Uncertainty Analysis Method Research Articles

An Interval Analysis Scheme Based on Empirical Error and MCMC to Quantify Uncertainty of Wind Speed

The widespread existence of uncertainty makes power system control and decision-making face various risks, especially the interconnection of modern large-scale power systems, which makes its influence range more extensive. In smart grid, the renewable energy prediction and uncertainty analysis methods represented by wind power generation have attracted extensive attention. In this study, an adaptive parameter selection method based on Kullback−Leibler divergence is designed to fully extract the potential information of data. The grey wolf optimization based on reverse learning is used for prediction optimization, identifying historical data, processing dynamic information, and improving the accuracy of the model. In interval prediction, an interval prediction method combining empirical error, quantile statistics, and Markov Chain Monte Carlo is creatively proposed to generate a large number of independent error cases, and form a series of interval predictions with given confidence according to the quantile of random simulation results. Taking wind speed forecasting as an example, it is verified that the method proposed in this article has good performance. Using interval analysis to quantify the volatility and randomness can provide new ideas for load forecasting in power system and renewable energy output forecasting.

Uncertainty analysis of vehicle-pedestrian accident reconstruction based on unscented transformation

In order to investigate the sensitivity of parameters and analyze the uncertainty of reconstructed results in traffic accident, the impact of correlations between parameters on accident reconstruction results was taken into account using uncertainty analysis. Based on unscented transformation (UT), a parameter sensitivity analysis method and an efficient uncertainty analysis method in accident reconstruction were proposed. Sensitivity analysis was performed through the sigma point sets generated by the UT method. A first-order response surface model was constructed to analyze the sensitivity of accident reconstruction parameters combined with regression analysis, which is more flexible and controllable than the general experimental design. For the uncertainty analysis of the reconstructed results, the other methods have been used to demonstrate the validity of the proposed method, including the first second-order method of moments (FOSM), the uncertainty theory, and the Monte Carlo (MC) methods, through analyzing the numerical and real-world cases. The results show that the presented method has high accuracy, significantly reduces the computational burden, and does not depend on the distribution type of variables. When considering the effect of the correlation between parameters of the vehicle-pedestrian crash on accident reconstruction results, the results show that the correlation coefficient between random variables had a much more significant impact on the standard deviation of vehicle speed than on the mean value of vehicle speed. Regardless of negative or positive correlations, the relative error of standard deviation of vehicle speed increased continuously as the correlation increased, reaching 52%. The proposed method is effective and reliable for vehicle collision accident reconstruction under uncertainty and correlation, which can provide more comprehensive information in accident reconstruction.

A method for using Josephson voltage standards for direct characterization of high performance digitizers to establish AC voltage and current traceability to SI

A method for traceability to SI for ac voltage and current based on high performance digitizers is presented. In contrast to the existing thermal-based methods, the proposed method utilizes direct traceability to quantum-based waveforms via the use of Josephson voltage systems. This allows not only a simplification of the traceability chain and reduced measurement times but also offers the potential for analysis of the ac voltage and current waveform spectral content, a feature which is not possible using thermal methods. Scaling of current and voltage is achieved by the use of current shunts and resistive voltage dividers respectively. Target operating ranges are up to 1 A and 100 V with a frequency range up to 1 kHz for both. The corresponding target uncertainty for this traceability route is 1 μV V−1 and 2 μA A−1 up to frequencies of 1 kHz. The traceability chain is described and various components are characterized to validate their suitability for this task. It is demonstrated that these uncertainty targets can be met under certain conditions. The use of multi-tone calibration waveforms is investigated to further reduce measurement time. An uncertainty analysis method based on simulation using real component performance data is demonstrated.

A new modified Bayesian method for measurement uncertainty analysis and the unification of frequentist and Bayesian inference

