Blade Design, Manufacture, and Testing - Small Wind Turbines: Analysis, Design, and Application - page 120

when tip losses are ignored. However, maximum efficiency is only one of the
parameters to be optimised on many blades. For example, large blades must
produce minimal noise, and as discussed in the last chapter, small blades should
usually start quickly in low winds. It is generally not possible to derive analytic
expression analogous to ( 5.12 ) and ( 5.13 ) for multi-dimensional optimisation.
Nevertheless, there are a vast number of multi-dimensional optimisation methods
available for blade design, a number of which are available as Matlab functions.
The strategy described here is not necessarily the best, but it (a) is the one the
author is most familiar with, (b) reproduces known optimum results such as
Eqs. 5.12 and 5.13 as special cases, (c) is very easy to code, (d) is easily extended
to higher dimensional optimisation, and (e) is computationally very fast.
This chapter considers optimisation methods using an ''evolutionary strategy'', [ 1 ].
These seek to mimic the process of natural selection to arrive at an ''optimum''
solution by ''evolving'' a population over a sufficient number of generations. They
start with a randomly-generated initial population, breed new members, determine
the fitness of existing and new members, and then decide which members live and
which ones die. The breeding must introduce the equivalent of mutation to the
''genes'' of each new member. In the present case, the genes are the twist and
chord of each blade element as it will be assumed that the same aerofoil profile is
used for the whole blade.
The particular method is called ''differential evolution'', DE. In its basic form,
DE generates a new population by the following process. For each member or
vector of the current generation, x i , say, a comparison vector, c i , is constructed
from a basis vector b i , or from a trial vector t i according to

t i ¼ b i þ w u i l i ð Þ ð 7 : 1 Þ

where w is a weighting factor and b i , u i and l i are randomly chosen members of
the current population that are different from each other and from x i . Following
[ 1 ], w is fixed at 0.8. A random number of genes from t i is added to the com-
plementing genes from b i to form c i . Genes are chosen for c i using a ''crossover
factor'', CR, whose function is best described by the following pseudo-code:

FOR j = 1, number of genes
IF (rand < CR)
c i (j) = t i (j)

! Take gene from trial vector

ELSE
c i (j) = b i (j) ! Take gene from base vector

END DO

Following the recommendation of Price et al. [ 1 ]CR = 0.10. The fitness of c i is
determined as

C p c ðÞ

Þ min T ðÞ
T s c ðÞ

max C p þ 1 a

fitness c ðÞ¼ a ð ð 7 : 2 Þ