# A Numerical Study of the Effects of Wind Direction on Turbine Wakes and Power Losses in a Large Wind Farm

^{*}

## Abstract

**:**

## 1. Introduction

_{wind}= 270° (from the west), 222° and 312°. For these wind directions, the wind is parallel to different “lines” of wind turbines, leading to what is commonly called full-wake conditions, with different streamwise distances between consecutive turbines (7.0-, 9.3- and 10.5-times the rotor diameter, respectively). In general, the longer that distance, the smaller the wake-induced power losses, particularly in the first rows of turbines. Barthelmie et al. [5] reported power data measured at the Horns Rev and the Nysted wind farms for seven wind directions, and showed an obvious increase in the power output from the downwind turbines when the incoming wind angle departs from the full-wake condition. Although all the above-mentioned studies provide valuable insights about the effects of wind direction on power deficits in wind farms, how these effects are related to the turbulent wake flow structure within the farms for a wide range of wind directions (including full-wake and partial-wake conditions) is not yet fully understood.

## 2. Numerical Simulations

#### 2.1. LES Framework

#### 2.2. Case Description

^{2}. Each turbine has a hub height H

_{hub}= 70 m (above sea level) and a rotor diameter d = 80 m. The wind farm layout has a rhomboid shape with a minimum spacing of seven rotor diameters between two consecutive turbines within each of its ten rows (approximately seven degrees turned from north-south) and eight columns (see Figure 1).

**Figure 1.**Schematic of the Horns Rev wind farm layout, consisting of 80 turbines arranged in ten rows (denoted as “R”) and eight columns (denoted as “C”). Distances are normalized by the turbine rotor diameter d = 80 m. Solid lines with arrows indicate selected wind directions involving full-wake conditions with different streamwise distances between consecutive wind turbines: (

**a**) θ

_{wind}= 312°, 295°, 288°, 284°, 270°; and (

**b**) θ

_{wind}= 270°, 256°, 251°, 242°, 222°.

_{z}= 1045.9 m in all the simulations. A constant streamwise pressure gradient is used to drive the flow within the boundary layer, which has a height of δ = 500 m. This value is consistent with the boundary-layer depth observed by Peña and Gryning [23] at Horns Rev. The domain of size (L

_{x}, L

_{y}, L

_{z}) is divided uniformly into N

_{x}× N

_{y}× N

_{z}= 640 × 640 × 128 grid points, with a grid resolution of Δ

_{x}, Δ

_{y}, and Δ

_{z}in the streamwise, spanwise and vertical directions, respectively. Table 1 summarizes the numerical set-up for some selected simulation cases. In this table, and throughout the paper, S

_{x}denotes the streamwise distance between consecutive wind turbines in the direction of the wind (under full-wake conditions), while S

_{y}refers to the spanwise distance between lines of wind turbines. It should be mentioned that in all the simulations, the turbine rotor diameter is covered by at least five points in the spanwise direction and nine points in the vertical direction. Based on previous resolution sensitivity studies presented in [19,20], the grid resolution over the rotor considered here is well suited for our LES framework to account for the most significant characteristics of wind-turbine wakes. The dynamic ADM-R requires the specification of the turbine blade airfoil data (including chord length, twist angle, lift and drag coefficients) and the turbine torque-speed relation. Here, we use the same turbine data (see Figure 2) that were recently used to validate the LES framework through the comparison with observed turbine power data [12].

**Table 1.**Numerical set-up for neutrally-stratified atmospheric boundary layer (ABL) simulations with different incoming wind directions under full-wake conditions.

Case | θ_{wind} (°) | S_{x} (d) | S_{y} (d) | L_{x} (m) | L_{y} (m) | L_{z} (m) | Δ_{x} (m) | Δ_{y} (m) | Δ_{z} (m) |
---|---|---|---|---|---|---|---|---|---|

