A numerical algorithm for the computation of the noncentral beta distribution function

Prediction interval of wind power using parameter optimized Beta distribution based LSTM model

The energy demand of electric vehicles (EVs) is expected to take a significant share in the electricity market in the future. As the charging power of EVs is much higher than any other individual household appliances, the power demand modeling of EVs needs to be analyzed. In this study, the charging profile of EVs is modeled by combining two probabilistic functions together: state of charge (SOC) before charging and start charging time (SCT). Analysis study shows that beta distribution function is selected as the more appropriate approach to represent the initial SOC of an EV. Furthermore, two charging scenarios are considered to model SCTs. Finally, demand profiles are worked out and compared between different scenarios, and among different penetration levels of EVs. The structure of this paper is presented as below. Two charging scenarios (methods) are introduced firstly. Next, fitness analysis of Beta distribution is performed. Thirdly, the SOC of EVs before charging is modeled. Fourthly, the SCT of EVs for each charging method is modeled. Fifthly, the power demand of an EV is modeled. Finally, the calculated new typical load profiles with EVs' charging are presented.

With the speed growth of financial technology (Fintech), modern electronic marketing has typically deployed the use of the World Wide Web. This has come with great challenges especially in decision making and in engaging the pre-tail for launching new products and services in an open environment susceptible to high risks and threats. A prodigious need to build a sellers reputation and trust between the seller and the buyer so as to diminish such risks and threats in online trading birthed the idea of reputation systems. The emergence of reputation systems has attracted a lot of researchers to propose rating aggregation methods such as simple mean and normal distribution based method. However, the existing methods cannot accurately produce reputation score in some cases. Hence, this paper proposes a new model aiming to producing even more accurate and effective reputation score. Our model uses the standard beta-distribution considering the received rating distribution, so as to generate the weights of each ratings and then derive the level weights of ratings. The final reputation score is the level weighted aggregation of the rating levels. The proposed model is innovative in the aspect that the ratings are not directly aggregated to the reputation score, but are treated as the samples in evaluating each respective rating levels. Through case studies, the model is demonstrated to achieve desired accuracy and effectiveness, and even performs better than the existing models.

This paper introduces generalised incomplete beta functions defined by the generalised beta function. Firstly, we provide some of the generalised beta function's basic properties, such as integral representations, summation formulas, Mellin transform, and beta distribution. We then present several fundamental properties, such as integral representations, summation formulas, and recurrence relations with the help of the generalised incomplete beta functions.

In this paper, the modeling of the liberation of scheelite is presented. A pattern of concentration experiments was performed to investigate the scheelite liberation and distribution density calculation procedure. In this work, one sample from a Mittersill tungsten ore was studied. This work describes a method for determining the downstream milling energy requirements for rod mill products based on a Bond mill test performance. The grade distribution of particles at a given size fraction was calculated using a predictive liberation model. The concentration behavior of these particles in size fractions was evaluated using batch concentrate tests. The recovery of particles in size/grade classes, image analysis using mineral liberation analysis (MLA), and function calculations were implemented for the modeling of the liberation. By describing the size, grade, and recovery data of particles in size/grade classes, a technique for the measurement of distribution functions was developed that relates beta distribution, a model for the function based on the incomplete beta function, and a solution to produce liberation modeling. It was shown that the predicted results agreed well with the observed results. With a procedure for measuring the liberation, it was possible to carry out the first experimental measurement of the beta distribution. This liberation/concentrate model has wide potential applications for metallurgy and plant design, where the liberation modeling is to be determined with the distribution density solution to the predictive mineral liberation function equation, which includes the liberation of ore samples and their liberation characteristics.

The beta distribution is seen as a suitable model in risk analysis because it provides a wide variety of distributional shapes over a finite interval. Unfortunately, the beta distribution is not easily understood and its parameters are not easily estimated. This paper explores the possibility of using the more intuitively obvious triangular distribution as a proxy for the beta distribution. It is shown that the differences between the two distribution functions are seldom significant and a procedure is proposed for estimating the parameters of a triangular distribution on the basis of two extreme percentiles and the median

Techno-economic energy management of micro-grid in the presence of distributed generation sources based on demand response programs

This paper presents measurements of the plasma turbulence with an emphasis on the ion temperature fluctuations in comparison to the electron temperature and the ion saturation current fluctuations in the edge of the COMPASS tokamak. The ion temperature values are obtained by using a fast swept ball-pen probe, with temporal resolution up to 10 µs, accompanied by independent 1 µs measurements of the electron temperature and ion saturation current. The experimental results obtained for a wide range of line-averaged densities show that statistical properties of the fluctuations are in good agreement with a form of beta or gamma distribution function. In addition, the dependence of the averaged ion and electron temperatures on the line-averaged density is studied in the near and main scrape off layer region.

