Assumed Probability Distribution Research Articles

Reliability analysis of deep tunnels in spatially varying brittle rocks using interval and random field modelling

Rock properties are estimated using objective lab/in-situ testing and subjective judgements invoking different types of uncertainties, i.e., aleatory, and epistemic, along them, often indicated by varying information levels. This study presents a unified reliability method to integrate the spatial variability of inputs modelled via alternate uncertainty models (intervals and probability-boxes (p-boxes)) with those modelled as stochastic variables via probability distributions. The methodology employs advanced Karhunen–Loève decomposition to generate interval and random fields of inputs modelled via alternate and stochastic models, respectively. Input properties are allocated to the zones of the numerical model in Fast Lagrangian Analysis of Continua-2D (FLAC-2D) based on their spatial dependency and correlation functions through a developed MATLAB-FLAC coupled code. The methodology is demonstrated for a deep tunnel in Canada to be constructed along a massive rock prone to brittle failures. Intact rock properties are modelled as stochastic variables due to objective estimation, while Geological Strength Index (GSI) and deformation modulus are modelled using alternate models (interval and p-box, respectively) due to subjective and hybrid estimation (double uncertainty propagation algorithm). The results of the methodology are compared with those of traditional deterministic and random field methods. The methodology reduces the subjectivity invoked by including unavailable additional information (e.g., assuming probability distributions of inputs based on literature) and propagates the originally available information of inputs accurately. The final outputs are the p-boxes of response parameters (i.e., displacements and damage zone) instead of their fixed values (deterministic analysis) and probability distributions (traditional reliability analysis), indicating the propagation of both impreciseness and variability of inputs by the method. For this case study, the p-boxes of outputs were bounding their values/distributions estimated via traditional analyses, verifying the accuracy of the methodology. Further, the impreciseness in the outputs, highest in the damage zone extent, was due to imprecision in the estimated GSI.

Mixing lint grades in relation to fiber and yarn quality of Egyptian cotton

This study was undertaken to investigate the effect of different mixing percentages of lint grades of Egyptian cotton varieties on fiber properties and yarn strength. Two lint grades within each variety of four Egyptian cottons from 2022 and 2023 crop seasons were used; Good to Fully Good (G/FG) and Fully Good Fair (FGF) for Giza 86, Good to Fully Good (G/FG) and Fully Good Fair to Good (FGF/G) for Giza 92, Good to Fully Good (G/FG) and Good Fair (GF) for Giza 94 and, Fully Good (FG) and Fully Good Fair (FGF) for Giza 95 with mixing percentage (30%-40%-50%) of the highest grades for each variety. Mixed lots of lint cotton grades were tested on Fiber Classifying System (FCS) instrument to measure fiber properties. Yarn strength was tested on Good Brand Lea tester at 60s and 3.6 twist factor. This experiment was conducted at Egyptian & International Cotton Classification Center (EICCC) in Cotton Research Institute (CRI), Agricultural Research Center (ARC), Giza, Egypt. Using traditional statistical analysis (normal probability distribution) with untraditional method (assumed probability distribution) illustrated that; It was remarkably increased upper half mean length,mean length, uniformity index, fiber strength, micronaire reading, maturity and linear density with increase of the percentage of the highest grade in the mixing for each studied variety. Meanwhile short fiber content showed the opposite trend, whereas, elongation had no obvious trend. Prominent improvement in yarn strength was observed when the highest lint cotton grade percentage increased from 30% through 40% and 50% in the mixing lots. Yarn strength increased from 2690 to 3170, from 2475 to 2850, from 2440 to 2845 and from 2135 to 2360 for Giza 92, Giza 86, Giza 94 and Giza 95, respectively. Generally, mixing method based on lint grades exhibited significant improvement in fiber and yarn quality properties. In the same time this method is easier than the experimental mixing methods based on micronaire reading or fiber strength. on the other hand, it is very important process in cotton production economy.