HR-173 | 173 | 7.00 | 6.95 | 14,537.1 | 9,614.5 | 1,045.9 | 22.71 | 15.02 | 8.24 |

HR-222 | 222 | 9.28 | 5.24 | 13,194.9 | 9,585.8 | 1,045.9 | 20.62 | 14.98 | 8.24 |

HR-242 | 242 | 15.00 | 3.24 | 15,089.4 | 9,302.7 | 1,045.9 | 23.58 | 14.54 | 8.24 |

HR-251 | 251 | 21.30 | 2.28 | 15,400.8 | 8,988.0 | 1,045.9 | 24.06 | 14.04 | 8.24 |

HR-256 | 256 | 28.90 | 1.68 | 14,467.1 | 9,873.4 | 1,045.9 | 22.60 | 14.43 | 8.24 |

HR-270 | 270 | 7.00 | 6.95 | 14,933.3 | 9,881.6 | 1,045.9 | 23.33 | 15.44 | 8.24 |

HR-284 | 284 | 28.90 | 1.68 | 14,518.3 | 9,323.3 | 1,045.9 | 22.68 | 14.57 | 8.24 |

HR-288 | 288 | 22.90 | 2.13 | 14,850.0 | 9,049.9 | 1,045.9 | 23.20 | 14.14 | 8.24 |

HR-295 | 295 | 16.30 | 2.99 | 14,756.0 | 9,491.9 | 1,045.9 | 23.06 | 14.83 | 8.24 |

HR-312 | 312 | 10.48 | 4.64 | 16,775.2 | 9,134.9 | 1,045.9 | 26.21 | 14.27 | 8.24 |

HR-353 | 353 | 7.00 | 6.95 | 14,537.1 | 9,361.5 | 1,045.9 | 22.71 | 14.63 | 8.24 |

**Figure 2.**(

**a**) Distributions of the chord length, c(r/R), and twist angle, θ(r/R), along the normalized rotor radius, r/R; (

**b**) measured and simulated power curves and thrust coefficient curves for the Vestas V-80 2 MW turbine; (

**c**) relationship between the total shaft torque and the rotor angular velocity; and (

**d**) distributions of lift and drag coefficients with respect to the angle of attack.

_{∗}= 0.442 m s

^{−1}and an aerodynamic roughness length z

_{0}= 0.05 m. The turbulence intensity (TI) is computed as follows:

^{−1}and a turbulence intensity of 7.7% at the same height. The aerodynamic roughness length used in this study was chosen to provide inflow characteristics that are in good agreement with the observed data at Horns Rev (TI < 8% at hub height when ${\overline{u}}_{hub}$ ≈ 8 m s

^{−1}) [4,5]. It should be noted that the LES lower boundary condition, based on Monin-Obukhov similarity, is likely to introduce some errors, as it is not able to capture all the complex interactions between the wind and the ocean wave field below. All the wind farm simulations were run for a period (physical time) of 100 min, and the flow statistics were computed during the last 60 min, which guarantees quasi-steady flow conditions, as well as the statistical convergence of the results presented in the next section.

**Figure 3.**Vertical profiles of the mean streamwise wind velocity, turbulence intensity (TI) and total (resolved plus subgrid) streamwise turbulent shear stress of the inflow used for the wind farm simulations. The inflow was obtained with a precursor simulation without turbines.

## 3. Simulation Results

_{wind}(from 173° to 353°), including full-wake and partial-wake conditions, is shown in Figure 4. Here, and throughout this paper, the simulated power is normalized by the power from an equivalent number of stand-alone wind turbines exposed to the same incoming wind condition. From Figure 4, it is clear that the minimum wind farm power occurs for the wind directions of 173°, 270° and 353°, which correspond to full-wake conditions with the shortest possible streamwise distance (S

_{x}= 7.0 d) between consecutive turbines. The farm power output is slightly lower in the 270° case because of the larger number of waked wind turbines in that case (72, compared with 70 for the 173° and 353° cases). Also evident in that figure is the presence of several local minima of power at other wind direction angles [e.g., 201°, 222°, 242° within the south-west (SW) 90° wind sector and 295°, 312°, 328° within the north-west (NW) 90° wind sector], associated with other full-wake conditions with different streamwise distances (S

_{x}) between consecutive downwind turbines. Some of those wind angles are also illustrated in Figure 1. It is also interesting to note that relatively higher power outputs are obtained for partial-wake cases in the SW wind sector, compared with those in the NW wind sector. This is due to the fact that, due to its geometry, the wind farm offers a larger frontal area to the wind coming from the SW sector. This larger wind farm frontal area can accommodate a larger number of unwaked turbines, which yields higher power outputs, for specific wind direction angles.

**Figure 4.**Distribution of the normalized wind farm power output obtained with large-eddy simulations (LES) for a wide range of wind directions, from 173° to 353°.

_{x}) between each wind turbine and its immediately downwind (fully-waked) neighbor. In particular, the streamwise turbine spacing is 7.0 d, 28.9 d, 22.9 d, 16.3 d and 10.5 d for the 270°, 284°, 288°, 295° and 312° cases, respectively. It should be noted that, due to the fixed turbine siting density, an increase in the streamwise turbine spacing leads to a decrease in the spanwise separation between the turbine wake axes. This, in turn, increases the lateral (partial-wake) interactions and, ultimately, the power losses. This explains why, for relatively large S

_{x}values (larger than about 20.0 d), the differences in power output between full-wake conditions and partial-wake conditions become very small, as depicted in Figure 4.

