The creation of wakes, with unique turbulence charac- teristics, downstream of turbines significantly increases the complexity of the boundary layer flow within a wind farm. In conventional wind farm design, analytical wake models are generally used to compute the wake-induced power losses, with different wake models yielding significantly different estimates. In this context, the wake behavior, and subsequently the farm power generation, can be expressed as functions of a series of key factors. A quantitative understanding of the relative impact of each of these factors is paramount to the development of more reliable power generation models; such an understanding is however missing in the current state of the art in wind farm design. In this paper, we quantitatively explore how the farm power generation, estimated using four different analytical wake models, is influenced by the following key factors: (i) incoming wind speed, (ii) land configuration, and (iii) ambient turbulence. The sensitivity of the maximum farm output potential to the input factors, when using different wake models, is also analyzed. The extended Fourier Amplitude Sensitivity Test (eFAST) method is used to perform the sensitivity analysis. The power generation model and the optimization strategy is adopted from the Unrestricted Wind Farm Layout Optimization (UWFLO) framework. In the case of an array-like turbine arrangement, both the first-order and the total-order sensitivity analysis indices of the power output with respect to the incoming wind speed were found to reach a value of 99%, irrespective of the choice of wake models. However, in the case of maximum power output, significant variation (around 30%) in the indices was observed across different wake models, especially when the incoming wind speed is close to the rated speed of the turbines.

1. Sensitivity of Array-like and Optimized Wind Farm
Output to Key Factors and Choice of Wake Models
Weiyang Tong*, Souma Chowdhury*, Ali Mehmani*, Jie Zhang#, and
Achille Messac**
* Syracuse University, Department of Mechanical and Aerospace Engineering
# National Renewable Energy Laboratory
** Mississippi State University, Bagley College of Engineering
ASME 2013 International Design Engineering Technical Conference
August 4-7, 2013
Portland, OR

2. Optimal Wind Farm Design
2
 The quality of a wind energy project is represented by several performance
objectives
 These performance objective(s) depend on multiple input factors
Net impact on
surroundings
Turbine
survivability
Grid
integration
Levelized cost
Energy
production
capability
Cost Investment
Energy production
Turbine survivability
Discount rate
Energy
production
Turbine features
Wind speed & direction
Land configuration
Quality of Wind
Energy Project

3. Optimal Wind Farm Design
3
Factors affecting wind
farm performance
Natural factors
(uncontrollable)
Wind speed
& direction
Mean Wind
Speed
Intermittency
Ambient
turbulence
Wind shear
Design factors
(controllable)
Land
configuration
Land area Farm layout
Turbine
features
Grid
connection
Energy
storage

4. Research Motivation
4
 The physical, socio-economical, and environmental objectives in wind farm
planning are highly complex
 Strong assumptions are made to quantify/optimize those objectives
• Neglecting some factors
• Assuming constant factors
 Establish an understanding of how each of the model inputs influences the
objectives
 Which assumptions are necessary
 What kind of errors/inaccuracies are being introduced
 How much does the optimal solution change if the input parameters
change

5. Research Objective
5
1. Apply sensitivity analysis to investigate how sensitive the wind farm output
to each of the following key factors is
I. Incoming wind speed
II. Ambient turbulence
III. Land area per MW installed
IV. Land aspect ratio
V. Nameplate capacity
2. Investigate the variation of sensitivity results with the choice of analytical
wake models
3. How the influence is translated to the optimal design – optimized wind farm
output

6. Outline
6
• Methods & Models
• Numerical Experiments
• Results & Discussion
• Sensitivity analysis of array-like farm output to key factors
• Sensitivity analysis of optimized farm output to key factors
• Sensitivity analysis of array-like and optimized farm output to the choice
of wake models
• Concluding Remarks