Small-sample linear profile error uncertainty assessment based on grey system

The uncertainty assessment of the profile error of the cam profile, as defined in the national standard method, is difficult to carry out under conditions of small sample size and absence of probability distribution assumptions. This paper proposes a small-sample assessment model for the uncertainty of the profile error based on grey system. Firstly, the coordinate transformation is conducted using Vector Alignment Method to reduce systematic errors, and the non-uniform rational B-splines curve interpolation is utilized to fit the cam profile curve and perform error assessment. Subsequently, based on the error assessment results, Grey Information Measurement Model (GIMM) for the uncertainty of the profile error in small samples is established. This model employs Grey Relational Analysis to eliminate outliers and evaluates the uncertainty of the profile error by solving grey correlation coefficients. Maximum-Minimum Information Measure Method is used to assess the optimal sample size. Finally, numerical experiments and experimental tests were conducted on the uncertainty of camshaft profile error in automobiles. A total of 15 sets of profile data were compared with Guide to the Representation of Uncertainty in Measurement (GUM) and Monte Carlo Method (MCM) under different sample sizes. The results showed that GIMM achieved evaluation with only 8 sets of data samples under small sample and poor information conditions, with an uncertainty of 0.6338 μm, compared to 0.6346 μm for GUM and 0.6391 μm for MCM. The acceptance rate of GIMM reached 95.2%. This model outperforms other methods, providing a simplified and reliable assessment of cam profile error uncertainty.

Enabling efficient regional seismic fragility assessment of multi‐component bridge portfolios through Gaussian process regression and active learning

AbstractRegional seismic fragility assessment of bridge portfolios must address the embedded uncertainties and variations stemming from both the earthquake hazard and bridge attributes (e.g., geometry, material, design detail). To achieve bridge‐specific fragility assessment, multivariate probabilistic seismic demand models (PSDM) have recently been developed that use both the ground motion intensity measure and bridge parameters as inputs. However, explicitly utilizing bridge parameters as inputs requires numerous nonlinear response history analyses (NRHAs). In this situation, the associated computational cost increases exponentially for high‐fidelity bridge models with complex component connectivity and sophisticated material constitutive laws. Moreover, it remains unclear how many analyses are sufficient for the response data and the resulting demand model to cover the entire solution space without overfitting. To deal with these issues, this study integrates Gaussian process regression (GPR) and active learning (AL) into a multistep workflow to achieve efficient regional seismic fragility assessment of bridge portfolios. The GPR relaxes the probability distribution assumptions made in typical cloud analysis‐based PSDMs to enable heteroskedastic nonparametric seismic demand modeling. The AL leverages the varying standard deviation to select the least but most representative bridge‐model‐ground‐motion sample pairs to conduct NRHA with much‐improved efficiency. Both independent and correlated multi‐output GPRs are proposed to deal with bridge portfolios with seismic demand correlations among multiple components (column, bearing, shear key, abutment, unseating, and joint seal). Considering a single benchmark highway bridge class in California as the case study, the AL‐GPR framework and the associated component‐level fragility results are investigated in terms of their efficiency, accuracy, and robustness. The fragility results show that 70 AL‐selected samples would enable the GPR to derive bridge‐specific fragility models comparable to the ones using the multiple stripes analysis approach with 1950 ground motions considered for each individual bridge. The AL‐GPR model also successfully captures the physics of how bridge span length, deck area, column slenderness, and steel reinforcement ratio would change the damage state exceedance probabilities of different bridge components. The efficiency of AL stems from the fact that, with the multi‐output independent GPR, a stable and reliable fragility model can be achieved using 50 AL‐selected samples compared to at least 270 randomly chosen samples. The proposed methodology advances the state of the art in enabling more efficient and reliable regional seismic fragility assessment of multi‐component bridge portfolios.

