Effect of the vertical wake deflection on the response of a 12MW semisubmersible FWT

The need to understand the interaction of floating wind turbines operating in the marine boundary layer with other surrounding turbines is of increasing importance as wind farms and rotor sizes get larger. For low wind speed conditions, the wake deflects upwards due to the floating wind turbine built-in rotor tilt and the platform pitch angle. Furthermore, based on field data from the North Sea, a convective atmosphere is likely to occur, especially for lower wind speeds. It is therefore key to understand the combined effect of meandering, wake deflection and atmospheric stability for low wind speed conditions on the floating wind turbines in a wind farm. In this work, to account for atmospheric stability, data from large-eddy simulation (LES) for a 7.5m/s wind speed scenario and three atmospheric stability conditions, are used as input to FAST. Farm to study the effect of the wake deflection upwards on a 12 MW semisubmersible placed 8 rotor diameters (D) downwind of a stationary wind turbine. Three built-in shaft tilt angles cases are considered for the fixed upwind turbine: 12°, 6° and no tilt angle. The main effects of the wake deflection are on the power output of the floating wind turbine in the wake, and on the tower top yaw moment. For stable conditions, the mean tower top yaw moment increases by more than three times. The findings of this work contribute to the investigation of the wake deflection as a control strategy to minimize wake losses in floating wind farms.


Introduction
Wake effects play a fundamental role in power production and load estimates on offshore wind turbines in offshore wind farms.These wake effects include deficit, meandering and horizontal and vertical deflection.The vertical and horizontal deflection are a consequence of the built-in shaft tilt (referred to as shaft tilt in the following) and vertical shear, and yaw misalignment, respectively.For floating wind turbines (FWTs), and specifically for low wind speed conditions, several authors [1][2][3] have reported the deflection of the wake not only because of the FWT shaft tilt, but also due to the platform pitch angle.This phenomenon, combined with the variation in the projected area of the rotor, affects both the produced power and the loads of the FWT in the wake.Doubrawa et al. [3] studied the effect of the vertical wake deflection on the structural response of a downwind, floating wind turbine, in neutral conditions.However, based on field data from the North Sea, non-neutral atmospheric conditions are likely to take place, especially for lower wind speeds [4,5].Therefore, in order to use vertical wake deflection as a floating wind farm control strategy, it is necessary to understand the combined effect of meandering, wake deflection and atmospheric stability for low wind speed scenarios.
In this work, to account for atmospheric stability, data from large-eddy simulation (LES) for stable, neutral, and unstable atmospheric stability conditions, for a mean wind speed of 7.5 m/s, are used.FAST.Farm [6], developed at NREL and which has the dynamic wake meandering model implemented, is used to study the effect of varying the shaft tilt of an upwind turbine on the turbine downstream.The characteristics of the 12 MW semisubmersible FWT and the numerical set-up in FAST.Farm are presented in Section 2. The main parameters of the turbulent wind fields are described in Section 3. The results for wake deficit and meandering and the effects of the shaft tilt angles on the FWT in the wake are presented in Section 4. Section 5 presents the conclusions.

Numerical set-up 2.1. INO WINDMOOR 12MW
The effects of the vertical wake deflection in non-neutral atmospheric conditions are studied for the INO WINDMOOR 12 MW semisubmersible FWT.A sketch of the set-up is presented in Figure 1.The wind turbine in free wind (indicated as T1 in Figure 1) is fixed, unless otherwise specified, so that the tilt is kept constant and equal to the prescribed shaft tilt angle β 1 .The wind turbine in the wake (T2), placed 8 rotor diameters (D) downwind of T1, has a shaft tilt angle equal to 6°and is floating.The semisubmersibles have three columns connected by three upper and three lower pontoons.The tower is installed on top of one of the columns.The mooring system consists of three hybrid catenary mooring lines [7].Three shaft tilt angles β 1 of T1 are considered, as presented in Table 1.Positive shaft tilt angles, β 1 for T1 and β 2 for T2, are sketched in Figure 1, together with a pitch positive angle θ, and the upwards wake deflection, for reference.able 2 presents the environmental conditions applied in this work to study the wake deflection and the effect on the structural responses of the wind turbine in the wake.Environmental condition 1 (EC1 in Table 2) consists of a turbulent wind field with a mean wind speed of 7.5 m/s.EC1 is run for stable, neutral and unstable conditions.Additionally, five constant wind fields (CW1-5 in Table 2), with a shear exponent characteristic of stable, neutral and unstable atmospheric stability conditions, are applied.The environmental conditions are based on the work of Li et al. [8], who defined a longterm joint distribution based on 10-year environmental hindcast data for a site in the Norwegian Sea with 200 m water depth (Site 14 in Li et al. [8]).The most likely significant wave height H s and peak period T p for the given mean wind speeds U w are retrieved in Table 2.  3, where dSLow is the resolution in y-and z-directions in the low-resolution domain, and dSHigh the corresponding one in the high-resolution domain.The resolution in the x-direction is 0.75 m in the high-resolution domain, and 0.24D in the low-resolution one.The time steps for both domains, DtLow and DtHigh, are indicated in Table 3.The work of Shaler et al. [9], who carried out a convergence study, is used as a reference to define the domains resolution.Table 3 shows the recommended values by Shaler et al. (indicated as Rec.) together with the ones used here., where C meand = 3 and D w ake = D r ot or .