This paper proposes a new modification of the traditional Bayesian method for measurement uncertainty analysis. The new modified Bayesian method is derived from the law of aggregation of information (LAI) and the rule of transformation between the frequentist view and Bayesian view. It can also be derived from the original Bayes Theorem in continuous form. We focus on a problem that is often encountered in measurement science: a measurement gives a series of observations. We consider two cases: (1) there is no genuine prior information about the measurand, so the uncertainty evaluation is purely Type A, and (2) prior information is available and is represented by a normal distribution. The traditional Bayesian method (also known as the reformulated Bayes Theorem) fails to provide a valid estimate of standard uncertainty in either case. The new modified Bayesian method provides the same solutions to these two cases as its frequentist counterparts. The differences between the new modified Bayesian method and the traditional Bayesian method are discussed. This paper reveals that the traditional Bayesian method is not a self-consistent operation, so it may lead to incorrect inferences in some cases, such as the two cases considered. In the light of the frequentist-Bayesian transformation rule and the law of aggregation of information (LAI), the frequentist and Bayesian inference are virtually equivalent, so they can be unified, at least in measurement uncertainty analysis. The unification is of considerable interest because it may resolve the long-standing debate between frequentists and Bayesians. The unification may also lead to an indisputable, uniform revision of the GUM (Evaluation of measurement data - Guide to the expression of uncertainty in measurement (JCGM 2008)).

Generalized isogeometric boundary element method for uncertainty analysis of time-harmonic wave propagation in infinite domains

This paper proposes a novel generalized nth order perturbation isogeometric fast multipole boundary element method for time harmonic wave propagation in infinite domains. The non-uniform rational B-splines are employed to construct structural geometries and discretize boundary integral equations. The randomness of wave number for incident plane wave is considered as the source of system uncertainty. The generalized nth order perturbation method is employed to model the physical field depending on the input random variable. The nth order derivatives of field functions and kernel functions in the boundary integral equations are derived by generalized nth order Taylor series expansions with a small perturbation parameter. The subtraction of singularity technique is used to evaluate the singular integrals and the fast multipole method is applied to accelerate the solution. The Monte Carlo simulations are conducted in numerical examples to demonstrate the validity and correctness of the proposed algorithm.

Numerical study on a coupled viscous and potential flow design method of axial-flow pump impeller

Axial-flow pump is widely used in water-jet propulsion, and its hydrodynamic and cavitation performance have an important effect on the performance of the vessel. Therefore, the study of pump design is very important. In this work, a coupled viscous and potential flow three-dimensional design method for impeller of axial-flow pump with pre-swirl stator is developed. The body-force model (BFM) is used to solve the effective inflow field for impeller design by using a viscous Reynolds-averaged Navier-Stokes equations (RANS) commercial software, and the lifting-surface vortex lattice method (VLM) is used to design the impeller. The feasibility of the design scheme is verified by quasi-steady numerical simulation, and the uncertainty analysis of the numerical simulation is carried out according to the uncertainty analysis method recommended by International Towing Tank Conference (ITTC). The design and verification of axial flow pump are carried out in this paper, and the results show that the design method developed in this work is feasible. In addition, the cavitation performance can be improved by adjusting the load distribution reasonably.

Analysis on Uncertainty of Field-to-Wire Coupling Model in Time Domain

This paper presents an approach for the time domain statistical simulation of field-to-wire coupling model with uncertain wire position in traverse direction. The parameters of governing equations of field-to-wire coupling model about the wire position are expanded to the augmented governing equations by using the generalized polynomial chaos technique with normalized orthogonal basis functions. And the coefficients of the polynomial chaos expansions are solved by combining with the stochastic Galerkin method. However, the solution of augmented governing equations is a complex problem. Therefore, a new finite-difference time-domain (FDTD) algorithm based on the implicit Wendroff difference format is proposed to solve the governing equations, and the general iterative equations are given in this paper. The proposed methods are verified by a field-to-wire coupling model. The effectiveness and advantages of the new FDTD algorithm and the uncertainty analysis method based on generalized polynomial chaos technique are verified, which are achieved by comparing with the conventional leapfrog fashion FDTD algorithm and the Monte Carlo technique, respectively.

Study of Stark broadening of krypton helium-β lines and estimation of electron density and temperature in NIF compressed capsules

The National Ignition Facility (NIF) diagnostic instrument manipulator (DIM) - based high resolution (dHIRES) x-ray spectrometer was used to measure the time evolution of the electron density (n e) and temperature (T e) in the hot spot of four NIF compressed capsules with 25 ps time resolution during the ‘stagnation’ phase. The electron density was inferred by comparing the measured Stark broadening of the krypton (Kr) Heβ spectral complex with theoretical calculations that include ion dynamic effects, and the electron temperature was inferred by comparing the measured ratio of the intensity of a dielectronically excited Li-like Kr line to the intensity of the Kr Heβ resonance line with calculations using the spectroscopic collisional radiative atomic model (SCRAM) and CRETIN collisional-radiative models. The inferred, time averaged n e values mainly agree with n e values from neutron diagnostics within uncertainties, but the neutron time-of-flight values of T ion are consistently higher than dHIRES T e values by 200–700 eV. The dHIRES measurements and measurement techniques, method of uncertainty analysis, and discussion of comparisons with measurements from neutron diagnostics are presented.

