Probabilistic Collocation Method Research Articles

Density estimation for time-dependent PDE with random input by a Legendre-based multi-element probabilistic collocation method

This paper proposed a Legendre-based multi-element probabilistic collocation method for time-dependent stochastic differential equations, used for density estimation of unknown functions. This method involves discretizing the stochastic space, and on each element, constructing Lagrange interpolation basis functions based on Legendre–Gauss–Lobatto collocation/quadrature nodes. The proposed method is applied to approximate one-dimensional/two-dimensional smooth/non-smooth functions and is tested for accuracy in approximating random function values, density estimations, and mathematical expectations. This method is applied to stochastic nonlinear Schrödinger equations and coupled stochastic nonlinear Schrödinger equations, and all numerical results are compared with Monte Carlo simulation.

Event-Triggered Guarantee Cost Control for Partially Unknown Stochastic Systems via Explorized Integral Reinforcement Learning Strategy.

In this article, an integral reinforcement learning (IRL)-based event-triggered guarantee cost control (GCC) approach is proposed for stochastic systems which are modulated by randomly time-varying parameters. First, with the aid of the RL algorithm, the optimal GCC (OGCC) problem is converted into an optimal zero-sum game by solving a modified Hamilton-Jacobin-Isaac (HJI) equation of the auxiliary system. Moreover, in order to address the stochastic zero-sum game, we propose an on-policy IRL-based control approach involved by the multivariate probabilistic collocation method (MPCM), which can accurately predict the mean value of uncertain functions with randomly time-varying parameters. Furthermore, a novel GCC method, which combines the explorized IRL algorithm and MPCM, is designed to relax the restriction of knowing the system dynamics for the class of stochastic systems. On this foundation, for the purpose of reducing computation cost and avoiding the waste of resources, we propose an event-triggered GCC approach involved with explorized IRL and MPCM by utilizing critic-actor-disturbance neural networks (NNs). Meanwhile, the weight vectors of three NNs are updated simultaneously and aperiodically according to the designed triggering condition. The ultimate boundedness (UB) properties of the controlled systems have been proved by means of the Lyapunov theorem. Finally, the effectiveness of the developed GCC algorithms is illustrated via two simulation examples.

Optimization and control strategy for wind turbine aerodynamic performance under uncertainties

Aerodynamic performance of wind turbine governs the overall energy efficiency, which has been an ever-lasting research focus in the field of wind power technology. Due to the coupling effect among the highly complex environmental and structural uncertainties, the practical aerodynamic performance may not be reliably predicted. To aggravate, this performance declines with time in service. It is of great significance to efficiently and reliably assess the impact of uncertain factors and reduce these influences on wind turbine aerodynamic performance. This paper establishes an uncertainty analysis and robustness optimization model of wind turbine aerodynamic performance considering wind speed and pitch angle error uncertainties. An approach combined the no-instrusive probabilistic collocation method is used, and the blade element momentum theory is applied to quantify influences of variable uncertainties on NREL 5 MW wind turbine aerodynamic performance. The optimization target is to reduce the sensitivity of wind turbine aerodynamic performance to uncertainties, as well as maintain capture power. The results show that the wind turbine aerodynamic and mechanical performance will be greatly affected with uncertain factors. By optimizing and adjusting wind turbine rotor speed and blade pitch angle, the wind turbine rotor power and thrust load variation can be reduced to 9.14% and 9.36%, respectively, which indeed reduces the uncertainty effects.

Robust optimization and uncertainty quantification of a micro axial compressor for unmanned aerial vehicles

