Normal Probability Density Function Research Articles

Power efficiency improvement in reactive power dispatch under load uncertainty

Nowadays, there is a significant rise in electricity demand, posing challenges for power grid operators due to inaccurate forecasting, leading to excessive power losses and voltage instability. This paper addresses these issues by focusing on solving optimal reactive power dispatch (ORPD) while considering load demand uncertainty. The main objective of solving ORPD is to reduce power losses by adjusting generator voltage ratings, transformer tap ratio, and shunt capacitors' reactive power. Monte Carlo simulation (MCS) is employed to generate load scenarios using the normal probability density function, while a reduction-based technique is implemented to decrease the number of those scenarios. The improved gray wolf optimization (I-GWO) algorithm is introduced for the first time to address the stochastic ORPD problem. Experimentation is conducted on an IEEE-30 bus system when results are contrasted with conventional gray wolf optimization (GWO) and five other algorithms as stated in the literature. The I-GWO algorithm's performance is assessed with and without considering load demand uncertainty. Through Friedman's statistical tests, a significant decrease of 20.96% in active power losses and 63.06% in the summation of expected power losses is observed. The I-GWO algorithm's results on the ORPD problem demonstrate its effectiveness and robustness.

Experimental research on the impact pressure of tidal bore fronts.

Tidal bore impact can be strong and destructive, placing estuarine infrastructures under great threat. However, there is a lack of research focusing on accurately estimating the impact pressure exerted by tidal bores. Herein new experiments were conducted to investigate the pressure of tidal bore fronts in a glass flume. Through analysis of instantaneous pressure of three forms of tidal bore, it was observed that the pressure fluctuation of weak and strong breaking bore fronts is characterized by impact pressure. The vertical distribution and maximum impact pressure of tidal bore were studied.The maximum impact pressure of breaking bore fronts appeared around 0.46 times height of it. The relationship between relative impact pressure and height of the tidal bore fronts was found to closely follow a normal probability density function. Through nonlinear regression analysis, an empirical equation was derived to calculate impact pressure, which was validated using observation data from the Qiantang River in China. This equation can be utilized to predict the impact pressure of tidal bore fronts and provide valuable support for estuarine engineering design.

Effects of Thermal Expansion and Pressure on Far-Field Fluctuations of Heterogeneous Propellants

In this work, we carry out three-dimensional mesoscale simulations of heterogeneous solid propellant combustion. We solve the reactive low-Mach-number equations in the gas phase with complete coupling to the solid phase. The model takes into account thermal expansion and deformation in the solid phase by using a hypoelastic law in the quasi-static limit. To account for morphology, we select two different propellant formulations with different particle size distributions and present the results as a function of pressure. The thermomechanical behavior is accessed by examining quantities such as strain and stress in the propellant as a function of pressure and propellant morphology. We also show that temperature and velocity fluctuations exist in the far field above the propellant surface and that these fluctuations can be significant. To better understand the nature of these fluctuations, we vary the pressure and make relevant plots of normal velocity and temperature probability density functions, as well as time autocorrelations. Such descriptions are necessary to account for the coupling between the mesoscale and the macroscale, where the fluctuations at the mesoscale can affect quantities at the macroscale, such as head-end pressure, trigger parietal vortex shedding, and aeroacoustics.

Underwater image enhancement using multi-task fusion.

Underwater images are often scattered due to suspended particles in the water, resulting in light scattering and blocking and reduced visibility and contrast. Color shifts and distortions are also caused by the absorption of different wavelengths of light in the water. This series of problems will make the underwater image quality greatly impaired, resulting in some advanced visual work can not be carried out underwater. In order to solve these problems, this paper proposes an underwater image enhancement method based on multi-task fusion, called MTF. Specifically, we first use linear constraints on the input image to achieve color correction based on the gray world assumption. The corrected image is then used to achieve visibility enhancement using an improved type-II fuzzy set-based algorithm, while the image is contrast enhanced using standard normal distribution probability density function and softplus function. However, in order to obtain more qualitative results, we propose multi-task fusion, in which we solve for similarity, then we obtain fusion weights that guarantee the best features of the image as much as possible from the obtained similarity, and finally we fuse the image with the weights to obtain the output image, and we find that multi-task fusion has excellent image enhancement and restoration capabilities, and also produces visually pleasing results. Extensive qualitative and quantitative evaluations show that MTF method achieves optimal results compared to ten state-of-the-art underwater enhancement algorithms on 2 datasets. Moreover, the method can achieve better results in application tests such as target detection and edge detection.

