Lognormal Probability Density Function Research Articles

Randomness of optical turbulence generated by Rayleigh–Bénard convection using intensity statistics

The experimental study of optical turbulence proves difficult due to challenges in generating controllable conditions in a laboratory environment. Confined water tanks that produce Rayleigh–Bénard (RB) convection are one method to generate optical turbulence using a controllable temperature gradient. It is of utmost concern to quantify the properties of the optical turbulence generated for characterization of other optical applications such as imaging, sensing, or communications. In this experimental study a Gaussian beam is propagated through a RB water tank where two intensity measurements are made at the receiver’s pupil and focal plane. The pupil and focal plane results include quantification of the intensity fluctuation distribution, scintillation distribution, and refractive index structure constant at various values of the temperature gradient. The angle of arrival fluctuations is also calculated at the focal plane to obtain a second estimate of C n 2. The pupil plane estimate for C n 2 using scintillation index and focal plane angle of arrival fluctuations is compared to preliminary predictions of C n 2 as a function of RB temperature gradient showing C n 2∼ΔT4/3. The outcomes of the study confirm that the RB process produces intensity fluctuations that follow gamma–gamma and log-normal probability density functions. Estimates of the refractive index structure constant C n 2 produce the same trends with different magnitudes when measured from the pupil and focal plane.

A novel derivative search political optimization algorithm for multi-area economic dispatch incorporating renewable energy

Multi-area economic dispatch (MAED) incorporating renewable energy has become an important issue in the power system optimization. Existing intelligent optimization algorithms often suffer from poor solution accuracy or slow convergence when dealing with MAED problems. In this paper, a novel derivative search-based political optimization (DSPO) algorithm is proposed to handle the MAED problem incorporating renewable energy including wind and solar energy. In the renewable energy modeling, the Weibull and log-normal probability density functions are used to calculate available wind and solar power respectively. In order to improve the search performance, DSPO adopts two strategies: leader guide strategy and derivative search mechanism. The former adds the leader’s global optimal information which can direct candidate solutions to more promising regions and speed up convergence. The latter derives neighborhood solutions around some high-quality solutions to improve the exploitation ability. The DSPO algorithm is applied to solve four MAED problems which take into account valve point effect, prohibited operating zone, power loss and so on. The simulation results show that the DSPO algorithm achieves the overall best results in terms of convergence speed, solution accuracy and stability when compared with several well-established algorithms.

Optimal power flow of thermal-wind-solar power system using enhanced Kepler optimization algorithm: Case study of a large-scale practical power system

In the current century, electrical networks have witnessed great developments and continuous increases in the demand for fossil fuels based energy, leading to an excessive rise in the total production cost (TPC), as well as the pollutant (toxic) gases emitted by thermal plants. Under this circumstances, energy supply from different resources became necessary, such as renewable energy sources (RES) as an alternative solution. This latter, however, characterized with uncertainty nature in its operation principle, especially when operator system wants to define the optimal contribution of each resource in an effort to ensure economic and enhanced reliability of grid. This paper presents an Enhanced version of Kepler optimization algorithm (EKOA) to solve the problem of stochastic optimal power flow (SOPF) in a most efficient way incorporating wind power generators and solar photovoltaic with different objective functions, the stochastic nature of wind speed and solar is modeled using Weibull and lognormal probability density functions respectively. To prove the effectiveness of the proposed EKOA, various case studies were carried out on two test systems IEEE 30-bus system and Algerian power system 114-bus, obtained results were evaluated in comparison with those obtained using the original KOA and other methods published in the literatures. Thus, shows the effectiveness and superiority of the efficient EKOA over other optimizers to solve complex problem. The incorporation of RES resulted in a significant 2.39% decrease in production cost, showcasing EKOA’s efficiency with a $780/h, compared to KOA’s $781/h, for IEEE 30-bus system. For the DZA 114-bus system revealed even more substantial reductions, with EKOA achieving an impressive 12.6% reduction, and KOA following closely with a 12.4% decrease in production cost.

