Joint Probability Density Function Research Articles

Probability density analysis of nonlinear stochastic dynamics of horizontal axis wind turbine blades

Wind turbine blades with horizontal axis are frequently subjected to cyclic and random loads during operation, which may decrease blade stability and impair power generation efficiency. In this paper, the finite element method (FEM) is mainly used to investigate the nonlinear stochastic dynamics of turbine blades and assess blade operation stability. The harmonic excitation and Gaussian white noise are combined to represent the external stimulation of the Mathieu-Duffing equation describing blade vibration, the Fokker-Planck (FP) equation is formulated and numerically solved using the FEM and the Crank Nicholson method to obtain the transient joint probability density function (PDF) and marginal PDF. The findings show that the blade's PDF exhibits an unstable split phenomenon in one vibration cycle, meaning that the blade's motion may jump randomly and has some uncertainty, potentially affecting blade operating stability and power generation efficiency. Furthermore, the sensitivity of blade stability to excitation strength is examined. The form and position of the PDF vary when the intensity of the harmonic excitation or noise is strong, increasing the random jump probability of blade vibration and compounding the instability and uncertainty of blade motion. Under severe circumstances, the blades may lose stability or even damage, endangering the safety and efficacy of blade operation. The current study might be used to evaluate blade stability and the overall safety of turbines.

Monte Carlo Simulation of Stochastic Differential Equation to Study Information Geometry.

Information Geometry is a useful tool to study and compare the solutions of a Stochastic Differential Equations (SDEs) for non-equilibrium systems. As an alternative method to solving the Fokker–Planck equation, we propose a new method to calculate time-dependent probability density functions (PDFs) and to study Information Geometry using Monte Carlo (MC) simulation of SDEs. Specifically, we develop a new MC SDE method to overcome the challenges in calculating a time-dependent PDF and information geometric diagnostics and to speed up simulations by utilizing GPU computing. Using MC SDE simulations, we reproduce Information Geometric scaling relations found from the Fokker–Planck method for the case of a stochastic process with linear and cubic damping terms. We showcase the advantage of MC SDE simulation over FPE solvers by calculating unequal time joint PDFs. For the linear process with a linear damping force, joint PDF is found to be a Gaussian. In contrast, for the cubic process with a cubic damping force, joint PDF exhibits a bimodal structure, even in a stationary state. This suggests a finite memory time induced by a nonlinear force. Furthermore, several power-law scalings in the characteristics of bimodal PDFs are identified and investigated.

Comprehensive Evaluation Index System of Distribution Network for Distributed Photovoltaic Access

Affected by the light intensity and multiple adjacent stations, the output power of photovoltaic power stations presents the characteristics of randomness and correlation, which puts forward new requirements for the safe operation and planning evaluation of the distribution network. First, in order to accurately describe the randomness and fluctuation of photovoltaic output, a Gaussian mixture model based on an improved optimal particle swarm optimization algorithm is proposed, and the joint probability density function of multiple photovoltaic outputs is solved. Then, the power flow equation is linearized, the linear expressions of bus voltage and line power flow are derived, and the joint probability distributions of multi-node voltage and multi-line power flow are obtained, respectively. Based on this, the reliability index and voltage quality index in the presence of the distributed renewable energy were constructed. Finally, the novel power grid planning evaluation index is tested in a real system in China, the combined weight is determined by the analytic hierarchy process, and the comprehensive evaluation results are obtained. The simulation results prove the feasibility of the evaluation index system.

A composite index coupling five key elements of water cycle for drought analysis in Pearl River basin, China

Drought, as a natural disaster, has widespread consequences and is notoriously difficult to manage. Critical to developing a drought management strategy is the identification and assessment of drought. To that end, this study developed a new composite index, called the standardized water cycle index (SWCI) based on the water cycle and water balance. The SWCI couplesd the key elements of the water cycle, including precipitation, evapotranspiration, leaf area index, surface runoff, and subsurface runoff, and requires the joint distribution of these elements which was determined using the D-vine copula. The Kendall transform was used to reduce the dimensionality of the five-element joint probability density function, which was then inversed to obtain the SWCI which was then evaluated with the data from the Pearl River basin obtained using the CMIP6. Results showed that the SWCI satisfactorily evaluated drought conditions, while reflecting the drought-mitigating effect of vegetation and subsurface runoff. The SWCI was also able to evaluate drought in areas with a high level of human activity.

