The numerical setup consists of five turbines aligned in a row perpendicular to the prevailing wind, similar to power performance test sites. In addition to the case of the wind approaching perpendicular to the row, θ=0∘, inflow angles between 5 and 45∘ are considered, as shown in Fig. . The modelled turbine is the NREL 5 MW with a diameter of 126 m , but any other turbine type could have been used as blockage is largely independent of turbine design . The effect of the turbine spacing (L) is evaluated by considering three different values: L=1.8, 2 and 3 D. The 1.8 D case is tested to evaluate whether the global-blockage effect changes dramatically for a spacing lower than 2 D, which is the lowest value currently accepted by the IEC standard. To highlight the effects of the rotors on the power output, the inflow is simplified as much as possible. Therefore, the inflow is uniform without turbulence, purely neutral (no buoyancy) and assumed as time invariant. Additionally, the ground is not modelled, so the flow is completely unconstrained. Virtual measurements from meteorological towers are simulated by extracting point-wise velocity values in front of the rotors at hub height and at 2, 2.5 and 3 D upstream, which are the distances prescribed in the IEC standard. Lidar measurements are simulated with a two-beam pulsed lidar mounted on the nacelle and pointing upstream with a half-opening angle of 15∘. The lidar is characterised by a range-gate length of 38.4 m and a full width at half maximum (FWHM) of 24.75 m. More details about the lidar simulator can be found in . As can be seen in Fig. , the mast measurements are taken at fixed locations, while the nacelle-mounted lidars yaw together with the rotors and their point of measurement changes with θ.

The numerical setup adopted here is the same as used and described in detail by , so here we will only briefly describe the simulation setup. All simulations are performed using the in-house incompressible finite-volume flow solver EllipSys3D . The simulations are carried out with steady-state Reynolds-averaged Navier–Stokes (RANS) equations using the k-ω shear-stress transport (SST) turbulence model by . The numerical domain is an ellipse-shaped cylinder with (Lx,Ly,Lz)=(95,84,25 D), where Lx and Ly denote the major and minor axes of the ellipse and Lz is the height of the cylinder. The turbines are placed as shown in Fig.  with T3 located in the centre of the domain. In the vicinity of the turbines, the grid cells are cubic with a side length of D/32 within an inner box of dimensions (15,4,2 D). From there, the mesh grows hyperbolically outwards. The turbines are modelled as actuator discs using the airfoil and blade data from the NREL 5 MW turbine . In contrast to , who prescribed a constant rotational speed and blade pitch angle, we instead use a controller that set these based on the velocity averaged over the rotor area at the rotor position . The accuracy of the computational fluid dynamics (CFD) model (numerical setup and actuator disc) over the wind turbine induction zone was validated using measurements from three lidars .

Measurements are available for a period of approximately 21 months from a site consisting of five turbines aligned perpendicularly to the predominant wind direction. The area is flat, and the surface characteristics within the analysed directions are the same for each of the turbines in the row (and rather homogeneous). The name of the site can not be disclosed due to proprietary reasons, but the layout is very similar to that in Fig. . The available dataset comprises the operational data from a turbine on one edge of the row (T1) together with measurements from “its power-performance” meteorological mast and a ground-based wind lidar aligned with the turbine along the predominant wind direction at distances of 2.3 and 2.5 D, respectively. The lidar is a WindCube WLS7 from Vaisala Leosphere. Additionally, the data include the operational status of the turbines T3–T5 (T2 operation and status are unknown). The five turbines are placed with a mutual distance L=2.3 D, with D being the diameter of T1. Although we do not know specifics on the turbines standing on the other four positions, considering the size of modern wind turbines, the spacing is likely to be lower than 3 D when normalised with the rotor diameters of the other turbines at the site.

Data from T1 are used to validate the numerical results. If the asymmetry due to wake rotation is neglected, the turbine can represent either turbine T1 or T5 from the simulations, as it is either the most upwind or downwind turbine of the row for θ>0∘ and θ<0∘, respectively.

Measurements from both the turbine and the mast are sampled at 35 Hz, while the wind lidar provides measurements at 11 different heights every 4 s, covering a vertical distance from -0.4 to +0.85 D relative to the wind turbine hub height. The analysis is performed by considering 10 min means for all examined variables.

In Fig. , the power output P of the five turbines is normalised by that of the isolated turbine Pref under the same inflow conditions. The normalised power varies with the free-stream velocity U∞ for different values of θ. At U∞≈8 m s-1, the turbine is within the region of the power curve where the turbine controller keeps a constant tip speed ratio and an optimal power output, i.e. a constant power coefficient (CP) and thrust coefficient (CT). We also show results for 7 and 11 m s-1 as they are the first two integer values out of this region . The highest variation from the reference case is found for the side turbines (T1 and T5) with an inflow angle θ=45∘. Although not shown, the difference between the power output of the five turbines and the isolated turbine decreases for 12 m s-1 (above the CP-constant interval), while it increases for 7 m s-1. The power output increases for all five turbines when θ=0∘, with the highest gain, when compared to the reference case, for T3 and U∞=7 m s-1 reaching nearly 2 %.

