Uncertainty Analysis Method Research Articles

An Uncertainty Analysis and Reliability-Based Multidisciplinary Design Optimization Method Using Fourth-Moment Saddlepoint Approximation

In uncertainty analysis and reliability-based multidisciplinary design and optimization (RBMDO) of engineering structures, the saddlepoint approximation (SA) method can be utilized to enhance the accuracy and efficiency of reliability evaluation. However, the random variables involved in SA should be easy to handle. Additionally, the corresponding saddlepoint equation should not be complicated. Both of them limit the application of SA for engineering problems. The moment method can construct an approximate cumulative distribution function of the performance function based on the first few statistical moments. However, the traditional moment matching method is not very accurate generally. In order to take advantage of the SA method and the moment matching method to enhance the efficiency of design and optimization, a fourth-moment saddlepoint approximation (FMSA) method is introduced into RBMDO. In FMSA, the approximate cumulative generating functions are constructed based on the first four moments of the limit state function. The probability density function and cumulative distribution function are estimated based on this approximate cumulative generating function. Furthermore, the FMSA method is introduced and combined into RBMDO within the framework of sequence optimization and reliability assessment, which is based on the performance measure approach strategy. Two engineering examples are introduced to verify the effectiveness of proposed method.

Uncertainty Analysis Method for Estimating Wave Parameters Based on Ship Responses (2nd Report- Representation of Uncertainty in Transfer Function Using Principal Component Analysis)

Uncertainty Analysis Method for Estimating Wave Parameters Based on Ship Responses (2nd Report- Representation of Uncertainty in Transfer Function Using Principal Component Analysis)

Uncertainty Analysis Method for Estimating Wave Parameters Based on Ship Responses (1st Report-Influence of the Ship Speed and Uncertainty in the measurement data)

Uncertainty Analysis Method for Estimating Wave Parameters Based on Ship Responses (1st Report-Influence of the Ship Speed and Uncertainty in the measurement data)

Reduced Space Optimization‐Based Evidence Theory Method for Response Analysis of Space‐Coiled Acoustic Metamaterials with Epistemic Uncertainty

Acoustic metamaterials have been widely concerned by researchers because of their excellent sound absorption properties. Traditional uncertainty analysis methods for response analysis of acoustic metamaterials with evidence variables have limitations. The excessive time spent on repetitive extreme analysis of focal elements severely impedes the practical application of evidence theory. To reduce the computational costs of uncertainty quantification for acoustic metamaterials under the evidence theory, a reduced space optimization‐based evidence theory method (RSO‐ETM) is proposed. In RSO‐ETM, a simplified surrogate model is first constructed by a modified adaptive arbitrary orthogonal polynomial (MAAOP) expansion method, and then, the monotonicity of the response surface is examined to reduce the space. Subsequently, the Newton iteration technique and a transformation boundary method are used to obtain the extreme points in the reduced space, through which the boundary of the response over each focal element can be readily obtained. By using RSO‐ETM, the optimization for each focal element can be avoided, and correspondingly the computational costs are reduced. Two mathematical examples and acoustic problems are employed to demonstrate the practicality of the methods.

The prediction of projectile-target intersection for moving tank based on adaptive robust constraint-following control and interval uncertainty analysis

To improve the hit probability of tank at high speed, a prediction method of projectile-target intersection based on adaptive robust constraint-following control and interval uncertainty analysis is proposed. The method proposed provides a novel way to predict the impact point of projectile for moving tank. First, bidirectional stability constraints and stability constraint-following error are constructed using the Udwadia-Kalaba theory, and an adaptive robust constraint-following controller is designed considering uncertainties. Second, the exterior ballistic ordinary differential equation with uncertainties is integrated into the controller, and the pointing control of stability system is extended to the impact-point control of projectile. Third, based on the interval uncertainty analysis method combining Chebyshev polynomial expansion and affine arithmetic, a prediction method of projectile-target intersection is proposed. Finally, the co-simulation experiment is performed by establishing the multi-body system dynamic model of tank and mathematical model of control system. The results demonstrate that the prediction method of projectile-target intersection based on uncertainty analysis can effectively decrease the uncertainties of system, improve the prediction accuracy, and increase the hit probability. The adaptive robust constraint-following control can effectively restrain the uncertainties caused by road excitation and model error.