Dynamic wake meandering (DWM) model
The dynamic wake meandering (DWM) model is included in the revised IEC standard [10] since 2019 as a recommended practice to account for wake effects from neighbouring turbines in a wind farm.It was proposed by Larsen et al. [11], and the main idea is that turbulent eddies smaller than two rotor diameters affect the wake deficit evolution and those larger than two rotor diameters influence the wake meandering.The model is divided into three submodels: the wake deficit one, the wake meandering and the added-wake turbulence.The wake deficit is based on the thin shear layer approximation of the Reynolds-averaged Navier-Stokes equations under quasi-steady-state conditions in axisymmetric coordinates.The velocity deficit in the far-wake region is modelled by: where U is the axial velocity component, V r the radial velocity component, r the radial coordinates and ν T the eddy viscosity.The eddy viscosity is modelled by the filter parameters F 1 and F 2 , described and calibrated by Madsen et al. [12], and extended by Larsen et al. [11] and Keck [13].

Inflow wind field generation
The input turbulent wind fields with a mean wind speed of 7.5 m/s (EC1) for stable, neutral and unstable conditions, are generated by the TIMESR model, a feature within TurbSim.This method uses the spectral amplitudes of a specific input time-series to generate time-series of wind input at additional locations.In the current case, the time-series input to TIMESR are the data of six LES realizations of one hour duration for each atmospheric stability condition for 870 points distributed over the rotor area.By modifying the phases of the generated time-series by TIMESR, coherence is included, based on Veers' method [14].Coherence gives information about the spatial variation of turbulence in the frequency space.Magnitude-squared co-coherence (referred to as coherence γ) of two spatially separated processes i and j , as a function of frequency, is defined as: where P i and P j are the power spectra of the two time-series separated by a specific lateral or vertical distance, and C i , j is the cross spectrum between these two time-series.In TIMESR, Davenport's coherence model, based on an exponential function with a decay parameter C K , is used as: To obtain the decay parameters for the three wind speed components, C u , C v and C w , the exponential function is fitted to the LES data.The fitted parameters are presented in Table 4, together with the power-law exponent to represent the mean vertical shear profile and the turbulence intensity T I .The height and width of the turbulent wind fields are 600 m, and 1304 m, with 76 and 164 points, respectively.The resolution in z and y is 8 m and the time step is 0.1 s.

