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Conference Paper

NREL/CP-5000-72924

January 2019

Comparison of Wind Farm Layout 

Optimization Results Using a Simple 

Wake Model and Gradient-Based 

Optimization to Large-Eddy 

Simulations

Preprint

Jared J. Thomas,1 Jennifer Annoni,2 Paul Fleming,2

and Andrew Ning1

1 Brigham Young University

2 National Renewable Energy Laboratory

Presented at the American Institute of Aeronautics and Astronautics (AIAA) 

SciTech Forum

San Diego, California

January 7–11, 2019
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Contract No. DE-AC36-08GO28308

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Conference Paper

NREL/CP-5000-72924

January 2019

Comparison of Wind Farm Layout 

Optimization Results Using a Simple 

Wake Model and Gradient-Based 

Optimization to Large-Eddy 

Simulations

Preprint

Jared J. Thomas,1 Jennifer Annoni,2 Paul Fleming,2

and Andrew Ning1

1 Brigham Young University

2 National Renewable Energy Laboratory

Suggested Citation

Jared J. Thomas, Jennifer Annoni, Paul Fleming, and Andrew Ning. 2019. Comparison of 

Wind Farm Layout Optimization Results Using a Simple Wake Model and Gradient-Based 

Optimization to Large-Eddy Simulations: Preprint. Golden, CO: National Renewable 

Energy Laboratory. NREL/CP-5000-72924. https://www.nrel.gov/docs/fy19osti/72924.pdf.
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Comparison of Wind Farm Layout Optimization Results Using

a Simple Wake Model and Gradient-Based Optimization to

Large-Eddy Simulations

Jared J. Thomas∗

Brigham Young University, Provo, UT 84602

Jennifer Annoni†and Paul Fleming‡

National Renewable Energy Laboratory, Golden, CO, 80401, USA

Andrew Ning§

Brigham Young University, Provo, UT 84602

The models used during wind farm layout optimization use simplifying assumptions that

can alter the design space. Some characteristics of the simple models may negatively influence

the resulting layouts. In this paper, we perform wind farm layout optimization and compare

the resulting improvements to large-eddy simulation (LES) results to confirm that the layout

was actually improved. We begin by describing the models used, including changes specific

for use with gradient-based optimization. We then compare our models’ output to previously

published model and LES results. Using the models described, we performed gradient-based

wind farm layout optimization using exact gradients. Power production for the original and

optimized layouts were recalculated using LES. The model and LES results were then compared.

The models predicted an improvement in annual energy production (AEP) of 7.4%, while the

LES reported an AEP improvement of 9.9%. We concluded that the improvements found

by optimizing with the simple models are not just an artifact of the models, but are real

improvements.

I. Introduction

Wind farm layouts are usually optimized using simple engineering models because the simple models require fewer

computational resources and less time than the higher fidelity models that are used for final analysis of a wind farm

design. The models commonly used for wind farm layout optimization problems are based on simplifying physical

assumptions that could potentially reduce the optimality of the final layout. Some of the common assumptions include

axisymmetric wakes and constant wake spread rates. In addition to the simplifying physical assumptions, the geometry

of the design space is often simplified by dividing it into a grid space of possible locations (as in [1–6]) or imposing a

regular structure by only allowing array-like layouts and optimizing with array parameters, such as turbine row spacing

and array rotation angle (as in [7]). However, regular and grid layouts impose artificial limitations, potentially forcing

the optimization algorithms away from the most optimal layouts [8]. Due to the limiting effect of grid and regular

layout approaches, there is interest in unstructured, or irregular, wind farm layout optimization [7–12]. However, with

the restrictions removed, unrestricted optimization could also be more susceptible to unrealistic characteristics of the

models used during optimization. Because of the limitations inherent in these simplified models, it is uncertain whether

the improvements reported by unstructured wind farm layout optimizations, using these simple models, are actual

improvements or if the improvements are merely products of the models’ weaknesses.

Because it is not practical to build and rebuild a wind farm to confirm the legitimacy of the improvements due to

optimization with simple models, we are left with wind tunnel experiments and higher-order models, such as large-eddy

simulation (LES). While many studies have compared simple wake models to LES and wind tunnel results (such as

[13–22]), few have investigated the validity of the optimization results obtained with the models. One study ([23])

∗PhD Student, Department of Mechanical Engineering, 360 EB, Provo, UT 84602, AIAA Student Member

†Research Engineer, National Wind Technology Center, 15013 Denver West Pkwy, Golden, CO 80401.

‡Senior Engineer, National Wind Technology Center, 15013 Denver West Pkwy, Golden, CO 80401.

§Assistant Professor, Department of Mechanical Engineering, 360 EB, Provo, UT 84602, AIAA Senior Member

1

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involved LES in hybrid, or multifidelity, optimization with the Jensen “top-hat” wake model. LES was used for

validation and for tuning during optimization using 30 turbines and up to 8 wind directions. This study showed similar

improvements according to both the LES and the Jensen model, but avoided some of the inherent inaccuracy of the

simple model by adjusting it iteratively based on LES results. While this approach worked well, it was still much more

costly computationally than optimization using only simple models. The only other validation activity we are aware of

for irregular or unstructured wind farm layout optimization problems was performed using wind tunnel experiments

and involved a small wind farm (nine turbines) and unidirectional flow [8]. The complexity of the wind farm layout

optimization problem comes from large numbers of wind turbines and probabilistic wind in many directions, so it is

important to investigate the results of unrestricted wind farm layout optimization problems involving larger wind farms

and many wind directions.

Large numbers of wind turbines cause a corresponding increase in the number of design variables and constraints

used by the optimization algorithm, which can reduce the effectiveness and efficiency of the algorithm. While the

wind farm layout optimization problem is typically solved using gradient-free optimization methods [24] (such as in

[2–7, 25, 26]), gradient-free optimization methods exhibit reduced performance when applied to high dimensional

problems [27]. In [27], most gradient-free solvers were unable to solve more than 15% of the test problems studied with

10–30 design variables, and even the best methods tested could solve only 28% of the test problems with 31–300 design

variables. New wind farms can include well over 100 turbines, meaning that unrestricted wind farm layout optimization

problems can have over 200 variables and thousands of constraints. In short, the variables and constraints of the

unrestricted wind farm layout optimization problems of interest exceed the abilities of most gradient-free optimization

methods.

