Estimation Of Weibull Parameters Research Articles

Accuracy Testing of Different Methods for Estimating Weibull Parameters of Wind Energy at Various Heights above Sea Level

The Weibull algorithm is one of the most accurate tools for forecasting and estimating wind energy potential. Two main parameters of the Weibull algorithm are the ‘Weibull shape’ and ‘Weibull scale’ factors. There are six different numerical methods to estimate the two Weibull parameters. These six methods are the empirical method of Justus (method 1), the empirical method of Lysen (method 2), the maximum likelihood method (method 3), the modified maximum likelihood method (method 4), the energy pattern factor method (method 5) and the graphical method (method 6). Many commercial wind energy software programs use the Weibull algorithm, and these six methods are used to calculate the potential wind energy at a given site. However, their accuracy is rarely discussed, particularly regarding wind data height. For this purpose, wind data measured for a long period (six years) at real sites are introduced. The wind data sites are categorized into three levels, i.e., low, medium, and high, based on wind data measurement height. The analysis shows that methods 1 and 2 are the most accurate methods among all six methods at low and medium heights. The number of errors increases with the height of these two methods. Methods 3 and 4 are the most suitable options for larger heights, as these scenarios have minimal error. The present study’s findings can be used in various fields, e.g., wind energy forecasting and wind farm planning.

Evaluating the lifetime distribution parameters and reliability of products using successive approximation method

AbstractThe Weibull distribution is an extensively used statistical model for analyzing the reliability of mechanical and electrical components. Due to the complexity of the nonlinear equations and the scarcity of failure data, the common method may not provide satisfactory results of the reliability. In this case, a new approach for Weibull parameter estimation and reliability analysis, based on successive approximation schemes, is presented. The shape and scale parameters are estimated by maximizing the likelihood functions, and the location parameter is obtained by constructing an approximate correction model between it and the failure data. In order to show the performance of the proposed method, an extensive Monte‐Carlo simulation study is conducted. Simulation results show that the proposed method provides better estimates and efficient confidence intervals for Weibull parameters. In addition, the proposed method works well in presence of small sample sizes. Finally, two real examples are analyzed to illustrate the application of the proposed method.

Analysis of wind characteristics and wind energy resource assessment for Tonga using eleven methods of estimating Weibull parameters

Analysisof wind characteristics and assessment of wind energy resource is carried out at a location in Tonga with the help of twelve months of measurements carried out at 34 m and 20 m heights above ground level. The daily, monthly and annual averages are computed. The wind shear analysis and its diurnal variation were studied and compared with the temperature variation. The turbulence levels at the two heights were estimated for the entire measurement period as well as for some typical days. For estimating the Weibull parameters, eleven methods were employed along with goodness of fit and error estimates to find the best method. The overall averaged wind speed for the entire period of study is estimated to be 4.41 m/s at 34 m above ground level. The predominant wind direction was south-east for Tonga. ‘Moments’ is seen to be the best method to determine accurate Weibull parameters. The average net annual energy production from one Vergnet 275 kW wind turbine is 198.57 MWh. A payback period of 8.95 years by installing five turbines near the measurement location was estimated, which is very encouraging in terms of investment. Installing wind turbines will lower the heavy reliance on the imported fossil fuels in the country and also help in achieving Sustainable Development Goal 7.

Evaluation of the performance of five distribution functions for estimating Weibull parameters for wind energy potential in Nigeria

Wind energy is one of the most sought after worldwide among renewable energy sources. In order to determine the wind potential of a particular site, the wind speed has to be characterized and the Weibull parameters accurately determined. Characterization of wind speed in terms of Weibull parameters for wind energy potential in Nigeria is scarce. Several methods of determining Weibull parameters exist in literature. In this study, five numerical methods, Weibull-two (W), Mixture Weibull (MW), Gamma (G), lognormal (LN) and normal (N), were used to analyse wind speed data from twelve stations measured at 10 m height in order to determine the best method for estimating Weibull parameters for each station and region. The methods were compared and their accuracies were determined by three goodness-of-fit (GOF) tests namely Maximum error of Kolmogorov–Smirnov (K-S), root mean square error (RMSE) and Chi square (χ2). The performances of the methods were ranked on a scale of 1–5 and the results revealed that the MW was ranked the best in all the stations by two of the GOFs tests. It was closely followed by LN. The Weibull and Gamma methods performed poorly. Thus the MW is selected as the best method for determining Weibull parameters accurately and hence wind energy potential for Nigeria.