On occurrence disaggregation of probabilistic seismic hazard

AbstractDisaggregation of probabilistic seismic hazard allows to quantify how much one or more earthquake scenarios contribute to the occurrence [exceedance] of a ground motion intensity measure threshold of interest (x) at the construction site. The scenario is usually defined in terms of magnitude (M), source‐to‐site distance (R), and possibly includes the standardized residual (ε) of the ground motion model considered in the hazard analysis. Analytically, in case occurrence is of interest, disaggregation provides the joint probability density function of or conditional on the event, that is, or . Occurrence disaggregation is important for a number of earthquake engineering applications, and it is typically addressed in the literature in an approximated manner, considering as the conditioning event , with being an arbitrary finite width of the interval. This short communication undertakes a deeper examination of occurrence disaggregation clarifying that: (i) no approximation is needed in the case of disaggregation in terms of magnitude and distance (i.e., when is sought); (ii) is theoretically degenerate, and as such, its approximation via finite can lead to misleading results; (iii) if is chosen coherently with the discretization of the domain used in the hazard integral, it leads to approximated , enabling the conclusion that occurrence disaggregation does not add information with respect to disaggregation.

Performance and Capacity Analysis of MDCSK-BICM for Impulsive Noise in PLC

To further mitigate the BER error floor of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> -ary differential chaos shift keying (MDCSK) modulation for impulsive noise, a low-cost high-reliability coded modulation scheme without equalizations is desirable for impulsive noise. In this paper, a protograph-based low-density parity-check coded MDCSK-based bit-interleaved coded modulation (MDCSK-BICM) scheme is proposed for the scenario with impulsive noise obeying Bernoulli-Laplace distribution. Considering the Gaussian distribution of protograph extrinsic information transfer (PEXIT) and the benchmark of coding design, the U-shaped non-coherent capacity is derived with optimal code rate for square MDCSK, showing that the code rates are stable with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$E_b/N_0$</tex-math></inline-formula> increasing but still much lower than that of DS-16DQAM at code rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$&gt;\!0.145$</tex-math></inline-formula> . The derived joint probability density function can be considered as approximately a Gaussian distribution, which is used to improve the impulsive noise fitted PEXIT. Finally, a new code under a novel principle is designed for better bit error rate performance based on the improved PEXIT analysis. Both PEXIT analysis and simulation results demonstrate that the proposed scheme achieves a better BER than that with a traditional scheme, and suggesting a research approach based on simple methods. This basic research work provides a guide to establish a framework and optimize the performance of transmission systems over practical power line communication.

Visual features as carriers of abstract quantitative information.

Four experiments investigated the extent to which abstract quantitative information can be conveyed by basic visual features. This was done by asking observers to estimate and discriminate Pearson correlation in graphical representations where the first data dimension of each element was encoded by its horizontal position, and the second by the value of one of its visual features; perceiving correlation then requires combining the information in the two encodings via a common abstract representation. Four visual features were examined: luminance, color, orientation, and size. All were able to support the perception of correlation. Indeed, despite the strikingly different appearances of the associated stimuli, all gave rise to performance that was much the same: Just noticeable difference was a linear function of distance from complete correlation, and estimated correlation a logarithmic function of this distance. Performance differed only with regard to the level of noise in the feature, with these values compatible with estimates of channel capacity encountered in classic experiments on absolute perceptual magnitudes. These results suggest that quantitative information can be conveyed by visual features that are abstracted at relatively low levels of visual processing, with little representation of the original sensory property. It is proposed that this is achieved via an abstract parameter space in which the values in each perceptual dimension are normalized to have the same means and variances, with perceived correlation based on the shape of the joint probability density function of the resultant elements. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Coupling Warm Rain With an Eddy Diffusivity/Mass Flux Parameterization: 1. Model Description and Validation

