Discretization of a Random Field – a Multiobjective Algorithm Approach

Karhunen-Loeve(KL) expansion of auto-covariance of input Gaussian random process is carried out using standard normal variables as the basis of KL expansion. Input random processes for many practicle problems have non-Gaussian random process description, so an algorithm is developed to find the basis random variables of KL expansion for non-Gaussian processes. The basis random variables of KL expansion are obtained via nonlinear transformation of marginal cumulative distribution function using standard deviation. Results are obtained for three known skewed distributions, Log-Normal, Beta, and Exponential. In all the cases, it is found that the proposed algorithm matches very well with the known solutions and can be applied to solve non-Gaussian process using Spectral Stochastic Finite Element Method.

Optimizing the operation of a multireservoir system is challenging due to the high dimension of the decision variables that lead to a large and complex search space. A spectral optimization model (SOM), which transforms the decision variables from time domain to frequency domain, is proposed to reduce the dimensionality. The SOM couples a spectral dimensionality‐reduction method called Karhunen‐Loeve (KL) expansion within the routine of Nondominated Sorting Genetic Algorithm (NSGA‐II). The KL expansion is used to represent the decision variables as a series of terms that are deterministic orthogonal functions with undetermined coefficients. The KL expansion can be truncated into fewer significant terms, and consequently, fewer coefficients by a predetermined number. During optimization, operators of the NSGA‐II (e.g., crossover) are conducted only on the coefficients of the KL expansion rather than the large number of decision variables, significantly reducing the search space. The SOM is applied to the short‐term operation of a 10‐reservoir system in the Columbia River of the United States. Two scenarios are considered herein, the first with 140 decision variables and the second with 3360 decision variables. The hypervolume index is used to evaluate the optimization performance in terms of convergence and diversity. The evaluation of optimization performance is conducted for both conventional optimization model (i.e., NSGA‐II without KL) and the SOM with different number of KL terms. The results show that the number of decision variables can be greatly reduced in the SOM to achieve a similar or better performance compared to the conventional optimization model. For the scenario with 140 decision variables, the optimal performance of the SOM model is found with six KL terms. For the scenario with 3360 decision variables, the optimal performance of the SOM model is obtained with 11 KL terms.

A stochastic framework is proposed to evaluate the effect of random effects on the overall performance of wireless links. A generalized polynomial chaos expansion is leveraged to relate the uncertainties in antenna geometry, orientation and position to the figures of merit characterizing the link. The stochastic testing procedure is proposed as a more efficient alternative to stochastic collocation, for a large number of random variables. The non-intrusive statistical framework is applied to evaluate the uncertainty on the efficiency of a wireless power transfer system.

We present a distributed active subspace method for training surrogate models of complex physical processes with high-dimensional inputs and function valued outputs. Specifically, we represent the model output with a truncated Karhunen–Loeve (KL) expansion, screen the structure of the input space with respect to each KL mode via the active subspace method, and finally form an overall surrogate model of the output by combining surrogates of individual output KL modes. To ensure scalable computation of the gradients of the output KL modes, needed in active subspace discovery, we rely on adjoint-based gradient computation. The proposed method combines benefits of active subspace methods for input dimension reduction and KL expansions used for spectral representation of the output field. We provide a mathematical framework for the proposed method and conduct an error analysis of the mixed KL active subspace approach. Specifically, we provide an error estimate that quantifies errors due to active subspace projection and truncated KL expansion of the output. We demonstrate the numerical performance of the surrogate modeling approach with an application example from biotransport.

Robustness of stochastic expansions for the stability of uncertain nonlinear dynamical systems – Application to brake squeal

This work explores generalized polynomial chaos (GPC) surrogate models and their effectiveness for general acoustics and vibration applications. GPC is primarily known as an uncertainty quantification (UQ) technique and in that context the underlying polynomial-based model has been shown to be effective in mapping input probability distributions to the corresponding output probability distributions. In this study, GPC surrogate models – including those generated by both quadrature and regression methods – are evaluated for their effectiveness in non-UQ focused analyses. As points of comparison, Krylov subspace and other more traditional reduced order modeling techniques are demonstrated and compared to GPC models so that the differences may be better understood. Example problems with several different levels of complexity are used to show how the computational burden as well as the overall effectiveness of the method changes as the number of input variables that must be considered grows.

This paper considers the model predictive control of dynamic systems subject to stochastic uncertainties due to parametric uncertainties and exogenous disturbance. The effects of uncertainties are quantified using generalised polynomial chaos expansions with an additive Gaussian random process as the exogenous disturbance. With Gaussian approximation of the resulting solution trajectory of a stochastic differential equation using generalised polynomial chaos expansion, convex finite-horizon model predictive control problems are solved that are amenable to online computation of a stochastically robust control policy over the time horizon. Using generalised polynomial chaos expansions combined with convex relaxation methods, the probabilistic constraints are replaced by convex deterministic constraints that approximate the probabilistic violations. This approach to chance-constrained model predictive control provides an explicit way to handle a stochastic system model in the presence of both model uncertainty and exogenous disturbances.

