Suitable Probability Distribution Function Research Articles

Optimal Allocation of Hybrid Renewable Energy System in Distribution Network using Arithmetic Optimization Algorithm

The main focus of this article is the optimal allocation of wind turbines, solar PV and storage in 33 bus radial distribution system. The arithmetic optimization algorithm (AOA) was employed in this work for the optimal sizing of wind turbine and solar PV units with batteries in 33 bus distribution networks. The 1 year period is divided into multiple time segments, and each time segment is evaluated independently. For each time slot solar irradiance and wind speed are generated using suitable probability distribution function. Sensitivity analysis was done to find potential buses that might be placed to reduce computation time and search space. The Backward-Forward sweep technique was used to conduct load flow analysis. For the sake of stability, a reasonable penetration level is selected. The overall energy loss is minimized by the AOA optimization method under equality and inequality constraints. The suggested technique was tested on 33 buses, and it was observed that correct sizing and placement of DG units results in a significant decrease in losses with improved voltage profile.

Novel Distance and Similarity Measures for Probabilistic Hesitant Fuzzy Set and Its Applications in Stock Selection Problems

The consensus of randomness and ambiguity exists in real world problems. To depict fuzziness, randomness and statistical ambiguity in a single framework, we develop an interactive approach to MCDM method, in which assessment of alternative over attributes are provided by probabilistic hesitant fuzzy elements (PHFEs). This method provides a tool to the decision makers for reasonable ranking of alternatives. The core intention of this paper is to define a series of novel distance and similarity measures and score function for PHFEs. To demonstrate the effectiveness of developed model, a real case study is taken as an example. To completely describe statistical and non-statistical uncertainty, suitable probability distribution function is associated with each element of constructed HFSs. The proposed method is more superior to other MCDM methods, because of introducing probabilistic information in HFEs, which can be helpful to ensure the integrality and accurateness of hesitant fuzzy information.

Reliability Evaluation of Nondispatchable Energy Sources in Generation Planning: A Wind Electrical System Case Study

Summary. Generation planning is an important aspect of the probabilistic production simulation (PPS) process for power systems. The inclusion of nondispatchable energy sources in the conventional system needs efficient algorithms for reliability evaluation. Moreover, it is crucial to represent the availability of nondispatchable energy sources with a suitable probability distribution function (PDF) that can be easily integrated with the conventional system. The present work extends the generation reliability to include nondispatchable energy sources in the reliability evaluation methodology. The wind electrical system (WS) is represented using a simple wind speed model based on the normal distribution function. The multistate wind speed model based on the mean and standard deviation of wind speed at a particular site is integrated with the conventional system for reliability study. The proposed fast Fourier transform (FFT)-based method eliminates extensive calculations encountered in the conventional approach due to cumbersome time-domain convolution. The efficacy of the methodology is validated with the case studies using the IEEE RTS-wind system with loss of load probability (LOLP) and expected energy not served (EENS) reliability indices.

Analysis of Wind Speed Characteristics Using Probability Distribution in Johor

Renewable energy and energy efficiency are the key factors to ensure a safe, reliable, affordable as well as sustainable energy system for a better future. One of the most congruous, environment-friendly, and renewable energy sources is wind energy. However, it is consequential to examine the suitable probability distribution function to study the wind speed characteristics before the element can be harnessed as a source of energy. In this study, five probability distributions, Gamma, Generalized Extreme Value (GEV), Lognormal, Rayleigh and Weibull distribution were selected to model the wind speed data from four wind stations in Johor in a ten-year period. In addition, the method of maximum likelihood estimation (MLE) was applied to obtain the parameter estimation for each selected distribution function, followed by the plotting the graphical representation of probability distribution function (PDF) and cumulative distribution function (CDF) for the theoretical distributions against the provided wind speed data. To determine the best-fitted model of the probability distribution, the Kolmogorov Smirnov (KS) test and Anderson Darling (AD) test were employed to assess the goodness-of-fit for each model distribution. Based on the plotted graph and calculated goodness-of-fit results, GEV distribution was found to be the best-fitted model for the wind speed dataset in Senai, Mersing, and Batu Pahat wind station, while Gamma distribution established the optimum model for the actual wind speed dataset in Kluang station.

