Probability Distribution Function Research Articles

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.

Stellar scintillation statistics and the impact of aperture averaging on space-to-ground optical communications.

Statistical probability distributions characterizing received optical power fluctuations, or scintillation, enable performance predictions of space-to-ground optical communication systems. In this paper, we present measurements of stellar scintillation over a wide range of elevation angles and turbulence conditions collected simultaneously with a 5 cm and 40 cm telescope aperture, which allows a comparison between minimal and significant aperture averaging conditions. The measured data is compared to a reasonable set of candidate probability distribution functions (PDFs), including lognormal, which is most often cited in the literature for weak to moderate scintillation. For scintillation indices (SIs) less than about 0.2, the Nakagami-m distribution provides the best representation of the collected data for both apertures and imposes a greater lasercom link penalty than a lognormal distribution, which has been inaccurately implemented as the default probability distribution in the literature. For larger values of the SI, the scintillation is best characterized by a Gamma-Gamma distribution. Additionally, the measured temporal covariance for weak to moderate scintillation conditions is found to be in reasonably good agreement with theoretical predictions.

A generalization of the Moore and Yang integral and interval probability density functions

ABSTRACT Based on an extension of Riemann sums, Moore and Yang have defined an integral notion for the context of continuous inclusion monotonic interval functions in which the limits of integration are real numbers. This integral notion generalizes the usual one for real-valued functions based on Riemann sums. In this paper we extend this approach by considering intervals as limits of integration and abolishing the inclusion monotonic restriction of the integrable interval functions. Also, such a new integration notion is used to define interval probability density functions and use it in interval probability distribution functions.

Identification of dynamic drought propagation from a nonstationary perspective and its application to drought warnings

Accurate drought warnings in changing environments remain challenging since the nonstationary behaviors of drought propagation. The novelty of this study is the presentation of a nonstationary framework for drought propagation analysis and the objective determination of drought propagation thresholds. The calculated propagation probabilities and thresholds are responsive to changing environments. The Generalized Addictive Models in Location, Scale and Shape (GAMLSS) and time-varying Clayton copula function were adopted to establish the nonstationary joint distribution between meteorological and hydrological drought indices. Several indices reflecting the climatic and anthropic impacts on hydrological process were selected as covariates. Combined with the conditional probability distribution function, the time-varying drought propagation probabilities can be estimated to describe the watershed drought tolerance variations. The receiver operator characteristic (ROC) curves were utilized to determine the propagation thresholds for drought warnings. The upper Yellow River Basin (UYRB) was selected as the study area. Results indicate that the nonstationary models show better fitting performance for two drought indices series. The Clayton copula function with dynamic parameters offers a more accurate description of the linkages between two types of drought. The analysis of watershed drought tolerance reveals a declining trend from June to October, contrasting with an increasing trend during other months. In comparison to the stationary approach, the hit rates of the proposed method improve by 0.18, reaching 0.88, while the false alarm rates decrease by 0.11, reaching 0.24. The criteria for propagation threshold determination in the UYRB is established as a conditional probability of 20%. These findings can provide a foundation for drought warning management in a changing environment.