Bayesian Risk Assessment Technique for Economic Stress-Strength Models

This paper explores two areas of risk assessment modelling in economics and business: the Stress-Strength model and Bayesian techniques. The model assumes that the probability of stress exceeding strength is a measure of risk. The interpretation of stress and strength largely depends on the particular event or phenomenon being modelled. The use of the Stress-Strength model is demonstrated through the Gaussian assumption of probability distributions for random model parameters, particularly in assessing the risk of not achieving a required margin value. The concept of the capability function, representing the difference between strength and stress, is introduced in the modelling process. The probability distribution for the capability function is initially determined based on the Gaussian distribution of the random variables used in the model, allowing for evaluating the risk metric. The Bayesian approach is then applied to generalise the problem statement when dealing with unknown parameters of probability distributions for the Stress and Strength models. The uncertainty of these parameters is modelled through uniform probability distributions, and equations for calculating prior and posterior estimates are consistently obtained. Since multidimensional integrals are involved in these calculations, and solutions cannot be obtained in closed analytical form, Monte Carlo simulation is used to solve this computation problem.

Mini-Batch Alignment: A Deep-Learning Model for Domain Factor-Independent Feature Extraction for Wi-Fi–CSI Data

Unobtrusive sensing (device-free sensing) aims to embed sensing into our daily lives. This is achievable by re-purposing communication technologies already used in our environments. Wireless Fidelity (Wi-Fi) sensing, using Channel State Information (CSI) measurement data, seems to be a perfect fit for this purpose since Wi-Fi networks are already omnipresent. However, a big challenge in this regard is CSI data being sensitive to ‘domain factors’ such as the position and orientation of a subject performing an activity or gesture. Due to these factors, CSI signal disturbances vary, causing domain shifts. Shifts lead to the lack of inference generalization, i.e., the model does not always perform well on unseen data during testing. We present a domain factor-independent feature-extraction pipeline called ‘mini-batch alignment’. Mini-batch alignment steers a feature-extraction model’s training process such that it is unable to separate intermediate feature-probability density functions of input data batches seen previously from the current input data batch. By means of this steering technique, we hypothesize that mini-batch alignment (i) absolves the need for providing a domain label, (ii) reduces pipeline re-building and re-training likelihood when encountering latent domain factors, and (iii) absolves the need for extra model storage and training time. We test this hypothesis via a vast number of performance-evaluation experiments. The experiments involve both one- and two-domain-factor leave-out cross-validation, two open-source gesture-recognition datasets called SignFi and Widar3, two pre-processed input types called Doppler Frequency Spectrum (DFS) and Gramian Angular Difference Field (GADF), and several existing domain-shift mitigation techniques. We show that mini-batch alignment performs on a par with other domain-shift mitigation techniques in both position and orientation one-domain leave-out cross-validation using the Widar3 dataset and DFS as input type. When considering a memory-complexity-reduced version of the GADF as input type, mini-batch alignment shows hints of recuperating performance regarding a standard baseline model to the extent that no additional performance due to weight steering is lost in both one-domain-factor leave-out and two-orientation-domain-factor leave-out cross-validation scenarios. However, this is not enough evidence that the mini-batch alignment hypothesis is valid. We identified pitfalls leading up to the hypothesis invalidation: (i) lack of good-quality benchmark datasets, (ii) invalid probability distribution assumptions, and (iii) non-linear distribution scaling issues.

A risk-averse day-ahead bidding strategy of transactive energy sharing microgrids with data-driven chance constraints

The rapid development of microgrids (MGs) with various prosumers promotes the accommodation of renewable distributed generation (RDG) and provides platforms for local energy sharing among prosumers. However, the operational uncertainties pose enormous challenges to the day-ahead bidding of MGs in the wholesale electricity market and there is an urgent need for a local market to facilitate the local energy sharing. Thus, this paper proposes a risk-averse day-ahead bidding strategy for MGs with full consideration of the multiple uncertainties originating from the wholesale electricity market, RDG and loads. Based on the transactive energy (TE) sharing concept, the local market is formulated as a Stackelberg game (SG) to effectively capture the strategic interaction among the MG and prosumers, where a distributed iterative algorithm with a bisection approach that only requires exchanging TE-related information is adopted to achieve the SG equilibrium without compromising privacy concerns. To handle the uncertainties of RDG and loads, the power balances are formulated as chance constraints and a data-driven quantile forecasting method is developed for achieving the computational tractability of chance constraints without any prior knowledge or probability distribution assumptions. Furthermore, a risk criterion of the conditional value-at-risk is incorporated in the day-ahead bidding model of MGs for risk aversion towards uncertainties of the wholesale electricity market. The effectiveness of the proposed solution is extensively demonstrated through numerical simulation.

