## 1. Introduction

Interactions among the wakes of individual turbines in a large wind farm have a significant impact on the overall performance of the wind farm. Depending on the wind direction and the layout of the turbines, the actual power produced can be significantly lower than the rated power of the wind farm (Archer et al. 2013; Wu and Porté-Agel 2013; Stevens et al. 2014). Optimal layout of wind turbines is, therefore, a crucial element in modern wind farm design considerations.

A variety of numerical and analytical techniques have been developed over the years to study wind turbine wakes. Computational fluid dynamics (CFD) studies of wind farms can be classified based on the type of turbulence closures employed. Studies such as Barthelmie et al. (2009) and Tian et al. (2014) use the Reynolds-averaged Navier–Stokes (RANS) approach for modeling the effect of turbulence. Numerical tools such as the Universidad Politécnica de Madrid version of the Park model (UPMPARK), the Technical University of Denmark “EllipSys” solver, and the Risø National Laboratory “Fuga” model (Archer et al. 2014) are all based on the RANS turbulence modeling approach. RANS simulations are not suitable for resolving the details of the unsteady features of turbulent flows, which are better captured by the large-eddy simulation (LES) technique (Churchfield et al. 2012a; Calaf et al. 2010, 2011; Stevens et al. 2014; Archer et al. 2013).

A number of studies have focused on evaluating the effects of wind farm layout on power generation. Meyers and Meneveau (2012) determined the optimal spacing in infinitely large wind farms, where the turbines were arranged in rows perfectly aligned with the incoming wind direction. The optimal spacing, assumed equal in the streamwise and spanwise directions, was found to be much larger than the spacing typically used in existing wind farms. The effect of staggering was studied by Wu and Porté-Agel (2013) in a downscaled wind farm comprising 30 miniature turbines. Staggering alternate rows was found to increase the angular speed of the turbines because of exposure to higher local wind speeds. Stevens et al. (2014) studied the alignment of rows of turbines in infinitely large wind farms with the wind direction. They found that the configuration with rows aligned at a small angle to the incoming wind yielded the maximum power. The optimal angle was smaller than the angle corresponding to perfect staggering of rows and was dependent on the turbine spacing in the spanwise direction. None of the above studies were carried out on actual wind farms, and most of them considered infinitely large wind farms, a necessary assumption for applying periodic conditions in the horizontal directions. A numerical investigation of layout effects in an actual, finite, wind farm offshore of Lillgrund (Sweden) was carried out by Archer et al. (2013). A number of layouts, generated by staggering rows, increasing streamwise and/or spanwise spacings, and simultaneously staggering and increasing spacings, were evaluated. Wind farm layouts were shown to affect the performance by up to 33%.

While useful in identifying certain features of layout effects, the above studies are limited in the combinations of layout design parameters and the number of layouts that can be evaluated. This is because CFD simulations, especially LES of large wind farms, are extremely computationally demanding. For example, each simulation of the Lillgrund wind farm reported in Archer et al. (2013) required on the order of 150 000 CPU hours. A simple and inexpensive method of investigating the effects of wind farm layout is needed, and this has motivated development of several analytical models.

Analytical models that are easy to implement and practical to use for layout studies follow from either the classical Park model (Jensen 1983; Katic et al. 1986; Peña et al. 2013a,b; Bastankhah and Porté-Agel 2014; Nygaard 2014) or the so-called top-down approach (Frandsen 1992; Emeis and Frandsen 1993; Frandsen et al. 2006; Calaf et al. 2010; Meyers and Meneveau 2012). The Park family of models accounts for lateral spread of wind turbine wakes and is able to account for the effects of turbine layout (Katic et al. 1986). However, these models ignore the vertical structure of the atmospheric boundary layer (ABL) and the interactions between wind turbine wakes and the ABL. The top-down models, on the other hand, are able to account for the ABL vertical characteristics on the wind farm performance by assuming the presence of two constant-stress regions above and below the wind turbine rotors (Calaf et al. 2010). These models depend only on the mean spacing between turbines and consequently do not account for the details of the horizontal layout of turbines. This assumption is justified in the fully developed regime, applicable to infinitely large arrays of wind turbines (Emeis and Frandsen 1993). Extensions of the top-down model to account for entrance effects in finite wind farms have been carried out by Meneveau (2012). Models coupling the above two contrasting approaches have been developed (Frandsen et al. 2006; Stevens et al. 2015), as have also models accounting for the effects of atmospheric stability (Emeis 2010; Peña et al. 2013a,b; Peña and Rathmann 2014).

All of the abovementioned analytical models use the principle of momentum conservation to predict the wake velocity, which is then coupled to a power curve to determine the generated power. Thus, these models attempt to take into account the physical processes occurring during turbine wake interactions and hence may be termed “physics based” models. In contrast, in the current study, an attempt is made to develop simple statistical models that rely only on the geometry of the wind farm to directly predict the generated power. Several geometric features associated with a given wind farm layout are defined in section 2. A series of LES of the Lillgrund wind farm, which are reported elsewhere (Archer et al. 2013), are used to calibrate models based on some of these geometric features. The models are validated in section 3 by their ability to reproduce the Lillgrund LES and other numerical simulation results, and difficulties in comparing with field observations are pointed out. The models are employed to study the effects of wind farm layout in hypothetical configurations in section 4. A number of design features, such as the streamwise and spanwise spacings, the staggering of alternate rows, the angle between the rows and the columns, and the wind farm orientation are varied. Some general conclusions regarding wind farm layout are drawn. Section 5 presents the summary and conclusions of this study demonstrating the utility of such geometry-based models in analyzing wind farm layout effects, as well as their limitations.

## 2. Geometry-based models

### a. Overview of LES results

*D*= 93 m and hub height of 63.4 m) arranged in an exceptionally tight (close spaced) layout (Dahlberg 2009). This tight spacing makes this farm site ideal for studying the effects of different layouts, such as varying the spacing between turbines along (streamwise) and across (spanwise) a given wind direction. The wind turbines in Lillgrund are represented by solid circles in Fig. 1a. A number of large-eddy simulations of the Lillgrund wind farm, with the turbine blades represented as rotating actuator lines, were carried out, as described in Archer et al. (2013), and are summarized in Table 1. For each case, the number of turbines, the wind direction, a brief description of the layout, and the total power generated

*P*

_{TOT}are listed. Also included is the averaged relative power, defined as

*P*

_{i}denotes the power generated by the

*i*th turbine,

*N*

_{T}denotes the number of turbines, and

*P*

_{max}denotes the maximum power among the

*N*

_{T}turbines (generally produced in the first upstream row of turbines). As shown in Table 1, the simulations include wind blowing from the southwest (marked 225), west (270), and northwest (315). The prevailing wind direction, or the direction from which the wind blows most often, is southwest at Lillgrund (Bergström 2009). With the same southwesterly wind, simulations of five additional layouts were carried out, with double the spacing in the streamwise (Spa-L), spanwise (Spa-X), and both (Spa) directions, with a staggered (Stag) layout and a staggered layout with double the spacing in both directions (Stag-Spa). All eight simulations were carried out under neutral atmospheric conditions, with a fixed wind speed of 9 m s

^{−1}at 90 m. Because of vertical wind shear, this meant a wind speed of 8.7 m s

^{−1}at hub height. All simulations had an incoming turbulent inflow with a turbulence intensity (TI) of approximately 7.3% at hub height.

Summary of LES results for the Lillgrund wind farm with neutral stability. Wind direction is in degrees from north, and *P*_{TOT} is in megawatts.

The Simulator for Wind Farm Applications (SOWFA), developed by the National Renewable Energy Laboratory, was used for these simulations. SOWFA uses the Open Source Field Operation and Manipulation (OpenFOAM; http://www.openfoam.org/index.php) computer-software toolbox to solve the incompressible Navier–Stokes equations. The equations were discretized using a finite-volume framework with collocated storage of variables and second-order accuracy in space and time. For these neutrally stratified simulations, the subgrid-scale stresses were computed using the classical Smagorinsky (Smagorinsky 1963) model. The simulations included precursor computations without wind turbines for generating a turbulent atmospheric boundary layer, followed by wind plant computations with wind turbines included. SOWFA has been previously validated for its ability to set up a turbulent atmospheric boundary layer in the absence of wind turbines (Churchfield et al. 2012a) and for its reasonable accuracy in simulating the Lillgrund wind farm (Churchfield et al. 2012b).