Polar Sea Ice Detection Using a Rotating Fan Beam Scatterometer

Scatterometers are dedicated to monitoring sea surface wind vectors, but they also provide valuable data for polar applications. As a new type of scatterometer, the rotating fan beam scatterometer delivers a higher diversity of incidence angles and more azimuth sampling. The paper takes the first rotating fan beam scatterometer, the China France Oceanography Satellite scatterometer (CSCAT), as an example to explore the effectiveness of this new type of scatterometer in polar sea ice detection. In this paper, a Bayesian method with consideration of geometric characteristics of CSCAT is developed for sea ice detection. The implementation of this method includes the definition of CSCAT backscatter space, an estimation of the sea ice Physical Model Function (GMF), a calculation of the sea ice backscatter distance to the sea ice GMF, a probability distribution function (PDF) estimation of the square distance to the GMF (sea ice GMF and wind GMF), and a calculation of the sea ice Bayesian posterior probability. This algorithm was used to generate a daily CSCAT polar sea ice mask during the CSCAT mission period (2019–2022) (by setting a 55% threshold on the Bayesian posterior probability). The sea ice masks were validated against passive microwaves by quantitatively comparing the sea ice edges and extents. The validation suggests that the CSCAT sea ice edge and extent show good agreement with the sea ice concentration distribution (i.e., sea ice concentration ≥ 15%) of the Advanced Microwave Scanning Radiometer 2 (AMSR2). The average Euclidean distance of the sea ice edges was basically less than 12.5 km, and the deviation of the sea ice extents was less than 0.3 × 106 km2.

A high-throughput technique to evaluate the probability distribution of strength of adhesively bonded joints after moisture absorption

ABSTRACT The reliability of adhesive bond strength influences the applicability of structural adhesives in industries. The statistical probability distribution of the strength of adhesive joints is an essential indicator when designing and choosing adhesives. In this study, we experimentally studied the probability distribution of the strength of adhesive joints. The strength data were collected using a novel high-throughput technique consisting of a sample preparation method and shear testing device. Numerous cylindrical butt shear-joint specimens were prepared by mechanical machining and tested using a self-developed shear testing device. The effect of moisture absorption by the adhesive was specifically considered. The probability distribution of the shear strength of the epoxy adhesive joints was characterized using five probability distribution functions: Normal, Lognormal, Exponential, Weibull, and Gamma. Quantile – quantile plots were employed to determine the suitability of each distribution function. The results suggested that the Weibull and Normal distributions were best suited for describing the probability distribution of the strength of the epoxy adhesive joints. The Weibull distribution is particularly suitable for brittle epoxy adhesives. Moisture absorption reduced both the mean and variance of the shear strength, which might be attributed to the plasticization of the adhesives. The high-throughput method proposed in this study significantly improved the efficiency of testing adhesive joints. It not only contributes to the study of the strength distributions of adhesive joints but also helps to shorten the research and development cycle of new adhesives by facilitating rapid strength evaluation.

A modification in Weibull parameters to achieve a more accurate probability distribution function in fatigue applications

Risk evaluation for fatigue failure of the engineering components is an important aspect of the engineering design. Weibull distributions are often used in preference to the log-normal distribution to analyze probability aspects of fatigue results. This study presents a probabilistic model for calculating Weibull distribution parameters to reduce the effect of percentage discretization error of experimental fatigue life and R–S–N curves for three reliability levels. By considering any result of standard fatigue test as an equivalent Weibull distribution, artificial data are generated and the accuracy of common Weibull distribution model can be improved. The results show error reduction in the Kolmogorov–Smirnov test and R-square values. Also, the Basquin model is used for different reliability levels with the same error order for risk evaluation of fatigue failure. The coefficient of variation for fatigue life increases at higher stress levels and has a linear relation with stress level for a high-cycle fatigue regime.

Noise-induced bistability and noise-enhanced stability of a stochastic model for resource production–consumption under crowding effect and sigmoidal consumption pattern

A resource production–consumption model with crowding effect and sigmoidal consumption pattern, subject to cross-correlated multiplicative and additive noises, is proposed in this paper to investigate noise-induced phenomena such as noise-induced bistability and noise-enhanced stability. Firstly, the stationary probability distribution and effective potential function are obtained through an approximate Fokker–Planck equation. Subsequently, P-bifurcation and noise-induced bistability are thoroughly investigated. Finally, the mean first passage time is utilized to calculate the average time for the system escaping from its attraction domain, while further exploring noise-enhanced stability. Our findings demonstrate that noise can induce P-bifurcation, especially bistability, regardless of the presence of bistability in the deterministic counterpart. Furthermore, appropriate noise parameters can cause a shift from an outbreak state to a shortage or depletion state. Additionally, both types of noise enhance the stability of the outbreak state when there is negative cross-correlation; however, for positive cross-correlation, they accelerate the transition from an outbreak state to a shortage or depletion state. The results obtained may offer preliminary or supplementary insights for the formulation of resource management policies.

