Estimation of Extreme Responses and Failure Probability of Wind Turbines under Normal Operation by Controlled Monte Carlo Simulation

Extreme value predictions for application in wind turbine design are often based on asymptotic results. Assuming that the extreme values of a wind turbine responses, i.e. maximum values of the mud-line moment or blades’ root stress, follow a certain but unknown probability density (mass) distribution, we are interested in estimating this distribution accurately far in its tail(s). This is typically done by assuming the epochal extremes in a 10 minute interval are distributed according to some asymptotic extreme value distribution with unknown parameters to be estimated based on simulated low order statistical moments. The results obtained by extrapolation of the extreme values to the stipulated design period of the wind turbine depend strongly on the relevance of these adopted extreme value distributions.

The problem is that this relevance cannot be decided from the data obtained by the indicated so-called crude Monte Carlo method. With failure probabilities of the magnitude 10−7 during a 10 min. sampling interval the tails of the distributions are never encountered during normal operations. To circumvent this problem the application of variance reduction Monte Carlo methods i.e. importance sampling (IS) might be considered. This suffers from strict requirement on the so called sampling density for a high dimensional parameter vector.

Newly developed advanced (controlled) Monte Carlo methods propose alternative solutions to this problem. Splitting methods such as Double & Clump (D&C), Russian Roulette & Splitting (RR&S) and finally distance controlled Monte Carlo (DCMC) are one class of these methods. The idea behind these methods is to artificially enforce “rare events” to happen more frequently. This can be done by distributing the statistical wight of the samples such that it is an estimate of their true probability density. Introducing sample weights allows increasing the number of samples which carry low statistical weights, i.e. low probability of occurrence, by further lowering their weights. This enables more accurate analysis of their behavior in the vicinity of the failure surface i.e. estimation of the probability of failure.

Another approach to this problem is to condition the Monte Carlo simulations using the so-called Markov Chain Monte Carlo (MCMC) technique. The standard method in this direction is the subset simulation (SS). Here the idea is to start by a standard Monte Carlo simulation with very low number of samples, compared to the true number required for estimation of the required probability. The next generations of samples are then simulated conditioned on those samples which have the least probability of occurrence in previous simulation.

Yet an alternative approach for estimation of the first excursion probability of any system is based on calculating the evolution of the Probability Density Function (PDF) of the process and integrating it on the specified domain. Clearly this provides the most accurate results among the three classes of the methods. The solution of the Fokker-Planck-Kolmogorov (FPK) equation for systems governed by a stochastic differential equation driven by Gaussian white noise will give the sought time variation of the probability density function. However the analytical solution of the FPK is available for only a few dynamic systems and the numerical solution is difficult for dynamic problem of more than 2-3 degrees of freedom. This confines the applicability of the FPK to a very narrow range of problems. On the other hand the recently introduced Generalized Density Evolution Method (GDEM), has opened a new way toward realization of the evolution of the PDF of a stochastic process; hence an alternative to the FPK. The considerable advantage of the introduced method over FPK is that its solution does not require high computational cost which extends its range of applicability to high order structural dynamic problems. The problem with method is that the number of basic random variables is rather limited.