Novel rough set theory-based method for epistemic uncertainty modeling, analysis and applications

Considering the multi-source of uncertainty and the complex correlation of uncertain parameters in many engineering practices, the bounded set describing uncertainties is sometimes irregular and lacks a precise mathematical expression. To overcome the limitations of existing methods in handling this issue under incomplete information, this paper proposes a novel uncertainty modeling and analysis strategy based on the rough set theory. Firstly, in terms of limited experimental points, a data-driven partitioning method is introduced to establish an adaptive knowledge base. By means of the equivalence classes in knowledge base, a dual-approximate quantification model composed of upper and lower approximation sets is constructed to describe an arbitrary bounded-but-irregular uncertain set from both internal and external perspectives. In the subsequent uncertainty propagation analysis, a concept of rough approximate accuracy of response prediction is defined by four extreme values to quantitatively characterize the influence of model approximation on system response. Meanwhile, to improve the computational efficiency of extreme-value prediction in engineering application, an adaptive Kriging model combined with rough set theory is developed as the surrogate model of the original time-consuming simulations. Finally, two numerical examples are investigated to substantiate the effectiveness of the proposed method.

Uncertainty analysis based on reduced-order models for composite structures with interval field variables

Composite structures exhibit significant spatial variabilities in their response due to the variations in the individual lamina properties. Considering the difficulty of obtaining accurate probability distributions of laminae properties in practical engineering, interval fields are proposed as a beneficial supplement to random fields and a novel uncertainty analysis method has been developed for composite structures. This study focuses on the development of model order reduction strategies to reduce the dimensionality of the problem and achieve faster computations. First, the bounds and autocorrelation are quantified based on measurement data of interval fields, which are expressed as interval Karhunen-Loève expansion. Next, considering the multilayer structure of the laminate, an equivalent reduced-order model is developed to reduce the computational cost. The specific focus of this strategy is on the equivalence of the constitutive matrix on the fine-scale and coarse-scale models, while the reduced-order model is constructed directly based on the data of the constitutive matrix. Finally, the results of the numerical examples present that the proposed reduced-order model can accurately predict the uncertainty response and significantly improve the convergence speed of the interval Monte Carlo results in the uncertainty analysis of composite structures.

A state-of-the-art review on uncertainty analysis of rotor systems

Uncertainty handling and analysis of rotor systems have been active research areas during the last two decades. This paper provides a state-of-the-art review on the research progress of uncertainty treatment in rotor systems. Firstly, sources of uncertainties are identified and then classified according to their nature. Secondly, the most popular uncertainty analysis methods are summarized based on different types of uncertainties, including their basic principles, strengths and limitations. Thirdly, the dynamic characteristics of rotor systems under various uncertainties are described in detail, i.e., the natural characteristics and unbalanced responses, faulted rotor dynamics, stability problems and parameter identifications. Then, the reliability analysis, optimizations and vibration control for uncertain rotor systems are reviewed. Finally, comments on the latest research progress are made and outlooks on future research directions are highlighted. It is expected that this review will provide useful guidelines for designers and researchers towards the more efficient handling of uncertainty and robust dynamic analysis for rotor systems as well as insightful prospects, showing where extra efforts are needed.

Comparison of GUM and Monte Carlo Methods for the Measurement Uncertainty Circular Runout Error of Shafts

Measurement uncertainty is one of the most important concepts. The ISO IEC 17025:2005 standard: describes harmonized policies and procedures for testing and calibration laboratories. Guide to the expression of uncertainty in measurement (GUM) is a direct uncertainty analysis method, which calculates the combined standard uncertainty and expanded uncertainty by law of propagation of uncertainty. Monte Carlo Method (MCM) as presented by the (GUM S1) involves the propagation of the distributions of the input sources of uncertainty by using a model to provide the distribution of the output. By random sampling, the probability density function of the input quantities. In this paper, present measurement uncertainty to circular runout error. By use shaft standard with a diameter of 32 mm., length 100 mm. From the experiment results, Comparison of GUM and MCM showed no differences. The cases the estimated uncertainty using the GUM approach slightly overestimated the results obtained with the MCM. However, the use of numerical methods such MCM as a valuable alternative to the GUM approach. The practical use of MCM it has proven to be a fundamental tool, being able to address more complex measurement problems that were limited by the GUM approximations.

