Page 248 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

238 Fundamentals of Ocean Renewable Energy

The objective function of an optimization problem depends on the inde-
pendent variables, which can also be called decision or control variables.
Independent variables in the ocean model parameterization example are physical
properties of the ocean, such as bottom and surface drag coefficients. The
objective function should be sensitive to independent variables. In some com-
plex problems, a sensitivity analysis is performed to eliminate parameters or
variables that do not affect the objective function. Independent variables can be
discrete or continuous. For example, finding the optimum number of blades for
a wind turbine is a discrete optimization problem; whereas finding the optimum
distance between two wind turbines is a continuous optimization problem.
Optimization problems are mostly formulated as deterministic problems, in
which independent variables and objective function(s) are treated determinis-
tically. In contrast, in stochastic optimization problems, due to uncertainties,
random variables are used to formulate the problem. The expected value of
the objective function is either maximized or minimized in such problems.
For instance, optimizing the operational cost of a renewable energy micro-
grid can be treated stochastically due to uncertainties in supply of energy (i.e.
intermittency of the resource [1]).
An optimization problem may be subjected to constraints. Referring to our
earlier example of finding the best mode of transport for a journey, assume that
minimizing the time is the objective function, but you may not exceed a certain
budget for the transportation cost. In that case, the problem becomes more
complicated. For instance, direct flights may exceed your budget. Technically,
any alternative that does not satisfy constraints is not feasible, or out of the
feasible region or search space. The best solution to the transportation problem
may be found by comparing indirect flights and trains that do not exceed your
budget. In the model parameterization example, an acceptable/feasible range for
each parameter is applied as a constraint. For instance, bottom friction cannot
be a negative number.
An optimization problem may have one, two, or multiple objective functions
that are conflicting. For instance, assume that you are trying to determine
the best type of energy system for an off-grid island community. Your first
objective function is to minimize the cost; your second objective function can be
minimizing pollution (i.e. emission of pollutants such as CO 2 ). These objective
functions are conflicting; the best solution may not be the least expensive or
the cleanest type of energy for the island. The majority of real life decision-
making problems, including renewable energy systems, have multiple objective
functions, and are subjected to various constraints.
Optimization problems are either static or dynamic. In dynamic optimization
problems, the parameters of the problem change with time. Finding the best
trajectory of an aeroplane over a fixed range (minimum fuel cost and minimum
travel time) is a dynamic optimization problem, because the environmental
conditions that affect the fuel consumption and travel time of the aircraft (e.g.
wind) change with time. In energy-related projects, several variables, such as