Input Random Variables Research Articles

Generalized likelihood ratio method for stochastic models with uniform random numbers as inputs

We propose a new unbiased stochastic gradient estimator for a family of stochastic models driven by uniform random numbers as inputs. Dropping the requirement that the tails of the density of the input random variables decay smoothly, the estimator extends the applicability of the generalized likelihood ratio (GLR) method. We demonstrate the new estimator for several general classes of input random variates, including independent inverse transform random variates and dependent input random variables governed by an Archimedean copula. We show how the new estimator works in settings such as density estimation, and we illustrate applications to credit risk derivatives. Numerical experiments substantiate broad applicability and flexibility in dealing with discontinuities in the sample performance.

A decentralized energy management scheme for a DC microgrid with correlated uncertainties and integrated demand response

Secure operation of a DC microgrid with high penetration of renewables and electric vehicle load is challenging. This paper proposes a decentralized energy management scheme for a grid-connected renewable integrated community DC microgrid considering the water-energy nexus. Uncertainties of renewable generation, plug-in electric vehicle load, electricity demand, electricity price in the day-ahead wholesale market, ambient temperature, and water demand are modelled using a probabilistic approach. Correlation between the input random variables is modelled using Copula theory. Flexibilities on the consumer side across multiple entities (temperature-dependent loads, electric vehicles, and water supply system storage) are coordinated to form an “integrated demand response entity”, which is further coordinated with the DC microgrid operator side flexibility (electrochemical storage) to support system operation. A distributed dynamic pricing scheme is used to implement the integrated demand response program. The objectives of the energy management scheme are to minimize the DC microgrid operator’s operating cost and consumers’ electricity cost. The decentralized algorithm is solved by the “alternating direction method of multipliers”. Simulation studies on a six-bus DC microgrid test system demonstrate that the proposed strategy reduces the operating cost of the DC microgrid operator by ∼19.28% and the electricity cost of the consumers by ∼13.82%.

Second-order reliability analysis of an energy pile with CPT data

As pile foundations and ground source heat producers, energy piles are becoming more and more popular. Even after much research, engineers find their design to be challenging. An effective way for using the Second-Order Reliability method (SORM) in foundation engineering scenarios including explicit limit state functions is shown in this article. The suggested SORM process is simple to implement in a spreadsheet and is based on an approximate paraboloid fitted to the limit state surface close to the design point. Using well-established closed-form SORM formulae, the failure probability is automatically determined using the primary curvatures of the limit state surface and the reliability index. An energy pile foundation engineering design examples are analyzed using the proposed method and discussed. Comparison with FORM, SORM and Genetic Programming (GP)-based FORM and SORM are made. In the case of determination of group capacity of pile and thermal load the input random variables are qc0, qc1, qc2, fs and E. The reliability index and probability of failure value are found to be 9.34 for FORM, 9.55 for SORM, 9.24 for GP-based FORM, and 9.45 for GP-based SORM. The probability of failure value for FORM 4.60E-21, SORM 3.24E-21, GP-based FORM 1.20E-20, and GP-based SORM is 8.50E-21. The relative percentage error for the average second-order approximation corresponding to FORM is also calculated, and found that Pf2 is 29.48 % in the case of simple SORM and 29.15 % in the case of GP-based SORM. The Breitung provides the most accurate estimation, and the Cai_Elishakoff provides the least accurate estimate.

Higher-order moments of spline chaos expansion

Spline chaos expansion, referred to as SCE, is a finite series representation of an output random variable in terms of measure-consistent orthonormal splines in input random variables and deterministic coefficients. This paper reports new results from an assessment of SCE’s approximation quality in calculating higher-order moments, if they exist, of the output random variable. A novel mathematical proof is provided to demonstrate that the moment of SCE of an arbitrary order converges to the exact moment for any degree of splines as the largest element size decreases. Complementary numerical analyses have been conducted, producing results consistent with theoretical findings. A collection of simple yet relevant examples is presented to grade the approximation quality of SCE with that of the well-known polynomial chaos expansion (PCE). The results from these examples indicate that higher-order moments calculated using SCE converge for all cases considered in this study. In contrast, the moments of PCE of an order larger than two may or may not converge, depending on the regularity of the output function or the probability measure of input random variables. Moreover, when both SCE- and PCE-generated moments converge, the convergence rate of the former is markedly faster than the latter in the presence of nonsmooth functions or unbounded domains of input random variables.