Uncertainty design and optimization of a hybrid rocket motor with mixed random-interval uncertainties

Uncertainties, which exist widely in the design of aerospace systems, have a great impact on system response and may lead to design failures. Both the probability and non-probability uncertainties exist in real engineering cases. Thus, it is imperative to incorporate the mixed uncertainties into the aerospace systems design and optimization. Present study proposed an efficient design optimization method considering the random and interval uncertainties to conduct the uncertainty analysis and optimize the design of a hybrid rocket motor with mixed uncertainties. A polynomial chaos expansion model was selected to deal with the random uncertainties, and an interval analysis method based on the first order Taylor expansion was adopted to calculate the response bounds of interval uncertainties. A simple structure was firstly used to verify the efficiency and accuracy of the proposed method. The random-interval analysis method was very efficient in searching the optimal results, showing considerable computational cost savings. Subsequently, a design model of hybrid rocket motor was introduced and verified by a long-time firing test. In the optimization of the upper-stage hybrid rocket motor, the comparison between uncertainty analysis and traditional methods showed that the design optimization considering the uncertainty in design variables was able to achieve more robust results. Compared with the deterministic scheme, the uncertainty scheme increased the total engine mass by nearly 18%, but the probabilities of all constraints considering parameter fluctuations remained above 0.9. Capable of solving complex problems with mixed uncertainties, this method shows its potential in aerospace engineering applications.

Short-Term Forecasting and Uncertainty Analysis of Photovoltaic Power Based on the FCM-WOA-BILSTM Model

Aiming to solve the problem that photovoltaic power generation is always accompanied by uncertainty and the short-term prediction accuracy of photovoltaic power (PV) is not high, this paper proposes a method for short-term photovoltaic power forecasting (PPF) and uncertainty analysis using the fuzzy-c-means (FCM), whale optimization algorithm (WOA), bi-directional long short-term memory (BILSTM), and no-parametric kernel density estimation (NPKDE). First, the principal component analysis (PCA) is used to reduce the dimensionality of the daily feature vector, and then the FCM is used to divide the weather into four categories: sunny, cloudy, rainy, and extreme weather. Second, the WOA algorithm is used to train the hyperparameters of BILSTM, and finally, the optimized hyperparameters were used to construct a WOA-BILSTM prediction model to train the four types of weather samples after FCM clustering. The NPKDE method was used to calculate the probability density distribution of PV prediction errors and confidence intervals for PPF. The RMSEs of the FCM-WOA-BILSTM model are 2.46%, 4.89%, and 1.14% for sunny, cloudy, and rainy weather types, respectively. The simulation results of the calculation example show that compared with the BP, LSTM, GRU, PSO-BILSTM, and FCM-PSO-BP models, the proposed FCM-WOA-BILSTM model has higher prediction accuracy under various weather types, which verifies the effectiveness of the method. Moreover, the NPKDE method can accurately describe the probability density distribution of forecast errors.

Study of nonlinear vibration of resonant gyroscope based on uncertainty analysis method

To study the nonlinear vibration characteristics of resonant gyroscope, the vibration model of the resonant beam is established primarily, aiming for the analytical formulas of the nonlinear stiffness coefficients and linear intrinsic frequencies of the gyroscope. With that foundation, the influence of the structural parameters of the resonant beam on the frequency tracking error caused by nonlinear vibration is analyzed thoroughly at first. Secondly, by random sampling, an uncertainty analysis model is established to evaluate the impact of each structural parameter of the resonant beam on the critical performance indexes of the gyroscope such as linear intrinsic frequency, nonlinear frequency offset and scale factor of the resonant beam. Finally, an equivalent circuit is constructed corresponding to the nonlinear vibration equations of the resonant gyroscope based on the electromechanical analogy method. By studying the nonlinearity of the circuit for equivalence, it turns out that the vibration characteristics of the equivalent circuit are consistent with that of the resonant gyroscope obtained by using the uncertainty analysis method. It effectively verifies the feasibility of the nonlinear vibration research method for the resonant gyroscope based on uncertainty analysis method.