An optimal power flow solution for a power system integrated with renewable generation

<abstract> <p>Integrating Green Renewable Energy Sources (GRES) as substitutes for fossil fuel-based energy sources is essential for reducing harmful emissions. The GRES are intermittent and their integration into the conventional IEEE 30 bus configuration increases the complexity and nonlinearity of the system. The Grey Wolf optimizer (GWO) has excellent exploration capability but needs exploitation capability to enhance its convergence speed. Adding particle swarm optimization (PSO) with excellent convergence capability to GWO leads to the development of a novel algorithm, namely a Grey Wolf particle swarm optimization (GWPSO) algorithm with excellent exploration and exploitation capabilities. This study utilizes the advantages of the GWPSO algorithm to solve the optimal power flow (OPF) problem for adaptive IEEE 30 bus systems, including thermal, solar photovoltaic (SP), wind turbine (WT), and small hydropower (SHP) sources. Weibull, Lognormal, and Gumbel probability density functions (PDFs) are employed to forecast the output power of WT, SP, and SHP power sources after evaluating 8000 Monte Carlo possibilities, respectively. The multi-objective green economic optimal solution consisted of 11 control variables to reduce the cost, power losses, and harmful emissions. The proposed method to address the OPF problem is validated using an adaptive IEEE bus system. The proposed GWPSO algorithm is evaluated by comparing it with PSO and GWO optimization algorithms in terms of achieving an optimal green economic solution for the adaptive IEEE 30 bus system. This evaluation is conducted within the confines of the same test system using identical system constraints and control variables. The integration of a small SHP with WT and SP sources, along with the proposed GWPSO algorithm, led to a yearly cost reduction ranging from <bold>$\$$19,368</bold> to <bold>$\$$30,081</bold>. Simulation findings endorsed that the proposed GWPSO algorithm executes fruitfully compared to alternative algorithms regarding a consistent convergence curve and robustness, proving its potential as a viable choice for achieving cost-effective solutions in power systems incorporating GRES.</p> </abstract>

Optimization of open optical communication with multiple UAV over non homogenous turbulent paths

The article is devoted to optimizing communication through open atmospheric channels with many UAV by using inhomogeneous turbulent paths. A probabilistic assessment of the total signal loss in the communication system between the central communication node and a group of UAV located at the same distance from this node is analyzed. The case is considered when the scintillation indices in the subzones of optical communication with the UAV are different, which leads to unequal signal losses in the corresponding optical channels. The target functional is compiled by using a certain integral of the lognormal probability density function, which includes the newly introduced function of the losses dependence from the scintillation index. An analytical dependence of these indicators, characterizing the worst case when the total losses in the network reach a maximum, is presented. Keywords optical communications, turbulence, UAV, optimization, atmosphere

Optimal power flow with distributed energy sources using whale optimization algorithm

<span lang="EN-US">Renewable energy generation is increasingly attractive since it is <br /> non-polluting and viable. Recently, the technical and economic performance of power system networks has been enhanced by integrating renewable energy sources (RES). This work focuses on the size of solar and wind production by replacing the thermal generation to decrease cost and losses on a big electrical power system. The Weibull and Lognormal probability density functions are used to calculate the deliverable power of wind and solar energy, to be integrated into the power system. Due to the uncertain and intermittent conditions of these sources, their integration complicates the optimal power flow problem. This paper proposes an optimal power flow (OPF) using the whale optimization algorithm (WOA), to solve for the stochastic wind and solar power integrated power system. In this paper, the ideal capacity of RES along with thermal generators has been determined by considering total generation cost as an objective function. The proposed methodology is tested on the IEEE-30 system to ensure its usefulness. Obtained results show the effectiveness of WOA when compared with other algorithms like non-dominated sorting genetic algorithm (NSGA-II), grey wolf optimization (GWO) and particle swarm optimization-GWO (PSO-GWO).</span>

