Considering induction factor using BEM method in wind farm layout optimization

J. Wind Eng. Ind. Aerodyn. 129 (2014) 31–39

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Journal of Wind Engineering and Industrial Aerodynamics journal homepage: www.elsevier.com/locate/jweia

Considering induction factor using BEM method in wind farm layout optimization A. Ghadirian a,n, M. Dehghan b, F. Torabi c a

DTU Wind, Denmark Technical University, Denmark Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Iran c Energy Department, Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Iran b

art ic l e i nf o

a b s t r a c t

Article history: Received 14 November 2013 Received in revised form 19 March 2014 Accepted 21 March 2014

For wind farm layout optimization process, a simple linear model has been mostly used for considering the wake effect of a wind turbine on its downstream turbines. In this model, the wind velocity in the wake behind a turbine is obtained as a function of turbine induction factor which was considered to be 0.324 almost in all the previous studies. However, it is obviously evident that this factor is a strong function of turbine blade geometry and operational conditions. In the present study, a new method is introduced by which the induction factor for wind turbines can be calculated based on the method of Blade Element Momentum theory. By this method, the effect of blade profile, wind speed and angular velocity of wind turbine on the induction factor can be easily taken into account. The results show that for different blade profiles and operational conditions, the induction factor differs from the single value used so far. Also it is shown that this difference has a very significant effect in calculated gained power from a wind farm. It is clearly seen that considering the new method for calculating an appropriate induction factor affects the total calculated power generation of a wind farm and consequently influences the farm layout in optimization process. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Induction factor Wind farm layout Wind farm optimization Wind turbine characteristics Blade element momentum Wind wake model

1. Introduction Daily increase of Wind Energy usage for electricity production and consequently boosting growth of wind farms' number around the world has made engineers and researchers attempt significantly to optimize wind farms' layout. Accurate estimation and optimization of wind farms total power is a complex problem because it is affected by several factors including (a) different wind speeds and directions in time, (b) different geographical conditions and (c) interactive effects of turbines on each other. This causes a percentage of error in every primary calculation of farm power. Nonetheless, because of particular importance of assessing expenditures and revenues there has been a significant endeavor to present a model for optimizing wind farm layout considering all the effective factors. To achieve this several models have been proposed by different researchers. Because of significant effect of wakes most of these researches consider it by some models. One of the most important and mostly used models is the model introduced by Jensen (1983). Jensen proposed a theory for wake distribution behind a wind

n

Corresponding author. E-mail addresses: [email protected] (A. Ghadirian), [email protected] (M. Dehghan), [email protected] (F. Torabi). http://dx.doi.org/10.1016/j.jweia.2014.03.012 0167-6105/& 2014 Elsevier Ltd. All rights reserved.

turbine based on distance, wind speed, gaining induction and entrainment constant from experiments using mass conservation theory. He used this theory in another article (Katic et al., 1987) to derive the total power of a wind farm. In the first article induction factor has been assumed a ¼ 13 and in the second one with considering some reformations it has been changed to a¼ 0.324. Mosetti et al. (1994) have used the theory to optimize a wind farm by genetic algorithm optimization method. After that many other researchers have tried to present different optimization algorithms using the proposed model of Jensen. Some of these articles (Grady et al., 2005; Kusiak and Song, 2010; Chowdhury et al., 2012) are concentrated on optimization algorithm and some others (Maroulis et al., 2001; Ozturk and Norman, 2004; Rajper and Amin, 2012) on the function that is going to be optimized; i.e. the objective function. There are few articles concerned about the wake model (Serrano Gonzalez et al., 2011; Gonzalez et al., 2010). In all the abovementioned articles the induction factor has been assumed to be a¼0.324 based on the Katic et al. (1987) and Mosetti et al. (1994). For example Wan et al. (2012) has used particle swarm optimization method based on the work of Mosetti et al. (1994) to optimize layout of a wind farm with known number of turbines. Also Chan et al. (2014), Pookpunt and Ongsakul (2013) and Chen et al. (2013) have used Jensen's model based on Jensen's original article while the induction factor has been considered constant. Erolu and Seckiner (2013, 2012) have used Jensen model too although in this model