Numerical simulation methods for reinforced concrete columns considering parameter stochasticity

Reinforced concrete column specimens with the same parameters often show differences in bearing capacity and deformation under the same loading path. Most of the existing studies attribute the causes to recording instrumentation errors or randomness in material and bond dimensions, but the method to improve the accuracy of numerical simulation of the reinforced concrete components is still unclear. For a given member, the randomness of material and geometry is an important cause of simulation error, which should be taken into account to improve the simulation accuracy. A weighted least squares estimation (WLS) method is used to fuse the test samples of reinforced concrete member section sizes, reinforcement and concrete with confidence factors as weights. The probability distribution functions of reinforced concrete member section sizes, reinforcement and concrete are calculated by a genetic algorithm method. Based on the existing seismic performance test data of reinforced concrete columns, a stochastic finite element model of reinforced concrete columns is developed for seismic performance analysis using Monte Carlo method through a mixture of two software programs, MATLAB and OpenSees. The numerical seismic performance indexes of eight types of reinforced concrete columns were compared with the test results to verify the importance of introducing stochastic parameters into the model. The results demonstrate the simulation accuracy of the finite element model considering parameter randomness is 88% higher than that of the finite element model without considering parameter randomness, which proves the effectiveness of the numerical simulation method for reinforced concrete columns considering parameter randomness.

Dynamic phase transition of black holes in massive gravity

The dynamical properties of small-large black hole phase transition in dRGT non-linear massive gravity theory are studied based on the underlying free energy landscape. The free energy landscape is constructed by specifying the Gibbs free energy to every state, and the free energy profile is used to study the different black hole phases. The small-large black hole states are characterized by probability distribution functions and the kinetics of phase transition are described by the Fokker–Planck equation. Further, a detailed study of the first passage process is presented which describes the dynamics of phase transitions. Finally, we have investigated the effect of mass and topology on the dynamical properties of phase transitions of black holes in dRGT non-linear massive gravity theory.

An enhanced arithmetic optimization algorithm for optimal control of reactive power

AbstractIn this article, an enhanced arithmetic optimization algorithm (EAOA) is utilized to resolve the optimal reactive power dispatch problem (ORPD) of power plants, which is a non‐linear, non‐smooth, complex optimization problem. Typically, it is formulated as a constrained optimal power flow problem. This paper utilizes the gamma distribution to generate initial random solutions, which improves the accuracy of the arithmetic optimization algorithm (AOA). The original mode of deviation of math optimizer accelerated (MOA) is replaced with the improved pattern of change in the prey's energy in Harris Hawk Optimization (HHO), which creates a superior balance between the exploration and exploitation of AOA. This paper implements logarithmic and exponential operators in place of division and multiplication operators, which improves the exploration ability of AOA and also utilizes the gamma distribution in the generation of random numbers, which are essential for the selection of operators in both search phase which enlarges the search area and improves the global optimum solution. A parametric analysis is performed on 10 benchmark functions for nine possible values before selecting the algorithm's initial parameters. A comparative statistical analysis along with Friedman's test is carried out on 23 benchmark functions to test the overall performance as compared to Sine‐Cosine Algorithm (SCA), Swarm‐Salp Algorithm, Grey Wolf Optimization (GWO), and basic Arithmetic Optimization Algorithm (AOA). EAOA is applied to the IEEE 57 bus system for optimal reactive power control. EAOA reports a significant reduction in a power loss of 23.2 MW, which is 18.596% smaller than the system's base case power loss of 28.5 MW as compared to Artificial Bee Colony (ABC), Firefly Algorithm (FA), Bat‐inspired Algorithm (BAT), Cuckoo Search Algorithm (CSA), and Modified Ant Lion Algorithm (MALO) which reports a reduction in a power loss of 14.73%, 15.43%, 15.43%, 16.14%, and 16.14% respectively. EAOA reduces the TVD to 0.76 pu, compared to ABC, FA, BAT, CSA, and MALO, which minimizes the TVD to 0.84, 0.82, 0.88, 0.82, and 0.79, respectively. This proves that EAOA performs better in resolving ORCP problems regarding the quality of solutions and convergence. A detailed comparative statistical analysis of EAOA with CSA, BAT, and WOA (Whale Optimization Algorithm) is conducted in terms of Probability Distribution Functions (PDF) and Cumulative Distribution Functions (CDF), along with the Wilcoxson sign rank test, which proves that EAOA gives the solution with a higher degree of precision.