Impact of Probabilistic Modelling of Wind Speed on Power System Voltage Profile and Voltage Stability Analysis

Wind speed characteristics may have a significant impact on the power system operation, stability and dynamics. Hence, appropriate modelling and quantification of wind probability distribution function (PDF) are critical for probabilistic power system analysis. Weibull distribution is the most widely used PDF for power system (particularly power system stability) studies, which might not be a valid PDF for different wind regimes having high, average, and low wind speed data. In this study, twenty different wind speed datasets (each dataset is different based on the location, duration, and resolution) have been collected and analysed with eight PDFs to model the wind speed to replicate different wind speed characteristics. The PDFs include Rayleigh, Weibull, Gamma, Lognormal, Normal, Inverse Gaussian, Generalized Extreme Value, and Exponential distributions. Root mean square error (RMSE) and the coefficient of determination (R2) are applied as the measure of accuracy. The obtained results show that the locations, duration, and resolution have a notable impact on the selection of suitable PDFs. It was found that the Gamma distribution is the best-suited PDF for representing low wind speed data. In comparison, the Weibull distribution represents the best PDF for high wind speed data. Also, the selection of the Gamma PDF for low wind speed data and Weibull distribution for high wind speed data is further validated by presenting voltage profile and voltage stability analysis. Study outcomes would assist power system planners to select the appropriate PDF for power system stability studies based on the characteristics of the available wind speed dataset.

Evaluation of Probability Distribution Functions for Modeling Forest Tree Diameters on Agricultural Landscapes in Ogun State, Nigeria

The use of probability distribution functions for describing tree diameter at breast height provides useful information for forest resource evaluation and quantification. A series of probability distribution functions have been widely developed and applied for managing forest trees in conventional forest reserves without much consideration for trees outside forest reserves. The aim of this study is to evaluate and propose a suitable probability distribution function for trees in Agricultural landscapes. The study examined 3-parameter lognormal, Lognormal, 3-parameter Gamma, Gamma, 3-parameter Weibull and Weibull distribution functions, using the Maximum Likelihood method for fitting tree diameter at breast height. Three hundred and thirty-two temporary farmlands were randomly selected from which stem diameter of all living trees, with diameter ≥ 10.0 cm, were measured. Results of the statistical analysis showed that the 3-parameter lognormal distribution gave a superior description of the stem diameter with the least values of Anderson Darling (1.627) and Akaike Information Criterion (5962.0) statistics. Hence, the 3-parameter lognormal distribution function was found suitable for the stem diameter of trees in Agricultural landscapes in the study area.

Spatial analysis of spring dust storms hazard in Iran

In this research, monthly data on the frequency of dust storms of 44 synoptic stations of Iran during a 50-year period from 1968 to 2017 and methods of Ward’s hierarchical cluster analysis, linear regression, Mann–Kendall test, and kriging geostatistic were used. After calculating the frequency of occurrence of dust storms and its related statistical characteristics, suitable probability distribution function was fitted with dust storm data and then, using cluster analysis, the classification of different regions of the country into homogeneous groups was done and by using the kriging method, the zoning of dust storms was performed. After examining the descriptive statistics of spring dust storms in Iran, it was found that the probability distribution of dust storm data occurrence is proportional to the probability distribution of the normal 3-parameter log normal. According to the results of cluster analysis, different regions of the country were clustered into five homogeneous regions comprising of (1) southeast regions, (2) southern and central regions, (3) western regions, (4) eastern and middle regions, and (5) the northern regions. In investigating the trend of time variations of dust storms, it was revealed that the frequency time series of dust storms in Iran had an increasing trend and these results were confirmed by Mann–Kendall statistics. After zoning, it was found that the main focus of this phenomenon was concentrated in the southeast and southern parts of the country.

Comparative analysis of probability distributions for the Standardized Precipitation Index and drought evolution in China during 1961–2015

As a representative index for monitoring and assessing drought, the Standardized Precipitation Index (SPI) relies on a suitable probability distribution function (PDF) to describe a precipitation series, which allows interregional comparisons following normalization. In this study, we considered nine PDFs (exponential (EXP), extreme value (EV), gamma (GAM), generalized Pareto (GP), logistic (LO), log-logistic (LOL), log-normal (LON), normal (NOR), and Weibull (WEI)) as candidates for use in SPI calculations. Based on monthly precipitation time series data (1961–2015) from 582 stations across China, together with the Kolmogorov–Smirnov (K–S) and Akaike Information Criterion (AIC) methods, differences in optimal PDFs for SPI calculations were compared comprehensively from the perspectives of timescale, record length, and index value. Based on the SPI calculated using the optimal PDF at the 6-month timescale (SPI6-opt), we analyzed the spatiotemporal characteristics of drought trends in China using the Mann–Kendall method. Results indicated both the timescale and the record length would affect the selection of the optimal PDF. The performance of the WEI and GAM distributions was superior to other distributions in describing monthly precipitation (especially for long precipitation records) at short and multiple timescales, respectively. During the entire study period, areas of China with high frequency of drought have transferred from the northwest (1960s), to the northeast (2000s), and to the southwest (most recent 5 years). Trend analysis revealed a noticeable wetting tendency confined mainly to Northwest, Northeast, and Southeast China and a significant trend toward drought in Southwest China and on the North China Plain.

