A modified differential evolution based solution technique for economic dispatch problems

Economic dispatch (ED) plays one of the major roles in power generation systems. The objective of economic dispatch problem is to find the optimal combination of power dispatches from different power generating units in a given time period to minimize the total generation cost while satisfying the specified constraints. Due to valve-point loading effects the objective function becomes nondifferentiable and has many local minima in the solution space. Traditional methods may fail to reach the global solution of ED problems. Most of the existing stochastic methods try to make the solution feasible or penalize an infeasible solution with penalty function method. However, to find the appropriate penalty parameter is not an easy task. Differential evolution is a population-based heuristic approach that has been shown to be very efficient to solve global optimization problems with simple bounds. In this paper, we propose a modified differential evolution based solution technique along with a tournament selection that makes pair-wise comparison among feasible and infeasible solutions based on the degree of constraint violation for economic dispatch problems. We reformulate the nonsmooth objective function to a smooth one and add nonlinear inequality constraints to original ED problems. We consider five ED problems and compare the obtained results with existing standard deterministic NLP solvers as well as with other stochastic techniques available in literature.Fundação para a Ciência e a Tecnologia (FCT

On Challenging Techniques for Constrained Global Optimization

This chapter aims to address the challenging and demanding issue of solving a continuous nonlinear constrained global optimization problem. We propose four stochastic methods that rely on a population of points to diversify the search for a global solution: genetic algorithm, differential evolution, artificial fish swarm algorithm and electromagnetism-like mechanism. The performance of different variants of these algorithms is analyzed using a benchmark set of problems. Three different strategies to handle the equality and inequality constraints of the problem are addressed. An augmented Lagrangian-based technique, the tournament selection based on feasibility and dominance rules, and a strategy based on ranking objective and constraint violation are presented and tested. Numerical experiments are reported showing the effectiveness of our suggestions. Two well-known engineering design problems are successfully solved by the proposed methods. © Springer-Verlag Berlin Heidelberg 2013.Fundação para a Ciência e a Tecnologia (Foundation for Science and Technology), Portugal for the financial support under fellowship grant: C2007-UMINHO-ALGORITMI-04. The other authors acknowledge FEDER COMPETE, Programa Operacional Fatores de Competitividade (Operational Programme Thematic Factors of Competitiveness) and FCT for the financial support under project grant: FCOMP-01-0124-FEDER-022674info:eu-repo/semantics/publishedVersio

Numerical study of augmented lagrangian algorithms for constrained global optimization

To cite this article: Ana Maria A.C. Rocha & Edite M.G.P. Fernandes (2011): Numerical study of augmented Lagrangian algorithms for constrained global optimization, Optimization, 60:10-11, 1359-1378This article presents a numerical study of two augmented Lagrangian algorithms to solve continuous constrained global optimization problems. The algorithms approximately solve a sequence of bound constrained subproblems whose objective function penalizes equality and inequality constraints violation and depends on the Lagrange multiplier vectors and a penalty parameter. Each subproblem is solved by a population-based method that uses an electromagnetism-like (EM) mechanism to move points towards optimality. Three local search procedures are tested to enhance the EM algorithm. Benchmark problems are solved in a performance evaluation of the proposed augmented Lagrangian methodologies. A comparison with other techniques presented in the literature is also reported

Convex optimization of launch vehicle ascent trajectories

This thesis investigates the use of convex optimization techniques for the ascent trajectory design and guidance of a launch vehicle. An optimized mission design and the implementation of a minimum-propellant guidance scheme are key to increasing the rocket carrying capacity and cutting the costs of access to space. However, the complexity of the launch vehicle optimal control problem (OCP), due to the high sensitivity to the optimization parameters and the numerous nonlinear constraints, make the application of traditional optimization methods somewhat unappealing, as either significant computational costs or accurate initialization points are required. Instead, recent convex optimization algorithms theoretically guarantee convergence in polynomial time regardless of the initial point. The main challenge consists in converting the nonconvex ascent problem into an equivalent convex OCP. To this end, lossless and successive convexification methods are employed on the launch vehicle problem to set up a sequential convex optimization algorithm that converges to the solution of the original problem in a short time. Motivated by the computational efficiency and reliability of the devised optimization strategy, the thesis also investigates the suitability of the convex optimization approach for the computational guidance of a launch vehicle upper stage in a model predictive control (MPC) framework. Being MPC based on recursively solving onboard an OCP to determine the optimal control actions, the resulting guidance scheme is not only performance-oriented but intrinsically robust to model uncertainties and random disturbances thanks to the closed-loop architecture. The characteristics of real-world launch vehicles are taken into account by considering rocket configurations inspired to SpaceX's Falcon 9 and ESA's VEGA as case studies. Extensive numerical results prove the convergence properties and the efficiency of the approach, posing convex optimization as a promising tool for launch vehicle ascent trajectory design and guidance algorithms

Notes on a PDE System for Biological Network Formation

We present new analytical and numerical results for the elliptic-parabolic system of partial differential equations proposed by Hu and Cai, which models the formation of biological transport networks. The model describes the pressure field using a Darcy's type equation and the dynamics of the conductance network under pressure force effects. Randomness in the material structure is represented by a linear diffusion term and conductance relaxation by an algebraic decay term. The analytical part extends the results of Haskovec, Markowich and Perthame regarding the existence of weak and mild solutions to the whole range of meaningful relaxation exponents. Moreover, we prove finite time extinction or break-down of solutions in the spatially onedimensional setting for certain ranges of the relaxation exponent. We also construct stationary solutions for the case of vanishing diffusion and critical value of the relaxation exponent, using a variational formulation and a penalty method. The analytical part is complemented by extensive numerical simulations. We propose a discretization based on mixed finite elements and study the qualitative properties of network structures for various parameters values. Furthermore, we indicate numerically that some analytical results proved for the spatially one-dimensional setting are likely to be valid also in several space dimensions.Comment: 33 pages, 12 figure