Sparse Polynomial Chaos Expansion Research Articles

Inconsistent evolutionary analysis of multi-level physical model for lithium-ion batteries considering calendering process uncertainties

Due to the variation in the calendering process, there are inconsistencies between battery cells that cannot be eliminated, which affects the energy utilization efficiency, safety, and reliability of the battery system. The deviations occur in the calendering process resulting in uncertainties in the structural parameters of the electrode. To analyze the evolutionary mechanism of inconsistency arising from the uncertainties in the calendering process, a stochastic, multi-level physical model approach combined with sparse polynomial chaos expansions is presented. The physical model is verified by voltage and temperature experiments. This approach enables to establish the relationship between calendering parameters variation and the inconsistency performance between cells. The performance of battery inconsistency has been quantified as a function of calendering process deviations at the manufacturing level. The most important uncertainty sources are identified by global sensitivity analysis. Based on the multi-level physical model approach, the battery module connected in series is constructed to study the effect of the calendering variations on the energy utilization efficiency of the battery module. The results suggest that the electrode thickness has the most significant effects on the battery capacity, resistance, and temperature. Meanwhile, the deviations of the mass load and electrode thickness need to be effectively controlled to achieve a better consistent performance between cells. Such information allows for more efficient design and targeted quality control, thereby reducing production costs.

An enhanced learning function for bootstrap polynomial chaos expansion-based enhanced active learning algorithm for reliability analysis of structure

Sparse polynomial chaos expansion (PCE) combined with the bootstrap resampling method is a viable alternative to obtain an active learning algorithm for reliability analysis. The existing learning functions in PCE-based active learning algorithms do not consider the joint probability density function (PDF) information. The present study explores a sparse PCE-based active learning algorithm based on a newly proposed learning function that maintains a balance between the misclassification probability and the joint PDF information of sample points. In doing so, the coefficients of the sparse PCE are estimated using a Bayesian compressive sensing regressor, as it is noted to be one of the best-performing regression solvers for PCE, irrespective of sampling schemes. The proposed learning function considers the weight of the joint PDF with the local accuracy measure of bootstrap PCE (bPCE) to add new samples iteratively in the existing training set. The convergence is achieved when the ten consecutive failure estimates are within a negligible discrepancy and also checks the confidence bounds of the bPCE estimates. The effectiveness of the proposed approach is demonstrated using two structural engineering examples and one well-known analytical test function and is found to be quite efficient and accurate in estimating reliability.

Multi-objective design optimization of a transonic axial fan stage using sparse active subspaces

In this paper, a multi-objective optimization strategy for efficient design of turbomachinery blades using sparse active subspaces is implemented for a turbofan stage design. The proposed strategy utilized sparse polynomial chaos expansion on a limited dataset to generate a function from which the differential and the covariance matrix can be obtained. Active subspace was used to compute the active variables via singular value decomposition and a hybrid polynomial correlated function expansion was used to construct a surrogate model on the active subspace. Coupled with freeform method and multi-objective genetic algorithm, an automated optimization loop was run at a single operating condition. An improvement in stage efficiency and total pressure ratio of 2.97% and 1.15% was achieved for the optimum design compared with the baseline. Additionally, total pressure loss coefficient decreased by 5.88%, exit flow angle by 34.65% and shock strength by 5.32%. The coupled effect of change in stagger angle, forward sweep, forward lean, and chord length reduced the recirculation at the hub, and blockage at the shroud due the tip leakage flow by decreasing the blade loading. The threshold value hyperparameter was found to be the most influential and must be accurately determined.

Amortized Bayesian Parameter Estimation Approach for WECC Composite Load Model

Calibrating the composite load model with distributed generation (CMPLDWG) is of a great challenge due to the presence of high-dimension parameters. This paper proposes an amortized Bayesian parameter estimation method for the CMPLDWG model. A prior distribution restrictor is developed via a classifier to filter out invalid simulations because of the unsuitable parameter combinations during the offline training stage. A sparse polynomial chaos expansion (SPCE) enabled global sensitivity analysis is also developed to identify the importance of each parameter's response to the captured system dynamics. This allows us to narrow down the problematic parameters and reduce the problem's complexity. Finally, the multi-fidelity trajectory statistics constructor is proposed to extract the representations from post-disturbance dynamic responses. The representations are further used by conditional masked autoregressive flow (CMAF) to learn the parameter posterior distribution. Comparative results with state-of-the-art approaches on the IEEE 39-bus power system demonstrate the high accuracy and robustness of the proposed method.