Reconstruction of groundwater levels to impute missing values using singular and multichannel spectrum analysis: application to the Ardabil Plain, Iran

ABSTRACT Groundwater-level time series often have a substantial number of missing values which should be taken into consideration before using them for further analysis, particularly for numerical groundwater flow modelling applications. This study aims to comprehensively compare two data-driven models, singular spectrum analysis (SSA) and multichannel spectrum analysis (MSSA), to reconstruct groundwater-level time series and impute the missing values for 25 piezometric stations in Ardabil Plain, northwest Iran. The reconstructed groundwater-level time series are assessed against the complete observed groundwater time series, while the imputed values are appraised against the artificially created gap values. The results show that both SSA and MSSA demonstrate a solid competency in imputation and reconstruction of groundwater-level data. However, depending on the spatial correlation between the piezometers, and the most suitable probability distribution function (pdf) fitted to the time series of each piezometer, the performance may vary from piezometer to piezometer.

Annual Daily Maximum Rainfall-Based IDF Curve Derivation Methodology

Intensity–duration–frequency (IDF) curves are among the most demandable information in meteorology, hydrology and engineering water resources design, planning, operation, and management works. This paper suggests their generation based on annual daily maximum rainfall (ADMR) records from Turkey, Ceylanpinar region meteorology stations next to Syrian border. First, available storm rainfall records are used for dimensionless–intensity–duration (DID) curve development, then, the suitable theoretical probability distribution function (PDF) is obtained for the ADMR record series, which provides return periods’ identification for a given set of risk values, and finally, the ADMR data combination with the DID curve results in IDF curve set. The application of the methodology is presented for four stations in the Southeastern Province of Turkey, Ceylanpinar drainage basin, next to the Syrian border. The comparison of the newly developed IDF curves with the available measurement-based IDF curves indicated the validity of the proposed approaches with less than 10% error.

Wind speed forecasting based on Quantile Regression Minimal Gated Memory Network and Kernel Density Estimation

As a renewable and clean energy, wind energy plays an important role in easing the increasingly serious energy crisis. However, due to the strong volatility and randomness of wind speed, large-scale integration of wind energy is limited. Therefore, obtaining reliable high-quality wind speed prediction is of great importance for the planning and application of wind energy. The purpose of this study is to develop a hybrid model for short-term wind speed forecasting and quantifying its uncertainty. In this study, Minimal Gated Memory Network is proposed to reduce the training time without significantly decreasing the prediction accuracy. Furthermore, a new hybrid method combining Quantile Regression and Minimal Gated Memory Network is proposed to predict conditional quantile of wind speed. Afterwards, Kernel Density Estimation method is used to estimate wind speed probabilistic density function according to these conditional quantiles of wind speed. In order to make the model show better performance, Maximal Information Coefficient is used to select the feature variables while Genetic Algorithm is used to obtain optimal feature combinations. Finally, the performance of the proposed model is verified by seven state-of-the-art models through four cases in Inner Mongolia, China from five aspects: point prediction accuracy, interval prediction suitability, probability prediction comprehensive performance, forecast reliability and training time. The experimental results show that the proposed model is able to obtain point prediction results with high accuracy, suitable prediction interval and probability distribution function with strong reliability in a relatively short time on the prediction problems of wind speed.

Performance Probability Distribution Function for Modelling Solar Radiation in South Southern Nigeria: A Case Study of Yenagoa

This study has attempted to assess the performance of the most suitable statistical distribution function for modelling solar radiation over Yenagoa, Bayelsa State in Nigeria. The probability distribution functions are tested based on eleven years (2007-2017) solar radiation data obtained from National Aeronautics and Space Administration (NASA). Six probability distribution functions are tested to ascertain the most appropriate one based on four different statistical tools and fitting accuracy. The associated parameters of the most appropriate fitted probability distribution function are calculated and the trends in the characteristic of the solar radiation are deduced. The result shows that logistic distribution presents the most suitable probability distribution function for modelling solar radiation over the selected environment with RMSE of 1.500 KWh/m2/day, MAE of 1.260 KWh/m2/day, MAPE of 22.000% and R2 of 0.880.When compared with the other five distribution functions, the same trend could be seen although with different values of RMSE, MAE, MAPE and R2. The estimated distribution location and scale parameters of the model vary with month and season. The overall result will be useful for predicting future solar radiation over the studied environment. It will also be a good reference point for the design of large solar power projects in Yenagoa in particular and south southern Nigerian environments at large.

