Application of Bayesian model averaging in modeling long-term wind speed distributions

Accurate estimation of wind speed distribution is critical to the assessment of wind energy potential, the site selection of wind farms, and the operations management of wind power conversion systems. This paper proposes a new approach for deriving more reliable and robust wind speed distributions than conventional statistical modeling approach. This approach combines Bayesian model averaging (BMA) and Markov Chain Monte Carlo (MCMC) sampling methods. The derived BMA probability density function (PDF) of the wind speed is an average of the model PDFs included in the model space weighted by their posterior probabilities over the sample data. MCMC method provides an effective way for numerically computing marginal likelihoods, which are essential for obtaining the posterior model probabilities. The approach is applied to multiple sites with high wind power potential in North Dakota. The wind speed data at these sites are the mean hourly wind speeds collected over two years. It is demonstrated that indeed none of the conventional statistical models such as Weibull distribution are universally plausible for all the sites. However, the BMA approach can provide comparative reliability and robustness in describing the long-term wind speed distributions for all sites, while making the traditional model comparison based on goodness-of-fit statistics unnecessary.     

Assessment of wind energy potentiality at Kudat and Labuan, Malaysia using Weibull distribution function

The wind resource is a crucial step in planning a wind energy project and detailed knowledge of the wind characteristic at a site is needed to estimate the performance of a wind energy project. In this paper, with the help of 2-parameter Weibull distribution, the assessment of wind energy potentiality at Kudat and Labuan in 2006-2008 was carried out. “WRPLOT” software has been used to show the wind direction and resultant of the wind speed direction. The monthly and yearly highest mean wind speeds were 4.76 m/s at Kudat and 3.39 m/s at Labuan respectively. The annual highest values of the Weibull shape parameter (k) and scale parameter (c) were 1.86 and 3.81 m/s respectively. The maximum wind power density was found to be 67.40 W/m² at Kudat for the year 2008. The maximum wind energy density was found to be 590.40 kWh/m²/year at Kudat in 2008. The highest most probable wind speed and wind speed carrying maximum energy were estimated 2.44 m/s at Labuan in 2007 and 6.02 m/s at Kudat in 2007. The maximum deviation, at wind speed more than 2 m/s, between observed and Weibull frequency distribution was about 5%. The most probable wind directions (blowing from) were 190° and 269° at Kudat and Labuan through the study years. From this study, it is concluded that these sites are unsuitable for the large-scale wind energy generation. However, small-scale wind energy can be generated at the turbine height of 100 m.     

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.     

A statistical analysis of wind power density based on the Weibull and Rayleigh models at the southern region of Turkey

The electric generating capacity of Turkey must be tripled by 2010 to meet Turkey’s electric power consumption, if the annual 8% growth in electric power consumption continues. Turkey has to make use of its renewable energy resources, such as wind and solar, not only to meet the increasing energy demand, but also for environmental reasons. Studies show that Iskenderun (36°35′N; 36°10′E) located on the Mediterranean coast of Turkey is amongst the possible wind energy generation regions. In the present study, the wind energy potential of the region is statistically analyzed based on 1-year measured hourly time-series wind speed data. The probability density distributions are derived from time-series data and distributional parameters are identified. Two probability density functions are fitted to the measured probability distributions on a monthly basis. The wind energy potential of the location is studied based on the Weibull and the Rayleigh models.     

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.     

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.     

Some statistical characteristics of monthly average wind speed at various heights

The article, which is a segment of a complex wind energy examination, uses statistical methods to analyze the time series of monthly average wind speed in the period between 1991 and 2000 measured on seven Hungarian meteorological stations. Empirical distribution of measured monthly average wind speeds is approximated by theoretical distributions to claim that certain distributions are universal, i.e. independent of orography. We used one of them, the Weibull distribution, to generate the distribution of monthly average wind speeds on levels different from anemometer altitude as well, then we calculate the averages for the entire period and we fit a power function on them. Thus we can demonstrate a correlation between Hellmann's wind profile law and the Weibull distribution.     

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.     

Wind resource evaluation in six selected high altitude locations in Nigeria

This paper presents an assessment of wind energy potentials of six selected high altitude locations within the North-West and North-East geopolitical regions, Nigeria, by using 36-year (1971–2007) wind speed data subjected to 2-parameter Weibull distribution functions. The results showed that the maximum mean wind speed is obtained in Katsina as 9.839 m/s while the minimum value of 3.397 m/s is got in Kaduna for all the locations considered. The annual wind power density and energy variation based on the Weibull analysis ranged from 368.92 W/m² and 3224.45 kWh/m²/year to 103.14 W/m² and 901.75 kWh/m²/year in Kano and Potiskum for the maximum and minimum values respectively. Furthermore, Katsina and Kano will be suitable for wind turbine installations while Gusau will only be appropriate for wind energy utilization using taller wind turbine towers whereas Kaduna, Bauchi and Potiskum will be considered marginal for wind power development based of their respective annual mean wind speeds and power densities.     

LiDAR‐based characterization of mid‐altitude wind conditions for airborne wind energy systems

Engineers and researchers working on the development of airborne wind energy systems (AWES) still rely on oversimplified wind speed approximations and coarsely sampled reanalysis data because of a lack of high‐resolution wind data at altitudes above 200 m. Ten‐minute average wind speed LiDAR measurements up to an altitude of 1100 m and data from nearby weather stations were investigated with regard to wind energy generation and impact on LiDAR measurements. Data were gathered by a long‐range pulsed Doppler LiDAR device installed on flat terrain. Because of the low overall carrier‐to‐noise ratio, a custom‐filtering technique was applied. Our analyses show that diurnal variation and atmospheric stability significantly affect wind conditions aloft which cause a wide range of wind speeds and a multimodal probability distribution that cannot be represented by a simple Weibull distribution fit. A better representation of the actual wind conditions can be achieved by fitting Weibull distributions separately to stable and unstable conditions. Splitting and clustering the data by simulated surface heat flux reveals substate stratification responsible for the multimodality. We classify different wind conditions based on these substates, which result in different wind energy potential. We assess optimal traction power and optimal operating altitudes statistically as well as for specific days based on a simplified AWES model. Using measured wind speed standard deviation, we estimate average turbulence intensity and show its variation with altitude and time. Selected short‐term data sets illustrate temporal changes in wind conditions and atmospheric stratification with a high temporal and vertical resolution.     