Quasi-periodic variability in the γ-Ray blazar PKS 0426–380

The identification of quasi-periodic oscillations (QPOs) inactive galactic nuclei (AGN) light curvescan shed light on supermassive black holes (SMBHs) and their surrounding central engines. They could signal the existence of SMBH binaries and reflect theirpreponderance through cosmic evolution. The Flat Spectrum Radio Quasar (FSRQ) at intermediate redshift (z=1.1) PKS 0426–380 was reported to possibly exhibit a quasi-periodic signal. Using the Fermi-LAT Pass 8 data, we generate an updated γ-ray light curve (LC) time series of this object from 2008 August to 2020 November. Aperture photometry γ-ray LCs were extracted at three energy ranges (E1: 200 MeV–1 GeV, E2: 1 GeV–10 GeV, and E3: 10 GeV–500 GeV). The cross-correlation function was applied to examine the variability properties within these bands. The source showed an E1-E2 high degree of correlation. Its flux distribution was noted to be closer a log-normal probability density function than a Gaussian. A search for QPOs in 100 MeV–500 GeV γ-ray LC was carried out using Lomb–Scargle periodogram, as well as the weighted wavelet Z-transform. Monte Carlo simulations were used to test the significance of the results. Furthermore, the REDFIT tool was used to precisely evaluate the power spectrum signal against the red-noise background. The results confirm the existence of QPO for the PKS 0426–380 blazar at high energy γ-rays, lasting for at least 12.2 yr with a period 3.65±0.45 yr at significance ∼3σ. They are consistent with this distant blazar being a binary system of SMBH with separation R∼0.0029pc, total mass M∼3×108M⊙, and timescale for SMBHs merging TGW∼6.9×104yr.

Stochastic Rational Method for Estimation of Flood Peak Uncertainty in Arid Basins: Comparison between Monte Carlo and First Order Second Moment Methods with a Case Study in Southwest Saudi Arabia

The flood peak is commonly estimated using the rational method for the design of hydraulic structures. The method is mainly used in a deterministic context. However, there is often uncertainty in flood predictions, which should be incorporated in the design of mitigation schemes. This research proposes a methodology to cope with uncertainty in the rational method via the application of a stochastic framework. Data from 158 storms, recorded in the period 1984–1987 in 19 subbasins in the southwestern part of Saudi Arabia, were used to implement the proposed methodology. A tri-variate log-normal probability density function was used to model the joint relationship between the rational method parameters. The model considered the parameters as random variables. The uncertainty in the rainstorms was represented by intensity or depth; the uncertainty in basin delineation (due to the use of different digital elevation model resolution) was represented by the basin area; and the uncertainty in the land use/land cover was represented by the runoff coefficient. The Monte Carlo method was used to generate realizations of the peak flow and runoff volume with 95% and 99% confidence levels from the input parameters. Although the correlation between the parameters was weak, the model was capable of simulating the rational model parameters and estimating the peak flow and runoff volume relatively well, and the generated realizations fell within the confidence levels, except for a few marginal cases. The model can be used to generate peak flows and the associated confidence limits in ungauged basins from the statistics of the input parameters using the equations developed in this study.

Global dynamics and density function in a class of stochastic SVI epidemic models with Lévy jumps and nonlinear incidence

<abstract><p>The paper studies the global dynamics and probability density function for a class of stochastic SVI epidemic models with white noise, Lévy jumps and nonlinear incidence. The stability of disease-free and endemic equilibria for the corresponding deterministic model is first obtained. The threshold criteria on the stochastic extinction, persistence and stationary distribution are established. That is, the disease is extinct with probability one if the threshold value $ R_{0}^{s} < 1 $, and the disease is persistent in the mean and any positive solution is ergodic and has a unique stationary distribution if $ R_{0}^{s} > 1 $. Furthermore, the approximate expression of the log-normal probability density function around the quasi-endemic equilibrium of the stochastic model is calculated. A new technique for the calculation of the probability density function is proposed. Lastly, the numerical examples and simulations are presented to verify the main results.</p></abstract>

Micromechanics of non-embedded spruce wood: Novel polishing and indentation protocol

