Multiple Uncertain Parameters Research Articles

A Least Squares Fitting Method for Uncertain Parameter Estimation in Solidification Model

This study proposes an automated method for estimating the uncertain parameters of the solidification model in response to the inefficient and time-consuming problem of manually estimating multiple uncertain parameters of the solidification model. The method establishes an uncertain parameter estimation model based on the relationship between the simulated images equiaxed crystal ratio and the uncertain parameters of the solidification model, fits the parameters of the model by the least squares method, and finally estimates the uncertain parameters in the solidification model using the parameters of the fitted model. In comparison with the traditional method of calculating uncertain parameters manually through empirical formulas, this method reduces the difficulty of tuning parameters and solves the problem of tuning multiple parameters simultaneously in the nonlinear solidification model. The experimental results show that the proposed method can accurately estimate the uncertain parameters of the solidification model, improve the efficiency and accuracy of the solidification model estimation parameters, and play a guiding role in simulating the solidification process of continuously casting billet to control the solidification structure.

Global sensitivity evolution equation of the Fréchet-derivative-based global sensitivity analysis

For stochastic dynamical systems with multiple uncertain parameters, it is often of interest to detect which parameters are dominant, in which the global sensitivity analysis may be one of the common means. To measure the global sensitivity in both qualitative and quantitative terms, it is of significant importance to adopt a global sensitivity index with sufficient quantification information. The Fréchet-derivative-based global sensitivity index (Fre-GSI) proposed by Chen et al. (2020) is appropriate to this goal. The present paper aims to provide new aspects of the Fre-GSI, including: (1) The numerical solution of the Fre-GSI given by Chen et al. (2020) is investigated in both analytical and numerical aspects; (2) A novel global sensitivity evolution equation is derived from the generalized density evolution equation, thus the Fre-GSI can be estimated by directly solving the global sensitivity evolution equation, rather than repeatedly solving the generalized density evolution equation as suggested in Chen et al. (2020). Numerical examples are studied to illustrate the efficiency and accuracy of the proposed approach. Some problems to be further studied are also outlined.

Developing an integrated PCE-ANOVA-RF method for uncertainty quantification of hydrological model – The Amu Darya River Basin in Central Asia

Uncertainty quantification of hydrological model parameters plays an important role in the streamflow prediction of river basins. It is still a challenge to evaluate the effect of parameter uncertainty on hydrological models in association with moderate computation and high accuracy. In this study, a PCE-ANOVA-RF (abbreviated as PAR) method is developed to analyze the individual and interactive effects of multiple uncertain parameters in the Soil and Water Assessment Tool (SWAT) model as well as generate probabilistic forecasts of daily streamflow. The proposed PAR method can not only unveil the effect of parameter uncertainty and save computation time without reducing the simulation accuracy, but also expand PCE’s ability to predict future streamflow processes. The Amu Darya River Basin in Central Asia is introduced to verify the feasibility and applicability of PAR. The results disclose that (i) compared with the Sequence Uncertainty Fitting ver-2 (SUFI-2) algorithm, the simulation accuracy of PAR is acceptable (e.g., NSE = 0.85 in calibration period and 0.81 in validation period); (ii) in terms of variance analysis and streamflow simulation, PAR is more efficient and can largely reduce the number of model runs (about 88%); (iii) the parameters related to snow melting process have significant effects on both average flow and peak flow; besides, the interaction among these parameters are more obvious than that of other sensitive parameters.

Torque coordinated control of the through-the-road (TTR) 4-wheel-drive (4WD) hybrid vehicle under extreme road conditions

Vehicular safety is of considerable significance to the intelligent development of hybrid vehicles. However, the real-time stability control or reasonable torque distribution under the extreme road conditions remain a huge challenge due to the multiple uncertain parameters and difficulties to reconcile the handling and stability performance. To address the above problems for a through-the-road (TTR) 4-wheel-drive (4WD) hybrid vehicle, this study provides a handling and stability management (HSM) approach by incorporating the offline optimization rules and on-line model predictive control (MPC). Firstly, the vehicle dynamic model with seven degrees of freedom (7-DOF) is used to offline extract torque distribution rules (Offline-ETDR), and the online MPC feedback (Online-MPCF) is utilized to compensate the extra torque requirements for the poor effect under the extreme conditions. Accordingly, the offline optimization results and online correction are fused to provide the total torque demand given the real-time road condition detection. Finally, the real vehicle test are implemented to validate the effectiveness of the proposed torque coordination strategy. In comparison to the vehicle with no torque control strategy, the proposed method significantly improves the vehicle's cornering ability while also ensuring the high stability performance.