Control variate Monte Carlo estimators based on sparse polynomial chaos expansions

We introduce two control variate Monte Carlo estimators where the control is based on the truncated sparse polynomial chaos expansion of the function in hand. We use the control variate estimators to estimate the lower and upper Sobol' indices in some applications, and compare them numerically with some of the best Monte Carlo estimators in the literature. The results suggest that in computationally expensive problems where a low-order polynomial chaos expansion is not an accurate approximation of the model but highly correlated with it, the control variate estimators are either the best or among the best in terms of efficiency.

Sparse polynomial chaos expansion for high-dimensional nonlinear damage mechanics

Finite Element Simulations in solid mechanics are nowadays common practice in engineering. However, considering uncertainties based on this powerful method remains a challenging task, especially when nonlinearities and high stochastic dimensions have to be taken into account. Although Monte Carlo Simulation (MCS) is a robust method, the computational burden is high, especially when a nonlinear finite element analysis has to be performed behind each sample. To overcome this burden, several “model-order reduction” techniques have been discussed in the literature. Often, these studies are limited to quite smooth responses (linear or smooth nonlinear models and moderate stochastic dimensions).This paper presents systematic studies of the promising Sparse Polynomial Chaos Expansion (SPCE) method to investigate the capabilities and limitations of this approach using MCS as a benchmark. A nonlinear damage mechanics problem serves as a reference, which involves random fields of material properties. By this, a clear limitation of SPCE with respect to the stochastic dimensionality could be shown, where, as expected, the advantage over MCS disappears.As part of these investigations, options to optimise SPCE have been studied, such as different error measures and optimisation algorithms. Furthermore, we have found that combining SPCEs with sensitivity analysis to reduce the stochastic dimension improves accuracy in many cases at low computational cost.

Extreme-oriented sensitivity analysis using sparse polynomial chaos expansion. Application to train–track–bridge systems

The use of sensitivity analysis is essential in model development for the purposes of calibration, verification, factor prioritization, and mechanism reduction. While most contributions to sensitivity methods focus on the average model response, this paper proposes a new sensitivity method focusing on the extreme response and structural limit states, which combines an extreme-oriented sensitivity method with polynomial chaos expansion. This enables engineers to perform sensitivity analysis near given limit states and visualize the relevance of input factors to different design criteria and corresponding thresholds. The polynomial chaos expansion is used to approximate the model output and alleviate the computational cost in sensitivity analysis, which features sparsity and adaptivity to enhance efficiency. The accuracy and efficiency of the method are verified in a truss structure, which is then illustrated on a dynamic train–track–bridge system. The role of the input factors in response variability is clarified, which differs in terms of the design criteria chosen for sensitivity analysis. The method incorporates multi-scenarios and can thus be useful to support decision-making in design and management of engineering structures.

Chance-constrained optimal dispatch of integrated energy systems based on data-driven sparse polynomial chaos expansion

In integrated energy systems, stochastic variations of different energy types of loads and the increasing penetration of renewable energy generation have resulted in considerable uncertainties. These uncertainties pose significant challenges to the economics and safe operation of integrated energy systems. Conventional deterministic methods of optimal dispatch overlook the effects of uncertainties, while stochastic optimization, though accounting for uncertainties, often yields conservative solutions that may adversely affect the economic operations. Chance-constrained optimization can effectively deal with uncertainties and provide more flexibility in balancing operational risks and benefits by expanding the feasible region. Hence, this paper proposes a chance-constrained optimal dispatch method for integrated energy systems and employs data-driven sparse polynomial chaos expansion method to enhance solving efficiency. First, the proposed chance-constrained optimization aims to ensure the optimal economic operation with an affordable security confidence level that balances safety and economics compared with deterministic and stochastic optimizations. The introduced data-driven sparse polynomial chaos expansion method enables the fast computing of the output response using only historical data, i.e., without knowledge pertaining to the distribution functions. Moreover, an improved iterative verification structure is proposed, which further improves the convergence speed and accuracy. Finally, the advantages and feasibility of the proposed method are verified using a test case and compared with those of deterministic and stochastic optimization. Results show that the proposed method successfully reduces the operational cost and controls violation probability.