Axial compressors are susceptible to uncertainties during their manufacturing and operation, resulting in reduced efficiency and performance dispersion. However, uncertainty quantification and robust design of compressors remains challenging due to the complexity of structure and internal flow. In this study, an automated framework for uncertainty quantification and robustness optimization of micro axial compressors is proposed. Ten geometrical uncertainties are propagated for the nominal design point and two off-design points, i.e., near stall and choke conditions, respectively. The main objective of this paper is to optimize the aerodynamic robustness performance at these operating points. The sparse grid-based probabilistic collocation method is used to propagate these uncertainties, and a multi-objective genetic algorithm is employed to perform robust optimization based on a novel constructed surrogate model.The results show that the optimal configuration achieves an improvement in aerodynamic robustness and mean performance across the entire characteristic map, with greater improvement at the design working point than at the off-design points. At the design working point, the mean isentropic efficiency and pressure ratio of the optimal configuration increase by 0.6% and 0.5%, respectively, while the standard deviation of isentropic efficiency, pressure ratio, and mass flow rate decreases by 32.4%, 41.2%, and 25.1%, respectively. This optimization framework proves to be both feasible and efficient and can be applied to aerodynamic robust optimization of turbomachinery. In the future, we will apply this framework to different aspects of the gas turbine life cycle to model and analyze uncertainties of larger orders of magnitude.

Stochastic analysis for in-plane dynamic responses of low-speed uniformity of tires due to geometric defects

Tire uniformity is defined as the dynamic mechanical characteristics of pneumatic tires due to the defects originating in tire components during manufacturing processes. These defects lead to deviations in the structural and material parameters of tires and, in turn, induce variations in the dynamic response. In this scenario, this paper proposes a probabilistic model of the low-speed uniformity analysis for predicting the impacts of the uncertainties in the geometric defects and the vertical load on the dynamic responses. The deterministic coupled rigid–flexible ring model, combined with the generalized polynomial chaos expansion theory, is developed to simulate the radial force variation due to radial run-out. In this approach, the non-uniform tire radius and the vertical load are selected as a set of random variables and are approximated by the generalized polynomial chaos expansions. The distributions of these input variables are identified from 200 measured tires by the Pearson model. A non-intrusive probabilistic collocation method is adopted to evaluate the coefficients of the generalized polynomial chaos expansions, and the selection of collocation points is also illustrated. Finally, the distributions of the radial force variation and its first harmonic due to the geometric defects and vertical loads are predicted in comparison with measurements. Moreover, the distributions of the dynamic response caused by each parameter and different variance levels are discussed and verified by the measurement data from 1130 tires.

Communication-Aware Formation Control of AUVs With Model Uncertainty and Fading Channel via Integral Reinforcement Learning

Most formation approaches of autonomous underwater vehicles (AUVs) focus on the control techniques, ignoring the influence of underwater channel. This paper is concerned with a communication-aware formation issue for AUVs, subject to model uncertainty and fading channel. An integral reinforcement learning (IRL) based estimator is designed to calculate the probabilistic channel parameters, wherein the multivariate probabilistic collocation method with orthogonal fractional factorial design (M-PCM-OFFD) is employed to evaluate the uncertain channel measurements. With the estimated signal-to-noise ratio (SNR), we employ the IRL and M-PCM-OFFD to develop a saturated formation controller for AUVs, dealing with uncertain dynamics and current parameters. For the proposed formation approach, an integrated optimization solution is presented to make a balance between formation stability and communication efficiency. Main innovations lie in three aspects: 1) Construct an integrated communication and control optimization framework; 2) Design an IRL-based channel prediction estimator; 3) Develop an IRL-based formation controller with M-PCM-OFFD. Finally, simulation results show that the formation approach can avoid local optimum estimation, improve the channel efficiency, and relax the dependence of AUV model parameters.

Evaluating grid strength under uncertain renewable generation

The increasing displacement of synchronous generators with renewable resources such as wind and solar via power electronic interfaces causes a reduction in short-circuit strength and weak grid issues. The variation and uncertainty of renewable energy increase challenges for identifying weak grid conditions. This paper proposes an efficient method to analyze the impact of uncertain renewable energy on grid strength. The proposed method uses the probabilistic collocation method (PCM) to approximate the results of grid strength assessment under uncertain renewable generation, in order to reduce computational burden without compromising result accuracy when compared with traditional Monte Carlo simulation (MCS). To improve the accuracy of the approximation results, the proposed method integrates the K-means clustering technique with PCM to select the approximation samples of input variables. The efficacy of the proposed method is demonstrated by comparison with MCS on the modified IEEE 9-bus system and modified IEEE 39-bus system with multiple renewable generators.