Of a total of eight simulations listed in Table 1, details regarding six simulations with a southwesterly wind are described in Archer et al. (2013). As an example of the LES results, the time-averaged horizontal velocity field obtained from the control case (i.e., original layout) is shown in Fig. 1b. The results of these eight LES are used here to formulate statistical models based on geometric measures, defined next.

### b. Geometric measures in wind farms

We assume a wind farm comprising *N*_{T} turbines located over flat terrain or over water. The domain is assumed to be a rectangular box, with the Cartesian *X* and *Y* axes aligned in the horizontal plane, along the east–west and north–south directions, respectively. The *Z* axis is aligned vertically. All turbines have identical hub heights *Z*_{h} and rotor diameters *D*. The corresponding area swept by the rotor blades (swept area) is also identically equal to *A* = *πD*^{2}/4. The farm layout is simply a list of the *X* and *Y* coordinates of each turbine *i*. Following the meteorological convention, the wind direction is described by the angle *θ*, varying from 0° to 360°, that it makes with the north, clockwise. It is further assumed that the nacelle yaw angles change with *θ* in such a way that the turbine swept areas are always perfectly normal to the wind.

Given a layout and a wind direction, a number of purely geometric quantities can be defined for each turbine. Five quantities, which can potentially be useful in modeling the power generated by the wind farm, are defined below. For any such measure *Q*_{i} defined for an individual turbine *i*, the corresponding farm-averaged quantity can also be defined as

*blockage ratio*BR

_{i}, defined as the fraction of the swept area of turbine

*i*that is blocked by the swept area of

*any*upstream turbine, for a given wind direction. Each turbine is assumed to influence (and be influenced by) only turbines within the cylinder with cross section equal to the swept area and aligned along the wind direction. There may be multiple upstream turbines that block the same or additional portions of the swept area of a given turbine. The BR

_{i}is a ratio, varying over the set [0, 1], that is determined by an integral over the swept area of the turbine as follows:

*χ*=

*χ*(

*x*,

*y*) is defined for each point (

*x*,

*y*) on the turbine rotor disk and is equal to 1 if point (

*x*,

*y*) is blocked by any upstream turbine and is zero otherwise. Note that

*χ*is a discontinuous function defined for every point (

*x*,

*y*) on the turbine rotor disk, whereas BR

_{i}is associated with the entire turbine swept area.

*blockage distance*BD

_{i}and is given by the equation

*χ*is as defined above. Similar to

*χ*,

*L*=

*L*(

*x*,

*y*) is a function of every point (

*x*,

*y*) on the turbine rotor disk and denotes the distance to the upstream blocking turbine. Wherever

*χ*= 0, that point on the swept area of turbine

*i*is not blocked by any other turbine swept area and

*L*is infinite. Term

*L*

_{∞}is an arbitrary large, but finite, length that is introduced to ensure that the integral remains finite. In this study, we set

*L*

_{∞}= 20

*D*since previous studies (Xie and Archer 2015; Wu and Porté-Agel 2011) have shown that the velocity profile in a turbine wake almost entirely recovers its original shape by 20

*D*downstream of the turbine. The blockage distance may be thought of as an average of the distances to the upstream blocking turbines, weighted by the fraction of the area blocked by each upstream blocking turbine. The BD

_{i}is therefore not entirely independent of BR

_{i}because BD

_{i}and BR

_{i}are functions of

*χ*. The BD

_{i}can be normalized by dividing it by

*L*

_{∞}.

*inverse blockage distance*IBD

_{i}, is similar to the blockage distance except that the reciprocal of the distance is used in place of the distance to the blocking turbine:

*L*is used, no arbitrary large value needs to be introduced for points on the swept area of turbine

*i*that are not blocked. Similar to BD

_{i}, IBD

_{i}can be considered to be a weighted average of the reciprocal of the distances to upstream blocking turbines, weighted by the fraction of areas blocked. Also, IBD

_{i}is not entirely independent of BR

_{i}because the function

*χ*is embedded in both definitions. IBD

_{i}can be normalized by multiplying it by

*L*

_{∞}.

*blockage distance without weight*DNW

_{i}, is simply the distance to the closest upstream turbine that blocks the swept area of turbine

*i*, without any weighting by the fraction blocked. It can be defined formally as a minimum over all points on the swept area of turbine

*i*, as

*x*,

*y*). As with BD

_{i}, a turbine that is completely unobstructed is assigned the arbitrary value of

*L*

_{∞}, which is set to be 20

*D*in this study. Unlike BD

_{i}and IBD

_{i}, this measure is truly independent of the blockage ratio.

*inverse blockage distance without weight*IDNW

_{i}, which is the inverse of the distance to the closest upstream blocking turbine without weightage by the blockage fraction:

*L*is defined above. Note that the definition of IDNW

_{i}does not contain any arbitrary length and that, similar to DNW

_{i}, IDNW

_{i}is independent of the blockage ratio.

A simple three-turbine wind farm, shown in Fig. 2, is used as an example to illustrate the five geometric measures. Plan views of the layout are shown in Figs. 2a and 2b, with the wind directed along 270° and 320°, respectively. The (*X*, *Y*) coordinates of the three turbines are (20, 30) for T1, (20, 0) for T2, and (60, 0) for T3. These locations are marked with circles. The diameters of the turbines are 20 units each. The swept areas of the turbines are seen as solid lines in Fig. 2a and as dashed lines in Fig. 2b. It is assumed that the turbine yaw angle changes with the wind direction so that the swept area is always perpendicular to the incoming wind. A rotated coordinate system, *x*–*y*, attached to the turbine rotor disks, also changes with the wind direction. As a consequence, *x* makes an angle of 90° with the positive *X* axis in Fig. 2a and an angle of 40° with the positive *X* axis in Fig. 2b. Figure 2c shows the wind farm from the *x*–*Z* plane, when the wind is directed along 320°. The turbine rotor disks are perfect circles in this plane and overlap with each other. The overlap areas are marked in Fig. 2c.

Illustration of geometric measures in a simple three-turbine layout. Plan view with wind from (a) 270° and (b) 320°. (c) View from the rotated plane *x*–*Z* with the wind coming from 320°.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Illustration of geometric measures in a simple three-turbine layout. Plan view with wind from (a) 270° and (b) 320°. (c) View from the rotated plane *x*–*Z* with the wind coming from 320°.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Illustration of geometric measures in a simple three-turbine layout. Plan view with wind from (a) 270° and (b) 320°. (c) View from the rotated plane *x*–*Z* with the wind coming from 320°.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

With the wind blowing in the direction of the positive *X* axis—that is, *θ* = 270°—the turbine swept areas are represented by the solid lines in Fig. 2a. For this configuration, there are no turbines upstream of T1 and T2, and, consequently, BR_{1} = BR_{2} = 0. Turbine T3 is completely blocked by T2, and hence BR_{3} = 1. Quantities BD_{i} and DNW_{i} are both equal to 20*D* for turbines T1 and T2 and are equal to the length *L*(P1, 270°) marked in Fig. 2a for turbine T3. Quantities IBD_{i} and IDNW_{i} are zero for T1 and T2 and are 1/*L*(P1, 270°) for T3. When the wind direction changes to *θ* = 320°, T1 remains unblocked (BR_{1} = 0), a tiny portion of T2 is blocked by T1 (BR_{2} is nonzero), and a large portion of T3 is blocked by T1 (BR_{3} ≠ 1). For this wind direction, the values of the four geometric measures remain unchanged from the 270° wind direction values for turbine T1. For turbine T3, the DNW_{i} and IDNW_{i} will be *L*(P1, 320°) and 1/*L*(P1, 320°), respectively, as marked in Fig. 2b. Terms BD_{i} and IBD_{i} will be weighted averages involving *L*(P1, 320°), *L*_{arb}, and their reciprocals, and with the blocked and unblocked areas as weights. Similarly, the geometric measures for turbine T2 with the wind directed along 320° will be different from the values with the wind along 270°.

