Page 249 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

Optimization Chapter | 9 239

the price of energy, demand, and interest rates, vary with time; therefore, many
renewable energy optimization problems are dynamic in nature.
Simple optimization problems were first introduced in calculus: finding the
maximum or minimum of a function. More realistic applications of optimization
theory were introduced in World War II in order to reduce the cost of
an army and maximize loss to the enemy. Since then, and mainly due to
advances in computing power, numerous methods/algorithms have emerged to
solve optimization problems, and have been applied across various disciplines.
These methods can generally be categorized as linear programming, nonlin-
ear programming, iterative methods, and metaheuristic techniques. In linear
programming problems, the objective function and constraints are linear—as
opposed to nonlinear programming. Metaheuristic approaches are the most
recent suite of optimization techniques, and are mainly based on artificial
intelligence and machine learning. Evolutionary algorithms, genetic algorithm,
and particle swarm optimization are popular metaheuristic approaches.
Solution techniques for optimization problems can be classified as
trajectory- and population-based. In the former method, a single solution is
used to search for the optimum. The optimum solution is also a single-optimized
solution at the end of the optimization process. Classical iterative schemes such
as Newton’s method fall into this category. In population-based techniques,
a population of solutions are used that evolve through each iteration of the
optimization. In general, a set of optimum solutions are provided at the end.
An example is genetic algorithms that provide a set of solutions that have the
best fitness (e.g. above an acceptable value). For decision-making applications,
population-based techniques are more attractive, because they give decision
makers a chance to select amongst a list of optimum solutions.
There are several excellent books describing various classical and modern
optimization techniques (e.g. [2,3]). Here, an overview of the basic concepts,
some examples, and tools to optimize ocean renewable energy projects are
introduced.

9.1.1 Mathematical Formulation of an Optimization Problem
An optimization problem can be mathematically formulated as follows:
minimize f(x 1 , x 2 , ... , x n ) (9.1)
subject to (x 1 , x 2 , ... , x n ) ∈ Ω
in which f is the objective (cost, fitness, etc.) function, x i are independent or
decision variables, and Ω represents the feasible region or search space. As
mentioned before, we could also maximize the objective function (e.g. fitness).
Any maximization problem can simply be converted into a minimization
problem by multiplying the objective function by −1. The feasible, or search
space, is controlled by the constraints of the problem.