Integrated Localization and Tracking for AUV With Model Uncertainties via Scalable Sampling-Based Reinforcement Learning Approach

This article studies the joint localization and tracking issue for the autonomous underwater vehicle (AUV), with the constraints of asynchronous time clock in cyberchannels and model uncertainty in physical channels. More specifically, we develop a reinforcement learning (RL)-based asynchronous localization algorithm to localize the position of AUV, where the time clock of AUV is not required to be well synchronized with the real time. Based on the estimated position, a scalable sampling strategy called multivariate probabilistic collocation method with orthogonal fractional factorial design (M-PCM-OFFD) is employed to evaluate the time-varying uncertain model parameters of AUV. After that, an RL-based tracking controller is designed to drive AUV to the desired target point. Besides that, the performance analyses for the integration solution are also presented. Of note, the advantages of our solution are highlighted as: 1) the RL-based localization algorithm can avoid local optimal in traditional least-square methods; 2) the M-PCM-OFFD-based sampling strategy can address the model uncertainty and reduce the computational cost; and 3) the integration design of localization and tracking can reduce the communication energy consumption. Finally, simulation and experiment demonstrate that the proposed localization algorithm can effectively eliminate the impact of asynchronous clock, and more importantly, the integration of M-PCM-OFFD in the RL-based tracking controller can find accurate optimization solutions with limited computational costs.

A meshfree peridynamic model for brittle fracture in randomly heterogeneous materials

In this work we aim to develop a unified mathematical framework and a reliable computational approach to model the brittle fracture in heterogeneous materials with variability in material microstructures, and to provide statistical metrics for quantities of interest, such as the fracture toughness. To describe the material responses such as the nucleation and growth of fractures in a uniform setting, we consider the peridynamics model. In particular, a stochastic state-based peridynamic model is developed, where the micromechanical parameters are modeled by a finite-dimensional random field, e.g., a combination of truncated random variables determined by the Karhunen–Loève decomposition or the principal component analysis (PCA). To solve this stochastic peridynamic problem, probabilistic collocation method (PCM) is employed to sample the random field that represents the micromechanical parameters. For each sample, the corresponding peridynamic problem is solved by an optimization-based meshfree quadrature rule. We present rigorous analysis for the proposed scheme and verify the convergence rate with a number of benchmark problems. The proposed scheme not only possesses the asymptotic compatibility spatially but also achieves an algebraic or sub-exponential convergence rate in the parametric random space when the number of collocation points grows. Finally, to validate the applicability of this approach on real-world fracture problems, we consider the problem of crystallization toughening in glass-ceramic materials, in which the material at the microstructural scale contains both amorphous glass and crystalline phases. The proposed stochastic peridynamic solver is employed to capture the crack initiation and its growth of the glass-ceramics with different crystal volume fractions, and the averaged fracture toughness are also calculated accordingly. The numerical estimates of fracture toughness show good consistency with data from experimental measurements.

An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems

In this paper we present an asymptotically compatible meshfree method for solving nonlocal equations with random coefficients, describing diffusion in heterogeneous media. In particular, the random diffusivity coefficient is described by a finite-dimensional random variable or a truncated combination of random variables with the Karhunen-Loève decomposition, then a probabilistic collocation method (PCM) with sparse grids is employed to sample the stochastic process. On each sample, the deterministic nonlocal diffusion problem is discretized with an optimization-based meshfree quadrature rule. In terms of analysis, we first show the analytic regularity of solutions with respect to parameters in the diffusion coefficients and then present the rigorous convergence analysis for the proposed numerical scheme, i.e., the scheme is asymptotic compatible spatially and achieves an algebraic or sub-exponential convergence rate in the random coefficients space as the number of collocation points grows. These results are further confirmed by a number of benchmark problems. Finally, to validate the applicability of this approach we consider a randomly heterogeneous nonlocal problem with a given spatial correlation structure, demonstrating that the proposed PCM approach achieves substantial speed-up compared to conventional Monte Carlo simulations.