Technical and performance assessments of wind turbines in low wind speed areas using numerical, metaheuristic and remote sensing procedures

The application of innovative technologies in the manufacture of wind turbines (WT) has produced more efficient WT that can operate successfully in low wind speed (LWS) environments. This technology has not been implemented in many LWS parts of the world due to the paucity of enabling technical information (wind resource availability and wind turbine configuration). This study uses ten years wind speed data from twelve Nigerian cities and their population densities, remote sensing, and the configuration of some commercially available LWS turbines in generating technical information suitable for data-backed decision-making on low-speed turbine deployability, operational conditions, and energy yield at 50 and 400 m. Five different numerical and metaheuristic procedures were randomly selected and utilized to estimate Weibull parameters used in computing wind energy development (WEDP) parameters (effective wind power density, EWPD, and wind available time). WT with low cut-in speeds (∼2 m.s−1) can be installed in all the cities. With the highest EWPD and WEDP ratings (100%) at both 50 and 400 m, Obudu is the most suitable location for the development of wind power infrastructure.

Evaluation of Weibull parameters by different methods for farms

One of the most prevalent clean and sustainable forms of energy produced worldwide is the power created from wind runoff. Wind turbines ought to be installed in areas with favorable circumstances to transform mechanical wind energy into electricity. Finding appropriate ways to predict the energy produced by a wind farm using the Weibull distribution is the main goal of this work. Theoretical techniques have been applied to calculate Weibull selected characteristics using experimental data gathered at the campus of Universidade Federal de Mato Grosso do Sul (UFMS) in Brazil. These data were gathered 10 meters above the surface. The effectiveness of four statistical techniques that are frequently used in the energy industry are compared: the standard energy factor method; the least squares regression method; the moment method; and the mean standard deviation method in estimating Weibull parameters. The root mean square error, Chi-square error, Kolmogorov-Smirnov test, and coefficient of determination are used to contrast the statistical methodologies. The results demonstrated that the least squares regression approach performs less well than other methods. The standard energy factor approach, the moment method, and the mean standard deviation method are the most effective techniques when modifying Weibull distribution curves for the assessment of wind speed data. The data analysis confirms that these three strategies are fully applicable if the wind speed distribution closely matches the Weibull distribution.

Wind Energy Assessment Using Weibull Distribution with Different Numerical Estimation Methods: A Case Study

The demand for electrical energy is increasing every day, which is one of the critical challenges facing the world today. Hence, the necessity of turning to clean renewable energy sources that are not harmful to the environment as an alternative to the traditional generation based on fossil fuels has become more important than ever before. Wind power is one of the renewable sources that provides a clean solution to generate electricity. In this context, the Kingdom of Saudi Arabia announces renewable energy projects to generate 9 GW from wind in 2032. Hence, the aim of this paper is to investigate the most suitable method of Weibull parameter estimation in order to predict wind characteristics and employ it for wind energy assessment in the Qassim region located in the center of the country. In this study, wind data is collected from NASA's forecasts of global energy resources for 2010–2015 based on their availability at altitudes of 10m and 50m and analyzed by using six different methods for Weibull parameter estimation: the graphical method (GM), standard deviation method (SDM), energy pattern factor method (EPF), moment method (MM), alternative maximum likelihood method (AMLM), and novel energy pattern factor method (NEPF). The efficiency of each method is tested by calculating the root mean square error (RMSE) and the relative wind power density error (RPDE). The comparison shows that the most appropriate method for estimating wind power density in the country is the Moment Method (MM), with the lowest RPDE ratio equal to 0.2018%. It has been found that the wind power density in the Qassim region falls into the class 1 category, as it is less than 100 W/m2 at a height of 10m and less than 200 W/m2at an altitude of 50m. The results show the region is only suitable for small off-grid projects. Doi: 10.28991/ESJ-2023-07-06-024 Full Text: PDF

Weibull parameter estimation and reliability analysis with small samples based on successive approximation method

Weibull parameter estimation and reliability analysis with small samples based on successive approximation method

Assessing the performance of several numerical methods for estimating Weibull parameters for Wind Energy Applications: A case study of Al-Hodeidah in Yemen

The estimation of wind potential is generally assessed using the two parameters of Weibull distribution (k, c). Relative Percentage Error (RPE), Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Relative Root Mean Square Error (RRMSE), Correlation Coefficient (R), and Index of Agreement (IA) are calculated to comparatively analyze five numerical methods of computing Weibull parameters. The five selected methods are (Empirical Justus (EMJ), Energy Pattern Factor (EPFM), Maximum Likelihood (MLM), Method of Moment (MOM), and L-moment Estimation Method (L-MOM)). With the daily wind speed dataset for the year 2014 measured at 10 m height, the five methods have been applied on the study site Al-Hodeidah city in Yemen. Therefore, compared to other methods, the EPFM method is the most accurate method for estimating the Weibull parameters, while the L-MOM method is the least accurate at 10, 30, and 50 m heights. Moreover, this study reveals that the parameter remains the same with height, while the c parameter increases with height. The values of the skewness and kurtosis of wind speed and power density do not change as height changes.