Impact of Vertical Scaling on Normal Probability Density Function Plots.

Probability density function (PDF) curves are among the few charts on a Cartesian coordinate system that are commonly presented without y-axes. This design decision may be due to the lack of relevance of vertical scaling in normal PDFs. In fact, as long as two normal PDFs have the same means and standard deviations (SDs), they can be scaled to occupy different amounts of vertical space while still remaining statistically identical. Because unfxed PDF height increases as SD decreases, visualization designers may fnd themselves tempted to vertically shrink low-SD PDFs to avoid occlusion or save white space in their fgures. Although irregular vertical scaling has been explored in bar and line charts, the visualization community has yet to investigate how this visual manipulation may affect reader comparisons of PDFs. In this paper, we present two preregistered experiments (n = 600, n = 401) that systematically demonstrate that vertical scaling can lead to misinterpretations of PDFs. We also test visual interventions to mitigate misinterpretation. In some contexts, we fnd including a y-axis can help reduce this effect. Overall, we fnd that keeping vertical scaling consistent, and therefore maintaining equal pixel areas under PDF curves, results in the highest likelihood of accurate comparisons. Our fndings provide insights into the impact of vertical scaling on PDFs, and reveal the complicated nature of proportional area comparisons.

Whole Life Cycle Cost Analysis of Transmission Lines Using the Economic Life Interval Method

With the large-scale construction and commissioning of transmission lines over the past two decades, the grid is facing a large-scale centralized decommissioning of transmission lines. The transmission line’s economic life is crucial to rationalizing its construction and reducing the grid’s development costs. Based on the minimum economic life calculation principle, the static and dynamic transmission line economic life calculation model is established, considering the whole life cycle for transmission line cost. The improved gray GM (1,1) model is applied to forecast cost data during the economic life assessment of transmission lines with fewer samples. Considering the cost uncertainty in life-cycle costing, the interval cost model based on the coefficient of variation wave amplitude is proposed to determine the economic life intervals under different guarantees by using the normal distribution probability density function, which reduces the influence of cost fluctuations on the economic life calculation error. The economic life analysis of a 500 kV transmission line is used as a case study to verify the model’s accuracy and effectiveness. The method shows the economic life intervals under different guarantee degrees based on the most probable economic life determination, which provides theoretical support for calculating the economic life elasticity of transmission lines.

Determination of measurement uncertainty by a Monte Carlo method for an RF power sensor calibration system using a VNA

This work proposes a systematic assessment of measuring type A uncertainty (caused by random errors) used in RF power sensor calibration. To reduce A type uncertainty, several successive measurements are repeated. The uncertainty arises from repeatability errors in connectors caused by changes in their electrical properties during repeated mating. The suitability of the METAS UncLib software was analysed and we concluded that software should be developed to take into account the shape of probability density function (PDF) using a Monte Carlo method (MCM), which was lacking in METAS UncLib. The self-developed software was then tested on an example taken from the literature and the superiority of the MCM over the analytical method (GUM) was confirmed. During the calibration of the RF sensor using a vector network analyzer (VNA), a series of repeated measurements were performed and, after applying our MCM software, it was found that the measurement uncertainties calculated by the MCM method were several times larger than those by the GUM. The reason for this was that the correlation between the measured input quantities was not taken into account. When this was done using a covariance matrix and assuming a normal PDF of the input quantities, the results obtained with the GUM and the MCM converged. Our main objective was to investigate the influence of the PDF shape of the input measurement samples on the measurement uncertainty. Taking more than a dozen measurements is too costly, on the other hand, the small sample size prevents a reliable determination of the PDF shape. Finally, to overcome this inconvenience, we have developed a special method that uses the histograms of standardized input data taken at all measurement frequencies under fixed conditions without disconnecting the connectors, to increasing the total number of results which were needed to create the PDF histograms of input quantities.

Modeling of truncated normal distribution for estimating hydraulic parameters in water distribution systems: taking nodal water demand as an example.