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Nomenclature a a0 A CP CT Cl Cd cn ct d D F r0 rd u u ui v0 v

axial induction factor, dimensionless rotational induction factor, dimensionless area, m2 power coefficient, dimensionless thrust coefficient, dimensionless lift coefficient, dimensionless drag coefficient, dimensionless normal load coefficient, dimensionless tangential load coefficient, dimensionless distance between two neighboring turbine, m turbine rotor diameter, m Prandtl tip loss factor, dimensionless turbine radius, m wake radius at distance x, m wind speed at infinity, m s  1 multiple wake velocity, m s  1 velocity affected by ith turbine, m s  1 wind velocity right after turbine, m s  1 wind speed at distance x, m s  1

induction factor a has been substitute by CT from (5). Similarly Prez et al. (2013) have used modified Jensen model with C T which has been assigned from the manufacturer provided data. Another wake model has been presented by Frandsen et al. (2006) which is more consistent with experimental results according to the writers acclaim. This work uses different theory and also shape of distribution. In this model the shape of the wake is a function of thrust coefficient which itself is a function of induction factor. Although this model can give a more realistic result, the authors have used the same value for induction factor used by Jensen i.e. a ¼0.324 and calculated thrust coefficient CT. Chowdhury et al. (2013) have used Frandsen model for wind farm layout optimization. Some researches in this area have been conducted using other wake models for example Song et al. (2013) have used CFD modeling for wake calculation in complex terrain and Son et al. (2014) have used eddy viscosity for calculating wind speed in wakes. Although these methods are not widespread and so not in the focus of this article. It can be clearly seen that a majority of work done in this area has used the same induction factor constant value which is obtained from experiments conducted by Jensen. It is evident that this value is reliable just for the wind turbine type which was used in Jensen's test and the specific wind speed of 8 m s  1 . Different turbines with different blade shapes, size, rotational speed and wind speed produce different induction factor which strongly affects the wake strength behind the wind turbine. As it can be seen in Fig. 1 a mite tolerance of 1% in induction factor significantly affects the velocity of wind behind the turbine which in turn affects the power of turbines in the wake zone. Moreover, these parameters affect the thrust coefficient hence we cannot use a single value for induction factor in wind farm optimization. This means that when using wake models (Katic et al., 1987; Frandsen et al., 2006) we have to calculate the induction factor or the thrust coefficient for each turbine geometry, operational conditions and wind velocity, separately. In this article a new method for calculation of induction factor is presented. The proposed method is based on Blade Element Momentum (BEM) theory which is capable to include all the geometrical and operational conditions of wind turbines. Therefore, this method can give a more realistic induction factor for wake models resulting into a more accurate wind farm power calculations (particularly for optimization algorithms). It is shown

xD z z0

dimensionless distance ratio, dimensionless hub hight, m terrain surface roughness, m

Greek

α ω ϕ s ρ

wake distribution constant, dimensionless rotational velocity, m s  1 angle of relative velocity with rotational plane, deg ratio of blades area to swept area, dimensionless air density, kg m  3

Subscripts/superscripts loc tot root tip e s

local total root tip end start

that the induction factor differs from turbine to turbine and in different wind velocity or operational conditions. Therefore the power calculation and optimization cannot be done using a single value for induction factor. This method proposes a more corresponding model to reality and therefore a more efficient layout for achieving higher energy production.

2. Methodology To understand the concept of the present work, a brief review of previous works is crucial. As Jensen (1983) explained in his proposed model the zone right after turbine blades has not been considered because blades rotation in this zone and consequently rotation of the wind behind it causes a special kind of wake. In spite of that, in a reasonable distance behind the turbine, a mass balance results

π r20 v0 þ π ðr 2d  r20 Þu ¼ π r2d v

ð1Þ

while u is the upstream wind speed, v0 is the wind speed right behind turbine and v is the wind speed at distance x after the

Fig. 1. Effect of induction factor on wind speed after turbine.