To illustrate the two quantities *χ* and *Ł*, consider two points P1 and P2, marked by circles on turbine T3. When the wind is directed along 270°, point P1 on turbine T3 is blocked by turbine T2 and has *χ* = 1; the distance to T2, *L*(P1, 270°), is shown in Fig. 2a. If the wind is directed along 320° (Figs. 2b,c), P1 is blocked by turbine T1. In this case, *χ* = 1 again, but *L*(P1, 320°) now changes to the distance to T1. The other point P2 is blocked by turbine T2 when *θ* = 270°, and *L*(P2, 270°) is identical to *L*(P1, 270°). With *θ* = 320°, P2 is unblocked (*χ* = 0). As a result, the distance *L*(P2, 320°) is infinite, and the arbitrary *L*_{∞} needs to be introduced in Eqs. (3) and (5).

A Fortran 90 code was written to compute the five geometric measures, given a layout and a wind direction. The swept area of each turbine is divided into a large number (on the order of 1 million) of discrete areas. The quantities *χ* and *L* are computed at the geometric center of each discrete area, and the integrals in Eqs. (2)–(6) are evaluated numerically to give the geometric measures. The computation time is dependent almost entirely on the number of turbines and was about 1 min for each of the 80-turbine layouts evaluated in section 4. To evaluate the large number of layouts in a reasonable amount of time, multiple instances of the code were launched simultaneously, with each instance evaluating its subset of layouts, independent of others. Another Fortran 90 program was written for concatenating the results of different layouts and for postprocessing the raw data.

### c. Construction of geometry-based models

*N*

_{T}in Table 1), which can be used for computing statistical averages.

*P*/

*P*

_{max}, with the five directional measures individually. The correlation coefficient between two quantities

*a*and

*b*is defined as

*ρ*= ±1 indicates perfect (positive or negative) correlation and that

*ρ*= 0 indicates no correlation between

*a*and

*b*. Since

*ρ*(

*a*,

*b*) = −

*ρ*(

*a*, −

*b*), for the purpose of determining whether two quantities correlate well with each other it is sufficient to look only at the magnitude of the correlation between them, without regard to its sign.

The second column in Table 2 shows that the blockage ratio has the highest correlation among all five measures, with a correlation-coefficient magnitude of 0.89. The two weighted distance measures, blockage distance and inverse blockage distance, are much better correlated with relative power than are the other two distance measures without weight.

Correlation coefficients *ρ* of geometric measures *Q* with relative power *P*/*P*_{max} extracted from LES of the Lillgrund wind farm [*ρ*(*P*/*P*_{max}, *Q*)].

For each wind farm layout, there are a few turbines that have a blockage ratio equal to zero. Depending on the wind direction, these turbines may or may not be located exclusively in the front row of the wind farm. Indeed, a front row can be identified only when the wind direction is perfectly aligned with the rows or columns—for example, 225° for Lillgrund. For all other layouts with rows/columns that are not perfectly aligned with the wind, it is impossible to identify unequivocally the front row. To avoid such ambiguity for layouts with arbitrary wind orientations, we refer to turbines with BR_{i} = 0 as “front row” turbines, even though in a few cases such turbines may actually be in the interior or at the edges of the wind farm.

Since these front-row turbines are not blocked by any upstream turbines, it is assumed that they encounter the undisturbed wind and generate the maximum power, which does not need to be modeled. We assign a relative power of 1 to these front-row turbines and only model those turbines for which BR_{i} ≠ 0. To evaluate the effectiveness of such a modeling strategy, the third column in Table 2 shows correlation coefficients between the relative power and the five geometric measures, with *P*/*P*_{max} enforced to be 1 for all data points that have BR_{i} = 0. The turbines with BR_{i} = 0 are, in a sense, excluded from the modeling, and we term this strategy as one with “front turbines excluded.” With the front turbines excluded, again the relative power is best correlated with the blockage ratio (|*ρ*| = 0.94), and the blockage distance and inverse blockage distance are better than the other two distances. We further note that the correlations in the third column are consistently better than those in the second column, which indicates that a strategy of modeling only those turbines that have BR_{i} ≠ 0 is likely to be better than a strategy of modeling all turbines.

The inverse correlation of the relative power generated by a turbine with its blockage ratio, characterized by a negative *ρ*, appears to be intuitively correct. A turbine whose swept area is blocked by upstream turbines to a lesser extent (smaller BR_{i}) produces more power than a turbine that is heavily blocked (larger BR_{i}). Furthermore, while the magnitudes of these correlations are consistently high (larger than 0.89), it would be logical to expect that the blockage ratio is an incomplete measure of the wind farm, since it does not include any information about the length scales available for the wakes behind turbines to diffuse and dissipate and for the velocity profile to recover its original shape.

*P*

_{i}/

*P*

_{max}and the blockage ratios BR

_{i}with front-row turbine data included yields model

*f*

_{1}(BR

_{i}). A multiple linear regression with

*P*

_{i}/

*P*

_{max}as the dependent variable and BR

_{i}and BD

_{i}/

*L*

_{∞}as the independent variables gives model

*f*

_{2}(BR

_{i}, BD

_{i}). Similarly, model

*f*

_{3}(BR

_{i}, IBD

_{i}

*L*

_{∞}) is obtained by a multiple linear regression comprising BR

_{i}and IBD

_{i}. Similar linear regression analyses based on data with the front (BR

_{i}= 0) turbines excluded yield models

*f*

_{4}(BR

_{i}),

*f*

_{5}(BR

_{i}, BD

_{i}) and

*f*

_{6}(BR

_{i}, IBD

_{i}). Note that the distance measures need to be nondimensionalized in order to combine them with the (nondimensional) blockage ratio.

The six models *f*_{1}–*f*_{6} are listed in Table 3 along with the correlations between the actual and predicted relative powers *ρ*(*P*_{i}/*P*_{max}, *f*_{m}) for *m* = 1, 2, 3, …, 6. All models show excellent correlations (larger than 0.89) with the relative powers obtained via LES. The models with front-row turbines excluded show even higher correlations, in excess of 0.95. As expected, the correlation coefficients *ρ*(*P*_{i}/*P*_{max}, *f*_{1}) and *ρ*(*P*_{i}/*P*_{max}, *f*_{4}) in Table 3 are identical in magnitude to the corresponding correlations listed for BR_{i} in Table 2. Furthermore, models based on a combination of BR_{i} with BD_{i} have higher correlations with the relative powers than do models based on BR_{i} alone. For example, *ρ*(*P*_{i}/*P*_{max}, *f*_{2}) > *ρ*(*P*_{i}/*P*_{max}, *f*_{1}) and *ρ*(*P*_{i}/*P*_{max}, *f*_{5}) > *ρ*(*P*_{i}/*P*_{max}, *f*_{4}). Similarly, models *f*_{3} and *f*_{6} are better correlated with *P*_{i}/*P*_{max} than are models *f*_{1} and *f*_{4}, respectively. This indicates that the blockage ratio supplemented by one of the distance measures is a better indicator of the wind farm power.

Geometry-based models for *P*_{i}/*P*_{max} in terms of geometric measures, based on linear regression of data from LES of the Lillgrund wind farm.

Figure 3 shows the raw data from the LES simulations, marked by the different symbols, along with the fitting lines corresponding to functions *f*_{1}–*f*_{6}. In all of the subfigures shown, the high correlation between the LES results and the model predictions is apparent. Also, the numerous data points corresponding to BR_{i} = 0 in Figs. 3a–c collapse to a single point in Figs. 3d–f. As a result, the “spread” around the fitting line is smaller in these figures as compared with Figs. 3a–c. This is also reflected in the correlation coefficients accompanying the plots and listed in Table 2.

Raw data (from neutral LES results) and linear regression fitting lines used in the construction of geometry-based models, with front-row turbines (a)–(c) included and (d)–(f) excluded.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Raw data (from neutral LES results) and linear regression fitting lines used in the construction of geometry-based models, with front-row turbines (a)–(c) included and (d)–(f) excluded.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Raw data (from neutral LES results) and linear regression fitting lines used in the construction of geometry-based models, with front-row turbines (a)–(c) included and (d)–(f) excluded.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Models combining BR_{i} with the two nonweighted distance measures (DNW_{i} and IDNW_{i}) were also investigated and were found to yield slightly smaller correlations than the models presented here in Table 3. For the sake of brevity, models based on DNW_{i} and IDNW_{i} are not explored further in this paper. We also note that only linear models of geometric measures have been considered in this paper. Along similar lines, it may be possible to develop nonlinear models based on either linear or nonlinear combinations of the different geometric measures. In particular, the correlations with inverse distances suggest the use of hyperbolic functions. However, this is not pursued in this study in view of retaining simplicity and in view of the sufficiently high correlations obtained using linear models.