Dynamic reliability analysis of angular contact ball bearing based on maximum entropy quadratic fourth moment method*

ABSTRACT Bearing failure occurs when the dynamic contact stress exceeds the yield strength of the bearing material during operation. To minimise the risk of bearing failure, a new reliability theory method is proposed based on the stress-strength model of bearings, incorporating random input variables to analyse the reliability of angular contact ball bearings (ACBBs) and quantify their dynamic performance. The maximum entropy quadratic fourth moment method is used to calculate the probability distribution of maximum orthogonal shear stress-strength of bearing, and the maximum entropy principle is used to obtain the approximation of the minimum human error factors. Additionally, reliability-based sensitivity indices are derived to investigate the parametric significance of random input variables. Using B218 bearing as an example, the engineering application of this method in dynamic reliability and reliability-based sensitivity analysis is illustrated. Monte Carlo simulation is performed to verify the accuracy and effectiveness of the method to provide benchmark results. The results show that the Poisson’s ratio have a greater influence on the dynamic reliability of ACBBs than other parameters. Improving parameters such as ball diameter, raceway groove curvature radius and contact angle can also improve the reliability of bearings to some extent.

Global sensitivity analysis and optimization of multi-stage TEG based on random and fuzzy mixed input variables

In practical engineering, uncertainties in thermoelectric power generation arise from various factors such as material properties, geometric dimensions, assembly processes, and operational fluctuations of the thermoelectric components. Addressing subjective and objective uncertainties in the design parameters of multi-stage thermoelectric generators (TEGs), this study establishes a robust TEG model incorporating non-equilibrium thermodynamics. Fuzzy and stochastic theories are employed for uncertainty representation, and a variance-based global sensitivity analysis is conducted. The results indicate that the uncertainties in the design variables have varying effects on the output responses of the three-stage TEG. Among them, the temperature of the hot and cold sources has the most significant impact on the output responses（STHP: 0.5018, STLP:0.2199, STHη: 0.5436, STLη: 0.3336）, Parameters like Seebeck coefficient, thermal resistance, thermal conductivity, and operating current exhibit relatively minor effects. The coefficient of variation of the output response varies consistently with changes in the standard deviation or coefficient of variation of input variables, following a similar order as the sensitivity indicators. The variations in the distribution parameters of the key design variable, i.e., the temperature of the hot and cold sources, do not affect the sensitivity index rankings of the variables concerning the system output responses. As the distribution parameters of the hot and cold source temperatures gradually increase, the sensitivity indices of the design variables to the system output responses exhibit different trends, indicating a change in the degree of sensitivity to the variations in distribution parameters of the design variables. Optimization based on global sensitivity analysis enhances power output by 43.37 %, reaching a maximum of 11.37 W under constraints compared to the original 7.93 W. This research provides guidance for designing and optimizing thermoelectric power systems under mixed uncertainties.