Since the start of nanoindentation in wood, cell walls have been supported by epoxy resin during the microtoming of surfaces. However, wood embedding in epoxy resin was initially developed to harvest thin sections destined for transmission electron microscopy. In nanoindentation, embedding can affect the identified mechanical properties and might be difficult for modified wood samples. For this reason, we propose a protocol that does not involve resin within the wood specimen, uses a more robust miller to achieve surfaces with root means squared roughness of around 35 nm, and can exclude invalid edge indents through a novel evaluation procedure based on fitting 2D histograms with a linear combination of bi-variate lognormal probability density functions. We examined twelve samples of Norway spruce (Picea abies) late wood and their counterparts soaked with a biological preservation pretreatment. A nanoindentation protocol on every sample led to a mean reduced modulus and mean hardness of around 12 % and 26 %, respectively, lower than those found in the literature for embedded Norway spruce. These results agree well with previous results found for Loblolly pine (Pinus taeda L.) when comparing embedded and non-embedded specimens.

Probability Distribution of SARS-Cov-2 (COVID) Infectivity Following Onset of Symptoms: Analysis from First Principles

The phasing out of protective measures by governments and public health agencies, despite continued seriousness of the coronavirus pandemic, leaves individuals who are concerned for their health with two basic options over which they have control: 1) minimize risk of infection by being vaccinated and by wearing a face mask when appropriate, and 2) minimize risk of transmission upon infection by self-isolating. For the latter to be effective, it is essential to have an accurate sense of the probability of infectivity as a function of time following the onset of symptoms. Epidemiological considerations suggest that the period of infectivity follows a lognormal distribution. This proposition is tested empirically by construction of the lognormal probability density function and cumulative distribution function based on quantiles of infectivity reported by several independent investigations. A comprehensive examination of a prototypical ideal clinical study, based on general statistical principles (the Principle of Maximum Entropy and the Central Limit Theorem) reveals that the probability of infectivity is a lognormal random variable. Subsequent evolution of new variants may change the parameters of the distribution, which can be updated by the methods in this paper, but the form of the probability function is expected to remain lognormal as this is the most probable distribution consistent with mathematical requirements and available information.

Exploring short-term optical variability of blazars using TESS

ABSTRACT We present a first systematic time series study of a sample of blazars observed by the Transiting Exoplanet Survey Satellite (TESS). By cross matching the positions of the sources in the TESS observations with those from Roma-BZCAT, 29 blazars including both BL Lacerate objects and flat-spectrum radio quasars were identified. The observation lengths of the 79 light curves of the sources, across all sectors on which the targets of interest have been observed by TESS, range between 21.25 and 28.2 d. The light curves were analysed using various methods of time series analysis. The results show that the sources exhibit significant variability with fractional variability spanning between 1.41 per cent and 53.84 per cent. The blazar flux distributions were studied by applying normal and log-normal probability density function models. The results indicate that optical flux histogram of the sources are consistent with normal probability density function with most of them following bimodal distribution as opposed to unimodal distribution. This suggests that the days-time-scale optical variability is contributed either by two different emission zones or two distinct states of short-term activity in blazars. Power spectral density analysis was performed by using the power spectral response method and the true power spectra of unevenly sampled light curves were estimated. The power spectral slopes of the light curves ranged from 1.7 to 3.2.

Investigation of optimal power flow solution techniques considering stochastic renewable energy sources: Review and analysis

Increased penetration of renewable energy sources (RESs) in power system networks poses several challenges in system planning and management due to their uncertain and non-dispatchable nature. Consequently, this paper presents a thorough and precise review of recent solution methodologies for solving the optimal power flow (OPF) problems incorporated with stochastic RESs based on multiple peer-reviewed research publications in reputed journals. The Teaching Learning Based Optimization algorithm has been discussed and implemented to solve the OPF problem considering solar photovoltaic, wind turbine, and tidal energy systems. Weibull, Lognormal, and Gumbel probability density functions representing the uncertainty associated with the availability of wind speed, solar irradiance, and tidal energy systems, respectively. The results obtained from the proposed technique validate its novelty regarding OPF problems like minimization of operating cost, power loss in transmission lines, enhancement of voltage profile, and voltage stability. The proposed solution technique for OPF problems is tested on a modified IEEE 30-bus test system. Thus, this study assists in understanding the OPF problem for new researchers concerned with this domain and also gives the idea of implementing nature-inspired optimization algorithms on a defined test system to solve the OPF problem.

