Some additional instrumentation was set up on the wind plant for the wake-steering experiment. First, all turbines were equipped with a supervisory control and data acquisition (SCADA) system allowing 1 Hz data for the most critical variables to be recorded. A Vaisala Triton sodar was installed in the proximity of turbines SMV5 and SMV6, and a Leosphere WindCube v1 profiling lidar was installed between turbines SMV2 and SMV3. Their precise locations are presented in Fig. . Although data from the sodar and profiling lidar were not used extensively during the analysis, they were used to cross-check and validate wind measurements from the turbines and to identify the best references for assessing the ambient wind conditions. Additionally, measurements from the WindCube lidar were used to estimate the turbulence intensity distribution at the site.

A WindCube Nacelle lidar was installed on top of the controlled turbine, SMV6, to measure the wind inflow, including the turbine's yaw misalignment with respect to the incoming wind direction. This sensor collects radial-wind-speed values from four beams at a sample frequency of 4 Hz at 10 range gates spanning 50–200 m upstream. For the lidar-based analyses presented here, we use the estimated horizontal wind speeds and wind directions at hub height provided by the lidar. We use measurements at a range of 150 m (1.8D) to determine wind speed and the average of measurements at 100, 150, and 200 m to estimate yaw misalignment.

Additionally, a Hemisphere GNSS compass was fixed on the nacelle of SMV6 to monitor the offset of the turbine's reported yaw orientation relative to north during the experiment. Indeed, this offset is known to deviate with time and must be calibrated properly when realizing wind farm control experiments. Figure  displays the evolution of this north offset with time by looking at the difference between SMV6's yaw position signal and the GNSS measurement of the nacelle orientation. It shows that even though some drifts were experienced at the end of 2019, a very stable offset was maintained for the full duration of the field experiment, 17 February–25 May 2020.

Before analyzing the data, all variables are downsampled to 1 min average values. As will be discussed further in Sect. , 1 min samples are intended to provide a balance between averaging small-scale turbulent variations and distinguishing between time-varying wind conditions.

Because the wind turbine controller could not be accessed or modified for this experiment, the wake-steering strategy was implemented following the same approach as in . A control box was installed to read the incoming relative wind direction signal from the nacelle wind vane installed on the turbine and to apply an offset before sending it to the turbine's existing yaw controller, thereby inducing the intended yaw offset. The control logic implemented in the control box is illustrated in Fig. . As shown in Fig. , because the lookup table defining the offset angles is dependent on both wind speed and direction, the nacelle wind speed and yaw position are also used as inputs to the control box. The measured wind direction – formed by combining the absolute yaw position and the relative wind vane direction – and wind speed are passed through low-pass filters with a time constant of 60 s before they are used to determine the corresponding target yaw offset in the lookup table. Finally, a toggle allows the yaw offsets to alternate between the target offsets and zero offset to analyze the effect of wake steering in wind conditions that are similar to the baseline yaw control case. Note that the unfiltered wind speed signal recorded through the control box is compared to the same signal measured by the 1 Hz SCADA system to remove any time lag between the two systems and to ensure that the two clocks are correctly synchronized.

The applied yaw offsets are represented by the offset schedule in Fig. . To avoid excessive yaw activity from switching between large positive and negative values of yaw misalignment, only positive yaw offsets are used in this experiment. Although energy improvements from wake steering are expected for both positive and negative yaw offsets, research suggests that, overall, wake steering is more effective with positive yaw misalignment . Further, results from a field experiment investigating the impact of yaw misalignment on loads show a reduction in blade loads for positive yaw offsets but an increase in loads for negative offsets . Adhering to an upper bound of 20∘, imposed to manage structural loads, SMV6 is misaligned by up to 20∘ for full-wake conditions (208–216∘), and then this angle is linearly reduced throughout the partial wake sector until it reaches zero at a wind direction of 236∘. The yaw offset schedule for wind speeds below 10 m/s shown in Fig.  is a simplified form of the optimal yaw offsets for maximizing the combined power of SMV5 and SMV6 determined using FLORIS. The FLORIS model used when designing the controller was calibrated using data from a previous wake-steering experiment performed on the same wind turbines. Additional considerations were included in the yaw offset schedule to provide some robustness to wind direction uncertainty, following the approach discussed by ; specifically, yaw offsets are applied for a wider sector of wind directions in the partial wake region than suggested using the original FLORIS model. Last, to further reduce the loading at higher wind speeds, the target offsets are reduced in four steps above 10 m/s until wake steering is stopped for wind speeds above 14 m/s. Note that the target yaw offsets are binned by wind speed in steps of 1 m/s rather than specified as a continuous function of wind speed to enable a simple lookup table implementation.

