting influences on the loads; for example, under stable conditions, loads induced by the wind profile are larger because of increased wind shear, whereas those induced by turbulence are lower because of less turbulent energy. The tower base loads are mainly influenced by diabatic turbulence, whereas the rotor loads are influenced by diabatic wind profiles. The blade loads are influenced by both, diabatic wind profile and turbulence, that leads to nullifying the contrasting influences on the loads. The importance of using a detailed boundary‐layer wind profile model is also demonstrated. The difference in the calculated blade and rotor loads is up to 6% and 8%, respectively, when only the surface‐layer wind profile model is used in comparison with those obtained using a boundary‐layer wind profile model. Finally, a comparison of the calculated loads obtained using site‐specific and International Electrotechnical Commission (IEC) wind conditions is carried out. It is observed that the IEC loads are up to 96% larger than those obtained using site‐specific wind conditions.Copyright © 2012 John Wiley & Sons, Ltd.     

WAsP in the Forest

This article compares mean wind estimates from a WAsP analysis for three forest sites and one site near a forest with measurements taken at the sites. By standard WAsP settings for forest, the mean wind speed at the sites was overestimated. Agreement between the estimates and the measurements improved significantly if displacement height and roughness length as calculated from the forest mast data were used or if a simple model estimate of roughness length and displacement height based on stand density (frontal area index) was used. The two estimates of displacement height and roughness length (mast data and simple model) did not agree well with each other. One reason for this may be that all evaluated sites are windy and that both d and z₀ depend on the wind speed. All analysed forest sites are dense, in which case the influence from the roughness sublayer is limited and the effect on mean wind speeds from this layer could not be evaluated. Copyright © 2005 John Wiley & Sons, Ltd.     

Evaluating winds and vertical wind shear from Weather Research and Forecasting model forecasts using seven planetary boundary layer schemes

The existence of vertical wind shear in the atmosphere close to the ground requires that wind resource assessment and prediction with numerical weather prediction (NWP) models use wind forecasts at levels within the full rotor span of modern large wind turbines. The performance of NWP models regarding wind energy at these levels partly depends on the formulation and implementation of planetary boundary layer (PBL) parameterizations in these models. This study evaluates wind speeds and vertical wind shears simulated by the Weather Research and Forecasting model using seven sets of simulations with different PBL parameterizations at one coastal site over western Denmark. The evaluation focuses on determining which PBL parameterization performs best for wind energy forecasting, and presenting a validation methodology that takes into account wind speed at different heights. Winds speeds at heights ranging from 10 to 160 m, wind shears, temperatures and surface turbulent fluxes from seven sets of hindcasts are evaluated against observations at Høvsøre, Denmark. The ability of these hindcast sets to simulate mean wind speeds, wind shear, and their time variability strongly depends on atmospheric static stability. Wind speed hindcasts using the Yonsei University PBL scheme compared best with observations during unstable atmospheric conditions, whereas the Asymmetric Convective Model version 2 PBL scheme did so during near‐stable and neutral conditions, and the Mellor–Yamada–Janjic PBL scheme prevailed during stable and very stable conditions. The evaluation of the simulated wind speed errors and how these vary with height clearly indicates that for wind power forecasting and wind resource assessment, validation against 10 m wind speeds alone is not sufficient. Copyright © 2012 John Wiley & Sons, Ltd.     

Assessing the suitability of wind speed probabilty distribution functions based on wind power density

Three functions have so far predominantly been used for fitting the measured wind speed probability distribution in a given location over a certain period of time, typically monthly or yearly. In the literature, it is common to fit these functions to compare which one fits the measured distribution best in a particular location. During this comparison process, parameters on which the suitability of the fit is judged are required. The parameters that are mostly used are the mean wind speed or the total wind energy output (primary parameters). It is, however, shown in the present study that one cannot judge the suitability of the functions based on the primary parameters alone. Additional parameters (secondary parameters) that complete the primary parameters are required to have a complete assessment of the fit, such as the discrepancy between the measured and fitted distributions, both for the wind speed and wind energy (that is the standard deviation of wind speed and wind energy distributions). Therefore, the secondary statistical parameters have to be known as well as the primary ones to make a judgement about the suitability of the distribution functions analysed. The primary and secondary parameters are calculated from the 12-month of measured hourly wind speed data and detailed analyses of wind speed distributions are undertaken in the present article.     