Global sensitivity analysis for multivariate outputs using generalized RBF-PCE metamodel enhanced by variance-based sequential sampling

The mathematical and computational models in engineering applications commonly have multiple outputs, so it is critical to develop global sensitivity analysis (GSA) for multivariate outputs, which can be used to explore the effect of input parameters on output responses. Amongst the existing sensitivity analysis, the covariance-based method is one of the most widely used methods due to its understandability and validity. However, traditional GSA is calculated by Monte Carlo Simulation (MCS), leading to huge computational cost and large sample demand. In this study, the generalized hybrid metamodel using radial basis function (RBF) and sparse polynomial chaos expansion (PCE) is applied for covariance-based GSA. To improve the efficiency and performance of RBF-PCE in mult-models, an efficient sequential sampling method is proposed based on the local variance density of sample points. Three analytical functions and two engineering problems are employed to demonstrate the accuracy and validity of the proposed method. The results indicate that the proposed method has a significant improvement in accuracy and efficiency compared with the existing GSA methods, and the sample demand is reduced by three orders of magnitude compared with MCS-based Sobol' indices.

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

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

An active sparse polynomial chaos expansion approach based on sequential relevance vector machine

Polynomial chaos expansion (PCE) is a popular surrogate modeling approach employed in uncertainty quantification for a variety of engineering problems. However, the challenges for full PCE lie in the “curse of dimensionality” of the expansion coefficients. In this paper, we propose a new sparse PCE approach by introducing active learning technique and sequence relevance vector machine (SRVM). As an active learning technique, efficient loss function based on expected improvement (EI-based ELF) is introduced to search for the best next sampling point and enrich the training samples. Relevance vector machine (RVM) is a superior machine learning technique due to the sparsity of its adopted model. SRVM leads to the maximization of marginal likelihood by sequentially selecting basis functions. To assess the performance of the proposed method, four examples are investigated, and the results show that the proposed method outperforms the classical sparse PCE in terms of both accuracy and efficiency.

Uncertainty Quantification for the Noah‐MP Land Surface Model: A Case Study in a Grassland and Sandy Soil Region

AbstractLand surface models rely on a multitude of parameters to simulate land‐atmosphere interactions, but the parameter uncertainty can limit the reliability of model predictions. This study utilizes a parameter uncertainty quantification (UQ) framework to quantify and reduce the parameter uncertainty of the Noah‐MP land surface model in a grassland and sandy soil region in the Midwest of the USA. First, the sparse polynomial chaos expansion method which can quantify the interaction effect of parameters, is employed. A relatively small parameter sample size (i.e., 20 times of the number of parameters) was sufficient to identify the sensitive parameters; an additional sensitive parameter, the saturated soil hydraulic conductivity, was screened out compared to previous study. Then, based on the selected sensitive parameters, the weighted multi‐objective adaptive surrogate modeling‐based optimization algorithm is used as the parameter optimization method. The optimization results showed that the root mean square error of flux of latent heat (FLH) on about 82% of the total grids was reduced, and the number was about 57% for gross primary production (GPP) compared to the results using the original parameter settings, indicating that the Pareto parameter set by the UQ framework improved the Noah‐MP model in simulating FLH and GPP in a grassland and sandy soil region in the Midwest of the USA.

Uncertainty Quantification for Thermodynamic Simulations with High-Dimensional Input Spaces Using Sparse Polynomial Chaos Expansion: Retrofit of a Large Thermal Power Plant

The assessment of the future thermodynamics performance of a retrofitted heat and power production unit is prone to many uncertainties due to the large number of parameters involved in the modeling of all its components. To carry out uncertainty quantification analysis, alternatives to the traditional Monte Carlo method must be used due to the large stochastic dimension of the problem. In this paper, sparse polynomial chaos expansion (SPCE) is applied to the retrofit of a large coal-fired power plant into a biomass-fired combined heat and power unit to quantify the main drivers and the overall uncertainty on the plant’s performance. The thermodynamic model encompasses over 180 components and 1500 parameters. A methodology combining the use of SPCE and expert judgment is proposed to narrow down the sources of uncertainty and deliver reliable probability distributions for the main key performance indicators (KPIs). The impact of the uncertainties on each input parameter vary with the considered KPI and its assessment through the computation of Sobol’ indices. For both coal and biomass operations, the most impactful input parameters are the composition of the fuel and its heating value. The uncertainty on the performance and steam quality parameters is not much affected by the retrofit. Key furnace parameters exhibit a skewed probability distribution with large uncertainties, which is a strong attention point in terms of boiler operation and maintenance.