Stability analysis of support systems using a coupled FEM-DFN model (Case study: a diversion tunnel at Lorestan dam site, Iran)

Various structural discontinuities, which form a discrete fracture network, play a significant role in the failure conditions and stability of the rock masses around underground excavations. Several continuum numerical methods have been used to study the stability of underground excavations in jointed rock masses but only few of them can take into account the influence of the pre-existing natural fractures. In this work, the pre-existing fractures are explicitly modeled as a Discrete Fracture Network (DFN) model, which is fully coupled with the FEM modeling for stability analysis of support systems in a diversion tunnel at the Rudbar Lorestan dam site. Hence, at first, using the surveyed data in the diversion tunnel and an estimation of the suitable probability distribution function on geometric characteristics of the existing joint sets in this region, the 3D DFN model was simulated using the stochastic discrete fracture networks generator program, DFN-FRAC3D. In the second step, a coupled 2D Finite Element Method and the prepared stochastic model were used for analysis of existent (based on technical reports) recommended support systems. The objective here is to grasp the role of the fracture networks on the results of the tunnel stability analysis using FEM modeling and also to compare the results with those obtained through stability analysis without considering the effect of fractures.

Optimal Placement and Sizing of Renewable Distributed Generations and Capacitor Banks into Radial Distribution Systems

In recent years, renewable types of distributed generation in the distribution system have been much appreciated due to their enormous technical and environmental advantages. This paper proposes a methodology for optimal placement and sizing of renewable distributed generation(s) (i.e., wind, solar and biomass) and capacitor banks into a radial distribution system. The intermittency of wind speed and solar irradiance are handled with multi-state modeling using suitable probability distribution functions. The three objective functions, i.e., power loss reduction, voltage stability improvement, and voltage deviation minimization are optimized using advanced Pareto-front non-dominated sorting multi-objective particle swarm optimization method. First a set of non-dominated Pareto-front data are called from the algorithm. Later, a fuzzy decision technique is applied to extract the trade-off solution set. The effectiveness of the proposed methodology is tested on the standard IEEE 33 test system. The overall results reveal that combination of renewable distributed generations and capacitor banks are dominant in power loss reduction, voltage stability and voltage profile improvement.

Uncertainty of Hydrological Drought Characteristics with Copula Functions and Probability Distributions: A Case Study of Weihe River, China

This study investigates the sensitivity and uncertainty of hydrological droughts frequencies and severity in the Weihe Basin, China during 1960–2012, by using six commonly used univariate probability distributions and three Archimedean copulas to fit the marginal and joint distributions of drought characteristics. The Anderson-Darling method is used for testing the goodness-of-fit of the univariate model, and the Akaike information criterion (AIC) is applied to select the best distribution and copula functions. The results demonstrate that there is a very strong correlation between drought duration and drought severity in three stations. The drought return period varies depending on the selected marginal distributions and copula functions and, with an increase of the return period, the differences become larger. In addition, the estimated return periods (both co-occurrence and joint) from the best-fitted copulas are the closet to those from empirical distribution. Therefore, it is critical to select the appropriate marginal distribution and copula function to model the hydrological drought frequency and severity. The results of this study can not only help drought investigation to select a suitable probability distribution and copulas function, but are also useful for regional water resource management. However, a few limitations remain in this study, such as the assumption of stationary of runoff series.

An assessment of using subsampling method in selection of a flood frequency distribution

In flood frequency analysis, a suitable probability distribution function is required in order to establish the flood magnitude-return period relationship. Goodness of fit (GOF) techniques are often employed to select a suitable distribution function in this context. But they have been often criticized for their inability to discriminate between statistical distributions for the same application. This paper investigates the potential utility of subsampling, a resampling technique with the aid of a GOF test to select the best distribution for frequency analysis. The performance of the methodology is assessed by applying the methodology to observed and simulated annual maximum (AM) discharge data series. Several AM series of different record lengths are used as case studies to determine the performance. Numerical analyses are carried out to assess the performance in terms of sample size, subsample size and statistical properties inherent in the AM data series. The proposed methodology is also compared with the standard Anderson–Darling (AD) test. It is found that the methodology is suitable for a longer data series. A better performance is obtained when the subsample size is taken around half of the underlying data sample. The methodology has also outperformed the standard AD test in terms of effectively discriminating between distributions. Overall, all results point that the subsampling technique can be a promising tool in discriminating between distributions.