Representing the Subgrid Surface Heterogeneity of Precipitation in a General Circulation Model

AbstractPrecipitation is highly variable on spatial scales smaller than a typical general circulation model (GCM) grid box. Neglecting this subgrid‐scale variability can impact the simulation of the underlying land surface and its coupling with the atmosphere. In this study we present a scheme to stochastically disaggregate precipitation from an atmospheric grid to an underlying mosaic of surface tiles, using an assumed probability distribution of subgrid precipitation derived from observations. Unlike previous GCM‐based schemes, our approach includes a treatment of memory to allow persistence of subgrid features. In single column model experiments, we demonstrate the ability of the scheme to reproduce observed precipitation statistics at the spatial scale of the surface tiles. The root mean square error of hourly spatial standard deviation of precipitation is reduced from 1.90 to 0.96 mm hr−1 when disaggregation is applied. The scheme increases the spatial standard deviations of surface heat fluxes, soil moisture and temperature, and we show that incorporating memory amplifies these increases by a factor of 2–4. We also document increases in mean precipitation runoff and the Bowen ratio.

Day-ahead bidding strategy of regional integrated energy systems considering multiple uncertainties in electricity markets

The regional integrated energy system (IES) is considered a promising paradigm for economic multi-energy provision while promoting the accommodation of renewable distributed energy resources (RDER). However, various uncertainties faced by the IES pose challenges to fulfilling its functionalities and economic promises. This paper proposes a day-ahead optimal bidding strategy of IES in electricity markets with multiple time scales of electricity settlement and energy dispatch for minimizing its expected operational cost based on the hybrid scenario-based stochastic and chance-constrained programming. It enables the IES to determine the optimal bidding electricity quantities in the day-ahead market in a multi-energy complementary manner by incorporating the potential real-time clearing price scenarios, with the uncertainties introduced by RDER generation and multi-energy loads efficiently addressed using chance constraints. Herein, a gradient boosted regression tree (GBRT) based quantile forecasting method is developed to equivalently reformulate the chance constraints to tractable deterministic constraints without any prior knowledge or probability distribution assumptions. The proposed day-ahead bidding strategy is extensively assessed through simulation experiments in the PJM and Guangdong electricity markets and the numerical results confirm the effectiveness of the proposed solution.

Adaptive client and communication optimizations in Federated Learning

Federated Learning (FL) is a very effective distributed machine learning framework that enables a large number of devices to jointly train models without sharing raw data. However, the process of iterative learning and data communication in FL can be time-consuming, depending heavily on the number of clients participating in training and the number of local iterations between two consecutive global aggregations (communication period). In this paper, we analyze the runtime of the FL framework and propose a joint optimization problem, which considers both the number of local clients involved in global training and the communication period to minimize the error-runtime convergence of FedAvg. By theoretical analysis, we obtain the optimal solutions of this optimization problem in closed forms under the assumption of different probability distributions of the client’s local computation time. In addition, we design an adaptive control algorithm that can dynamically select the number of clients and communication period during FL training to speed up the convergence of the model. Experimental results validate our theoretical analysis and show that the proposed algorithm has outstanding performance compared with other related FL control algorithms.