A bivariate subinterval method for dynamic analysis of mechanical systems with interval uncertain parameters

Dynamic analysis involving interval uncertainties can provide accurate reference to performance evaluation and reliability optimization. However, methodologies for dynamic problems under high dimensional interval uncertainties in mechanical systems need further exploration and development. To this end, this paper proposes a bivariate subinterval method (BSM) to handle interval dynamic problems, in which multiple uncertain parameters with small uncertainty levels in mechanical systems are involved. The BSM is developed by combining the subinterval technique and the bivariate decomposition of response functions expanded by the second-order Taylor series. Additionally, the subinterval-division-based bivariate subinterval method (SBSM) is developed to obtain a tight bound for the dynamic analysis of mechanical systems considering interval parameters with large uncertainty levels. To verify the efficiency and accuracy of the developed methods, numerical examples and engineering applications with consideration of interval uncertainties are employed and discussed. Results show that compared with the existing bivariate Chebyshev method, the BSM presents a better tradeoff between the accuracy of results and computational cost, which is more suitable to interval dynamic problems for the analysis of mechanical systems with uncertainties of high dimensions and small uncertainty levels. Furthermore, the SBSM can obtain a tight bound of dynamic responses for the interval dynamic problems with uncertain parameters of low dimensions and large uncertainty levels in mechanical systems.

Uncertainty propagation with B-spline based interval field decomposition method in boundary value problems

In this paper, uncertainty propagation problems are addressed by modelling the non-deterministic parameters as an interval field to account for spatial dependency. The interval field is constructed using a recently proposed B-spline based interval field decomposition method, which is related to an explicit formulation composed of B-spline basis functions and corresponding interval field coordinates, which can be incorporated directly into the governing equation of the boundary value problems. The solution to the governing equation can be approximated by a B-spline basis expansion using the collocation method taking advantage of high-degree continuity of B-spline basis functions. In this way, the crisp bounds of the output can be effectively accounted for. Numerical cases are provided to illustrate the effectiveness of the proposed method. The impact of the influence radius, the results obtained using an interval variable model and the combined impact of multiple uncertain parameters are also studied. Furthermore, for discretised problems, the interval field finite element formulation is presented and the resulting bounds of the output are determined by the Neumann expansion method, by which the extreme values can be effectively approximated.

Interaction effects of multiple input parameters on the integrity of safety instrumented systems with the k‐out‐of‐n redundancy arrangement under uncertainties

AbstractA safety instrumented system (SIS) is extensively used to prevent or reduce risk. The Probability of Failure to perform its intended functions on Demand (PFD) of a practical SIS may vary within multiple safety integrity levels (SIL) due to uncertainties relevant to input parameters. An SIS will be considered to be unsafe when its PFD is greater than a prescribed value, and unsafety probability (UP) is employed to measure the unsafety degree of the investigated SIS in this work. Redundancy architecture is commonly employed to improve the reliability of an SIS. This paper investigates the effects of multiple uncertain input parameters on the UP of an SIS with the k‐out‐of‐n redundancy arrangement. We derive the detailed formulations of the sensitivity for such effects and we also discuss the physical meaning of the proposed sensitivity. An example is employed to demonstrate the proposed sensitivity and the results show that the estimated sensitivity values will be kept to be unchanged with respect to the redundancy when the redundancy is high, whereas the results will vary with the redundancy when the redundancy is low. Meanwhile, we provide a comparison of the results for the truncated uncertain parameters and the non‐truncated uncertain parameters. The results for truncated uncertain parameters and non‐truncated uncertain parameters will approach the same values as the truncated region of the uncertain parameter reduces to zero. The results show that we can approximate the sensitivities of the truncated parameters by the ones of the non‐truncated parameters with less computation time when the truncated region is less than 10−3.

Development of a multi-regional factorial optimization model for supporting electric power system's low-carbon transition – A case study of Canada

In this study, a multi-regional factorial optimization model (MRFO) has been developed for supporting nation-wide transitions to low-carbon electric power systems (EPS) under the commitment to reduce greenhouse gas emissions. Through integrating non-deterministic optimization methods (interval linear, chance-constrained, and mixed-integer linear programming) with factorial analysis, MRFO can address multiple uncertainties stated as intervals and probability distribution in system parameters and objectives; it can also unveil the effects of multiple uncertain parameters and their interactions on system performance. A Canadian case study is provided to demonstrate the applicability of the proposed approach. Optimal schemes of electricity generation, capacity expansion, and inter-regional trades at different risk levels are examined with the objective of minimizing the total system costs. Results indicate that renewable energy (i.e., wind and solar) would play an important role in facilitating the transitions to low-carbon EPS, which would contribute to approximately 13% of the total national electricity generation by 2050. In addition, increasing the collaboration among regional EPS would have positive effects on the national penetration of low-carbon power generation in the light of the diversities in regional generation mixes. The findings can support the national efforts in formulating desired long-term power generation expansion plans and emission reduction policies.