Bayesian method for estimating Weibull parameters for wind resource assessment in a tropical region: a comparison between two-parameter and three-parameter Weibull distributions

Abstract. The two-parameter Weibull distribution has garnered much attention in the assessment of wind energy potential. The estimation of the shape and scale parameters of the distribution has brought forth a successful tool for the wind energy industry. However, it may be inappropriate to use the two-parameter Weibull distribution to assess energy at every location, especially at sites where low wind speeds are frequent, such as in tropical regions. In this work, a robust technique for wind resource assessment using a Bayesian approach for estimating Weibull parameters is first proposed. Secondly, the wind resource assessment techniques using a two-parameter Weibull distribution and a three-parameter Weibull distribution, which is a generalized form of two-parameter Weibull distribution, are compared. Simulation studies confirm that the Bayesian approach seems a more robust technique for accurate estimation of Weibull parameters. The research is conducted using data from seven sites in the tropical region from 1∘ N of the Equator to 21∘ S of the Equator. Results reveal that a three-parameter Weibull distribution with a non-zero shift parameter is a better fit for the wind data with a higher percentage of low wind speeds (0–1 m s−1) and low skewness. However, wind data with a smaller percentage of low wind speeds and high skewness showed better results with a two-parameter distribution that is a special case of the three-parameter Weibull distribution with a zero shift parameter. The proposed distribution can be incorporated into commercial software like WAsP to improve the accuracy of wind resource assessments. The results also demonstrate that the proposed Bayesian approach and application of a three-parameter Weibull distribution are extremely useful for accurate estimation of wind power density.

Assessment of wind power potential in the North region of Malaysia, Chuping Perlis

The wind turbines is a main device that convert the kinetic energy from blades to electrical energy. Before installing wind turbines, the Weibull probability distribution must be calculated to determine the certain wind speed probability. Many problems will come if there no analysis the characteristics of wind in selected location, such as wind speed that not suitable for building wind farm to supply the population in that area. Shape and scale factors, which be controlled in a variety of ways, influence the Weibull distribution. Many studies have looked into which of the various Weibull parameter estimation methods is the most dependable. However, because the results of these investigations were inconsistent, research into more trustworthy Weibull parameter estimation methods is still ongoing. An analysis of data collected Chuping, Perlis for two years was conducted in this study (from 2018 to 2019). By using statistical analysis to evaluate the Weibull distribution method, this study used three methods to compared the Weibull parameters and identified the most reliable and effective method to obtain the Weibull probability distribution by using a three approach that compares the variances of RMSE, MSE and R2, which provides comprehensive insight into level error and volatility. Modified maximum likelihood method, graphical method, and power density method are the three methods used in this study. Therefore, the graphical method has the best accuracy in the wind speed distribution prediction, several methods such as the modified maximum likelihood method, and the power density method have the worst prediction of the wind speed distribution based on all the statistical method variances for this region.

Estimation of Weibull Distribution Parameters for Wind Speed Characteristics Using Neural Network Algorithm

Harvesting the power coming from the wind provides a green and environmentally friendly approach to producing electricity. To facilitate the ongoing advancement in wind energy applications, deep knowledge about wind regime behavior is essential. Wind speed is typically characterized by a statistical distribution, and the two-parameters Weibull distribution has shown its ability to represent wind speeds worldwide. Estimation of Weibull parameters, namely scale and shape parameters, is vital to describe the observed wind speeds data accurately. Yet, it is still a challenging task. Several numerical estimation approaches have been used by researchers to obtain c and k. However, utilizing such methods to characterize wind speeds may lead to unsatisfactory accuracy. Therefore, this study aims to investigate the performance of the metaheuristic optimization algorithm, Neural Network Algorithm (NNA), in obtaining Weibull parameters and comparing its performance with five numerical estimation approaches. In carrying out the study, the wind characteristics of three sites in Saudi Arabia, namely Hafer Al Batin, Riyadh, and Sharurah, are analyzed. Results exhibit that NNA has high accuracy fitting results compared to the numerical estimation methods. The NNA demonstrates its efficiency in optimizing Weibull parameters at all the considered sites with correlations exceeding 98.54.

