Scheduling of power generation via large-scale nonlinear optimization

We investigate methods for solving high-dimensional nonlinear optimization problems which typically occur in the daily scheduling of electricity production: problems with a nonlinear, separable cost function, lower and upper bounds on the variables, and an equality constraint to satisfy the demand. If the cost function is quadratic, we use a modified Lagrange multiplier technique. For a nonquadratic cost function (a penalty function combining the original cost function and certain fuel constraints, so that it is generally not separable), we compare the performance of the gradient-projection method and the reduced-gradient method, both with conjugate search directions within facets of the feasible set. Numerical examples at the end of the paper demonstrate the effectiveness of the gradient-projection method to solve problems with hundreds of variables by exploitation of the special structure.     

Pure parametric excitation of a micro cantilever beam actuated by piezoelectric layers

This paper deals with the stability analysis of transverse motions of a cantilever microbeam sandwiched by two piezoelectric layers located on the lower and upper surfaces of the microbeam. Application of same DC and AC voltages to the upper and lower piezoelectric layers creates an axial force with steady and time-varying components. The eigenfunction expansion of the transverse motion equation leads to the creation of a Mathieu type parametric equation which is mostly seen in the stability analysis of the structures in the literature; using Floquet theory for single degree of freedom systems the stable and unstable regions of the problem are investigated. The effect of viscous damping and DC voltage on the stability region of the problem is also studied. The results show the stabilizing effect of the viscous damping and positive DC voltage on the behavior of the microbeam. The achieved results are finally compared with those reported in the literature.     

A column generation algorithm for nonlinear programming

An algorithm using column generation and penalty function techniques is presented. A linear program with a uniformly bounded number of columns, similar to the restricted master in generalized programming, is used to reduce the number of constraints included in forming a penalty function. The penalty function is used as a Lagrangian in an unconstrained subproblem.This work was supported in part by National Science Foundation Grant GS-3032.     

Thermoelastic damping effect analysis in micro flexural resonator of atomic force microscopy

In the design of high-Q micro/nano-resonators, dissipation mechanisms may have damaging effects on the quality factor (Q). One of the major dissipation mechanisms is thermoelastic damping (TED) that needs an accurate consideration for prediction. Aim of this paper is to evaluate the effect of TED on the vibrations of thin beam resonators. In particular, we will focus on cantilever beam resonator used in atomic force microscopy (AFM). AFM resonator is actually a cantilever with a spring attached to its free end. The end spring is considered to capture the effect of surface stiffness between tip and sample surface. The coupled governing equations of motion of thin beam with consideration of TED effects are derived. In general, there are four elastic equations that are coupled with thermal conduction equation. Based on accurate assumptions, these equations are simplified and the various boundary conditions have been used in order to validate the computational procedure. In order to accurately determine TED effects, the coupled thermal conduction equation is solved for the temperature field by considering three-dimensional (3-D) heat conduction along the length, width and thickness of the beam. Weighted residual Galerkin technique is used to obtain frequency shift and the quality factor of the thin beam resonator. The obtained results for quality factor, frequency shift and sensitivity change due to thermo-elastic coupling are presented graphically. Furthermore, the effects of beam aspect ratio, stress-free temperature on the quality factor and the influence of the surface stiffness on the frequencies and modal sensitivity of the AFM cantilever with and without considering thermo-elastic damping effects are discussed.     

On eigenfunctions of nonlinear heat generation

Summary We consider the equation u+ expu=0, >0,u(boundary)0 in the formv= exp (K,v), whereK ^–1=–. We give bounds on for the latter equation to be solvable by the contraction mapping principle, and estimate theL ² norm of the solution so obtained. We also give a bound on for the topological index of the solution to be non-zero and apply Krasnoselskii's results to the least squares method of approximating the solution.

---

Sommario Consideriamo l'equazione u+ expu=0, >0,u(frontiera)=0 nella formav= exp (Kv), doveK ^–1=–. In questo lavoro diamo limitazioni per per cui la seconda equazione e risolubile col metodo delle contrazioni, e diamo una stima della norma inL ² della soluzione cosi ottenuta. Diamo anche una limitazione per per cui l'indice topologico della soluzione diventa non zero, e applichiamo i risultati di Krasnoselskii al metodo dei minimi quadrati per approssimare la soluzione.

     

On generation of nonlinear operator semigroups and nonlinear evolution operators

Generation of nonlinear operator semigroups and nonlinear evolution operators are proved by a different method, based on the theory of difference equations.     

Characterization of the shape stability for nonlinear elliptic problems

We characterize all geometric perturbations of an open set, for which the solution of a nonlinear elliptic PDE of p-Laplacian type with Dirichlet boundary condition is stable in the L^∞-norm. The necessary and sufficient conditions are jointly expressed by a geometric property associated to the γp-convergence.If the dimension N of the space satisfies N−1<p?N and if the number of the connected components of the complements of the moving domains are uniformly bounded, a simple characterization of the uniform convergence can be derived in a purely geometric frame, in terms of the Hausdorff complementary convergence. Several examples are presented.     