**Figure 5.**Spatial distribution of the normalized simulated power output for the five wind angles shown in Figure 1a: (

**a**) power output as a function of turbine row (averaged over columns C2, C3 and C4); and (

**b**) power output distribution along the wind direction lines (lines of turbines) shown in Figure 1a. The symbols indicate the position of the wind turbines. The streamwise distance is normalized by the turbine rotor diameter d = 80 m.

**Figure 6.**Contour plot of the time-averaged streamwise velocity on a horizontal plane at hub level for different incoming wind directions of (

**a**) 270°; (

**b**) 284°; (

**c**) 295°; and (

**d**) 312°. Distances are normalized by the turbine rotor diameter d = 80 m.

**Figure 7.**Contour plot of the time-averaged vertical velocity on a horizontal plane at hub level for different incoming wind directions of (

**a**) 270°; (

**b**) 284°; (

**c**) 295°; and (

**d**) 312°.

**Figure 8.**Contour plot of the streamwise turbulence intensity (resolved part) on a horizontal plane at hub level for different incoming wind directions of (

**a**) 270°; (

**b**) 284°; (

**c**) 295°; and (

**d**) 312°.

**Figure 9.**Contour plot of the horizontal momentum flux [the resolved part plus the subgrid-scale (SGS) part] on a horizontal plane at hub level for different incoming wind directions of (

**a**) 270°; (

**b**) 284°; (

**c**) 295°; and (

**d**) 312°.

_{x}= 7.0 d for θ

_{wind}= 270°); it also corresponds to the case of the largest spanwise spacing between lines of turbines (also S

_{y}= 7.0 d). Here, and throughout this paper, the term ‘line of turbines’ is used to refer to a straight line parallel to the wind direction that connects the hubs of a series of wind turbines under full-wake conditions. It is important to note that, with a small change of wind direction of 14° (from 270° to 284°), the power outputs from all the turbines inside the farm increase substantially, leading to an increment of 43% in normalized wind farm power output (from 0.60 to 0.86), as shown in Figure 4. There are three reasons that explain this behavior:

- The relatively short streamwise distance (S
_{x}= 7.0 d) between consecutive wind turbines in the 270° case limits the wake recovery, leading to relatively large velocity deficits at the turbine locations (Figure 6a) and, in turn, large power deficits (see Figure 5) compared with the other wind direction cases. - The relatively large spanwise distance (S
_{y}= 7.0 d) between lines of turbines in the 270° case leaves “high-speed channels” (Figure 6a), inside which a large fraction of the kinetic energy of the incoming wind is convected through the wind farm without being extracted by the turbines. For larger wind direction angles (particularly, 284°), those channels do not exist or are much less evident, thus allowing the wind farm to extract more of the kinetic energy available in the incoming wind. - The wind direction also determines at which row number the turbines begin to be affected by the upwind turbine wakes, which, in turn, defines the total number of waked and unwaked turbines. For example, in the 270° case, a power deficit of about 50% is found already at the second turbine row and affects 90% of the wind farm, while for the 284° case, the first four turbine rows, which make up about 40% of the Horns Rev wind farm, are minimally affected by turbine wakes.

_{x}, increases from 7.0 d for 270° to 28.9 d for 284°. In general, the power output from the turbines keeps decreasing along the wind direction. Unlike the 270° case, for which the power output drops sharply at the second turbine row due to the short wake recovery distance (as shown in Figure 6), and remains nearly constant further downwind, the power reduction is more progressive as one moves downwind inside the wind farm for the other cases with longer streamwise distances between turbines S

_{x}(particularly for S

_{x}> 15 d). This can be explained by the fact that the smaller spanwise distance between lines of turbines, S

_{y}, facilitates lateral wake interactions. The accumulated wakes become wider as they expand downwind (see Figure 6), leading to increasingly larger partial-wake effects and, thus, power losses when they interact laterally with the neighboring lines of turbines. In order to further illustrate the effect of wind direction on the turbine-wake structure and associated power losses, the supplementary material contains an animation showing the wind farm power output and the mean streamwise velocity at hub height for all the 67 simulated wind directions (see supplementary material).