Study on Rolling Bearing Life Based on Weibull Distribution and Correlation Coefficient Optimization and Maximum Likelihood Estimation

For the reliability evaluation of rolling bearing life, this paper proposes a parameter estimation method based on the correlation coefficient optimization method and the maximum likelihood estimation method (Cor-MLE), which can obtain more accurate Weibull distribution parameter estimation. Subsequently, the feasibility of Cor-MLE method for Weibull parameter estimation was verified by Monte Carlo simulation data and rolling bearing life test data. Compared with the parameter estimation results of least square method, grey model method and correlation coefficient optimization method, it is proved that the estimation results of this method have high accuracy and can meet the life analysis of rolling bearings.

Comparative evaluation of optimal Weibull parameters for wind power predictions using numerical and metaheuristic optimization methods for different Indian terrains.

The accurate selection of the wind speed distributions is crucial for a better utilisation of wind energy. The Weibull distribution is most commonly used distribution and hence its parameters need to be optimized. In this study five numerical methods, namely, maximum likelihood method (MLM), graphical method (GM), empirical method of Justus (EMJ), modified maximum likelihood method (MMLM) and wind atlas analysis and application program (WAsP) and three metaheuristic optimization algorithms, namely, social spider optimization (SSO), particle swarm optimization (PSO) and genetic algorithm (GA) are applied for estimating Weibull distribution parameters at three different locations (onshore-Kayathar, nearshore-Jafrabad and offshore-Gulf of Khambhat (GOK) in India and also comparison of numerical and optimization methods are employed to tune the optimal parameters. The accuracy of the methods was evaluated using three different statistical analysis techniques. As per the results, GOK has the maximum wind power density of 450.2W/m2 compared to Jafrabad and Kayathar. It was observed that among the five methods used for Weibull parameters estimation, WAsP method presented a better curve fit with the histogram of the wind speed. The results shows that SSO and PSO presents a comparably better performance than GA in the term of accuracy on the basis of closeness to converged solution.

Comparison of Weibull parameter estimation methods using LiDAR and mast wind data in an Indian offshore site: The Gulf of Khambhat

This paper gives a proper study of 11 types of Weibull parameters estimation methods and their performance comparison for wind potential assessment on India's first offshore project in the Gulf of Khambhat, Gujarat. The Lidar (up to 200 m) and met mast (up to 100 m) are deployed to record wind data from Nov-2017 to Dec-2019 by the National Institute of Wind Energy (NIWE), Government of India. The estimation methods such as, modified maximum likelihood method (MMLM), maximum likelihood method (MLM), empirical method of Justus (EMJ), empirical method of Lysen (EML), graphical method (GM), first method of moment (MOM-1), second method of moment (MOM-2), third method of moment (MOM-3), alternative maximum likelihood method (AMLM), WAsP or equivalent energy method (EEM), energy pattern factor method (EPFM), found in literature survey are compared to evaluate the performance using statistical methods in the offshore region of India. The mathematical expressions of methods have been coded and implemented by MATLAB software through a proper data analysis. There is a need to explore more appropriate probabilistic distribution than Weibull distribution. The MMLM, MLM, and MOM versions perform nicely, followed by the empirical method of Justus and Lysen. The GM method underperformed and that needs modification.

Wind Resource Assessment: Optimizing the Weibull Distribution Curve

Abstract: In this paper, it was analyze the wind speed characteristics and provide projections of optimal wind resources by comparing data at different hub heights using wind power law. We have acquired data from an operational project site in the western part of India using data points acquired from the site from August 2017 to February 2018.Weibull distribution probability distribution function and cumulative distribution function for wind speed and other parameters involved in an accurate wind resource assessment methodology. To predict wind regimes, the Weibull probability function is commonly used. A comparison of alternative methods for estimating Weibull parameters of a wind regime is presented in this research. For the estimation, five alternative methods are discussed and employed. The study examined wind speed time series data at two sites over the course of a year. Although all five methods produce similar results when plotting error versus wind speed, the standard deviation method produces the best results.

