The wind turbine examined in this study is one of five floating offshore wind turbines (FOWTs) that form the Hywind Scotland wind farm. The wind turbine positions (labelled HS1, HS2, HS3, HS4, and HS5) are arranged in a W-shaped configuration, rotated 30° clockwise relative to true north, as illustrated in Fig. . The configuration is described within a right-handed coordinate system, where the y axis is oriented towards the north; and the origin is defined at the location of the HS4 turbine, which is the one used in this study. The FOWTs (SWT-6.0-154, Siemens Gamesa Renew. Energ.) have a hub height of 98.6 m and a rotor diameter of D=154 m. They are mounted on floating platforms based on a ballasted spar buoy concept and have been operating since 2017 . Below the rated wind speed, blade pitch is regulated in the same manner as in a conventional bottom-fixed wind turbine. Above the rated wind speed, however, the blade pitch control operates in coordination with the floater motion control system.

The most notable motion experienced by the wind turbines during operation occurs along the pitch axis, with angles ranging from 0 to 7° at wind speeds between cut-in and rated . Above the rated wind speed, the pitch angle decreases until it reaches a mean value close to 2°. Less significant variations are observed in the roll angle, which on average ranges between 0 and 0.5° for wind speeds between cut-in and rated. In contrast to the pitch angle, the roll angle continues to increase above rated speed, reaching approximately 1°. When comparing the magnitude of the mean pitch and roll angles, the roll angle is 60 %–90 % smaller than the pitch angle. In addition to the mean rotation about the pitch and roll axes, the operation of the Hywind Scotland turbines is characterized by wind-speed-dependent dynamic rotations around the yaw, pitch, and roll axes . The standard deviation of the yaw and roll angles follows a similar trend, increasing with hub-height wind speed and reaching a maximum of about 0.8 and 0.4°, respectively. In contrast, the maximum standard deviation of the pitch (i.e. 0.8°) occurs near the rated wind speed, while at lower and higher wind speeds the standard deviation of the pitch is less than 0.4°. Furthermore, the magnitude of the dynamic motion of the Hywind Scotland turbines depends on atmospheric stability, with lower dynamic motion in all yaw, pitch, and roll rotations observed under stable conditions . Overall, the observed mean and dynamic pitch motion indicates that the Hywind Scotland wind turbines tilt away from the wind in a stable manner, as the standard deviation of the pitch angle remains small. This is not necessarily the case for other floater types. For example, reports FOWT pitch angles with a lower mean magnitude but higher dynamic fluctuations.

The inflow wind conditions experienced by the HS4 wind turbine were monitored using a nacelle-mounted Doppler lidar. The Doppler lidar (Wind Iris Turbine Control, Vaisala Oyj) acquires radial wind speed measurements along four separate lines of sight. The direction of each line of sight is defined by a three-dimensional unit vector n={n1,n2,n3}, with azimuth angles of -15 or +15° and tilt angles of -5 or +5°, relative to the instrument's optical axis (see Figs.  and ). Radial wind speeds are acquired at 10 distances from the lidar: 50 m (0.32D), 80 m (0.52D), 120 m (0.78D), 160 m (1.04D), 200 m (1.30D), 240 m (1.56D), 280 m (1.82D), 320 m (2.08D), 360 m (2.34D), and 400 m (2.60D) – all distances measured along the optical axis of the Doppler lidar. The sampling rate of 0.25 Hz results in 150 measurements per range gate per a 10 min period. The transceiver of the lidar was tilted by 2.5° relative to a levelled nacelle. Although the Wind Iris is a fixed-pattern-scanning

Definition according to

The wind turbine was instrumented with a motion reference unit (MRU) installed on the nacelle to monitor its mean and dynamic responses. The MRU measures the pitch and roll angles of the nacelle, corresponding to rotations about the longitudinal and transverse axes relative to the turbine's yaw direction. Data from the MRU were logged alongside active power and wind speed measurements from a nacelle-mounted anemometer via the supervisory control and data acquisition (SCADA) system at 1 Hz. Unlike the other parameters, the yaw direction was recorded only when changes occurred. Based on the sampling rate of the Doppler lidar data, a complete set of radial wind speed observations for all lines of sight was available every 4 s. Synchronization between the two data acquisition systems was verified by comparing the internal accelerometer readings of the Doppler lidar with those from the MRU in the nacelle.

In this study, we examine data from January 2019 to October 2020. The dataset does not cover the entire period but includes three intervals: 1–30 January 2019, 1 September–29 November 2019, and 1 June–1 October 2020. Discontinuities were due either to missing turbine data or periods when the nacelle-mounted lidar was not operating. Overall, the dataset contains SCADA measurements equivalent to approximately 6 months.