Comparison of two methods for assessment of the rod positioning uncertainty and consequences on the evaluation of correlation factors

In this paper two major families of methods to deal with the assessment of the rod positioning uncertainty in a lattice are tested: a traditional one described in the International Handbook of Evaluated Criticality Safety Benchmark Experiments (IHECSBE) Handbook and the other one consisting in sampling the position of rods with Monte Carlo techniques (ISO Uncertainty Guidelines). They are applied on a benchmark with tight-packed lattice of UO2 rods that is sensitive to the rod positioning as it is clearly under-moderated. It is shown that the choice of the method has a great impact on the propagated uncertainty, the traditional one leading to a significant overestimation of the overall uncertainty and can also contribute to a bias in the correlation factors that are used for assessing biases due to nuclear data using GLLSM methodologies. The paper briefly describes the tight-packed lattice experimental program performed at the Valduc Research Centre, which is at the origin of these concerns. Then it proposes a simple model on which to apply simulations of rod positioning to be performed with MORET 5 Monte Carlo code using the Prométhée tool. Results demonstrate that use of Monte Carlo methodologies provide more realistic uncertainty estimates in fuel pitch that are consistent with repeatability/reproducibility experiments. The current comparisons use light water reactor systems, which is directly relevant to some small modular reactor designs. However, accurate prediction and estimate of uncertainties in pitch for advanced reactor systems is also relevant. The application of unrealistic uncertainty analysis methods can incur larger margins in advanced reactor design, safety, and operation than are necessary.

Uncertainty analysis of photovoltaic cells to determine probability of functional failure

In engineering practice, extensive uncertainties exist in photovoltaic (PV) cell material parameters owing to manufacturing process errors and in PV cell working environment parameters owing to environmental uncertainties. Ignoring the uncertainty of these parameters will greatly reduce the reliability of photovoltaic cells. In this study, a PV cell model was used to conduct an uncertainty analysis based on functional failure. Functional failure is defined as output power fluctuation beyond the specified range, and functional safety region is defined as the allowable fluctuation range of output power during operation. A global sensitivity analysis method based on the Monte Carlo method was employed to study the influence of parameter fluctuation on the functional failure probability of a PV cell. An uncertainty analysis method based on Latin hypercube sampling was applied to study the influence of parameters on the probability of PV cell operation in the functional safety domain. The results indicate that the uncertainty of surface temperature had the most significant influence on the failure probability, followed by the ideality factor, radiation intensity, and current temperature coefficient, the series and parallel resistances had the smallest impact. Optimal values of 510.200 W/m2, 284.600 K, and 1.446 were obtained for radiation intensity, surface temperature, and ideality factor, respectively, which ensured that system functional failure probability remained in the safety zone with values of 93.763 %, 99.211 %, and 98.856 %, respectively. Additionally, the lower series resistance and the higher parallel resistance, the more likely the system to operate in the functional safety domain. The results of study expand the evaluation standard for PV cell operation and ensure the desired system output power.

Short-term wind power forecasting and uncertainty analysis based on FCM–WOA–ELM–GMM

