Weibull Probability Distribution Function Research Articles

Estimation of Weibull Probability Distribution Parameters with Optimization Algorithms and Foça Wind Data Application

In this study, the scale and shape parameters of the Weibull probability distribution function (W.pdf) used in determining the profitability of wind energy projects are estimated using optimization algorithms and the moment method. These parameters are then used to estimate the wind energy potential (WEP) in Foça region of İzmir in Turkey. The values of Weibull parameters obtained using Particle Swarm Optimization (PSO), Sine Cosine Algorithm (SCA), Social Group Optimization (SGO), and Bat Algorithm (BA) were compared with the estimation results of the Moment Method (MM) as reference. Root mean square error (RMSE) and chi-square (χ^2) tests were used to compare the parameter estimation methods. The wind speed measurement values of the observation station in Foça were used. As a result of Foça speed data analysis, the annual average wind speed was determined as 6.15 m/s, and the dominant wind direction was found as northeast. Wind speed frequency distributions were compared with the measurement results and calculated with the estimated parameters. When RMSE and χ^2 criteria are evaluated together; it can be concluded that each used method behaves similarly for the given parameter estimation problem, with minor variations. As a result, it has been found that the optimization parameters produce very good results in wind speed distribution and potential calculations.

Modelling and fatigue reliability investigation on wear prediction of piston/cylinder pair based on friction fatigue mechanism

This paper aims to assess the wear reliability characteristics of the piston/cylinder pair based on the wear failure prediction using numerical and experimental techniques for the water hydraulic state. This is due to the complex forces and contact states acting in the piston, hence, making it difficult to establish a numerical model for wear failure prediction. The Hertzian contact and Elastic-Plastic mechanical equivalence are proposed in numerically modelling the friction fatigue mechanism based on asperity damage for piston/cylinder pair. An experimental analysis is conducted for the reciprocating ceramic peek-2507 steel tribopair to validate the effectiveness of the prediction model. The error between the numerical and experimental models is assessed to be less than 10% with a coefficient of determinant R2 of 0.99. The numerical wear data were scattered within the 10% confidence interval, indicating the high survivability of the predicted data. In addition, the reliability characteristics based on the volume of wear predicting in risk monitoring using the Weibull probability distribution function are proposed to model the reliability and probability of failure for the piston/cylinder pair. From the volume of wear prediction from the computed Mean-Volume-To-Failure (MVTF) = 17.8 mm3, the probability of failure based on the numerical modelling is estimated to be at 0.52. Hence, the proposed numerical model is suitable for predicting the wear volume in the water hydraulic state for various conditions.

Optimal stochastic power flow using enhanced multi-objective mayfly algorithm

For the classical multi-objective optimal power flow (MOOPF) problem, only traditional thermal power generators are used in power systems. However, there is an increasing interest in renewable energy sources and the MOOPF problem using wind and solar energy has been raised to replace part of the thermal generators in the system with wind turbines and solar photovoltaics (PV) generators. The optimization objectives of MOOPF with renewable energy sources vary with the study case. They are mainly a combination of 2–4 objectives from fuel cost, emissions, power loss and voltage deviation (VD). In addition, reasonable prediction of renewable power is a major difficulty due to the discontinuous, disordered and unstable nature of renewable energy. In this paper, the Weibull probability distribution function (PDF) and lognormal PDF are applied to evaluate the available wind and available solar power, respectively. In this paper, an enhanced multi-objective mayfly algorithm (NSMA-SF) based on non-dominated sorting and the superiority of feasible solutions is implemented to tackle the MOOPF problem with wind and solar energy. The algorithm NSMA-SF is applied to the modified IEEE-30 and standard IEEE-57 bus test systems. The simulation results are analyzed and compared with the recently reported MOOPF results.

A multi-objective two-stage stochastic programming approach for wind farms and demand side provider decisions using the weighted metric method

The rapid expansion of wind-based renewable energy sources highlights the critical need to enhance reserve capacity within power systems. This research aims to develop an effective model for harnessing the combined potential of flexible demand-side resources, acting as demand-side providers (DSPs), to address the inherent unpredictability of wind power. The DSP framework comprises three demand response aggregators: the capacity market program (CAP), energy storage systems (ESSs), and plug-in electric vehicle parking lots (PEVs PLs). To achieve this, a multi-objective two-stage stochastic programming model is introduced. It not only minimizes operational costs for system operators but also maximizes returns for demand-side resource owners supervised by DSPs in the energy and reserve market environment. The problem is tackled using a mixed-integer linear programming (MILP) approach employing a weighted metric methodology. Within this framework, the Weibull probability distribution function (WPDF) is utilized to model wind power uncertainty, while the truncated Gaussian distribution function (GDF) encapsulates the uncertainty regarding electric vehicle owner behavior. The results demonstrate that employing this method for simultaneous programming of energy and reserve not only reduces system operator costs but also enhances profits for resource owners.

