Parametric Problems in Power System Analysis:Recent Applications of Polynomial Approximation Based on Galerkin Method

In power systems, there are many uncertainty factors such as power outputs of distributed generations and fluctuations of loads. It is very beneficial to power system analysis to acquire an explicit function describing the relationship between these factors(namely parameters) and power system states(or performances). This problem, termed as parametric problem(PP) in this paper, can be solved by Galerkin method,which is a powerful and mathematically rigorous method aiming to seek an accurate explicit approximate function by projection techniques. This paper provides a review of the applications of polynomial approximation based on Galerkin method in power system PPs as well as stochastic problems. First, the fundamentals of polynomial approximation and Galerkin method are introduced. Then, the process of solving three types of typical PPs by polynomial approximation based on Galerkin method is elaborated. Finally, some application examples as well as several potential applications of power system PPs solved by Galerkin method are presented, namely the probabilistic power flow, approximation of static voltage stability region boundary, and parametric time-domain dynamic simulation.

Inequalities for the Extended Best Polynomial Approximation Operator in Orlicz Spaces

In this paper we pursue the study of the best approximation operator extended from L~Φ to L~φ, where φ denotes the derivative of the function Φ. We get pointwise convergence for the coefficients of the extended best approximation polynomials for a wide class of function f, closely related to the Calder′on–Zygmund class t_m^p(x) which had been introduced in 1961. We also obtain weak and strong type inequalities for a maximal operator related to the extended best polynomial approximation and a norm convergence result for the coefficients is derived. In most of these results, we have to consider Matuszewska–Orlicz indices for the function φ.

On the Choice of Design Points for Least Square Polynomial Approximations with Application to Uncertainty Quantification

In this work,we concern with the numerical comparison between different kinds of design points in least square(LS)approach on polynomial spaces.Such a topic is motivated by uncertainty quantification(UQ).Three kinds of design points are considered,which are the Sparse Grid(SG)points,the Monte Carlo(MC)points and the Quasi Monte Carlo(QMC)points.We focus on three aspects during the comparison:(i)the convergence properties;(ii)the stability,i.e.the properties of the resulting condition number of the design matrix;(iii)the robustness when numerical noises are present in function values.Several classical high dimensional functions together with a random ODE model are tested.It is shown numerically that(i)neither the MC sampling nor the QMC sampling introduce the low convergence rate,namely,the approach achieves high order convergence rate for all cases provided that the underlying functions admit certain regularity and enough design points are used;(ii)The use of SG points admits better convergence properties only for very low dimensional problems(say d≤2);(iii)The QMC points,being deterministic,seem to be a good choice for higher dimensional problems not only for better convergence properties but also in the stability point of view.

ON CONSERVATIVE APPROXIMATION BY LINEAR POLYNOMIAL OPERATORS AN EXTENSION OF THE BERNSTEIN'S OPERATOR

In this work we slwly linear polynomial operators preserving some consecutive i-convexities and leaving in-verant the polynomtals up to a certain degree. First we study the existence of an incompatibility between the conservation of cenain i-cotivexities and the invariance of a space of polynomials. Interpolation properties are obtained and a theorem by Berens and DcVore about the Bernstein’s operator ts extended. Finally, from these results a genera’ized Bernstein’s operator is obtained.

Bifurcation Analysis of a Nonlinear Vibro-Impact System with an Uncertain Parameter via OPA Method

In this paper,the bifurcation properties of the vibro-impact systems with an uncertain parameter under the impulse and harmonic excitations are investigated.Firstly,by means of the orthogonal polynomial approximation(OPA)method,the nonlinear damping and stiffness are expanded into the linear combination of the state variable.The condition for the appearance of the vibro-impact phenomenon is to be transformed based on the calculation of themean value.Afterwards,the stochastic vibro-impact systemcan be turned into an equivalent high-dimensional deterministic non-smooth system.Two different Poincarésections are chosen to analyze the bifurcation properties and the impact numbers are identified for the periodic response.Consequently,the numerical results verify the effectiveness of the approximation method for analyzing the considered nonlinear system.Furthermore,the bifurcation properties of the system with an uncertain parameter are explored through the high-dimensional deterministic system.It can be found that the excitation frequency can induce period-doubling bifurcation and grazing bifurcation.Increasing the randomintensitymay result in a diffusion-based trajectory and the impact with the constraint plane,which induces the topological behavior of the non-smooth system to change drastically.It is also found that grazing bifurcation appears in advance with increasing of the random intensity.The stronger impulse force can result in the appearance of the diffusion phenomenon.

A Novel Approach for Enhancement of Geometric and Contrast Resolution Properties of Low Contrast Images

The present work encompasses a new image enhancement algorithm using newly constructed Chebyshev fractional order differentiator. We have used Chebyshev polynomials to design Chebyshev fractional order differentiator. We have generated the high pass filter corresponding to it. The designed filters are applied for decomposing the input image into four bands and low-low(L-L) sub-band is updated using correction coefficients. Reconstructed image with updated L-L sub-band provides the enhanced image. The visual results obtained are encouraging for image enhancement. The applicability of the developed algorithm is illustrated on three different test images.The effects of order of differentiation on the edges of images have also been presented and discussed.

A Matrix-Vector Operation-Based Numerical Solution Method for Linear m-th Order Ordinary Differential Equations:Application to Engineering Problems

Many problems in engineering sciences can be described by linear,inhomogeneous,m-th order ordinary differential equations(ODEs)with variable coefficients.For this wide class of problems,we here present a new,simple,flexible,and robust solution method,based on piecewise exact integration of local approximation polynomials as well as on averaging local integrals.The method is designed for modern mathematical software providing efficient environments for numerical matrix-vector operation-based calculus.Based on cubic approximation polynomials,the presented method can be expected to perform(i)similar to the Runge-Kutta method,when applied to stiff initial value problems,and(ii)significantly better than the finite difference method,when applied to boundary value problems.Therefore,we use the presented method for the analysis of engineering problems including the oscillation of a modulated torsional spring pendulum,steady-state heat transfer through a cooling web,and the structural analysis of a slender tower based on second-order beam theory.Related convergence studies provide insight into the satisfying characteristics of the proposed solution scheme.

Uniform Parallel-Machine Scheduling with Time Dependent Processing Times

We consider several uniform parallel-machine scheduling problems in which the processing time of a job is a linear increasing function of its starting time.The objectives are to minimize the total completion time of all jobs and the total load on all machines.We show that the problems are polynomially solvable when the increasing rates are identical for all jobs;we propose a fully polynomial-time approximation scheme for the standard linear deteriorating function,where the objective function is to minimize the total load on all machines.We also consider the problem in which the processing time of a job is a simple linear increasing function of its starting time and each job has a delivery time.The objective is to find a schedule which minimizes the time by which all jobs are delivered,and we propose a fully polynomial-time approximation scheme to solve this problem.