Data-driven sparse polynomial chaos expansion for models with dependent inputs

Polynomial chaos expansions (PCEs) have been used in many real-world engineering applications to quantify how the uncertainty of an output is propagated from inputs by decomposing the output in terms of polynomials of the inputs. PCEs for models with independent inputs have been extensively explored in the literature. Recently, different approaches have been proposed for models with dependent inputs to expand the use of PCEs to more real-world applications. Typical approaches include building PCEs based on the Gram–Schmidt algorithm or transforming the dependent inputs into independent inputs. However, the two approaches have their limitations regarding computational efficiency and additional assumptions about the input distributions, respectively. In this paper, we propose a data-driven approach to build sparse PCEs for models with dependent inputs without any distributional assumptions. The proposed algorithm recursively constructs orthonormal polynomials using a set of monomials based on their correlations with the output. The proposed algorithm on building sparse PCEs not only reduces the number of minimally required observations but also improves the numerical stability and computational efficiency. Four numerical examples are implemented to validate the proposed algorithm. The source code is made publicly available for reproducibility.

Probabilistic analysis of width‐limited 3D slope in spatially variable soils: UBLA enhanced with efficiency‐improved discretization of horn‐like failure mechanism

AbstractReliability analysis of earth slopes considering soil spatial variability has garnered significant attention from researchers. However, previous studies have predominantly focused on long slopes with infinite or ample width, such as dams and subgrades, but rare research was dedicated to slopes with width restricted by boundary constraints. In this regard, this paper proposes an efficient reliability framework suitable for width‐limited slopes in spatially variable soils. The proposed framework employed spatial discretization‐based upper bound limit analysis (UBLA) to conduct the deterministic slope stability analysis. The established failure mechanism can well capture the width‐limited three‐dimensional slip surface characteristics and satisfies the kinematically admissible conditions in spatially variable soils. Two innovative strategies are proposed to boost the efficiency of the discretized mechanism. They reduce the computational time to determine safety factors from over 20 min to less than 2 min. For the uncertainty modeling and slope reliability computing procedure, the Sparse Polynomial Chaos Expansion (SPCE) and Monte Carlo Simulation (MCS) methods are combined to generate the distribution of safety factors and failure probabilities. This combination addresses the high‐dimensional stochastic problem arising from 3D spatial variability. Using the proposed framework, a parametric analysis of a 3D spatially variable slope is conducted to investigate the impact of various factors. The findings of this paper are beneficial to the risk evolution of width‐limited slopes, and the efficiency enhancement strategies have the potential to expedite the limit analysis of other geo‐structures, such as tunnels, retaining walls, and foundations.

Uncertainty Quantification of Human Electromagnetic Exposure From Electric Vehicle Wireless Power Transfer System

With the rapid development of electric vehicles (EVs), the wireless power transfer (WPT) technology of EV is becoming more and more mature. The safety of human electromagnetic exposure caused by the WPT system of EV has become the focus of research. Considering the practical application of EV WPT system, this paper studies the electromagnetic exposure of human body based on the uncertainty quantification theory. A sparse polynomial chaos expansion method is used to establish the surrogate model of the mean value of some human organs’ SAR, and the probability distribution function of SAR is calculated. The Sobol is used to quantify the impact of relevant variables on SAR. The final calculation results show that when the person is on the side of the EV, the offset between the magnetic coupling coils of the WPT and the lateral displacement of the human body on the side of the EV have a greater impact; When a person is at the rear of the EV, the vertical displacement of the magnetic coupling coils of the WPT has a greater impact on the longitudinal displacement of the human body at the rear of the vehicle. Combined with the above analysis, this paper can provide reasonable suggestions for the safety protection of electromagnetic exposure of human body in the WPT of EVs.

