Weibull Probability Distribution Research Articles

Freeze–thaw damage analysis and life prediction of modified pervious concrete based on Weibull distribution

An assessment of the freeze–thaw durability of pervious concrete was conducted using a rapid freezing method to investigate the durability of modified pervious concrete within a freeze–thaw environment. The optimal proportion of the modified material was determined by evaluating the mass-loss rate and the relative dynamic elastic modulus as indicators of freeze–thaw performance. Scanning electron microscopy (SEM) was employed to observe microcrack propagation and reaction products. A freeze–thaw damage model, based on the Weibull probability distribution, was formulated. Freeze–thaw damage analysis and life prediction assessment were carried out using curves such as freeze–thaw damage velocity curves, freeze–thaw damage acceleration curves, failure rate curves, and reliability life curves. The outcomes revealed that the inclusion of polyacrylonitrile (PAN) and silica ash had a favourable impact on improving the frost resistance of pervious concrete at later stages. The optimal proportions for PAN fibre and silica ash were determined as 0.6 % and 8 %, respectively. As the number of freeze–thaw cycles approached approximately 52.03 times, the freeze–thaw cycle velocity peaked. Subsequently, the freeze–thaw damage velocity was categorised into three stages based on the freeze–thaw damage acceleration curve. The degradation of pervious concrete due to freeze–thaw cycles followed a typical loss-type failure pattern. Notably, a 50 % increase in PAN fibre content resulted in a 62.92 % reduction in failure rate. Furthermore, a predictive analysis was conducted to estimate the lifespan of pervious concrete in a representative freeze–thaw region in China was performed.

Wind energy feasibility and wind turbine selection studies for the city Surat, India

Abstract 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.

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.

Simplified Approach for Wind Uncertainty Cost Functions using a Mixture of Uniform Probability Distribution

Wind energy generation is an individual source of energy considered in renewable energy resources. It is completely dependent on the natural process of air blown in the area or past forecasting of air blown data. It is considered an uncertain source of energy and less reliable in the context of real-time energy provisions. To make this wind energy more reliable and consistent, we are introducing a simplified approach for wind uncertainty cost functions based on uniform probability distributions and its evaluation for usage. We have developed analytical mathematics cost functions based on several assumptions and they are constructed using Weibull and Rayleigh probability distributions. The proposed method produces the required results to make this energy reliable and consistent with the distribution networks. Computing time and complexity are less as compared to the other methods to minimize the uncertainty of wind energy.

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.

Weibull Technique for Evaluation of Swelling: Composite Graphite Resin Electrode for Electrochemical Treatment of Gold Mining Wastewaters

This paper evaluated the swelling of graphite resin electrodes developed for utilization in the electrochemical treatment of gold mining wastewater. Graphite-resin electrodes were developed from used dry cells and resin using non-heat treatment processes (segregation). The Microstructure of the electrode was determined using a scanning electron microscope (Carl Zeiss Smart Evo 10) to ascertain the composition of the electrode. The swelling property of the electrodes was measured using the standard method through a combination of gold mining wastewater and chloride salt solutions. Effects of operational factors (particle size, percentage binder and compressive “compacting” pressure) on the swelling of the electrodes were monitored and evaluated statistically (using analysis of variance). Weibull probability distribution (2 and 3 parameters) was applied to the swelling through Microsoft Excel Solver and Moment Likelihood Method to ascertain the usefulness of the electrode in environmental pollution control through computation of reliability. The study revealed that the swelling was in the range of 1.48 % to 2.24 %, particle size (F5,20 =196.48, p = 2.76 x 10-16), percentage binder (F4,12 =181.58, p = 1.27 x 10-10), and compressive pressure (F3,12 = 106.69, p = 6.43 x 10-9) were significant factors that influence swelling of graphite-resin electrode at 95 % confidence level. the values of α and β for 2-parameters Weibull distribution are 63.162 and 15.098, and 1.265 and 10.089 for MSE and MLM methods, respectively. The Table shows that the values of α, β and θ for 3-parameters Weibull distribution are 3.679, 8.097 and 0.168, and 4.350, 7.165 and 0.198 for MSE and MLM methods, respectively. It was concluded that particle size and compacting pressure are significant factors that had an effect on the swelling of graphite resin electrodes for treatment water and wastewater.

Impact of Operational Configuration Onboard a Drillship on Main Generator Engine Subcomponents Interval and Maintenance Strategies

Maintenance intervals and approaches for marine equipment have been the subject of numerous studies, focusing on previous failure patterns and associated risks. However, in vessels like drillships, operating under high reliability requirements and possessing significant redundancy, maintenance strategies have a critical impact on overall performance. This study investigates the influence of operational configurations on maintenance intervals and approaches for generator engine subcomponents onboard a drillship. Using data on injector failures onboard drillships, we employ the maintenance concept adjustment and design (MA-CAD) method to analyse and determine maintenance intervals across various operational configurations. Initially, we analyse the failure pattern of fuel injectors and propose a reduction in maintenance intervals through Weibull probability distribution fitting. Additionally, a risk analysis is conducted to assess the impact of operational configurations on risk indices related to fuel injector reliability and safety consequences. Our findings reveal that different operational configurations of the power generation plant can lead to changes in risk indices, shifting injector reliability status from unacceptable to undesirable limits. Consequently, maintenance intervals need to be adjusted as well to ensure optimal performance and safety. Furthermore, considering various engine subcomponents and their reliability under different operational configurations suggests the need for tailored maintenance approaches. This research provides insights into optimising maintenance strategies for drillship, ensuring reliability and safety across diverse operational scenarios.

