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Improved modelling of fatigue loads in wind farms under non-neutral ABL stability conditions

The purpose of this study is improve the predictive capability of the Dynamic Wake Meandering (DWM) model generalized to non-neutral atmospheric boundary layer (ABL) conditions in general and under stable ABL stratification in particular. The emphasis is on rotating wind turbine components, and the model improvement in focus is intimately linked to a newly developed refinement of the classic Monin-Obukhov theory, which, for stable ABL stratification, primary results in less pronounced mean wind shear outside the surface layer, where most modern wind turbines are operating. The model improvements are validated against a huge set of full-scale data, which allows for a one-to-one comparison of wind turbine load simulations and measurements conditioned on ABL stability conditions.


Introduction
For wind farm (WF) production estimation, stationary WF flow field modeling as provided by e.g. full CFD RANS models or fast linearized CFD RANS models [1] may suffice. However, for load estimation of wind turbines (WT's) exposed to wake affected inflow conditions, a non-stationary WF flow field description is inevitable. Insisting on models reflecting the basic physics of the problem, and considering direct numerical simulation (DNS) precluded due to its excessive computational demand, we are thus left with either high-fidelity CFD LES actuator disc or line type of approaches or, alternatively, the medium-fidelity Dynamic Wake Meandering (DWM) [2], [3] type of model approach. CFD LES models must -like the DWM model -be linked to an aeroelastic model of each and every WT in a WF to provide a complete picture of the load conditions inside a WF for all design load cases. This is extremely CPU demanding and considered unrealistic for design purposes even with the capacity of nowadays very powerful state-of-the-art super computer clusters. This challenge is further enhanced when including ABL stability as an additional design load case dimension and/or considering optimization of WF layout.
Being a medium-fidelity approach the DWM model offers significant savings in CPU-demands compared to the above mentioned non-stationary CFD alternatives, and it thereby facilitates detailed WF production/load design simulations to be conducted even when including the extra design load case dimension constituted by ABL stability. The "classic" (i.e. associated with neutral ABL conditions) DWM model is well established and about to be included in the revised IEC standard [19] as a recommended practice. The model has been extensively validated both directly against full-scale wake affected flow measurements [5], [6], [7] and indirectly [8], [9] by using DWM together with the aeroelastic code HAWC2 [10] to compare predictions of WT production and loading with full-scale WF data.
In [9] a huge scatter in the measured load data was observed, and to investigate whether at least part of this scatter can be attributed to ABL stability effects, the DWM model was generalized to non-neutral ABL stability conditions [13], [14]. The capability of the generalized DWM model was subsequently investigated using the same set of full-scale WF data in [23], [24]. These studies have demonstrated the importance of ABL stability regarding fatigue loading of WTs operating in WF conditions. For fatigue load prediction of non-rotating WT components, the validation of the generalized DWM model against the full-scale load data was fully satisfactory. For the investigated rotating WT component (i.e. the blade), however, some deviations between measurements and predictions were seen.
The purpose of the present study is therefore to further improve the generalized DWM model with special emphasis on its predictive capability regarding rotating WT components, as well as to get further fundamental -qualitative and quantitative -insights in ABL stability effects on WT loading in WF's.
The paper is structured as follows. First, the generalization of the DWM model to non-neutral ABL conditions is briefly described in Section 2. Then follows in Section 3 a description of the impact of non-neutral ABL conditions on the vertical mean wind speed profile, which is a significant load contributor to the fatigue loading of WT blades under stable ABL conditions. Section 4 describes the Lillgrund case study, and in Section 5 results from comparing model simulations with the Lillgrund fullscale measurements are presented and discussed. Finally, conclusions are drawn in Section 5.