7. Methods & Models
7
• Extended Fourier Amplitude of Sensitivity Test (eFAST) is a variance-based
global sensitivity analysis method
• Based on Fourier analysis, the first-order index is defined as the ratio of the
conditional variance of each input parameter to the variance of the model output
𝑆𝑖 =
𝜎 𝑌 𝑋𝑖
2
𝜎 𝑌
2
• The total-order index estimates the sum of all effects involving the associated
input parameter
𝑆 𝑇 𝑖
= 1 −
𝜎 𝑌 𝑋≠𝑖
2
𝜎 𝑌
2
• Power Generation Model adopted from the UWFLO framework[1]
• A function of incoming wind conditions, farm layout, and turbine features
• considers variable induction factor, wake merging, and partial wake overlap
[1]: Chowdhury et al , 2011

8. 8
Denmark's Horns Rev 1 wind farm
 Two major impacts:
 Power loss due to the reduction of wind speed on downstream
turbines
 Reduction of turbine lifetime due to increased turbulence-
induced structural load
Factors regulating the wake behavior in turn affect the power generation of a
wind farm
– the reliability of power estimation is based on the accuracy of the wake
model used
Methods & Models
Wind speed
Topography
Ambient
turbulence
Turbine
features
Land
configuration
Wake
Model

9. Numerical Experiments
9
Assumptions:
 The incoming wind speed is unidirectional
 The incoming wind speed is uniformly distributed over the entire rotor area
 Identical turbines (GE 1.5MW – 82.5m ) are used
 The ambient turbulence over the farm site is constant everywhere
 The wind farm has a rectangular shape
Wake
model
Incoming
wind speed
Downstream
spacing
Radial
spacing
Induction
factor
Rotor
Diameter
Hub
height
Ambient
turbulence
Jensen √ √ √ √
Frandsen √ √ √ √
Larsen √ √ √ √ √ √ √
Ishihara √ √ √ √ √ √
Inputs to each wake model
Input factors Lower bound Upper bound
Incoming wind speed
Region I 3.5 𝑚/𝑠 10.35 𝑚/𝑠
Region II 10.35 𝑚/𝑠 12.65 𝑚/𝑠
Region III 12.65 𝑚/𝑠 20 𝑚/𝑠
Ambient turbulence 1% 25%
Land area/MW installed 45,000 𝑚2
/𝑀𝑊 135,000 𝑚2
/𝑀𝑊
Land aspect ratio 0.1 20
Nameplate Capacity 15 𝑀𝑊 75 𝑀𝑊

10. Sensitivity of Array-like farm Output to key Factors
10
 An array-like wind farm with 12 GE 1.5 MW – 82.5m turbines is considered.
 Uniform spacing between rows/columns.
Layout is controlled by land area/MW installed and land aspect ratio

11. Sensitivity Analysis of Array-like Farm Output
(Region I: incoming wind speed 3.5 m/s – 10.35 m/s)
11
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming wind
speed
Ambient
turbulence
Land area/MW
installed
Land aspect
ratio
Jensen model
first-order index total-order index
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming wind
speed
Ambient
turbulence
Land area/MW
installed
Land aspect
ratio
Frandsen model
first-order index total-order index
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming wind
speed
Ambient
turbulence
Land area/MW
installed
Land aspect
ratio
Larsen model
first-order index total-order index
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming wind
speed
Ambient
turbulence
Land area/MW
installed
Land aspect
ratio
Ishihara model
first-order index total-order index

12. Sensitivity Analysis of Array-like Farm Output
(Region II: incoming wind speed 10.35 m/s – 12.65 m/s)
12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming wind
speed
Ambient
turbulence
Land area/MW
installed
Land aspect
ratio
Jensen model
first-order index total-order index
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming wind
speed
Ambient
turbulence
Land area/MW
installed
Land aspect
ratio
Frandsen model
first-order index total-order index
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming wind
speed
Ambient
turbulence
Land area/MW
installed
Land aspect
ratio
Larsen model
first-order index total-order index
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming wind
speed
Ambient
turbulence
Land area/MW
installed
Land aspect
ratio
Ishihara model
first-order index total-order index