Deficit and wake deflection
Figure 2 presents the time-averaged wake deficit over the last 600 s for one realization, for EC1, at a YZ-plane of the high-resolution domain, placed 7.5 D downstream T1, for stable (a), neutral (b) and unstable (c) atmospheric conditions.For the three conditions, the mean pitch angle of the semisubmersible of T1 is almost 4°.The shaft tilt angle is 6°.The deficit here is the ratio between the mean wind speed at 7.5 D downwind T1, u 1 , and the incoming undisturbed wind speed at T1, u 0 .The black circle outlines the initial position of the rotor.
The center of the wake is defined here as the location where the deficit is the greatest.Figure 2 shows that the wake center deflects half a diameter upwards from the hub height.For the unstable case, the wake deflects in the direction of the positive y-axis.A higher meandering in an unstable atmosphere causes this non-zero mean of the transverse deflection, which does not exceed 0.1 D. Furthermore, the deficit is the lowest in unstable conditions due to the higher wake recovery rates  related to a higher T I .In stable conditions, turbulence intensity is lower than in neutral conditions, and so should be the wake recovery rate; however, the deficit in stable conditions is very similar to the one in neutral conditions (and even slightly smaller).The reason for this slightly smaller deficit in stable conditions is the larger vertical shear of the incoming wind speed.Due to the larger shear, the wind speed variation with height is larger.Since the wake deflects upwards, it (the wake) experiences larger wind speeds in stable conditions.These larger wind speeds yield a faster wake recovery.This effect is compensated in neutral conditions by the higher T I ; consequently, both conditions present a similar deficit.
To illustrate the fact that the vertical wake deflection mainly takes place at low wind speeds, Figure 3, a), shows the statistics of the vertical center (z c ) position of the wake for below-, close to-and above-rated scenarios, for stable, neutral and unstable conditions.The vertical wake center displacements are measured at 7.5 D, and depicted in rotor diameters.T1 has a shaft tilt angle β 1 of 6°.In this specific case, T1 is free to float, and therefore the combined total tilt angle yields 10°, i.e., 6°, in addition to the mean pitch angle of 4°.The middle of the rectangles represents the average position of the vertical meandering of the wake center, for six realizations, and the height of the rectangles is equal to two standard deviations, computed as the average of the standard deviations of the six realizations.The points over and below the box show the six maxima and minima, respectively, for each realization.For the 7.5 m/s wind speed scenario, in unstable conditions, a few non-physical spikes in the wake center output were removed and replaced by a linear interpolation from the immediately previous and following values.The vertical positions of the wake center meander the most under unstable conditions, and for the 7.5 m/s wind speed scenario.The reason is that coherent structures are larger in unstable conditions, resulting in increased meandering.Figure 3, b), shows the statistics of the vertical center position of the wake for the 7.5 m/s wind speed scenario, for stable, neutral and unstable conditions, for the case of a shaft-tilt angle of T1 of 12°, 6°and 0°, but fixed (zero pitch).It can be seen how the mean z c decreases as the shaft tilt angle β 1 of T1 approaches 0°.
The decreased meandering for increasing mean wind speed is explained by the higher longitudinal component of the wind speed compared to the vertical or lateral components, with which the meandering of the wake is highly correlated.Because of the larger u-component of the flow parallel to the wake longitudinal advection, the wake is prevented from meandering laterally and vertically as easily.Figure 4 illustrates this effect by showing the vertical-to-horizontal wind speed components ratio (w-to-u) for the 7.5 m/s wind speed case (a) and the 16 m/s case (b), for T1 fixed, and a shaft tilt angle β 1 of 6°. Figure 4 also shows the non-zero vertical component of the thrust that the flow experiences when encountering a rotor with a specific tilt with respect to the ground.The wind speed vertical component induces a vertical force in the wake, which results in an increased Stable, Neutral, Unstable     2), for the three shaft tilt angles β 1 (12°, 6°and 0°), for three shear levels, each of them with a characteristic α shear exponent of stable (a), neutral (b) and unstable (c) atmospheric conditions, as indicated in Table 4. From Figure 5, top, it is seen that the mean vertical deflection is barely affected by shear, reaching a maximum difference of 0.05 D between stable and unstable conditions, for the case of 12°tilt angle, for 4.5 m/s up to 9 m/s.However, the effect of the tilt on the wake deflection yields a difference of between 0.35 and 0.4 D, for neutral and unstable conditions, respectively, if the shaft tilt angle is 12°, compared to 0°shaft tilt angle.For wind speeds up to 9 m/s, mean wind speed effect is not noticeable.From 9 m/s on, the deflection is decreased considerably.The INO WINDMOOR controller includes peak shaving for the thrust, which implies that the thrust coefficient is reduced before the rated wind speed (10.6 m/s) is reached, and the induced tangential wind speeds are lower compared to lower mean wind speeds.This effect, combined with a higher longitudinal wind speed component, results in a more pronounced decrease in the vertical wake deflection from 9 m/s wind speeds on, i.e., when the peak shaving starts affecting the turbine performance.Jiménez et al. [15], and Gebraad et al. [16] suggested an empirical equation to estimate the horizontal wake deflection due to yaw.The black lines in Figure 5 show these estimations applied to the vertical deflection due to tilt, which show good agreement with the deflections from FAST.Farm.
The effect of the wake at T2 decreases as the wake deflects further; consequently, the averaged rotor wind speed increases, as does the power output of T2.The effect of the wake deflection on the generated power of T2 is depicted in Figure 5, bottom.The output power of T2 is normalized by the output power of the turbine in free wind, with the default shaft tilt angle of 6°and free to float.Since the wake deflection decreases as the wind speed approaches the rated speed of the wind turbine, the influence of the tilt angle decreases.For the wind speed scenarios below rated, the normalized power output of T2 decreases with increasing wind speed, regardless of the stability condition.The reason behind this decrease is the smaller rotor averaged wind speed ratios between T1 and T2 as wind speed increases due to a larger T2 platform pitch (and a smaller rotor projected area) as thrust increases with higher mean wind speeds before reaching rated.The higher power output ratio observed for unstable conditions for lower wind speeds than 10.5 m/s for a shaft tilt angle of 12°is related to the higher recovery ratios of the wind speed in the wake inherent to unstable atmospheric conditions, combined with the slightly larger vertical deflection.