A. Ghadirian et al. / J. Wind Eng. Ind. Aerodyn. 129 (2014) 31–39

33

Some researchers including Serrano Gonzalez et al. (2011) and Gonzalez et al. (2010) have used this model for wind farm design. In Eq. (4) the thrust coefficient CT has been assumed to be 0.88. It worth mentioning that CT is a function of induction factor where this relation under one-dimensional momentum theory can be obtained as C T ¼ 4að1  aÞ

ð5Þ

In order for CT to become 0.88, one should use a¼0.324 which is clearly the same value used by Jensen (1983) and Mosetti et al. (1994). Therefore it is again evident that the value for induction factor is 0.324. 2.1. 3D blade element momentum Fig. 2. Schematic wake model of Jensen (1983).

turbine, r 0 is the turbine radius and rd is the wake radius at distance x after the turbine. In this model, wake distribution behind the turbine has been assumed to be linear (Fig. 2). In addition, the velocity right after wind turbine has been assumed to be 13u based on previous theories. Therefore the velocity in the wake region is obtained from
2 ! 2 r0 v ¼ u 1 ð2Þ 3 r 0 þ αx in which, α is the wake distribution constant and has always been assumed to be around 0.1. This has been speculated and approved by Jensen (1983) comparing the results with the data presented by Højstrup (1983) for specific conditions. Mosetti et al. (1994) have provided the following formula for deriving wake distribution constant: 0:5

α¼   ln

z z0

BEM theory helps us to calculate loads, axial and tangential induction factor and other characteristics on blade elements in different wind speeds, turbine angular speeds and pitch angles for various airfoil profiles. As Hansen (2008) explained, in one dimension momentum theory, rotor geometry, blade number, blade twist, blade chord and generally airfoil characteristics are neglected and not considered in calculations. BEM theory does consider these specification effects on momentum theory. As shown in Fig. 3 in this model control volumes are annular tubes with a thickness of dr assuming that there is no flow between these control volumes. Furthermore, Prandtl's tip loss factor is defined to correct the assumption of load homogeneity on flow from the blade elements in the control volume (infinite number of blades). Hansen (2008) introduced an 8-step procedure for calculating a and a0 (axial and tangential induction factors) for each element and then local loads on that segment. This 8-step algorithm shown in Fig. 4 is:

ð3Þ

where z is the hub hight and z0 is the terrain surface roughness. The reason that a has been assumed 13 is similarity with Betz theory and reaching maximum CP based on one dimensional momentum theory. The amendment to a ¼0.324 is to reach more consistency with real site data conducted by Vermeulen et al. (1979) and Højstrup (1983). These data were gathered from a Nibe-A turbine with 20 m diameter and in 8:1 m s  1 wind velocity in 100 m hub hight. All the coefficients are selected and modified by this method and still the results are near 10dimensionless13% away from real experiments (Jensen, 1983). On the other hand, even though most of optimization algorithms have used Jensen (1983) model, Frandsen believed that this model is far from reality and cannot provide admirable results specially for large wind farms. He believed that interaction between the atmosphere and the turbine should also be considered. Then he proposed another wake model for investigating the small-scale and large-scale flow of wind over offshore wind farms. This model consists of three regimes counting from upwind. The first regime is the zone where turbine wakes have been combined linearly; one turbine is in the wake of several wakes of upwind turbines. The second regime happens when the parallel wake cones merge. In this zone the wake grows only vertically and is similar and not identical to a change in surface roughness. The third regime happens when the wind farm is infinitely large and wind flow is in balance with boundary layer flow (Frandsen et al., 2006). Frandsen firstly describes his model of single wake. Based on this model wind speed in the wake behind the turbine can be calculated using the following formula: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 v 1 1 r0 ¼ þ ð4Þ 1  2C T u0 2 2 rd where rd is the wake radius at distance d downstream the turbine.