Investigating the magnetic field of the quiet Sun internetwork

Abstract We analyze the magnetism of the quiet Sun internetwork (IN) using high-spatial-resolution data obtained by the spectropolarimeter (SP) of the Solar Optical Telescope aboard the Hinode satellite near the disk center of the Sun. The SP data were inverted using the Stokes Inversion based on Response functions (SIR) inversion code with a single-component atmosphere with depth dependent in the solar photosphere, assuming gradients in physical parameters along the line of sight (LOS). To avoid the effect of noise, only pixels with Stokes U and/or Q signals above 4.5 times the noise level are considered. The inversion results show that the magnetic field of the IN has mainly hG field strength and the inclination distribution is quasi-isotropic at the solar surface. The field strength decreases with height and becomes predominantly horizontal at the upper layers. At the mid photosphere, the distributions of field strength and inclination are consistent with those derived by Milne–Eddington inversion. The mean transverse and longitudinal flux densities are 66 Mx cm−2 and 13 Mx cm−2 at log τ = −1.0; we also study the ratio between the transverse and longitudinal components in the IN region as a function of depth in the photosphere, finding that the ratio is almost 2.7 in the deep layer, increasing to 5.1 in the upper layer. The mean field strength is greater than 100 G in the upper photosphere, which is consistent with the results based on the Hanle effect. We present the LOS velocity probability distribution function for IN at different optical depths, where its distribution is mainly associated with upflow velocities of VLOS = 1.2 and 0.6 km s−1 in the deeper and upper layers, respectively. In addition, there exists a reliable inversion analysis, which is obvious from the comparison between the observed and calculated area asymmetries in both Fe i lines of Hinode SP data.

Probability analysis and rainfall forecasting using ARIMA model

A 34-year rainfall data from 1976 to 2009 of ten sub-basins of the Vaigai River in Tamil Nadu were collected and analysed statistically using various probability distribution functions. The best-fit probability distributions for the annual, monthly and seasonal rainfall for the study area were found using two goodness-of-fit tests. The Box-Jenkins Autoregressive Integrated Moving Average (ARIMA) methodology has been adopted for model identification, diagnostic checking and forecasting the study area's annual rainfall. The best ARIMA models were selected for each sub-basin and the average annual precipitation for 2010, 2015, 2020 and 2025 has been forecasted. The forecasted result compared well with observed dataup to 2020, which indicates the appropriateness of the model.

Influence of welded pores on fatigue behavior of TC17 titanium alloy welded joints subjected to gigacycle regime at room and high temperatures

Welded structures in aero-engines commonly operate in high-temperature environments, making them susceptible to reduced fatigue life and premature failure due to welding defects within the structure. Thus, the gigacycle fatigue behaviors of titanium alloy welded joints at both room temperature (RT) and 400 °C were investigated, aiming to uncover the mechanism behind the formation of fine granular area (FGA) surrounding welded pores. The research findings demonstrate that the S-N curves of TC17 titanium alloy electron beam welded joint undergo a transition from a single linear decline at RT to a bilinear decline at 400 °C. However, the fatigue failure mode remains unaffected by temperatures, and crack initiation is attributed to welded pores. By utilizing the Chapetti model curve to modify the Kitagawa–Takahashi (K–T) diagram, the lower threshold stress amplitude is introduced, enabling the determination of a safe size for welded pores at 400 °C, which is calculated to be 11.3 µm. Additionally, the Gumbel probability distribution function is employed to assess the maximum size of welded pores. Finally, based on dislocation interactions, the formation mechanism of the FGA consisting of discontinuous nanograins with high-density dislocations is elucidated.