AbstractA new version of the stochastic multiplume Jet Propulsion Laboratory Eddy‐Diffusivity/Mass‐Flux (JPL‐EDMF) parameterization which consistently couples the simplified Khairoutdinov and Kogan (2000), https://doi.org/10.1175/1520-0493(2000)128&lt;0229:ANCPPI&gt;2.0.CO;2, warm phase cloud microphysical parameterization with the parameterization of cloud macrophysical and subgrid scale dynamical processes is described. The new parameterization combines the EDMF approach with an assumed shape of a joint probability density function of thermodynamic and kinematic variables which provide the basis for the computation of all parameterized processes. As far as we are aware this is the first attempt to consistently couple all of these parameterized processes in the EDMF framework. This paper is part one of a two paper series. Here, the JPL‐EDMF parameterization is described and benchmark simulations of precipitating stratocumulus and cumulus convection are performed in a single‐column‐model framework. The parameterization results compare favorably to the reference large‐eddy‐simulation results. In the second part (Smalley et al., 2022, https://doi.org/10.1029/2021MS002729) the JPL‐EDMF parameterization is validated for a wide range of observation‐based scenarios covering the continuous transition from subtropical stratocumulus to cumulus convection derived from global reanalysis, and parameterization uncertainties are studied in detail.

Statistical modelling of the joint probability density function of air density and wind speed for wind resource assessment: A case study from China

This paper proposes a joint probability density function of air density and wind speed, aiming to integrate the air density distribution into statistical wind resource models, thereby helping better understand the correlation between air density and wind speed at different spatiotemporal scales. Hourly air pressure, temperature, and wind speed time series data from 1745 meteorological stations distributed over China for the period 2008–2019 were used to calculate the empirical distributions of air density and wind speed. Four one-component probability density functions and ten two-component mixture probability density functions were fitted to the empirical distributions. It is indicated that the five-parameter Weibull–Gamma distribution is able to achieves more accurate and stable effects in both unimodal and bimodal air density empirical distribution, and the six-parameter Weibull–Burr distribution is the most appropriate distribution for describing wind speed regimes. The parameterised air density and wind speed marginal distributions were linked using five joint copulas. The goodness-of-fit evaluation of the copula models demonstrates that the Gumbel copula most accurately reproduces the bivariate empirical distribution of air density and wind speed. The application of joint distribution has the capability to simplify the evaluation procedures for wind resource assessment while significantly increasing the accuracy of the evaluation results, and also achieve a systematic assessment of available wind resources, considering the temporal patterns of air density. The results of this study will contribute to offering critical information for making better decisions on wind energy projects to realise full development and effective utilisation of wind resources.

The adaptive weight learning function for reliability analysis and its application to multiple active learning methods

AbstractIn engineering, evaluating the failure probability of structures with complex performance functions is a challenging task. Some active learning methods based on the Kriging model gain considerable attention for structural reliability analysis. The existing learning functions offer new learning criterions to select sample points, but the probability density function (PDF) of sample points is ignored in some learning functions. To decrease selecting sample points in low PDF regions, this study presents a new adaptive weight learning function (WLF). The learning function WLF constituted by the adaptive orient function and the joint PDF can be applicable to multiple learning functions. Depending on the importance degree of candidate sample points, the sample points can be assigned different weight by learning function WLF. When the leaning function WLF is applicated to the existing active learning methods, the sample point with a higher PDF in neighborhood of the limit state function (LSF) can be given a larger weight to preferentially select into design of experiments (DoE). Therefore, the learning function WLF can help the multiple learning functions to select informative sample points with high PDF, which can further improve the efficiency of these learning functions. Four examples are used to illustrate the accuracy and efficiency as well as the suitability of the learning function WLF.