To avoid including cases in which wakes from adjacent turbines distorted the inflow conditions at the HS4 turbine, only data acquired during turbine operation with yaw directions between 90 and 270° were selected, resulting in 13 247 10 min periods of SCADA data. The wind lidar data were filtered based on the data availability at each range gate. Only periods with at least 50 % data availability were selected, which reduced the initial wind lidar data set by 10 % to 17 057 10 min periods. The SCADA data were post-processed to estimate statistics synchronized with the wind lidar measurements. However, concurrent lidar and SCADA data were available for only 6729 periods. Additionally, some periods exhibited large yaw standard deviations. To focus on stable wind directions, only periods where the yaw standard deviation within a 10 min interval was ≤ 10° were retained. Applying this criterion resulted in a dataset of 6659 cases (i.e. 10 min periods). The selected dataset represents climatological conditions typical of summer and autumn.

Based on the radial speed measurements acquired at a height z but from different lines of sight, it is possible to estimate the longitudinal and transverse components of the wind vector. For this calculation, an estimation of the induction factor a of the wind turbine is required. The two horizontal wind components are then equal to 4u∞(z)=n2(j)vr(i)-n2(i)vr(j)find(n1df)(-n1(j)n2(i)+n1(i)n2(j))andv∞(z)=-n1(j)vr(i)+n1(i)vr(j)-n1(j)n2(i)+n1(i)n2(j), where the superscripts i and j denote the index of two line-of-sight vectors over which radial speeds are acquired at height z, and n1 and n2 are the two horizontal components of the line-of-sight vectors. The pitch and the roll angle of the nacelle of the FOWT are considered for the determination of line-of-sight vector n. Equation () is used to reconstruct the wind profile at the top part of the rotor.

The first objective of this study was to assess how well the radial speed model presented in Eq. () reproduces the trends observed in the lidar data. Figure  shows the mean radial speeds for each line of sight during four separate 10 min periods, all characterized by the same hub-height wind speed (8 m s⁻¹). Each case represents different inflow conditions and corresponds to periods where lidar measurements were available for all 10 range gates along each line of sight. Figure a illustrates the simplest wind conditions: a positive gradient of u ∂u‾∂z>0 and negligible gradient of v ∂v‾∂z∼0. The radial speed model (solid lines), based on the assumption of constant velocity vertical gradients, adequately describes the observations (dots). In general, when the turbine yaw is aligned with the wind direction, the two lower beams measure the same line-of-sight velocity at any distance. Vertical shear results in higher radial speeds for the upper beams compared to the lower beams, while vertical veer causes relative high values of ∂v‾∂z which leads to larger differences between the upper beams' radial speeds because wind direction changes with height. This effect is evident in Fig. b, where the wind veer causes the wind direction to change by 18° between hub height and blade tip height. Agreement between observations and the radial speed model is not limited to cases of positive shear. An example is shown in Fig. c, where negative shear at the top of the rotor results in lower radial speeds for the upper beams compared to the lower ones. Even when shear is zero, differences between upper and lower beams are expected due to the tilt angle affecting the projection of the horizontal wind vector on the upper beam's line of sight, typically resulting in a 1.5 %–2.5 % difference. However, the observed differences range from 5 % to 20 %, depending on the upper beam and measurement range, which indicate the presence of negative shear. In all three cases, a clear reduction in radial wind speed is observed at around 160 m (∼1D) towards the rotor, caused by turbine operation. This reduction corresponds to high induction factors (0.38–0.41), consistent with velocity deficits in the wake when the turbine operates at below-rated wind speed . Similar high induction factors have been reported by , who investigated the aerodynamic induction of a full-scale wind turbine using two scanning Doppler lidars.

In contrast to the good agreement observed in Fig. a–c, which corresponded to cases where the RMSE εu of Eq. () has values of less than 0.2 m s⁻¹, 11 % of the examined 10 min periods show that the inflow model could not reproduce the trends in the lidar data (see Fig. d). This discrepancy is attributed to wind profiles with wind speed inversions, as occurs in LLJs. In such cases, simplified parameterizations of the vertical wind profile (i.e. constant shear and veer coefficients) cannot predict the observed radial speed distribution.

The analysis of wind profile characteristics was conducted only for 10 min periods, where the model in Eq. () performed adequately. These cases were identified by inspecting the RMSE εu between the inflow model and the 10 min mean radial speeds. Values above 0.2 m⁻¹ were considered unsatisfactory for representing the spatial distribution of the lidar measurements. This threshold was empirically chosen and corresponds to mean absolute differences of less than 5 % between hub-height wind speed measurements from the nacelle-mounted lidar and the turbine anemometer . As already stated in the previous section, based on this criterion, 757 cases (11 % of the dataset) were excluded. As shown in Fig. d, these cases are likely associated with the presence of LLJs. A similar occurrence of LLJs (approximately 12 %) during spring and summer months was reported in an offshore field campaign in the North Sea by . During these events, the turbine yaw direction spanned a sector from 90 to 256°, indicating that the observed LLJs were not caused by flow crossing a coastal-sea interface.