## 3. Validation of geometry-based models

The geometry-based models developed in the previous section are based on linear regression of data from LES of the Lillgrund wind farm. Before employing these models for general layout studies, they are evaluated for their ability to reproduce the results of LES and field observations in the Lillgrund and Horns Rev wind farm configurations.

### a. Lillgrund wind farm

Any statistical model must, at a minimum, reproduce the data that went into building it. The geometry-based models developed are first validated by comparing their predictions for individual turbines with the LES data. Figure 4 compares the model predictions with the actual relative power obtained from the LES of the original Lillgrund layout, with the wind blowing from the southwest (case 225 in Table 1). Since the wind turbines in Lillgrund are aligned along southwest–northeast, it is natural to present the results for this wind direction along columns B–E as shown in Fig. 1a.

Comparisons of relative power of turbines in the Lillgrund wind farm along columns (a),(b) B; (c),(d) C; (e),(f) D; and (g),(h) E for models based on LES data with front-row turbines (left) included and (right) excluded. LES results were reported by Archer et al. (2013).

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Comparisons of relative power of turbines in the Lillgrund wind farm along columns (a),(b) B; (c),(d) C; (e),(f) D; and (g),(h) E for models based on LES data with front-row turbines (left) included and (right) excluded. LES results were reported by Archer et al. (2013).

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Comparisons of relative power of turbines in the Lillgrund wind farm along columns (a),(b) B; (c),(d) C; (e),(f) D; and (g),(h) E for models based on LES data with front-row turbines (left) included and (right) excluded. LES results were reported by Archer et al. (2013).

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

It is apparent that the geometry-based models capture the patterns of variability of turbine power along the columns very well. While there are underpredictions and overpredictions at some individual turbines, overall the model results compare very well to the LES results. Furthermore, the models with the front excluded (*f*_{4}–*f*_{6}; panels in the right column of Fig. 4) appear to be slightly more accurate than the models with front turbines included in their construction (*f*_{1}–*f*_{3}; panels in the left column of Fig. 4), as expected.

The ability of the models to reproduce field observations is evaluated next. As compared with predicting simulation data, reproducing observational data is a more rigorous test for the geometry-based models, because field observations invariably involve uncertainty regarding the wind speed, direction, and atmospheric stability conditions. For example, the observations at the Lillgrund wind farm reported in Dahlberg (2009) are averaged over several years and include different wind speeds. Most significant is that the observations lack information regarding atmospheric stability. It is thus likely that the averaged profiles obtained from observations include various stability conditions and cannot be reproduced precisely by models trained on data from well-controlled numerical simulations under neutral stability.

The power generated by turbine C2, normalized by the power generated by turbine C1 (see Fig. 1a for their positions), as predicted by several models, is compared in Fig. 5a with the observations reported in Dahlberg (2009). Wind directions varying from 200° to 240° are considered. Figure 5a shows that the geometry-based models overpredict *P*(*C*_{2})/*P*(*C*_{1}) over most angles. The overprediction is larger for wind directions in the vicinity of 220°. Nevertheless, the overall shape of the profile is preserved. Also, as before, models with the front-row turbines excluded (*f*_{4} and *f*_{5}) are more accurate than the models that include front-row turbines (*f*_{1} and *f*_{2}).

Comparisons of (a) power of turbine C2 normalized by power of turbine C1 in the Lillgrund wind farm for various wind directions with field observations reported in Dahlberg (2009) and (b) power of turbine G2 normalized by power of turbine G1 in the Horns Rev wind farm for various wind directions with field observations reported in Hansen et al. (2012).

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Comparisons of (a) power of turbine C2 normalized by power of turbine C1 in the Lillgrund wind farm for various wind directions with field observations reported in Dahlberg (2009) and (b) power of turbine G2 normalized by power of turbine G1 in the Horns Rev wind farm for various wind directions with field observations reported in Hansen et al. (2012).

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Comparisons of (a) power of turbine C2 normalized by power of turbine C1 in the Lillgrund wind farm for various wind directions with field observations reported in Dahlberg (2009) and (b) power of turbine G2 normalized by power of turbine G1 in the Horns Rev wind farm for various wind directions with field observations reported in Hansen et al. (2012).

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Figure 6 plots the average relative power *P*_{rel} reported in Table 1 against the farm-averaged blockage ratio. The figure is composed of eight data points, one corresponding to each of the LES cases. Although the number of data points is limited, these global quantities can be seen to be almost perfectly correlated, with *ρ* ≈ −1. The correlations between *P*_{rel} and the global averages of the independent variables corresponding to the other five geometry-based models have also been examined, and they too are found to be consistently and significantly higher than the correlations based on individual turbines. This indicates that these geometry-based models are likely to predict the global average power generated by a wind farm much more accurately than the power generated by individual turbines.

Farm-averaged relative power as a function of farm-averaged blockage ratio, from the LES results. Fitting line and correlation coefficient is also shown.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Farm-averaged relative power as a function of farm-averaged blockage ratio, from the LES results. Fitting line and correlation coefficient is also shown.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Farm-averaged relative power as a function of farm-averaged blockage ratio, from the LES results. Fitting line and correlation coefficient is also shown.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Over the eight LES cases, the farm-averaged power predictions of models *f*_{4} and *f*_{5} are found to have a bias error of approximately −6% and an rms error of approximately 4%. The negative value for the bias error indicates that these models underpredict the farm-averaged power by about 6% on average. The rms error provides a threshold value that can be useful while evaluating different layouts (see section 5). A comparison between two layouts can be considered meaningful only if the predicted powers differ by more than 4%.

### b. Horns Rev wind farm

The performance of the geometry-based models is next assessed for a layout other than that of the Lillgrund wind farm. The Horns Rev wind farm, located in the North Sea off the coast of Denmark, comprises 80 turbines arranged in an almost rectangular layout of 10 numbered rows and eight lettered columns (Hansen et al. 2012; Gaumond et al. 2012). Figure 5b shows the power generated by turbine G2 over that of turbine G1 as a function of wind direction. The geometry-based model predictions are compared with field observations reported in Hansen et al. (2012), where the turbines are referred to as 17 and 7, respectively. The geometry-based models capture the overall trend; that is, the power is reduced in the vicinity of 270°. However, it can be seen that the power is underpredicted on either side near 270° and overpredicted elsewhere. Recall, however, that, while the geometry-based models may not predict the power generated by individual turbines accurately, they are expected to be much more accurate for the farm-averaged quantities.

The relative power generated by the Horns Rev wind farm under different wind directions, as given by an LES study by Porté-Agel et al. (2013), is compared with geometric model predictions in Fig. 7. These LES simulations assumed a velocity of 8 m s^{−1} and a turbulence intensity of 7.7% at hub height (Porté-Agel et al. 2013). Predictions of *f*_{4} and *f*_{5} are shown in Fig. 7. As expected, the prediction of the farm-averaged relative power is more accurate than the prediction of the individual turbines seen in Fig. 5b. All the major trends in the LES results are reproduced by the two models. The magnitudes of relative powers are also captured reasonably accurately, although the geometry-based models display more variability than the LES results. Model *f*_{5}, based on the weighted distance BD_{i}, displays some overshoots and undershoots, while *f*_{4}, based on simply the blockage ratio mainly underpredicts the relative power. The LES results and the model predictions are well correlated, with coefficients 0.81 for *f*_{4}(BR_{i}) and 0.87 for *f*_{5}(BR_{i}, BD_{i}). The improvement in the relative power prediction offered by *f*_{5} over *f*_{4} also demonstrates the contribution of the distance measure BD_{i} over and above that of the blockage ratio BR_{i}. Overall, the geometry-based models predict the relative power with 80%–90% accuracy. Figure 7 also demonstrates the power of geometry-based models, since the broad features of the LES results are generated at a fraction of the cost associated with the LES.