Evaluating the impact of wind induced roughness change and tidal range on extrapolation of offshore vertical wind speed profiles

Wind energy resource estimation frequently requires extrapolation of wind speeds from typical measurement heights to turbine hub‐heights. However, this extrapolation is uncertain, and this uncertainty is exacerbated in the offshore environment by the effect of the dynamic surface (i.e. surface roughness and height respond to wind speed or vary over time). This paper examines the impact of roughness variations and small tidal ranges on mean predicted wind speeds in near‐neutral conditions. Roughness variations offshore are in the range 0.002 and 0.00002 m. This range of roughnesses gives a difference in predicted wind speed extrapolated from 10 to 50 m of less than 8%. For a more typical range of 0.0005 tp 0.00005 m, the difference will be smaller (～3%). With a tidal range of 4 m the difference in mean wind speed extrapolated from 10 to 50 m height is about 1%. Copyright © 2001 John Wiley & Sons, Ltd.     

Errors and uncertainties associated with missing wind data and short records

A near‐complete 4 year data set of 10 min average 80 m wind speeds is used to examine the impact of missing data on monthly and yearly estimates of mean wind speed and energy production from a generic wind turbine. Missing data is a source of uncertainty in wind energy resource assessment studies. Quantifying that uncertainty can improve the reliability of P90 and related wind farm energy production estimates. An empirical relationship between missing data percentage and relative uncertainty in monthly mean wind speed is derived. Relationships between uncertainties in monthly average wind speed and uncertainties in monthly energy production are also explored. In many cases with monthly data losses of 10% or less the contribution to the overall uncertainty in annual energy production will be small (<1%), but with substantial losses in cold winters, typically caused by icing; the uncertainties can become more significant. The data set is also used to indicate uncertainties associated with short data periods. Annual average wind speed estimates based on less than a complete year's data also add significant uncertainty to wind resource assessments. Copyright © 2013 John Wiley & Sons, Ltd.     

Comparison between the bivariate Weibull probability approach and linear regression for assessment of the long-term wind energy resource using MCP

A detailed investigation of a measure–correlate–predict (MCP) approach based on the bivariate Weibull (BW) probability distribution of wind speeds at pairs of correlated sites has been conducted. Since wind speeds are typically assumed to follow Weibull distributions, this approach has a stronger theoretical basis than widely used regression MCP techniques. Building on previous work that applied the technique to artificially generated wind data, we have used long-term (11 year) wind observations at 22 pairs of correlated UK sites. Additionally, 22 artificial wind data sets were generated from ideal BW distributions modelled on the observed data at the 22 site pairs. Comparison of the fitting efficiency revealed that significantly longer data periods were required to accurately extract the BW distribution parameters from the observed data, compared to artificial wind data, due to seasonal variations. The overall performance of the BW approach was compared to standard regression MCP techniques for the prediction of the 10 year wind resource using both observed and artificially generated wind data at the 22 site pairs for multiple short-term measurement periods of 1–12 months. Prediction errors were quantified by comparing the predicted and observed values of mean wind speed, mean wind power density, Weibull shape factor and standard deviation of wind speeds at each site. Using the artificial wind data, the BW approach outperformed the regression approaches for all measurement periods. When applied to the real wind speed observations however, the performance of the BW approach was comparable to the regression approaches when using a full 12 month measurement period and generally worse than the regression approaches for shorter data periods. This suggests that real wind observations at correlated sites may differ from ideal BW distributions and hence regression approaches, which require less fitting parameters, may be more appropriate, particularly when using short measurement periods.     

A comparison between Advanced Scatterometer and Weather Research and Forecasting wind speeds for the Japanese offshore wind resource map

This paper investigates the validity of the method used in the Japanese offshore wind map (NeoWins) to seamlessly connecting satellite‐derived wind speed for open oceans to mesoscale model‐simulated wind speed for coastal waters. In the NeoWins, the former was obtained from the satellite‐borne Advanced Scatterometer (ASCAT), and the latter was obtained from numerical simulations using the Weather Research and Forecasting (WRF) model. In this study, the consistency of the ASCAT and WRF 10‐m height wind speeds is examined in their overwrapping areas. The comparison between ASCAT and WRF model reveals that their differences in annual mean wind speed are mostly within ±5% and that the ASCAT annual mean wind speed is, as a whole, slightly higher than the WRF annual mean wind speed. The results indicate that there are no large wind speed gaps between WRF and ASCAT in most parts of the Japanese offshore areas. It is moreover found that the discrepancies between the two wind speeds are due to two factors: one is the existence of winter sea ice in the ASCAT observation in the Sea of Okhotsk in ASCAT observation and the other is that the accuracy of the WRF wind speed depends on atmospheric stability.     

大型风力机叶片在不同平均风速作用下的挠度及应力分析

利用有限元分析软件对风力机叶片在不同平均风速作用下的挠度和应力进行了分析。采用Davenport脉动风速谱模拟出不同平均风速,并计算出叶片各部位的挠度及其应力分布。计算结果表明,叶片的振动主要以挥舞振动为主,随着工作风速的增大,叶片挥舞振幅增大,叶片承受的mises应力主要集中在叶片迎风面的中部位置。该研究为风力机叶片的安全设计提供了技术参考。