The distributions of the wind directions, wind speeds, and turbulence intensities analyzed during the wake-steering experiment for the baseline and the controlled periods are shown in Fig.  for the wind direction sector between 195 and 241∘ investigated in this paper. Specifically, Fig.  shows the number of 1 min samples obtained for each wind direction, wind speed, and turbulence intensity bin. Note that the reference wind direction and wind speed measurements are obtained from nacelle-based wind turbine sensors, as will be discussed in Sect. . Turbulence intensity is estimated using measurements from the WindCube profiling lidar at the hub height of 80 m.

The wind direction histogram shown in Fig. a reveals a relatively uniform distribution of wind directions across the sector of interest for both the baseline and the controlled periods. Compared to the long-term wind speed distribution illustrated in Fig. , the wind speed histogram provided in Fig. b indicates above-average wind speeds during the experiment period; however, higher wind speeds are expected because most of the data were collected during the winter (in February and early March, as illustrated in Fig. ), when the wind resource is strong at the site. Figure c shows that similar turbulence intensity distributions were sampled during the baseline and the controlled periods. Last, the joint distribution of wind direction and wind speed during the experiment period is shown in Fig. , considering the total amount of data analyzed for the baseline and the controlled periods. For almost the entire wind direction sector, data were collected for wind speeds from 3–13 m/s.

Wake interactions are simulated for the SMV wind plant using the default Gauss–curl hybrid (GCH) model within FLORIS . The GCH model is built on the Gaussian wake deficit model presented by and , as well as the wake deflection model developed by . To capture the effects that large-scale trailing vortices generated through yaw misalignment have on wake deflection, however, the GCH model includes a computationally efficient approximation of the curl model developed by . The curl-specific elements of the GCH model include secondary steering – whereby the vortices created by a misaligned wind turbine deflect the wakes of downstream turbines with which they interact – and yaw-added recovery, in which the vortices increase wake recovery through mixing with higher velocity flow.

The FLORIS model of the SMV wind plant relies on the turbines' theoretical power and thrust coefficients, shown in Fig. , to determine wake behavior, and it is further tuned using field measurements. As illustrated in Fig. , the theoretical power curve used in FLORIS closely matches the observed power curve for SMV6 during baseline operation using wind speed measurements from the WindCube Nacelle lidar. The FLORIS model is tuned to match the depth of the measured baseline wake losses for SMV5 during the experiment by adjusting the turbulence intensity input, which affects the rates of wake recovery and expansion; we found that when using the “Gauss” velocity model, a turbulence intensity of 11 % represents the overall wake losses during the experiment reasonably well. We treat turbulence intensity as a tuning parameter rather than using the measured turbulence intensity as an input to FLORIS because (1) the turbulence intensity measurements provided by the ground-based and nacelle lidars do not represent traditional turbulence measurements (e.g., from a cup or sonic anemometer) because of volume averaging and line-of-sight measurement limitations  and (2) further work is required to validate the relationship between turbulence intensity and wake deficits  that is used in the GCH model. Last, the power loss suffered as a result of yaw misalignment is modeled in FLORIS by scaling the rotor-averaged wind speed, vavg, used to determine power and thrust as follows: 1vavg′=vavgcos⁡γpv/3, where γ is the yaw misalignment, following the approach suggested by . We use a cosine exponent of pv= 1.61, estimated using WindCube ground-based lidar measurements and power data for SMV6 from previous experiments at the wind plant .

Examples of the hub-height flow fields generated by FLORIS for the SMV wind plant are provided in Fig.  for a wind speed of 8 m/s. Figure a shows the flow field with a wind direction of 195∘, which is the southernmost wind direction investigated in this study. Figure b highlights the impact of SMV6 operating with a 20∘ yaw misalignment for a wind direction of 208∘, which is the first wind direction (moving clockwise) for which yaw offsets are implemented (see Fig. ). Finally, Fig. c shows the FLORIS flow field corresponding to the northernmost wind direction of 241∘ investigated here.