On the Distribution of Magnetic Moments in a System of Magnetic Nanoparticles

Particle size distribution carries out a substantial role in the magnetic behavior of nanostructured magnetic systems. In fact, a vast literature on superparamagnetism has been reported, suggesting that the particle size distribution in a system of magnetic nanoparticles (MNPs) corresponds to a lognormal probability density function, and several works have properly considered their magnetic moments following a similar distribution, as a universal rule. In this manuscript, it is demonstrated that alternative probability distribution functions, such as the gamma and Weibull ones, can be used to obtain useful parameters from the analysis of the magnetization curves, indicating there is no universal model to represent the actual magnetic moment distribution in a system of magnetic nanoparticles. Inspired by this observation, a reliable method to properly identify the actual magnetic moment distribution in a given nanostructured magnetic system is proposed and discussed.

Arctic Sea-Ice Surface Elevation Distribution from NASA’s Operation IceBridge ATM Data

In this paper, we characterize the sea-ice elevation distribution by using NASA’s Operation IceBridge (OIB) Airborne Topographic Mapper (ATM) L1B data over the Arctic Ocean during 94 Spring campaigns between 2009 and 2019. The ultimate objective of this analysis is to better understand sea-ice topography to improve the estimation of the sea-ice freeboard for nadir-looking altimeters. We first introduce the use of an exponentially modified Gaussian (EMG) distribution to fit the surface elevation probability density function (PDF). The characteristic function of the EMG distribution can be integrated in the modeling of radar altimeter waveforms. Our results indicate that the Arctic sea-ice elevation PDF is dominantly positively skewed and the EMG distribution is better suited to fit the PDFs than the classical Gaussian or lognormal PDFs. We characterize the elevation correlation characteristics by computing the autocorrelation function (ACF) and correlation length (CL) of the ATM measurements. To support the radar altimeter waveform retracking over sea ice, we perform this study typically on 1.5 km ATM along-track segments that reflect the footprint diameter size of radar altimeters. During the studied period, the mean CL values range from 20 to 30 m, which is about 2% of the radar altimeter footprint diameter (1.5 km).

Probability density function of whole-body vibration in passenger car

This study identified the probability density function (PDF) model of the driver/passenger whole body vibration (WBV) in the passenger car based on wide in-situ measurements. The WBV was evaluated at the seat surface and feet of twenty-eight test vehicles travelled on test sections of different quality of total travel distance over 2,600 km. The WBV was quantified by the frequency-weighted acceleration at the driver’s/passenger’s seat surface and feet according to the ISO 2631-1 in three perpendicular axes. The lognormal PDF was identified as the most suitable model to fit empirical WBV data. Vibration data were divided in the groups according to the road roughness, vehicle speed, road and vehicle category. The results showed the marked influence of the road roughness and the lower influence of the vehicle speed, road and vehicle category on the PDF parameters. The linear relation between scale and shape parameters of PDF and road roughness level was estimated. The identified parameters of lognormal PDF can be useful for the basic statistics estimation of the WBV in passenger car and estimation of human ride comfort and health.

Comparative analysis of underwater wireless optical communication system

The performance of an underwater wireless optical communication (UWOC) system is affected by many factors, such as channel conditions, modulation and detection techniques, type of optical source, etc. The UWOC system performance was investigated in the presence of weak and strong turbulence. Pure sea was the water type used for the performance analysis. Binary phase shift keying, quadrature phase shift keying (QPSK), 16-quadrature amplitude modulation, and non-return to zero-on–off keying were used for comparison. Further, a continuous wave laser was used as the optical source, and photodetectors are at the receiver. The performance of the UWOC system was analyzed in terms of the bit error rate (BER). The BER performance was evaluated using the log-normal probability density function (PDF) in weak turbulence and the gamma-gamma PDF in strong turbulence. In all modulation techniques, QPSK had the best BER performance. The BER was reduced further using avalanche photodiode (APD) as the photodetector instead of positive-intrinsic-negative diode for the same transmitter power and channel conditions. In weak turbulence, a BER of 10 − 8 was achieved at a transmitter power of 12.5 dBm using the QPSK modulation technique and APD as the photodetector. For the same conditions, a BER of 9 × 10 − 7 was achieved in strong turbulence. The multiplication gain, dark current, and carrier ionisation ratio of the photodetector were optimized for better BER performance in the UWOC system.