In contrast to gradient-free optimization methods, some gradient-based methods can solve problems with thousands

of variables and constraints [28]. However, gradient-based methods are prone to premature convergence when used with

multimodal problems, such as the wind farm layout optimization problem. Because of the tendency to converge to

local optima in multimodal problems, gradient-based optimization methods have not historically been seen as viable for

solving the wind farm layout optimization problem [24]. However, recent studies have demonstrated that significant

improvements to wind farm layouts, if not global optimality, can be obtained by using gradient-based methods either

alone [10–12, 29, 30], or with gradient-free methods in a hybrid approach [9].

Solving a wind farm layout optimization problem requires several different types of models. Two of the most

important of these are the wake model and the farm model. Generally speaking, wind turbine wake models define

the wind speed in a turbine wake given a set of inflow conditions, while farm models combine the wakes of multiple

turbines to determine the cumulative effect of their wakes. Many different wind turbine wake and wind farm models

have been presented, all with varying levels of accuracy and computational cost. Many of the wake models, such as

the “top-hat” Jensen/Park model [31], the Frandsen model [32], and the original FLOw Redirection and Induction in

Steady State (FLORIS) model [33], have regions of nonphysical zero-valued gradients, where their gradients go to zero

while the gradients of the real design space do not. If a model with nonphysical zero-valued gradients is used with

gradient-based optimization, the optimization may converge to an especially poor local optimum due to the lack of

information provided in the gradients [29].

The Bastankhah and Porté-Agel (BP) wake model presented in [34, 35] is well suited to gradient-based optimization,

with the exception of the near-wake region. Most of the BP model is smooth, continuous, and free from flat regions,

but the near-wake definition of the BP model can be either flat or undefined. The near-wake region is rarely needed

for evaluating a wind farm since wind turbines are usually placed far enough from each other that they are in the far

wake. While the final wind farm design may not place turbines in the near-wake, it is important to have the near-wake

region defined and nonflat during gradient-based optimization because optimization algorithms may attempt infeasible

solutions during the optimization process [36] and the near-wake region of the BP model may sometimes be within the

defined optimization constraints. The BP wake model uses a Gaussian distribution to define the far wake, and is easily

applied to different turbines since it only has one independent parameter. Building on the BP wake model, Niayifar

and Porté-Agel (NP) have proposed a farm model [37, 38] to be used in concert with the BP wake model as published

in 2014. The NP farm model makes the one independent variable dependent on turbulence intensity [34]. In this

study, we use the NP farm model with the 2016 version of the BP wake model. While the NP model seems to provide

good agreement with LES data, there are some minor problems with the NP farm model for use with gradient-based

optimization that need to be addressed.

In this work, we address the lack of near-wake definition in the BP wake model for the purposes of using gradientbased optimization methods to solve the wind farm layout optimization problem, apply the NP wind farm model

with some adjustments, and obtain exact gradients across the entire system. We then optimize a test case using a

2

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gradient-based approach. Finally, we perform an LES study of the base case and optimized wind farm layouts and

compare the LES results with the model results. We found that the models predicted 7.4% improvement in annual

energy production (AEP) for the optimized layout as compared to the base case layout, while the LES predicted a

9.9% improvement. We conclude that, in this case at least, the improvements found through unstructured optimization

are not merely a result of the optimization algorithm exploiting unrealistic model characteristics, but are legitimate

improvements to the wind farm layout design.

II. Methods

A. Wake Model

While the BP wake model exhibits many characteristics necessary for use with gradient-based optimization, there

are some small problems that must be overcome. First, the main BP model is undefined from the turbine location up to

several diameters downstream of the wind turbine. Bastankhah and Porté-Agel suggested using regions of constant

velocity in the near-wake [35], but using regions of constant velocity in the wind turbine wake has been shown to cause

premature convergence to poor local optima when optimizing with gradient-based methods [29]. The standard BP

model cannot be used in the near-wake region because there is a discontinuity due to a negative value in the square-root

term of the velocity deficit calculation of the BP wake model as shown in Eq. (1).

∆ui j

u ̄i

=

"

1 −

s

1 −

CT cos γ

8σyσz /d

2

#

exp − 0.5



y − δ

σy

2

exp − 0.5



z − zh

σz

2

(1)

In Eq. (1), ∆ui j is the velocity deficit in the wake due to turbine i at the point j, u ̄iis the average inflow velocity of

turbine i, CT is the thrust coefficient, γ is the yaw relative to the inflow wind direction, d is the rotor diameter, y is the

horizontal distance from the line through the rotor hub to the point of interest perpendicular to the wind direction, δ is

horizontal wake offset, z and zh are the heights of the point of interest and the rotor hub, respectively, and σy and σz are

the horizontal and vertical wake spread as defined in Eqs. (2) and (3).

σy = ky[x − x0] +

d cos γ

√

8

(2)

σz = kz [x − x0] +

d

√

8

(3)

In Eqs. (2) and (3), ky and kz are the horizontal and vertical wake growth rates (assumed to be equal), x is the downstream

distance from the turbine, and x0 is the length of the potential core as defined in Eq. (4).

x0

d

=

cos γ

f

1 +

√

1 − CT

g

√

2

f

α∗I + β

∗

f

1 −

√

1 − CT

g g (4)

In Eq. (4), I is the turbulence intensity, α

∗ = 2.32, and β∗ = 0.154.

Wind shear was added to the wake model using a power law defined as shown in Eq. (5).

u = ur



z − zo

zr − zo

ψ

(5)

In Eq. (5), ur is the reference wind speed, z is the height of interest, zr is the height at which ur was measured, zo is the

height of local ground, and ψ is the shear exponent. The value of ψ for comparison to the LES was determined by

fitting Eq. (5) to a series of wind speeds in the LES precursor. To fit the curve, zr was set to 80 m to focus on matching

the LES values in the rotor area. The appropriate value of ψ for comparison to the LES was determined to be 0.31. The

shear exponent fitting plot is shown in Fig. 1.

The discontinuity mentioned previously can occur for various combinations of ky, kz , and (x − x0). When local

turbulence intensity is calculated in the NP wind farm model, as in [38] (discussed further in Section II.B), the values of

ky and kz vary and so the discontinuity is even more mobile than in the original model. We can determine where the

model is undefined as follows. First, we know the model is undefined when the condition in Eq. (6) is met.