With large-scale wind power connected to the power grid, accurate short-term wind power forecasting has become a key technology for safe, economic power grid operation. Therefore, a short-term wind power forecasting and uncertainty analysis method based on the FCM-WOA-ELM-GMM was proposed. Fuzzy C-means (FCM) was used to cluster the numerical weather prediction (NWP) and wind farm power data, and the data points with similar meteorological information are classified into one class. Using the rapid convergence and high convergence accuracy of the whale optimization algorithm (WOA), the input weight and hidden layer threshold of the extreme learning machine (ELM) model were optimized to improve the ELM calculation speed and forecasting accuracy. The WOA-ELM model was trained using the clustered NWP and wind farm power data and the short-term wind power was projected using the trained forecasting model. To accurately calculate the forecasting error probability density distribution, the Gaussian mixture model (GMM) was applied and the wind power forecast confidence intervals under different climatic conditions and time scales were calculated. The forecasting accuracies of the WOA-ELM, ELM, PSO-LSSVM, LSSVM, LSTM, PSO-BP, and WNN models were compared and analyzed, and the RMSE values of the 4-h forecasting results in April were as follows: WOA-ELM, 5.95%; ELM, 26.73%; PSO-LSSVM, 3.78%; LSSVM, 5.19%; LSTM, 23.71%; PSO-BP, 15.63%; and WNN, 23.23%. The RMSE values of 24-h forecasting results in April were: WOA-ELM, 6.62%; ELM, 19.86%; PSO-LSSVM, 9.91%; LSSVM, 13.73%; LSTM, 23.69%; PSO-BP, 14.08%; and WNN, 20.11%. The RMSE values of 72-h forecasting results in April were as follows: WOA-ELM, 5.24%; ELM, 13.64%; PSO-LSSVM, 12.03%; LSSVM, 13.67%; LSTM, 16.61%; PSO-BP, 15.46%; and WNN, 20.22%. According to the calculation results, under different climatic conditions and forecasting time scales, the forecasting accuracy of the FCM-WOA-ELM-GMM model is higher than those of the other models.

Engineering uncertainty analysis of the heat transfer of fuel plates

Uranium-containing aluminum-cladding dispersion plate fuels are widely used in various research reactors owing to their excellent heat-transfer capability. In the nuclear design stage, both the maximum temperature of the cladding surface and the minimum departure from nucleate boiling ratio (MDNBR) must satisfy safety acceptance criteria to ensure the performance of the fuel plates. Many engineering-related uncertainties lead to deviations between the actual and design values. For safety considerations, these engineering-related uncertainties must be considered when analyzing the maximum temperature of the cladding surface and MDNBR. In this study, the various uncertainties of the engineering hot channel and hot spot factors, three methods of uncertainty analysis, and sensitivity analysis of these uncertainty factors were investigated. The results show that the square root of the sum of squares method is both conservative and economical. The sensitivity analysis results show that the flow channel gap deviation, produced by fuel plate manufacturing, has the greatest influence on the maximum temperature of the fuel cladding surface.

A non-intrusive frequency normalisation approach for uncertain response analysis of nonlinear dynamic systems

This paper proposes a non-intrusive technique based on the frequency normalisation for non-intrusive propagations of parameter uncertainties in nonlinear mechanical systems. The multi-solution dilemma found in resonance areas, which prohibits successful applications of uncertainty quantification methods, is alleviated by the additional normalised frequency measure. The spurious peaks around the nonlinear peaks and discontinuous points, known as the Gibbs phenomenon, due to parameter variabilities are resolved as well. The proposed technique coupled with the non-intrusive surrogate modelling is applied to two deliberately constructed nonlinear dynamic systems, i.e., a mass-spring system with interactive cubic nonlinearity and a piece-wise rotor/stator contact problem in rotating machines. The two systems exhibit complex amplitude-frequency response characteristics featured by the softening and hardening effects. Numerous case investigations show the effectiveness of the proposed normalisation method for uncertainty analysis of complex nonlinear vibration systems and its working principle is demonstrated in detail via examples. Accuracy tests against the traditional sampling methods are carried out in both systems. The proposed technique can be easily generalised to other nonlinear systems for random and non-random uncertainty propagations because it works non-intrusively and permits users to choose arbitrary nonlinear tools.

Application of Microwave Transmission Sensors for Water Cut Metering under Varying Salinity Conditions: Device, Algorithm and Uncertainty Analysis.