Uncertainty propagation and sensitivity analysis of three-phase flow in porous media using polynomial chaos expansion

In this study, polynomial chaos expansion (PCE) will be utilized to investigate propagation and evolution of uncertainty from uncertain model parameters to model outputs. Unknown PCE coefficients are calculated using probabilistic collocation method (PCM), which is a non-intrusive technique. The proposed approach is evaluated on a 2D reservoir under three-phase flow condition. By use of PCEs, dependent model outputs, such as well bottomhole pressure and production rates, are represented as functions of uncertain model parameters, including parameters defining the capillary pressure (Pc) and relative permeability (kr) curves. Accuracy, compatibility and efficiency of the proposed approach are compared with Monte Carlo (MC) method, which is considered as the reference case. Mean values, standard deviations and histograms for well bottomhole pressure and production rates estimated from the PCM-based PCE are closely matched to those obtained from MC method. This discloses the fact that the proposed approach can successfully restore the simulation model to represent propagation of uncertainty from uncertain model parameters toward model outputs and to perform sensitivity analysis (SA). Unlike the MC approach which requires an extremely large number of realizations to obtain accurate results, the PCM-based PCE requires to perform model evaluations only at the selected collocation points, thus the computational cost is significantly reduced. Subsequently, using the proposed PCE as the surrogate model of the reservoir, Sobol's sensitivity indices are calculated analytically. Results of SA reveal that those parameters defining Pc curves have negligible effect on the model outputs compared with those parameters defining kr curves, and thus may be ignored during history matching process. Moreover, it is observed that the effects caused by interactions between uncertain kr and Pc parameters have negligible contribution compared to the individual effect of each parameter.

Uncertainty Evaluation Algorithm in Power System Dynamic Analysis With Correlated Renewable Energy Sources

The variation and uncertainty of renewable energy potentially impact power system stability. This article proposes an efficient uncertainty evaluation algorithm to evaluate the influence of renewable energy uncertainty on power system dynamic performance. This algorithm applies probabilistic collocation method (PCM), which greatly reduces the simulation burden without compromising the result accuracy, to approximate the dynamic simulation results. The proposed algorithm also considers the correlation among adjacent renewable generators by using the Copula function, which is suitable to model nonlinear correlations. Additionally, this article extends the utilization of PCM. The actual historical data of renewable generation productions can be utilized directly, and kernel density estimation (KDE) is used to capture the nonparametric distribution of renewable generation production. To improve the accuracy of the approximation results, the proposed method adopts K-means clustering technique to select the approximation samples of input variables. The proposed probabilistic dynamic simulation algorithm is compared with Monte Carlo simulations (MCS) with the probabilistic results on the IEEE 39-bus system with multiple renewable generators, and the accuracy and efficiency of the proposed algorithm are validated.

Development of clustered polynomial chaos expansion model for stochastic hydrological prediction

This study introduced a clustered polynomial chaos expansion (CPCE) model to reveal random propagation and dynamic sensitivity of uncertainty parameters in hydrologic prediction. In the CPCE model, the random characteristics of the streamflow simulations resulting from parameter uncertainties are characterized through the polynomial chaos expansion (PCE) model based on the probabilistic collocation method. At the same time, a multivariate discrete non-functional relationship between PCE coefficients and hydrological model inputs is established based on stepwise cluster analysis. Therefore, compared with traditional PCE method, the developed CPCE model cannot only reflect uncertainty propagation in stochastic hydrological simulation, but also have the capability of random forecasting. Moreover, the dynamic sensitivities of model parameters are investigated through the multilevel factorial analyses. The developed approach was applied for streamflow forecasting for the Ruihe watershed, China. Results showed that with effective quantification for the random characteristics of hydrological processes, the CPCE model can directly predict runoff series and generate the associated probability distributions at different time periods. The dynamic sensitivity analysis indicates that the maximum soil moisture capacity within the catchment plays a key role in the accuracy of the low-flow forecasting, while the degree of spatial variability in soil moisture capacities has a remarkable impact on the accuracy of the high-flow forecasting in the studied watershed.