Evolutionary hybrid approaches for generation scheduling in power systems

Generation scheduling (GS) in power systems is a tough optimisation problem which continues to present a challenge for efficient solution techniques. The solution is to define on/off decisions and generation levels for each electricity generator of a power system for each scheduling interval. The solution procedure requires simultaneous consideration of binary decision and continuous variables. In recent years researchers have focused much attention on developing new hybrid approaches using evolutionary and traditional exact methods for this type of mixed-integer problems. This paper investigates how the optimum or near optimum solution for the GS problem may be quickly identified. A design is proposed which uses a variety of metaheuristic, heuristics and mathematical programming techniques within a hybrid framework. The results obtained for two case studies are promising and show that the hybrid approach offers an effective alternative for solving the GS problems within a realistic timeframe.     

Convex quadratic relaxations for mixed-integer nonlinear programs in power systems

This paper presents a set of new convex quadratic relaxations for nonlinear and mixed-integer nonlinear programs arising in power systems. The considered models are motivated by hybrid discrete/continuous applications where existing approximations do not provide optimality guarantees. The new relaxations offer computational efficiency along with minimal optimality gaps, providing an interesting alternative to state-of-the-art semidefinite programming relaxations. Three case studies in optimal power flow, optimal transmission switching and capacitor placement demonstrate the benefits of the new relaxations.     

New exact solutions for some power law nonlinear diffusion equation

An extended auxiliary equation method for exact traveling wave solutions of constant coefficient nonlinear partial differential equations of evolution is proposed. This, together with a convenient characterization, affords new exact traveling wave solutions of some classes of nonlinear power law diffusion equations to be obtained.     

A participatory multi-criteria approach for power generation and transmission planning

It is well-known that reinsurance can be an effective risk management solution for financial institutions such as the insurance companies. The optimal reinsurance solution depends on a number of factors including the criterion of optimization and the premium principle adopted by the reinsurer. In this paper, we analyze the Value-at-Risk based optimal risk management solution using reinsurance under a class of premium principles that is monotonic and piecewise. The monotonic piecewise premium principles include not only those which preserve stop-loss ordering, but also the piecewise premium principles which are monotonic and constructed by concatenating a series of premium principles. By adopting the monotonic piecewise premium principle, our proposed optimal reinsurance model has a number of advantages. In particular, our model has the flexibility of allowing the reinsurer to use different risk loading factors for a given premium principle or use entirely different premium principles depending on the layers of risk. Our proposed model can also analyze the optimal reinsurance strategy in the context of multiple reinsurers that may use different premium principles (as attributed to the difference in risk attitude and/or imperfect information). Furthermore, by artfully imposing certain constraints on the ceded loss functions, the resulting model can be used to capture the reinsurer’s willingness and/or capacity to accept risk or to control counterparty risk from the perspective of the insurer. Under some technical assumptions, we derive explicitly the optimal form of the reinsurance strategies in all the above cases. In particular, we show that a truncated stop-loss reinsurance treaty or a limited stop-loss reinsurance treaty can be optimal depending on the constraint imposed on the retained and/or ceded loss functions. Some numerical examples are provided to further compare and contrast our proposed models to the existing models.     

Analytical solution in 2D domain for nonlinear response of piezoelectric slabs under weak electric fields

Piezoceramic materials exhibit different types of nonlinearities depending upon the magnitude of the mechanical and electric field strength in the continuum. Some of the nonlinearities observed under weak electric fields are: presence of superharmonics in the response spectra and jump phenomena etc. especially if the system is excited near resonance. In this paper, an analytical solution (in 2D plane stress domain) for the nonlinear response of a rectangular piezoceramic slab has been obtained by use of Rayleigh–Ritz method and perturbation technique. The eigenfunction obtained from solution of the differential equation of the linear problem has been used as the shape function in the Rayleigh–Ritz method. Forced vibration experiments have been conducted on a rectangular piezoceramic slab by applying varying electric field strengths across the thickness and the results have been compared with those of analytical solution. The analytical solutions compare well with those of experimental results. These solutions should serve as a method to validate the FE formulations as well as help in the determination of nonlinear material property coefficients for these materials.     

Reliable analysis for nonlinear Schrödinger equations with a cubic nonlinearity and a power law nonlinearity

The tanh method and the sine–cosine method are effectively used for reliable analysis for the nonlinear Schrödinger equations with cubic and power law nonlinearities. A variety of exact solutions with distinct structures are formally derived for each equation. The study reveals the power of the two proposed algorithms.     

Analysis of a quasistatic contact problem for piezoelectric materials

We consider a mathematical model which describes the frictional contact between a piezoelectric body and an electrically conductive foundation. The process is quasistatic, the material behavior is modeled with an electro-viscoelastic constitutive law and the contact is described with subdifferential boundary conditions. We derive the variational formulation of the problem which is in the form of a system involving two history-dependent hemivariational inequalities in which the unknowns are the velocity and electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on a recent result on history-dependent hemivariational inequalities obtained in Migórski et al. (submitted for publication) [16].     

A power series method for computing singular solutions to nonlinear analytic systems

Summary Given a system of analytic equations having a singular solution, we show how to develop a power series representation for the solution. This series is computable, and when the multiplicity of the solution is small, highly accurate estimates of the solution can be generated for a moderate computational cost. In this paper, a theorem is proven (using results from several complex variables) which establishes the basis for the approach. Then a specific numerical method is developed, and data from numerical experiments are given.