Optimal Reactive Power Dispatch problem integrating probabilistic model of wind power for uncertainties in load using JAYA algorithm

The optimal reactive power dispatch (ORPD) is a method for optimising power system engineering to reduce power loss and voltage deviation. This work addresses the ORPD problem considering load uncertainty, where load is varied in ten different instances. The optimal conditions have been determined for the system with minimum power loss and minimum total voltage deviations for different loading instances. The problem has also been integrated with the probabilistic nature of wind power, calculated using the Weibull Probability Distribution Function (WPDF). The aim is to obtain the optimal values of wind power using the WPDF for two separate cases: overestimation and underestimation, integrate these power individually, and minimise both the power loss and voltage deviation for the ten different loading scenarios. This hybrid wind power integrated ORPD problem, considering load uncertainty, has been tested on the IEEE 30 and IEEE 57 bus systems using the JAYA algorithm and other optimisation techniques. This tuning-parameter-free JAYA algorithm has obtained the best solutions for power loss of 1.8641 MW and TVD of 0.123 p.u. for the overestimated wind power integration at the IEEE 30 bus system at 100% loading compared to the other algorithms. It also acquired the best power loss and TVD solutions of 1.9144 MW and 0.1245 p.u., respectively, for the underestimated wind power at the same bus system for 100% loading with the highest convergence rate. The overall study establishes the JAYA algorithm as a superior optimising technique for solving this hybrid ORPD problem compared to the other techniques.

Modelling diameter at breast height distribution of jack pine and black spruce natural stands in eastern Canada

Individual tree diameter at breast height (DBH) distribution is an important information for forest management planning. Forest managers obtain the DBH data either by field measurements or estimations using predictive models. However, probability distribution models are still lacking or need improvement. Therefore, we aimed to construct and fit diameter distribution models that reflect forest structure and composition change. We evaluated gamma, log-normal, and Weibull probability distribution functions (PDFs) for two commercially important tree species, black spruce ( Picea mariana (Mill) B.S.P.) and jack pine ( Pinus banksiana Lamb), grown in natural stands across Ontario, Canada. We modelled the parameters of the distributions as a function of stand-level variables for these species. We used DBH data from 735 permanent sample plots. Our results showed that all three evaluated PDFs reflected observed DBH distribution. We demonstrated that the moment-based recovered parameters could represent the maximum likelihood-estimated parameters precisely, and parameters of the PDFs can be modelled as a function of stand-level dynamic covariates. The models unbiasedly predicted the PDF parameters DBH means and DBH classes. The R2 of the model fit ranged between 0.35 and 0.98 for the predicted parameters and 0.90 and 0.97 for the predicted DBH.

Integrating renewable energy and V2G uncertainty into optimal power flow: A gradient bald eagle search optimization algorithm with local escaping operator

AbstractHere, a new approach is proposed for solving the optimal power flow (OPF) problem in transmission networks using a Gradient Bald Eagle Search Algorithm (GBES) with a Local Escaping Operator (LEO). The method takes into account uncertainty of the renewable energy sources (wind energy and photovoltaic systems) and Vehicle‐to‐Grid (V2G) in the stochastic OPF problem. To improve the efficiency of the proposed technique and enhance its local exploitation capability, the LEO method's selection features are utilized. Monte Carlo methods are employed to estimate the generation costs of the renewable sources and PEVs and study their feasibility. The uncertainty of the renewable sources and PEVs is represented by Weibull, lognormal, and normal probability distribution functions (PDFs). The GBES approach is experimentally compared with well‐known meta‐heuristics using twenty‐three different test functions, and the results indicate its superiority over BES and other recently developed algorithms. Furthermore, the proposed method's effectiveness is evaluated using IEEE 30‐bus test system under various scenarios, and the simulation results demonstrate that it can effectively address OPF issues considering renewable energy sources and V2G, providing superior optimal solutions compared to other algorithms.