Uncertainty Quantification in Reservoir Simulation Using Modern Data Assimilation Algorithm

Production forecasting using numerical simulation has become a standard in the oil and gas industry. The model construction process requires an explicit definition of multiple uncertain parameters; thus, the outcome of the modelling is also uncertain. For the reservoirs with production data, the uncertainty can be reduced by history-matching. However, the manual matching procedure is time-consuming and usually generates one deterministic realization. Due to the ill-posed nature of the calibration process, the uncertainty cannot be captured sufficiently with only one simulation model. In this paper, the uncertainty quantification process carried out for a gas-condensate reservoir is described. The ensemble-based uncertainty approach was used with the ES-MDA algorithm, conditioning the models to the observed data. Along with the results, the author described the solutions proposed to improve the algorithm’s efficiency and to analyze the factors controlling modelling uncertainty. As a part of the calibration process, various geological hypotheses regarding the presence of an active aquifer were verified, leading to important observations about the drive mechanism of the analyzed reservoir.

Multi-objective risk-constrained optimal performance of hydrogen-based multi energy systems for future sustainable societies

Hydrogen is one of the leading energy sources under special consideration due to its sustainability and technological development. The main concept of integrated energy system is applied in this article. The goal is to maintain a coordination between hydrogen cycle where hydrogen is produced and converted with renewable energy (RE). Three energy storage for charging/discharging hydrogen, hot and cold water are employed enhance the energy efficiency. The demand-side energy system could rely on this combination that enables the increase in wind and solar energy utilization and uses it to generate hydrogen in future smart cities. This paper uses a scenario-based stochastic optimization to model the associated uncertainties. The epsilon-constraint and max-min fuzzy methods are used as multi-objective approach to satisfy environmental and economic goals, simultaneously. To analyze the associated risk imposed by multiple uncertain parameters, the downside risk constraint (DRC) is implemented to render several risk-averse strategies to the decision-maker based on various constraint level. The results show that the final solution in risk neutral strategy satisfies both emission and operation costs. The ultimate associated risk in conservative strategy will tend to zero at the 4.2% operation cost increment and 3% emission reduction.

Data-driven scenario generation for two-stage stochastic programming

Optimisation under uncertainty has always been a focal point within the Process Systems Engineering (PSE) research agenda. In particular, the efficient manipulation of large amount of data for the uncertain parameters constitutes a crucial condition for effectively tackling stochastic programming problems. In this context, this work proposes a new data-driven Mixed-Integer Linear Programming (MILP) model for the Distribution & Moment Matching Problem (DMP). For cases with multiple uncertain parameters a copula-based simulation of initial scenarios is employed as preliminary step. Moreover, the integration of clustering methods and DMP in the proposed model is shown to enhance computational performance. Finally, we compare the proposed approach with state-of-the-art scenario generation methodologies. Through a number of case studies we highlight the benefits regarding the quality of the generated scenario trees by evaluating the corresponding obtained stochastic solutions.

Interval Uncertainty Quantification for the Dynamics of Multibody Systems Combing Bivariate Chebyshev Polynomials with Local Mean Decomposition

Interval quantification for multibody systems can provide an accurate dynamic prediction and a robust reliability design. In order to achieve a robust numerical model, multiple interval uncertain parameters should be considered in the uncertainty propagation of multibody systems. The response bounds obtained by the bivariate Chebyshev method (BCM) present an intensive deterioration with the increase of time history in the interval dynamic analysis. To circumvent this problem, a novel method that combines the bivariate Chebyshev polynomial and local mean decomposition (BC-LMD) is proposed in this paper. First, the multicomponent response of the system was decomposed into the sum of several mono-component responses and a residual response, and the corresponding amplitude and phase of the mono-component were obtained. Then, the bivariate function decomposition was performed on the multi-dimensional amplitude, phase, and residual to transform a high-dimensional problem into several one-dimensional and two-dimensional problems. Subsequently, a low order Chebyshev polynomial can be used to construct surrogate models for the multi-dimensional amplitude, phase, and residual responses. Then, the entire coupling surrogate model of the system can be established, and the response bounds of the system can be enveloped. Illustrative examples of a slider-crank mechanism and a double pendulum are presented to demonstrate the effectiveness of the proposed method. The numerical results indicate that, compared to the BCM, BC-LMD can present a tight envelope in the long time-dependent dynamic analysis under multiple interval parameters.