FLORIS is designed to model wake interactions for fixed wind directions and yaw positions. But in realistic dynamic wind environments with imperfect wake-steering control, uncertainty exists in the yaw position a turbine achieves for a particular wind direction. Further, after the turbine settles on a specific yaw position, the wind direction will vary until the yaw error is large enough for the turbine to yaw again, causing wind direction uncertainty. and investigated the impact of yaw position uncertainty on optimal wake-steering performance by performing FLORIS simulations with a distribution of possible yaw positions for a given wind direction. Similarly, and included wind direction uncertainty when optimizing wake-steering control using FLORIS by assuming a distribution of possible wind directions about the intended wind direction. Here, we model the uncertainty resulting from wind direction variability and controller limitations using the approach presented by , wherein FLORIS simulations are performed assuming uncertainty in both yaw position and wind direction.

As described in depth by and , we model the impact of wind direction variability on wake-steering performance by creating a joint probability mass function (PMF) of the wind directions and yaw positions for the controlled turbine (discretized using 1∘×1∘ bins). First, the ideal PMF is established by assigning a probability of one to the intended yaw position corresponding to each wind direction using the yaw offset schedule. Next, the ideal PMF is convolved with a zero-mean joint PMF representing the uncertainty in the wind direction and yaw position (approximated as a bivariate normal distribution). The predicted mean yaw offsets can then be calculated from the resulting PMF by finding the expected value of the yaw position for a particular wind direction. Similarly, the predicted mean power production can be determined by calculating the expected value of the power from FLORIS across all possible yaw positions corresponding to the wind direction of interest.

The distribution of wind direction and yaw position uncertainty is characterized using the standard deviations of the wind direction uncertainty, σϕ, and the yaw position uncertainty, σθ. As explained by , we assume a value of σθ=1.75∘, which the authors found to closely approximate the yaw position uncertainty observed in simulations using a standard yaw controller. Assuming independent wind direction and yaw position uncertainty variables, σϕ can be estimated from the standard deviation of the measured yaw misalignment, σγ, as 2σϕ=σγ2-σθ2.

We explore different methods for estimating the yaw error standard deviation for SMV6 during baseline operation as a function of wind speed, as shown in Fig. . Specifically, we compare yaw error measurements using the turbine's nacelle wind vane, the WindCube Nacelle lidar, and the difference between the reference wind direction – defined as the average wind direction measured by turbines SMV1, SMV2, SMV3, and SMV7, as will be discussed in Sect. , and the turbine's nacelle position. Whereas the wind vane and nacelle lidar sample the more variable wind directions local to SMV6, the reference wind direction approximates the slowly varying mean wind direction across the wind plant; therefore, the latter method yields the lowest yaw misalignment standard deviation. Because the wind direction input to FLORIS is intended to represent the mean wind direction across the wind plant, we use the yaw error standard deviation determined from the reference wind direction as the input to the wind direction variability model. As shown in Fig. , the yaw misalignment standard deviation increases for wind speeds below 8 m/s because (1) the wind direction tends to be more variable at lower wind speeds and (2) this turbine model's yaw controller is less responsive to wind direction changes for wind speeds below 7 m/s; therefore, we use wind-speed-dependent yaw misalignment standard deviation values in the wind direction variability model, but we approximate the standard deviation as σγ=5.2∘ for all wind speeds greater than 8 m/s.

When analyzing the yaw offsets during the wake-steering experiment in Sect.  as well as the change in energy from wake steering predicted by FLORIS in Sect. , the following nomenclature will be used to distinguish different methods for determining the yaw offsets:

Measured offsets are the yaw offsets measured using SMV6's nacelle wind vane or the WindCube Nacelle lidar.

To improve the likelihood that observed differences in power production for the baseline and the controlled periods are caused by wake steering rather than abnormal turbine operation, the data are filtered using the following steps. First, periods for which the reference wind speed (which will be discussed in Sect. ) is less than 4 m/s are removed because relatively few samples were collected for these wind speeds (see Fig. ) and little energy production is expected during these conditions (as shown in Fig. ). Periods with known derating, curtailment, or other forced downtime (aside from periods with wind speeds below the cut-in speed) are then removed from the data set by examining the wind turbine status codes. Next, any remaining periods with anomalous power production for any of the test or reference turbines are removed using power curve filtering functions available in NREL's OpenOA (Open Operational Assessment) software  as follows (see Fig.  for context):

Samples for which the nacelle wind speed measurement is greater than 6 m/s and power is less than 1 % of rated power or greater than 101.5 % of rated power are removed.

When investigating the yaw offsets achieved by the controlled wind turbine, SMV6, in Sect.  or the impact of yaw misalignment on the power production of SMV6 in Sect. , additional filtering steps are performed. First, samples for which SMV6 is generating less than 10 kW are removed. This step ensures that the turbine is operating and capable of responding to yaw offset commands. Next, because the analyses in Sects.  and rely on the WindCube Nacelle lidar measurements, the data are additionally filtered to remove 1 min samples in which the availability of the underlying 4 Hz lidar measurements for any of the four beams is less than 35 %. Note that because of periodic blade blockage, a significantly higher availability threshold would result in too much data removal.

