Weibull Parameters Research Articles

Constraint Shock in Diameter Distribution Models: The Cost of Ignoring Negative Bounds

Abstract Diameter distributions are invariably fitted with constrained zero or positive lower bounds to prevent negative diameters. Goodwin (2021) found that the bound for the 3-parameter Weibull could be negative for negatively skewed data, and that constraining it to zero reduced model accuracy. The loss of model accuracy due to a bound constraint is referred to here as constraint shock. This article shows that the 4-parameter Kumaraswamy and Johnson’s SB distributions can also have negative lower bounds and exhibit constraint shock. A 3-step parameter recovery method was used to fit these distributions to plots in unthinned mixed species eucalypt plantations and results were compared with the less flexible Weibull. Based on Kolmogorov-Smirnov statistics, mean and maximum constraint shock for the 4-parameter distributions were 19% and 62%, respectively, compared with 19% and 51% for the Weibull, which indicated that constraint shock was not affected by model flexibility. Constraint shock was largely avoided by truncating and normalizing distributions with negative bounds. This work introduces a paradigm shift in diameter distribution modelling and adds clarity to a field that has not previously recognized the existence of constraint shock. Study Implications: Constraint shock is the loss of accuracy incurred by constraining the negative lower bound of a distribution to a nonnegative value. For a plantation eucalypt dataset in which 83% of plot diameters were negatively skewed, maximum constraint shock was 62% for the Kumaraswamy and Johnson’s SB distributions, and 51% for the less flexible 3-parameter Weibull, based on Kolmogorov-Smirnov statistics. Using distribution expectations of kurtosis, skewness, maximum diameter, and minimum diameter—all independent of bound parameters—a tractable 3-step parameter recovery method is described for unconstrained, constrained, and truncated 3- and 4-parameter distributions. This work introduces a paradigm shift in the treatment of distribution bounds that will result in substantial model improvements for negatively skewed data.

A simple model to predict the energy loss due to leading edge roughness

In this paper a simple model to predict losses in annual energy production due to leading edge roughness or leading edge erosion from wind turbines is presented. Because detailed information about wind turbines not always are available, the input to this model requires only a few parameters: rated power, rotor radius, two Weibull parameters and categories of local damages. Therefore, some assumptions are made in the model. It was validated against a common implementation of a BEM code and showed good agreement in the loss of annual energy production.Also, a categorisation methods was proposed. The categories ranged between a and e corresponding to an undamaged blade and a severely damaged blade, respectively. This categorisation was made locally on the blade. The blade is divided into annular elements and each annular element is graded with a category. By summing up the annular elements on the blade, each with their categories, the total loss in energy production was predicted.

On transmuted skewed Laplace distribution and its application

The aim of this study is to add a shape parameter into the existing 3-parameter skewed Laplace distribution using the method of transmutation and apply the proposed distribution on two life simulated data generated using R-statistical software. The parameters of the proposed distribution are obtained using the method of maximum likelihood estimation. The fitness of the proposed distribution was compared with that of existing skewed Laplace distribution and four parameter Weibull distribution using measures of fitness criteria, and the results obtained showed that the proposed model can be used effectively to model real life situations than the Skewed Laplace Distribution. The results also showed that the Four Parameter Weibull Distribution which is a competing model performed better than the Skewed Laplace Distribution and Transmuted Skewed Laplace Distribution.

Rüzgar enerji uygulamaları için Weibull dağılım parametrelerinin tahmini: Türkiye’nin Dinar bölgesi için bir çalışma

In this study, the wind energy characteristics of Dinar region were researched. Required wind data was provided by the Dinar Meteorology Station of the Turkish State Meteorology Service. Wind data was recorded hourly between June 2015 and May 2020. The two variable Weibull probability distribution function was used to research wind characteristics of the region. Six different methods were used to calculate shape (k) and scale (c) variables of the probability distribution function. These are moment, graph, Justus empirical, energy trend, energy pattern and maximum likehood estimation methods. The performance of these methods were compared with four different statistical error analysis methods. These are root mean square error (RMSE), coefficient of determination (R2), chi-square error (X2) and mean absolute error (MAE). According to the results, maximum likehood estimation and energy trend methods have been the most successful methods in the calculation of Weibull parameters. In addition, this study is an important preliminary study in determining the wind energy characteristic of the region.