Anomalous features in internal cylinder flow instabilities subject to uncertain rotational effects

We study the flow dynamics inside a high-speed rotating cylinder after introducing strong symmetry-breaking disturbance factors at cylinder wall motion. We propose and formulate a mathematically robust stochastic model for the rotational motion of the cylinder wall alongside the stochastic representation of incompressible Navier–Stokes equations. We employ a comprehensive stochastic computational fluid dynamics framework combining the spectral/hp element method and the probabilistic collocation method to obtain high-fidelity realizations of our mathematical model in order to quantify the propagation of parametric uncertainty for dynamics-representative quantities of interests. We observe that the modeled symmetry-breaking disturbances cause a flow instability arising from the wall. Utilizing global sensitivity analysis approaches, we identify the dominant source of uncertainty in our proposed model. We next perform a qualitative and quantitative statistical analysis on the fluctuating fields characterizing the fingerprints and measures of intense and rapidly evolving non-Gaussian behavior through space and time. We claim that such non-Gaussian statistics essentially emerge and evolve due to an intensified presence of coherent vortical motions initially triggered by the flow instability due to the symmetry-breaking rotation of the cylinder. We show that this mechanism causes memory effects in the flow dynamics in a way that noticeable anomaly in the time-scaling of enstrophy record is observed in the long run apart from the onset of instability. Our findings suggest an effective strategy to exploit controlled flow instabilities in order to enhance the turbulent mixing in engineering applications.

Ship Robust Design Optimization Based on Polynomial Chaos Expansions

Considerable parameter perturbations occur owing to the influence of uncertain factors in actual ship transportation, resulting in a substantial decline in ship performance. These parameters should not be regarded as certain values but uncertain variables. Ship robust design optimization (RDO) is a method in which various uncertainties are fully considered in the early stages of ship design to ensure that the optimal case adapts to the perturbation of the uncertain parameters. In this study, instead of the commonly used Monte Carlo method, polynomial chaos expansions (PCEs) are adopted to quantify the uncertainty, and an improved probabilistic collocation method (PCM) based on the linear independence principle is proposed to select sample points for calculating polynomial coefficients of PCE, which not only reduces the number of collocation points compared with the traditional statistical sampling method but also avoids the problem that arises with the traditional PCM, which cannot maintain high calculation accuracy even with considerable collocation points. Finally, to ensure ship robustness, in comparison with deterministic optimization design, the proposed RDO framework is applied to minimum Energy Efficiency Design Index (EEDI) KRISO Container Ship hull form design.

Nonintrusive Uncertainty Quantification of Dynamic Power Systems Subject to Stochastic Excitations

Continuous-time random disturbances (also called stochastic excitations) due to increasing renewable generation have an increasing impact on power system dynamics; However, except from the Monte Carlo simulation, most existing methods for quantifying this impact are intrusive, meaning they are not based on commercial simulation software and hence are difficult to use for power utility companies. To fill this gap, this paper proposes an efficient and nonintrusive method for quantifying uncertainty in dynamic power systems subject to stochastic excitations. First, the Gaussian or non-Gaussian stochastic excitations are modeled with an It\^{o} process as stochastic differential equations. Then, the It\^{o} process is spectrally represented by independent Gaussian random parameters, which enables the polynomial chaos expansion (PCE) of the system dynamic response to be calculated via an adaptive sparse probabilistic collocation method. Finally, the probability distribution and the high-order moments of the system dynamic response and performance index are accurately and efficiently quantified. The proposed nonintrusive method is based on commercial simulation software such as PSS/E with carefully designed input signals, which ensures ease of use for power utility companies. The proposed method is validated via case studies of IEEE 39-bus and 118-bus test systems.

Dynamic reliability sensitivity analysis for inter-story isolation structure under stochastic excitations

Considering the uncertainties of seismic excitation and stochastic parameters of the Inter-story Isolation Structure, the implicit structural response function is transformed into an explicit one by the Stochastic Response Surface Method (SRSM), combined with a probabilistic collocation method based on the linear independence principle. By using the reliability analysis method based on the first-passage failure, the reliability of isolation structure is analyzed and the sensitivity of the reliability of each sub-structure is further analyzed. The results show that conducting sensitivity research based on reliability analysis can provide an important theoretical basis for an optimized and robust design of a random vibration system. The changes of the stiffness and damping ratio of the Isolation Layer have a great influence on the reliability of the system and have different effects on the reliability of different parts of the system. The analysis results of reliability sensitivity could provide system reliability indexes to rank stochastic parameters by their importance, thus improving the efficiency of the structure reliability analysis and optimization design.

