0
Research Papers

# Application of Constrained Stochastic Simulation to Determine the Extreme Loads of Wind Turbines

[+] Author and Article Information
Wim Bierbooms

Wind Energy Research Group, Delft University of Technology, 2629 HS Delft, The Netherlandsw.a.a.m.bierbooms@tudelft.nl

J. Sol. Energy Eng 131(3), 031010 (Jul 09, 2009) (9 pages) doi:10.1115/1.3142726 History: Received July 18, 2008; Revised January 29, 2009; Published July 09, 2009

## Abstract

Via so-called constrained stochastic simulation, gusts can be generated, which satisfy some specified constraint. In this paper, it is used in order to generate specific wind gusts, which will lead to local maxima in the response of wind turbines. The advantage of constrained simulation is that any gust amplitude (no matter how large) can be chosen. By performing simulations for different gust amplitudes and mean wind speeds, the distribution of the response is obtained. This probabilistic method is demonstrated on the basis of a generic 1 MW stall regulated wind turbine. By considering a linearized dynamic model of the reference turbine, the proposed probabilistic method could be validated. The determined 50 year response value indeed corresponds to the theoretical value (based on the work of Rice on random noise). Next, both constrained and (conventional) unconstrained simulations have been performed for the nonlinear wind turbine model. For all wind speed bins, constrained simulation results in 50 year estimates closer to the real value. Furthermore, via constrained simulation a lower uncertainty range of the estimate is obtained. The involved computational effort for both methods is about the same.

<>

## Figures

Figure 2

Example of constrained stochastic simulation; 5σ gust at t=0 s. Top: turbulence (input) in m/s (mean value of 12 m/s). Bottom: blade root flapping moment (response) in MN m.

Figure 3

The distributions of the maxima in the response (linear model) obtained via constrained stochastic simulation for several gust amplitudes (for a mean wind speed of 12 m/s and turbulence intensity of 12%). In the bottom graph, the uncertainty bounds of the convolution are given by dashed lines.

Figure 4

The (fitted) distribution of the maxima in the response (linear model) obtained via constrained stochastic simulation for one gust amplitude (5σ); a mean wind speed 12 m/s and a turbulence intensity of 12%. In the bottom graph, the uncertainty range of the fit is indicated by dashed lines.

Figure 13

The distributions of the maxima in the response obtained via constrained stochastic simulation of maximum amplitude gusts (for several gust amplitudes and a mean wind speed of 12 m/s and a turbulence intensity of 12%); for comparison the result obtained by unconstrained simulations (from Fig. 9) are shown as well.

Figure 14

Example of a constrained gust (with respect to the mean value) for a pitch regulated wind turbine

Figure 1

Relation between turbulence intensity and maximum in response with respect to steady value (for one particular turbulence time series)

Figure 5

The distributions of the maxima in the response (linear model) obtained via constrained stochastic simulation for several mean wind speeds (and a turbulence intensity of 12%).

Figure 11

The distributions of the maxima in the response obtained via constrained stochastic simulation for several mean wind speeds (and a turbulence intensity of 12%); for comparison the result obtained by unconstrained simulations (from Fig. 9) is shown as well.

Figure 12

The distributions of the maxima in the response obtained via unconstrained stochastic simulation (for a mean wind speed of 12 m/s and a turbulence intensity of 12%); obtained via ten different sets of 100 simulations each.

Figure 6

Top: The value of the conditional distribution (exceedance probability) for y=0.89 MN m (linear model) for mean wind speeds 6 m/s, 9 m/s, 12 m/s, 15 m/s, 18 m/s, and 21 m/s. Middle: the fraction of time nx for each mean wind speed. Bottom: contribution of each mean wind speed to the tail estimation of the response (i.e., the normalized product of the values of the top and middle graphs).

Figure 7

Top: The value of the conditional distribution (exceedance probalitity) for y=0.89 MN m (linear model), a mean wind speed of 21 m/s; gust amplitudes of 3σ–7σ. Middle: the fraction of time nx for each gust amplitude. Bottom: contribution of each gust amplitude to the tail estimation of the response (i.e., the normalized product of the values of the top and middle graphs).

Figure 8

Top: the value of the conditional distribution (exceedance probalitity) for y=0.91 MN m (linear model), a mean wind speed 21 m/s; gust amplitudes of 2σ–5.9σ. Middle: the fraction of time nx for each gust amplitude. Bottom: contribution of each gust amplitude to the tail estimation of the response (i.e., the normalized product of the values of the top and middle graphs).

Figure 9

The distributions of the maxima in the response obtained via normal (unconstrained) stochastic simulation for several mean wind speeds (and a turbulence intensity of 12%).

Figure 10

The distributions of the maxima in the response obtained via constrained stochastic simulation for several gust amplitudes (for a mean wind speed of 12 m/s and a turbulence intensity of 12%); in the upper graph the uncertainty bounds in the determination of the convolution are shown by dashed lines. For comparison, the result obtained by unconstrained simulations (from Fig. 9) is shown as well.

## Errata

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Topic Collections