Probabilistic Identification of Multi-DOF Structures Subjected to Ground Motion Using Manifold-Constrained Gaussian Processes

Bayesian uncertainty quantification has a pivotal role in structural identification, yet the posterior distribution estimation of unknown parameters and system responses is still a challenging task. This study explores a novel method, named manifold-constrained Gaussian processes (GPs), for the probabilistic identification of multi-DOF structural dynamical systems, taking shear-type frames subjected to ground motion as a demonstrative paradigm. The key idea of the method is to restrict the GPs (priorly defined over system responses) on a manifold that satisfies the equation of motion of the structural system. In contrast to widely used Bayesian probabilistic model updating methods, the manifold-constrained GPs avoid the numerical integration when formulating the joint probability density function of unknown parameters and system responses, hence achieving an accurate and computationally efficient inference for the posterior distributions. An eight-storey shear-type frame is analyzed as a case study to demonstrate the effectiveness of the manifold-constrained GPs. The results indicate the posterior distributions of system responses, and unknown parameters can be successfully identified, and reliable probabilistic model updating can be achieved.

Voltage Restoration After Unforeseen Disturbances in Weakly Observable Distribution Systems

Following an unforeseen disturbance in a power system, the system state should be known to extract the emergency voltage restoration strategy. Such a condition does not usually hold in distribution systems due to the lack of enough measurements. Here, the joint probability density function ( <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pdf</small> ) of uncertain parameters is extracted from historical data. This <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pdf</small> is updated to comply with the measurements. A set of samples is then generated that best models the updated <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pdf</small> . These data reconciliation and sampling techniques enable the functioning of the emergency voltage control ( <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">evc</small> ) with measurement scarcity. They are designed to be fast enough to meet the quasi-real-time requirements of the intended application. The <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">evc</small> problem is cast as a stochastic programming problem. The resultant problem is a mixed-integer non-convex (and hence, NP-hard) optimization problem. It is solved using five distinct <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sm</small> s based on relaxation/approximation of the non-convex constraints. The results show that a novel combination of relaxation and approximation aimed at respectively mitigating the emergency under- and over-voltages outperforms the others. The propounded stochastic <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">evc</small> is validated through numerical studies.

An active learning Kriging-based method combining the weight information entropy function and the adaptive candidate sample pool

The reliability analysis methods based on the Kriging model have been explored to update the design of experiments (DoE), but the number of calls to the performance function is high. In this paper, a new active learning method combining the weight information entropy function (WH) and the adaptive candidate sample pool is proposed to improve the reliability analysis efficiency. Based on the information entropy function (H), the learning function WH is constructed to consider Kriging variance and the joint probability density function (PDF). In this proposed method, the sample points can be assigned different weight by the learning function WH according to the important degrees of sample points. The point which not only nears the limit state function (LSF), but also has a high probability density function value and large Kriging variance is assigned more weight than others. To select the sample points with lower confidence level, the adaptive candidate sample pool is generated by Markov Chain Monte Carlo (MCMC) simulation. The Kriging model can be updated efficiently by the proposed method. Four numerical examples and an engineering example with implicit performance function are used to verify the efficiency and accuracy of the proposed method. The results show that the proposed method can significantly improve the computational efficiency of the reliability analysis without losing accuracy.

Velocity perturbations and Reynolds stresses in Holmboe instabilities

The velocity perturbations and Reynolds stresses associated with finite-amplitude Holmboe instabilities are investigated using linear stability analysis, numerical simulations, and laboratory experiments. The rightward and leftward propagating Holmboe instabilities are separated, allowing for a direct comparison of the perturbation fields between the numerical simulations and the linear stability analysis. The decomposition and superposition of the perturbation fields provide insights into the structure and origin of Reynolds stresses in Holmboe instabilities. Shear instabilities in stratified flows introduce a directional preference (anisotropy) in velocity perturbation fields, thereby generating Reynolds stresses. Here, we investigate this anisotropy by comparing pairs of horizontal and vertical velocity perturbations (u′,w′), obtained from the simulations and the laboratory experiment, with predictions from linear stability analysis. For an individual Holmboe mode, both the simulations and linear theory yield elliptical (u′,w′)-pairs that are oriented toward the second and fourth quadrants (u′w′&amp;lt;0), corresponding to the tilted elliptical trajectories of particle movement. Combining the leftward and rightward Holmboe modes yields (u′,w′) ellipses whose orientation and aspect ratio are phase-dependent. When averaged over a full cycle, the joint probability density functions of (u′,w′) in the linear theory and single wavelength simulations exhibit “steering wheel” structures. This steering wheel is smeared out in multiple wavelength simulations and the laboratory experiment due to varying wavelengths, resulting in an elliptical cloud. All of the approaches adopted in the present study yield Reynolds stresses that are comparable to those reported in previous laboratory and field investigations.

