Abstract
This work proposes a hybrid genetic algorithm (GA) to address the unit commitment (UC) problem. In the UC problem, the goal is to schedule a subset of a given group of electrical power generating units and also to determine their production output in order to meet energy demands at minimum cost. In addition, the solution must satisfy a set of technological and operational constraints. The algorithm developed is a hybrid biased random key genetic algorithm (HBRKGA). It uses random keys to encode the solutions and introduces bias both in the parent selection procedure and in the crossover strategy. To intensify the search close to good solutions, the GA is hybridized with local search. Tests have been performed on benchmark large-scale power systems. The computational results demonstrate that the HBRKGA is effective and efficient. In addition, it is also shown that it improves the solutions obtained by current state-of-the-art methodologies.

Abstract
The unit commitment (UC) problem is a well-known combinatorial optimization problem arising in operations planning of power systems. It involves deciding both the scheduling of power units, when each unit should be turned on or off, and the economic dispatch problem, how much power each of the on units should produce, in order to meet power demand at minimum cost while satisfying a set of operational and technological constraints. This problem is typically formulated as nonlinear mixed-integer programming problem and has been solved in the literature by a huge variety of optimization methods, ranging from exact methods (such as dynamic programming and branch-and-bound) to heuristic methods (genetic algorithms, simulated annealing, and particle swarm). Here, we discuss how the UC problem can be formulated with an optimal control model, describe previous discrete-time optimal control models, and propose a continuous-time optimal control model. The continuous-time optimal control formulation proposed has the advantage of involving only real-valued decision variables (controls) and enables extra degrees of freedom as well as more accuracy, since it allows to consider sets of demand data that are not sampled hourly.

Abstract
The environmental concerns are having a significant impact on the operation of power systems. The traditional Unit Commitment problem (UCP), which minimizes the total production costs is inadequate when environmental emissions need to be considered in the operation of power plants. This paper proposes a metaheuristic approach combined with a non-dominated sorting procedure to find solutions for the multi-objective UCP. The metaheuristic proposed, a Biased Random Key Genetic Algorithm, is a variant of the random-key genetic algorithm, since bias is introduced in the parent selection procedure, as well as in the crossover strategy. (C) 2017 The Authors. Published by Elsevier Ltd.

Abstract
We consider a Unit Commitment Problem (UCP) addressing not only the economic objective of minimizing the total production costs-as is done in the standard UCP-but also addressing environmental concerns. Our approach utilizes a multi-objective formulation and includes in the objective function a criterion to minimize the emission of pollutants. Environmental concerns are having a significant impact on the operation of power systems related to the emissions from fossil-fuelled power plants. However, the standard UCP, which minimizes just the total production costs, is inadequate to address environmental concerns. We propose to address the UCP with environmental concerns as a multi-objective problem and use a metaheuristic approach combined with a non-dominated sorting procedure to solve it. The metaheuristic developed is a variant of an evolutionary algorithm, known as Biased Random Key Genetic Algorithm. Computational experiments have been carried out on benchmark problems with up to 100 generation units for a 24 h scheduling horizon. The performance of the method, as well as the quality, diversity and the distribution characteristics of the solutions obtained are analysed. It is shown that the method proposed compares favourably against alternative approaches in most cases analysed.

Abstract
The Unit Commitment Problem (UCP) is a well-known combinatorial optimization problem in power systems. The main goal in the UCP is to schedule a subset of a given group of electrical power generating units and also to determine their production output in order to meet energy demands at minimum cost. In addition, a set of technological and operational constraints must be satisfied. A large variety of optimization methods addressing the UCP is available in the literature. This panoply of methods includes exact methods (such as dynamic programming, branch-and-bound) and heuristic methods (tabu search, simulated annealing, particle swarm, genetic algorithms). This paper proposes two non-traditional formulations. First, the UCP is formulated as a mixed-integer optimal control problem with both binary-valued control variables and real-valued control variables. Then, the problem is formulated as a switching time dynamic optimization problem involving only real-valued controls.

Abstract
In this work we address the Hop-Constrained Minimum cost Flow Spanning Tree (HMFST) problem with nonlinear costs. The HMFST problem is an extension of the Hop-Constrained Minimum Spanning Tree problem since it considers flow requirements other than unit flows. We propose a hybrid heuristic, based on ant colony optimization and on local search, to solve this class of problems given its combinatorial nature and also that the total costs are nonlinearly flow dependent with a fixed-charge component. We solve a set of benchmark problems available online and compare the results obtained with the ones reported in the literature for a Multi-Population hybrid biased random key Genetic Algorithm (MPGA). Our algorithm proved to be able to find an optimum solution in more than 75 % of the runs, for each problem instance solved, and was also able to improve on many results reported for the MPGA. Furthermore, for every single problem instance we were able to find a feasible solution, which was not the case for the MPGA. Regarding running times, our algorithm improves upon the computational time used by CPLEX and was always lower than that of the MPGA.