Wind speed Weibull distribution and wind energy potential of Chandpur, Bangladesh

Weibull distribution goes particularly well with wind speed data and is necessary to assess wind energy harnessing potential at a site. For the best estimation of the statistical distribution parameters at Chandpur (23.21116° N; 90.64237° E), located at the southern part of Bangladesh, six models: graphical method, method of moments, power density method, Justus method, Lysen method and maximum likelihood method have been analyzed for 18.8 m, 40.2 m and 59.9 m above ground level (AGL) using measured wind speed data for 2014-2017. Out of the six models, Justus method has showed the best result for estimation of Weibull parameters at all heights AGL with average RMSE of 0.0082 and R2 of 0.9650. Further more to evaluate the wind energy potential at modern wind turbine hub height of 100 m, firstly the wind shear exponent at the site (the average value is 0.37 considering the 10 m elevation of the site above sea level) is estimated from the wind data of 3 heights; secondly average wind speed (4.88 m/s), wind power density (108.8 Wm-2), Weibull distribution parameters (shape factor 2.29 and scale factor 5.33 m/s) and wind turbine power curve related wind speeds (cut in 2.28 m/s, rated 9.33 m/s and cut out 21.03 m/s)have been extrapolated at the hub height. The average wind speed and wind power density values show that the class of the site is 1 in the worldwide accepted wind power classification, based on long term practical wind farm installation experiences; the low cut in speed (<3 m/s) shows that suitable wind turbines for the site are commercially unavailable for the site and turbines with 3 m/s cut in speed have average annual capacity factor of 16.62 %, less than the commercially viable capacity factor >25 %. The wind power class (< class 3) and capacity factor (<25 %) strictly reveals that Chandpur is not suitable for wind energy exploration in near future until a high technical improvement of wind turbines and reduction of relevant costs. DUJASE Vol. 6 (2) 99-108, 2021 (July)

A comparative study of five numerical methods for the estimation of Weibull parameters for wind energy evaluation at Eastern Jerusalem, Palestine

Wind power provides a clean and feasible solution to generate electricity. The development of wind power applications requires a deep analysis of wind profiles and an accurate prediction of wind energy at a study site. This work explores the distribution of wind speed to estimate the two Weibull parameters (shape and scale) that are widely utilized for modeling and providing an accurate and efficient estimation of wind resource and power. These two parameters are calculated based on measured daily wind speed data from 2008 to 2018, collected in Jerusalem, Palestine. Three assessment criteria were used to assess the goodness of fit; they are Root Mean Square Error (RMSE), Mean Absolute Percentage Error (MAPE), and chi-square. The findings revealed that of the five estimation methods being considered in this study, both the Empirical Method (EM) and the Method of Moment (MoM) were the most accurate in determining the values of the Weibull shape and scale parameters to approximate wind speed distribution at the study site. Based on the goodness-of-fit tests, both methods provide lower values of the used assessment criteria. The statistical performance tests rejected the Energy Pattern Factor Method (EPFM) as an adequate method due to the higher values of chi-square and revealed that the Maximum Likelihood (MLM) and the Modified Maximum Likelihood (MMLM) methods ranked third and fourth, respectively.

Comperative Analysis of Weibull Parameter Estimation Using HPSOBA and Standard PSO and Classical Bat Algorithms

The paper benevolences a hybrid metaheuristic optimization algorithm developed for the computation of Weibull probability and cumulative density functions (PDF) and (CDF) respectively. The algorithm is designed through espousing protruding topographies wich are used in the Bat algorithm (BA) and in the Particle Swarm Optimization (PSO) and is used to analyze the figure and measure constraints of Weibull Distribution (WD). The established algorithm is also verified on target test functions. The results and computational time are compared with the results obtained through the implementation of standard PSO and BA. The computed values of the parameters of WD are also verified through the calculation of, index of agreement (AI), mean absolute percentage error (MAPE) coefficient of determination (R2) and root mean square error (RMSE). The results of parameter estimation are also compared with standard PSO and BA for the mentioned region. The comparative analysis of the results of benchmark functions and Weibull parameter estimation reveal the robustness and verstality of the developed algorithm.

Elaboration and Charpy Test of Bioplastics Reinforced by Renewable Fibers: Starch /Diss

This work is based on an investigation study to develop bio-composite materials that are renewable, biodegradable, and environmentally safe. The fibers used in this work are extracted from the plant Ampelodesma Mauritanica, It is a wild plant that is produced in abundance in the Mediterranean regions. Through this work, an overview of Diss fibers was provided, developing bio-composite using different starch matrices reinforced by Diss fibers, and evaluate their mechanical behavior using Charpy-tests to determine standard test specimens to estimate Weibull parameters suitable for the composite using statistical methods based on Weibull distribution. The obtained results, it was found that the bio-composite starch/Diss 40% Glycerol and 5% fiber reinforcement (SG40/RF5) had better results compared to the rest of the bio-composite, The Charpy impact energy modulus was about 31.25 (KJ/m2), which is 2.1 times higher than that achieved Measured from SG40 matrix (40% glycerol), and 1.3 times higher than those fortified with 10% fiber SG40/RF10 (40% glycerol reinforcement 10% fiber), and the statistical study confirmed the distribution of the results obtained, especially Weibull, which has three parameters.