An active multiphase probabilistic power flow based on a clustering approach

Probabilistic methods for analyzing power systems have become more widespread, mainly due to the increase in load and generation uncertainties and the current need for planning problems that are sufficiently accurate to take appropriate actions. However, some characteristics need to be better investigated in the Distribution Systems (DS) analysis, such as the input random variables’ modeling using real databases and, mainly, the probabilistic treatment of unbalance and scalability studies. This paper proposes an active Multiphase Probabilistic Power Flow (MPPF) evaluation, considering real measurement databases to represent customer demand uncertainties. It is called Multiphase Probabilistic Power Flow Based on K-Means (MPPFKM), and it encompasses a mixture of a clustering technique and an analytical method. The K-Means algorithm divides the input database into a set of clusters that models the scenarios to be evaluated in the multiphase power flow problem, which considers the distribution network with its inherent unbalance. Subsequently, the output variables’ higher order statistical moments are calculated using the obtained responses for each cluster. Finally, their PDFs and CDFs are recovered using the Gram-Charlier Expansion analytical method. While the clustering algorithm allows working with real data, giving the proposed work greater proximity to real-world results, the analytical tool provides an accurate recovery of the probability functions, because it detects their distortions caused by the nature of the multiphase problem. Estimating the correct number of clusters is a key factor in obtaining the precision required by power systems engineers and achieving significant computational time reduction, especially for large-scale unbalanced systems. Simulations are carried out using several DS feeders to test the proposed method, which specifically addresses DS issues not yet well explored in the literature, such as probabilistic simulations under different unbalance situations, applications with single-phase DGs, and scalability studies are conducted using the IEEE 8500 test feeder. Comparisons are made with usual MPPF algorithms, including Monte Carlo Simulation (MCS) and Point Estimation Methods (PEM). Simulations using large-scale unbalanced systems proved that the output variables' probability functions might present distortions, becoming more challenging for other methods. Under this perspective, MPPFKM presents better exactness in the calculation of higher order statistical moments and, consequently, can estimate the PDFs of the MPPF output variables with great precision, which is crucial for statistical models that use actual input data.

Fatigue reliability and sensitivity analysis of aero‐disk considering correlation

AbstractAero‐disk is a key component of aero‐engine. Due to its complex working conditions, aero‐disk is prone to structural failures. Therefore, it is essential to analyze the reliability and importance input variables of aero‐disk. In this paper, a framework of aero‐disk reliability analysis and global sensitivity analysis (GSA) was established. First, the hazardous regions of aero‐disk were determined by finite element analysis, and the limit state function of aero‐disk was defined by the life interference model. Then D‐Vine model was utilized to establish the correlation model for aero‐disk hazardous regions to evaluate the reliability of aero‐disk. In addition, two GSA methods based on Copula and space partition were proposed to identify important input random variables considering underlying correlation, and the accuracy of the proposed method was verified by two examples. Finally, the proposed method was applied to the GSA of aero‐disk. The results show that the established framework fills in the gap of uncertainty analysis of aero‐disk, which can be extended to other engineering fields. The proposed GSA methods have both high efficiency and accuracy and can realize multi‐dimensional GSA when the correlation of input variables is considered.

An adaptive data-driven subspace polynomial dimensional decomposition for high-dimensional uncertainty quantification based on maximum entropy method and sparse Bayesian learning

Polynomial dimensional decomposition (PDD) is a surrogate method originated from the ANOVA (analysis of variance) decomposition, and has shown powerful performance in uncertainty quantification (UQ) accuracy and convergence recently. However, complex high-dimensional problems result in a large number of polynomial basis functions, leading to heavy computational burden, and the probability distributions of the input random variables are indispensable for PDD modeling and UQ, which may be unavailable in practical engineering. This study establishes an adaptive data-driven subspace PDD (ADDSPDD) for high-dimensional UQ, which employs two types of data for modeling the PDD basis function and the low-dimensional subspace directly, namely, the data of input random variables and the input-response samples. Firstly, we propose a data-driven zero-entropy criterion-based maximum entropy method for reconstructing the probability density functions (PDF) of input variables. Then, with the aid of the established PDFs, a data-driven subspace PDD (DDSPDD) is proposed based on the whitening transformation. To recover the subspace of the function of interest accurately and efficiently, we put forward an approximate active subspace method (AAS) based on the Taylor expansion under some mild premises. Finally, we integrate an adaptive learning algorithm into the DDSPDD framework based on the sparse Bayesian learning theory, obtaining our ADDSPDD; thus, the real subspace and the significant PDD basis functions can be identified with limited computational budget. We validate the proposed method by using four examples, and systematically compare four existing dimension-reduction methods with the AAS. Results show that the proposed framework is effective and the AAS is a good choice when the corresponding assumptions are satisfied.