Farm-averaged relative power as a function of wind direction in the Horns Rev wind farm, obtained from the geometry-based models and from LES reported in Porté-Agel et al. (2013).

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Farm-averaged relative power as a function of wind direction in the Horns Rev wind farm, obtained from the geometry-based models and from LES reported in Porté-Agel et al. (2013).

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Farm-averaged relative power as a function of wind direction in the Horns Rev wind farm, obtained from the geometry-based models and from LES reported in Porté-Agel et al. (2013).

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

### c. Translating geometry-based models to other conditions

The applicability of geometry-based statistical models trained on one set of data (LES of the Lillgrund wind farm, in this case) to other conditions is considered here. The power generated by a wind farm can be thought to depend on six parameters: the number and type of turbines, turbine layout, wind speed, atmospheric TI, and atmospheric stability. Section 3b showed that the models trained on LES of Lillgrund translated well to LES of Horns Rev. The two farms differed in the first four parameters, that is, number, type and layout of turbines, and wind speeds (8.7 m s^{−1} in Lillgrund vs 8 m s^{−1} in Horns Rev at their respective hub heights). Both sets of LES were conducted under neutral stability and had similar levels of incoming turbulence intensities (7.3% in Lillgrund and 7.7% in Horns Rev at hub height). This indicates that the geometry-based models are able to account for differences in the first four parameters, as long as the last two are fixed.

The dependence on TI and atmospheric stability needs to be built into the geometry-based models. We anticipate that the correlations between the relative powers and the geometric quantities will still hold under different TI and stability conditions, albeit with different coefficients for the fitting functions. This issue will be considered in future studies.

It is always challenging to compare model predictions with field observations. First, field observations are often averaged over large periods of time; for example, the data in Dahlberg (2009) are averaged over a 2-yr period and therefore include flow conditions with widely varying TI and atmospheric stability levels. Furthermore, it is difficult to isolate conditions that match a few given criteria—for example, a certain wind speed at hub height—yet yield a sufficient number of measurements for meaningful averaging. Comparisons between model predictions of farm-averaged power data and field observations for different directions can only be carried out if good-quality observations are available and once the effects of TI and stability have been incorporated in the geometry-based models. For these reasons, we do not present comparisons with farm-averaged observations in this study.

In short, we conclude that geometry-based models calibrated on a few LES of a specific wind farm can reproduce the LES results of that wind farm, including different layouts and wind speeds, with a good level of accuracy. Furthermore, the geometry-based models can predict the farm-averaged power obtained from LES of other wind farms with reasonable accuracy. The models, however, cannot be assumed to translate well to wind farms operating under different levels of TI and stability or to reproduce field observations. In their present form, though, the geometry-based models can be used to study the effects of different design parameters under neutral conditions, and this is considered in the rest of the paper.

## 4. Evaluation of wind farm layouts using geometry-based models

The geometry-based models developed and evaluated above are used here to study the effects of wind turbine layout in a hypothetical wind farm comprising 80 turbines, arranged in 10 rows and eight columns, similar to Horns Rev. We introduce four design parameters, as shown in Fig. 8: the spacing SX between consecutive turbines in the *X* direction, the spacing SY between consecutive turbines in the *Y* direction, the staggering SDY of alternate rows in the *Y* direction, and the angle *β* between rows and columns. When *β* = 90°, the turbines are laid out in a regular rectangular grid and a rectangle bounding the entire layout can be identified easily (Fig. 8a), even when SDY ≠ 0 (Fig. 8b). We term all layouts of the types shown in Figs. 8a,b as “rectangular.” Layouts characterized by a bounding rhombus or a bounding diamond, as exemplified in Fig. 8c, are referred to as “nonrectangular,” and the spacing between columns is denoted SY, although this is not exactly along the *Y* direction (Fig. 8c).

Examples of wind farm layouts with different design variables. Solid circles represent individual turbines. (a) Rectangular layout without staggering of alternate rows, (b) rectangular layout with alternate rows staggered, and (c) rhomboidal layout, with nonorthogonal “rows” and “columns.”

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Examples of wind farm layouts with different design variables. Solid circles represent individual turbines. (a) Rectangular layout without staggering of alternate rows, (b) rectangular layout with alternate rows staggered, and (c) rhomboidal layout, with nonorthogonal “rows” and “columns.”

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Examples of wind farm layouts with different design variables. Solid circles represent individual turbines. (a) Rectangular layout without staggering of alternate rows, (b) rectangular layout with alternate rows staggered, and (c) rhomboidal layout, with nonorthogonal “rows” and “columns.”

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

*D*, 12

*D*], SY over [2

*D*, 8

*D*], and SDY over [0, 5SY/6] in steps of

*D*,

*D*, and SY/6, respectively (10 × 7 × 6 = 420). In addition, 576 nonrectangular layouts are investigated in section 4d, with SX ∈ [3

*D*, 10

*D*], SY ∈ [3

*D*, 10

*D*], and

*β*∈ [70°, 90°] in steps of

*D*,

*D*, and 5°, respectively. For each rectangular layout, 37 wind directions are assumed, with

*θ*varying from 270° to 360° in steps of 2.5°. The range of wind directions is confined to only 90° since there is a four-way symmetry associated with rectangular layouts. Given a generic geometric measure

*G*(

*θ*),

*θ*∈ [270°, 360°], the measure for other wind angles can be determined using

The different wind directions considered here are assumed to be independent of each other. Keeping the farm layout fixed and varying the wind directions is equivalent to fixing the wind direction and rotating the entire layout. Thus, evaluating the effect of different wind directions, in effect, allows for a study of layouts with different orientations with respect to a fixed wind direction, for example, the prevailing one. Throughout this paper, we refer to wind directions and wind farm orientations interchangeably. The effect of frequency distributions of wind directions (described by wind roses) will be considered in a subsequent study.

For each layout and each wind direction, the three geometric measures for each turbine as well as the farm-averaged measures and the relative power predicted by the six models are evaluated. While it is impractical, and indeed unnecessary, to present all of the results, a few general observations and important trends are discussed below. The results of the six models are found to be very similar, and therefore only *f*_{5} is discussed.

### a. Effect of wind direction

The effect of wind direction on the power production in a nonstaggered wind farm is shown in Fig. 9. The results of four combinations of streamwise and spanwise spacings are shown. For all four combinations of spacings, the power production curves show local minima at *θ* = 270°, 315°, and 360°. The first of these angles corresponds to the wind directed exactly along the columns. As a result, only the eight turbines at the extreme west of the wind farm are exposed to the undisturbed wind and all other turbines are completely shadowed (BR_{i} = 1) by these eight turbines, resulting in a very low value of the predicted power. Similarly, the other two angles correspond to the wind directed precisely along the rows and the diagonals of the wind farm, respectively, which again leads to BR_{i} = 1 for many of the turbines and results in small values of the predicted power.

Effect of wind direction on the relative power in a hypothetical wind farm with SDY = 0 and (a) SY = 4*D* and (b) SY = 8*D*.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Effect of wind direction on the relative power in a hypothetical wind farm with SDY = 0 and (a) SY = 4*D* and (b) SY = 8*D*.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Effect of wind direction on the relative power in a hypothetical wind farm with SDY = 0 and (a) SY = 4*D* and (b) SY = 8*D*.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Figure 9 further shows that the generated power increases drastically if the wind is slightly misaligned with respect to the three perfectly aligned directions discussed above. In all four curves, the power increases rapidly around *θ* = 270°, 315°, and 360°. In comparing the curves with squares and diamonds in each panel, it is seen that the increase in power in the vicinity of 270° is much more dramatic if the spacing SX is larger. In the vicinity of 360°, however, the increase is dependent on the spacing SY rather than on SX.

