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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.

Rockfall events are a common geological hazard in the mountain areas of western China and frequently occur on unstable slopes. The purpose of this study was to establish a statistical model of random rockfall–slope collision using the impulse moment theorem considering that the impacted stones are moving. The rebound velocity of rockfall is random because the microstructure of the ground and slope can be described with statistical properties. The analytical solutions of the rebound particles are obtained as formulas for the incident velocity, incident angle, impacted rock movement, slope angle, and ground surface microstructure. The means and standard deviations of rebound velocity are fitted with slope angle to calculate formulas based on probability theory, which agrees with the previous results. The probability distribution functions of the horizontal distance and vertical distance follow a Gaussian distribution and a negative exponential distribution under the given slope angle, respectively. A formula of trajectory is also given.

Seismic properties play a fundamental role in the geological and petrophysical modeling of reservoirs due to their dependence on petrophysical properties. Most existing stochastic seismic inversion methods are based on Gaussian probability distribution functions and assume linear dependence. Examples include sequential Gaussian co-simulation (SGCS) and direct sequential simulation (DSS). In contrast, spatial stochastic co-simulation methods based on Bernstein copulas (BCCS) have recently been developed. These methods do not require a specific distribution type or linear dependence, thereby overcoming the limitations of traditional approaches. In this context, we propose a novel approach for the joint seismic inversion of elastic and petrophysical properties using parametric copulas within a Bayesian inference framework. A joint probability distribution is constructed using well-scale petrophysical and elastic property data, fitted to parametric copula functions and treated as prior information. The model parameters are then updated a posteriori using petrophysical properties scaled by a moving window averaging method and seismic properties upscaled using the Backus averaging method. The resulting posterior model is used within the inversion process to generate elastic property realizations at the seismic scale. The inverse problem is solved using a simulated annealing algorithm that minimizes a global objective function combining the root-mean-square (RMS) error between synthetic and observed seismic traces, and the semivariogram error between the simulated and target variogram models. For each elastic realization, a reflectivity series is computed and convolved with a seismic wavelet to generate a synthetic seismic trace. The best-fitting elastic realization is then used to simulate the corresponding petrophysical property using the same joint probability distribution. The proposed method was applied to a deepwater reservoir case study to estimate total porosity and acoustic impedance at the seismic scale. Results demonstrate that the use of parametric copulas reduces computational cost and execution time while enabling effective integration of nonlinear dependencies. The synthetic traces exhibit RMS errors below 8%, validating the accuracy and robustness of the copula-based inversion framework.

The rising use of renewable energy sources, particularly those that are weather-dependent like wind and solar energy, has increased the uncertainty of supply in these power systems. In order to obtain considerably more accurate results in the analysis of power systems, such as in the planning and operation, it is necessary to tackle the stochastic nature of these sources. Operators require adequate techniques and procedures to mitigate the negative consequences of the stochastic behavior of renewable energy generators. Thus, this paper presents a modification of the original probability distribution functions (PDFs) where the original PDFs are insufficient for wind speed and solar irradiance modeling because they have a significant error between the real data frequency distribution and the estimated distribution curve. This modification is using a mixture of probability distributions, which can improve the fitting of data and reduce this error. The main aim of this paper is to model wind speed and solar irradiance behaviors using a two-component and a three-component mixture of PDFs generated from the integration of the original Weibull, Lognormal, Gamma, and Inverse-Gaussian PDFs. Three statistical errors are used to test the efficiency of the proposed original and mixture PDFs, which are the root mean square error (RMSE), the coefficient of correlation (R2), and the Chi-square error (X2). The results show that the mixture of PDFs gives better fitting criteria for wind speed and solar irradiance frequency distributions than the original PDFs. The parameters of the original and the mixture of PDFs are calculated using the innovative metaheuristic Mayfly algorithm (MA). The three-component mixture of PDFs lowered the RMSE by about 73% and was 17% more than the best original and the two-component mixture distributions.

Climate change is altering flood risk globally, with local variations prompting the necessity for regional assessments to guide the planning and management of water-related infrastructures. This study details an integrated framework for assessing future changes in flood frequencies, using the case of Bitlis Creek (Turkey). The precipitation and temperature simulations of 21 global circulation models (GCMs) from the coupled model intercomparison project phase 6 (CMIP6) are used to drive the developed soil and water assessment tool (SWAT) model in generating daily streamflow projections under the CMIP6 historical experiment and the shared socio-economic pathway (SSP) scenarios of SSP245 and SSP585. Five probability distribution functions are considered to calculate the 5-, 10-, 25-, 50-, 100-, and 500-year flood discharges for the historical period 1955–2010 and the future periods 2025–2074 and 2025–2099. The quantification of climate change impacts on the design discharges is based on the medians of the flood discharges obtained for the climate data of each GCM, using the best-fitted distribution functions. The findings illustrate significant increases in discharge rates, ranging from 21.1 to 31.7% for the 2025–2099 period under the SSP585 scenario, highlighting the necessity of considering changing climate conditions in designing water-related infrastructures.