Analyzing Wind Energy Potential Using Efficient Global Optimization: A Case Study for the City Gdańsk in Poland

Wind energy (WE), which is one of the renewable energy (RE) sources for generating electricity, has been making a significant contribution to obtaining clean and green energy in recent years. Fitting an appropriate statistical distribution to the wind speed (WS) data is crucial in analyzing and estimating WE potential. Once the best suitable statistical distribution for WS data is determined, WE potential and potential yield could be estimated with high accuracy. The main objective of this paper is to propose a novel approach for calculating wind energy potential. For this purpose, the Efficient Global Optimization (EGO) technique was proposed for fitting a statistical distribution to WS data and the performance of the technique was compared with genetic algorithm (GA), simulated annealing (SA), and differential evolution (DE). Performance metrics showed that EGO is providing better estimations compared with GA, SA, and DE. Based on Weibull parameters obtained by using EGO, potential WE and potential annual revenue were estimated for Gdańsk, which is the capital of Pomerania Voivodeship in Poland, in the case of having city-type wind turbines in the city center. Estimations for Gdańsk showed that city-type wind turbines might be helpful for producing electricity from WE in the city without being limited by constraints such as having a long distance between wind turbines and buildings. If such wind turbines were erected on the roofs of residential buildings, malls, or office buildings, there is a possibility that part of the electric energy needed for such buildings could be generated using WE. However, this topic should be further investigated from technical and financial perspectives.

Probabilistic analysis of cleavage fracture in commercial polycrystalline tungsten

Polycrystalline tungsten is a promising candidate as plasma-facing armour material in future fusion reactors. Below the DBTT tungsten shows brittle or semi-brittle behaviour with no or little plastic deformation before cleavage fracture. Moreover, tungsten has an inherent low fracture toughness as well as a large scatter within the strength properties. This requires a statistical treatment of the strength values. Therefore, four-point bending tests in the temperature range of room temperature (RT) to 400 °C were performed to allow the Weibull analysis in brittle failure regime and/or the Beremin model analysis in case of onset of plastic behaviour while approaching the DBTT temperature. The analyses are supported by FE simulations. Weibull and Beremin parameter values are determined from the simulation results using post-processing codes.

Probability distribution of wind speed and wave height in Nowshahr Port using the data acquired from wave scan buoy

The present study aimed to determine probability density function (Pdf) of wind speed and wave height in Nowshahr Port using the data obtained from wave scan buoy. Three Pdfs, i.e. Weibull, Rayleigh and Lognormal, were adopted for estimation of data frequency. The shape (k) and scale (c) parameters of the Weibull function were calculated by four methods, i.e. Graphical method (Gm), Moment method (Mm), Maximum likelihood method (Mlm) and Empirical method (Em) whereas the parameters of the other two functions were estimated only through Mlm. In order to analyze the efficiency of pdfs root mean square error (Rmse), determination coefficient (R2) and chi-square test (χ2) were considered. The results revealed that Mm, Em and Mlm presented almost similar ‘k’ and ‘c’ values. However, Gm yielded smaller values. Accuracy of Weibull Pdf was generally higher than those in Rayleigh one. However, it was seen in the present study that Rayleigh Pdf is better only when Weibull parameters in wind data were measured via Gm. Lognormal Pdf was the most efficient in both wind and wave data. Finally, cumulative distribution function was estimated based upon Weibull and Lognormal probability models and the former was again found to be stronger and more accurate.

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.