1. The axial and tangential flow induction factors assumed to be zero initially (a ¼ a0 ¼ 0). 2. Computing ϕ which is the angle between relative velocity and rotation plane using the following equation: tan ϕ ¼

ð1  aÞu ð1 þ a0 Þωr

ð6Þ

where ω is angular velocity of turbine and u is the wind speed in infinity. 3. Computing relative angle of attack, using the following equation:

α ¼ ϕ  θlocal twist  θpitch

ð7Þ

4. Obtaining drag and lift coefficients, C l ðαÞ and C d ðαÞ, from relevant tables or experimental data. 5. Computing normal and tangential load coefficients, Cn and Ct: C n ¼ C l cos ϕ þC d sin ϕ

ð8Þ

C t ¼ C l sin ϕ C d cos ϕ

ð9Þ

Rotor plane Control Volume R r dr

Fig. 3. Control volume shaped as an annular element.

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By this procedure the induction factor is calculated for each blade segment and not for the whole turbine. In the following section we will use these values for calculation of the total induction factor.

3. Calculation of overall induction factor for a wind turbine Total axial induction factor can be calculated using the conservation of mass as is shown in Fig. 5. In this figure a single blade and the relative velocity for each blade segment is shown. The induced velocity on each segment causes reduction of relative velocity on itself. It should be noted that the relative velocity on each blade segment depends on local rotating velocity. Consequently, a distribution of wind relative velocity is formed on the blade from root to tip. Then the overall induction factor can be obtained using balance of mass on the whole swept area on the turbine surface right behind it (Fig. 5). Obviously the total air volume exiting from behind the turbine is equal to the sum of exiting air from each element. This can be written as Ns

_ total ¼ ∑ m _ seg m

ð14Þ

i¼1

_ total and m _ seg are the total and segment mass flow rates, in which m respectively and Ns is the total number of blade segments. The segment mass flow rate can be calculated from the following equation: _ seg ¼ ρð1  aloc Þv0 π ðr 2e  r 2s Þ m

ð15Þ

In the above equation, aloc is the local induction factor and re and rs are the radii of the end and the start of the blade segment, respectively. Note that each element segment area has an annular shape as shown in Fig. 3. On the other hand, the total mass flow rate is equal to the mass of air passing the wind turbine with the total reduced velocity, i.e. _ total ¼ ρAtot V m

ð16Þ

in which V is the reduced velocity: V ¼ ð1 atot Þv0

ð17Þ

equating Eqs. (14) and (16) results Fig. 4. Blade element momentum theory algorithm.

Ns

ðr 2tip  r 2root Þð1  atot Þv0 ¼ ∑ðr 2e  r 2s Þð1  aloc Þv0 1

6. The axial induction factor a is calculated using Prandtl's tip loss factor, 8 1 if a oac > 2 > ϕ > < 4F ssin þ1 Cn  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ a¼ 1 > > X  X 2 þ 4ðKa2c  1Þ if a 4ac > : 2 while X ¼ 2 þ Kð1 2ac Þ and 4F sin ϕ sC n The tangential induction factor, a0 , from 1 a0 ¼ 4F sin ϕ cos ϕ þ1 sC n

ð11Þ

w flo

Segm ent m Segm ass fl ent m ow ass fl ent m Segm ow ass fl e nt m ow Segm a ss flo ent m Segm w ass fl en S ow Segm egment m t mass fl o ent m w ass fl ass fl o ow w Segm

2

K¼

a ss lm a t To

ð12Þ

ð13Þ

7. We use these values and repeat steps 2dimensionless6 until the a and a0 are equal to the guest values (convergence occurs) then we proceed to step 8. 8. Finally local loads can be calculated using the obtained data.

Relative velocity of each segment Fig. 5. Illustration of conservation of mass on a blade.

ð18Þ

A. Ghadirian et al. / J. Wind Eng. Ind. Aerodyn. 129 (2014) 31–39

From this equation the total induction factor can be obtained: atot ¼ 1 

2 2 s ∑N 1 ðr e  r s Þð1  aloc Þ 2 r tip  r 2root

ð19Þ

Finally, assuming a linear distribution of loads on each element length, thrust and momentum coefficients of each element can be calculated and the turbine shaft power and total thrust coefficients, CP and CT, can be derived accordingly.