CT cos γ

8σyσz /d

2

> 1 (6)

3

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0 2 4 6 8 10 12

Wind Speed, m/s

0

50

100

150

200

250

Height, m

LES

Shear Func

Fig. 1 The calculated wind speed using the power law (Eq. (5)) with shear exponent ψ = 0.31 compared with

the values obtained from the LES precursor. The reference values used were a height of 80 m with a wind speed

of 8 m/s.

By substituting Eqs. (2) and (3) into Eq. (6), and separating the terms based on the powers of [x − x0], we obtain Eq. (7).

ky kz [x − x0]

2 +

d[ky + kz cos γ]

√

8

[x − x0] −

d

2

cos γ[CT − 1]

8

< 0 (7)

To find the exact downstream location where the model begins to be defined, xd, we apply the quadratic formula to

Eq. (7), select the relevant root, and then solve for x to obtain Eq. (8).

xd = x0 +

d

"

ky + kz cos γ −

q

[ky + kz cos γ]

2 − 4ky kz [CT − 1] cos γ

#

2ky kz

√

8

(8)

The model, then, is undefined for regions where x < xd.

To remove the discontinuity, we used linear interpolation on both the velocity value and the wake spread. First, the

magnitude of the Gaussian curve at xd was calculated, as in Eq. (9).

∆Udm = 1 −

s

1 −

CT cos γ

8σydσzd

/d

2

(9)

In Eq. (9), σydand σzdrepresent the wake spread at xd. Then, using the magnitude of the Gaussian curve, and keeping

the wake spread constant for the velocity calculation, the normalized velocity difference is calculated as a function of

the downstream position of the point of interest (x), as shown in Eq. (10).

∆ui j

u ̄i

=



x

∆Udm − ∆Ur

xd



+ ∆Ur



exp − 0.5



y − δ

σyd

2

exp − 0.5



z − zh

σzd

2

(10)

In Eq. (10), ∆Ur is a predetermined normalized velocity deficit at the rotor hub. To avoid overpredicting the velocity

deficits and potentially causing an unrealistic flow reversal, we used a value of ∆Ur = ∆Udm .

The results of the model changes discussed above are shown in Fig. 2. These changes make the model continuous

throughout the domain and differentiable everywhere except at the exact turbine location. With a separation constraint

in place, the probability of turbines landing exactly on top of each other is extremely small, as noted in [29]. With the

linear interpolation method, optimizations succeed even when turbines are placed less than 1 diameter from each other.

While the linear interpolation makes little attempt to be physically accurate in the near-wake, the accuracy of the model

for AEP calculation purposes should be unaffected since turbines will almost never be placed close enough together

to be in the linear interpolation region in a final design. If greater accuracy is desired in the near-wake, it would be

feasible to use the linear interpolation during optimization and then switch to a more accurate near-wake model for final

calculations, such as the one proposed in [39].

B. Wind Farm Model

NP proposed a wind farm model that used the BP model as published in 2014 [34] for calculating the wind speed in

the wakes [37, 38]. We applied the same wind farm model to the 2016 version of the BP model, with some variation as

discussed in the following sections.

4

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0 5 10 15

Downstream Distance, x=d

0.0

0.5

1.0

1.5

2.0

Height, z=d

1.7

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

1.0

1.5

2.0

2.5

wind speed, m/s

(a)

0 5 10 15

Downstream Distance, x=d

0.0

0.5

1.0

1.5

2.0

Height, z=d

1.7

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

1.0

1.5

2.0

2.5

wind speed, m/s

(b)

Fig. 2 Bastankhah and Porté-Agel wake model (a) without and (b) with the added linear interpolation for the

near-wake. The heavy black line represents the turbine rotor. This figure was included for comparison with Fig.

6 of Bastankhah and Porté-Agel (2014) [34], and so model values were chosen accordingly (small wind turbine

for use in a wind tunnel). We used a shear exponent of 0.15 for this figure.

1. Wake Combination

NP found the most accurate method of wake combination to be a linear velocity deficit superposition that uses the

inflow velocity of the upstream turbine as shown in Eq. (11).

uj = u ̄∞ −

X

NT

i=1

[  ̄ui − ui j] = u ̄∞ −

X

NT

i=1

∆ui j (11)

In Eq. (11), ujis the flow velocity at the point of interest, u ̄iis the average inflow velocity of upstream turbine i, ui j is

the wind speed that would exist at the point j if only turbine i were accounted for, and u ̄∞ is the average free-stream

velocity. For the upstream turbine velocity to be used, the inflow velocity of the turbines must be calculated in order

from upstream to downstream. We used the heapsort algorithm to obtain a sorted index list of the turbines prior to

solving the system for each wind direction. Using the sorted index list allowed turbine sorting without overcomplicating

the gradients of the system. Our tests of the accuracy of various wake combination methods also found this method to

be the most accurate in comparison to the LES results used by Niayifar and Porté-Agel in [38].

2. Turbulence Intensity

Niayifar and Porté-Agel calculated local turbulence intensity using the model presented by Crespo and Hernández in

[40]. The local turbulence intensity is then applied to the BP model through an empirical model for calculating the

wake expansion coefficient as shown in Eq. (12) [38].

k

∗ = 0.3837I + 0.003678 (12)

In Eq. (12), I is the turbulence intensity (TI). This model is only valid for TI values between 0.065 and 0.15. While the

initial BP wake model does not include TI [34], TI was included by Bastankhah and Porté-Agel in their 2016 version of

the model [35]. We used the same TI values for the BP 2016 wake model and the NP wind farm model.

The algorithm for calculating local turbulence intensity using the hard maximum function is given in Algorithm 1.

In our code implementation, we combined the nested loops used for the velocity and TI calculations.