Probabilistic model for cattail and canola fibers: effect of environmental conditions, structural parameters, fiber length, and estimators

Biomass fibers are being widely investigated for industrial applications as an alternative to synthetic fibers using a standard humidity condition. In this study, the mechanical properties of two waste biomass fibers – canola and cattail – have been investigated when subjected to different environmental conditions, fiber length, and type of estimators used during analysis. The effect of different environmental conditions and structural variations were investigated by measuring the tensile properties after exposing them to eight different relative humidity conditions using a fixed fiber length of 25 mm. Further investigation was conducted using fiber lengths of 25, 35 and 45 mm using the most conservative relative humidity condition. The data were analyzed by a Weibull distribution model using four different estimators. The results revealed that Weibull strength ( σavg) and modulus (Eavg) closely followed experimental values for cattail and canola fibers. The different relative humidity conditions and fiber lengths resulted in different Weibull parameters with 11% relative humidity and the mean rank estimator predicted the most conservative tensile strength for both waste biomass fibers. The experimental and characteristic Weibull strength decreased when fiber gauge length increased from 25 to 45 mm. The tensile strength and modulus of both waste biomass fibers at 50% reliability lie within the range of average experimental values. However, these values are reduced to 155 MPa (strength) and 20 GPa (modulus) for cattail fiber at 90% reliability. The survival probability of the tensile strength and modulus were found to be the highest at 75% and 100% relative humidity for cattail and canola fibers, respectively.

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.

Water as Veneering Modeling Liquid Affects Microhardness of Glassy Matrix Veneering Ceramic.

To evaluate the effect of different veneering liquids used for modeling on microhardness, fracture toughness and biaxial flexural strength of a glass-veneering ceramic. The manufacturer recommended modeling liquid (ML), distilled water (DW), isopropyl alcohol (IA), 0.5% (P05), 1% (P1), and 2% (P2) polyethylene glycol solutions were mixed with feldspathic ceramic powder to form disc-shaped samples (n=20, 15 mm × 1.2 mm). After sintering, samples were mirror-polished and subjected to Vickers indentation (n=5) for measurement of microhardness and fracture toughness. The remaining 15 samples from each group were subjected to biaxial flexural strength. Data were subjected to one-way ANOVA and Weibull analysis. The microhardness was affected by veneering liquid (p=0.002): DW promoted higher microhardness values than ML and IA. Fracture toughness (p=0.301) and flexural strength (p=0.930) were not affected by the veneering liquid but Weibull parameters were affected. All groups presented surface pores under high magnification. Even though the use of DW led to higher values of surface microhardness than the ML, all obtained values are inside the range of enamel microhardness values reported in the literature. Such parameters may affect antagonist wear and should be reported in clinical trials.

Modeling the Effects of Asphalt Overlay Design on Pavement Distress Initiation with Random Parameters Hazard-Based Duration Models

Pavement deterioration model is a key input to a pavement management system that optimizes pavement maintenance and rehabilitation (M&R) planning. As a typical pavement M&R technique, asphalt overlay is commonly implemented to restore pavement performance. However, the impact of its design factors on pavement deterioration is not well documented in existing publications. This study aimed to investigate the effects of asphalt overlay design on the initiation of various types of cracking and severe rutting with consideration of endogenous bias and unobserved heterogeneity. A series of random parameters Weibull duration models were developed based on 317 asphalt overlay projects, which were collected from the Long-Term Pavement Performance program. Endogenous overlay design, overlay thickness, milling indicator, and overlaid pavement deflection were identified to represent their heterogeneity caused by unobserved factors over highway sections. Based on analysis results, the endogenous overlay design was shown to underestimate the service life of overlaid pavement. The occurrence probability of overlaid pavement distress would significantly decrease with an increase of asphalt overlay thickness. The application of 30% reclaimed asphalt pavement materials would retard the occurrence of severe rutting, but accelerate the occurrence of all types of cracking. Relative to asphalt overlay, milling and overlay can significantly reduce the hazard rate of severe rutting and all types of cracking. In addition, existing pavement condition, overlaid pavement deflection, traffic volume, and climate factors were identified to have significant effects on the distress initiation in overlaid pavements.