In recent years, the non-invasive load monitoring (NILM) method based on sparse coding has shown promising research prospects. This type of method learns a sparse dictionary for each monitoring target device, and it expresses load decomposition as a problem of signal reconstruction using dictionaries and sparse vectors. The existing NILM methods based on sparse coding have problems such as inability to be applied to multi-state and time-varying devices, single-load characteristics, and poor recognition ability for similar devices in distributed manners. Using the analysis above, this paper focuses on devices with similar features in households and proposes a distributed non-invasive load monitoring method using Karhunen–Loeve (KL) feature extraction and an improved deep dictionary. Firstly, Karhunen–Loeve expansion (KLE) is used to perform subspace expansion on the power waveform of the target device, and a new load feature is extracted by combining singular value decomposition (SVD) dimensionality reduction. Afterwards, the states of all the target devices are modeled as super states, and an improved deep dictionary based on the distance separability measure function (DSM-DDL) is learned for each super state. Among them, the state transition probability matrix and observation probability matrix in the hidden Markov model (HMM) are introduced as the basis for selecting the dictionary order during load decomposition. The KL feature matrix of power observation values and improved depth dictionary are used to discriminate the current super state based on the minimum reconstruction error criterion. The test results based on the UK-DALE dataset show that the KL feature matrix can effectively reduce the load similarity of devices. Combined with DSM-DDL, KL has a certain information acquisition ability and acceptable computational complexity, which can effectively improve the load decomposition accuracy of similar devices, quickly and accurately estimating the working status and power demand of household appliances.

Our ultimate goal is a topology optimization for a permanent-magnet (PM) machine, while including material uncertainties. The uncertainties in the output data are, e.g., due to measurement errors in the non-/linear material laws. In the resulting stochastic forward problem, these uncertainties are stochastically modeled by random fields. The solution of the underlying PDE, which describes magnetostatics, is represented using the generalized polynomial chaos expansion. As crucial ingredient we exploit the stochastic collocation method (SCM). Eventually, this leads to a random-dependent bi-objective cost functional, which is comprised of the expectation and the variance. Subject to the optimization of the PM machine are the shapes of the rotor poles, which are described by zero-level sets. Thus, the optimization will be done by redistributing the iron and magnet material over the design domain, which allows to attain an innovative low cogging torque design of an electric machine. For this purpose, the gradient directions are evaluated by using the continuous design sensitivity analysis in conjunction with the SCM. In the end, our numerical result for the optimization of a two-dimensional model demonstrates that the proposed approach is robust and effective.

This paper presents a complete real-time implementation of a Non-Intrusive Appliance Load Monitoring (NIALM) system that, is robust under residential voltage level fluctuations. Existing NIALM techniques rely on multiple measurements taken at high sampling rates, but, only have been proven in simulated environments without even considering the effect of residential voltage level fluctuations — which is a severe problem in power systems of most developing countries like Sri Lanka. In contrast, through the NIALM method proposed in this paper, accurate load monitoring results were obtained in realtime using only smart meter measurements taken at a low sampling rate from a real appliance setup under residential voltage level fluctuations. In the proposed NIALM method, initially in the learning phase, a properly constructed MATLABTM Graphical User Interface (GUI) was used to acquire signals of each appliance active power consumption and voltage levels. Then, obtained active power measurements were separated into subspace components (SCs) via the Karhunen Loeve' Expansion (KLE) while also taking the voltage variations into account. Using those SCs, a unique information rich appliance level signature database was constructed and it was then used to obtain the signatures for all possible device combinations. Next, a separate GUI was designed to identify the turned ON appliance combination in the current time window using the pre-constructed signature databases, after reading the total residential active power consumption and the supply voltage. To validate the proposed real-time NIALM implementation, data from a laboratory arrangement consisting of ten household appliances was used. From the results, it was found that the proposed method is capable of accurately identifying the turned on appliances even under severe residential supply voltage level fluctuations.

Parametric uncertainty assessment in hydrological modeling using the generalized polynomial chaos expansion

A certain general class of polynomials is defined and its properties are considered. By specializing the definition in one direction we are led to Bernoulli's polynomials of order n regarded as generalizations of xv, by specializing in another direction xv leads to Euler's polynomials of order n. The same methods are apphed to Hermite's polynomials and thus two new types of polynomial are found.

Two squeeze‐film gas damping models are proposed to quantify uncertainties associated with the gap size and the ambient pressure. Modeling of gas damping has become a subject of increased interest in recent years due to its importance in micro‐electro‐mechanical systems (MEMS). In addition to the need for gas damping models for design of MEMS with movable micro‐structures, knowledge of parameter dependence in gas damping contributes to the understanding of device‐level reliability. In this work, two damping models quantifying the uncertainty in parameters are generated based on rarefied flow simulations. One is a generalized polynomial chaos (gPC) model, which is a general strategy for uncertainty quantification, and the other is a compact model developed specifically for this problem in an early work. Convergence and statistical analysis have been conducted to verify both models. By taking the gap size and ambient pressure as random fields with known probability distribution functions (PDF), the output PDF for the damping coefficient can be obtained. The first four central moments are used in comparisons of the resulting non‐parametric distributions. A good agreement has been found, within 1%, for the relative difference for damping coefficient mean values. In study of geometric uncertainty, it is found that the average damping coefficient can deviate up to 13% from the damping coefficient corresponding to the average gap size. The difference is significant at the nonlinear region where the flow is in slip or transitional rarefied regimes. Copyright © 2010 John Wiley & Sons, Ltd.

A polynomial chaos expansion in dependent random variables

A new multiple objective evolutionary algorithm is proposed for reliability optimization of series-parallel systems. This algorithm uses a genetic algorithm based on rank selection and elitist rein...