K-Medoids clustering applications for high-dimensionality multiphase probabilistic power flow

Actual power systems’ planning studies require the stochastic modeling of networks due to their increasing penetration of uncertain parameters, such as probabilistic-nature loads and renewable generation features. Although numerous Probabilistic Power Flow (PPF) methods have been presented in the literature, the scalability of such tools is still under study due to the ‘curse of dimensionality’, which states that the probabilistic methods lose their accuracy with the increase of random input variables. Since many authors are using clustering algorithms to solve the PPF, mainly because of their easy implementation and good computational precision, this paper proposes for the first time the K-Medoids clustering method to model uncertainties in unbalanced distribution systems. Modifications in the clustering algorithm are included to improve its accuracy in view of a large set of input variables. The multiphase modeling of networks is also considered to represent the inherent unbalance of distribution systems correctly. Simulations were carried out using the IEEE unbalanced test-feeders IEEE 123 and the large-scale system IEEE 8500, which includes more than 2000 input variables. Results show that K-Medoids are a better alternative in relation to other numerical/approximation approaches, such as Monte Carlo Simulation or Point Estimate Methods, because it is faster and presents great accuracy, especially for large-scale applications. Comparisons were also made regarding other clustering approaches, highlighting the benefits of the proposed one in scalability.

A simulation model to analyze the behavior of a faculty retirement plan: a case study in Mexico

AbstractThe main goal in this study was to determine confidence intervals for average age, average seniority, and average money-savings, for faculty members in a university retirement system using a simulation model. The simulation—built-in Arena—considers age, seniority, and the probability of continuing in the institution as the main input random variables in the model. An annual interest rate of 7% and an average annual salary increase of 3% were considered. The scenario simulated consisted of the teacher and the university making contributions, the faculty 5% of his salary, and the university 5% of the teacher’s salary. Since the base salaries with which teachers join to university are variable, we considered a monthly salary of MXN 23 181.2, corresponding to full-time teachers with middle salaries. The results obtained by a simulation of 30 replicates showed that the confidence intervals for the average age at retirement were (55.0, 55.2) years, for the average seniority (22.1, 22.3) years, and for the average savings amount (329 795.2, 341 287.0) MXN. Moreover, the risk that a retiree of 62 years of age and more of 25 years of work, is alive after his savings runs out is approximately 98% and this happens at 64 years of age.

Probabilistic Load Flow Analysis Using Nonparametric Distribution

In the pursuit of sustainable energy solutions, this research addresses the critical need for accurate probabilistic load flow (PLF) analysis in power systems. PLF analysis is an essential tool for estimating the statistical behavior of power systems under uncertainty. It plays a vital part in power system planning, operation, and dependability studies. To perform accurate PLF analysis, this article proposes a Kernel density estimation with adaptive bandwidth for probability density function (PDF) estimation of power injections from sustainable energy sources like solar and wind, reducing errors in PDF estimation. To reduce the computational burden, a Latin hypercube sampling approach was incorporated. Input random variables are modeled using kernel density estimation (KDE) in conjunction with Latin hypercube sampling (LHS) for probabilistic load flow (PLF) analysis. To test the proposed techniques, IEEE 14 and IEEE 118 bus systems are used. Two benchmark techniques, the Monte Carlo Simulation (MCS) method and Hamiltonian Monte Carlo (HMC), were set side by side for validation of results. The results illustrate that an adaptive bandwidth kernel density estimation with the Latin hypercube sampling (AKDE-LHS) method provides better performance in terms of precision and computational efficiency. The results also show that the suggested technique is more feasible in reducing errors, uncertainties, and computational time while depicting arbitrary distributions of photovoltaic and wind farms for probabilistic load flow analysis. It can be a potential solution to tackle challenges posed by sustainable energy sources in power systems.

Probabilistic Modeling for Conditional Statements

A new mathematical framework is proposed in this study to comprehend the impact of program architecture on input random variables, the IF statement was the main topic. The primary idea that is theoretically and experimentally supported in this study is that the part of the joint pmf of a collection of random variables that represents the condition will be shifted to the part that represents the action. After sorting two random variables, the framework is used with four random variables, and the theoretically produced results were realistically validated. The study's equations can be applied to assessing probabilistic models of various sorting algorithms or other intricate program structures. This may also result in future investigations formalizing more precise execution time expectations.