5

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Algorithm 1 Turbulence Intensity Calculation (for more information, see [38])

Ensure: Turbines are sorted by relative downstream position

for j = 1...N do . Downstream turbines

Arj ← 1

4

πD

2

rj

. Rotor swept area of turbine j

I+j ← 0

for i = 1...N do . Upstream turbines

∆X ← x j − xi . Turbine separation distance

if ∆X > 0 then

β ← 0.5

f f1 +

p

1 − CTi

g

/

p

1 − CTi

g

 ← 0.2

√

β

σ = k

∗

i

∆X + Dri

Dw ← 4σ

Awi j ← Wake overlap area of wake i and rotor j . Assume circular wake

if CTi > 0.96 then

ai ← 0.143 +

p

0.0203 − 0.6427[0.889 − CTi] . Glauert correction [12]

else

ai ← 1

2

f

1 −

p

1 − CTi

g

end if

I+i j ← 0.73a

0.8325

i

I

0.0325

i

[∆X/Dri]

−0.32

I+j ← max [I+i j Awi j /Arj, I+j] . We used Eq. (15) during the final optimization

end if

end for

Ij ←

q

I

2

0

+ I

2

+j

. Directly influenced by only one upstream turbine

k

∗

j ← 0.3837Ij + 0.003678 . To be used in the wake model calculations

end for

We used three different approaches to calculating TI at various points in the optimization process: (1) ambient

TI only, (2) local TI with a smooth maximum function, and (3) local TI with a hard maximum function as done by

Niayifar and Porté-Agel [38] and shown in Algorithm 1. The reasons for using the three methods for calculating TI are

discussed in Section II.D. The smooth maximum function we used is shown in Eqs. (13) to (15) and is derived from the

LogSumExp function [41].

tmax = min(t1, t2) (13)

tmin = max(t1, t2) (14)

maxsmooth (tmax, tmin, s) =

ln(1 + exp(s[tmin − tmax])) + stmax

s

(15)

In Eqs. (13) to (15), s controls the smoothness of the transition between terms and influences the error of the calculation

and tmax and tmin represent the respective maximum and minimum of the two values being compared. In calculating

local turbulence intensity, Eq. (15) is applied, as in Eq. (16), to compare the product of the added local turbulence

intensity and the wake overlap ratio.

I+j = maxsmooth 

I+i j Awi j

Arj



(16)

In Eq. (16), Awi j is the area of the wake of turbine i at the downstream location of turbine j, Arjis the area of the

rotor of turbine j, and I+i j is the added turbulence intensity from turbine i at the downstream location of turbine j. In

our implementation, only two values are compared at a time: the current value and the previous result of the smooth

maximum function. The total turbulence intensity is then calculated using Eq. (17).

Ij =

q

I

2

0

+ I

2

+j

(17)

The maximum error of Eq. (15) occurs when tmin = tmax, and can be calculated as in Eq. (18).

Errormax =

ln (2)

s

(18)

To keep the error from the smooth maximum below 0.001, or about 1% of the total turbulence intensity, we set s = 700.

6

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3. Sampling Over the Rotor-Swept Area

Various methods can be used to approximate the effective velocity at the rotor. The ideal method is a full integration

across the rotor-swept area. We approximated the integral by sampling the velocity at various points and taking the

average. When comparing our code to LES and other data, we used 100 sampling points across the rotor-swept area

arranged in the sunflower seed pattern as shown in Fig. 3(a). We found that 100 sample points on the rotor-swept

area were more than sufficient to achieve the level of accuracy demonstrated by Niayifar and Porté-Agel in [38]. For

efficiency during optimization, we approximated the effective velocity of each wind turbine by using a single sampling

point located at the rotor hub as shown in Fig. 3(b).

0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8

Horizontal Distance From Hub, ¢y=d

0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

Vertical Distance From Hub, 

¢z=d

Sampling points

Rotor swept area

(a)

0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8

Horizontal Distance From Hub, ¢y=d

0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

Vertical Distance From Hub, 

¢z=d

Sampling point

Rotor swept area

(b)

Fig. 3 (a) Sampling locations used to approximate effective hub velocity for LES comparisons. (b) Location of

sampling point for approximating effective hub velocity during optimization.

4. AEP and Wind Turbine Power

AEP was calculated as in Eq. (19).

AEP =



hour s

day

  days

year  X

ND

k=1



fk

X

NT

j=1

Pjk



(19)

In Eq. (19), ND is the number of wind directions, NT is the number of wind turbines, and fk is the probability of wind

in direction k. The power of turbine j in direction k, Pjk, is calculated based on the definition of the power coefficient

as done in [33] and shown in Eq. (20).

Pjk =









0  ̄uj < ucutin

0.5ρAjCPju ̄

3

j

u ̄j ≥ ucutin and Pj < Pr ated

Pr ated Pj ≥ Pr ated

(20)

In Eq. (20), ρ is the air density, Aj represents the rotor-swept area of turbine j, and CPjis the power coefficient of

turbine j for the given effective wind speed at the rotor (u ̄j).

5. Model Verification

To verify our implementation of the models described, we compared the output of our code with data for the Horns

Rev wind farm from Niayifar and Porté-Agel [38] as shown in Figs. 4 and 5. We calculated error, in normalized power,

based on the LES data values from [38] and reported the error in Table 1. For the Horns Rev row-averaging comparison

using 100 samples on the rotor-swept area as shown in Fig. 4(a): the model values shown by Niayifar and Porté-Agel

had an average error of 0.034 and a maximum error of 0.057; ignoring local turbulence intensity resulted in an average

7

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error of 0.179 and a maximum error of 0.253; calculating local turbulence intensity using either the hard maximum

or smooth maximum both resulted in an average error of 0.018 and a maximum error of 0.035. For the Horns Rev

directional power comparison using 100 samples on the rotor-swept area as shown in Fig. 5(a): the model values shown

by Niayifar and Porté-Agel had an average error of 0.022 and a maximum error of 0.055; ignoring local turbulence

intensity resulted in an average error of 0.058 and a maximum error of 0.206; calculating local turbulence intensity

using either the hard maximum or smooth maximum both resulted in an average error of 0.017 and a maximum error

of 0.053. The high directional errors when local turbulence intensity is ignored are in the directions of highest wake

interactions as shown in Fig. 5 and do not alter the general trend of the directional power. The Horns Rev comparisons

using one sample on the rotor-swept area are shown in Fig. 4(b) and Fig. 5(b). Using only one sample on the rotor-swept

area resulted in slightly larger errors as reported in Table 1. Based on our model error being within the range reported

in [38] when we used 100 samples and local TI (less than 0.057 and 0.055 for the row and directional comparisons,

respectively), we determined that our implementation of the BP wake and NP farm models was sufficient and could be

used to perform wind farm layout optimization case studies.