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.

Effect of PVA fibers on durability of nano-SiO2-reinforced cement-based composites subjected to wet-thermal and chloride salt-coupled environment

Abstract Marine engineering structures are often faced with complex environmental factors. It is the focus of current research to modify cement-based composites (CBCs) to achieve their high durability in complex environments such as seawater. In this study, the effect of polyvinyl alcohol (PVA) fibers on durability of nano-SiO2 (NS)-reinforced cement-based composites was investigated by simulating seawater environment and taking PVA fiber content as variable. In addition, based on the Weibull probability distribution model, the damage degree of NS and PVA fiber-reinforced cement-based composites (NFRCCs) subjected to wet-thermal and chloride salt-coupled environment (WTCSE) after 300 freeze–thawing cycles (FTCs) was predicted. The test results demonstrated that the NFRCC exhibited the most excellent durability subjected to WTCSE when the content of PVA fibers was 1.2%. Compared with the reference group only doped with NS subjected to WTCSE, its impermeability pressure increased by 150%, the chloride ion electric flux decreased by 31.71%, the compressive strength loss rate decreased by 19.00% after 125 FTC, and the compressive strength corrosion resistance coefficient of chloride salt erosion increased by 9.15% after 25 wetting–drying cycles. The predicted results of the Weibull probability distribution model indicated that the damage degree of NFRCC subjected to WTCSE after 300 FTC would not exceed 0.35. The microscopic test analysis showed that the incorporation of PVA fibers reduced the proportion of large pores and the overall porosity of NFRCC subjected to WTCSE. PVA fibers bridged microcracks while adsorbing NS and its hydration products, thus enhancing the adhesion of the substrate. This study provides a reference for the research of high-performance CBC in complex environment.

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.

A novel approach to the investigation and quantification of the stop/start process for pedestrian traffic using motion capture devices

Characterising the stop/start walking process of individuals in a crowded and congested space is an important consideration in modelling pedestrian movement. However, the reaction of pedestrians to speed changes, especially to the person in front, has not been fully characterised nor quantified for full adoption in computer models. This study, therefore, explored the different phases of the stop/start process through a series of novel experiments conducted at University College Dublin (UCD) in which individual movements were captured precisely using motion capture equipment. The overall aim of the study was to develop a novel methodology (inspired from vehicle traffic flow) to break down and quantify the components of the stop/start walking process, i.e., the perception-reaction time, slow-down time, and start-up time of individuals walking in a single-file. These times, together with the total stopping distance and inter-person distances of each follower to their leader at the beginning and the end of each phase were quantified successfully and their inter-relationships were explored. The results showed the mean perception-reaction time, slow-down time, and start-up time delay were 0.48, 0.58, and 0.39 s, respectively. Where applicable, normal, lognormal, Weibull, gamma, and log-logistic probability distributions were fitted to the data to determine the best fit. The novel methodology developed in this study can be used in the future to investigate pedestrians’ behaviour in response to any changes in leaders’ speed, i.e. quantify the reaction of individuals in different phases. The results of this study can inform the representation of the stop/start process in microscopic pedestrian models.

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.

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.

Mechanical Characteristics of Structural Thick Plates Depending on Vacuum Time of RH Process

The structural thick plates were manufactured by varying the vacuum time at a constant argon gas flow rate and degree of vacuum during the Ruhrstahl Heraeus (RH) refining process. The inclusion and mechanical properties of the structural thick-plate specimens were evaluated according to vacuum time. The nitrogen in the converter had an effect after the RH process, and as the vacuum time of the RH process increased, the nitrogen content also increased. The nitrogen in the continuous casting (CC) decreased more than that in the converter, and as the vacuum time increased, the hydrogen decreased. Al2O3·CaO, MnS and Al2O3·MgO inclusions were observed in the structural thick plate. The average size of the oxides was (6.1 to 33.46) μm, while the average size of the Al-O-based inclusions was (4.26 - 6.3) μm. The mechanical properties, such as tensile strength, yield strength, and elongation, were affected by the vacuum time of the RH process with 10 min being the best. The micro-Vickers hardness showed dispersion regardless of the vacuum time of the RH process, which can be explained by the Weibull probability distribution. From the shape and scale parameters, although the specimen with the vacuum time of the RH process of 10 min showed a large dispersion in the micro-Vickers hardness, the characteristic value of 63.2 % was the best.

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.