Systematic framework for handling uncertainty in probabilistic failure analysis of corroded concretes

Due to ambiguous correlations between input random variables and multi-source uncertainties from the electrochemical model, significant differences between deterministic corrosion prediction models and actual measurements are often observed. A systematic framework of quantification of uncertainties is developed for structures with correlated random variables originated from multiple sources, which allows efficiently estimating the failure probability distribution of the steel corrosion over time considering the randomness of the cover depth, the surface chloride concentration, and the chloride diffusion coefficient. After partitioning correlated random variables into different groups based on their uncertain sources, the Morris one-step-at-a-time and Sobol model is established to rank with respect to the importance of each correlated random variable. Based on polynomial chaos expansions and genetic programming methods, a more condensed set of random variables is created to propagate parametric problems. The unknown probability distribution of the input random variables is formulated by the Markov chain Monte Carlo to realize rigorous uncertainty quantification of the structural reliability. The application of the systematic framework to a set of numerical examples of steel corrosion includes experimental validation and uncertainty quantification and propagation of environmental, material and geometric properties. The results show that the framework can be integrated with parametric electrochemical models to allow robustness and reliability of corrosion prediction.

A generalized isogeometric boundary element method for the uncertain analysis of infinite domain two-dimensional acoustic problems

The key aim of this paper is to provide a new nth generalized order perturbed isogeometric fast multistage technique of boundary elements to compute the propagation of time harmonics in an infinite region. Structural geometry and boundary integral equations are constructed by using non-uniform rational B-splines. The source of system uncertainty is believed to be the incident plane wave number’s unpredictability. The actual field, depending on the input random variables, is simulated using the extended nth-order perturbation method. The field and kernel values for boundary integral formulas are generated via the nth-order generalized series of Taylor expansions using perturbation parameters. The fast multipole method (FMM) is utilized to speed up the process. The effectiveness and correctness of the proposed algorithm are verified by Monte Carlo simulations (MCs) with numerical examples.

Risk-constrained stochastic scheduling for energy hub: Integrating renewables, demand response, and electric vehicles

This research introduces a stochastic scheduling approach that incorporates risk constraints for an energy hub (EH), considering uncertainties related to renewable generation and load demands. The proposed method utilizes the Conditional Value at Risk (CVaR) technique to assess and quantify risks. By striking a balance between reducing operational and emissions costs and increasing risk aversion, the approach presents a trade-off. The EH comprises various components such as a wind turbine (WT), photovoltaic (PV) cells, a fuel cell power plant (FCPP), a combined heat and power generation unit (CHP), and plug-in electric vehicles (PEVs). Uncertain variables encompass factors such as wind speed, solar irradiation, different demands, and market prices. To optimize profits and enhance the consumption curve, demand response programs (DRPs) for electrical, thermal, and cooling demands are implemented. To address the uncertainties associated with input random variables, the efficient k-means data clustering method is employed. A new slime mold algorithm, based on coughing and chaos theory, has been proposed to enhance the problem's solution. The algorithm incorporates innovative operators to improve its capabilities. By utilizing the coughing mechanism and chaos theory, the algorithm explores the solution space more effectively, resulting in improved outcomes for the problem at hand. The results demonstrate significant flexibility in EH management and are extensively discussed. Simulation results indicate that integrating PEVs, FCPP, and DRPs can lead to reductions of 2 %, 7 %, and 11 % in the EH's operating costs, respectively. Furthermore, considering PEVs, FCPP, and DRPs can improve the EH's risk cost by 1.98 %, 6.7 %, and 10.5 %, respectively. Based on the numerical results, in Case 4 led to a remarkable 12.65 % reduction in operational costs while simultaneously achieving a 15.43 % decrease in emission costs, showcasing the effectiveness of the proposed approach in optimizing energy management in an energy hub system.