Explainable probabilistic deep learning framework for seismic assessment of structures using distribution‐free prediction intervals

AbstractA new probabilistic framework is proposed for providing a distribution‐free prediction interval (PI) of seismic responses required for various earthquake engineering applications. The framework overcomes the limitation of point prediction models and avoids the complexity of traditional probabilistic methods. The framework utilizes a few assumptions of probability distributions and requires no prior assumed statistical distribution for the PI. Ensemble probabilistic deep learning models (DLMs) are used to provide quality‐driven PIs of seismic responses for low‐ to mid‐rise buildings with limited irregularity. Considering these systems and ground motions with the aid of Monte Carlo simulation and nonlinear time‐history analysis (NLTHA), huge datasets are generated for training. To have an insight into the probabilistic DLM, explainable artificial intelligence techniques are used. The superiority of the proposed framework in quantifying uncertainties is validated by comparison with the conventional Bayesian method. In addition, its applicability is investigated by providing bounds of seismic fragility curves, life cycle cost, and resilience index obtained by NLTHA for a benchmark case study model. The results showed that the proposed framework is robust and outperforms the conventional Bayesian method in uncertainty quantification for the considered dataset.

Single-Location and Multi-Locations Scenarios Generation for Wind Power Based On WGAN-GP

To address the randomness of renewable energy, scenario generation can simulate the random process of renewable energy. Still, most of the previous scenario generation algorithms are based on assumed probability distribution models, which are difficult to grasp the dynamic characteristics of renewable energy accurately. Based on an improved generative adversarial network algorithm, this paper generates single-site and multi-site scenarios for wind power generation data. The temporal and spatial correlations of the generated data were judged based on the maximum mean difference and the Pearson coefficient. Finally, multiple sets of wind power generation data are used to verify that the proposed method can reflect not only the volatility of renewable energy but also ensure the temporal and spatial correlation between the data.

Modelling parametric uncertainty in large-scale stratigraphic simulations

We combine forward stratigraphic models with a suite of uncertainty quantification and stochastic model calibration algorithms for the characterization of sedimentary successions in large scale systems. The analysis focuses on the information value provided by a probabilistic approach in the modelling of large-scale sedimentary basins. Stratigraphic forward models (SFMs) require a large number of input parameters usually affected by uncertainty. Thus, model calibration requires considerable time both in terms of human and computational resources, an issue currently limiting the applications of SFMs. Our work tackles this issue through the combination of sensitivity analysis, model reduction techniques and machine learning-based optimization algorithms. We first employ a two-step parameter screening procedure to identify relevant parameters and their assumed probability distributions. After selecting a restricted set of important parameters these are calibrated against available information, i.e., the depth of interpreted stratigraphic surfaces. Because of the large costs associated with SFM simulations, probability distributions of model parameters and outputs are obtained through a data driven reduced complexity model. Our study demonstrates the numerical approaches by considering a portion of the Porcupine Basin, Ireland. Results of the analysis are postprocessed to assess (i) the uncertainty and practical identifiability of model parameters given a set of observations, (ii) spatial distribution of lithologies. We analyse here the occurrences of sand bodies pinching against the continental slope, these systems likely resulting from gravity driven processes in deep sea environment.

A unified low-order information-theoretic feature selection framework for multi-label learning

The approximation of low-order information-theoretic terms for feature selection approaches has achieved success in addressing high-dimensional multi-label data. However, three critical issues exist in such type of approaches: (1) existing approaches are devised based on single heuristic variable correlation assumption, which biases towards some specific scene; (2) high-order variable correlations are ignored by cumulative summation low-order information-theoretic terms; (3) abundant approaches confuse researchers to devise and utilize appropriate approaches. To address these issues, two types of probability distribution assumption in terms of candidate features and labels are derived based on low-order variable correlations. Afterwords, clearing up all information-theoretic terms, we propose a unified feature selection framework including three low-order information-theoretic terms for multi-label learning named Selected Terms of Feature Selection (STFS). STFS contains high-order variable correlations in the form of low-order information-theoretic terms. Furthermore, many previous multi-label feature selection approaches can be reduced to special forms of STFS. Finally, extensive experiments conducted on twelve benchmark data sets in comparison to seven state-of-the-art approaches demonstrate the classification superiority of STFS.