The best offset angle at which the power curves display local maxima depends on the ratio *S*_{al}/*S*_{ac}, where *S*_{al} stands for spacing along the wind direction and *S*_{ac} stands for the spacing across the wind direction. Note here that the directions represented by the labels “along” and “across” vary with the wind direction. For example, with the wind directed along 270°, SX represents the spacing along the wind direction and SY represents the spacing across the wind direction. Conversely, with the wind directed along 360°, the labels are reversed, with SY and SX representing spacings along and across, respectively. For the curve for SX = 12*D* and SY = 4*D* (Fig. 9a), the local maximum in the vicinity of *θ* = 270° occurs for the offset Δ*θ* = 5°. In the vicinity of *θ* = 360°, local maxima of power occur for Δ*θ* = 15°–20°. The values of the ratio *S*_{al}/*S*_{ac} for these two directions are 3 and ⅓, respectively. Similarly, for the curve for SX = 4*D* and SY = 8*D* (Fig. 9b), the local maxima of power in the vicinities of *θ* = 270° (*S*_{al}/*S*_{ac} = ½) and *θ* = 360° (*S*_{al}/*S*_{ac} = 2) occur at Δ*θ* = 15° and 7.5°, respectively. This indicates the general trend that the best offset angle, with respect to a layout with rows or columns perfectly aligned with the wind direction, varies from approximately 15°–20° for smaller *S*_{al}/*S*_{ac} to around 5°–7.5° for larger values of *S*_{al}/*S*_{ac}.

These findings regarding the benefit of slight misalignment of rows of turbines with the wind direction are similar to those made by Stevens et al. (2014). The geometry-based models confirm that the findings made via LES of infinite arrays of turbines continue to hold for large, but finite, wind farms. The geometry-based models, however, incur a fraction of the computational cost of the LES of Stevens et al. (2014).

### b. Effect of turbine spacing

From Fig. 9, the power with SX = 12*D* is smaller than that with SX = 4*D* in a few cases; for example, *θ* = 287.5° for SY = 4*D* and *θ* = 302.5° − 305° for SY = 8*D*. Thus, the general impression that larger spacing between turbines usually leads to smaller wake losses is found to be violated in a few stray cases. To examine this issue further, in Fig. 10 the variation of the power with the spacing SX is plotted for different values of SY without staggering, SDY = 0. Three wind directions are considered, corresponding to the offsets rated best according to Fig. 9.

Effect of axial spacing on the normalized power production in a hypothetical wind farm with SDY = 0, varying SY, and wind direction (a) 277.5°, (b) 287.5°, and (c) 345°.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Effect of axial spacing on the normalized power production in a hypothetical wind farm with SDY = 0, varying SY, and wind direction (a) 277.5°, (b) 287.5°, and (c) 345°.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Effect of axial spacing on the normalized power production in a hypothetical wind farm with SDY = 0, varying SY, and wind direction (a) 277.5°, (b) 287.5°, and (c) 345°.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

With the wind directed along *θ* = 270° + 7.5°, Fig. 10a shows that increasing SX leads to a more or less monotonic increase in the power prediction, for all SY. Figure 10b shows complicated interaction between SX and SY with the wind from *θ* = 270° + 17.5°. For three of the four SY, the maximum spacing SX = 12*D* is found to be suboptimal. For SY = 4*D* and SY = 8*D*, the largest SX in fact yields the smallest power, smaller than even when SX is as small as 3*D*.

Looking at the effect of the spacing with the wind directed along the third of the optimal angles reported in Fig. 9—that is, *θ* = 345°—the curves in Fig. 10c display the general trend of increasing power with increasing SX, followed by a saturation beyond a certain point. For SY = 12*D* the saturation is reached for SX as low as 5*D*, whereas for SY = 4*D* the power does not increase beyond SX = 9*D*. The power generated with SY = 6*D* and SY = 8*D* does not saturate with increasing SX. Another striking observation in Fig. 10c is that for SX > 6*D* the power generated is maximum when SY = 4*D* and that for SX > 9*D* the model *f*_{5} predicts zero wake losses when SY = 4*D*.

The reasons for the complicated interactions among SX, SY, and *θ* are purely geometric. Turbines that are not blocked by any upstream turbine in a layout with smaller spacing can, under certain wind directions, move into the shadow of an upstream turbine when the spacing is increased. Consequently, layouts with larger spacings produce less power than layouts with smaller spacings, in a few instances seen (Figs. 9, 10).

Although predictions of *f*_{5} alone are shown in Fig. 10 (and all other figures in this section), the predictions of other models, particularly *f*_{4}, are qualitatively similar with slight quantitative differences.

### c. Effect of staggering of alternate rows

The numerical simulations by Archer et al. (2013) showed that the power at Lillgrund increased by almost 33% over the perfectly aligned configuration when alternate rows were staggered. In that study, alternate rows were displaced by SDY = SY/2 to yield a configuration that may be termed “perfectly staggered.” Figure 11 shows the effect of two additional steps between the perfectly aligned and perfectly staggered configurations (SDY = SY/6 and SDY = SY/3) and two steps beyond the perfectly staggered configuration (SDY = 4SY/6 and SDY = 5SY/6). Four different combinations of turbine spacings and two different wind directions are considered. With the wind directed along 270°, Fig. 11a shows that, for small values of SY (solid lines), the power first increases monotonically with increasing SDY and that the perfectly staggered configuration yields the optimal power. The relative power prediction then decreases as staggering increases beyond SDY = SY/2. For larger values of SY (dashed lines), however, the optimum power is reached at SDY = SY/6, well before the perfectly staggered configuration. In other words, staggering of alternate rows is beneficial only up to the point at which the turbines in the second row step out of the shadow of the turbines in the first row. Increasing SDY beyond this point does not lead to an increase in power.

Effect of staggering of alternate rows of turbines in a hypothetical wind farm for four combinations of axial and spanwise spacings, with the wind from (a) 270° and (b) 277.5°. Also shown is the change in the normalized power production due to staggering in a hypothetical wind farm for various wind directions and staggerings, and with (c) SX = 8*D* and SY = 4*D* and (d) SX = 4*D* and SY = 8*D*. SDY = 0 denotes the unstaggered configuration, and SDY = SY/2 denotes the perfectly staggered configuration.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Effect of staggering of alternate rows of turbines in a hypothetical wind farm for four combinations of axial and spanwise spacings, with the wind from (a) 270° and (b) 277.5°. Also shown is the change in the normalized power production due to staggering in a hypothetical wind farm for various wind directions and staggerings, and with (c) SX = 8*D* and SY = 4*D* and (d) SX = 4*D* and SY = 8*D*. SDY = 0 denotes the unstaggered configuration, and SDY = SY/2 denotes the perfectly staggered configuration.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Effect of staggering of alternate rows of turbines in a hypothetical wind farm for four combinations of axial and spanwise spacings, with the wind from (a) 270° and (b) 277.5°. Also shown is the change in the normalized power production due to staggering in a hypothetical wind farm for various wind directions and staggerings, and with (c) SX = 8*D* and SY = 4*D* and (d) SX = 4*D* and SY = 8*D*. SDY = 0 denotes the unstaggered configuration, and SDY = SY/2 denotes the perfectly staggered configuration.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Figure 11b shows that the staggered configurations are not always beneficial as compared with the nonstaggered configurations when the wind is directed along *θ* = 270° + 7.5°. Note that, irrespective of the wind direction, the staggering of alternate rows is always defined in the north–south direction along *Y* in this study. Figure 11b shows that different values of SDY yield the optimum power, depending on the turbine spacings SX and SY.

The advantage offered by configurations with nonzero SDY over configurations with SDY = 0, over all wind directions for two combinations of turbine spacings, is shown in Figs. 11c and 11d. The quantity plotted is Δ*f*_{5} = *f*_{5}(SDY > 0) − *f*_{5}(SDY = 0). A complex interplay between the turbine spacings, staggering, and wind directions is seen. For example, with the wind directed along 315°, staggering is beneficial (Δ*f*_{5} > 0) in a wind farm with SX = 8*D* and SY = 4*D* but not beneficial in a wind farm with SX = 4*D* and SY = 8*D*. Staggering leads to increased power in both of these farms with the wind directed along 270°, but it leads to a reduction in power in both wind farms with the wind along 285°. Since the staggering is in the north–south direction, it has no effect on the power production when the wind is directed along 360°. Also, there might be instances in which intermediate staggering, SDY = SY/6 or SY/3, is preferable to perfect staggering, SDY = SY/2, for example, as seen in Fig. 11c for *θ* = 285° and in Fig. 11d for *θ* = 315°.