1 2 3 4 5 6 7 8 9 10

Row Number

0.0

0.2

0.4

0.6

0.8

1.0

Normalized Average Power

NP LES

NP Model

Hard max TI

Smooth max TI

No local TI

(a)

1 2 3 4 5 6 7 8 9 10

Row Number

0.0

0.2

0.4

0.6

0.8

1.0

Normalized Average Power

NP LES

NP Model

Hard max TI

Smooth max TI

No local TI

(b)

Fig. 4 Averaged power of turbines in columns two, three, and four of the Horns Rev wind farm by turbine row.

(a) Comparing our code using 100 points to approximate the effective wind speed at the rotor. (b) Comparing

our code using a single sample at the rotor hub to approximate the effective wind speed at the rotor. NP refers

to LES and model results from [38]. All three methods of calculating TI refer to results obtained using our code.

The three methods for calculating TI are to address various complications during our optimization process (see

Sections II.B.2 and II.D for details).

Table 1 Normalized model power error as compared to Horn Rev LES results [38].

No Local TI Hard Max TI Smooth Max TI

Samples Samples Samples

NP Model 100 1 100 1 100 1

Directional

Max 0.055 0.206 0.266 0.053 0.072 0.053 0.072

Average 0.022 0.058 0.074 0.017 0.025 0.017 0.025

Row

Max 0.057 0.253 0.312 0.035 0.161 0.035 0.161

Average 0.034 0.179 0.243 0.018 0.058 0.018 0.058

8

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160 180 200 220 240 260 280 300 320 340 360

Wind Direction, degrees

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Normalized Directional Power

NP LES

NP Model

No local TI

Hard max TI

Smooth max TI

(a)

160 180 200 220 240 260 280 300 320 340 360

Wind Direction, degrees

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Normalized Directional Power

NP LES

NP Model

No local TI

Hard max TI

Smooth max TI

(b)

Fig. 5 Power by direction for the Horns Rev wind farm. (a) Comparing our code using 100 points to approximate

the effective wind speed at the rotor. (b) Comparing our code using a single sample at the rotor hub to

approximate the effective wind speed at the rotor. NP refers to LES and model results from [38]. All three

methods of calculating TI refer to results obtained using our code. The three methods for calculating TI are to

address various complications during our optimization process (see Sections II.B.2 and II.D for details).

C. Test Case

For the LES comparison, we used the National Renewable Energy Laboratory (NREL) 5-MW reference turbine

[42]. We used the same thrust and power coefficient curves for the NREL 5-MW turbine as used in [12], with linear

interpolation between points. Our test wind farm was constrained by the domain of the LES simulation precursor. We

limited the LES simulation to a 5-km square area to keep necessary computational resources to a reasonable level. In

order to avoid edge effects in the simulation, we needed at least 0.5 km between the edge of wind farm and the edge of

the simulation space. To limit the computational cost, we used only a single wind direction and rotated the wind turbine

locations within the simulation space to calculate the power production in different directions so that a single precursor

could be used. The resulting available space for the wind farm was a circle with a 2-km radius in the middle of the

simulation space. Within the resulting circular region, we placed as many turbines as possible in a concentric circular

pattern with a minimum allowable distance between turbines of five times the rotor diameter. The resulting baseline

layout is shown in Fig. 6.

For the wind frequencies, we chose to use the Nantucket wind rose [43]. To reduce the computational cost of the

LES simulations, we generated a wind rose with 12 directions by binning every 3 directions from the Nantucket wind

rose starting with wind from the north. We used the 12-direction wind rose for both the optimizations and the LES. The

original and final wind roses are shown in Fig. 7.

D. Optimization

We optimized the wind farm layout using AEP as the objective. The optimization problem was formulated as shown

in Eq. (21).

maximize

xi,yi

AEP(xi, yi, ) i = 1...38

subject to Si j ≥ 2d i, j = 1...38 i , j

[xc − xi]

2 + [yc − yi]2 ≤ r2

b

i = 1...38 (21)

In Eq. (21), (xi, yi) is the position of each turbine i, Si j represents the separation distance between each pair of turbines

i and j, (xc, yc ) is the location of the center of the wind farm, and rb is the radius of the wind farm boundary.

The optimization problem was built in OpenMDAO, a Multidisciplinary Design Analysis and Optimization platform

[44]. Gradients of the wake model were obtained using Tapenade, an algorithmic differentiation tool [45]. Gradients of

all other system components were derived by hand. The final system gradients were compiled by OpenMDAO.

9

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1000 0 1000 2000 3000 4000 5000 6000

Turbine X Position, m

0

1000

2000

3000

4000

5000

Turbine Y Position, m

0 1

23

4

5 6

7

8

9

10 11

12

13

14

15 16 17

18

19

20

21

22

24 23 25

26

27

28

29

30

31

32 33 34

35

36

37

LES Domain Farm Boundary

Fig. 6 Base case wind farm layout. The circles marking turbine locations are to scale, with diameters equal to

the rotor diameter.

The gradients of both the objective function and constraints were scaled to be close to order one. The final problem

was then solved using the Sparse Nonlinear OPTimizer (SNOPT), a gradient-based optimization algorithm that uses

a sequential quadratic programming approach. SNOPT was used in this case because it is well suited to nonlinear

problems with high dimensionality [28].

Along with SNOPT, we used the wake expansion continuation (WEC) method for reducing the multimodality of the

wind farm layout optimization problem [30]. WEC relaxation factors ran from at 3.00 down to, and including, 1.00 in

steps of 0.25. After completing the WEC series, we ran one final optimization, from the optimized layout found using

WEC. In the final optimization, local turbulence intensity was calculated with the smooth maximum function as in

Eq. (15). For final wind farm assessment, we calculated local turbulence intensity using the hard maximum as done in

[38].

As mentioned in Section II.B.2, we used three methods for calculating TI: (1) constant ambient TI only, (2) local

TI calculated with a smooth max, and (3) local TI calculated with a hard maximum as done in the NP wind farm

model. Because the hard maximum (3) creates discontinuities in the model and is not differentiable, we used a smooth

maximum function (2) in the final step of our optimization process. Calculating local TI as done in the NP wind farm

model, with either the hard or smooth maximum functions, adds additional local optima that are exaggerated when

using WEC. The additional local optima drastically reduce the effectiveness of the WEC method, so we neglected local

turbulence intensity during the WEC optimization series.