Multi-objective optimal power flow of thermal-wind-solar power system using an adaptive geometry estimation based multi-objective differential evolution

Sustainable energy is a key component of sustainable development. The current grid can be supplied by fossil fuel generators and renewable energy sources (RESs)-based generators, such as solar photovoltaic (PV) and wind power generators. In an electrical network, power generation from several sources must be optimally coordinated to ensure efficient and economical operation. However, the intermittent and uncertain nature of RESs complicate the operation of power systems. In this study, an adaptive geometry estimation-based multi-objective differential evolution (AGE-MODE) method is proposed for multi-objective optimal power flow in a hybrid power system of thermal, wind, and solar energy sources (MOOPF-TWS). In the proposed approach, wind and solar PV power outputs are predicted based on Weibull and lognormal probability distribution functions, respectively. Therefore, the generation costs for solar and wind power can be divided into direct costs, penalty costs for underestimation, and reserve costs for overestimation. Furthermore, the emissions, voltage deviation, and real power loss are considered in particular cases. AGE-MODE is applied to modified IEEE 30-bus and 57-bus systems, where different case studies are simulated with combinations of two-, three-, and four-objective optimizations in MOOPF-TWS problems. Comparisons between AGE-MODE and other recently developed multi-objective methods demonstrate its effectiveness in resolving MOOPF-TWS problems, particularly for cases with more than two objectives.

Accuracy of wind speed forecasting based on joint probability prediction of the parameters of the Weibull probability density function

This work aims to evaluate different error estimations of the shape and scale parameters of the Weibull probability density function of wind speed measured at the Fujairah site over a 1-year period. This study estimates trends in the variation of Weibull parameters using moving averages and Markov series methods. The focus is on the scale and shape factors, which are evaluated by mapping monthly mean wind speeds into a Weibull probability distribution function. Due to the imprecise nature of these factors, multiple data simulations are used to predict Weibull factors based on data measuring interpolations. A procedural algorithm is proposed to select the overall best forecast based on several estimation methods that evaluate raised prediction errors. A probabilistic analysis is followed to predict future wind speed and wind energy based on variations in the scale and shape factors. This study focuses on the scale factor variation as it is found to be more dominant than the Weibull shape factor. The forecasted wind speed is checked with the measured value in future months and found to be within trend values. The results suggest that the proposed algorithm provides an accurate and reliable method for predicting future wind speed and energy output.

Dynamic scheduling of market price-based combined heat–power-constrained renewable microgrid

Abstract In this paper, a market price-based combined heat–power dynamic dispatch model for a microgrid is presented. The microgrid comprises cogeneration units and wind and solar power-generation units. A battery and a heat storage tank are incorporated to optimally balance variations in heat-and-power load demands. The proposed model explores the impact of market prices of electricity, heat supply and load variability on the optimal schedule such that profit maximizes and emission, loss and waste heat are minimized. The Weibull probability distribution function is applied to characterize the uncertain renewable power variable in the model and to find the over- and under-scheduling costs. The problem is solved using an improved differential evolution algorithm in which a fuzzy membership module is appended to obtain a solution having the highest attainment for the selected multiple objectives. The results show that the proposed model can handle uncertain heat–power demand and price scenarios to produce feasible and optimal schedules with owner profits, heat utilization and renewable share varying between 10.55–115.97%, 72.51–90.39% and 26.82–38.05%, respectively.

Northern Goshawk Optimization Algorithm for Optimal Power Flow with FACTS Devices in Wind Power Integrated Electrical Networks

Power-operational problems must be resolved with flexible AC transmission systems (FACTS) and optimum power flow (OPF). The complexity of power flow analysis in modern power network models is increased by the unpredictability of renewable energy generating sources in addition to the challenge of determining the appropriate locations and ratings for FACTS devices. FACTS devices can increase the thermal capacity of power transmission lines, enabling the network to accommodate more electricity from renewable sources. The major goals of this study are to lower production costs, decrease power losses, enhance the voltage profile of the system, and enhance load capability. To solve a variety of optimization issues optimally, optimization meta-heuristics approaches have been created in response to the increased emphasis on renewable energy sources, such as wind power. Wind energy is an important component of modern power systems due to its affordability and clean nature. This article presents an OPF method for efficiently scheduling energy from wind farms using the Northern Goshawk Optimization (NGO) meta-heuristic technique. The Weibull probability distribution function is used to represent wind speed and estimate the cost of electricity produced by wind turbines based on the planned power. To validate the NGO approach and its results compared with those of other meta-heuristic optimization techniques, such as Chef-Based Optimization Algorithm, Dingo Optimization Algorithm, and Chameleon Swarm Algorithm, the IEEE system 30-bus is used to solve the OPF problem.