新疆达坂城风电场风能资源特性分析

对新疆达坂城风电场的风能资源特性进行了详细的研究。基于在达坂城风电场实测的10m和24m高程的10min平均风速数据,分析了原始风速的分布特性。根据地表风速沿高度呈风剪指数分布的特性,计算了在各个轮毂高度上的风速分布。采用最小误差逼近算法原理,计算了风速韦布尔分布的参数以及平均风速和分布方差。通过对韦布尔分布的分析,计算了各个高度上风电场的平均风功率密度、有效平均风功率密度和可利用小时数等风能资源特性参数,为当地的风能开发提供分析基础。     

对福建省陆地风能资源的评估

对我国福建省福州和厦门2座城市进行了风能开发潜力的评估.基于对该地区近15年的日平均风速的统计分析,计算了各月的风能密度,拟合出了Weibull分布密度函数的特征参数.用Weibull分布密度函数预测了各月的风能密度.并与实测值进行了对比及相关性分析,结果证明了Weibull函数对实测数据有很好的拟合性,同时也表明福建省陆地风力资源的不足,对该地区风力资源的调查重点应放在沿海滩涂及浅海.     

Wind power resource in the south-western region of Algeria

In this study, we present a statistical analysis of wind speeds at Tindouf in Algeria using Risoe National Laboratory's Wind Atlas Analysis and Application Program (WAsP). It requires information related to the sheltering obstacles, surface roughness changes and terrain height variations in order to calculate their effects on the wind. Wind data, consisting of hourly wind speed records over a 5-year period, 2002–2006, were obtained from SONELGAZ R&D Office; the average wind speed at a height of 17 m above ground level was found to range from 7.19 to 7.95 m/s. The Weibull distributions parameters (c and k) were found to vary between 8.0 and 8.9 m/s and 2.54–3.23, respectively, with average power density ranging from 318 to 458 W/m². The dominant wind directions and the frequency distributions were also determined.     

A wind energy analysis of Grenada: an estimation using the ‘Weibull’ density function

The Weibull density function has been used to estimate the wind energy potential in Grenada, West Indies. Based on historic recordings of mean hourly wind velocity this analysis shows the importance to incorporate the variation in wind energy potential during diurnal cycles. Wind energy assessments that are based on Weibull distribution using average daily/seasonal wind speeds fail to acknowledge that wind speed probabilities can vary significantly during day and night. In particular where wind energy estimation is linked to electricity loads neglecting diurnal wind patterns can result in significant under/overestimation of wind power potential.     

Potential wind power generation in the State of Kuwait

The wind characteristics of six locations in the State of Kuwait have been assessed. The annual average wind speed for the considered sites ranged from 3.7 to 5.5 m/s and a mean wind power density from 80 to 167 W/m² at standard height of 10 m. The Weibull parameters and power density of each station have been determined using Weibull distribution. The wind data at heights 15, 20, 25 and 30 m were obtained by extrapolation of the 10 m data using the Power-Law. The potential wind energy at different heights was estimated using Weibull parameters. Maximum power density is found at 30 m height which varies between 130 and 275 W/m² with 70% increase from the standard height indicating fairly potential wind energy especially in the northern part of the country. The highest potential wind power was found during the summer season which is the peak demand season of electricity in Kuwait.     

基于PWMM的三参数Weibull分布参数外推方法的研究

为提高低风速区分散式风电项目的风资源评估精度,降低测风成本,在对三参数Weibull分布参数估计和外推的研究基础上,提出基于概率加权矩法(PWMM)的三参数Weibull分布参数垂直外推方法。利用较低高度处风速统计的概率加权矩,经垂直外推得到平坦地形、较高高度处风速Weibull分布的参数,进而得到Weibull分布函数和风功率密度。算例分析表明:基于PWMM的三参数Weibull分布参数垂直外推法在平坦地形不同测风点处有一定的适用性外推较高高度处风速Weibull分布的参数,可有效体现平坦地形低风速区的风速分布特征,提高风功率密度评测精度。     

Wind power statistics and an evaluation of wind energy density

In this paper the statistical data of fifty days' wind speed measurements at the MERC-solar site are used to find out the wind energy density and other wind characteristics with the help of the Weibull probability distribution function. It is emphasized that the Weibull and Rayleigh probability functions are useful tools for wind energy density estimation but are not quite appropriate for properly fitting the actual wind data of low mean speed, short-time records. One has to use either the actual wind data (histogram) or look for a better fit by other models of the probability function.     

Wind energy potential in Aden-Yemen

The wind energy resource is very large and widely distributed throughout the world as well as in Yemen. Aden possesses a very good potential of wind energy. In this article a number of years data on wind speed in Aden has been studied and presented. A statistical analysis was carried out from which the annual wind speed was found to be 4.5 m/s and most of the time the wind speed is in the range of 3.5–7.5 m/s. The wind speed distributions were represented by Weibull and Rayleigh distributions. It was found that the Rayleigh distribution is suitable to represent the actual probability of wind speed data for Aden. The wind speed data showed that the maximum monthly wind speed occurs in the month of February with the maximum in the month of June. It is concluded that Aden can be explored for wind energy applications.