The probability distribution function (PDF)-based unit hydrographs (UHs) are gaining momentum in an application for more accurate rainfall-runoff transformation. Employing seven statistical performance indices, R2, NSE, MSE, RMSE, MAE, MAPE, and SE in generalized reduced gradient nonlinear programming (GRG-NLP) optimization, 18 known and 12 adaptable PDF-based UHs were assessed against UHs derived from 18 storms in 7 basins across the United States, Turkey, and India. To this end, 27 Maple codes were proposed for UH-application requiring only peak discharge (qp), time to peak (tp), and time base (tb) for derivation. The introduced PDFs, such as Dagum, Generalized Gamma, Log-Logistic, Gumbel Type-I, and Shifted Gompertz, replicated the observed data-derived UHs more closely than did the known PDFs, like Inverse Gaussian, two-parameter gamma distribution (2-PGD), Log-Normal, Inverse-Gamma, and Nagakami. Among the three-parameter (6 nos.), two-parameter (21 nos.), and single-parameter (3 nos.) PDFs, the Dagum, Log-Logistic, and Poisson consistently outperformed their respective counterparts in replication.

The determination of the probability distribution function (PDF) of uncertain input and model parameters in engineering application codes is an issue of importance for uncertainty quantification methods. One of the approaches that can be used for the PDF determination of input and model parameters is the application of methods based on the maximum entropy principle (MEP) and the maximum relative entropy (MREP). These methods determine the PDF that maximizes the information entropy when only partial information about the parameter distribution is known, such as some moments of the distribution and its support. In addition, this paper shows the application of the MREP to update the PDF when the parameter must fulfill some technical specifications (TS) imposed by the regulations. Three computer programs have been developed: GEDIPA, which provides the parameter PDF using empirical distribution function (EDF) methods; UNTHERCO, which performs the Monte Carlo sampling on the parameter distribution; and DCP, which updates the PDF considering the TS and the MREP. Finally, the paper displays several applications and examples for the determination of the PDF applying the MEP and the MREP, and the influence of several factors on the PDF.

Over the past decades, the mathematical formulation of wind turbines (WTs) has been handled using different methodologies to model the probabilistic nature via different distribution functions. Many recently published articles have applied either the wind speed or the obtained active power from the WT on various probabilistic curves, such as Weibull, log-normal, and Gamma. In this work, the wind speed was modeled at five different locations in Egypt via a novel mixture probability distribution function (MPDF) that included four well-known distribution functions used to imitate the probabilistic nature of wind speed. Moreover, a decision-making multiple objective formulation was developed to optimally fit the MPDF with a minimum root mean square error (RMSE) and ensure reliable fitting by two other effective indices. Two methodologies, namely, equal and variable class widths, were investigated to model the density of wind speed and obtain a more realistic model for the tested wind speed profiles. The results showed the effectiveness of the proposed MPDF model as the RMSE was effectively minimized using multiobjective particle swarm optimization (MOPSO), showing nearly 10% improvement compared to the nondominated sorting genetic algorithm (NSGA-II).

Wind energy represents a clean, abundant and cost-effective power source, fostering job growth and environmental mitigation. Although wind energy harnesses several gigawatts today, its availability hinges on diverse factors, with geographical location standing out. Commercial turbines, with varying capacity ranges, saturate the market. Locating site-specific suitability and matching the appropriate turbine to meet specific requirements are of paramount importance. This study aims to assess the feasibility of wind energy in Surat, Gujarat, India and select an optimal small commercial turbine for residential use. The research involves Rayleigh and Weibull probability distribution functions based on yearlong velocity data. These distributions are fitted with actual data, revealing the most probable velocity (vmf = 3 m/s) and velocity at maximum power (vpmax = 5 m/s). The power availability of the site has been assessed as 42.6 W/m2 using both graphical and analytical methods. Several commercial turbines have been shortlisted based on on-site power criteria and their specifications are evaluated against site power availability. A comparative analysis culminates in identifying the most suitable turbine for the location. The best suitable turbine for the site with an annual energy yield of 8 MW has been suggested amongst selected turbines for small-scale residential applications.

The paper discusses service load measurements (weight of construction materials, small equipment and workers) conducted on 120 frame scaffoldings all over Poland in 2016‒2018. Despite the fact that the scaffolding should ensure the safety of its users, most accidents on construction sites are caused by fall from height. Service loads are one of the elements affecting the safety of scaffolding use. On the basis of the studies, maximum load on one platform and maximum load on a vertical scaffolding module for one day were obtained. They were treated as the random variables of the maximum values. Histograms and probability density functions were determined for these variables. The selection of a probability distribution consisted in the selection of a probability density function by means of fitting curves to the study result histograms using the method of least squares. The analysis was performed for distribution Weibull and Gumbel probability density functions which are applied for maximum values of random variables. Parameters of these functions can be used for the purposes of the reliability analysis to calibrate partial safety factors in simulation of service load during the scaffolding failure risk assessment. Besides, the probability of not exceeding the standard loads provided for frame scaffoldings for 120 weeks was established on the aforementioned basis. The results of the presented research show that in Poland there is a high probability of exceeding the permissible service loads in one year and thus there is a high risk of scaffolding damage.