DWM for non-neutral conditions
The core of the DWM model is a split of scales in the wake affected flow field, with large turbulence scales being responsible for stochastic wake meandering and small scales being responsible for wake attenuation and expansion in the meandering frame of reference as caused by turbulent mixing [2]. Thus, the DWM model essentially assumes that the transport of wakes in the ABL can be modeled by considering the wakes to act as passive tracers driven by a combination of large-scale turbulence structures and a mean downstream advection velocity, adopting the Taylor hypotheses [4]. This basic split in scales further facilitates generalization of the DWM model to non-neutral ABL stability conditions in a simplistic way, as ABL stability is known primary to affect the low wave number turbulence regime.
The large scale turbulence structures, used to describe wake dynamics in the DWM model, are traditionally provided by a fast Navier-Stokes (NS) consistent turbulence model [11] which, however, assumes neutral ABL stability and thereby neglect the effects of buoyancy on turbulence production and thus turbulence characteristics. As ABL stability mainly affects the large scale turbulence structures [4], the effect of buoyancy on wake meandering cannot be neglected. In addition, the significance of buoyancy effects on small turbulence structures relevant for wake deficit expansion and attenuation in the meandering frame of reference must be clarified. The crucial question in this respect is whether the effect of buoyancy can be "decomposed" in analogy with the description of wake dynamics.
The starting point is the following fundamental conjecture launched in [12]: In a wake context, ABL stability affects primary wake meandering driven by large (lateral and vertical) turbulent scales, whereas wake expansion in the meandering frame of reference is a second order effect only. This conjecture has been investigated for a variety of stability conditions in [13] and [14], respectively. Based on full-scale LiDAR and sonic measurements, as well as detailed numerical simulations, the wake affected flow field behind a 500kW turbine was examined with focus on both organized wake deficit flow structures (resolved in the meandering frame of reference) and wake dynamics. These extensive investigations have justified the above stated conjecture, and we will therefore generalize the applicability of the DWM model to non-neutral ABL stability regimes by adopting this conjecture. For consistency, we need a generalization of the classical spectral tensor to the non-neutral regime. Such a model has been developed and validated against full-scale data in [15], [16] and [17]. Being a generalization of the classical Mann spectral tensor, this model share some of the basic features of the Mann spectral tensor (i.e. the linear vertical mean wind speed profile and the eddy life time model), but includes in addition a linear vertical temperature profile as well as buoyancy terms from the linearized NS equations used in the Rapid Distorsion approach. The inclusion of buoyancy comes with a cost of two additional input parameters. For consistency reasons, the generalized spectral tensor is used to model both the large-and small scale ABL turbulence used with the DWM model, although the primary effect on the turbulence structure is on the big spatial scales (i.e. the low wave number regime).

Vertical mean wind speed profile for non-neutral conditions
Apart from turbulence intensity and turbulence structure, non-neutral ABL stability conditions also affect the vertical mean wind speed profile and the wind veer significantly -especially in the stable regime [4]. This is not essential for fatigue loading of non-rotating WT components as e.g. the tower, but for rotating WT components like blades, especially the shear profile is of crucial importance. In the present study we will neglect the effect of ABL stability on the wind veer.
The vertical mean wind speed profile is stability dependent through the impact of buoyancy on mixing in the ABL layer -in stable conditions with little/suppressed mixing the gradient of vertical mean wind speed profile tend to increase, whereas in unstable conditions, with increased mixing, the vertical mean wind speed profile tend to become more uniform.
Within the surface layer the Monin-Obukhov (M-O) scaling theory has proven very successful and is widely used. Adopting the physically consistent logarithmic profile for the neutral case and neglecting the influence of boundary layer height, the M-O formulation of the non-neutral vertical mean wind speed profile, U, may be expressed as [4] where u* is the friction velocity, z0 is the roughness length, LM is the M-O scaling length, and κ = 0.41 is the von Karman constant. The celebrated stability function, ψm, has unstable-and stable branches, respectively. The vertical mean wind speed profile may, under unstable conditions, be formulated as [4] , [21] ( , 0 , ) = * [ ( The analog expression for the vertical mean wind speed profile under stable conditions is given by [4] ( , 0 , ) = * [ ( Classic M-O scaling is in principle limited to the surface layer. This has been challenged in a recent Danish study [25], where a thorough investigation of three Danish offshore sites (Horns Reef; Nysted; and Laesø) outside the surface layer has been conducted. The investigation demonstrates that mean wind speeds in unstable cases are remarkably well predicted by the classic surface layer M-O similarity theory even outside the surface layer. In stable cases, however, some large over-predictions were observed, especially for low mean wind speeds under strongly stable conditions (cf. Figure 1a).
The is re-formulated. For derivation of the extended M-O theory we refer to [25]. The developed extended M-O theory has been validated against full-scale meteorological observations. Convincing agreement between model predictions and full-scale data has been demonstrated in altitude the regime ranging from 62m and up to 900m using data from three Danish offshore sites as well as from the German Fino platform, and it is concluded that the model is behaving surprisingly well even for heights elevated high above the surface layer (cf. Figure 1b).  eq. (4) may be reformulated as The extended M-O theory, developed in [25], expresses the vertical mean wind speed profile under stable conditions as with μ being the Monin-Kazanski parameter (the non-dimensional Coriolis parameter) defined by where f is the Coriolis parameter defined by Ω is the rotation rate of the Earth, and φ is the latitude. At the Lillgrund location f = 1.914⨯10 -5 Hz.  Whereas the Egmond aan Zee WF, previously investigated in [8], is characterized by a "conventional" WT inter spacing, the layout of the Lillgrund WF is characterized by very small WT inter spacing'si.e. down to 3.3 D. This makes the Lillgrund WF especially interesting and challenging as a wake-load validation case, because the close spacing magnifies wake generated load effects compared to more traditional spaced WF's.