The reference wind direction, wind speed, and power variables are derived from turbines SMV1, SMV2, SMV3, and SMV7. The reference wind directions and wind speeds are used to represent the wind conditions encountered by the test turbines, whereas the reference power acts as an unbiased reference to which the power produced by the test turbines can be compared. Measurements from SMV6 are not used to estimate the wind direction or the wind speed to avoid the potentially confounding effects of yaw misalignment on the estimated values. SMV4 is excluded from the set of reference wind turbines because of potential influences from wake steering, given its close proximity to the test turbines (see Fig. ). Note that the WindCube ground-based lidar is not used to provide reference measurements because of the impact of wakes on measurement accuracy for southerly flow.

The reference wind direction is calculated as the mean wind direction measured by SMV1, SMV2, SMV3, and SMV7 using their nacelle position sensors and wind vanes. Measurements from multiple turbines are averaged to smooth small-scale wind direction variations that might be encountered at particular locations. The reference wind direction is calibrated to true north by first identifying the measured wind direction where the ratio between the mean power produced by SMV5 and SMV6 during baseline periods reaches a minimum, representing the direction where the wake losses suffered by SMV5 reach their peak. Assuming negligible wake deflection relative to the true wind direction during baseline operation, the offset between this observed wind direction and the known direction of alignment between SMV6 and SMV5 is then subtracted from the reference wind direction.

Similarly, the reference wind speed is based on the mean wind speed measured by SMV1, SMV2, SMV3, and SMV7 using nacelle anemometry. We then apply additional steps to estimate the free-stream equivalent wind speed encountered by the test turbines. First, to account for sensor bias, wake effects, and the impact of terrain and surface roughness on local wind conditions (e.g., the forest south of SMV7), a wind direction and wind-speed-dependent transfer function (i.e., a multiplier that is a function of wind direction and wind speed) is applied to the reference wind speeds to remove any bias from the wind speeds measured by SMV6 during baseline operation. This transfer function is estimated as the ratio between the mean wind speed measured by SMV6 and the mean uncorrected reference wind speed for the baseline periods, binned by the reference wind direction (in overlapping 3∘ bins) and uncorrected reference wind speed (in 1 m/s bins). Next, a nacelle transfer function (a wind-speed-dependent multiplier in this case) is applied to estimate the free-stream wind speed from the nacelle-anemometer-based reference wind speed. The nacelle transfer function is calculated as the ratio between the mean wind speed measured by the WindCube Nacelle lidar at a range of 150 m (1.8D) upstream of the rotor and the mean reference wind speed – considering only periods with baseline control and wind directions with free-stream inflow – binned by the reference wind speed in 1 m/s bins.

Last, the reference power is formed by averaging the power production of SMV1, SMV2, SMV3, and SMV7. Following an approach similar to the reference wind speed derivation, a transfer function is applied to the average power produced by the four reference wind turbines to remove any bias from the power generated by SMV6 during baseline operation (e.g., caused by differences in turbine performance, wake effects, or the impact of local terrain and surface roughness). Again, this transfer function is a wind direction and wind-speed-dependent multiplier that is estimated by dividing the data into overlapping 3∘ reference wind direction bins as well as 1 m/s reference wind speed bins, then calculating the ratio between the mean power produced by SMV6 and the mean uncorrected reference power for periods with baseline control.

To quantify uncertainty in the wake-steering metrics presented in Sect.  through Sect. , we provide 95 % confidence intervals to accompany the estimates. Because many of the metrics require complicated calculations, analytic expressions for the confidence intervals are difficult to derive; therefore, we use bootstrapping, wherein the collection of 1 min data samples used to derive a particular metric is randomly resampled with replacement many times to obtain a distribution of the estimates of the metric . From this distribution, which we derive using at least 2000 bootstrap samples, the confidence interval containing 95 % of the estimates is used as a measure of uncertainty. Many of the results presented in Sect.  through Sect.  are shown as a function of wind direction; for these cases, bootstrapping is performed using data from each wind direction bin individually. Similarly, for metrics based on data from both the baseline and the wake-steering periods, the data corresponding to each control period are resampled independently before the final metric is calculated.