A reliability estimation method for spring group balanced system at different degradation temperatures

The failure process of a balanced system consisting of several independent springs with the same structure is more complicated than that of a single spring. The balanced system of spring group should not only satisfy that the performance of all springs in the spring group is greater than the failure threshold, but also ensure that the performance difference between springs is not big. In this paper, the degradation parameters of spring compression ratio related to ambient temperature are estimated and the degradation parameters of other temperatures are extrapolated by Arrhenius model. Then the joint probability density function of the maximum and minimum order statistics of the spring compression ratio is used to estimate the reliability of the spring group balanced system at different degradation ambient temperatures. The results show that the Arrhenius model is effective in fitting the parameters of spring degenerate drift. The model established in this paper realizes the reliability evaluation of the spring balance system at different temperatures.

An Optimization Model and Computer Simulation for Allocation Planning of Hospital Bed Resources

Aim. This study intends to improve the arrangement of hospital beds, make rational use of limited beds to minimize the time for patients to seek medical care, improve service quality, and reduce hospital operating costs. Methods. This study determines the reasonable evaluation indices, established a reasonable evaluation index system, and assesses the merits and demerits of the bed arrangement model for this problem. After observing and dealing with various types of patients’ conditions in the ophthalmology outpatient department of the hospital, some effective hospitalization rules are proposed. The coefficients of each evaluation index under the FCFS (first-come first-service) and hospitalization rules are obtained, and a mathematical model of evaluation indices is constructed. The merits and demerits of the evaluation index system under each rule are found with MATLAB programming. At the same time, considering that the five types of patients are in a certain serial sequence during the queuing process for medical treatments, a random planning model with the least average stay time of patient is established via introducing the corresponding joint probability density function for the stay time of each patient. The proportional allocation of beds is obtained by genetic algorithms with the speed and convenience of computer calculations. Results. The proportions of beds allocated to the five categories of patients with trauma, cataract in one eye, cataract in both the eyes, glaucoma, and retina are 15%, 16%, 28%, 11%, and 29%, respectively. Conclusion. The results are in good agreement with the actual situation, and the optimization model has a certain degree of reference significance for the reasonable allocation of beds.

Uncertainty Evaluation of Stochastic Structural Response with Correlated Random Variables

It has been realized that the influence of system parameter uncertainties may be very significant, even dominant, in stochastic response evaluation. Nevertheless, in reality, this evaluation process may be difficult to conduct due to these parameter variables (viz. structural property parameters, such as stiffness, damping, and strength, and excitation characteristics parameters, such as frequency content and duration) that are usually correlated with each other. Therefore, this study devotes to develop a method for evaluating stochastic response uncertainty involving correlated system parameter variables. In this method, the evaluation expression for the mean and standard deviation of the maximum response including uncertainty parameter variables are provided first; subsequently, a third-moment pseudo-correlation normal transformation is able to be performed for converting the correlated and non-normal system parameter variables with unknown joint probability density function (PDF) or marginal PDF into the mutually independent standard normal ones; ultimately, a point estimate procedure (PEP) based on univariate dimension reduction integration can be carried out for evaluating the structural stochastic response including uncertainty system parameters. Several numerical examples with an engineering background involving correlated system parameter variables are analyzed and discussed under stochastic excitation, and their results are compared with those yielded by Monte Carlo simulation (MCS) so as to demonstrate the effectiveness of the approach proposed. It indicated that the method proposed, in this study, provides an effective path to deal with uncertainty evaluation of stochastic structural response involving correlated random variables.