The normal probability density function (PDF) is widely used in parameter estimation in the modeling of dynamic systems, assuming that the random variables are distributed at infinite intervals. However, in practice, these random variables are usually distributed in a finite region confined by the physical process and engineering practice. In this study, we address this issue through the application of truncated normal PDF. This method avoids a non-differentiable problem inherited in the truncated normal PDF at the truncation points, a limitation that can limit the use of analytical methods (e.g., Gaussian approximation). A data assimilation method with the derived formula is proposed to describe the probability of parameter and measurement noise in the truncated space. In application to a water distribution system (WDS), the proposed method leads to estimating nodal water demand and hydraulic pressure key to hydraulic and water quality model simulations. Application results to a hypothetical and a large field WDS clearly show the superiority of the proposed method in parameter estimation for WDS simulations. This improvement is essential for developing real-time hydraulic and water quality simulation and process control in field applications when the parameter and measurement noise are distributed in the finite region.

Effects of surfactants on bubble-induced turbulence

We use experiments to explore the effect of surfactants on bubble-induced turbulence (BIT) at different scales, considering how the bubbles affect the flow kinetic energy, anisotropy and extreme events. To this end, high-resolution particle shadow velocimetry measurements are carried out in a bubble column in which the flow is generated by bubble swarms rising in water for two different bubble diameters (3 and 4 mm) and moderate gas volume fractions (0.5 %–1.3 %). We use tap water as the base liquid and add 1-Pentanol as an additional surfactant with varying bulk concentration, leading to different bubble shapes and surface boundary conditions. The results reveal that with increasing surfactant concentration, the BIT generated increases in strength, even though bubbles of a given size rise more slowly with surfactants. We also find that the level of anisotropy in the flow is enhanced with increasing surfactant concentration for bubbles of the same size, and that for the same surfactant concentration, smaller bubbles generate stronger anisotropy in the flow. Concerning the intermittency quantified by the normalized probability density functions of the fluid velocity increments, our results indicate that extreme values in the velocity increments become more probable with decreasing surfactant concentration for cases with smaller bubbles and low gas void fraction, while the effect of the surfactant is much weaker for cases with larger bubble and higher void fractions.

R] Inferential Statistics IV: Uniform and (Standard) Normal Probability (Density) Functions

R] Inferential Statistics IV: Uniform and (Standard) Normal Probability (Density) Functions

A method to consider a maximum admissible risk in decision-making procedures based on measurement results

Measurement uncertainty plays a very important role ensuring validity of decision-making procedures, since it is the main source of incorrect decisions in conformity assessment. The guidelines given by the actual Standards allow one to take a decision of conformity or non-conformity, according to the given limit and measurement uncertainty associated to the measured value. Due to measurement uncertainty, a risk of a wrong decision is always present, and the Standards also give indications on how to evaluate this risk, although they mostly refer to a normal probability density function to represent the distribution of values that can be reasonably attributed to the measurand. Since such a function is not always the one that best represents this distribution of values, this paper considers some of the most-often used probability density functions and derives simple formulas to set the acceptance (or rejection) limits in such a way that a pre-defined maximum admissible risk is not exceeded.

A Novel Adaptive Manta-Ray Foraging Optimization for Stochastic ORPD Considering Uncertainties of Wind Power and Load Demand

The optimal control of reactive powers in electrical systems can improve a system’s performance and security; this can be provided by the optimal reactive power dispatch (ORPD). Under the high penetration of renewable energy resources (RERs) such as wind turbines (WTs), the ORPD problem solution has become a challenging and complex task due to the fluctuations and uncertainties of generated power from WTs. In this regard, this paper solved the conventional ORPD and the stochastic ORPD (SORPD) at uncertainties of the generated power from WTs and the load demand. An Adaptive Manta-Ray Foraging Optimization (AMRFO) was presented based on three modifications, including the fitness distance balance selection (FDB), Quasi Oppositional based learning (QOBL), and an adaptive Levy Flight (ALF). The ORPD and SORPD were solved to reduce the power loss (PLoss) and the total expected PLoss (TEPL), the voltage deviations (VD) and the total expected VD (TEVD). The normal and Weibull probability density functions (PDFs), along with the scenario reduction method and the Monte Carlo simulation (MCS), were utilized for uncertainty representations. The performance and validity of the suggested AMRFO were compared to other optimizers, including SCSO, WOA, DO, AHA, and the conventional MRFO on the IEEE 30-bus system and standard benchmark functions. These simulation results confirm the supremacy of the suggested AMRFO for the ORPD and SORPD solution compared to the other reported techniques.