Application of the Standardised Streamflow Index for Hydrological Drought Monitoring in the Western Cape Province, South Africa: A Case Study in the Berg River Catchment

In many regions around the world, drought has been recurrent, more frequent, and more intense over time. Hence, scientific research on drought monitoring has become more urgent in recent years. The aim of this study was to test the applicability of the Standardised Streamflow Index (SSI) for hydrological drought monitoring in the Berg River catchment (BRC), Western Cape (WC) province, South Africa (SA). Using various methods described in this study, the sensitivity of the SSI to the commonly used Gamma, Log-normal, Log-logistic, Pearson Type III, and Weibull Probability Distribution Functions (PDFs) was tested. This study has found that all the tested PDFs produced comparable results for mild to severe drought conditions. The SSI calculated using the Gamma, Log-Normal, and Weibull PDFs is recommended for the BRC because it consistently identified extreme drought conditions during the 1990–2022 study period and identified the 2015–2018 droughts as the worst during the study period. Although more studies are required to test other PDFs not considered, this study has shown that the SSI can be applicable in the BRC. This study has provided a foundation for more research on the application of the SSI in the BRC and other catchments in SA.

Graphical interface for wind resource estimation in a region, case study: Cuatrocienegas, Coahuila de Zaragoza

Mexico has an estimated wind potential of 70 GW, however, until 2022 only 7.3 GW were used. This is due, among other factors, to the lack of analysis of the wind resource in many of the country's regions, especially in those where the wind potential is not obvious. With the aim of promoting wind power throughout the country, a free application has been developed that analyzes the resource in detail in any region using the annual records of its wind speed and direction. To demonstrate its functionality, data from the year 2022 from Cuatrocienegas, Coahuila de Zaragoza were used, obtaining the average energy density per hour (MED) using two analytical methods and the Weibull probability distribution function (WPDF). The instantaneous power curve, the histogram of the occurrence of the wind and the wind rose that shows the predominant direction of the wind and its trend of change were also presented. In conclusion, the developed application provides the necessary information to evaluate the technical feasibility of installing horizontal axis wind turbines in the analyzed region.

Chloride ion transport and service life prediction of aeolian sand concrete under dry–wet cycles

Abstract In this study, the dry–wet cycle test of chloride salt was carried out on aeolian sand concrete with different contents, and the chloride ion content in aeolian sand concrete was determined by taking powder from different depths on one side of the test block. And then combined with Monte Carlo stochastic statistical simulation and Weibull probability distribution function, the service life prediction model of aeolian sand concrete against chloride ion erosion is established. The results show that the free chloride ion content in aeolian sand concrete decreases with the increase in the depth from the surface of the specimen. At the same depth from the surface of the specimen, the free chloride ion content gradually increases with the increase in the number of dry–wet cycles. Through the analysis of life prediction, it is concluded that with the increase in aeolian sand content, the service life of aeolian sand concrete increases first and then decreases. The service life value of concrete with 75% aeolian sand content is the largest, and the greater the thickness of the protective layer, the more favorable the service life value.

Machine Learning Classification and Prediction of Wind Estimation Using Artificial Intelligence Techniques and Normal PDF

Estimating wind energy at a specific wind site depends on how well the real wind data in that area can be represented using an appropriate distribution function. In fact, wind sites differ in the extent to which their wind data can be represented from one region to another, despite the widespread use of the Weibull function in representing the wind speed in various wind locations in the world. In this study, a new probability distribution model (normal PDF) was tested to implement wind speed at several wind locations in Jordan. The results show high compatibility between this model and the wind resources in Jordan. Therefore, this model was used to estimate the values of the wind energy and the extracted energy of wind turbines compared to those obtained by the Weibull PDF. Several artificial intelligence techniques were used (GA, BFOA, SA, and a neuro-fuzzy method) to estimate and predict the parameters of both the normal and Weibull PDFs that were reflected in conjunction with the actual observed data of wind probabilities. Afterward, the goodness of fit was decided with the aid of two performance indicators (RMSE and MAE). Surprisingly, in this study, the normal probability distribution function (PDF) outstripped the Weibull PDF, and interestingly, BFOA and SA were the most accurate methods. In the last stage, machine learning was used to classify and predict the error level between the actual probability and the estimated probability based on the trained and tested data of the PDF parameters. The proposed novel methodology aims to predict the most accurate parameters, as the subsequent energy calculation phases of wind depend on the proper selection of these parameters. Hence, 24 classifier algorithms were used in this study. The medium tree classifier shows the best performance from the accuracy and training time points of view, while the ensemble-boosted trees classifier shows poor performance regarding providing correct predictions.