Evaluation of Eleven Numerical Methods for Determining Weibull Parameters for Wind Energy Generation in the Caribbean Region of Colombia

The two-parameter Weibull probability density function (PDF) is widely utilized by different researchers and engineers to fit wind speed data for statistical analysis and modeling. The characterization of wind resources in the frequency and probability domain is necessary to estimate the power output potential of new wind energy projects. Considering that exist a variety of Weibull equations evidenced in the literature review, this article evaluates 11 different methods to calculate the shape and scale parameters of the Weibull PDF. In this sense, it was written an algorithm within a Matlab function that solves the 11 methods for calculating the Weibull PDF parameters. Wind speed data extracted from the ERA5 database was used as input data for applying the proposed algorithm, and statistical parameters such as the Root Mean Square Error (RMSE), the Relative Root Mean Square Error (RRMSE), and chi-square test (X2) we utilized for assessing the performance of each one of the 11 methods for modeling the wind distribution. The statistical results pointed that the numerical iteration methods (e.g. maximum likelihood method) showed better results than parameterized equations such as the Graphical Method, hence, this research recommends the implicit methods for determining Weibull PDF parameters of wind speed data.

WIND CHARACTERISTICS ANALYSIS IN FOUR POTENTIAL TOWNSHIPS IN MYANMAR

Myanmar has set renewable energy aspirations in the energy mix of the country to meet growing energy demand and to increase clean access to electricity as indicated in Myanmar’s Intended Nationally Determined Contribution. Among renewable energy resources, solar and wind energy are expected to contribute to 9% of total energy mix. Although there have been initiatives on the implementation of solar photovoltaic in Myanmar, implementation of wind energy has not been reported. Few studies on wind energy in Myanmar focused on resource assessment investigating the spatial variation of wind speed and power density. Little has been studied on the seasonal nature and persistence of wind in Myanmar from the perspective of energy generation. This study aims at generating wind speed and power density maps of Myanmar using most recent wind data from 2010 to 2017 to identify potential townships with favourable wind conditions. Prominent wind direction, seasonal variations and wind speed persistence were analyzed for four townships with favourable wind conditions. Weibull parameters suggest that frequency of wind is also favourable for wind energy generation. Hence, this study has identified Chauk, Kyaukpadaung, Meiktila and Natogyi townships as potential regions for wind energy development with estimated power producing time of 53.17% and 65.91% of a year at average wind speed of 4 m/s above. This study serves as a basis for further resource assessment for micrositing of wind turbines to identify feasible sites for wind energy generation in Myanmar.

Assessment of onshore wind energy potential under temperate continental climate conditions

Using the Weibull distribution function, the study presents a statistical evaluation of wind characteristics and onshore wind power potential under temperate continental climate conditions. Wind energy projects are more expensive to install than conventional energy sources, but they have lower operating and maintenance costs (zero costs for fuel). Because the initial investment is so large, conducting as precise and reliable feasibility studies as possible is critical because they play a significant role in lowering the cost of energy production. As a result, wind speed measurements are critical for conducting accurate feasibility studies. The availability and accuracy of weather data are critical for determining wind potential at a specific location. Wind speed is an important factor to consider when analyzing the wind speed distribution function. The monthly and yearly values of Weibull parameters, and wind power density at the analyzed location, were estimated using wind speed data. The yearly value of the k parameter for the Weibull analysis was 1.153, and for the c parameter was 5.602, respectively. Also, the yearly average wind speed was found to be greater than the maximum wind speed. Another important conclusion is that the maximum energy-producing wind speed, rather than the average or most frequent wind speed, has the greatest influence on wind potential. The final outcome of this study indicate that the location analyzed is adequate for the exploitation of wind energy.