ESMA-OPF: Enhanced Slime Mould Algorithm for Solving Optimal Power Flow Problem

In this work, an enhanced slime mould algorithm (ESMA) based on neighborhood dimension learning (NDL) search strategy is proposed for solving the optimal power flow (OPF) problem. Before using the proposed ESMA for solving the OPF problem, its validity is verified by an experiment using 23 benchmark functions and compared with the original SMA, and three other recent optimization algorithms. Consequently, the ESMA is used to solve a modified power flow model including both conventional energy, represented by thermal power generators (TPGs), and renewable energy represented by wind power generators (WPGs) and solar photovoltaic generators (SPGs). Despite the important role of WPGs and SPGs in reducing CO2 emissions, they represent a big challenge for the OPF problem due to their intermittent output powers. To forecast the intermittent output powers from SPGs and WPGs, Lognormal and Weibull probability density functions (PDFs) are used, respectively. The objective function of the OPF has two extra costs, penalty cost and reserve cost. The penalty cost is added to formulate the underestimation of the produced power from the WPGs and SPGs, while the reserve cost is added to formulate the case of overestimation. Moreover, to decrease CO2 emissions from TPGs, a direct carbon tax is added to the objective function in some cases. The uncertainty of load demand represents also another challenge for the OPF that must be taken into consideration while solving it. In this study, the uncertainty of load demand is represented by the normal PDF. Simulation results of ESMA for solving the OPF are compared with the results of the conventional SMA and two further optimization methods. The simulation results obtained in this research show that ESMA is more effective in finding the optimal solution of the OPF problem with regard to minimizing the total power cost and the convergence of solution.

Probability density function estimation for filament creation in lossy, turbulent, nonlinear media.

Optical Kerr effects induced by the propagation of high peak-power laser beams through real atmospheres have been a topic of interest to the nonlinear optics community for several decades. Previous work has focused on estimating the Filamentation Onset Distance (FOD) in real atmospheres but not its statistical variance. This paper describes two ad hoc engineering models for predicting the FOD Probability Density Function (PDF) for lossy, turbulent, nonlinear media. Specifically, these models characterize the FOD variation with turbulence. One model uses a log-normal PDF with mean and variance proportional to the Rytov Variance. The other uses a gamma PDF employing the same mean and variance equations. These two PDFs will be compared to previous computer simulation results. Both show reasonable agreement between PDF predictions and computer simulation results for long-range filamentation. In fact, both give similar results, and there is no preference given to the data comparisons presented.

Procena čvrstoće snopa materijala metalnog zupčanika

Expressions for the pulsating or beam strengths of many popular metallic gear materials are derived based on the tensile strength and endurance ratio. The strength values predicted are for a reliability of 99% at load cycles corresponding to that of the endurance strength of the materials. The expressions are based on the consideration of the revised Lewis gear root stress formula by treating the design parameters as random variables associated with the lognormal probability density function and application of the Gerber fatigue failure rule. Pulsating strength predictions are compared with those of AGMA estimates for through-hardened steels and other materials. The variances between model predictions and AGMA values for steel and ductile cast iron materials are reasonably low. Low variances between model and AGMA values for high-strength gray cast iron and cast bronze were also observed. However, high variances between model and AGMA values for low-strength gray cast iron and cast bronze were found. Overall, the model estimates are considered sufficiently accurate for preliminary design applications where initial sizes of gears are generated. The study showed that for many metallic gear materials, the average pulsating strength ratio is 0.36 at 99% reliability. Therefore, the suggestion by Buckingham, that the fatigue strength of a gear tooth is approximately one-third (0.333) of the tensile strength of the material is justified.