13. 13
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ambient
turbulence
Land area/MW
installed
Land aspect ratio
Jensen model
first-order index total-order index
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ambient
turbulence
Land area/MW
installed
Land aspect ratio
Larsen model
first-order index total-order index
Models w/o turbulence
Models with turbulence
Sensitivity Analysis of Array-like Farm Output
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ambient
turbulence
Land area/MW
installed
Land aspect ratio
Jensen model
first-order index total-order index
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ambient
turbulence
Land area/MW
installed
Land aspect ratio
Larsen model
first-order index total-order index
Fixed incoming wind speed at 7 m/s Fixed incoming wind speed at 11 m/s

14. Sensitivity of Optimized Farm Output to Key Factors
14
 The farm layout is optimized using Mixed-discrete Particle Swarm
Optimization (MDPSO) algorithm
max 𝑓(𝑉) = 𝐶𝐹
subject to
𝑔 𝑉 ≤ 0
𝑉 𝑚𝑖𝑛 ≤ 𝑉 ≤ 𝑉𝑚𝑎𝑥
𝑉 = {𝑋1, 𝑋2, ⋯ , 𝑋 𝑁, 𝑌1, 𝑌2, ⋯ , 𝑌𝑁}
Inter-Turbine Spacing
determined by farm configuration
Side constraints
0
500
1000
1500
2000
0 500 1000 1500 2000 2500 3000
Optimal farm layout of 20 turbines

15. Sensitivity Analysis of Optimized Farm Output
(Region I: incoming wind speed 3.5 m/s – 10.35 m/s)
15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming wind
speed
Ambient
turbulence
Land area/MW
installed
Land aspect
ratio
Nameplate
capacity
Ishihara model
first-order index total-order index
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming wind
speed
Ambient
turbulence
Land area/MW
installed
Land aspect
ratio
Nameplate
capacity
Larsen model
first-order index total-order index
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming wind
speed
Ambient
turbulence
Land area/MW
installed
Land aspect
ratio
Nameplate
capacity
Frandsen model
first-order index total-order index
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming wind
speed
Ambient
turbulence
Land area/MW
installed
Land aspect
ratio
Nameplate
capacity
Jensen model
first-order index total-order index

16. Sensitivity Analysis of Optimized Farm Output
(Region II: incoming wind speed 10.35m/s – 12.65 m/s)
16
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming wind
speed
Ambient
turbulence
Land area/MW
installed
Land aspect
ratio
Nameplate
capacity
Ishihara model
first-order index total-order index
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming wind
speed
Ambient
turbulence
Land area/MW
installed
Land aspect
ratio
Nameplate
capacity
Larsen model
first-order index total-order index
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming wind
speed
Ambient
turbulence
Land area/MW
installed
Land aspect
ratio
Nameplate
capacity
Frandsen model
first-order index total-order index
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming wind
speed
Ambient
turbulence
Land area/MW
installed
Land aspect
ratio
Nameplate
capacity
Jensen model
first-order index total-order index

17. Concluding Remarks
17
 For an array-like wind farm, the incoming wind speed has a dominant
impact on power generation, irrespective of the choice of wake models.
 For a wind farm with optimized farm layout, incoming wind speed still a
dominant factor for incoming wind speed lower than the rated speed;
however, design factors start affecting the power generation when
incoming wind speed .varies around the rated speed
 Overall, natural factors have the most contribution to the farm output
when comparing to design factors, which emphasize the importance of
wind resource assessment.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Incoming wind
speed
Ambient
turbulence
Land area/MW
installed
Land aspect
ratio
Nameplate
capacity
Ishihara model

18. Future work
18
 Quantify the uncertainties introduced by lack of understanding of the
wind resource, how it affect the optimal wind farm planning.