Effect on the semisubmersible structural response
The main effect of the shaft tilt for EC1, i.e., a mean wind speed of 7.5 m/s (see Table 2), is observed for the tower top yaw moment (TTYM).The tension at fairlead 1 (FL1T) follows a similar trend to the tower base fore-aft bending moment and the blade root out-of-plane moments, dominated by the thrust.For this reason, the latter are not presented here.Bottom: mean and deviation of the six realizations for the TTYM.All the subfigures correspond to EC1, for the three stability conditions (stable in blue, neutral in green, unstable in red) and the three shaft tilt angles.Figure 6, top, shows the mean and standard deviation of the fairlead 1 tension for T2 for EC1 (see Table 2), for the three stability conditions, and the three shaft tilt angles of T1, with T1 fixed.For stable and neutral conditions, the mean decreases as the tilt angle approaches 0°by approximately 20%, due to the decrease in wind speed, i.e., in thrust, if the rotor is completely in waked conditions.In general, the shaft tilt angle does not present a notable effect on the standard deviation, compared to the effect that atmospheric stability has.The standard deviation in unstable conditions is more than three times that of stable conditions.Doubrawa et al. [3] investigated the effect of the wake deflection on the blade root out-of-plane moment, which follows the same trend as the fairlead tension, since it is mainly influenced by thrust.They found that for a 6 MW spar buoy FWT, in neutral conditions, the mean and damage equivalent loads of the blade root out of plane moment slightly decreased with a shaft tilt of 0°, compared to a shaft tilt of 6°, which is consistent with the findings in this work.
Figure 6, bottom, presents the mean and standard deviation of the TTYM for EC1, for the three stability conditions, and the three shaft tilt angles of T1.For stable conditions, the mean increases remarkably as the shaft tilt angle decreases due to the larger vertical shear, and to the wider range of wind speeds that the rotor experiences, as depicted in Figure 7. Figure 7, top, shows the time-averaged u-component of the wind speed over the rotor for the last 2000 s for EC1, with a mean wind speed of 7.5 m/s (see Table 2), for stable conditions.Figure 7, bottom, shows the wind speed as a function of the vertical coordinate, for the 32 locations in y, in rotor diameters: the darker the lines, the more towards the negative y-direction they lay; if there were no horizontal wake deflection, the vertical distribution of the wind speeds would be symmetric with respect to the XZ-plane.From this figure, column c), the larger shear for the case of T1 not having any shaft tilt angle is seen, together with the more dispersed range of wind speeds, which is reduced for the positive tilt angles (columns a) and b)).Regarding the standard deviation of the TTYM, it increases as the atmospheric stability decreases due to the increased TI.The effect of the shaft tilt angle on the standard deviation is reduced compared to the effect on the mean and compared to the effect that stability has.In neutral conditions, the increase in standard deviation between the 12°and 0°tilt angles reaches 37% due to the reduced effect of the wake for the 12°tilt case.

Conclusions and recommendations
As the installation of floating offshore wind farms gains momentum, the need to investigate methods to decrease wake losses increases.This work investigates the combined effect of upwards wake deflection and atmospheric stability for the INO WINDMOOR 12 MW semisubmersible floating wind turbine [7] placed 8 D downwind in the wake.To account for atmospheric stability, TIMESR is used to generate turbulent wind fields based on input time-series from LES.The turbulent wind fields are input to FAST.Farm, which has the DWM model implemented.Three shaft tilt angles β 1 (12°, 6°and 0°) are used to study the vertical wake deflection for a mean wind speed of 7.5 m/s.Positive shaft tilt angles imply that the wake deflects upwards, as indicated in Figure 1 and, consequently, the rotor of the FWT placed downwind (T2) is in partially waked conditions.Five additional constant wind fields, from 4.5 m/s to 10.5 m/s (see Table 2) are used to study the effect of the shaft tilt on the wake deflection and on the output power of the turbine in the wake.The effect of the upwards wake deflection on the tower top yaw moment and the fairlead 1 tension is studied for the case of having a turbulent wind field with a mean wind speed of 7.5 m/s (EC1 in Table 2), for stable, neutral and unstable atmospheric conditions.
The constant wind fields study shows that the wake deflection is relatively constant up to close to rated wind speed conditions, when the ratio between the vertical and longitudinal wind speed components is reduced.The power output of T2 is directly affected by the upwards deflection of the wake.This effect is observed regardless of the stability condition.Regarding the structural response of T2, in the wake, the upwards wake deflection for the 7.5 m/s case affects the mean of the TTYM the most, in stable conditions, due to the decrease of the shear effect when the rotor of T2 barely experiences the wake effects.The current work suggests further investigation into wake deflection as a control strategy to minimize wake losses in the power output of floating offshore wind farms.Furthermore, it is key to investigate other potential control strategies for offshore floating wind farms, such as yaw control or power derating.