The ideal, expected, and measured mean yaw offsets for all baseline and wake-steering control periods are shown in Fig.  as a function of wind direction. As expected, the mean yaw offsets measured by the wind vane with baseline control are close to zero. But the mean offsets measured by the nacelle lidar show a bias of 2–3∘ during baseline periods, suggesting that the wind vane might be poorly calibrated (as will be explored in more detail in Sect. ). During wake-steering control periods, similar mean yaw offsets are measured by the wind vane and nacelle lidar. Whereas the measured yaw offsets fall short of the ideal offsets for wind directions between 209 and 229∘, they are reasonably well represented by the expected yaw offsets using the wind direction variability model. Specifically, the wind direction variability model predicts a reduction in the peak yaw offsets accompanied by the unintended yaw offsets outside of the target wake-steering sector. Although the measured yaw offsets match this predicted trend, they exhibit a higher, more pronounced peak near the wind direction of 210∘. Differences between the expected and measured yaw offsets could be partly explained by biases in the wind vane measurements (see Sect. ), which propagate to the estimated wind direction signal used by the wake-steering controller.

Whereas Fig.  revealed the mean yaw offsets aggregated among all wind conditions, Fig.  highlights the wind speed dependence of the ideal, expected, and measured yaw offsets. The ideal yaw offsets reflect the yaw offset schedule shown in Fig. ; for wind speeds above 10 m/s, the target yaw offsets are gradually phased out, until no offsets are applied when the wind speed reaches 14 m/s. The more spread-out expected yaw offsets show a similar reduction for wind speeds above 10 m/s, but they also vary for lower wind speeds because of the impact of the wind-speed-dependent yaw error on the wind direction variability model, as explained in Sect. . Namely, because the standard deviation of the wind direction uncertainty increases for low wind speeds, the expected yaw offsets reach a lower peak offset and are spread out over a wider wind direction sector as wind speed decreases below 8 m/s.

Despite the large scatter and uncertainty in the measured yaw offsets for certain wind speed bins – caused by the relative lack of data for wind speeds below 8 m/s and above 12 m/s (see Fig. ) as well as the greater wind direction variability for wind speeds below 8 m/s, potentially causing the reference wind direction measurements to poorly represent the wind conditions at SMV6 – Fig.  reveals several trends. First, the positive wind vane bias observed during the baseline control periods, as measured by the nacelle lidar, is apparent for wind speeds below 10 m/s, but it disappears for higher wind speeds, suggesting wind-speed-dependent vane error. Additionally, when wake-steering control is active, the yaw offsets measured by the wind vane and nacelle lidar closely agree for the 8–10 m/s wind speed bin. But relative to the lidar-measured yaw misalignment, the wind vane appears to underestimate the yaw offsets at lower wind speeds and to overestimate the offsets at higher wind speeds.

Focusing on the wind speed dependence of the measured and expected yaw offsets with wake-steering control, Fig.  shows that the measured offsets for wind speeds below 8 m/s generally agree with the expected offsets; because of the high wind direction variability at these wind speeds, the measured offsets are spread among a large wind direction sector. Uncertainty in the reference wind direction measurements stemming from wind direction variability could further contribute to the broadening of the measured offset curves. For wind speeds between 8–10 m/s, the measured yaw offsets closely match the expected offsets. This wind speed bin is also favorable from a measurement perspective because of the large amount of data collected and the relatively low wind direction variability. Within the 10–12 m/s wind speed bin, the measured yaw offsets tend to be higher than predicted, possibly because of lower wind direction variability or larger yaw offsets persisting from operation in lower wind speeds. Last, for wind speeds between 12–14 m/s, measurement uncertainty caused by the relative lack of data obscures the yaw offset trends. But, as expected, the measured offsets are relatively low (the maximum target yaw offset for this wind speed bin is only 5∘).

Given the biases between the yaw offsets measured by the wind vane and the WindCube Nacelle lidar observed in Figs.  and , in this section we estimate wind-speed-dependent transfer functions to correct the vane measurements. These transfer functions are determined by treating the yaw misalignment measured by the lidar as unbiased for all yaw offsets, although we assume that zero-mean lidar measurement errors exist (e.g., from the limitations of the wind field reconstruction based on line-of-sight wind speed measurements). To increase the amount of data that can be analyzed, the transfer functions are determined using reference wind directions up to 270∘. But to reduce the chance that wake effects from SMV7 influence the yaw misalignments measured by the vane or lidar (see Fig. ), only data corresponding to wind directions above 210∘ are included in the analysis.