Figure shows the normalised power output for cases with the same free-stream velocity (8 m s-1) and a number of turbine inter-spacings (1.8, 2 and 3 D). The normalised power varies with turbine inter-spacing for all five turbines and decreases the larger the turbine inter-spacing. For the largest turbine inter-spacing (3 D), the normalised power is still larger than 2 % for the side turbines when θ=30 and 45∘. The reduction of the turbine inter-spacing from 2 to 1.8 D results in small variations in the power output; the highest variation (0.75 %) is for turbine T5 when θ=45∘.

The higher power output of the five turbines relative to that of the isolated case cannot be explained with upstream velocity measurements. The upstream induction on the row of turbines is higher than that of the single turbine, so that there is a higher velocity reduction in front of the rotors, as expected because of the global-blockage effect. This is shown in Fig. , where the vertical velocity profile in front of T1 and T3 is compared to that of an isolated turbine for U∞=8 m s-1 and θ=0∘. Lower velocities correspond to higher power production, with T3 producing the most despite the lowest incoming wind speed at both 2 and 1 D. It is only very close to the rotor (closer than 0.2 D) that the incoming wind speed in front of T3 is higher than in the isolated case. Although not shown, the same trend is found for all values of U∞ and θ.

already showed that these counter-intuitive power deviations of the turbines on the row relate to the downstream induced velocity caused by the neighbouring turbines. Particularly, a positive downstream induced velocity results in faster advection of the wake and lower induction upstream of the turbine. The “local” blockage at the rotor is thus lower compared to the isolated case, which results in higher power output. Likewise, a negative downstream induced velocity results in lower power output compared to the isolated case.

Figure shows the velocity induced by the isolated turbine at T3 along the rotor axis at the locations T2 and T4 (but without other rotors than T3) for θ=45∘ and U∞=7, 8 and 11 m s-1. For -1.3⪅yi⪅1.5, the induction is positive along y4 and negative along y2. This explains the results in Fig. , where the downstream turbines (T4 and T5) produce more than the upstream ones (T1 and T2). It should be noted that a velocity increase of ≈1 % at the rotor (case with U∞=8 m s-1 in Fig. ) is not negligible and it could definitely be enough to explain the power variations observed in Figs.  and . For example, assuming the same air density and power coefficient values, such a velocity increase can result in a power increase of ≈3 %. Additionally, the magnitude of the induction decreases the higher the wind speed, also in agreement with the results in Fig. , where the power variation decreases for higher wind speeds. Furthermore, as shown in Fig. , the magnitude of the induced velocities varies with the turbine inter-spacing so that stronger inductions are observed for an inter-spacing of 2 D compared to those of the 3 D case, which is in agreement with the power variations in Fig. . The variation in induced velocities with both turbine inter-spacing and wind speed further confirms the relation between downstream induced velocities and power variations.

From the previous results, one might expect biases in power performance measurements carried out for non-isolated turbines. Particularly, we would like to quantify whether the effects shown for specific θ values in Figs.  and  cancel out when averaging over an inflow sector typical for power performance measurements.

A series of simulations are performed for both the reference case and the turbine row with an inter-spacing of 2 D for a number of U∞ and θ values. The free-stream velocity varies from 7 to 11 m s-1 with a step of 1 m s-1, while θ varies from -45 to +45∘ with a step of 5∘. A normal distribution Nθ(μθ,σθ2) (with μθ and σθ as mean and standard deviation) is assumed for the wind direction, and the mean power output is calculated for each free-stream velocity as P¯=∫P(U∞,θ)Nθ(μθ,σθ2)dθ. The effect of averaging over the whole inflow sector is shown in Fig.  for a distribution given by μθ=0∘ and σθ=41∘. A standard deviation value of 41∘ is chosen in order to get a nearly uniform distribution of wind directions within the interval [-45, 45∘]. Although not shown here, a narrower Gaussian distribution would enhance the increase in power for T3, and the results would not be representative of power performance tests in general, but rather of tests conducted with that specific and narrow wind direction distribution. As illustrated in Fig. , there is a difference with respect to the reference case; the five turbines in the row produce more than in isolation for all values of U∞. Since Nθ(μθ,σθ2) is symmetric and centred in θ=0∘, the power output of T5 is exactly the same as T1, and the same applies to T2 and T4. The central turbine T3 shows the largest increase in power relative to the reference case, with a power gain higher than 1% for most wind speeds. Furthermore, the highest and lowest power variations are observed for U∞=7 and 11 m s-1, respectively. The power variations are nearly constant for free-stream velocities within the range 8–10 m s-1. These results further confirm that the global-blockage-related power variations depend on the power curve region of operation of the wind turbines; they are CT-dependent and consequently steady in the constant-CP region of the power curve, while they decrease for U∞ closer to the rated wind speed and increase for U∞ closer to the cut-in value.