This paper presents an optimization approach based on mixed-integer programming (MIP) to maximize the profit of the Microgrid (MG) while minimizing the risk in profit (RIP) in the presence of demand response program (DRP). RIP is defined as the risk of gaining less profit from the desired profit values. The uncertainties associated with the RESs and loads are modeled using normal, Beta, and Weibull distribution functions. The simulation studies are performed in GAMS and MATLAB for 5 random days of a year. Although DRP increases the total profit of the MG, it can also increase the risk. The simulation results show that RIP is reduced when downside risk constraint (DRC) is considered along with DRP implementation. Considering DRC significantly reduces the percentage of the risk while slightly decreasinz the profit.

This research takes on a crucial task- exploring the optimal placement of Renewable Distributed Generators such as Solar Photovoltaic, wind turbines and Electric Vehicles into the Radial Distribution System (RDS). This is a strategic move aimed at minimising power loss (PLoss) and improving the voltage profile and stability index. The RDGs are integrated into RDS with and without considering the uncertainty of the different load demands for 24 h. The probability function of Beta and Weibull distribution functions are employed to attain the solar irradiance and wind speed in a particular region. In addition, EVs are also integrated into RDS, employing meta-heuristic algorithms intended to reduce power loss (PLoss) and improve the voltage profile. The study uses an Indian 28-bus test system mimicking a balanced radial distribution network to integrate distributed generators (DGs) and EV charging stations. The simulated results demonstrate that integrating DGs into power systems has offered considerable benefits, including reduced PLoss, heightened efficiency, decreased dependency on centralised generation, and improved environmental sustainability. It is discovered that the Multi-population Evolution Whale Optimization Algorithm (MEWOA) produces better results than other methods in the literature and is valuable and practical for handling these nonlinear optimisation situations.

A fair electricity market strategy for energy management and reliability enhancement of islanded multi-microgrids

In this study, we undertake a systematic study of historic market volatility spanning roughly five preceding decades. We focus specifically on the time series of the realized volatility (RV) of the S&P500 index and its distribution function. As expected, the largest values of RV coincide with the largest economic upheavals of the period: Savings and Loan Crisis, Tech Bubble, Financial Crisis and Covid Pandemic. We address the question of whether these values belong to one of the three categories: Black Swans (BS), that is, they lie on scale-free, power-law tails of the distribution; Dragon Kings (DK), defined as statistically significant upward deviations from BS; or Negative Dragons Kings (nDK), defined as statistically significant downward deviations from BS. In analyzing the tails of the distribution with RV>40, we observe the appearance of “potential” DK, which eventually terminate in an abrupt plunge to nDK. This phenomenon becomes more pronounced with the increase in the number of days over which the average RV is calculated—here from daily, n=1, to “monthly”, n=21. We fit the entire distribution with a modified Generalized Beta (mGB) distribution function, which terminates at a finite value of the variable but exhibits a long power-law stretch prior to that, as well as a Generalized Beta Prime (GB2) distribution function, which has a power-law tail. We also fit the tails directly with a straight line on a log-log scale. In order to ascertain BS, DK or nDK behavior, all fits include their confidence intervals and p-values are evaluated for the data points to check whether they can come from the respective distributions.

A simple recursive algorithm is provided for evaluating the noncentral beta distribution function. This algorithm basically reduces the problem to that of evaluating a single central beta distribution function. The algorithm is summarized in a step-by-step form. Since the noncentral F distribution function can be evaluated directly from the noncentral beta distribution function, the algorithm has broad practical use for evaluating the power of analysis of variance and other F tests. The method of this algorithm was used in Lenth. The latter article, however, focuses on a FORTRAN program and on an error bound for the procedure. It does not provide a full development of the procedure. The present article provides a complete development enabling one to readily produce a noncentral beta distribution function program in any desired language, with understanding for each step.

This paper describes evaluation procedures and experimental results for the estimation of Cumulative Distribution Functions (CDF) giving best-fit to the sample data in the Probability based risk Evaluation Techniques (PET) which is to assess the risks of a small-sized sea floater. The CDF in the PET is to provide the reference values of risk acceptance criteria which are to evaluate the risk level of the floater and, it can be estimated from sample data sets of motion response functions such as Roll, Pitch and Heave in the floater model. Using Maximum Likelihood Estimates and with the eight kinds of regulated distribution functions, the evaluation tests for the CDF having maximum likelihood to the sample data are carried out in this work. Throughout goodness-of-fit tests to the distribution functions, it is shown that the Beta distribution is best-fit to the Roll and Pitch sample data with smallest averaged probability errors of 0.024 and 0.022, respectively and, Gamma distribution is best-fit to the Heave sample data with smallest of 0.027. The proposed method in this paper can be expected to adopt in various application areas estimating best-fit distributions to the sample data.

Determining control points of a production system under a chance constraint