DDOFM: Dynamic malicious domain detection method based on feature mining

The domain name system is an essential part of the network, and target hosts are often attacked by malicious domain names to steal resources. Some traditional detection methods have low accuracy, poor generalization ability, and high resource overhead on model construction to deal with complex and variable malicious domain names. A three-level dynamic malicious domain detection method (DDOFM) is proposed in this paper. DDOFM only needs to combine a few high-order statistical features of benign domains with some DNS features, without flagging malicious samples and involving them in training. Firstly, the boundary recognition of passive DNS (PDNS) features extracted from DNS traffic is carried out to conduct an early- warning for some domains. Second, the Hidden Markov Model (HMM) forward algorithm and normal distribution probability density function are used to calculate the formation probabilities of the warned domains and their probability density values. Then the probabilities of every character in the warned domain name and the standard deviation between these probabilities are counted. Further, the probability density values and the standard deviations of these warned domain names are compared with their respective thresholds to identify the attribution of the warned domain names. Finally, if the domain name is not warned, the Jensen–Shannon divergence (JS divergence) between it and the previous domain name will be calculated. Then the local iterative threshold finding algorithm (LLTFA) proposed in this paper will be combined to determine its attribution and identify whether the host is connected to the command and control (C&C) server. Experiments show that the detection indexes of this method exceed 99% for multiple types of malicious domain names. The C&C servers can also be identified by DDOFM faster than similar methods.

Imitating human responses via a Dual-Process Model

Advancements in autonomy are leading to an increased need for machines capable of collaborative effort with humans to achieve team goals. One way of enhancing these human-autonomous system work arrangements leverages the concept of a shared mental model. The idea being that when the human and autonomous teammate have aligned models, the team is more productive due to an increase in trust, predictiveness, and apparent understanding. An open issue is how to have autonomous teammates learn a user aligned mental model. This research presents a dual-process learning model that leverages multivariate normal probability density functions (DPL-MN) to extrapolate state-responses into system 2. By leveraging dual-process learning concepts, an autonomous teammate is able to rapidly align with a user and extrapolate their consistencies into longer term memory. Evaluation of DPLM with user responses from a game called Space Navigator shows that DPL-MN accurately responds to situations similarly to each unique user.

On the properties of residential rooftop azimuth and tilt uncertainties for photovoltaic power generation modeling and hosting capacity analysis

One of the essential epistemic uncertainties that has not yet been studied enough for distributed photovoltaic systems is the azimuth and tilt of rooftop photovoltaic panels, as previous studies of grid impacts and hosting capacity have tended to assume uniform and optimal roof facet conditions. In this study, rooftop facet orientation distributions are presented and analyzed for all single-family buildings in the Swedish city of Uppsala, based on LiDAR-based data that consist of every roof facet from the around 13,500 single-family buildings in the city. From these distributions, novel methods to proportionally include less suitable roofs for every penetration level are proposed using a simple method based on normal and uniform probability density functions, and are tested for both time-series and stochastic hosting capacity analysis. The results show that under the assumption that the best roof facets are utilized first, a uniform distribution for rooftop facet azimuth and a normal distribution for rooftop facet tilt with parameters that depend linearly on the penetration level were shown to be accurate. The hosting capacity simulations demonstrate how the proposed methods perform significantly better in estimating the photovoltaic hosting capacity than the more common simplified methods for both time-series and stochastic hosting capacity analysis. The proposed model could help distribution system operators as well as researchers in this area to model the rooftop facet orientation uncertainty better and improve the quality of aggregated photovoltaic generation models and hosting capacity analyses.

