## Selecting a method

Based on an investigation of the main properties of both the unconstrained and constrained optimization methods, the most suitable method for updating the physical parameters of a structural wind turbine model must be selected. The first step is to investigate the smoothness of the objective function. After all, most algorithms exclude all problems for which the objective function is not smooth. This arises when finding best solutions to over-determined systems (m > n, where m number of data points and n number of tunable variables) as in data fitting applications. In general, non-smooth (i.e. non-differentiable) objective functions are more difficult to minimize than smooth (i.e. at least twice continuously differentiable) objective functions.

The general rule in selecting a method for optimization of smooth objective functions is to make use of as much of derivative information as possible [79]. In our situation it is impossible to analytically determine the gradient vector of Fobj, since the function is the result of a simulation. Consequently, either the values of the derivatives are to be approximated by a finite difference method or a direct method must be used. Finite difference methods use the first term of the Taylor series expansion to compute the gradient vector.

Non-smooth problems are generally solved by direct methods, since indirect optimization routines assume that the objective function has continuous first and second derivatives. However, if the objective function has just a "few" discontinuities in its first derivative, and these discontinuities do not occur in the neighbourhood of the solution, methods designed for smooth problems are likely to be more efficient [79].

It is decided to use a time-domain model parameter updating technique. Fig. 5.3 shows a schematic of the implemented model parameter updating procedure. Measured input (force) and output (accelerations) data of a modal test are used as reference. The optimization performs successive simulations on a DAWIDUM model and changes the tunable parameters in an attempt to minimize the error between the measured (ymeas) and the simulated (ysim) response (i.e. output error). That is, the parameters are determined in such a way that the objective function

t=0 v eoe is minimal. || • || denotes the Euclidian norm and 0 the physical parameter vector (0 e IRn with n the number of tunable parameters). Note that for the situation that the system is present in the model set and assuming noise-free data, the output error will be zero for the true parameter values (i.e. Fobj(00) = 0).

The procedure uses either the Nelder-Mead simplex method or a combined Gauss-Newton and modified Newton algorithm. Both algorithms require function values only. Both objective functions are defined in a

MATLAB ® M-file (ObjFun.m and lsfun1.m respectively). The optimization routine (either fmins [91] or e04fdf [197]) is invoked in NMstart.m/LSstart.m. Examples of these files can be found in Molenaar [191].

In the sequel it is assumed that the mass, inertia and length of the rigid bodies within the superelements are estimated fairly accurately at the design stage, but that calculation of the stiffness and damping parameters may be difficult. Consequently, the tunable parameters are the torsional spring and damper constants. When the uncertainties in mass, inertia or length can not be neglected, however, SD/FAST® offers the possibility of leaving systems parameters unspecified in the System Description file (using a "?" instead of a number). In addition, a default numerical value can be specified immediately before the question mark. For example, the en-

Figure 5.3: Schematic of the time domain model parameter updating procedure, with Vmeas the measured output, ysim the simulated output and eoe the output error.

try: mass = 500? indicates that the initial mass is equal to 500 kg. but that its value can be changed at run time. Thus mass properties and system geometry can also be treated as variable.

## Solar Stirling Engine Basics Explained

The solar Stirling engine is progressively becoming a viable alternative to solar panels for its higher efficiency. Stirling engines might be the best way to harvest the power provided by the sun. This is an easy-to-understand explanation of how Stirling engines work, the different types, and why they are more efficient than steam engines.

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