4. Results In this section, the application of the present work is discussed in more details. Firstly, we check the dependency of the induction factor on different geometrical and operational conditions. It is shown that the induction factor is a function of the abovementioned factors. Secondly, the effect of induction factor on the downstream wind turbines is studied. Finally, the effect of variation of induction factor on the total power gained from a wind farm is investigated. 4.1. Dependency of the induction factor It has been discussed that in wind farm layout optimization models, induction factor has been assumed to be 0.324 so far while this factor depends on wind speed, blade profile and blade rotational velocity. This dependency can be better understood by studying Fig. 6. This figure shows a cross section of a typical turbine blade together with the far-filed wind velocity, rotational speed, induced velocities and the relative wind velocity. As it can be seen, according to this figure, it is clear that the relative wind velocity, which is seen by the profile, absolutely differs from the far-field wind velocity. In fact, the rotational wind speed, axial induction factor, a, and rotational induction factor, a0 , play an important role in the magnitude and direction of the relative wind. It should be emphasized that the total lift and drag produced by an airfoil depends on the relative wind velocity and not the far-field wind. Therefore, it is completely evident that the far-field wind velocity and the rotational wind speed strongly affect the induction factor. It can be also concluded that even in a uniform far-field wind velocity, different sections of a blade have different relative wind speed and direction, due to the different rotational speed. On the other hand, the calculation of induction factor, according to the pre-mentioned 8-step procedure, depends on the lift and drag curve of a specific airfoil; hence the induction factor is also a function of the blade profile. The dependency of the induction factor on the above-mentioned parameters can be studied using the following three scenarios.

35

(1983) used the experimental results for wind velocity of 8 m s  1 and obtained the induction factor (i.e. a ¼0.324) accordingly. Now we want to investigate what happens if the wind velocity differs from that value. To investigate this effect, a sample turbine is used for simulation. This turbine was introduced by Hansen (2008) whose geometry is listed in Table 1. In this study, the blade profile is assumed to be NACA 634-18. Fig. 7 shows the variation of induction factor versus upstream wind speed. From the figure it can be seen that the induction factor is 0.45 for wind speed 4 m s  1 and while wind speed increases it declines and reaches to about 0.07 at 18 m s  1 wind speed. This means that the lower the wind speed, the higher the induction factor. In other words, induction factor plays more important role in low-speed rather than high-speed winds. 4.1.2. Effect of blade profile The second scenario is related to the dependency of induction factor on blade profile. To investigate this effect, in the same turbine used above, we keep all the parameters intact except the blade profile. Fig. 8 shows the results of the present simulation. From the figure one can see that induction factor for different blade profiles differs at the same wind speeds. Note that for all these samples, other parameters such as twist, pitch, radius, chord and angular velocity have been assumed the same. Table 1 Blade characteristics (Hansen, 2008). r (m)

Twist (deg)

Chord (m)

4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5 16.5 17.5 18.5 19.5 20.3

20 16.3 13 10.05 7.45 5.85 4.85 4 3.15 2.6 2.02 1.36 0.77 0.33 0.14 0.05 0.02

1.63 1.597 1.54 1.481 1.42 1.356 1.294 1.229 1.163 1.095 1.026 0.955 0.881 0.806 0.705 0.545 0.265

4.1.1. Effect of upstream wind velocity The first scenario happens when the wind speed in farm varies from cut-in to cut-off velocities. It was mentioned that Jensen

ωr(1+a/ ) Rotor plane

Vrel

v0(1-a) Fig. 6. Illustration of relative wind, angle of attack and induced velocities on a blade profile.

Fig. 7. Effect of wind speed on induction factor.

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A. Ghadirian et al. / J. Wind Eng. Ind. Aerodyn. 129 (2014) 31–39

Fig. 8. Effect of blade profile on induction factor.

For instance at 8 m s  1 wind speed the induction factor for a turbine with NACA 634-18 profile is about 0.28 while at the same wind speed for profiles FFA-W3-24 is 0.26, for profile FFA-W3-36 is 0.24 and for profile FFA-W3-30 is 0.28. This shows that the blade profile affects the induction factor and we cannot use a single value of the induction factor for all the possible blade profiles. 4.1.3. Effect of rotor angular velocity Angular velocity also has a strong effect on induction factor. In the third scenario, this dependency is shown quantitatively. The angular velocity produces a tangential velocity which varies from root to tip because the blade profiles are subjected to different rotational speed. This causes that the blade sees a distribution of relative wind velocity (even in a uniform upstream wind velocity) from root to tip. This distribution is schematically shown in Fig. 5. To investigate the effect of angular velocity on induction factor, we use the original turbine used in the first scenario. Again all the parameters are kept intact except for the angular velocity. The results of simulation are shown in Fig. 9. It is clearly seen that in 8 m s  1 of wind speed when blades angular velocity raises from 10 rpm to 40 rpm the induction factor increases almost five times. This shows that the angular velocity has a very strong effect on induction factor. It is also interesting to mention that for the angular velocity of 34 rpm which was used by Katic et al. (1987), the induction factor for all the blade profiles (although not the same) is about a ¼0.33 which is absolutely compatible with the results of that study. 4.2. Effect of induction factor on downstream turbines The above sensitivity analysis shows that the induction factor differs for different turbines and operational conditions. Here the effect of induction factor on downstream turbines is investigated. In other words, we are going to check how much the assumption of using a single value for induction factor is valid. For investigating the effect of such a delinquency, we consider a hypothetical problem and solve it with and without this assumption. Assume that four wind turbines are in a row in a wind stream while each two sequential turbines are positioned at a distance of 6 time of their diameter (Fig. 10). The upstream wind speed is u and the blade profile is NACA 634-18. The other parameters are the same as what used above and the values of Table 1. The total power output and other farm characteristics are calculated using two different strategies. In the first strategy, a single value induction factor, a¼ 0.324, is set for all the turbines as

Fig. 9. Effect of rotor angular velocity on induction factor.

u

6D

6D

6D

Fig. 10. Illustration of a single row wind farm.

discussed in this article. In the second strategy, the induction factor for each individual turbine is calculated using the method discussed in the present work. For each strategy, the power of the first turbine is calculated from the BEM theory while the turbine is operated at upstream wind velocity. The second turbine is in the wake zone of the first one; hence the wind speed at its position can be calculated from the following equation (Katic et al., 1987):
2 ! r0 v ¼ u 1  2a ð20Þ r 0 þ αx The third turbine is in the wake of the first and second turbines and the wind speed at its position is obtained from the following formula which has been proposed by Katic et al. (1987):
2  ui u  r0 ¼ 1  1  ð1  2ai  1 Þ i  1 u u r 0 þ αx

ð21Þ

where ui is the wind speed at the location under effect of ði 1Þ th turbine and ai  1 is the induction factor at the upstream neighboring turbine. This equation can be used to determine the wind speed at the location of any successive turbine. Fig. 11 shows the calculated power of the farm for different upstream wind velocities. As it can be seen, the total output power of the farm differs in different strategies. The difference between the calculations increases as the upstream wind velocity increases. Because, as discussed before, the induction factor decreases when the upstream wind velocity increases. This results into different wind velocity at the position of each turbine. To have a better understanding, the relative error is also plotted in the same figure. It is clear from the figure that the relative error can reach above 20% in higher upstream wind velocities which is very important in wind farm layout optimization process. Even in lower wind speeds, the relative error is about 12% which is remarkable and cannot be ignored. Fig. 12 shows how the induction factor for each turbine varies with respect to the upstream wind velocity. As it can be seen, in the most of the time for all the turbines, the induction factor is far from a ¼0.324 (shown in the figure with the horizontal line). In some cases, the induction factor (specially for the third and forth

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18

30 Constant Induction Factor Variable Induction Factor Error Percentage

Local Wind Speed / ms-1

3

20 2 10 1

0

10

12

14

2nd Turbine 3rd Turbine 4th Turbine

16

Error / %

Output Power / MW

4

16

18

14 12 10 8 6 4

0

2

Wind Velocity / ms-1

10

12

14

16

Upstream Wind Speed / ms

Fig. 11. Error analysis between the two strategies.

18 -1

Fig. 13. Local wind speed for each turbine versus upstream wind velocity. Lines without symbols are obtained assuming a ¼0.324 and lines with symbols show the results using variable induction factor.

0.5 Turbine 1 Turbine 2 Turbine 3 Turbine 4

0.4

1200 nd

2 Turbine 3rd Turbine th 4 Turbine

1000

a = 0.324

0.3 800

Power / kW

Induction Factor

37

0.2

0.1

10

12

14

16

18

600 400 200

Wind Velocity / ms-1

0

Fig. 12. Calculated induction factor for each turbine.

10 turbines) is greater than that value because in this situation, the wake effect of the first and the second turbines has lowered the local wind speed. To have a better comparison, the local wind speed for each turbine is also plotted in Fig. 13. It is clear (not shown in the figure) that the first turbine meets the upstream far field velocity. From the figure also one conclude that assuming a constant value for induction factor is more crucial for the turbines located behind the others. The graphs of Fig. 13 also predict that if the upstream wind velocity is lower than u ¼ 10 m s  1 , the local wind speed at the fourth turbine (or even third one) reaches cut-in velocity which is considered to be ucutin ¼ 4:5 m s  1 hence produces no power. Different local wind velocity results into different output power. Fig. 14 presents this difference for the turbines located at the wake of the others. We expect that the turbines located behind the others produce less power which is clearly shown in the figure. However, the comparison shows that the error arose from different strategies for power calculation, increases when the turbines located behind the others. 4.3. The effect of variation of induction factor on total power gained from a wind farm In this section, gained power of a hypothetical wind farm with 16 wind turbines has been investigated. This farm is assumed to be

12

14

16

18

Upstream Wind Speed / ms-1 Fig. 14. Turbines' power versus upstream wind velocity. Lines without symbols are obtained assuming a¼ 0.324 and lines with symbols show the results using variable induction factor.

square shaped with 4 by 4 turbines placed with equal distance from each other (Fig. 15). The distance between each two neighboring turbines can be varied from 4 to 10 times of their diameter. Hence, shown in the figure, it is indicated by dimensionless distance ratio, xD, which is defined as xD ¼

d D

ð22Þ

where d is the physical distance between two neighboring turbine and D is the turbine rotor diameter. The gained power of the wind farm is the summation of the power of each individual turbine. Therefore, just like the previous subsection, we calculate the power of each turbine, considering the effect of wake from all the turbines which are in front of it. Then the total power of the farm can be easily calculated by summing up all the powers. This method is applied using two different cases; namely with the assumption of a¼ 0.324 which is called Case-I and calculating the induction factor from the present model which is called Case-II. The comparison of the results, once again, shows the importance of the calculation of real induction

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A. Ghadirian et al. / J. Wind Eng. Ind. Aerodyn. 129 (2014) 31–39

40

xD D

Error Percentage / %

Northings [Distance Ratio]

35

D

3xD

2xD

xD

0

u=10 u=12 u=14 u=16

30 25 20 15 10 5

xD

0

2xD

0

3xD

4

5

6

Eastings [Distance Ratio] Fig. 15. The diagram of the hypothetical wind farm.

u=10 u=12 u=14 u=16

12 10

7 xD

8

9

10

Fig. 17. Relative error arose from different strategies of power calculation.

m s-1 m s-1 m s-1 m s-1

u=10 u=12 u=14 u=16

12

Total Power / MW

Total Power / MW

m s-1 m s-1 m s-1 m s-1

8 6 4

m s-1 m s-1 m s-1 m s-1

10 8 6 4 2

2 4

5

6

7 xD

8

9

10

Fig. 16. Calculated power of the wind farm. Lines without symbols are obtained assuming a ¼0.324 and lines with symbols show the results using variable induction factor.

factor in wind farm design. In all the calculations, the rotor angular velocity is kept constant, ω ¼ 34 rpm, and the turbine characteristics are the same used in the previous subsection. The results of the present study are shown in Fig. 16. In this study, the distance ratio is varied from 4 to 10. The wind velocity (blown from the North to the South) also varies from 10 to 16 m s  1 . As it can be seen, the error between the two cases increases when the wind velocity increases. This is because the induction factor strongly depends on the wind velocity (see Fig. 7). The relative error for these calculations is plotted in Fig. 17. When the unperturbed upstream velocity is about 10 m s  1 , the relative error of calculations is less than 15%. However, when the wind velocity reaches 16 m s  1 , the error is more than 10% for xD ¼10 and about 30% for xD ¼ 4. It is obvious that the closer the turbines are, the more effective the wake becomes. Hence the wake modeling becomes more and more important. This fact can be clearly seen in Fig. 17; the larger xD becomes, the less the relative error becomes. It means that in a very long distance, the relative error may tend to zero which means that the results are independent of the wake modeling. In

0

4

5

6

7

8

9

10

xD Fig. 18. Power output of the farm with different blade profiles. Lines without symbols are obtained using NACA634-18 and lines with symbols are obtained using FFA-W3-24.

reality, for a distance far enough from a turbine, the wind velocity reaches its unperturbed upstream velocity. Finally, the effect of blade profile is studied and the results are plotted in Fig. 18. In this study, the same wind farm with the same characteristics and operating conditions is solved using two different blade profiles: namely NACA-634-18 and FFA-W3-24. It is clear from the figure that changing the blade profile has made some differences in power calculations. This is also attributed to the differences between the calculated induction factor for the two different blade shape. 4.4. Using CT instead of induction factor As it was mentioned before, some researchers (Erolu and Seckiner, 2013, 2012; Prez et al., 2013) use CT instead of induction factor to obtain the velocity deficit in the wake region. Also it was explained that the thrust coefficient, CT, is a function of induction factor where this relation under one-dimensional momentum theory can be obtained from Eq. (5). The value used for the thrust coefficient in many articles is CT ¼ 0.88 which is the result of assuming a ¼0.324.

A. Ghadirian et al. / J. Wind Eng. Ind. Aerodyn. 129 (2014) 31–39

6

Total Power / MW

5

u=12 ms

which in turn significantly affect the total gained power of a wind farm. The wind farm calculations for a hypothetical wind farm also showed that the assumption of a typical constant value for the induction factor does not give accurate results. It was also shown that using CT with a constant value for induction factor does not give a good prediction of the farm total gained power. Hence the present study provides a powerful tool for wind farm power calculations.

-1

4 3

References

2 u=10 ms

1 0

39

-1

CT obtained from variable a CT obtained from BEM Constant CT

4

5

6

7 xD

8

9

10

Fig. 19. Power output comparison using different methods for wake velocity calculation.

Here we try to calculate the total power output of the farm described in Section 4.3 with the following three different scenarios and compare the results: 1. Calculating CT from Eq. (5) with constant a¼ 0.324. 2. Calculating CT from Eq. (5) with variable a discussed in the present work. 3. Calculating CT directly from BEM. Fig. 19 shows the results for different unperturbed upstream wind velocities: u ¼10 and u ¼ 12 m s  1 . From the figure it is obvious that assuming a constant value for thrust coefficient results into some error (about 11.5–19%) while the other two methods more or less predict the same values. The relative error in these methods is less than 4%. It is quite logical that the more the total induction factor deviates from 0.324, the higher the error becomes. For example, the higher the upstream velocity, the higher the error becomes. This fact is clearly seen from Fig. 19 where the farm power calculation has more error in u ¼ 12 m s  1 . Other parameters influencing the induction factor discussed in the present article have the same influence accordingly. 5. Conclusions In the present study, a new approach for calculation of induction factor was introduced. This approach is based on the BEM method which is extensively used for calculation of the power and thrust coefficient of a wind turbine. Since in BEM method all the geometrical and operational conditions are taken into account, the present model is also able to consider those effects, as well. For instance, calculation of induction factor by the present work takes into account the effect of blade profile, wind velocity and angular velocity of the turbine. Through different scenarios it was shown that these factors significantly affect the value of induction factor

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