Corrective wind vane transfer functions are estimated using least-squares regression by fitting a line to the lidar-measured yaw offsets as a function of the wind-vane-measured offsets. Although the actual relationship between the vane-measured and true offsets could be nonlinear, we use a simple linear approximation to reveal (1) the wind vane bias when zero yaw misalignment is reported and (2) the overall degree to which the wind vane tends to underestimate or overestimate the magnitude of the yaw misalignments. The best-fit slope and intercept could then be applied to the wind vane measurements to remove measurement biases; however, applying linear regression to the individual 1 min yaw offset measurements poses a problem because we assume that both the lidar-measured yaw misalignments and the offsets measured by the wind vane contain random measurement errors. When the predictor variable in a linear regression contains measurement errors, the regression slope tends to be underestimated through a process called “regression dilution” . To overcome this challenge, we attempt to remove zero-mean measurement errors by first binning and averaging the yaw offset measurements. Specifically, we bin the measured yaw offsets by a separate reference yaw offset, given by the difference between the reference wind direction and the nacelle position of SMV6 (see Fig. a). With the random measurement errors significantly reduced, the bin-averaged yaw offsets measured by the lidar and wind vane can then be compared to reveal biases in the wind vane measurements. An example of a linear regression applied to the bin-averaged yaw offsets for all wind speeds is provided in Fig. b, revealing an intercept of only 0.2∘ but a slope of ∼0.92, indicating the tendency for the wind vane to overestimate the magnitude of the yaw misalignment.

Scatterplots of the bin-averaged yaw offsets measured by the WindCube Nacelle lidar and the wind vane, along with the corresponding best-fit lines, are provided in Fig.  for different wind speed bins. Beginning with the wind vane bias when the measured yaw misalignment is zero, indicated by the intercept, a positive bias of ∼2∘ is observed for wind speeds below 8 m/s. For the 8–10 m/s wind speed bin, a bias of only 1∘ is observed; however, for wind speeds above 10 m/s, the wind vane measurement contains a negative bias of approximately -3∘. This wind speed dependence of the mean wind vane measurement error has been previously reported by . Despite the measurement bias for individual wind speed bins, it is likely that the wind vane measurements are calibrated to achieve an average measurement bias close to zero, as indicated by the results in Fig. . In contrast to the intercepts, the slopes of the best-fit lines in Fig.  are similar across different wind speeds; slopes between 0.84 and 0.92 are observed, suggesting that the wind vane measurements overestimate the true yaw misalignment by roughly 10 %.

Although we did not apply any corrections to the wind vane signals used by the wake-steering and yaw controllers in this experiment, the identified transfer functions could be used to help ensure that the intended yaw offsets are achieved in future controller implementations. Note that because of the somewhat nonlinear relationship between the vane-measured and true yaw offsets shown in Figs.  and , more sophisticated transfer functions (e.g., higher-order polynomial functions) might be more appropriate than the linear approximations presented here. Further, the amount of wind vane bias likely depends on the wind turbine model.

The overall energy ratios and the change in energy ratio for the baseline and the wake-steering control periods for the downstream turbine, SMV5, are plotted in Fig.  as a function of wind direction, along with 95 % confidence intervals. The measured energy ratios and the change in energy ratio with wake steering are compared to the same metrics based on FLORIS simulations using the three different FLORIS modeling assumptions discussed in Sect. . First, FLORIS estimates of power production are calculated for the observed distribution of wind directions, wind speeds, and yaw offsets measured using SMV6's nacelle wind vane (labeled “measured offsets”). Next, the ideal FLORIS estimates are calculated using the intended yaw offsets for SMV6 as a function of the observed wind direction and wind speed according to the yaw offset schedule shown in Fig.  (labeled “ideal offsets”). Last, the realistic expected energy ratios based on FLORIS are calculated by combining the ideal yaw offsets for SMV6 with the wind direction variability model discussed in Sect.  (labeled “expected offsets”).

As shown in Fig. , improvements in the energy production of SMV5 from wake steering are observed for wind directions from roughly 205–225∘, with a peak gain of nearly 0.15 (i.e., 15 % of the average energy production of the unwaked reference turbines) at 213∘. The measured energy gains are generally greater than the FLORIS estimates based on the measured and expected yaw offsets, but, as anticipated, they are lower than the FLORIS gains using the ideal offsets. A loss in energy is observed in the 203∘ wind direction bin, as predicted to a lesser extent using the expected yaw offsets based on the wind direction variability model. Because of unintended yaw offsets for wind directions below 208∘ (see Fig. ) the wake of SMV6 can potentially be redirected back toward SMV5, causing a loss in energy production (i.e., “wrong-way steering”). Note that although a similar loss in power for wind directions immediately below the intended wake-steering sector was observed in a previous experiment , the power loss shown in Fig.  coincides with a wind direction bin where the baseline energy production appears anomalously high; therefore, the loss could potentially be an artifact of significantly different operating conditions experienced by the wind turbines during the baseline and the controlled periods. Finally, as evidenced by energy ratios greater than one for both the baseline and the controlled periods, a slight speedup effect at the edge of the wake likely exists for wind directions between approximately 221–233∘.