Optimal allocation of solar based distributed generators in distribution system using Bat algorithm

Summary With increased demand of electrical energy, limited availability of fossil fuels and environmental concerns, it is necessary to consider renewable energy based generation in a power system network. Optimal allocation of renewable based distributed generators in the distribution system is a challenging task in the recent years. In this paper an effective technique is proposed for optimal allocation of solar based distributed generators in the distribution network using a Bat algorithm (BA) is presented. The objective is to minimize power loss of radial distribution system. Different operating constraints related to the distribution network are considered. The stochastic nature of solar irradiance is modeled by using suitable probability distribution function (PDF). The proposed method is tested and validated on IEEE 33 bus test system.

Strategic approach for reinforcement of intermittent renewable energy sources and capacitor bank for sustainable electric power distribution system

To meet ever increasing load demand in a sustainable way, reinforcement of photovoltaic (PV) array, wind turbine (WT) and capacitor bank in distribution network is proposed in this paper. A comprehensive planning model is presented to determine location and required installation capacity of multiple PV array, WT and capacitor units in an electric power distribution network under heavy load growth situation. Intermittent power generation of renewable energy sources (RESs) are quantified with suitable probability distribution functions and incorporated in the planning model. The planning approach considers several welfare areas in the distribution systems, viz., increment of profit margin, reduction of carbon-di-oxide emission, minimization of distribution power losses, enhancement of voltage stability level and improvement of the network security considering power flow, voltage limit, line capacity, RES penetration, capacitor penetration and utility economy constraint. Non-dominated sorting based multi-objective particle swarm optimization algorithm along with fuzzy decision making criteria is used to find the best allocation alternative for mix RES and capacitor planning problem. The effectiveness of the proposed model has been tested on a typical 28-bus Indian rural distribution network. The results show that more efficient techno-eco-environmental optimization can be obtained from combined RES and capacitor planning model.

Optimal wind energy penetration in power systems: An approach based on spatial distribution of wind speed

Contributing in power system expansions, the present study establishes an efficient scheme for optimal integration of wind energy resources. The proposed approach highly concerns the spatial distribution of wind speed at different points of a wind farm. In mathematical statements, a suitable probability distribution function (PDF) is well-designed for representing such uncertainties. In such conditions, it is likely to have dissimilar output powers for individual and identical wind turbines. Thus, the overall aggregated PDF of a wind farm remarkably influences the critical parameters including the expected power and energy, capacity factor, and the reliability metrics such as loss of load expectation (LOLE) and expected energy not supplied (EENS). Furthermore, the proposed approach is deployed for optimal allocation of wind energy in bulk power systems. Hence, two typical test systems are numerically analyzed to interrogate the performance of the proposed approach. The conducted survey discloses an over/underestimation of harvestable wind energy in the case of overlooking spatial distributions. Thus, inaccurate amounts of wind farm’s capacity factor, output power, energy and reliability indices might be estimated. Meanwhile, the number of wind turbines may be misjudged to be installed. However, the proposed approach yields in a fair judgment regarding the overall performance of the wind farm. Consequently, a reliable penetration level of wind energy to the power system is assured. Extra discussions are provided to deeply assess the promising merits of the founded approach.

Seismic fragility assessment of existing sub-standard low strength reinforced concrete structures

An analytical seismic fragility assessment framework is presented for the existing low strength reinforced concrete structures more common in the building stock of the developing countries. For realistic modelling of such substandard structures, low strength concrete stress-strain and bond-slip capacity models are included in calibrating material models. Key capacity parameters are generated stochastically to produce building population and cyclic pushover analysis is carried out to capture inelastic behaviour. Secant period values are evaluated corresponding to each displacement step on the capacity curves and used as seismic demand. A modified capacity demand diagram method is adopted for the degrading structures, which is further used to evaluate peak ground acceleration from back analysis considering each point on the capacity curve as performance point. For developing fragility curves, the mean values of peak ground acceleration are evaluated corresponding to each performance point on the series of capacity curves. A suitable probability distribution function is adopted for the secant period scatter at different mean peak ground acceleration values and probability of exceedance of limit states is evaluated. A suitable regression function is used for developing fragility curves and regression coefficients are proposed for different confidence levels. Fragility curves are presented for a low rise pre-seismic code reinforced concrete structure typical of developing countries.