Page 252 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

242 Fundamentals of Ocean Renewable Energy

A simple analogy to optimization is climbing towards the summit of a
mountain (maximum elevation). Here, the objective function is elevation, and
latitude and longitude are the decision variables. Assume that a helicopter leaves
you at a random location on the mountain. Based on the best local information,
you move towards a direction for a specific distance (step size), and then you
change your direction. The distance that you move in each direction is not
necessarily the same: the step size can also vary within each iteration. An
obvious direction is going up, and in particular in the steepest ascent direction.
The steepest ascent is mathematically the direction of the gradient, which can
be written as

∂f ∂f ∂f
∇f = , , ... , (9.4)
∂x 1 ∂x 1 ∂x n
and the method of steepest ascent becomes

x k+1 = x k + α k ∇f k (9.5)
This method almost guarantees that you reach a summit that is closest to
your initial location but, as you can imagine, it does not guarantee that you
will reach the highest summit of a mountain range. Many classical optimization
techniques are based on gradient. The method of steepest ascent (or descent in
minimization) is not very effective, especially near the solution. Referring to
Fig. 9.1, it is clear that the gradient of the function (slope of the tangent) near
the local and global minimums approaches zero. This causes some issues for
methods that only use gradient for the direction. Other techniques are called
conjugate gradient methods that use second derivatives (Hessian matrix) as well
as first derivative (gradient) for direction.

9.2 INTRAARRAY OPTIMIZATION
Although understanding the natural (undisturbed) resource, and the operation of
single devices, is essential, it is only when devices are installed in arrays that
significant levels of electricity generation can be achieved. In this section, we
discuss how devices within such arrays can be optimized for wind, tidal stream,
and wave energy conversion.

9.2.1 Micro-Siting of Offshore Wind Farms
Referring to Section 4.4, marine spatial planning is the standard procedure used
to find suitable and potential sites for the development of offshore wind farms in
a region. Marine spatial planners, after a comprehensive study of technical, eco-
nomic, environmental, sociopolitical, legal, and regulatory aspects, recommend
areas that have minimum conflicts with other users of the ocean, in addition to
a feasible energy resource. Macro-siting is the selection of a location for a wind