On New Probabilistic Hermite Polynomials

In the theory of differential equation and probability, Probabilistic Hermite polynomials Hr(x) = {r=0,1,2,…,n} are the polynomials obtained from derivatives of the standard normal probability density function (pdf) of the form α(x)=1/√2π e^(-1/2 x^2 ). These polynomials played an important role in the Gram-Charlier series expansion of type A and the Edgeworth’s form of the type A series (see [18]). In this paper, we obtained new Probabilistic Hermite polynomials by considering a standard normal distribution with probability density function (pdf) given as β(x)=1/(2√π) e^(-1/4 x^2 ). The generating function, recurrence relations and orthogonality properties are studied. Finally, a differential equation governing these polynomials was presented which enables us to obtain the expression of the polynomial in a closed form.

Managing Geotechnical Uncertainty With Simulation Models: An Introduction

In a standard deterministic analysis discrete scenarios are considered, and a moderately conservative “characteristic” value is used as a design basis. However, fixed or exact values in a real-world geotechnical site seldom occurs. Deterministic approaches may not explicitly consider the ground uncertainty. Simulations using various probabilities provides for this uncertainty as each parameter input is treated as a random variable within certain measured ranges or ability to evaluate. Monte Carlo (MC) sampling is a traditional technique for generating random numbers to sample from a probability distribution. When low probability events occur, a small number of MC iterations might not sample sufficient quantities of these outcomes for inclusion in the simulation model. Latin Hypercube (LH) sampling uses stratification of the input probability distributions, to overcome the limitations of Monte Carlo sampling. The simulation results show low probability outcomes are included in the sampling for the simulation model. At a high number of simulation iterations both provide similar outputs, but at low simulation iterations the LH is more reliable. However, both the MC and LH sampling suffer from impractical values at low or high probability events when the normal probability density function (PDF) is adopted. The normal PDF is commonly used in statistical modelling. Non-normal PDFs often represent the best fit PDF when a goodness of fit test is carried out. The errors associated with using the common normal PDF are shown with the above-mentioned simulation models. This best fit PDF applies whether simulation models as described above is used or even with simple “what if” sensitivity models in traditional analysis.

Probabilistic Interval Predictor Based on Dissimilarity Functions

This work presents a new methodology to obtain probabilistic interval predictions of a dynamical system. The proposed strategy uses stored past system measurements to estimate the future evolution of the system. The method relies on the use of dissimilarity functions to estimate the conditional probability density function of the outputs. A family of empirical probability density functions, parameterized by means of two scalars, is introduced. It is shown that the proposed family encompasses the multivariable normal probability density function as a particular case. We show that the presented approach constitutes a generalization of classical estimation methods. A validation scheme is used to tune the two parameters on which the methodology relies. In order to prove the effectiveness of the presented methodology, some numerical examples and comparisons are provided.

Markovian Statistical Model of Cloud Optical Thickness. Part I: Theory and Examples

Abstract We present a generalization of the binary-value Markovian model previously used for statistical characterization of cloud masks to a continuous-value model describing 1D fields of cloud optical thickness (COT). This model has simple functional expressions and is specified by four parameters: the cloud fraction, the autocorrelation (scale) length, and the two parameters of the normalized probability density function of (nonzero) COT values (this PDF is assumed to have gamma-distribution form). Cloud masks derived from this model by separation between the values above and below some threshold in COT appear to have the same statistical properties as in binary-value model described in our previous publications. We demonstrate the ability of our model to generate examples of various cloud-field types by using it to statistically imitate actual cloud observations made by the Research Scanning Polarimeter (RSP) during two field experiments.

Model Validation of a Single Degree-of-Freedom Oscillator: A Case Study

In this paper, we investigate a validation process in order to assess the predictive capabilities of a single degree-of-freedom oscillator. Model validation is understood here as the process of determining the accuracy with which a model can predict observed physical events or important features of the physical system. Therefore, assessment of the model needs to be performed with respect to the conditions under which the model is used in actual simulations of the system and to specific quantities of interest used for decision-making. Model validation also supposes that the model be trained and tested against experimental data. In this work, virtual data are produced from a non-linear single degree-of-freedom oscillator, the so-called oracle model, which is supposed to provide an accurate representation of reality. The mathematical model to be validated is derived from the oracle model by simply neglecting the non-linear term. The model parameters are identified via Bayesian updating. This calibration process also includes a modeling error due to model misspecification and modeled as a normal probability density function with zero mean and standard deviation to be calibrated.