Due to large variability of the offshore environment, the load analysis of an offshore wind turbine is a complex task. It is normally performed in the time domain, by running stochastic simulations. This is usually very time consuming due to the necessity of having long time series in order to obtain results with little bias, or to sample events with a low probability of happening. An alternative is to perform the analysis in the frequency domain. The drawback is that this method is only valid for linear systems, which makes it rather inaccurate for fatigue damage results. Also, there are well-known inaccuracies when estimating fatigue damage from a response spectrum. This paper investigates a novel approach based on probability evolution methods, which can obtain accurate results for linear, as well as non-linear systems. The method is not very well known, and a drawback is that it can be numerically challenging, especially for high-dimensional problems. The benefits of the method are that it evaluates all possible states of the wind turbine without generating a long signal in the time domain. We show how this can be used to efficiently evaluate fatigue damage.
This paper presents a probability density evolution method in the time domain for a simplified wind turbine model with one mode. The cell mapping approach discretizes the two-dimensional phase space (Hsu, 1987). The method is evaluated using both deterministic and random loads and compared to time domain simulations. Different sizes in the discretization of the mesh and in the area of state space covered are investigated, to determine when the method converges towards the final solution. The algorithm has been programmed in Python with the Numba Just-In-Time compiler to help speed up the calculations. How to implement a simple stochastic load model by the cell-mapping method is discussed. The result is a joint response probability density that contains information about the extrema of the structure motion, and how often these occur. This allows for calculating Markov transition probabilities between minima and maxima from the response density, which then makes it possible to estimate fatigue damage. Determining the response probability density is done by many short time-domain integrations, instead of one or more long simulations. The main novelty introduced and demonstrated here is that as soon as one has calculated the cell mapping, one can easily obtain the peak transition matrix and use this to obtain fatigue damage estimates.