An effective implementation of reliability methods for Bayesian model updating of structural dynamic models with multiple uncertain parameters

The implementation of reliability methods in the framework of Bayesian model updating of structural dynamic models using measured responses is explored for high-dimensional model parameter spaces. The formulation relies on a recently established analogy between Bayesian updating problems and reliability problems. Under this framework, samples following the posterior distribution of the Bayesian model updating problem can be obtained as failure samples in an especially devised reliability problem. An approach that requires only minimal modifications to the standard subset simulation algorithm is proposed and implemented. The scheme uses an adaptive strategy to select the threshold value that determines the last subset level. Due to the basis of the formulation, the approach does not make use of any problem-specific information and, therefore, any type of structural model can be considered. The approach is combined with an efficient parametric model reduction technique for an effective numerical implementation. The performance of the proposed implementation is assessed numerically for a linear building model and a nonlinear three-dimensional bridge structural model. The results indicate that the proposed implementation represents an effective numerical technique to address high-dimensional Bayesian model updating problems involving complex structural dynamic models.

Uncertainty Analysis of multi-site damage Crack Propagation Based on the Monte Carlo Method

Abstract Widespread fatigue damage is one of the main damages of aging aircraft structures, and the problem of multi-site damage has always been a research hotspot in engineering analysis. The traditional probabilistic model for calculating the growth rate of multiple cracks takes into account the multiple uncertain parameters that affect each other’s cracks. However, when solving the multi-site damage (MSD) crack growth problem, the traditional random model needs to solve multiple nonlinear equations. This method becomes no longer applicable. Based on the statistical point of view, this study gives the mutual influence factors of the crack growth process in the form of distribution, and then calculates the probability density function of the crack length under different cycles by sampling with the Monte Carlo (MC) method. Finally, compared with the experimental data, the results show that the calculation results of this method are more accurate and are in good agreement with the experimental data. This method is feasible and accurate for multi-crack propagation analysis of random parameter structures.

A new perspective on the simulation of cross-correlated random fields

Cross-correlated random fields are widely used to model multiple uncertain parameters and/or phenomena with inherent spatial/temporal variability in numerous engineering systems. The effective representation of such fields is therefore the key element in the stochastic simulation, reliability analysis and safety assessment of engineering problems with mutual correlations. However, the simulation of such fields is generally not straightforward given the complexity of correlation structure. In this paper, we develop a unified framework for simulating non-Gaussian and non-stationary cross-correlated random fields that have been specified by their correlation structure and marginal cumulative distribution functions. Our method firstly represents the cross-correlated random fields by means of a new general stochastic expansion, in which the fields are expanded in terms of a set of deterministic functions with corresponding random variables. A finite element discretization scheme is then developed to further approximate the fields, so that the sets of deterministic functions reflecting the cross-covariance structure can be straightforwardly determined from the spectral decomposition of the resulting discretized fields. For non-Gaussian random fields, an iterative mapping procedure is developed to generate random variables to fit non-Gaussian marginal distribution of the fields. By virtue of the remarkable property of the presented stochastic expansion, i.e., various random fields share an identical set of random variables, the framework we develop is conceptually simple for simulating non-Gaussian cross-correlated fields with arbitrary covariance functions, which need not be stationary. In particular, the developed method is further generalized to a consistent framework for the simulation of multi-dimensional random fields. Five illustrative examples, including a spatially varying non-Gaussian and nonstationary seismic ground motions, are used to demonstrate the application of the developed method.

Dynamic analysis of Jeffcott rotor under uncertainty based on Chebyshev convex method

Rotor system is often affected by various uncertain factors during work, which may affect the normal working process of the rotor system. In this paper, aiming at the uncertainty of the Jeffcott rotor system, a non-probabilistic convex model is constructed to describe the uncertain parameters. By introducing the non-embedded Chebyshev expansion function, the Chebyshev Convex Method (CCM) is proposed. Then, the CCM is employed to calculate the rotor response when the mass and bearing stiffness are respectively uncertain parameters. Meanwhile, by changing the mass of the rotor disc and the bearing support position on the Bently RK4 rotor rig, the Jeffcott rotor with uncertain parameters is simulated. The numerical and experimental results show that the proposed method is valid and effective for quantifying the uncertain parameters of the Jeffcott rotor system. In addition, the Jeffcott rotor response with multiple uncertain parameters is discussed in detail. The results show that compared with the parameters related to the material properties of the rotor such as mass and density, when the parameters related to the bearing support such as stiffness and damping are uncertain, it will have a greater impact on the response of the rotor system.