To arrive at the final optimized solution, we used a multistart optimization approach. Starting locations we used were

the baseline layout shown in Fig. 6, as well as 199 pseudo-random layouts meeting the constraints of the optimization

problem.

E. Large-Eddy Simulation

The Simulator fOr Wind Farm Applications (SOWFA) is a high-fidelity, large-eddy simulation tool that was

developed at NREL for wind plant studies [46–48]. It is a computational fluid dynamics solver based on OpenFOAM

[49] and can model turbines as actuator disks or actuator lines. This study uses turbines modeled as actuator disks to

reduce computational cost. Separate studies have been conducted that demonstrate that the steady-state power is similar

in actuator disk and actuator line cases [50].

10

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E

NE

N

NW

W

SW

S

SE

1.0 %

2.0 %

3.0 %

4.0 %

(a)

E

NE

N

NW

W

SW

S

SE

3.0 %

6.0 %

9.0 %

12.0 %

(b)

Fig. 7 (a) Nantucket frequency wind rose [43]. (b) Nantucket frequency wind rose binned into 12 directions.

SOWFA solves the three-dimensional, incompressible, Navier-Stokes equations and transport of potential temperature

equations, which take into account the thermal buoyancy and Earth rotation (Coriolis) effects in the atmosphere. The

inflow conditions for these simulations are generated using a periodic atmospheric boundary layer precursor with no

turbines.

SOWFA calculates the unsteady flow field to compute the time-varying power, velocity deficits, and aerodynamic

loads at each turbine in a wind plant. This level of computation, with high-fidelity accuracy, takes on the order of hours

to days to run on a supercomputer using a few hundred to a few thousand processors, depending on the size of the wind

plant. The simulations run for this study were performed on Peregrine, NREL’s high-performance computer [51].

It should be noted that studies have been performed to validate SOWFA. For example, it has been compared with

48-turbine Lillgrund wind plant field data and shows good agreement through the first five turbines in a row aligned

with the wind direction [52]. In addition, SOWFA has been tested to verify that it captures the inertial range in the

turbulent energy spectra and log layer in the mean flow, both of which characterize a real atmospheric boundary layer

[47]. Further validation studies are ongoing.

Actuator disk simulations of the 38-turbine layout described in Section II.C, as well as the optimized layout, were

performed using SOWFA. To avoid needing a different precursor for each of the 12 directions, the layout was rotated to

a reference wind direction of 270◦. The scenarios were simulated under neutral atmospheric conditions with an 8 m/s

mean wind speed at 80 m and approximately 10.8% turbulence intensity at hub height.

The simulations each ran for approximately 2000 s of simulated time. A structured mesh was used in this study with

a grid spacing of 10 m in the x, y, and z directions, resulting in a grid that is 500 × 500 × 100 (i.e., 25,000,000 grid

cells). Using 500 cores, simulations took on the order of 1 day to complete.

III. Results and Discussion

Due to the multimodal nature of the wind farm layout optimization problem, optimizing with the 200 different

starting layouts resulted in a range of AEP values. While the optimizations were performed using one sample on the

rotor-swept area, we used 100 samples across the rotor-swept area for the analysis. The distributions of AEP values

as calculated with one and 100 samples across the rotor-swept area are shown in Fig. 8. Note that the distribution of

AEP results is shifted lower when calculated using 100 samples on the rotor-swept area due to the denser sampling

11

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accounting for more wake effects. The next step was to determine the best layout. Calculating with 100 samples or one

sample both indicate that the best layout is the one that started from the base case layout (shown in Fig. 6). The final

optimized layout is shown in Fig. 9.

3 4 5 6 7 8 9 10

AEP Improvement, %

0

5

10

15

20

25

30

Count

Fig. 8 Distribution of AEP improvement calculated with one rotor sample point (black) and 100 rotor sample

points (blue) from the 200 optimization runs. The high outlier started from the baseline layout. Note that the

optimizations were performed using only one rotor sample point.

1000 0 1000 2000 3000 4000 5000 6000

Turbine X Position, m

0

1000

2000

3000

4000

5000

Turbine Y Position, m

0 1

3 2

4

5

6

7

8

9 11 10

12

13

14

15 16 17

18

19

20

21

22 24 23

25

26

27

28

29

30

31

32

33 34

35

36

37

LES Domain Farm Boundary

Fig. 9 Optimized wind farm layout. The circles marking turbine locations are to scale, with diameters equal

to the rotor diameter.

We will now investigate the error with more granularity. The directional power production for both SOWFA and the

BP model are shown in Fig. 10. The optimized power production in each direction for the optimized layout was nearly

equal according to BP, with a spread of only 0.3 MW across all directions. More variance appeared in the SOWFA

results, which had a spread of 3.4 MW across all directions for the optimized layout. The SOWFA predictions for the

12

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base case directions had more variation than BP as well. When we dive a little deeper we find that the directional power

predictions had errors ranging from −5.4% to 5.0% for the base case layout and −4.5% to 0.8% for the optimized layout.

The difference in these error ranges indicates that the BP model is underestimating the power more for the optimized

layout than the base case layout. The directional power errors for the base case layout have generally larger magnitudes

than the directional errors for the optimized layout, but the directional errors for the base case layout are more equally

distributed in the positive and negative than the directional errors for the optimized layout. Despite the directional

errors, both the BP model and SOWFA predicted increased power production for every direction for the optimized

layout as compared to the base case layout. The directional power, AEP, AEP improvement, and corresponding errors

are reported in Table 1.

104070

100130160190220250280310340

Wind Direction, degrees

0

10

20

30

40

50

60

70

80

Power, MW

SOWFA BP

(a)

104070

100130160190220250280310340

Wind Direction, degrees

0

10

20

30

40

50

60

70

80

Power, MW

SOWFA BP

(b)

Fig. 10 Power production in each direction for SOWFA and the BP wake model using the hard maximum and

100 samples across the rotor for (a) the base layout (Fig. 6) and (b) the optimized layout (Fig. 9).