In addition to two intermediate steps between the perfectly aligned and perfectly staggered configurations, layouts with SDY = 4SY/6 and SDY = 5SY/6 were also investigated. Figures 11a and 11b show that, as may be expected from geometry, the relative power predictions for layouts with SDY = 4SY/6 were found to be almost exactly equal to the relative power predictions for layouts with SDY = SY/3. Similarly, layout configurations with SDY = 5SY/6 were found to be almost identical to configurations with SDY = SY/6. This was found to hold for all combinations of spacings and wind directions examined. The staggered configurations, thus, show symmetry around the corresponding perfectly staggered (SDY = SY/2) configurations.

### d. Nonrectangular layouts

The effect of changing *β*, the angle between the rows and the columns of the wind farms, is considered in this section. This angle was fixed to be *β* = 90° in sections 4a–c. As explained earlier, orthogonal layouts involve a four-way symmetry whereas nonorthogonal layouts exhibit only two-way symmetry. Thus, in order to examine the influence of wind blowing over all directions, a sector of width 180° needs to be considered for nonorthogonal layouts, as opposed to a 90°-wide sector for orthogonal layouts. We consider the sector [270°, 450°], which may also be referred to as ranging over [−90°, 90°].

Figures 12a and 12b plot the changes in the power brought about by changing *β* to 70° and 110° for different wind directions. Two combinations of the spacings SX and SY are considered. The dotted line marks the zero level, corresponding to *β* = 90° (rectangular layout). The power produced can either increase or decrease [*f*_{5} − *f*_{5}(*β* = 90°) > 0 or < 0, respectively] in nonorthogonal layouts, in comparison with orthogonal layouts, depending on the wind direction. For both combinations of SX and SY considered, the largest benefit occurs for a wind direction around 360°, whereas there is no benefit as *θ* → ±90°. This is easily understood, since changing *β* to nonorthogonal values leads to an increase in the number of turbines directly encountering the undisturbed wind when the wind is along 360°, whereas the number of front-row turbines does not change for wind directions of ±90°.

Effect of nonorthogonal arrangement of rows and columns (*β* ≠ 90°) in a hypothetical wind farm: The difference between the predictions of *f*_{5} for *β* ≠ 90° and *β* = 90° for varying wind directions and (a) SX = 8*D* and SY = 4*D* and (b) SX = 4*D* and SY = 8*D*. Also shown is the effect of *β* on (c) the power predicted and (d) the difference between the predicted power and that predicted for the corresponding orthogonal layout, for various SX and a fixed SY = 4*D*, with the wind directed along 315°.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Effect of nonorthogonal arrangement of rows and columns (*β* ≠ 90°) in a hypothetical wind farm: The difference between the predictions of *f*_{5} for *β* ≠ 90° and *β* = 90° for varying wind directions and (a) SX = 8*D* and SY = 4*D* and (b) SX = 4*D* and SY = 8*D*. Also shown is the effect of *β* on (c) the power predicted and (d) the difference between the predicted power and that predicted for the corresponding orthogonal layout, for various SX and a fixed SY = 4*D*, with the wind directed along 315°.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Effect of nonorthogonal arrangement of rows and columns (*β* ≠ 90°) in a hypothetical wind farm: The difference between the predictions of *f*_{5} for *β* ≠ 90° and *β* = 90° for varying wind directions and (a) SX = 8*D* and SY = 4*D* and (b) SX = 4*D* and SY = 8*D*. Also shown is the effect of *β* on (c) the power predicted and (d) the difference between the predicted power and that predicted for the corresponding orthogonal layout, for various SX and a fixed SY = 4*D*, with the wind directed along 315°.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

The effect of *β* for one particular wind direction (*θ* = 315°), fixed SY = 4*D*, and different SX is shown in Figs. 12c and 12d. For SX = 4*D* and SX = 8*D*, all of the nonorthogonal layouts evaluated are better than the orthogonal layout. This is seen in Fig. 12c as well as in Fig. 12d, which plots the difference between the power predicted and that predicted for the corresponding orthogonal (*β* = 90°) layout. For SX = 10*D*, however, most of the layouts generate less power as compared with the orthogonal layout. In particular, layouts with highly obtuse *β* > 100° generate much less power as compared with layouts with *β* ≤ 95°. This illustrates that the effect of *β* can be different for different turbine spacings.

### e. Layouts with zero blockage ratio

Of the 15 540 combinations of rectangular layouts and wind directions evaluated in this section, 225 had a global blockage ratio equal to zero. In other words, in these configurations, none of the turbines’ swept area was blocked by that of other upstream turbines. Evidently, all of the geometry-based models predicted very high values of power for these cases. It is instructive to examine these layouts more closely.

All the layouts with BR = 0 had SX at least equal to 9*D*. The number of cases with BR = 0 increased with increasing SX. For SX = 9*D*, BR = 0 was found in only 15 cases, whereas the number of cases with BR = 0 increased to 96 when SX increased to 12*D*. For a given (sufficiently large) SX, layouts with BR = 0 were found for all values of SY—even as small as 2*D* or 3*D*. This indicates that wind farms that keep wake losses to very low values, while occupying very small areas, can be constructed by keeping spacing in one direction very small and making appropriate choices of the spacing in the other direction and of the alignment with the wind direction.

A few layouts that yield zero BR are shown in Fig. 13. The differences in the *X* and *Y* extents of different panels should be noted. As mentioned above, Figs. 13a and 13c have a very small SY = 3*D*, although the spacing SX = 9*D* is large. The wind directions for which zero BR is obtained are shown in each panel. For (SX, SY) = (9*D*, 3*D*) and for (SX, SY) = (9*D*, 8*D*), zero blockage is obtained for only one wind direction each. As the spacing SX increases, the sectors of wind directions for which zero blockage is obtained increase steadily. As shown in Figs. 13c and 13d, zero blockage ratios are obtained for 7.5°- and 5°-wide sectors for layouts with (SX, SY) = (12*D*, 3*D*) and (SX, SY) = (12*D*, 8*D*), respectively. Comparing Figs. 13a and 13c with Figs. 13b and 13d, respectively, it is seen that the effect of increasing spacing SY leads to a change in the wind direction that yields zero blockage. The width of the sector over which BR = 0 is only modestly affected by the changes in SY.

Examples of layouts with global blockage ratio BR = 0. Arrows and labels indicate wind directions over which BR = 0. Rectangular layouts, *β* = 90°, with (a) SX = 9*D* and SY = 3*D*; (b) SX = 9*D* and SY = 8*D*; (c) SX = 12*D* and SY = 3*D*; and (d) SX = 12*D* and SY = 8*D*. Also shown are nonrectangular layouts with SX = 9*D* and SY = 8*D* and (e) *β* = 70° and (f) *β* = 110°.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Examples of layouts with global blockage ratio BR = 0. Arrows and labels indicate wind directions over which BR = 0. Rectangular layouts, *β* = 90°, with (a) SX = 9*D* and SY = 3*D*; (b) SX = 9*D* and SY = 8*D*; (c) SX = 12*D* and SY = 3*D*; and (d) SX = 12*D* and SY = 8*D*. Also shown are nonrectangular layouts with SX = 9*D* and SY = 8*D* and (e) *β* = 70° and (f) *β* = 110°.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Examples of layouts with global blockage ratio BR = 0. Arrows and labels indicate wind directions over which BR = 0. Rectangular layouts, *β* = 90°, with (a) SX = 9*D* and SY = 3*D*; (b) SX = 9*D* and SY = 8*D*; (c) SX = 12*D* and SY = 3*D*; and (d) SX = 12*D* and SY = 8*D*. Also shown are nonrectangular layouts with SX = 9*D* and SY = 8*D* and (e) *β* = 70° and (f) *β* = 110°.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-14-00199.1

Note that in an actual wind farm the wind rarely blows from one fixed direction and can be described more accurately by considering a small sector centered around one direction. In view of this variability of the wind direction, layouts shown in Figs. 13c and 13d should be considered to be better than those in Figs. 13a and 13b. This is because the blockage is zero over small sectors of wind directions in the former set of layouts, as opposed to being zero for only a single wind direction in the latter. Layouts that offer zero blockage over a range of wind directions should be considered to be more robust than those that yield zero blockage for only a single wind direction.