The energy ratios for the upstream turbine, SMV6, shown in Fig. , reveal energy losses from the yaw misalignment of up to ∼ 5 % when wake steering is active. Losses are observed for wind directions from 199–237∘, corresponding to the sector where nonzero-mean yaw offsets are observed, as shown in Fig. . The greatest energy losses occur between 211–217∘, near where the highest measured yaw offsets occur. The measured reductions in energy production are roughly in line with the FLORIS losses using the expected yaw offsets based on the wind direction variability model, but they are generally lower than the FLORIS losses using the ideal yaw offsets; however, significant measurement uncertainty resulting from the relatively small changes in energy production precludes a direct comparison between the observed and the modeled energy losses as a function of wind direction.

To reveal the net impact of wake steering on energy production as a function of wind direction, the energy ratios for the baseline and the controlled periods along with the change in energy ratio for the average power produced by SMV5 and SMV6 are provided in Fig. . Despite the energy loss at SMV6 from the yaw misalignment, a net increase in energy production of up to 3 %–5 % of the energy produced by the unwaked reference turbines is measured for wind directions from 205–225∘. The increases in energy production in this sector are generally greater than the FLORIS gains using both the expected and the ideal yaw offsets. As predicted by FLORIS using the expected yaw offsets based on the wind direction variability model, losses are observed near 207∘ – possibly because of unintentional wrong-way steering – and above 225∘. In the latter case, a minor loss in energy is predicted by FLORIS because the gains at SMV5 are not large enough to outweigh the loss in energy from the yaw misalignment at SMV6.

Because increases in energy production are observed for the combined upstream and downstream turbines for most wind directions, as shown in Fig. , but losses are measured as well, we assess the net impact of wake steering on energy production over the entire wind direction sector from 195–241∘. We quantify the net impact by estimating the percentage of wake losses in the sector that are reduced by wake steering, as discussed by . Wake losses are calculated for the baseline and the controlled periods separately by first binning the difference between the reference power and the average power produced by SMV5 and SMV6 by wind direction (in 2∘ bins) and wind speed (in 1 m/s bins). Next, the power losses are weighted by the fraction of the total samples contained in each bin, including the baseline and the controlled periods. The weighted power losses are then summed to determine the average wake losses over the entire wind direction sector. Using this method, wake steering was found to reduce wake losses by 5.6 % but with a large 95 % confidence interval of -0.4 % to 11.2 %, estimated through bootstrapping.

The changes in the energy ratios with wake steering for wind turbines SMV5 and SMV6 combined for different wind speed bins are provided in Fig. , followed by the wind-speed-dependent energy ratio changes for the downstream turbine, SMV5, and upstream turbine, SMV6, separately, in Figs.  and , respectively. For reference, the energy ratios for the combined upstream and downstream wind turbines during the baseline and the controlled periods are provided in Fig.  in Appendix . Note that because fewer data samples comprise the energy ratios for the individual wind speed bins, the uncertainty is larger than that of the overall energy ratios shown in Sect. . Nevertheless, trends in wake-steering performance as a function of wind speed can be observed.

Mixed wake-steering performance is observed for wind speeds below 8 m/s, as shown in Fig. . For the 4–6 m/s wind speed bin, significantly larger increases in energy production are achieved with wake steering than predicted by FLORIS, with energy ratios increasing by as much as 0.1–0.3 for wind directions from 195–233∘. The large energy gains for wind speeds below 6 m/s are potentially caused by wake steering preventing SMV5 from shutting down by allowing higher velocity inflow to interact with the rotor, as discussed by . As shown in Figs.  and , however, significant energy improvements are observed for the upstream wind turbine, SMV6, in addition to SMV5 for the 4–6 m/s wind speed bin; therefore, additional sources of the apparent energy improvement for this wind speed bin might exist. For wind speeds between 6–8 m/s, wake steering appears to cause little net change in energy production for the combined wind turbines despite the large energy gains predicted using FLORIS with all three yaw offset calculation methods. Specifically, as illustrated in Figs.  and , lower-than-predicted energy gains at SMV5 are roughly canceled by losses at SMV6. Poor wake-steering performance for wind speeds between 6–8 m/s is likely related to relatively high wind direction variability, as explained in Sect. .

