Blade Design, Manufacture, and Testing - Small Wind Turbines: Analysis, Design, and Application - page 121

where the two ''objective functions'' to be optimised—the power coefficient, C P ,
and the inverse of the starting time, T s —are calculated at user-input wind speeds.
The factor a is also input by the user. Note that the form of ( 7.2 ) maximises the
fitness when searching for minimal starting times. The maximum C p and minimum
T s are determined from the current population.
The fitness of x i is also found from ( 7.2 ), and the blade (x i or c i ) with the higher
fitness is retained for the next generation. To encourage continuing genetic diver-
sity, no member of the population is allowed to survive more than 20 generations.
Obviously a controls the relative importance of efficiency and starting time in
determining the fitness. It has been found from many calculations that a = 1
always gives the most efficient blade design in terms of closely approximating
( 5.12 ) and ( 5.13 ), and this blade is often the slowest to start. Both these features
will be demonstrated in the example design considered later in this chapter. On the
other hand there is no minimal starting time equivalent to the Betz-Joukowsky
limit. A moment's thought will indicate the reason for this: for any R, the blade
inertia depends on the distribution of (cr) 2 from ( 6.25 ) whereas Q depends on cr
from ( 6.17 ). In the absence of Re effects, the smaller the chord the faster the blade
will start. Thus the fastest blade has infinitely small chord and is not useful in
practice. All judgements of starting time must be relative.
Provided the number of generations and the size of the population are appro-
priate, calculations for varying a should provide a good estimate of the ''Pareto
front'' defined as the subset of blades for which at least one objective function is
larger than that for every other blade. The members of the Pareto front, alterna-
tively called the ''optimal fitness front'', are the ''non-dominated'' blades. An
example Pareto front is given later. There is no single optimum blade for a two-
dimensional optimisation, and the best that a blade designer can do is chose one
blade on the Pareto front.
In the present application, it was found that assigning a single random chord
and separate single random twist to each member of the initial population, gave the
best results. In addition, the parameters listed in Table 7.1 are required as input by
the user.
7.3 Matlab Programs for Optimisation
The objective functions in the programs to be described are modifications of the
blade element methods for power extraction ( Chap. 5 ) and starting time ( Chap. 6 ).
These programs are modified as described below but are not listed. All the pro-
grams and scripts are available from the online materials http://extras.springer.
comand further description is given in the information file accompanying them. In
particular there is a Matlab script file blade_opt_setup.m used to define the many
important blade parameters which are held in the data structure data . The function
that runs the optimisation is listed below: