**2. Wake fundamentals**

Wake influences from one turbine to the next have received the majority of attention in this field of study due to the significant influence this has on performance and reliability. Down‐

stream wake effects are frequently quantified through the use of rotor disc theory and the conservation of linear momentum. The rotor disc refers to the total swept area of the rotor as shown in Figure 1. The expanding wake downstream of the turbine in conjunction with a decrease in wind speed U is also shown.

**Figure 1.** Swept area of wind turbine rotor with expanding wake section [1].

For conservation of mass:

$$
\rho A\_1 \mathcal{U}\_1 = \rho A\_2 \mathcal{U}\_2 \tag{1}
$$

where ρ is the air density, *A* is the cross sectional area, *U1* is the free stream wind speed and *U2* is the wind speed downstream from the turbine. A decrease in wind speed across the ro‐ tor area results in a greater downstream area. From elementary energy conservation princi‐ ples, it can be shown that a high pressure area is formed upstream of the rotor disc and a lower pressure area is formed downstream. This pressure change is due to the work of the rotor blades on the air passing over them. The force of the air on the blades results in an opposing force on the air stream causing a rotation of the air column. This low pressure col‐ umn of rotating air expands as it moves downstream of the turbine and eventually dissi‐ pates as equilibrium is reached with the surrounding airflow [2, 3]. This simplified explanation constitutes what is known as the "wake effect" of a wind turbine [1]. An in‐ crease in downstream turbulence is caused by wake rotation, disruption of the air flow across the rotor blades and the vortices formed at the blade tips. This results in less power being available for subsequent turbines.

The Bernoulli equation can also applied to wind turbine wake analysis. The equation fol‐ lows the concept of conservation of energy:

$$\frac{\rho U\_1^2}{2} + \text{ } p = H \tag{2}$$

where *p* is pressure and *H* is the total energy for the constant streamline. Using the Bernoulli equation and conservation of momentum together the following equation can be developed:

stream wake effects are frequently quantified through the use of rotor disc theory and the conservation of linear momentum. The rotor disc refers to the total swept area of the rotor as shown in Figure 1. The expanding wake downstream of the turbine in conjunction with a

U2 U1

*AU AU* 11 22

 r

where ρ is the air density, *A* is the cross sectional area, *U1* is the free stream wind speed and *U2* is the wind speed downstream from the turbine. A decrease in wind speed across the ro‐ tor area results in a greater downstream area. From elementary energy conservation princi‐ ples, it can be shown that a high pressure area is formed upstream of the rotor disc and a lower pressure area is formed downstream. This pressure change is due to the work of the rotor blades on the air passing over them. The force of the air on the blades results in an opposing force on the air stream causing a rotation of the air column. This low pressure col‐ umn of rotating air expands as it moves downstream of the turbine and eventually dissi‐ pates as equilibrium is reached with the surrounding airflow [2, 3]. This simplified explanation constitutes what is known as the "wake effect" of a wind turbine [1]. An in‐ crease in downstream turbulence is caused by wake rotation, disruption of the air flow across the rotor blades and the vortices formed at the blade tips. This results in less power

The Bernoulli equation can also applied to wind turbine wake analysis. The equation fol‐

*ρU*<sup>1</sup> 2 = (1)

<sup>2</sup> <sup>+</sup> *<sup>p</sup>* <sup>=</sup>*<sup>H</sup>* (2)

r

decrease in wind speed U is also shown.

66 Advances in Wind Power

For conservation of mass:

being available for subsequent turbines.

lows the concept of conservation of energy:

**Figure 1.** Swept area of wind turbine rotor with expanding wake section [1].

$$\mathcal{U}\mathcal{U}\_2 = \mathcal{U}\_1 \sqrt{1 - \mathcal{C}\_T} \tag{3}$$

Here CT refers to the thrust coefficient of the wind turbine. In this way we have a simple wake model for the representation of downstream wind velocity based on the free stream wind speed and the characteristics of the turbine being considered. For more information re‐ fer to [4]. While this does include wake expansion it does not consider other factors such as wake rotation. Several other models are currently in use and under development.

The study of wind turbine wakes is broken into two parts: near wake and far wake. The near wake region is concerned with power extraction from the wind by a single turbine, whereas the far wake is more concerned with the effect on the downstream turbines and the environment [7]. Opinions on near wake length have varied, but can be considered to fall in the range of 1 to 5 rotor diameters (1D to 5D) downstream from the rotor disc [5-6], with far wake regions dependent on terrain and environmental conditions. The full extent of far wake length is currently still under study, but may range from up to 15D for onshore sites [8] and up to 14 km for offshore [9]. The 5D to 15D wake region has been defined as an intermediate wake region by some [10], with the far wake pertaining to distances farther than 15D.

Data from an array of turbines within a commercial scale wind farm are given in Figure 2. The turbines are in a straight line with a separation of 4 rotor diameters. Time series data for nacelle position, wind speed and power are given for a wake event affecting 4 turbines. This specific event has a wind direction moving from 120 to 170 degrees from north, clockwise as positive, over a time span of approximately 10 hours. This can be seen in the nacelle position plot for all four turbines in Figure 2a. During this time period the wind direction passes through the alignment condition of 145 degrees from north. An alignment condition refers to a wind direction measured by the lead (upwind) turbine that is coincident with the straight line formed by the turbine row. The nacelle position plot shows the turbines track‐ ing the wind direction while the wind speed plot (Figure 2b) reveals a drop in wind speed for the downstream turbines between the nacelle position range of TA +/- 15 degrees, where TA refers to direct turbine alignment. In addition, the power is shown to drop along the with wind speed (Figure 2c). This is evidence of wake interaction between the four ma‐ chines. Vermeer et al. [5] found that the wake velocity recovers more rapidly after the first turbine leaving the most dominant effect between the upstream and primary downstream turbine.

The profile of the measured velocity deficit caused by the wake in an array of 4 wind tur‐ bines is given in Figure 3. The upstream nacelle position was used as the reference wind di‐ rection. The wind speeds of the downstream turbines are given with respect to this wind direction and show the wake centerline as well as the profile of the outer edges of the re‐ gion. Upstream wind speeds less than 5 m/s are not included due to the added complexity of low wind conditions and cut-in behaviour of the turbines. Wind speeds greater than 11 m/s are also neglected due to the lack of data at these higher speeds and the reduction in wake pronunciation. The wake region extends across a range of 30 degrees (TA +/- 15 de‐ grees) on average for a wind speed range of 5-11 m/s. A number of features are evident in this Figure. The first downstream turbine exhibits the greatest drop in wind speed at ap‐ proximately 35 %. The second downstream turbine appears to recover by approximately 5 % with respect to free stream velocity under direct turbine alignment. The third in the row shows similar behaviour to the second.

m/s are also neglected due to the lack of data at these higher speeds and the reduction in wake pronunciation. The wake region extends across a range of 30 degrees (TA +/- 15 de‐ grees) on average for a wind speed range of 5-11 m/s. A number of features are evident in this Figure. The first downstream turbine exhibits the greatest drop in wind speed at ap‐ proximately 35 %. The second downstream turbine appears to recover by approximately 5 % with respect to free stream velocity under direct turbine alignment. The third in the row

shows similar behaviour to the second.

68 Advances in Wind Power

**Figure 2.** Time series SCADA data for a period containing a case of turbine alignment. a) nacelle position, b) wind speed and c) power [11].

**Figure 3.** Array wake profile for an upstream turbine wind speed range of 5-11 m/s, considered her e as the free stream wind speed. The data are averaged over 6 months. Wind speed is normalized by the upstream turbine (free stream) wind speed: downstream wind speed/upstream (free stream) wind speed [11].

This recovery is also observed by Barthelmie et al. [12] where momentum drawn into the wake by lateral or horizontal mixing of the air external to the wake region is attributed to the recovery of wind velocity. In the case of Barthelmie et al. the offshore wind farms of Nysted and Horns Rev in Denmark were used to profile the wake regions in the farm's grid style arrangement. The findings revealed the largest wind velocity deficit after the first tur‐ bine with a smaller relative wind speed loss after the initial wake interaction with the first downstream machine. Here the velocity continues to decrease for downstream turbines due to wake mixing from neighboring turbine rows.

It has proven useful to concentrate analysis on more narrow bands of wind speeds as each wind speed tends to produce a measurably different result in turbine behaviour. The wind response for the wind speed range of 8-9 m/s is given in Figure 4. This exhibits the same trends as shown in Figure 3 for a more narrow wind speed range. The power coefficient pro‐ file for the same wind speeds is shown in Figure 5 where the power coefficient is taken as:

$$C\_P = \frac{8P}{\pi \rho U\_\sigma^3 D^2} \tag{4}$$

The values were estimated using the upstream wind speed reading as free stream velocity,, with P representing power produced by the turbine, ρ is the air density, and D is the rotor diameter. The coefficients are plotted for each turbine including the upstream lead turbine. The wake boundaries produce a power coefficient in the range of 0.40 or greater with a min‐ imum at the wake center of 0.14.

**Figure 4.** Normalized wind speed for an upstream turbine (free stream) wind speed of 8-9 m/s averaged over 6 months [11].

In Figure 6, turbulence intensity is implied through consideration of the standard deviation of wind speed. Deviations were calculated by the wind farm SCADA system and provided at 10 minute intervals. For the wind turbines discussed above, the downstream wind speed standard deviation is shown. A clear peak in deviation occurs approximately at turbine alignment +/- 10 degrees with a trend in nominal deviation towards the outer edges of the wake region. Turbulence increases are not excessive and are much less than observed for some special weather events; however, the trend shown in Figure 6 is consistent and may have potential to cause issues over the long term life of the turbine. This is due to the in‐ creased fatigue loading caused by the frequent fluctuations. In addition, increased variation in wind speed along the length of the rotor may contribute to damaging loads. It can also be seen that the greatest wind speed standard deviation does not necessarily correspond with the greatest loss in power. For example, downstream 3 experiences the highest standard de‐ viation under wake conditions but shows the lowest power deficit in Figure 5.

It has proven useful to concentrate analysis on more narrow bands of wind speeds as each wind speed tends to produce a measurably different result in turbine behaviour. The wind response for the wind speed range of 8-9 m/s is given in Figure 4. This exhibits the same trends as shown in Figure 3 for a more narrow wind speed range. The power coefficient pro‐ file for the same wind speeds is shown in Figure 5 where the power coefficient is taken as:

> 3 2 8

The values were estimated using the upstream wind speed reading as free stream velocity,, with P representing power produced by the turbine, ρ is the air density, and D is the rotor diameter. The coefficients are plotted for each turbine including the upstream lead turbine. The wake boundaries produce a power coefficient in the range of 0.40 or greater with a min‐

**Figure 4.** Normalized wind speed for an upstream turbine (free stream) wind speed of 8-9 m/s averaged over 6

In Figure 6, turbulence intensity is implied through consideration of the standard deviation of wind speed. Deviations were calculated by the wind farm SCADA system and provided at 10 minute intervals. For the wind turbines discussed above, the downstream wind speed standard deviation is shown. A clear peak in deviation occurs approximately at turbine alignment +/- 10 degrees with a trend in nominal deviation towards the outer edges of the wake region. Turbulence increases are not excessive and are much less than observed for some special weather events; however, the trend shown in Figure 6 is consistent and may have potential to cause issues over the long term life of the turbine. This is due to the in‐

= (4)

*P <sup>P</sup> <sup>C</sup>* pr*U D*¥

imum at the wake center of 0.14.

70 Advances in Wind Power

months [11].

**Figure 5.** Array power coefficient profile for an upstream turbine wind speed of 8-9 m/s averaged over 6 months [11].

**Figure 6.** Wind speed standard deviation for upstream wind speeds of 8-9 m/s averaged over 6 months [11].

It is evident that there are other external factors that contribute to the definition of these pro‐ files as they show irregularities and do not exhibit a smooth shape. It is expected that the 6 month averaged data has reduced the effects of short term, isolated fluctuations in wind speed, humidity, temperature, air density and inhomogeneous wake at the downstream tur‐ bine and so there is a consistent fluctuation in the wind speed under turbine alignment con‐ ditions. Seasonal and site specific wind conditions are likely to contribute to the small scale unpredictability of wake velocity deficit and turbulence intensity. Inter-turbine wake effects are quantifiable and are accounted for in all major wind farms projects. However, there is still a large amount of uncertainty and error in the modeling of wind turbine wakes and as‐ sociated power losses. The next section discusses wind farm siting and its importance in minimizing uncertainty in the planning stages of a wind project.