Uncertainty Quantification on the Cooling Performance of a Transonic Turbine Vane With Upstream Endwall Misalignment

Abstract With increasing aerodynamic and thermal loads, film cooling has been a popular technology integrated into the design of the modern gas turbine vane endwall, especially for the first-stage vane endwall. A staggering amount of research has been completed to quantify the effect of operating conditions and cooling hole geometrical properties. However, most of these investigations did not address the influence of the manufacturing tolerances, assembly errors, and operation degradations on the endwall misalignment. In this paper, uncertainty quantification (UQ) analysis was performed to quantify the impacts of upstream endwall misalignment uncertainties on the endwall film cooling performance and vane surface phantom cooling performance. The upstream endwall misalignment, step geometry with various step heights, commonly exists between the combustor exit and the first-stage vane endwall. Based on the non-intrusive polynomial chaos expansion (NIPC) and the uniform probability distribution assumption, the deviation (step height) uncertainties of the upstream endwall misalignment were quantified. To predict the endwall secondary flow and film cooling effectiveness in the transonic linear vane passage, the commercial computational fluid dynamic solver ANSYS FLUENT was used to numerically solve the three-dimensional steady-state Reynolds-Averaged Navier–Stokes (RANS) equations. The robustness analysis of endwall film cooling performance and phantom cooling performance to the upstream endwall misalignment was conducted for three design upstream step heights (ΔH): a baseline configuration (ΔH = 0 mm), two misaligned configurations with forward step (ΔH = −5 mm) and backward step (ΔH = 5 mm), respectively. Results show that the actual cooling performance has a high probability of deviating from the nominal value for the baseline configuration. The critical regions that are most sensitive to the upstream step misalignment are also identified by variances. The UQ results also show that the design geometry with a forward step has a more robust film cooling performance on endwall and phantom cooling performance on the vane pressure side surface, which means a smaller variance and a better expectation than the no-step configuration. In contrast, the design geometry with a backward step induces the reductions in the expectation of the film cooling effectiveness and coolant coverage and the amplification of performance fluctuations. This work provides a certain guiding direction for the optimization design for the upstream step geometry.

Chance-constrained co-optimization for day-ahead generation and reserve scheduling of cascade hydropower–variable renewable energy hybrid systems

The rapid development of variable renewable energy (VRE), such as wind and solar energy, has stimulated the complementary operation of VRE with flexible cascade hydropower stations in China. Due to the uncertainties of VRE power generation and complex constraints of cascade hydropower stations, formulating reliable day-ahead generation and reserve scheduling plans is a real challenge in the actual operation stage of the cascade hydropower–VRE hybrid systems (CHVHS). In this paper, we propose a tractable chance-constrained co-optimization model for day-ahead generation and reserve scheduling of a CHVHES. First, compared with existing models, the relationship between the hydropower reserve capacities and water-electricity conversion efficiency is finely modeled. Accordingly, the economic allocation of total VRE reserve requirements among cascade hydropower stations is for the first time considered in our proposed model. This can save more water resources for cascade hydropower stations when compensating for VRE, further improving hydro–VRE complementary profits. Second, we propose a solution approach for chance constraints by incorporating the nonparametric probabilistic forecasting of VRE power based on quantile regression into the chance-constrained model, ensuring that the stochastic dependence between VRE power output and its point forecast can be effectively captured. Importantly, this solution approach does not require prior knowledge or any probability distribution assumptions of VRE power and does not introduce any additional computational burden. With the help of three-dimensional interpolation technology for nonlinear constraints, the proposed scheduling model is finally cast as a mixed-integer linear programming model that is computationally tractable. Numerical tests implemented on a real CHVHES located in Southwest China verify the effectiveness and advantages of the proposed methods.