Table 2 BP and SOWFA Results

Base Case Optimized Improvement

Dir. SOWFA BP Error SOWFA BP Error SOWFA BP

10◦ 54.6 MW 57.3 MW 5.0 % 63.7 MW 61.4 MW −3.6 % 16.7 % 7.2 %

40◦ 56.0 MW 55.5 MW −0.9 % 60.9 MW 61.4 MW 0.8 % 8.7 % 10.5 %

70◦ 57.0 MW 57.5 MW 0.9 % 62.6 MW 61.4 MW −1.8 % 9.8 % 6.8 %

100◦ 60.9 MW 57.6 MW −5.4 % 62.8 MW 61.2 MW −2.6 % 3.2 % 6.4 %

130◦ 57.5 MW 58.4 MW 1.6 % 61.6 MW 61.2 MW −0.7 % 7.2 % 4.8 %

160◦ 56.9 MW 56.9 MW 0.0 % 61.5 MW 61.2 MW −0.5 % 8.0 % 7.5 %

190◦ 54.1 MW 56.9 MW 5.3 % 63.7 MW 61.4 MW −3.7 % 17.9 % 7.9 %

220◦ 54.8 MW 55.8 MW 1.8 % 62.3 MW 61.3 MW −1.6 % 13.8 % 9.9 %

250◦ 59.5 MW 57.7 MW −2.9 % 61.9 MW 61.3 MW −0.9 % 4.1 % 6.2 %

280◦ 57.9 MW 56.9 MW −1.8 % 62.8 MW 61.2 MW −2.6 % 8.5 % 7.5 %

310◦ 58.8 MW 58.8 MW −0.1 % 60.8 MW 61.1 MW 0.5 % 3.3 % 3.9 %

340◦ 55.1 MW 56.4 MW 2.4 % 64.2 MW 61.3 MW −4.5 % 16.5 % 8.6 %

AEP 478 GWh 481 GWh 0.5 % 526 GWh 516 GWh −1.8 % 9.9 % 7.4 %

Just as the directional errors are larger than the AEP errors, the wind turbine power errors are, in general, much

13

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greater than the directional errors. Power production errors for individual turbines ranged from −27.6% to 96.6% for the

base case layout and −32.9% to 118.4% for the optimized layout. The BP model accurately predicted the direction of

the change in power for only 61.6% of the wind turbines. Power predictions for individual wind turbines have too much

error to be trusted individually. It may be argued that if we cannot trust the individual wind turbine power predictions,

then the wind farm layout optimization cannot be trusted either since the optimizer moves the turbines to improve their

power and therefore the power of the farm as a whole. However, despite the turbine errors, the BP model still allows the

optimizer to reduce wake effects, which is the primary driver for increasing power production in a real wind farm. The

errors for power production for each turbine in each wind direction are shown in the appendix in Figs. 11 and 12.

We have found that each time we average turbine powers to get directional power, or average directional power to

calculate AEP, the error decreases. The maximum errors at each level are as follows: 118.4% for wind turbine power,

5.4% for directional power, and 1.8% for AEP. This is as expected since we know from the law of large numbers that the

more samples that are combined to get an averaged result, the more precise that result tends to be. The directional

power and AEP values can be seen as scaled means of the power production of the individual wind turbines. We can

then place some confidence in AEP and directional power predictions from the BP model. We have shown that the

overall improvements in directional power for this case are not just a factor of the simplified wake model since the AEP

improvement is shown to be even greater according to SOWFA simulations than predicted by the BP model.

IV. Conclusion

In this work, we have made some adjustments to the 2016 Bastankhah and Porté-Agel wake model and the 2016

Niayifar and Porté-Agel wind farm model for improved compatibility with gradient-based optimization methods. We

then demonstrated that our implementation of our altered models met or exceeded the accuracy of the models presented

by Bastankhah, Niayifar, and Porté-Agel as compared to the LES data presented by Niayifar and Porté-Agel in [38]. We

described our test case and gradient-based optimization methods, as well as the SOWFA setup used to produce the

LES results that we used to check the model and optimization results. We found that the individual turbine predictions

based on the models had large errors as compared to our SOWFA simulations, up to 118.4%. The directional and AEP

errors were much less, up to only 5.4% and 1.8%, respectively. The models predicted an improvement in AEP of 7.4%,

while SOWFA predicted that the AEP improvement should be 9.9%. Based on these results, we conclude that while the

individual turbine power predictions may not be accurate, the improvements in directional power and AEP, at least for

this case, can be trusted and are not simply an artifact of the simplified models being used during optimization.

Future work should include further analysis of the individual wind turbine power errors, an investigation of methods

for sampling wind speeds across the rotor-swept area, and demonstration of optimization improvements in other wind

farms and wind conditions using simulations. More wind tunnel tests of optimized wind farms are needed to validate

the results of wind farm layout optimization using simplified models.

Acknowledgments

This work was authored [in part] by the National Renewable Energy Laboratory, operated by Alliance for Sustainable

Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. Funding provided

by the U.S. Department of Energy Office of Energy Efficiency and Renewable Energy Wind Energy Technologies Office.

The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government. The U.S.

Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government

retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work,

or allow others to do so, for U.S. Government purposes.

A portion of the research was performed using computational resources sponsored by the DOE Office of Energy

Efficiency and Renewable Energy and located at the National Renewable Energy Laboratory.

A portion of the research was performed using computational resources of the Fulton Supercomputing Lab located

at Brigham Young University.

Appendix

The power production error for individual turbines in each direction for the 2016 Bastankhah and Porté-Agel wake

model and the 2016 Niayifar and Porté-Agel wind farm model as compared to LES simulation performed using SOWFA

are shown in Fig. 11(a) and Fig. 12(a). Directional power production errors for the models compared to the SOWFA

14

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results are shown in Fig. 11(b) and Fig. 12(b).