Elucidation of ‘Cosmic Coincidence’

In the standard cosmological model the dark energy (DE) and nonrelativistic (NR) matter densities are observationally determined to be comparable at the present time, in spite of their greatly different evolution histories. This ‘cosmic coincidence’ enigma – also referred to as the ‘why now? problem’ – relies, by its very definition, on the implicit prior expectation for our ‘typicality’ in the cosmic (expanding) spacetime volume. Otherwise, this conundrum does not exist in the first place. It is shown here that this apparent coincidence could be explained as a non-anthropic observational selection effect: for us to be typical observers in the comoving (static) spacetime volume, the cosmic energy budget must contain a non-vanishing DE component. In addition, it is shown that irrespective of the cosmological initial conditions and assuming no ‘new physics’, the Universe is most likely to be observed at a time when the conformal Hubble radius, H−1, attains a maximum. The latter takes place at the epoch when ρDE and ρm, the energy densities of DE and NR matter, respectively, are comparable. Specifically, our presumed ‘typicality’ along the conformal timeline, coupled to a few other plausible assumptions, implies that R≡ρDE/ρm is ‘sampled’ from a Beta Prime probability distribution function. A priori 68% (95%) confidence range for the ratio is 0.20<R<3.46 (0.033<R<17.20), with an expectation value of R̄=3.5. These are in agreement with the observationally inferred value, Robs=2.23.

A statistics view of contact pressure distribution for normal contact of fractal surfaces

Due to inherent nonlinear stiffness and damping characteristics, dry friction interfaces have a significant impact on the dynamics of jointed structures. When two surfaces are brought into purely normal contact, contact pressure distribution is of high concern. This work focused on the statistics of the contact pressure distribution between a rough surface and a smooth rigid plane, which provides new insight into the interface contact behaviour. First, the fractal rough surface is generated using the Weierstrass–Mandelbrot function with measured roughness parameters. With meshed rough surfaces, an elastic–plastic finite element contact analysis is performed to determine the contact pressure distribution. Then, the results of contact pressure are statistically analysed. The effects of roughness and contact load on contact area, contact stiffness and mean contact pressure are thoroughly investigated. The probability distribution of contact pressure is determined by fitting a continuous function using a twofold Weibull mixture model. The proposed probability distribution function is found to be capable of describing contact pressure. The contact pressure distribution is affected by the surface fractal characteristics and evolves with the contact load.

Peridynamic simulations of rock indentation

AbstractThis paper presents an ordinary state‐based peridynamic model to simulate failure in porous rocks and its application in the rock excavation processes. Rock indentation is a fundamental process for mechanical rock fracturing, which is frequently encountered in rock excavation. To model crushing occurring under high triaxial loads observed in the rock indentation experiments, peridynamic model has been extended to consider the pore‐collapse phenomenon. Rock heterogeneity is considered in the simulations by sampling the strength parameters from a probability distribution function. The model is qualitatively validated using the acoustic emission data from the indentation experiments. The developed model is able to form a crushed zone, which serves as the point of initiation for tensile cracks, which split the specimen. Quantitative validation relies on the comparison of the force‐penetration data as well as the indentation pressure‐penetration data obtained from the simulations and experiments performed on Bentheim Sandstone specimens.