Figure a and b present a bar chart of the estimated wind speed gradients for each month using Eq. (). The values of ∂u‾∂z within the height layer corresponding to the upper section of the FOWT rotor range from -0.06 to 0.06 s⁻¹ (Fig. a). The vertical gradient of the transverse wind component ∂v‾∂z is generally found to be negative, as expected in the location of the FOWT (Fig. b), with the higher values having been observed in cases where the gradient ∂u‾∂z is strong. Using the estimated values of the vertical gradients of the wind components, we compute the shear and veer of the wind profile by estimating the horizontal wind speed and direction at the hub height and at the top of the rotor, which are presented in Fig. c and d. Periods with negative shear are found during the months from June to September and account for 22 % of the dataset, a percentage similar to that reported by , who studied wind profile characteristics over the North Sea. These profiles, which may include LLJs, typically occur in areas with variations in topography . For negative shear cases, it remains unclear whether the core of an LLJ is found within the lower half of the rotor or if a wind speed inversion occurs at very low heights, as has been observed in the North Sea . When focusing on data acquired between June and September, negative shear does not consistently occur when the wind originates from a specific sector. Only during July and September is negative shear observed in a narrow wind direction sector between 200 and 250°. These values appear in at least one 1 h period on 13 d in July and almost every day (i.e. 30 d) in September. In contrast, during June and August, negative shear is not associated with a specific wind direction sector but occurs across the entire selected sector (90–270°), as shown in Appendix . The observed negative shear values are independent of the time of day.

To further investigate wind profiles with negative shear and/or wind speed inversions, we examine as a case study a 12 h period between 08:40 and 20:40 on 26 June 2020. Figure a shows the corresponding wind conditions. The figure presents the mean horizontal wind speed U∞ at different heights, corresponding to the magnitude of the two horizontal components (i.e. u‾∞2+v‾∞2), which are calculated using Eq. (). The estimation of these components is based on pairs of radial speed measurements acquired at different ranges and heights, and represents the spatial average over 10 m vertical layers. This period was chosen because it is characterized by vertical profiles with either negative shear or a speed inversion (highlighted in red or black, respectively, in the bar below Fig. a).

In the cases of negative shear, the wind speed decreases from hub height (98.6 m) towards the top of the rotor (∼170 m) at rates varying between 0 and -0.05 s⁻¹. On these occasions, a speed inversion occurs somewhere below hub height, but, due to the absence of measurements in the lower half of the rotor, its exact location cannot be determined. For wind profiles with a speed inversion, the profiles typically exhibit one inflection point around a maximum value. In a few cases, profiles such as that shown in Fig. b display two inflection points – around a maximum and a minimum – within the vertical range of 90–180 m, which form a core in the wind profile. The presence of two inflection points enables the identification of LLJs based on the difference (ΔU‾) between the minimum (U‾min) and maximum (U‾max) wind speeds. In addition to these vertical profile features, this period is noteworthy for its highly variable hub-height wind speed, ranging from 6 to 14 m s⁻¹, with a yaw direction between 140 and 180°. This makes it an interesting case for studying turbine power production under different inflow conditions. Figure c shows the turbine's normalized power output. The normalization is performed based on the power curve of a fixed-bottom wind turbine using the hub-height wind speed – either the wind lidar (UHub) or the sonic anemometer (UNac) installed on the nacelle. We note here that a transfer function for correcting the effect of the rotor has been applied by the wind turbine manufacturer. Deviations from nominal power reach up to 50 %, with smaller deviations observed when using the nacelle-mounted sonic anemometer as the reference wind speed. These results indicate that inflow conditions with negative shear or speed inversions can significantly impact FOWT power production. Therefore, it is important to quantify the height at which wind speed inversions occur, the magnitude of the associated speed difference, and the duration of these events. Among the entire dataset, a local maximum in the vertical wind speed profile could be detected in 10 % of the cases. Figure  presents (a) the wind speed difference between the maximum and minimum wind speeds within the examined height range (100–200 m), (b) the inversion height, and (c) the duration of those events. The wind speed difference was usually small (0.25–0.5 m s⁻¹), although in 10 % of the selected cases, differences greater than 2 m s⁻¹ were observed. For a slight majority of cases, the inversion height was around 130 m, but overall, the range over which inversions were observed was between 100 and 160 m. This height range is consistent with studies that have been performed in different locations in the North Sea . To identify the duration of the events characterized by such wind conditions, hourly periods where at least 30 min of either velocity inversions or negative shear in the wind profile were selected. Using this criterion, we find that most profiles persist for about 60 min; however, eight periods were identified where these characteristics lasted for an extended duration (700–900 min). Similar LJ duration statistics have been reported by for the wind conditions over the North Sea.

