Wind Energy

39 Multi-Criteria Shape Optimization of VAWT Blade Profile 217
39.3.2 Reaching the Global Optimum
On the hyper-surface computed by the RSM method, classical optimization
algorithms are applied to reach the global optimum. The main deterministic
method used in Optimus is the Sequential Quadratic Programming approach.
It consists in formulating the continuous optimization problem in term of
Lagrangian and in solving the Karush–Kuhn–Tucker conditions at each iteration
[4]. In case of several local optima, a stochastic method is preferable. The
simulated annealing is a very simple one which consists in a random research
where a less fitness of the parameters is not always rejected, but with a certain
probability. On the other hand, in case of highly irregular RSM, Genetic
Algorithms are very efficient to reach the global optimum but imply a higher
number of cost function evaluations. The optimization approach has to be
chosen considering RSM shape and, in a general case, the last evolutionary
method is always convenient.
39.4 Numerical Results
Before emphasizing the present method capabilities, each part of the
optimization process is validated by comparison with reference studies.
39.4.1 Validation Results
Considering a reference test case (12 ◦ of incidence, Re =10 5 , NACA0012
profile), the comparison of the lift and drag coefficients provided by Prostar
with previous experimental and numerical values [5] underlines the efficiency
of the present method. With ∆t =10 −2 s as time-step, the whole computation
lasts approximately 40 min on a XEON 2.4 GHz mono-processor/RAM
3 GB and the evaluation of the complete efficiency graphe lasts 7 h. The comparison
between NACA0012 power graphe computed by the present method
and experimental results [6] � shows a real agreement for the same “solidity”
value (0.2), considering the static approximation.
� number of blades× blade chord
rotor radius
39.4.2 Optimization Results
The optimization presented is based on the two parameter shape design
space. The evaluated profiles associated to each experiment are illustrated
on Fig. 39.1. The objective of the present optimization process is to maximize
the nominal power production and the range efficiency under an inequality
constraint on blade weight. An important fact is that the RSM related to
nominal production presents a local optimum. As a consequence the best way
to solve is the Genetic Algorithms approach, which converge in nine iterations.