Reliability sensitivity analysis for RBSMC: A high-efficiency multiple response Gaussian process model

In modern architectural design, reduced beam section moment connections (RBSMC) play a crucial role. However, quantifying the importance of random input variables and failure modes has not been performed to improve RBSMC reliability. Thus, we propose a sensitivity study that evaluates the effect of geometric dimensions on the reliability of the RBSMC. The analysis measures the significance of each random input variable on the reliability of the RBSMC. Then, a reliability predictor, which combines the multiple response Gaussian process (MRGP) with the Monte Carlo simulation (MCS), is introduced to efficiently produce surrogate models for failure surfaces. Based on the well-established surrogate models, the reliability is estimated without additional calling of the original limit state functions. The quadratic regression and analysis of variance are conducted to statistically evaluate the contribution of different variables to the reliability of the RBSMC. The proposal methodology accurately predicts reliability and identifies variables that significantly contribute to reliability. The results show that the proposed methodology is more efficient than existing methods at the early design stage. Furthermore, we applied this methodology to a radius cut RBSMC, which can minimize stress concentration and reduce the possibility of fracture to identify significant input variables for reliability. The computational results and discussions provide useful insights for the design of RBSMC.

Data-Driven Nonparametric Probabilistic Optimal Power Flow: An Integrated Probabilistic Forecasting and Analysis Methodology

With large-scale integration of renewable energy such as wind power, probabilistic analysis of optimal power flow becomes crucial for the decision-making of power systems. This paper proposes a novel data-driven integrated probabilistic forecasting and analysis (IPFA) methodology for the nonparametric probabilistic optimal power flow (N-POPF), which internalizes the probabilistic forecasting and nonparametric distributional description forms into uncertainty analysis. The proposed IPFA methodology fully utilizes the uncertainty analysis method to guide the model-free nonparametric probabilistic forecasting of wind power, and then conducts the N-POPF analysis effectively based on the uncertainty information contained in historical data. A comprehensive uncertainty evaluation criterion based on point estimate method and differential entropy is proposed to assess both the inherent uncertainty and uncertainty influence of input random variables. Then a model-free multivariate probabilistic forecasting method is established to directly support the solving of N-POPF problems with similar historical measurements. Finally, with deterministic optimal power flow problems corresponding to the selected historical samples, a weighted combination approach for the power flow results is developed to derive the quantiles of the output random variables. Comprehensive experiments on IEEE 24-bus and 118-bus test systems validate the superiority of the proposed IPFA methodology in estimation accuracy and computational efficiency.

Interval-Probabilistic Electricity-Heat-Gas Flow Calculation by Dual-Level Surrogate Structure

The study of highly efficient methods for performing multiple energy flow (MEF) computation with epistemic and aleatory uncertainties, known as interval-probabilistic energy flow (IPEF) calculation, is a fast-growing area of research. However, existing approaches are easily caught in a quandary among accuracy, efficiency, and scope of applicability. This paper presents a dual-level surrogate structure (DLSS) for dealing with the IPEF calculation of an integrated energy system (IES) in an accurate and efficient manner. The structure accommodates electricity, heat, and gas, attaining a high-level balance among precision, speed, and range of applicability. At the lower level of DLSS, a sparse polynomial chaos expansion (sPCE) model is employed to directly map random input variables to output boundary values, saving massive repetition of deterministic calculation. At the upper level of DLSS, a heterogeneous-learning-based multiple energy flow (HL-MEF) model is proposed to accelerate the indispensable deterministic MEF calculation for online sampling, thereby easing the computational burden of sPCE training. Finally, numerical results on a real-world electricity-heat-gas system demonstrate that the proposed DLSS can solve the IPEF problem at nearly 400 times the speed of the standard double-layer Monte Carlo simulation (DLMCS) with guaranteed accuracy.

Stochastic nonlinear inelastic analysis for steel frame structure using Monte Carlo sampling

This work comprehensively investigates the realistic ultimate resistance of steel frame structures based on the probability study model of input random variables due to uncertainty of material properties and geometric parameters. Four independent parameters and thirteen of their dependent components vary randomly in a variational range and seven scenarios of random combination are investigated by the probabilistic modeling, which incorporates the advanced inelastic analysis with direct Monte Carlo simulation. The analysis results for a typical example shows that the stochastic analysis result of steel frame's realistic ultimate resistance is lower than its deterministic analysis result, indicating that a stochastic analysis is needed to comprehensively qualify the safety of the structure. Moreover, a spectrum response, which is produced based on stochastic analysis, provides a better scientific solution.