MODIFIED INVERSE GENERALIZED EXPONENTIAL DISTRIBUTION: MODEL AND PROPERTIES

A three parameter continuous probability distribution Modified Inverse Generalized Exponential Distribution: Model and Properties, is introduced in this article. To study the properties of the introduced model, probability distribution, density, survival and hazard rate functions are introduced. A data of real life is used for checking the application. Some important methods of estimation are used for estimation of the constants. Model validation is checked using Akakie’s information, Bayesian Information, Corrected Akaike’s information and Hannan Qiunan Information Criteria as well as by plotting the P-P and Q-Q plots. For testing the goodness of fit Kolmogrov Smirnov test, Anderson darling test and Cramer-von Mises test are used. All the data analysis is performed using R-language programming.

Orbital Uncertainty Propagation Based on Adaptive Gaussian Mixture Model under Generalized Equinoctial Orbital Elements

The number of resident space objects (RSOs) has been steadily increasing over time, posing significant risks to the safe operation of on-orbit assets. The accurate prediction of potential collision events and implementation of effective and nonredundant avoidance maneuvers require the precise estimation of the orbit positions of objects of interest and propagation of their associated uncertainties. Previous research mainly focuses on striking a balance between accurate propagation and efficient computation. A recently proposed approach that integrates uncertainty propagation with different coordinate representations has the potential to achieve such a balance. This paper proposes combining the generalized equinoctial orbital elements (GEqOE) representation with an adaptive Gaussian mixture model (GMM) for uncertainty propagation. Specifically, we implement a reformulation for the orbital dynamics so that the underlying state and the moment feature of the GMM are propagated under the GEqOE coordinates. Starting from an initial Gaussian probability distribution function (PDF), the algorithm iteratively propagates the uncertainty distribution using a detection-splitting module. A differential entropy-based nonlinear detector and a splitting library are utilized to adjust the number of GMM components dynamically. Component splitting is triggered when a predefined threshold of differential entropy is violated, generating several GMM components. The final probability density function (PDF) is obtained by a weighted summation of the component distributions at the target time. Benefiting from the nonlinearity reduction caused by the GEqOE representation, the number of triggered events largely decreases, causing the necessary number of components to maintain uncertainty realism also to decrease, which enables the proposed approach to achieve good performance with much more efficiency. As demonstrated by the results of propagation in three scenarios with different degrees of complexity, compared with the Cartesian-based approach, the proposed approach achieves comparable accuracy to the Monte Carlo method while largely reducing the number of components generated during propagation. Our results confirm that a judicious choice of coordinate representation can significantly improve the performance of uncertainty propagation methods in terms of accuracy and computational efficiency.

Estimation Fuzzy Reliability of the Distribution Data (Topp leone- Kumaraswamy)With Application

The process of mixing distributions is one of the vital and important processes that contribute to increasing the flexibility and efficiency of the basic distributions. In this research، a new idea was identified، which is the idea of building a new probability distribution function by the function generating the distributions (Topp - Leone) through the single distribution mode (Kumaraswamy). In the function generated and the derivation of the statistical properties of the resulting new mixed distribution (Kumaraswamy- Topp Leone)، which is characterized by efficiency and flexibility in data representation، as the more parameters of the distribution، the greater the accuracy، reliability and preference، and then estimate the parameters of the aforementioned distribution using the informational standard Bayes estimation methods in light of Different loss functions، and the method of optimization algorithm for the gray wolf (GWO)، the failure times were studied، which are often random and fuzzy mixture in it، and this is expressed in fuzzy numbers، and this leads us to estimate the fuzzy reliability function (Fuzzy Reliability) within certain ranges of belonging to the fuzzy group. Practically applying the (Kum - TL) distribution to the real data obtained from the Ministry of Electricity / South Baghdad Station - Al-Zafaraniya، which numbered (70) engines، representing the operating time of the engine until the failure in order to estimate the expected times of future failures. The results showed that the proposed distribution achieved a clear preference in representing real data. In addition، the results showed the superiority of the (BLS_10) method at all different cut-off levels.