Analysis of Dynamic Mesh Stiffness and Dynamic Response of Helical Gear Based on Sparse Polynomial Chaos Expansion

This paper presents an efficient method for obtaining the dynamic mesh stiffness and dynamic response of a helical gear pair. Unlike the traditional dynamic model that utilizes a time-dependent sequence, the mesh stiffness using the presented method is updated according to the gear displacement vector at each sub-step of the numerical calculation. Three-dimensional loaded tooth contact analysis (3D LTCA) is used to determine the mesh stiffness, and a surrogate model based on sparse polynomial chaos expansion (SPCE) is proposed to improve the computational efficiency, which is achieved by reducing the number of coefficients in the polynomial chaos expansion (PCE) model though a quantum genetic algorithm. During the calculation, the gear displacement vector at each sub-step is converted into the changes in center distance, misalignment angle, and mesh force, which are then introduced into the SPCE model to update the mesh stiffness for subsequent calculations. The results suggest that the SPCE model exhibits high accuracy and can significantly improve the computational efficiency of the PCE model, making it suitable for dynamic calculations. Upon updating the mesh stiffness during the dynamic calculation, the mesh stiffness declines, the dynamic transmission error (DTE) increases, and the frequency components of the responses change significantly.

A surrogate-assisted uncertainty-aware Bayesian validation framework and its application to coupling free flow and porous-medium flow

Existing model validation studies in geoscience often disregard or partly account for uncertainties in observations, model choices, and input parameters. In this work, we develop a statistical framework that incorporates a probabilistic modeling technique using a fully Bayesian approach to perform a quantitative uncertainty-aware validation. A Bayesian perspective on a validation task yields an optimal bias-variance trade-off against the reference data. It provides an integrative metric for model validation that incorporates parameter and conceptual uncertainty. Additionally, a surrogate modeling technique, namely Bayesian Sparse Polynomial Chaos Expansion, is employed to accelerate the computationally demanding Bayesian calibration and validation. We apply this validation framework to perform a comparative evaluation of models for coupling a free flow with a porous-medium flow. The correct choice of interface conditions and proper model parameters for such coupled flow systems is crucial for physically consistent modeling and accurate numerical simulations of applications. We develop a benchmark scenario that uses the Stokes equations to describe the free flow and considers different models for the porous-medium compartment and the coupling at the fluid–porous interface. These models include a porous-medium model using Darcy’s law at the representative elementary volume scale with classical or generalized interface conditions and a pore-network model with its related coupling approach. We study the coupled flow problems’ behaviors considering a benchmark case, where a pore-scale resolved model provides the reference solution. With the suggested framework, we perform sensitivity analysis, quantify the parametric uncertainties, demonstrate each model’s predictive capabilities, and make a probabilistic model comparison.

Efficiency Optimization of Wireless Power Transfer System for Electric Vehicle Based on Improved Marine Predators Algorithm

Electric vehicle (EV) is the core part of future automobile technology, and safe and reliable wireless power transfer (WPT) technology is the key link to improve the intelligent driving technology of EV. In this paper, the uncertainty quantification method is proposed to guide the optimization design of WPT structure, so as to improve the efficiency of WPT. First this paper establishes a surrogate model of WPT efficiency based on the adaptive sparse polynomial chaos expansion, and the uncertainty of EVs WPT transmission efficiency is quantified, the computational efficiency is improved by about 8.4 times. Then the surrogate model is combined with the global sensitivity analysis method to quantify the impact of different variables in WPT on efficiency and screen out the variables with greater impact. Finally, this paper uses the improved marine predators algorithm to optimize the selected WPT system structure parameters. Considering the uncertainty, the average efficiency of the optimized WPT system is increased from 73.43% to 94.64%. Compared with other optimization methods, it proves that the method in this paper can optimize WPT more efficiently, and significantly improve the transmission efficiency.

Sparse polynomial chaos expansions for uncertainty quantification in thermal tomography

This contribution presents the identification strategy of thermal parameters relying solely on data measured on boundaries — thermal tomography. The idea is to obtain crucial information about the thermal properties inside the domain under consideration while keeping the test sample intact. Such methodology perfectly fits into historic preservation where it is of particular interest to perform only non-destructive surface measurements. We propose an advanced, accelerated, and reliable inverse solver for thermal tomography problems. Here, Bayesian inference is addressed as a method, where unknown parameters are modeled as random variables regularizing the inverse problem. The obtained results are probability distributions – posterior distributions – summarizing all available information and any remaining uncertainty in the values of thermal parameters. Novelties of our approach consist in the combination of (i) formulation of parameter identification in a probabilistic setting, and (ii) use of the surrogate models based on the sparse polynomial chaos expansion. This new sparse formulation significantly reduces the total number of polynomial terms and represents the main achievement of this paper.