_{u}, and the mean velocity of the undisturbed inflow at hub height, ${\overline{u}}_{hub}$. From these results, it is clear that, as expected, turbulence intensity levels are enhanced inside the farm with respect to the incoming wind conditions for all cases. Different wind directions, however, lead to different levels of turbulence enhancement. In particular, the largest turbulence levels at the location of the downwind turbines are found for the 270° case, and they become smaller as the streamwise distance between consecutive turbines increases. Therefore, not only the most severe full-wake cases lead to the strongest velocity deficit and power losses in the wind farm, but they also yield the highest turbulence levels at turbine level and, consequently, the largest fatigue loads on the downwind turbines.

_{x}, which is consistent with the larger turbulence intensities reported in Figure 8.

**Figure 10.**Zoom-in of the total normalized farm power output shown in Figure 4 for a wind sector between 250° and 290°.

**Figure 11.**Contour plots of the time-averaged streamwise velocity $\overline{u}$ (m s

^{−1}) around the third, fourth and fifth turbine columns (C3, C4 and C5 in Figure 1) at hub height for wind angles of (

**a**) θ

_{wind}= 280° and (

**b**) θ

_{wind}= 284°. The black lines show the edge of the wakes (defined using the 1% velocity deficit from the undisturbed flow).

**Figure 12.**Schematic of the wake from a wind turbine from the first row for two selected wind direction angles: 270°, corresponding to the full wake condition (top); and the minimum angle (270° + ϕ, where ϕ is 14° in this case) required for the wake to have no effect on the next turbine within the same column. The blue dot-dashed line represents the wake edge, and r

_{w}is the radius of the wake.

_{w}= 1.2 d (see Figure 12) at the position of the rotor plane of the second wind turbine at a distance of approximately 6.9 d downwind. In those studies, the turbine wake edge was defined based on the position at which the wake velocity deficit reaches 1% of the incoming wind velocity. That lateral position corresponds to the location of the turbine edge in the 284° case. This explains why the power output from rows 2 and 3 is larger for the 284° wind case compared with the 280° case, for which those rows are affected by partial-wake effects (see Figure 13). Interestingly, however, this trend is reversed in rows 4–6 (see Figure 13). This is due to the fact that the larger wind angle increases the effect of the turbine wakes on the neighbor turbine columns, which are affected after the fourth row for θ

_{wind}= 284°, while this effect is not evident until the sixth row for the 280° case (see also Figure 11). The combination of these two opposing wake effects (on the downwind turbines within the same row vs. on the neighboring turbine columns) makes that, for the layout and size of this wind farm, the two effects cancel out.

**Figure 13.**Normalized power output as a function of turbine row for wind angles of θ

_{wind}= 280° and θ

_{wind}= 284°. The results are averaged over the seven turbine columns (C1 to C7 in Figure 1).

## 4. Conclusions

_{x}= 7.0 d) between turbines in the direction of the wind. Moreover, in that case, part of the kinetic energy in the incoming wind passes through the wind farm through the large “channels” that separate the turbine columns, thus not being used by the wind turbines. Other angles also leading to full-wake conditions yield much smaller power losses, because the distance that the wakes have to recover before encountering the next downwind turbine is longer and the number of unwaked turbines is also larger. Furthermore, in these cases, like in many partial-wake condition cases, a reduction in the lateral distance between turbine lines (S

_{y}) leads to a reduction of the high-speed (untapped) channels and an increase in lateral wake interactions. These interactions, which become larger as the accumulated wakes expand downwind, lead to a more progressive increase of the power deficit inside the farm, compared with the full-wake condition cases with small S

_{x}(and correspondingly, large S

_{y}), for which lateral wake interactions are minimal and the power deficit reaches its maximum value already at the second row of turbines.

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Porté-Agel, F.; Wu, Y.-T.; Chen, C.-H.
A Numerical Study of the Effects of Wind Direction on Turbine Wakes and Power Losses in a Large Wind Farm. *Energies* **2013**, *6*, 5297-5313.
https://doi.org/10.3390/en6105297

**AMA Style**

Porté-Agel F, Wu Y-T, Chen C-H.
A Numerical Study of the Effects of Wind Direction on Turbine Wakes and Power Losses in a Large Wind Farm. *Energies*. 2013; 6(10):5297-5313.
https://doi.org/10.3390/en6105297

**Chicago/Turabian Style**

Porté-Agel, Fernando, Yu-Ting Wu, and Chang-Hung Chen.
2013. "A Numerical Study of the Effects of Wind Direction on Turbine Wakes and Power Losses in a Large Wind Farm" *Energies* 6, no. 10: 5297-5313.
https://doi.org/10.3390/en6105297