The measurement of water cut in crude oil is an essential procedure in petroleum production and it is desirable to obtain these data through an automatic and real-time method. Microwave sensors can be used for the task, and they are safe, robust and can cover the whole water cut range. However, they are relatively susceptible to the water conductivity and temperature, and the algorithms for addressing these problems are still rare in the literature. In this paper, a microwave transmission sensor that can measure the water cut under varying salinity conditions is proposed, and the algorithm for solving the water cut and salinity simultaneously with the measured amplitude and phase is described in detail. Experiments under different water cut and salinity conditions are conducted, and the results are used to verify the model and algorithm. Finally, a simplified and fast method for uncertainty analysis is proposed and applied to the iteration algorithm under test conditions. It can be concluded that accuracy higher than 95% in the water cut measurements can be expected under the 0~100% water cut range, and an error of about 10% in the water conductivity is achievable under water-continuous flow conditions. The uncertainty analysis shows that the calculated water cut and salinity results are negatively correlated, and the water salinity uncertainty tends to be larger than the water cut uncertainty. When the water salinity is high, the water cut uncertainty tends to be high whereas the water salinity uncertainty tends to be low.

An amplitude-nested surrogate model for nonlinear response using double-layer Hilbert–Huang transform

Uncertainty parameters are ubiquitous in engineering and have a significant influence on the performance of mechanical systems. When evaluating the effect of uncertainty parameters on time-dependent response, a previous method is based on building surrogate models at each time instant for the amplitude, phase, and trend of the response, which can be obtained through signal decomposition techniques such as Hilbert–Huang transform. However, for a nonlinear oscillatory response, not only the response itself but also its amplitude might contain vibration components. Therefore, when modeling nonlinear responses, the previous method encounters an approximation error due to the increasingly complicated relationship between uncertainty parameters and the amplitude over time. To reduce this error, this paper proposes an amplitude-nested surrogate model using double-layer Hilbert–Huang transform through two steps. First, the response and its amplitude are successively decomposed through Hilbert–Huang transform to obtain the first- and second-layer decomposition results, respectively. Then, the polynomial surrogate models are built for the amplitude, phase, and trend in the first- and second-layer results at each time instant. The proposed method can improve the accuracy of the previous single-layer surrogate model without increasing the number of samples. Numerical examples have verified the effectiveness of the proposed method. This method can expand uncertainty analysis methods of dynamics.

An accident diagnosis method of pressurized water reactor based on BI-LSTM neural network

Due to the lack of accident data, previous researchers used accident data obtained from system code simulation to train and verify the machine learning models. In this study, a long short-term memory (LSTM) neural network was used to construct a pressurized water reactor (PWR) accident diagnosis model. The model was trained and tested using simulation data of a PWR based on the deterministic analysis method and uncertainty analysis method. The accuracy of the LSTM model based on deterministic analysis and uncertainty analysis are 96.67% and 98.77%, respectively. The LSTM model developed in the present study is superior to traditional mathematical statistics and support vector machine (SVM) models in accident diagnosis, especially for diagnoses with uncertain data. The results of this study show that the LSTM model has practical value in reactor accident diagnosis.

Sensitivity Analysis of Influencing Factors of Underground Civil Air Defense Construction Vibration under Vehicle Load

Grey correlation theory is an uncertainty analysis method, which can find the relevance of various factors studied through certain data processing in incomplete information and find the main influencing factors. In order to study the influence of vehicle load on the vibration of underground civil air defense construction, taking the civil air defense construction at the intersection of Zhongzhou Middle Road and Shachang South Road in Luoyang as the study subject, based on the field measured data, and using ANSYS to expand the working conditions, this paper studies the influence of such factors as elastic modulus of concrete, thickness of overburden layer, density of soil layer, elastic modulus of soil layer, vehicle speed, and vehicle mass on the vibration of the civil air defense construction through the grey correlation sensitivity analysis method. The field measurement and numerical simulation results show that the displacement, velocity, and acceleration values decrease gradually from the mid−span position of the roof to the wall. Displacement, velocity, and acceleration are important indexes for evaluating vibration. The grey correlation degree of the influencing factors of displacement and velocity from large to small is density of soil layer, vehicle mass, vehicle speed, elastic modulus of concrete, elastic modulus of soil layer, and thickness of overburden layer. The grey correlation degree of the influencing factors of acceleration from large to small is vehicle mass, layer of density of soil layer, vehicle speed, elastic modulus of concrete, elastic modulus of soil layer, and thickness of overburden layer. The grey correlation degree of soil layer density, vehicle mass and vehicle speed to displacement, velocity, and acceleration is 0.7951–0.9993, which indicates that their influence is significant. This study can provide reference for the design and construction of similar civil air defense constructions.