Measurements
One of the Lillgrund WT's (C-8) is instrumented with strain gauges resolving respectively the blade root flap bending moment, two main shaft bending moments and the tower base for-aft bending moment. These measurements have been made available by Siemens Wind Power, and they constitute probably one of the most comprehensive sets of wake affected wind turbine load measurements ever recorded. The measurement period extends from 2008-06-03 to 2013-03-19 -i.e. over a period of almost 5 years. In addition to these high-frequency data, WT SCADA data (pitch setting, rotational speed and electrical power) were available for the WF WT's during the measuring period.
Unfortunately no meteorological mast data was available within the recording period, but as the WT power and pitch angle are directly correlated with the inflow wind speed, the ambient undisturbed wind speed has been determined as based on power and pitch angle recordings for corner placed WT's located in free inflow conditions. A similar philosophy was used for estimating the ambient undisturbed wind direction, which was determined from nacelle orientations of corner placed WT's. These nacelle orientations were initially calibrated against power deficit polar's constrained to directions, where wake losses were dominated by the closest neighboring WT's. To facilitate stability classification, meteorological data from the nearby Drogden light tower was offered for the investigation. This light tower is located a few kilometers WNW of the WF, and with the distance being of the same order of magnitude as the extension of the WF, the light tower meteorological data are deemed applicable for stability classification of the WF structural data. The data set holds approximately 15.600 hours of measurements covering the period 2008-2013. Like the structural data, the light tower data are organized as 10-minute recordings, and 10-minute mean values of wind speed and temperature at 22m altitude as well as water temperature 1m below mean sea level (MSL) was available.
As for the structural data, full polar load cases, associated with normal WT operation, are available for mean wind speeds ranging from 6m/s to 16m/s. Data binning in three dimensions -mean wind speed; mean wind direction; and ABL stability class -is quite demanding and requires a huge set of data. The desire of smallest possible bins has to be balanced against sufficient 'data population' of each individual bin, and in the end it was decided to use 2m/s bins for the mean wind speed and 4 deg. bins for the wind direction. The ABL stability was classified according to Table 1, which is obtained from the suggested stability classification in [26] by collapsing these stability classes into only 3 classes.
The blade root flap moments and the tower bottom for-aft moments have been post processed to fatigue equivalent moments using the Palmgren-Miner [18] approach and subsequently normalized with the respective fatigue equivalent moments associated with an inflow wind speed of 9m/s -i.e. here represented by mean equivalent moments associated with the velocity bin [8;10]m/s. Wöhler exponents of 4 and 10 were assumed for the tower and blade composite structures, respectively.

Simulations
For the DWM/HAWC2 validation study the load response of WT C-8 is simulated for mean wind speeds reflecting the median of the velocity bins defined for the measurements. Measured wind speed dependent turbulence intensities (TI's) are used, reflecting the offshore wind speed dependent "surface" roughness. However, no attempt is done to resolve TI as function of upstream fetch (i.e. direction). Thus, in the mean wind speed regime 6m/s-14m/s a TI of 5.8% is used -gradually increasing to 6.2% at 16m/s.

Mean wind shear
Because site meteorology measurements, facilitating derivation of the site mean wind speed vertical profile, doesn't exists, it was decided to use the IEC offshore wind shear profile as a surrogate for the neutral profile in the analysis. The IEC offshore profile is specified in terms of a power exponent α equal to to 0.14 [20]. To recast this profile into the more physically sound log-formulation described in Section 3, we approximated a log-profile to the IEC power law profile. Specifically, the best possible agreement of respectively the power law profile and log profile over the vertical extent of the rotor was requested, and this 'best' fit was defined in terms of minimum least square metric Π. Thus where the requested roughness, z0, are defined the particular value that minimizes Π; viz.
Adopting the roughness defined in eq. (11), the vertical mean wind speed profile for the non-neutral ABL stability conditions was evaluated using eqs. (2) and (7) for the unstable-and the stable ABL regime, respectively.