19. Questions
and
Comments
19
Thank you

20. Acknowledgement
 I would like to acknowledge my research adviser
Prof. Achille Messac, and my co-adviser Prof.
Souma Chowdhury for their immense help and
support in this research.
 I would also like to thank my friends and colleagues
Dr. Jie Zhang and Ali Mehmani for their valuable
contributions to this paper.
 Support from the NSF Awards is also
acknowledged.
20

21. 21
Layout-based Power Generation Model
 Adopted from the Unrestricted Wind Farm Layout Optimization (UWFLO)
methodology*
 In this power generation model, the induction factor is treated as a function of the
incoming wind speed and turbine features:
U: incoming wind speed; P: power generated, given by the power curve
kg, kb: mechanical and electrical efficiencies, Dj: Rotor Diameter, 𝜌: Air density
 A generalized power curve is used to represent the approximate power response of
a particular turbine
𝑈𝑖𝑛, 𝑈 𝑜𝑢𝑡, and 𝑈𝑟: cut-in speed, cut-out speed, and rated speed
𝑃𝑟: Rated capacity, 𝑃𝑛: Polynomial fit for the generalized power curve
*: Chowdhury et al , 2011

22. Numerical Experiments
22
Input parameters Lower bound Upper bound
Incoming wind speed
Region I 3.5 𝑚/𝑠 10.35 𝑚/𝑠
Region II 10.35 𝑚/𝑠 12.65 𝑚/𝑠
Region III 12.65 𝑚/𝑠 20 𝑚/𝑠
Ambient turbulence 1% 25%
Land area/MW installed 45,000 𝑚2
/𝑀𝑊 135,000 𝑚2
/𝑀𝑊
Land aspect ratio 0.1 20
Nameplate Capacity 15 𝑀𝑊 75 𝑀𝑊
 Incoming wind speed is divided into 3 regions
based on the power curve
 Region I starts from cut-in speed to -10%
of the rated speed
 Region II ranges from -10% to +10% of
the rated speed
 Region III starts from +1-% of the rated
speed to cut-out speed

23. Numerical Experiments
23
Input parameters Lower bound Upper bound
Incoming wind speed
Region I 3.5 𝑚/𝑠 10.35 𝑚/𝑠
Region II 10.35 𝑚/𝑠 12.65 𝑚/𝑠
Region III 12.65 𝑚/𝑠 20 𝑚/𝑠
Ambient turbulence 1% 25%
Land area/MW installed 45,000 𝑚2
/𝑀𝑊 135,000 𝑚2
/𝑀𝑊
Land aspect ratio 0.1 20
Nameplate Capacity 15 𝑀𝑊 75 𝑀𝑊
 Turbulence intensity’s range is determined
based on the majority data collected in reference
 For simplicity, some wake models do not
account for ambient turbulence

24. Numerical Experiments
24
Input parameters Lower bound Upper bound
Incoming wind speed
Region I 3.5 𝑚/𝑠 10.35 𝑚/𝑠
Region II 10.35 𝑚/𝑠 12.65 𝑚/𝑠
Region III 12.65 𝑚/𝑠 20 𝑚/𝑠
Ambient turbulence 1% 25%
Land area/MW installed 45,000 𝑚2
/𝑀𝑊 135,000 𝑚2
/𝑀𝑊
Land aspect ratio 0.1 20
Nameplate Capacity 15 𝑀𝑊 75 𝑀𝑊
 Land use of a single turbine is difficult to depict
due to constraints, such as turbine’s geometry
and complex terrain
 Using Land area/MW installed (LAMI) makes
the land use of a wind farm independent from
those constraints
 The range of LAMI is determined using the data
collected by NREL
10
𝐷2
𝑃𝑟
< 𝐴 𝑀𝑊 < 30
𝐷2
𝑃𝑟

