## 1 INTRODUCTION

Wind speed statistics are usually denoted as *resource data*, and such statistics comprise information on *mean* wind speed, wind speed *standard deviation* and *maximal*/*minimal* wind speed realizations within each of the recording periods defining the measuring campaign. In the present paper, the wind speed standard deviation—or in its normalized form, the turbulence intensity—associated with such resource data is in focus.

Wind speed fluctuations in the atmospheric boundary layer (ABL) contain virtually all scales and are traditional classified in micro-scales (turbulence), convective scales (buoyancy-driven diurnal variations) and macro-scales (weather systems and planetary waves). Real (finite length) wind speed time series are therefore indeed non-stationary, and the concept of stationary wind speed time series is therefore an abstraction. However, for practical reasons, the wind speed standard deviation is nevertheless traditionally computed as based on assumptions of *ergodicity* and *stationarity* of the wind speed process in question. Likewise, turbulent wind fields are traditionally described and modelled as a *stationary* zero-mean multi-dimensional stochastic process imposed on a *time-invariant* mean wind speed field using Reynolds de-composition. From a mathematical point of view, this formulation has the advantage of being significantly more tractable compared with a non-stationary formulation.

The assumptions of ergodicity and stationarity basically mean that the ideally requested ensemble statistics are approximated by the statistics of sampled time traces of the process. As a consequence, a possible time-varying (ensemble) mean is replaced by its *time average* over the time span considered; and a time-varying (ensemble) variance is replaced by the sum of the *time-averaged* (ensemble) variance over the time span considered and the variance of the time-varying (ensemble) mean over the same time span. Thus, analysing non-stationary time series uncritically, pretending stationarity, eventually leads to erroneous results; e.g. distortion of the ‘true’ turbulence spectra,[1] which in turn may be detrimental for prediction of turbulent-driven fatigue and ultimate loading of structures exposed to fluctuating ABL wind fields.

In the general non-stationary case, all statistical characteristics are time dependent. However, an often used simplifying approach is to consider the high frequency fluctuating part of the time series as stationary ‘around’ a trend, which essentially means that the inherent non-stationarity is assumed to be attributed to a time variation in the (ensemble) mean value only, whereas the (ensemble) variability around this mean value is assumed time independent. This in turn means that the directly derived variance, apart from the true (micro-scale) turbulence variance, contains a macro-scale contribution originating from the assumed mean wind speed being off the true ensemble mean value. With this setting, a proper analysis of micro-scale turbulence characteristics—e.g. in relation to short-term wind gust magnitude prediction or assessment of wind turbine fatigue loading—requires elimination of possible macro-scale contributions in the measured time series characteristics.

For simulation of fatigue loading of wind turbines erected at a given site, it is particularly important to separate the directly measured wind speed variability in a micro-scale contribution and a macro-scale contribution, respectively.[2, 3] This is because the ‘total’ turbulence standard deviation (i.e. the directly measured standard deviation including both the micro-scale and macro-scale variabilities), when used as input to turbulence generators that assume stationary turbulence, will be distributed over the entire frequency range according to the presumed turbulence spectrum. However, in reality, only the stationary part of the turbulence (i.e. the micro-scale part) displays the prescribed frequency behaviour, whereas the contribution originating from mean level changes (i.e. the macro-scale variability) affects the low-frequency fluctuations only.[2, 3] Neglecting de-trending may consequently result in significant errors in the prediction of fatigue lifetime consumption.

An illustrative example is the computation of fatigue damage of a wind turbine blade. Fatigue damage, and thereby fatigue lifetime, is a highly non-linear process[3] dictated by material stress cycles imposed by the external fluctuating loading. For blade flapwise fatigue loading, atmospheric turbulence is by far the dominating load driver. Fatigue modelling associated with such stochastic loading is a challenging task, and failure criteria as well as fatigue degradation assessment models are in general purely empirical. The most widely used approach for fatigue damage estimation is the linear Palmgren–Miner damage accumulation approach, combined with an S–N curve characterizing the material fatigue performance. This is a crude approach, basically neglecting effects of load cycle sequencing and assuming that partial damages, each characterized by a specific load cycle amplitude, can be linearly accumulated.[3, 4] Applying this approach for fatigue damage estimation, the blade fatigue damage is roughly proportional to the turbulence intensity raised to the power *m*, where *m* denotes the Wöhler exponent[3] characterizing the relevant S–N curve. For blade composite materials, *m* may easily exceed 10. With *m* equal to 10 and a realistic relative trend contribution to the true micro-scale turbulence intensity of the order of 10%, the *trend induced* increase in turbulence intensity causes the blade flapwise fatigue loading to increase by a factor of approximately 2.5, corresponding to a reduction of the blade fatigue lifetime of the order of 60%.

Similar results were obtained in Hansen[5] using a simple heuristic fatigue load model, where only the fatigue loading caused by the stochastic part of the wind field was considered. This analysis demonstrated reductions in estimated tower fatigue life consumption of the order of 40% when using de-trended turbulence characteristics computed from measured full-scale high sampled time series. As no existing fatigue models are able to account for non-stationary stochastic load processes, de-trending is a mandatory pre-processing step when using fatigue simulation approaches based on a stationarity assumption.

In the present context, we adopt the simplification of stationary* (ensemble) *variance* and further *define* the trend as composed of the sum of all Fourier modes present in the signal with wavelengths longer than the time span of the time signal considered. This is equivalent with the trend definition proposed in Kaimal and Finnigan.[1]

In case high sampled data of the time series are available, the type of trend within each particular recording period can in principle be identified using a Fourier de-composition technique including Fourier modes with wavelengths longer than the observation period[6] and therefore consistent with the trend definition or, alternatively, by using more classical de-trending approaches[7] like first differencing approaches (e.g. autoregressive integrated moving average models), higher order differencing approaches, curve-fitting approaches in which the trend is described as a deterministic function in time (e.g. obtained from a least-squares fit to a parameterized curve or from physical considerations) and digital filtering in which the trend is described as a suitable linearly filtered version of the original time series. Based on the identified trend, the raw standard deviation estimate, as computed based on the time-averaged ensemble mean, is easily corrected to give the true micro-scale ensemble turbulence standard deviation. In practice, a *linear* variation (trend) of the ensemble mean wind speed over each considered recording period is often assumed and estimated using a least-squares approach.

In case high sampled time series data are not available, traditional de-trending approaches as described earlier are not applicable. De-trending of *resource data* therefore calls for alternatives, which is the topic of the present paper. This type of de-trending has previously been treated in Hansen[5] using a heuristic approach involving an empirically based calibration to certain terrain types. Contrary to the approach in Hansen,[5] the present contribution deals with models, which quantify the effect of de-trending on estimated wind speed standard deviation as based on the available statistical data only.

Two models are presented. The first model quantifies the effect of non-stationary characteristics of the ensemble mean wind speed on the estimated wind speed standard deviation as based on observed mean wind speeds only. The second model uses the full set of information and includes, in addition, estimated (directly computed) raw wind speed standard deviations to estimate the effect of ensemble mean non-stationary characteristics on the estimated wind speed standard deviations. The developed techniques allow in principle for identification of both linear and non-linear trends, as well as for quantification of the associated consistent estimates of the de-trended micro-scale turbulence standard deviations.

The capabilities of the two models are analysed by comparing their predictions with predictions obtained from a traditional linear de-trending of high sampled time series data. A huge amount of wind speed time series data, extracted from ‘Database on Wind Characteristics’,[8] is used for this comparative analysis, and the selected data represent a broad variety of different terrain types.