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To study the nonlinear vibration characteristics of resonant gyroscope, the vibration model of the resonant beam is established primarily, aiming for the analytical formulas of the nonlinear stiffness coefficients and linear intrinsic frequencies of the gyroscope. With that foundation, the influence of the structural parameters of the resonant beam on the frequency tracking error caused by nonlinear vibration is analyzed thoroughly at first. Secondly, by random sampling, an uncertainty analysis model is established to evaluate the impact of each structural parameter of the resonant beam on the critical performance indexes of the gyroscope such as linear intrinsic frequency, nonlinear frequency offset and scale factor of the resonant beam. Finally, an equivalent circuit is constructed corresponding to the nonlinear vibration equations of the resonant gyroscope based on the electromechanical analogy method. By studying the nonlinearity of the circuit for equivalence, it turns out that the vibration characteristics of the equivalent circuit are consistent with that of the resonant gyroscope obtained by using the uncertainty analysis method. It effectively verifies the feasibility of the nonlinear vibration research method for the resonant gyroscope based on uncertainty analysis method.

Gaussian Mixture Model for Multivariate Wind Power Based on Kernel Density Estimation and Component Number Reduction

The Gaussian mixture model (GMM) is a powerful tool to establish the probability distributions of random variables in power system analyses. GMM can model arbitrary probability distributions by increasing the number of its Gaussian components, but the commonly used expectation-maximization (EM) algorithm fails to obtain accurate GMM for large component numbers, which limits the application of GMM to multivariate wind power modeling. In this letter, a parameter estimation method for GMM with large component numbers is proposed based on kernel density estimation (KDE) and the improved density-preserving hierarchical EM algorithm. Then, the closed-form solution to probabilistic power flow (PPF) calculation is derived based on piecewise linearization and GMM, which validates the importance of large component numbers. Finally, the proposed uncertainty modeling method is compared with EM-based GMM, Copula functions, KDE, k-nearest neighbors and block neural autoregressive flow on actual wind speed data to validate its superiority in describing the details of probability densities of wind power. PPF calculation is performed to show the efficiency and accuracy of the proposed uncertainty analysis method.

Uncertainty analysis of linear vertical bending moment in model tests and numerical prediction

Due to the complexity of model tests of wave loads, some random factors affect the accuracy of measurement results, such as model processing, coefficient calibration, counterweight, and wave creation quality. Based on the uncertainty expansion theory, a set of wave load measurement accuracy and uncertainty analysis methods for ship model in wave basin were established. The significant factors affecting accuracies of wave load tests were assessed by the linear multi-segment model tests of wave loads. Two numerical methods were used to evaluated the prediction errors and uncertainties of the linear vertical moment, horizontal moment and torsional moment. The research shows that with the updating of test instruments and approaches, the standard uncertainty of model test decreased. The uncertainty caused by the calibration coefficient was the largest, followed by the moments of inertia of ship models. As the wave direction changes from head seas to oblique seas, the errors and uncertainties of numerical predictions begin to increase. The numerical predicting errors of vertical bending moments was the smallest, followed by horizontal bending moment and torsion error was the largest.

Uncertainty analysis of MSD crack propagation based on polynomial chaos expansion

Widespread fatigue damage (WFD) is one of the main damages in aging aircraft structures, and the prediction of multi-site damage (MSD) crack propagation plays an important role in the safety assessment of aging aircraft. However, the existing multi-crack propagation stochastic models need to calculate multiple integrals, and the calculation efficiency is low. Based on the Polynomial Chaos Expansion (PCE) method, this paper proposes a new uncertainty analysis method to effectively solve the problem of multi-crack mutual interference. The rivet area of aircraft skin mostly appears randomly in the form of short cracks. Therefore, 30 groups of multi-crack fatigue propagation experiments are carried out to obtain the data of multi-cracks at different stages. On the basis of the existing multi-crack mutual interference model, a new semi-analytical multi-crack propagation model is designed by considering the uncertainty of multi-crack initiation and the interaction factors between cracks. The random variables are substituted into the improved multi-crack propagation model, and the Hermite polynomials are introduced and expanded into orthogonal polynomial series, and the time-varying polynomial coefficients, moments and probability densities are solved by stochastic regression method. The accuracy and effectiveness of the method are verified by comparing with the experimental and Monte Carlo methods.

Comparison of Rough Sets and Local Rough Sets in Data Analysis

As it is well known the rough set is a beneficial method for rough data uncertainty analysis. However, this is a time- consuming task for many big data sets. So we utilized the concept of local rough sets in data analysis of children addicted to social media to handle big data efficiently and give some of the properties. With the results, we proved that local rough sets gave more concrete and clear information than rough sets in data analysis.