Evaluation of Wind Energy Potential in Omu Aran, Nigeria Using Weibull and Rayleigh Models

The depletion of resources and emission of hazardous gases have been identified with conventional sources of energy. The negative influence of conventional sources of energy on the environment necessitates the call for the use of renewable and sustainable energy sources, such as wind. Wind power is one of the available renewable energy sources in Nigeria with huge potential that can be tapped in order to contribute to its energy mix. Wind energy utilization in Nigeria is poor because the available data in all six geopolitical political regions for system design have not been fully analysed and implemented. Wind energy projects are liable to failure if proper analysis is not done. Therefore, before any location could be considered suitable or unsuitable for wind power generation, the power density must be determined using the standard approach. This study, therefore evaluated the wind energy potential of Omu Aran, Nigeria using Weibull and Rayleigh models. Five years data collected from the metrological station of the Landmark University on Lat. 8.14 oN; Long. 5.10 oE were processed and analysed in Matlab computer software using a code developed for two statistical modelling methods (Weibull and Rayleigh). The actual mean yearly wind speed of 3.964 m/s for Kwara falls in the low wind speed. Although, the power density for hours of the day, months, and seasonal variation ranged from 24 to 141 W/m2. More than 50% of the power density for daily hours was less than 100 W/m2 which indicated that Omu Aran, Nigeria belongs to class 1. The coefficient of efficiency (COE) for Weibull probability distribution ranged from 39.95 to 94.9 while the coefficient of determination (COD) R2 ranged from 0.66 to 0.98. This range of performance values for the Weibull model, when compared to the Rayleigh model, were within the acceptable limits for prediction accuracy, hence the Weibull probability distribution function can be used for the preliminary design of wind power plants for Kwara State, Nigeria. Therefore, it would help the relevant stakeholders in wind power project investment to make the appropriate decision.

Optimal Sizing and Sitting of Distributed Generation in Distribution Network considering Power Generation Uncertainty

This paper presents an application of the recent metaheuristic Geometric Mean Optimizer (GMO) for the allocation of renewable energy sources (RES), including wind turbine (WT) and biomass-based Distributed Generation (DG) units in the distribution network (DN). The primary objective function is to minimize the total power and energy losses. The Weibull probability distribution function (PDF) is employed to describe the uncertainty of wind speed. The high penetration of RES with intermittent availability and demand variations has introduced many challenges to DN, such as power fluctuations, voltage rise, high losses, and low voltage stability. Therefore, the use of dispatchable biomass is considered to smooth out supply fluctuations and maintain supply continuity. A standard IEEE 69-bus test system is used to verify the performance of the proposed approach. The simulation results and comparison with other techniques demonstrate the significant energy loss reduction achieved by the proposed technique.

Accurate and efficient forecasted wind energy using selected temporal metrological variables and wind direction

The aim of this work is to find the most efficient and suitable input features to be selected for forecasting monthly wind energy accurately. Machine learning is employed for a modular pipelined neural network, composed of time-delayed and feedforward networks with features of metrological variables such as atmospheric temperature, humidity, wind direction, and wind speed frequency distribution parameters. Logged data over a year’s period at a UAE site are analyzed on daily and monthly bases depending on their variation characteristics, in which standard Weibull probability distribution function is used for the feedforward neural network together with wind direction data, while daily average ambient temperature and humidity are attempted for the composite time delay networks. Different network abstractions of input features are compared, and it is found that wind direction data offer a better wind speed forecast. Wind energy is calculated based on monthly forecasting. A detailed adaptive probabilistic analysis is conducted to predict thresholds in variations of the forecast analysis. Error estimation tools are performed for adopting this method

The Mixture of Probability Distribution Functions for Wind and Photovoltaic Power Systems Using a Metaheuristic Method

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.

Wind-wave characteristics and extremes along the Emilia-Romagna coast

Abstract. This study examines the wind-wave characteristics along the Emilia-Romagna coasts (northern Adriatic Sea, Italy) with a 10-year wave simulation for the period 2010–2019 performed with the high-resolution unstructured-grid WAVEWATCH III (WW3) coastal wave model. The wave parameters (significant wave height, mean and peak wave period, and wave direction) were validated with the in situ measurements at a coastal station, Cesenatico. In the coastal belt, the annual mean wave heights varied from 0.2–0.4 m, and the seasonal mean was highest for the winter period (>0.4 m). The Emilia-Romagna coastal belt was characterized by wave and spectra seasonal signals with two dominant frequencies of the order of 10 and 5–6 s for autumn and winter and 7–9 and 4 s for spring and summer. The wavelet power spectra of significant wave height for 10 years show considerable variability, having monthly and seasonal periods. This validated and calibrated data set enabled us to study the probability distributions of the significant wave height along the coasts and define a hazard index based on a fitted Weibull probability distribution function.