A number of nonorthogonal layout–wind direction combinations also yield zero blockage, as long as the spacing in at least one direction is 9*D* or greater. A couple of nonorthogonal layouts and associated wind directions with BR = 0 are shown in Figs. 13e and 13f. As expected, the wind directions that yield zero blockage change with the angle *β*. One of the wind directions (*θ* = 347.5° for *β* = 70° and *θ* = 372.5° for *β* = 110°), in each case, consistently makes an angle of 7.5° with the columns. The nonorthogonal layouts also exhibit a symmmetry with respect to the north–south direction. For example, the *θ* = 365° wind direction in the layout with *β* = 70° is mirrored by the *β* = 110° layout with the wind along *θ* = 355°. We also mention without details that, similar to the orthogonal layout cases, the number of nonorthogonal layouts and the wind direction sector widths that yield zero blockage ratios increase with increasing spacings.

### f. Discussion

The findings in section 4 point to the difficulty involved in formulating a good wind farm layout. The turbine spacings in the *X* and *Y* directions, the staggering of alternate rows in the *Y* direction, the angle *β* between the rows and the columns, and the wind farm orientation all interact in a complex manner to determine the wake losses in a wind farm. This study is restricted to wind farms that comprise a finite number of identical turbines and is based on neutral stability conditions at a single TI level.

One way of applying the results in this paper is to calculate averages of relative power, weighted by the wind direction frequency distributions given by wind roses, for each combination of SX, SY, *β*, and SDY. The appropriately weighted relative power can then be optimized over different combinations of the design parameters. Such a study involving combinations of different wind directions will be discussed in a subsequent paper. Even with the limitation of considering wind directions individually, a few general rules can be formulated for determining efficient wind farm layouts that reduce wake losses:

The columns of the wind farm should make a small angle with the wind. The precise value of the angle that yields the maximum power can vary between 7.5° and 15°, or even 20°, depending on the spacings between the columns.

In general, the larger the spacing between turbines is, the higher is the power generation. However, the spacing along both the axes of the rectangular grid need not be large. If oriented correctly, wind farms with highly elongated rectangular layouts yield larger power than wind farms with square layouts.

Staggering of alternate rows along one of the axes of the rectangle is beneficial if the wind is directed mostly perpendicular to the direction of staggering, as this serves to move the turbines out of the shadow of upstream turbines. However, for certain combinations of farm orientation with respect to the wind and the spacings, staggering may lead to turbines moving into the shadow of upstream turbines, resulting in less power.

Depending on the orientation of the farm with respect to the wind, nonorthogonal layouts might yield greater power than orthogonal layouts by increasing the number of turbines that encounter the undisturbed wind. Changing the angle between rows and columns from orthogonal to nonorthogonal may not necessarily be beneficial.

These general principles have been derived for wind farms with a regular grid of turbines. The combinations of the four design variables considered here by no means cover the entire gamut of different layouts possible. The geometry-based models developed, however, are general and can be easily applied to other arbitrary layouts as well.

## 5. Summary and outlook

Wake effects in wind farms can lead to a significant reduction in the power produced as compared with the installed capacity in a wind farm. Hence, layout studies are critical components in the design of large wind farms. In contrast to analytical and CFD-based models, statistical models based on geometric quantities associated with the farm layout have been developed here. Results of large-eddy simulations of the Lillgrund wind farm, reported in Archer et al. (2013), are used for calibrating the models.

The basic idea is that the geometric blockage of wind turbine disks by other upwind turbines overwhelmingly controls the wake losses and the power generation. The geometric quantities that we term blockage ratio, blockage distance, and inverse blockage distance, and their linear combinations, are found to be very well correlated with the relative power generated by the turbines. The correlations between the relative power and the geometric measures are found to be even better (in excess of 0.95 in most cases), when the variables are averaged over the entire wind farm. Inspired by these high correlations, six simple models have been developed based on the geometric quantities and the Lillgrund LES results. Although all six models are of comparable accuracy, model *f*_{5} may be considered to be the most complete, since it contains elements of the blockage ratio as well as the distance to the upstream blocking turbines.

The models are evaluated by comparison with observations and LES results of the Lillgrund and Horns Rev wind farms. It is shown that models trained on the results of LES of the Lillgrund wind farm translate reasonably well to other wind farms with the same stability and incoming turbulence intensity. In particular, the broad features of the LES results of the Horns Rev wind farm are reproduced to a reasonable level of accuracy. Comparisons with observations show that there may be some discrepancies between the model predictions and individual turbine power. However, the models are expected to be reasonably accurate with respect to prediction of the global, farm-averaged power. Comparisons with actually observed farm-averaged power values are difficult because the observations are averaged over long time periods, comprising effects of differing levels of turbulence intensity and atmospheric stability. Further work incorporating the effects of these parameters in the geometry-based models needs to be carried out in the future to perform effective and fair comparisons.

The geometry-based models are used to study different layouts in hypothetical wind farms, with turbines arranged in regular rectangular or rhomboidal layouts. A total of 15 540 rectangular combinations and approximately 42 000 nonrectangular combinations are obtained by varying the spacings in the *X* and *Y* directions, the staggering of alternate rows in one direction, the angle between the rows and the columns, and the orientation of the wind farms with respect to the wind direction. It is found that the optimum layout is a complicated and nonintuitive function of the different design parameters.

The orientation of the wind farm, defined as the angle between the columns and the prevailing wind direction, is found to be the most critical design parameter. This angle should be small, between 7° and 20°, but should not be zero, which would correspond to a perfectly aligned orientation and thus one with maximum losses. Small changes in this orientation angle can, in some cases, lead to drastic improvement or deterioriation in the generated power.

The spacing between turbines is another important design consideration. It is found that the spacing along at least one direction should be fairly large (approximately 9*D* or more). The spacing along the other direction can be as small as 2*D* if the wind varies within a very narrow sector. Slightly larger spacings in this second direction can be used in order to account for the variability in the wind direction usually found in the field.

The effect of staggering alternate rows along one of the axes of the rectangle is found to depend heavily on the orientation of the wind farm with respect to the wind direction and can lead to significant increases or decreases of the generated power. The effect of the angle between the rows and the columns too is dependent largely on the farm orientation with respect to the wind.

In addition to the general principles outlined above, this study also demonstrates the advantages of the geometry-based models over other physics-based numerical simulation techniques. Using the geometry-based models, a large number of layouts can be evaluated at a fraction of the cost associated with CFD-based tools. An LES or RANS study covering a similar number of layouts is prohibitively expensive. The geometry-based models can be very useful when used in conjunction with high-resolution numerical simulations, since a large number of layouts can be screened very quickly with these models. As mentioned in section 3a, a criterion to select the subset of best layouts would be to include only those that increase the power production by 4% or more over a control layout. The details of a few near-optimal layouts so identified can then be investigated using high-resolution numerical simulations.

There are several limitations to the current study. This study does not aim at minimizing the levelized cost of electricity (LCOE), which is a more complete measure of farm performance. Minimizing the LCOE requires estimates of the lifetime of a wind farm and operation and maintenance costs, among other things, and is beyond the scope of the present study. Maximizing the power generated by minimizing the losses due to interactions among turbines is, nevertheless, an important step in achieving an optimal wind farm. This study is concerned with determining the optimal layout of turbines so as to minimize wake losses, without considering operation and maintenance costs.

Another limitation of the current work is the fact that only regular rectangular or rhomboidal layouts of wind turbines have been investigated. More complex and seemingly random turbine layouts can potentially be more efficient than the regular layouts examined here. The level of incoming atmospheric turbulence can also impact the spread of the turbine wakes, thus affecting the overall farm performance. The sensitivity of these models to the turbulence level will be investigated in the future. Furthermore, since TI is known to be affected directly by the atmospheric stability, future work also includes extension of the models to account for unstable and stable stratification. It is expected that the incorporation of these effects will also make possible accurate and fair comparisons with field observations. Evaluation of layouts under the influence of multiple wind directions (with wind distributions given by wind roses) and the incorporation of geometry-based models in advanced optimization algorithms will also be pursued in the future.

## Acknowledgments

Partial support for this research was provided by the National Science Foundation (Grant 1357649). All simulations were conducted on the Mills High Performance Computer cluster of the University of Delaware.

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