Due to the increase in power and reduction of the upstream wind speed, the differences in power coefficient CP compared to the isolated case are higher than those of the power output. Results for CP are shown in Fig.  for the same case of Fig. . The estimated free-stream velocity U∞ is extracted at hub height and 2.5 D upstream of the rotor by both the virtual met mast and the virtual nacelle-mounted lidar. CPs for the turbines of the row are up to 4 % higher than that of the reference when measuring with a mast and higher when measuring with the nacelle lidar. This is due to the masts measuring at fixed locations, while the nacelle-mounted lidars yaw together with the turbine. The volume-averaging effect of the lidar is considered negligible due to the nearly uniform velocity within the probe volume.

To compute the power variation P(θ) observed at the site, the inflow sector θ=0∘±50∘ is divided into three different intervals θ=0∘±16.5∘, +33∘±16.5∘ and -33∘±16.5∘. These are selected to characterise the three conditions of operation (“upwind”, “downwind” and inflow perpendicular to the row) and obtain the largest possible amount of data within each interval. The wind direction is taken from measurements of a wind vane installed on the met mast 4 m below hub height. Additionally, the data are filtered according to the wind speed measured by the hub height cup anemometer (UHH) and corrected for air density, even though the correction leaves the data nearly unchanged due to the flat and sea level terrain. Only wind speeds within the constant-CP range are considered, as this is also the range providing a constant CT (the manufacturer's CT curve is not available). Therefore, after determining the CP curve of the turbine, the interval UHH=[5.5,8.5] m s-1 is selected.

After selecting the data according to 10 min mean values of both θ and UHH, other meteorological conditions are imposed to both increase the compliance with the numerical setup and avoid biases due to extreme conditions. Conditions of both very low and very high turbulence are filtered out by considering only 10 min intervals where the turbulence intensity at hub height is between 2 % and 10 %. Additionally, thresholds are set for both the wind veer (γ) and the wind direction standard deviation (σθ). Measurements with either γ>10∘ or σθ>10∘ are filtered out. γ is the difference between the 10 min mean wind directions given by the WindCube at heights of -0.4 and 0.85 D relative to hub height.

We also consider only power-law-like wind profiles to avoid biases due to different profiles among different wind directions. For all the 10 min intervals, the power law U(z)/Uref=(z/zref)α is fitted to the WindCube measurements at the 11 different heights via a least-squares fit. Then, the mean absolute error (MAE) between the measured wind speeds and the values estimated by the power law is calculated. Only the profiles providing a MAE lower than 0.03 are taken for the analysis to avoid reducing the amount of data excessively. Additionally, to avoid conditions of very strong shear, profiles with a shear exponent α higher than 0.35 are discarded. Finally, to increase the amount of data, we select all the intervals when at least two of the other four turbines are operating.

Due to the substantial differences between the numerical setup and the real site, the objective of the inter-comparison with the measurements is to evaluate the trends of power variations. Thus, we evaluate the power output for each of the three wind direction bins (θ=0∘±16.5∘, θ=+33∘±16.5∘ and θ=-33∘±16.5∘) and make sure that the same meteorological conditions are in place in all three bins, so that the power differences are mainly explained by the effect of the other four turbines. However, we could have different wind speed distributions among the bins, since the wind speed interval is relatively large (3 m s-1). Therefore, we normalise the 10 min mean power values Pi with the power value derived from the power curve for the related 10 min mean wind speed measured at hub height, resulting in the normalised power values P^. The power curve is derived from the dataset filtered for meteorological conditions, without including the operational status of the other turbines. Different atmospheric stability conditions might be associated with different wind directions. To decrease the effect of stability, data are sampled so that the three bins present the same number of measurements within each interval α=α¯±0.02, for α¯=0.01,0.03,0.05,…,0.33,0.35. This sampling assures that all the inflow sectors have the same distribution of α values, and it results in 534 10 min mean data for each of the three sectors. The distributions for the normalised power P^ are shown in Fig. . Additionally, the sampling for α is repeated for 50 random seeds, and the results are nearly constant (standard deviations lower than 0.1 % of the means), proving that the findings are not affected by the random sampling.

The highest mean power output is observed for θ¯=-33∘ (θ¯ stands for the mean of all the 10 min wind directions), i.e. when the turbine is the most downwind in the row. The lowest power output is observed for θ¯=32∘, when the wind turbine is the most upwind. The comparison between measurements and simulations is shown in Fig. , where the numerical results, indicated as red squares, represent the means of the power distributions obtained for the same wind direction distribution of the measurements and a free-stream velocity of 8 m s-1. To ease the comparison, as the simulations do not correspond to the same turbine, the means and uncertainties are normalised by the mean power output for the central sector (θ¯=1∘). As illustrated, both simulations and observations show the same trend. Also, the measurements show that the differences among mean power values for different inflow angles are larger than the uncertainties, which represent 95 % confidence intervals. The statistical significance of the results is also tested through null hypothesis significance testing, resulting in p values below 0.05 for all the inflow angles.