Figure 1 :
Figure 1: Sketch of the INO WINDMOOR 12MW models separated 8 D, and the upwards deflection of the wake, due to the positive shaft tilt angle β 1 .β 2 indicates the positive shaft tilt angle of T2.A positive platform pitch angle is indicated by θ.The fairlead connection at the upwind column of T2 is indicated by FL1.

Figure 3 :
Figure 3: Left (a): statistics of the vertical position of the wake center 7.5 D downwind T1, for the six seeds and stable, neutral and unstable atmospheric stability, for 7.5 m/s, 12 m/s and 16 m/s wind speed scenarios, and for T1 floating and a shaft tilt angle β 1 of 6°.Right (b): statistics of the vertical position of the wake center 7.5 D downwind T1 for the six seeds and stable, neutral and unstable stability, for 7.5 m/s, for T1 fixed, with 12°, 6°and 0°tilt angles.The mid-point of the rectangles is the mean of the six seeds and its height two average standard deviations of the six seeds.The points along the vertical lines represent the maxima and minima for the six seeds for each condition.

Figure 4 :
Figure 4: w-to-u ratio for the 7.5 m/s mean wind speed (a) and the 16 m/s mean wind speed (b) scenarios.T1 in this case is floating, and has a shaft tilt angle β 1 of 6°and a mean pitch angle of 2.5°, for both cases.

Figure 5 ,
Figure 5, top, shows the averaged wake vertical deflection as a function of mean wind speed for the five constant mean wind speed cases (CW1-5 in Table2), for the three shaft tilt angles β 1 (12°, 6°and 0°), for three shear levels, each of them with a characteristic α shear exponent of stable (a), neutral (b) and unstable (c) atmospheric conditions, as indicated in Table4.From Figure5, top, it is seen that the mean vertical deflection is barely affected by shear, reaching a maximum difference of 0.05 D between stable and unstable conditions, for the case of 12°tilt angle, for 4.5 m/s up to 9 m/s.However, the effect of the tilt on the wake deflection yields a difference of between 0.35 and 0.4 D, for neutral and unstable conditions, respectively, if the shaft tilt angle is 12°, compared to 0°shaft tilt angle.For wind speeds up to 9 m/s, mean wind speed effect is not noticeable.From 9 m/s on, the deflection is decreased considerably.The INO WINDMOOR controller includes peak shaving for the thrust, which implies that the thrust coefficient is reduced before the rated wind

Figure 5 :
Figure 5: Top: wake deflection as function of constant horizontal mean wind speed for 12°, 6°and 0°s haft tilt angles, with vertical shear α exponents characteristic of stable (a), neutral (b) and unstable (c) conditions.The black lines, dashed, dash-dotted and solid, represent the estimated vertical deflection based on the formulae from Gebraad et al. and Jiménez et al.[15,16].Bottom: power generated by T2, normalized by T1, floating and with a shaft tilt angle of 6°.

Figure 6 :
Figure6: Top: mean and standard deviation of the six realizations for the FL1T.Bottom: mean and deviation of the six realizations for the TTYM.All the subfigures correspond to EC1, for the three stability conditions (stable in blue, neutral in green, unstable in red) and the three shaft tilt angles.

Figure 7 :
Figure 7: Top: time-averaged u-component u 1 of the wind speed at 7.5 D downwind T1, for EC1.Bottom: vertical distribution of the wind speed for every horizontal distance, for EC1.The figures correspond to stable conditions, for a shaft tilt angle β 1 of T1 of 12°(a), 6°(b) and 0°(c).

Table 2 :
Environmental conditions used in this work.EC1 is run for stable, neutral and unstable conditions.

Table 3 :
Recommended and used values for the spatial and time resolution in the FAST.Farm set-up.

Table 4 :
Davenport coherence model parameters C K for 7.5 m/s mean wind speed.The rightmost column indicates the α shear exponent parameter from the power-law shear model.