Uncertainty Propagation of Fission Product Yields from Uranium and Plutonium in Pebble-Bed HTGR Burnup Calculation

Quantifying fission product yield uncertainty contribution to reactor burnup calculation is an important aspect of pebble-bed High Temperature Gas-cooled Reactor (pebble-bed HTGR) uncertainty analysis. In this work, uncertainty propagation of fission product yield to pebble-bed HTGR burnup calculation is conducted. Uncertainty of fission product yields from four fissile isotopes, namely 233U, 235U, 239Pu and 241Pu, are considered. The stochastic sampling-based uncertainty analysis method is adopted and fission product yield covariance matrices are estimated from ENDF/B-VII.1. The covariance matrix for each fissile actinide is estimated based on the Bayesian method and fission product yields are assigned with log-normal distribution in the sampling process with the Latin Hypercube Sampling (LHS) method. Since the fission fraction from 239Pu plays an important role in fissions of fuels with high burnup value in pebble-bed HTGR, the fission product yield uncertainty contribution from 239Pu is highlighted in this work. The result shows that, in the burnup equilibrium state of pebble-bed HTGR, fission product yield uncertainty contributions from 235U and 239Pu to relative uncertainty of keff are 0.027% and 0.026%, respectively. The overall uncertainty contribution from four fissile isotopes (233U, 235U, 239Pu and 241Pu) to relative uncertainty of equilibrium core keff is 0.038%. Furthermore, fission product yield uncertainty has an important contribution to the nuclide density uncertainty of fission products. The most relative uncertainty, 10.82%, is observed in 109Ag contributed from the fission product yield uncertainty of 239Pu at the burnup equilibrium state. This indicates the uncertainty contribution from the fission product yield of 239Pu cannot be neglected in pebble-bed HTGR burnup uncertainty analysis.

Evaluation of the temporal reconstruction methods for MODIS-based continuous daily actual evapotranspiration estimation

Monitoring continuous daily actual evapotranspiration (ET) is very useful for managing regional water resources and understanding vegetation water consumption and water use efficiency. However, the research on remote sensing-based temporal reconstruction methods for mapping continuous daily ET is still limited, lacking comparative evaluations and mechanism discussions. In the study, four reconstruction methods, including two interpolation methods (the reference evaporative fraction (ETrF) and evaporative fraction (EF) method) and two calibration methods (the advection-aridity (AA) and modified Katerji-Perrier (KP) method), were systematically evaluated to reconstruct continuous daily ET from discrete daily ET based on MODIS products and atmospheric data. Meanwhile, an additional test using in situ measurements as a perfect proxy for the retrievals on satellite overpass dates was added as a baseline case that is independent of remotely sensed ET error. The internal coupling mechanism between each model’s simulations and the underlying surface physical and atmospheric driving factors were also analyzed using the sensitivity and uncertainty analysis methods. The results demonstrated that the interpolation methods performed better and had a higher potential for improvement as the quality of remote sensing ET inputs improved. Although the underestimation errors in the remote sensing-based ET estimates propagated to subsequent reconstructions and caused underestimation to a certain extent, the four methods could still effectively simulate continuous daily ET. Among them, the EF method with a simple calculation process, which used reference quantity directly related to solar radiation, had the best performance based on in situ with an R2 of 0.8233 and an NRMSE of 0.0889. Given remote sensing estimates, the AA method was the best due to the compensation effects, with an R2 of 0.7267 and an NRMSE of 0.1442. As all methods showed high sensitivity to ET of satellite overpass days, its accuracy should be a top priority for continuous daily ET reconstruction work.

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)).