Nonparametric Probabilistic Optimal Power Flow

With the increasing penetration of renewable energy, accurate and efficient probabilistic optimal power flow (POPF) calculation becomes more and more important to provide decision support for secure and economic operation of power systems. This paper develops a novel nonparametric probabilistic optimal power flow (N-POPF) model describing the probabilistic information by quantiles, which avoids any parametric probability distribution assumptions of random variables. A novel critical region integral method (CRIM) which combines the multiparametric programming theory and discrete integral is proposed to efficiently solve the N-POPF problem. In the CRIM, the critical region partitioning algorithm is firstly introduced into the POPF model to directly establish the mapping relationship from wind power to optimal solutions of the POPF problem. Besides, a discrete integral method is developed in the CRIM to achieve the probability convolution calculation based on quantiles. Comprehensive numerical experiments verify the superior performance of the proposed CRIM in estimation accuracy and computational efficiency, and demonstrate that N-POPF model significantly improves the accuracy of uncertainty analysis. In general, the proposed N-POPF model and CRIM form a new framework of POPF problem for power system analysis and operation.

Second‐level global sensitivity analysis of numerical simulators with application to an accident scenario in a sodium‐cooled fast reactor

AbstractNumerical simulators are widely used to model physical phenomena and global sensitivity analysis (GSA) aims at studying the global impact of the input uncertainties on the simulator output. To perform GSA, statistical tools based on inputs/output dependence measures are commonly used. We focus here on the Hilbert–Schmidt independence criterion (HSIC). Sometimes, the probability distributions modeling the uncertainty of inputs may be themselves uncertain and it is important to quantify their impact on GSA results. We call it here the second‐level global sensitivity analysis (GSA2). However, GSA2, when performed with a Monte Carlo double‐loop, requires a large number of model evaluations, which is intractable with CPU time expensive simulators. To cope with this limitation, we propose a new statistical methodology based on a Monte Carlo single‐loop with a limited calculation budget. First, we build a unique sample of inputs and simulator outputs, from a well‐chosen probability distribution of inputs. From this sample, we perform GSA for various assumed probability distributions of inputs by using weighted HSIC measures estimators. Statistical properties of these weighted estimators are demonstrated. Subsequently, we define 2nd‐level HSIC‐based measures between the distributions of inputs and GSA results, which constitute GSA2 indices. The efficiency of our GSA2 methodology is illustrated on an analytical example, thereby comparing several technical options. Finally, an application to a test case simulating a severe accidental scenario on nuclear reactor is provided.

Prediction from Transmuted Rayleigh Distribution in the Presence of Outliers

The quality of the procedures used in statistical analysis depends largely on the assumed probability distribution. However, there are still many problems with data that do not follow any of the classical distributions; therefore, researchers have developed many standardized probability distributions by generalizing or transforming them. Transmuted Rayleigh distribution extends the Rayleigh distribution in the analysis of data and provides larger flexibility in modeling real data. In this article, Bayesian predictive intervals for order statistics of future observations from this distribution are obtained in the presence of outliers when the scale parameter is unknown. The slippage outlier model is utilized in addition to the two-sample prediction scheme. We shall consider two cases: (i) a single outlier in the informative sample and (ii) multiple outliers in the future sample. Numerical computations are obtained to illustrate the effect of outliers on the Bayesian predictive intervals.

Tractable Convex Approximations for Distributionally Robust Joint Chance-Constrained Optimal Power Flow Under Uncertainty

Uncertainty arising from renewable energy results in considerable challenges in optimal power flow (OPF) analysis. Various chance-constrained approaches are proposed to address the uncertainty within OPF models. However, most existing approaches either assume that the uncertainty distributions are known <i>a priori</i> or consider individual chance constraint modeling. This paper proposes a distributionally robust (DR) joint chance-constrained OPF model, which ensures that the operational constraints are simultaneously satisfied with a given probability and does not require an assumption of specific probability distributions. An ambiguity set built on the first two moments is used to model the uncertainty. An optimized Bonferroni approximation (OBA) is first introduced to decompose the DR joint chance constraint into DR individual chance constraints. The resulting OBA formulation is strongly nonconvex. Different convex approximations are then proposed to formulate the OBA formulation as tractable forms. The proposed convex approximations can be easily extended to incorporate the structural information associated with uncertainty, and correlations among reserve chance constraints. Case studies demonstrate the effectiveness of the proposed convex approximation methods and their extensions.