Statistical analyses of the energy demand and thermal comfort for multiple uncertain input parameters, performed using transformed variable and perturbation method

In the calculations of buildings’ thermal comfort, the input parameters are usually considered as strictly determined values. Numerous of them may be characterized by certain probability density functions. In the energy related problems, the uncertainty analyses are usually performed using the Monte Carlo method. However, this method requires multiple calculations and, therefore, may be very time-consuming. In the proposed work, two approaches are applied for the probabilistic studies: the stochastic perturbation method and the transformed random variables method. The stochastic analysis is based on the response functions and their derivatives with respect to all random input parameters. The relation between the thermal comfort and the input (random) variables have been calculated using the Energy Plus software. Afterwards, the response functions were estimated using the polynomial regression. The expected value and central moments of the response functions were calculated by means of the perturbation method and the transformed random variable theorem. The latter method allowed to obtain, using the same response functions, the implicit form of probability distributions function of the output parameter.

A scenario discovery approach to least-cost electrification modelling in Burkina Faso

This paper presents the first application of the scenario discovery approach in geospatial electrification modelling. 1944 electrification simulations were constructed for Burkina Faso from a combination seven input levers, including four grid-extension strategies. The scenario discovery analysis identifies a scenario described by a high grid electricity generation cost in combination with an intensification strategy for grid-extension, as most likely to lead to a high cost of electricity in Burkina Faso. Thus, to avoid such a high cost, decisions in the country could be targeted either at lowering grid electricity generation costs or to choose one of the other two grid-extension strategies, or both. For each of the grid-extension strategies, a number of drivers causing a high LCOE were identified. Common drivers for all strategies were the grid electricity generation cost and discount rate. The scenario discovery approach was used to identify the key drivers of high electrification costs and their interactions, providing useful information that might not be gained from a traditional scenario-axes approach. This approach provided a structured way to analyze more parameters than found in previous electrification studies for Burkina Faso. The paper discusses on the pros compared to a traditional scenario-axes approach, such as reduced risk of perceived bias and improved ability to deal with multiple uncertain parameters, but also notes the additional computational requirements.

Adaptive Backstepping Robust Coordinated Control with Consideration of Parameter Uncertainty for Generator Excitation and STATCOM

In order to improve the transient stability of the coordinated control system of generator excitation and static synchronous compensator (STATCOM), an adaptive backstepping robust coordinated control strategy based on immersive and invariant (I&I) adaptive backstepping control and L2‐gain control is proposed with consideration of the multiple uncertain parameters and external disturbance in the system modeling. The parameter adaptive law and coordination of control law are designed independently. The parameter adaptive laws for the two uncertain parameters: damping coefficient and mechanical power in the system are designed simultaneously by adopting I&I adaptive control method; the L2‐gain control is introduced in the backstepping design to eliminate the effects of the external disturbance on the system transient stability. The results of simulation show that the proposed strategy has strong adaptability to parameter fluctuations and robustness to external disturbances. © 2021 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Design optimization and uncertainty analysis of multi-energy complementary system for residential building in isolated area

Due to extreme climatic and geographical factors, conventional power sources, which typically use large power-grids, have difficulties to supply energy to isolated areas. In addition, many isolated areas rely mainly on fossil fuel to ensure adequate energy-supply, which involves transport across long distances and high costs. Thus, finding a multi-energy complementary system (MECS) for residential buildings can help ensure the survival and improve the well-being of about 1.5 billion people that are currently living in isolated areas worldwide. In this study, a rough interval-Copula stochastic planning (RI-CSP) programming model is proposed to address this problem. Its purpose is to optimize the MECS in isolated areas, with the aim to reduce cost and provide high-quality energy. RI-CSP can deal with multiple uncertain parameters that are characterized by rough intervals and probability distributions in the MECS. It can also reveal the combined effect of different random variables. To prove the feasibility of RI-CSP, it was applied to a residential building on Yongxing island in the South China Sea. 24 different scenarios were selected that consider four renewable energy scarcity levels and 2 constraint-violation levels (p). Then, the optimal scenarios with minimum total system costs were selected. This study can help decision makers find better configurations for many alternative power-generation scenarios. The results show that the uncertainties that exist in the MECS have a significant effect on the power-generation configuration and total system cost. The results indicate that the total cost of MECS decreases with the increase in scarcity levels. Our results also revealed that, as the scarcity levels for renewable energy decrease, the fraction of PV power-generation gradually increases.