Understanding the Microphysical Control and Spatial‐Temporal Variability of Warm Rain Probability Using CloudSat and MODIS Observations

By combining measurements from MODIS and the CloudSat radar, we develop a parameterization scheme to quantify the combined microphysical controls by liquid water path (LWP) and cloud droplet number concentration (CDNC) of the probability of precipitation (PoP) in marine low cloud over tropical oceans. We demonstrate that the spatial‐temporal variation of grid‐mean in‐cloud <PoP> can be largely explained by the variation of the joint probability density function of LWP and CDNC in the phase space specified by the bivariate PoP (LWP and CDNC) function. Through a series of sensitivity tests guided by this understanding, we find that in the Southeastern Pacific and Atlantic the stratocumulus to cumulus transition of the <PoP> is mainly due to the variation of CDNC while the annual cycle is mainly due to the variation of LWP. The results of this study provide a viable way to diagnose the root cause of warm rain problems in global climate models.

The evolution of turbulent micro-vortices and their effect on convection heat transfer in porous media

New insight into the contribution of the microscale vortex evolution to convection heat transfer in porous media is presented in this paper. The objective is to determine how the microscale vortices influence convection heat transfer in turbulent flow inside porous media. The microscale temperature distribution is analysed using flow visualization in two dimensions using streamlines and in three dimensions using the Q-criterion. The pertinent observations are supplemented with a comparison of surface skin friction and heat transfer using: (i) surface skin-friction lines and (ii) the joint probability density function of the pressure and skin-friction coefficients, along with the Nusselt number. The microscale flow phenomena observed are corroborated with the features of the frequency spectra of the drag coefficient and macroscale Nusselt number. The large eddy simulation technique is used in this study to investigate the flow field inside a periodic porous medium. The Reynolds numbers of the flow are 300 and 500. The porous medium consists of solid obstacles in the shape of square and circular cylinders. Two distinct flow regimes are represented by using the porosities of 0.50 and 0.87. The results show that the surface Nusselt number distribution is dependent on whether the micro-vortices are attached to or detached from the surface of the obstacle. The spectra of the macroscale Nusselt number and the pressure drag are similar, signifying a correlation between the dynamics of heat transfer and the microscale turbulent structures. Both vortex shedding and secondary flow instabilities are observed that significantly influence the Nusselt number. The fundamental insight gained in this paper can inform the development of more robust macroscale models of convection heat transfer in turbulent flow in porous media.

Bottom wall shear stress fluctuations in shallow-water Langmuir turbulence

In neutrally stratified shallow water, full-depth Langmuir cells (LCs) can interact with the turbulent benthic boundary layer and, thus, influence bottom wall shear stresses. In this paper the impacts of full-depth LCs on the streamwise and spanwise wall shear stresses are systematically studied using the database obtained from wall-resolved large-eddy simulation of shallow-water Langmuir turbulence. Analyses focus on the instantaneous wall shear stress fluctuations and the joint probability density functions between the stress fluctuations and the LCs parts of the velocity fluctuations, which show that the linear superimposition effect and nonlinear modulation effect of LCs are responsible for the spanwise organized distribution of wall shear stress fluctuations. Compared with the statistics in pure shear-driven turbulence without LCs, the mean square values of wall shear stress fluctuations in shallow-water Langmuir turbulence are enhanced by the strong linear superimposition effect of LCs, while the skewness and kurtosis are reduced by the combination of the linear superimposition effect and nonlinear modulation effect of LCs. Based on the scalings of these effects, a new predictive model of wall shear stress fluctuations is proposed for shallow-water Langmuir turbulence. The proposed model can predict the spatial distribution and statistics of wall shear stress fluctuations using the LCs parts of velocity fluctuations measured above the water bottom. Owing to the persistence of the spanwise inhomogeneity of wall shear stresses induced by full-depth LCs, the new predictive model will be useful for improving the wall-layer modelling for shallow-water Langmuir turbulent flows.