A Three-Dimensional Ring Model for Uncertainty Quantification in Natural Frequencies and Sound Radiation Characteristics of Tires

Material and geometrical parameters of tires involve some degree of uncertainty mainly related to production processes. Accordingly, the associated structural responses are affected by these uncertainties. In this study, a novel theoretical ring model is presented to describe the in-plane and out-of-plane vibrations as well as the steady-state response of tires, and then to evaluate the influence of the uncertainties in structural parameters on the natural frequencies and the sound radiation characteristics under uncertain excitations. The Hamilton principle is applied here to derive the governing equations. The modal superposition method is used to calculate the steady-state response of the tire. In the sound radiation analysis, the in-plane and out-of-plane bending and torsional vibrations under a set of harmonic unit forces and moments are treated as the source of noise generation. On this basis, the generalized polynomial chaos expansion method is then adopted to evaluate the influence of the uncertainty on the natural frequencies and the sound power. To obtain the unknown coefficients of the expansions, the nonintrusive probabilistic collocation method is employed. Moreover, considering the concept of linear independence of vectors, the number of collocation points is reduced. It is applied to investigate the impacts of the elastic and structural uncertainties on the natural frequencies of the tire. This yields an efficient simulation in terms of computational costs. Finally, the distributions of the sound power due to the forced vibration under the random concentrated line forces are given.

Optimal design of probabilistic robust damping controllers to suppress multiband oscillations of power systems integrated with wind farm

This paper proposes a general optimal design method of probabilistic robust damping controllers (PRDCs) to suppress multiband oscillations in the power system integrated with wind farm. The proposed optimal design method consists of the following two steps. In the first step, owing to the high efficiency of the probabilistic collocation method (PCM), it is adopted to investigate the probabilistic small signal stability analysis (PSSSA) of power system integrated with wind farm. In the second step, a novel adaptive compass search (ACS) with an adaptive sequence of exploration directions is proposed to enhance the global searching ability through the previous searching results, and the proposed ACS is used to obtain the parameters of the PRDCs via solving an optimization problem based on the results of PSSSA obtained from the first step. Case studies are conducted on 16-machine 68-bus system to verify the accuracy and computational efficiency of the PCM. Moreover, simulation studies are also conducted to verify the advantages of the control performances of ACS in the design of damping controllers compared with that of the traditional residue method, particle swarm optimization (PSO), grey wolf optimizer (GWO), and teaching learning-based optimization (TLBO), respectively. Finally, the effectiveness of the proposed method is validated by the time-domain simulation.

Adaptive multi-fidelity probabilistic collocation-based Kalman filter for subsurface flow data assimilation: numerical modeling and real-world experiment

The ensemble Kalman filter (EnKF) has received substantial attention in hydrologic data assimilation due to its ease of implementation. In EnKF, a large enough ensemble size is often required to ensure accuracy, which may result in considerable computational overhead, especially for large-scale problems. Motivated by recent developments in multi-fidelity simulation, we develop a novel data assimilation method that provides an alternative to EnKF, namely adaptive multi-fidelity probabilistic collocation-based Kalman filter (AMF-PCKF). The appealing feature is to approximate the system response with polynomial chaos expansion (PCE) using the adaptive multi-fidelity probabilistic collocation method, which improves the computational efficiency without sacrificing accuracy. This constitutes the forecast step of AMF-PCKF, while the analysis step is established by sequentially updating the PCE coefficients. As demonstrated by a synthetic numerical case of heat transport in unsaturated flow and a real-world two-phase flow experiment, AMF-PCKF can provide more accurate estimations than EnKF under the same amount of computation, even when the number of unknown parameters is as high as 100.