The transceiver of the nacelle-mounted wind lidar was installed above the rotor centre. Consequently, Eq. () estimates the two components of the horizontal free wind vector at a position vertically displaced relative to hub height. To verify the installation height of the wind lidar (zL), we calculated the mean minimum absolute difference between the hub-height wind speed reported by the nacelle-mounted anemometer and a wind speed estimated using the function u‾∞+∂u‾∂zzL. Different values of zL were tested for all periods, ranging from -10 to 10 m in steps of 0.1 m, and the minimum difference was found when the lidar was translated vertically by 4 m. This height offset is therefore used to estimate hub-height wind speed.

In the case presented in Fig. , vertical wind profiles with speed inversions or negative shear at the top of the rotor are linked to a reduction in turbine power production (as a reference, we used the power curve of a bottom-fixed turbine of the same type as the FOWT examined in this study). To assess the impact of inflow conditions on FOWT power production, we performed a power curve verification (PCV) analysis. For this purpose, an accurate hub-height wind speed estimate is required. Although the range gates at 2.6D of the two lower beams are nearly horizontal at a 5° turbine pitch, the mean turbine angle places them between approximately 60 and 125 m above sea level. In addition, the mean roll angle induces a vertical displacement between the two that can reach 7 m (see Appendix ). Therefore, we examine three hub-height wind speed estimation methods that can be used in the context of a PCV:

Wind field reconstruction (2.6D). The hub-height wind speed is calculated using Eq. () with measurements only at 2.6D (see Fig. a). Note that the nacelle pitch angle causes the measurement plane at a given distance to be tilted rather than vertical. The tilt is proportional to the nacelle pitch angle. However, because nacelle-mounted lidars typically use low-elevation angles for their lines of sight, this tilt has minimal impact on the horizontal distance from the rotor. Considering that measurements are acquired over a probe length, the horizontal displacement can be treated as negligible.

The purpose of testing the three wind speed estimates is to investigate how sensitive the power curve verification is to the choice of reference wind speed. In this context, power and wind speed data were grouped in 0.5 m s⁻¹ bins, and mean and standard deviation statistics of the produced power were estimated which are presented in Fig. . The mean power is compared to a reference power curve model of a fixed-bottom wind turbine of the same type as the one installed in Hywind Scotland. We observe a notable difference in the power curve between positive and negative wind shear conditions (Fig. a–b). The estimation of REWS in the case of the profiles with positive shear, shown in Fig. c, results in a power curve similar to those obtained using the other two hub-height wind speed definitions. Typically, using REWS reduces scatter in the power curve depending on the magnitude of wind shear . For low wind shear values, only small differences are observed when REWS is applied . In our dataset, offshore wind conditions produce power curve scatter, expressed by the standard deviation of power, that is comparable across all three hub-height wind speed estimates in the cases of wind profiles with positive shear. However, a significant difference is observed in the case of negative shear profiles, where the standard deviation is nearly twice that of positive shear cases below rated speed. Here we have to note that if the estimation of the REWS was possible in the case of the wind profiles with a negative shear, then this could result in a decreased estimated value of the hub-height wind speed since an inversion would occur closer to the ground. This could lead to a closer agreement between the power curves of the positive and negative shear cases, and could explain the differences that we see in the standard deviation of the mean produced power presented in Fig. d and e.

To quantify the variations between the estimated power curve and the reference curve, we calculate the relative difference using three different wind speed estimates. Figure  shows the relative mean difference between cases with negative and positive shear, and the reference power curve for different wind speeds. The differences observed in Fig.  are quantified by calculating the relative percentage difference between the measured and modelled power curves. We find that when the wind profile exhibits negative shear, power production is reduced by up to 18 %. Smaller differences are observed in cases with positive shear. Interestingly, the difference between positive and negative shear ranges from 5 % to 10 %, depending on whether all range gates are used or only measurements at 2.6D are selected. The variation of the produced power is attributed to both the shear and veer values of the wind profile. In the examined dataset, negative shear values are usually related to negative veer values which results in reduction to the power production (see Appendix ). This result is in agreement with the findings reported by and for the cases of onshore and offshore fixed-bottom wind turbines, respectively.