Complementarity Problem

The modeling and control problem for a grid-connected photovoltaic (PV) power electronic system, which includes a dc/dc boost converter, an inverter and a filter are considered. A linear complementarity (LC) dynamic model of the PV system allows the design of a model predictive controller (MPC). Dynamic models of the subsystems are obtained and merged in order to represent the whole PV system in a compact and comprehensive LC model, which is valid for all operating modes of power converters and PV cells involved in the energy conversion process. A finite-control-set MPC problem is formulated as a mixed-integer quadratic program subject to the dynamic LC model and pulse width modulators. The minimization of an objective function aimed at tracking dc voltage and grid current references provides directly the commands for the switches of the boost converter and inverter. Numerical results show the effectiveness of the proposed strategy for maximum power point tracking and synchronization to the grid under dynamic scenarios characterized by variations of the solar irradiance.

We investigate some properties related to the generalized Newton method for the Fischer-Burmeister (FB) function over second-order cones, which allows us to reformulate the second-order cone complementarity problem (SOCCP) as a semismooth system of equations. Specifically, we characterize the B-subdifferential of the FB function at a general point and study the condition for every element of the B-subdifferential at a

By using the topological degree we introduce the concept of ’’exceptionalfamily of elements‘‘ specifically for continuous functions. This has importantconsequences pertaining to the solvability of the explicit, the implicit andthe general order complementarity problems. In this way a new direction forresearch in the complementarity theory is now opened.

Vector general quasi-optimization problems are formulated and some sufficient conditions on the existence of solutions for these problems are shown. These concern the existence of solutions, the stability and the structure of solution set of general vector problems. As special case, we obtain results on the existence of solutions of vector quasi-optimization problem. Some relationships between these problems and others, as vector quasi-equilibrium problems, quasi-variational inequalities, complementarity problems,..., are shown. From these we extend some well-known results obtained by Blum and Oettli [4], Park [21], Chan and Pang [6], Parida and Sen [20], Browder and Minty [18], Fan [8], etc.

We consider a setting of a two settlement power market where firms compete in the forward market and an uncertain real-time market. A recourse-based framework is proposed where firms make simultaneous bids in the forward market and take recourse in the real-time market contingent on the realization of uncertainty. The market participants include both generation firms as well as the independent system operator (ISO), the latter of which is assumed to maximize wheeling revenue. The resulting stochastic game-theoretic problem is seen to be a Nash game with coupled strategy sets, often referred to as a generalized Nash game. In general, the primal variational conditions of such problems are given by quasi-variational inequality. Yet, the associated complementarity problem in a primal-dual space admits a monotonicity property that allows us to derive an appropriate existence statement. Computation of equilibria is complicated by the challenge arising from the size of the sample-space. We present a distributed iterative regularization technique that is shown to scale well with the size of the sample-space. Finally, the paper concludes with the application of this model on an electrical network and provides insights on market design and operation.

In this paper, we suggest and analyze some three-step iterative schemes for finding the common elements of the set of the solutions of the Noor variational inequalities involving two nonlinear operators and the set of the fixed points of nonexpansive mappings. We also consider the convergence analysis of the suggested iterative schemes under some mild conditions. Since the Noor variational inequalities include variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. Results obtained in this paper may be viewed as an refinement and improvement of the previously known results.