10.0 40.0 70.0 100.0 130.0 160.0 190.0 220.0 250.0 280.0 310.0 340.0

Direction, degrees

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

Turbine

-12 -16 20 -12 -1 -24 -20 -10 -9 6 8 11

-19 -19 -10 -25 -9 -15 -13 -5 24 -22 19 -16

-11 -6 -14 7 6 -8 48 42 44 15 -14 15

34 -16 14 -18 5 -6 -4 -21 -19 32 75 46

8 -3 17 -15 15 -22 -12 18 1 22 12 5

-7 -20 5 -23 10 42 -12 8 -13 44 -2 4

12 -28 33 12 -0 -21 3 2 2 24 17 -22

7 40 -6 8 -9 -14 58 -4 -5 -17 -17 38

-16 56 -9 23 97 -3 10 14 -0 -22 12 9

-5 32 1 4 70 34 -11 -27 -16 63 -1 -12

11 39 9 -9 8 41 11 -21 -20 -9 -23 4

3 -9 28 6 10 37 -21 2 -7 1 -4 -19

-3 23 -10 16 25 4 0 46 1 -0 -2 -1

14 -12 -10 1 -16 -18 -3 -2 9 7 0 -17

8 42 12 3 27 3 -9 -1 -22 2 -7 72

16 -23 -11 -13 -21 -16 56 4 6 51 19 -15

42 -2 3 -27 -3 -4 50 0 -9 -5 10 23

-9 25 16 -20 -14 24 41 14 13 -7 14 -25

28 13 -17 -18 -11 19 -14 8 -1 -6 19 -10

-9 -15 1 -12 4 -9 82 -22 3 -10 28 16

5 3 -11 15 -6 -6 5 35 6 -5 0 3

-8 4 -0 -18 -10 39 62 -10 1 -18 26 3

-8 -3 8 -10 11 -1 -4 41 -16 -9 -15 -0

-13 -12 -13 12 -7 33 -18 52 -6 4 -2 -11

16 11 -14 -5 39 79 20 -6 24 1 4 -1

3 0 -1 -4 3 38 29 36 -13 -1 7 0

-10 -3 18 -22 13 -19 22 53 1 -3 -13 4

16 4 -2 15 4 39 44 10 -8 -14 -4 5

-8 -9 76 56 11 7 16 -7 0 -1 -11 -1

24 97 4 -5 -11 20 8 6 -4 -14 -5 -8

30 -2 38 -2 -24 -10 -12 -7 4 3 -11 68

30 -9 33 -9 5 18 23 -1 -6 -12 -19 18

23 4 -21 4 -10 -14 1 -8 -9 16 26 -7

39 6 -16 -16 -6 5 -4 1 5 46 -14 37

75 19 33 21 29 -14 13 19 0 -23 11 30

48 -9 -10 0 -4 2 16 5 2 -23 -2 28

21 -7 24 -11 -5 -2 12 17 -14 28 6 33

53 3 6 -6 28 -11 9 -22 19 -24 7 2 120

100

80

60

40

20

0

20

40

60

80

100

120

Turbine Power Error, 

%

(a)

10.0 40.0 70.0 100.0 130.0 160.0 190.0 220.0 250.0 280.0 310.0 340.0

Direction, degrees

5 -1 1 -5 2 -0 5 2 -3 -2 -0 2

(b)

Fig. 11 (a) Power production error of individual turbines in each direction as compared to SOWFA for the

base case layout (Fig. 6). (b) Directional power production error as compared to SOWFA for the base case layout

(Fig. 6). The color bar applies to both (a) and (b).

15

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10.0 40.0 70.0 100.0 130.0 160.0 190.0 220.0 250.0 280.0 310.0 340.0

Direction, degrees

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

Turbine

-17 2 -2 31 -7 -6 27 -11 -3 -4 14 2

-4 -5 -3 -18 2 10 -10 46 7 -1 42 -11

-20 6 3 16 0 30 -8 -3 14 -4 -3 -2

4 -4 34 26 -1 12 -23 9 -8 -0 -19 10

-0 54 5 -9 50 -6 -16 -1 -6 -10 -3 -26

5 3 -6 -10 -16 -12 7 -17 3 13 -4 13

26 -13 -17 -11 -0 -13 -5 2 14 -21 -12 -14

-1 -4 -15 -10 -3 -2 -2 16 30 -11 22 -19

9 2 -2 -7 -3 -2 32 -8 12 11 3 -23

-20 -23 -5 -12 -11 3 -5 -20 43 7 -4 -13

-18 -4 -19 -7 22 -6 0 4 13 -15 -8 -7

-14 -11 -5 28 18 4 -5 62 -11 -2 -15 -14

-16 10 40 10 16 3 -22 -16 -13 -8 -5 -8

20 29 10 -3 22 3 -21 -26 -21 -10 -15 7

16 5 12 55 5 -20 -11 -3 4 4 118 -12

-13 -11 21 5 -0 14 -15 3 -7 5 22 22

-3 25 7 -10 -1 -1 -11 -5 -14 10 -2 10

4 14 -12 -22 -10 -15 10 -14 -6 14 -6 -9

-10 10 -23 -9 -16 -10 -22 15 24 11 27 23

-5 7 -2 3 -8 3 -10 -20 28 -26 -16 -24

-4 -8 -7 -9 3 -3 -1 30 -1 9 -4 -7

9 -7 -16 6 -14 10 5 27 1 -18 20 -3

-1 27 -1 -8 -19 -30 30 -16 -19 -4 2 -18

-4 5 6 -17 -7 1 4 21 20 -2 -12 16

-8 -8 -14 3 23 9 22 2 -13 -16 20 -8

22 2 1 8 -9 13 -5 19 -5 -5 -5 -9

-4 -9 -8 1 8 27 48 -11 2 5 -8 -7

-3 36 39 -9 10 -22 6 -5 -17 -1 -5 18

-25 40 33 -18 17 22 -11 2 -7 9 11 -8

26 2 -6 65 27 -8 -16 -6 -5 3 -1 -11

-30 30 13 28 7 14 -10 7 -2 17 3 -17

17 8 26 -21 -11 1 10 6 -8 -3 -9 31

-5 -3 -5 -23 -4 -4 -6 2 -5 -2 -2 -23

-12 11 8 -11 7 8 -5 -7 6 -25 -23 30

8 -24 -19 -4 -21 -5 12 7 -12 -10 12 3

-1 -33 -5 4 -6 -2 10 -14 -0 11 9 7

20 -10 -3 -5 -3 18 -15 -19 -3 17 10 21

-5 3 -14 -2 1 12 -1 1 30 -4 17 10 120

100

80

60

40

20

0

20

40

60

80

100

120

Turbine Power Error, 

%

(a)

10.0 40.0 70.0 100.0 130.0 160.0 190.0 220.0 250.0 280.0 310.0 340.0

Direction, degrees

-4 1 -2 -3 -1 -1 -4 -2 -1 -3 0 -5

(b)

Fig. 12 (a) Power production error of individual turbines in each direction as compared to SOWFA for the

optimized layout (Fig. 9). (b) Directional power production error as compared to SOWFA for the optimized

layout (Fig. 9). The color bar applies to both (a) and (b).

16

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19

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