25. Numerical Experiments
25
Input parameters Lower bound Upper bound
Incoming wind speed
Region I 3.5 𝑚/𝑠 10.35 𝑚/𝑠
Region II 10.35 𝑚/𝑠 12.65 𝑚/𝑠
Region III 12.65 𝑚/𝑠 20 𝑚/𝑠
Ambient turbulence 1% 25%
Land area/MW installed 45,000 𝑚2
/𝑀𝑊 135,000 𝑚2
/𝑀𝑊
Land aspect ratio 0.1 20
Nameplate Capacity 15 𝑀𝑊 75 𝑀𝑊
 The Land Aspect Ratio affects both the
streamwise and the spanwise spacing between
turbines, causing different levels of wake effect
on downstream turbines.
 It also determines the optimal farm layout.

26. Numerical Experiments
26
Input parameters Lower bound Upper bound
Incoming wind speed
Region I 3.5 𝑚/𝑠 10.35 𝑚/𝑠
Region II 10.35 𝑚/𝑠 12.65 𝑚/𝑠
Region III 12.65 𝑚/𝑠 20 𝑚/𝑠
Ambient turbulence 1% 25%
Land area/MW installed 45,000 𝑚2
/𝑀𝑊 135,000 𝑚2
/𝑀𝑊
Land aspect ratio 0.1 20
Nameplate Capacity 15 𝑀𝑊 75 𝑀𝑊
 Since identical turbines are used, the sensitivity
of farm output to nameplate capacity is
equivalent to the number of turbines installed

27. 27
UWFLO Power Generation Model
 Turbine-j is in the influence of the wake of Turbine-i, if and only if
Considers turbines with differing rotor-diameters and hub-heights
 The Katic model* is used to account for wake merging and partial wake
overlap
𝑢𝑗: Effective velocity deficit
𝐴 𝑘𝑗: Overlapping area between Turbine-j
and Turbine-k
Partial wake-rotor overlap *: Katic et al , 1987

28. 28
Incoming wind speed = 8 m/sIncoming wind speed = 12.5 m/s
Capacity Factor variation with the Land Area per Turbine
Rated speed: 12.5 m/s
Turbine Power Curve
0
0.2
0.4
0.6
0.8
1
1st Row 2nd Row 3rd Row
POWER TREND

29. 29
Capacity Factor variation with the Incoming Wind Speed
LAT = 25 ha
Single wake test

30. 30
Layout-based Power Generation Model
 In this power generation model, the induction factor is treated as a
function of the incoming wind speed and turbine features:
U: incoming wind speed; P: power generated, given by the power curve
kg, kb: mechanical and electrical efficiencies, Dj: Rotor Diameter, 𝜌: Air density
 A generalized power curve is used to represent the approximate power
response of a particular turbine
𝑈𝑖𝑛, 𝑈 𝑜𝑢𝑡, and 𝑈𝑟: cut-in speed, cut-out speed, and rated speed
𝑃𝑟: Rated capacity, 𝑃𝑛: Polynomial fit for the generalized power curve*
*: Chowdhury et al , 2011

31. Sensitivity Analysis
31
• Objective – to determine the amount and type of change occurred in the model
predictions due to a change in the input parameter of the model
a) factors that mostly contribute to the output variation;
b) factors (or parts of the model itself) that are insignificant;
c) any regions in the space of input factors where the output variation is the maximum; and
d) if and which factors interact with other ones.
• Extended Fourier Amplitude of Sensitivity Test (eFAST) is a variance-based
global sensitivity analysis method
• Based on Fourier analysis, the first-order index is defined as the ratio of the
conditional variance of each input parameter to the variance of the model output
𝑆𝑖 =
𝜎 𝑌 𝑋𝑖
2
𝜎 𝑌
2
• The total-order index estimates the sum of all effects involving the associated
input parameter
𝑆 𝑇 𝑖
= 1 −
𝜎 𝑌 𝑋≠𝑖
2
𝜎 𝑌
2
Sensitivity Analysis Flow Chart#
#: Saltelli, Chan and Scott, 2000