A comparative analysis of wind characteristics for distinct terrains of India

The accurate wind potential assessment is the critical and inevitable process before wind farm planning. This paper aims to evaluate and compare the wind potential between the evolutionary algorithms and numerical methods using the wind data measurement data on flat, complex, coastal offshore, and offshore sites from the mast and remote sensing method. The maximum likelihood method, modified maximum likelihood method, and WAsP (wind atlas analysis and application program) are employed to estimate the optimum Weibull parameters. Furthermore, the Jaya algorithm is compared with particle swarm optimization and genetic algorithms to evaluate Weibull function for different topological terrains of India. The four-performance metrics, root mean square error, normalized root mean square error, normalized error, and root mean square difference are estimated to analyze the accuracy of various methods. Considering numerical methods, the WAsP method was found better than other numerical techniques. While the Jaya algorithm and particle swarm optimization were found to be efficient as compared to the genetic algorithm. The mean power density is maximum in the coastal area followed by offshore, complex terrain, and flat terrain with 687 W/m2, 485 W/m2, 301 W/m2, and 243 W/m2, respectively at 100 m height.

Machine learning‐based statistical downscaling of wind resource maps using multi‐resolution topographical data

AbstractThe need for high‐resolution wind resource maps is increasing with the increase in the supply and development of wind power. Many physical downscaling models have been developed and applied to make these maps. However, as the existing models require extensive computations and time, statistical models with higher efficiency are being studied. Statistical models such as regression and machine learning models can quickly calculate wind resource maps, but they have a problem of low accuracy. This study proposes a machine learning model with new topography‐derived variables to interpret the physical characteristics of the wind. As the shape of topography, which was unable to be interpreted in previous studies, can be considered with new derived variables, a significant performance improvement was identified. The analysis was conducted using 1 km Weather Research and Forecasting (WRF) results and ERA5 reanalysis data from South Korea. Two Weibull distribution parameter maps were calculated and used as input and output data. Three collections of derived variables were devised and compared. Therefore, the multi‐resolution topography data showed the highest improvements with approximately 15% reduction in root mean square error (RMSE) for both the linear regression and machine learning models. In particular, the land area showed a decrease of 20%. The best proposed models showed an RMSE of 7% and 8% for two Weibull parameters. The results are expected to serve as a reference for continuing research and utilization of statistical models.

Human–Machine Systems Reliability: A Series–Parallel Approach for Evaluation and Improvement in the Field of Machine Tools

Machine workshops generate high scrap rates, causing non-compliance with timely delivery and high production costs. Due to their natural characteristics of a low volume, high-mix production batches, and serial and parallel configurations, generally the causes of their failure are not well documented. Thus, to reduce the scrap rate, and evaluate and improve their reliability, their system characteristics must be considered. Based on them, our proposed methodology allows us to evaluate the system, subsystem, and component–subsystem relationship by using either the Weibull and/or the exponential distribution. The strategy to improve the system performance includes reliability tools, expert interviews, cluster analysis, and root-cause analysis. In the application case, the failure sources were found to be mechanical and human errors. The component maintenance/setup, institutional conditions/attitude, and subsystem process/operation were the machine factors that presented the lowest reliability indices. The improved activities were monitored based on the Weibull β and η parameters that affect the system reliability. Finally, by using a life–effort analysis, and the method of comparative analysis of two sequential periods, we identified the causes that generated a change in the Weibull parameters. The contribution of this methodology lies in the grouping of the tools in the proposed application context.

Influence of the Random Data Sampling in Estimation of Wind Speed Resource: Case Study

In this study, statistical analysis is performed in order to characterize wind speeds distribution according to different samples randomly drawn from wind speed data collected. The purpose of this study is to assess how random sampling influences the estimation quality of the shape (k) and scale (c) parameters of a Weibull distribution function. Five stations were chosen in West Africa for the study, namely: Accra Kotoka, Cotonou Cadjehoun, Kano Mallam Aminu, Lomé Tokoin and Ouagadougou airport. We used the energy factor method (EPF) to compute shape and scale parameters. Statistical indicators used to assess estimation accuracy are the root mean square error (RMSE) and relative percentage error (RPE). Study results show that good accuracy in Weibull parameters and power density estimation is obtained with sampled wind speed data of 30% for Accra, 20% for Cotonou, 80% for Kano, 20% for Lomé, and 20% for Ouagadougou site. This study showed that for wind potential assessing at a site, wind speed data random sampling is sufficient to calculate wind power density. This is very useful in wind energy exploitation development.