As revealed in Fig. , significant energy gains from wake steering are observed for wind speeds from 8–12 m/s. For the 8–10 m/s wind speed bin, the energy improvement is roughly in line with the energy gains predicted using FLORIS with the expected yaw offsets. Notably, large gains at SMV5, shown in Fig. , outweigh the relatively high losses at SMV6, as illustrated in Fig. . Note that the individual changes in energy ratio from wake steering for SMV5 and SMV6 closely match the FLORIS-based predictions using the expected yaw offsets for this wind speed bin as well. For the 10–12 m/s wind speed bin, the energy gains at SMV5 are slightly greater than the improvements predicted using FLORIS with all three yaw offset calculation methods, yet very little change in energy is observed at SMV6 from yaw misalignment. Consequently, a greater-than-predicted net energy gain is observed for the wind turbine pair.

Last, the performance of wake steering for the 12–14 m/s wind speed bin is unclear based on the data collected. Figure  suggests that energy ratio increases of up to 0.1 are achieved for wind directions between 213–221∘. But these gains appear to be largely canceled by losses for wind directions from 201–211∘, which could be caused by wrong-way steering. On the other hand, the target yaw offsets are limited to a maximum of 5∘ for wind speeds between 12–14 m/s; consequently, the FLORIS predictions show very little change in energy production across the wind direction sector investigated. Further, because the yaw controller is less responsive to small yaw offset commands, it might not be able to effectively implement the intended offsets of 5∘ or less. Finally, the large variations in the observed changes in energy ratio for different wind directions could be caused by the relative lack of data collected for wind speeds above 10 m/s, as indicated in Fig. b.

As discussed in Sect. , above-average wind speeds were observed during the experiment period because of the typically stronger wind resource at the site in the winter. To estimate the expected impact of wake steering on energy production during a typical year, we compute long-term corrected energy ratios for the baseline and the wake-steering periods based on the long-term wind rose frequencies shown in Fig. . The calculation of the long-term corrected energy ratio requires only a slight modification to the energy ratio definition in Eq. (); instead of weighting the mean power for the test and reference turbines in a particular 2∘ × 1 m/s wind direction and wind speed bin by the total number of samples measured in that bin, we weight the mean power by the long-term frequency of occurrence of the wind conditions within the bin.

Using the modified energy ratio calculations, the long-term corrected energy ratios for the baseline and the wake-steering periods together with the change in energy ratio with wake steering for SMV5 and SMV6 combined are provided in Fig. . Energy gains of up to 3 %–5 % of free-stream energy production are measured for wind directions between 207–225∘, generally matching the predicted energy improvements using FLORIS with the expected yaw offsets. Slight losses in energy production are observed for wind directions below 207∘ and above 225∘, as also predicted by FLORIS using the expected offsets. Overall, the long-term corrected change in energy ratio from wake steering follows the same trends as the change in energy ratio for the experiment period shown in Fig. ; however, the peak gains and losses are less extreme when the long-term corrected wind rose is applied.

To estimate the long-term net impact of wake steering for the combined upstream and downstream wind turbines over the entire wind direction sector from 195–241∘, we repeat the analysis of the wake loss reduction from wake steering introduced in Sect. . But similar to the method of the long-term corrected energy ratio, we modify the procedure for calculating the wake losses by weighting the difference between the mean power of the reference and test turbines in each 2∘ × 1 m/s wind direction and wind speed bin by the long-term frequencies of occurrence of the wind conditions within the bin rather than by the frequencies observed during the experiment period. Based on this procedure, we estimate a long-term corrected wake loss reduction of 9.3 % from wake steering but with a large 95 % confidence interval spanning from 1.3 %–16.2 %. Note that this confidence interval is estimated through bootstrapping by randomly resampling the data used to calculate the mean power values, as explained in Sect. ; however, the long-term wind condition frequencies are fixed at the values determined from the long-term wind rose shown in Fig. . The expected long-term corrected wake loss reduction from wake steering is greater than the value of 5.6 % calculated for the experiment period in part because wind speeds from 4–6 m/s occur more frequently in the long-term wind rose. Although the wind turbines' power production is relatively low for this wind speed bin, the large energy improvements from wake steering contribute to a significantly higher reduction in overall wake losses.