Turbulence
The coupled DWM-HAWC2 platform [3] requires simulation of three different turbulence fields -a large scale turbulence field dictating the wake meandering, a traditional turbulence field accounting for conventional WT turbulence loading, and an isotropic and inhomogeneous small scale turbulence field accounting for WT loading caused by wake self-generated turbulence. The wake self-generated turbulence field is assumed to be invariant with respect to the ABL stability condition, and it is simulated using the classic Mann turbulence simulator [11]. The two remaining turbulence fields, which are highly dependent on ABL stability conditions, are generated based on the generalized buoyancy dependent spectral tensor [15], [16], [17], which degenerates to the classic spectral tensor for neutral stability conditions. In all cases the turbulence fields are simulated as three dimensional fields resolved in suitable Cartesian grid configurations.
The parameters of the buoyancy dependent spectral tensor are, for each stability class, obtained from fitting model auto-and cross spectra to respective spectra obtained from full-scale sonic measurements from the Høvsøre site in Denmark. Data associated with the [8 ; 9]m/s mean wind speed bin were used for the parameter fitting due to the good data coverage. Of the available sonic measurements, it was decided to base the fitting on data from a sonic located at 40m altitude, because this sonic is more likely to be within the surface layer, where LM is defined, than the sonic's located at higher alternative altitudes (especially for stable stratification). Moreover, the model fitting to data at this altitude seems to slightly outperform the fits to alternative sonic data.
Except for the turbulence intensity (described by the αε 2/3 parameter of the spectral tensor when the remaining parameters have been fixed), it was further decided to adopt the (neutral) turbulence input specifications from the IEC code to [19] mimic the turbulence conditions at the Lillgrund site, where no high-frequency meteorological measurements, as mentioned, are available. The IEC spectral tensor input parameters were originally obtained by fitting the spectral tensor spectra to a target Kaimal spectra [19]. Since we have decided to base the neutral turbulence generation partly on IEC specifications of turbulence tensor input (i.e. turbulence length scale L and eddy life time parameter Γ), the spectral tensor input parameters for neutral conditions do not match the fitted parameter values from the Høvsøre fullscale data. As a consequence, it was decided to scale (L, Γ) for the non-neutral (i.e. diabatic) conditions accordingly, although this approximation might not be completely true. However, it is believed to be a fair approximation -alternatively the neutral case could defined by the directly fitted values for L and Γ at the Høvsøre site, which is just another approximation to the Lillgrund conditions.

Results
The results for the blade root equivalent moments and for the tower base for-aft equivalent moments are presented for the investigated ABL stability regimes in Figure 3 and in Figure 4, respectively. For all wind speeds it can be seen that the tower bottom fatigue bending moments are largest in unstable and smallest in stable ABL conditions. This is explained by the difference in turbulence intensity associated with the investigated ABL stability conditions, where unstable conditions causes a turbulence level of 9.8%, neutral conditions corresponds to a turbulence level of 6.0%, and stable conditions corresponds to a turbulence level of 1.99%. The measured load levels are in general captured reasonable well by the simulations. However, the difference in fatigue load level caused by ABL stability effects are typically larger under multiple wake inflow situations in the simulations than seen from the measurements.
In the inflow sector 150deg. -180deg. the investigated WT is not affected by upstream wakes, and it is thus possible to evaluate the impact from ABL stability on solitary WT loading. When investigating the blade flapwise fatigue loads in this free sector, the results are significantly different from the tower fatigue loading. Contrary to tower loading, the blade loads are sensitive to both shear-and turbulence levels, as both cause varying wind speeds over the rotor area. For rotating WT components these are counter acting effects, because stable conditions result in large shear and low turbulence intensity and vice versa for unstable conditions. It is seen, that with the mean wind speed profile modeled based the extended M-O theory, the impact from wind speed shear on the blade fatigue loading is coming close to the measured blade data. This is contrary to the case, where the mean wind speed profile is based on the classic M-O theory, where the blade fatigue loading in the stable ABL regime were significantly overestimated [23]. For most wind speeds, it is seen that the fatigue load level 'inside the WF' (i.e. corresponding to multiple wake inflow situations) is quite well predicted.

Conclusion
A medium-fidelity DWM/HAWC2 framework for modelling of wind turbine component fatigue loading, when the wind turbines are subjected to wake affected non-stationary flow fields under nonneutral atmospheric stability conditions, has been developed and demonstrated on full-scale load data from the Lillgrunden WF. Compared to previous studies [22], [23], a newly developed refinement of the classic M-O scaling theory was included in this study to account for un-realistic wind speed shears above the ABL surface layer under (strongly) stable stratification. This has eliminated previous overpredictions of blade fatigue loading of solitary WT's in (strongly) stable stratification, when using classic M-O theory to describe the vertical mean wind shear.
A comparative analysis of measured and simulated blade and tower fatigue loads has revealed that ABL stability effects can explain a significant part of the large measurement scatter observed in fullscale data. The study has further gained insight in important load mechanism, and shown that increased vertical mean wind speed shear, to a large extend, is balancing the decrease in turbulence driven fatigue loading for rotating WT components in stable ABL conditions as also observed in the measurements. Contrary to this, turbulence (including its dictating wake meandering mechanism) is the dominating load driver for tower fatigue loading in general. Similar results were found in [22].
In a future perspective, the mean wind speed veer -also predicted by the extended M-O theory -will be included in the HAWC2/DWM model platform, as well as selected meso-scale effects. Finally, it is considered to consolidate the present Drogden based stability classification with WRF-simulations.