The planning of a wind farm, which minimizes the project costs and maximizes the power generation capacity, presents significant challenges to today’s wind energy industry. An optimal wind farm planning strategy that accounts for the key factors (that can be designed) influencing the net power generation offers a powerful solution to these daunting challenges. This paper explores the influences of (i) the number of turbines, (ii) the farm size, and (iii) the use of a combination of turbines with differing rotor diameters, on the optimal power generated by a wind farm. We use a recently developed method of arranging turbines in a wind farm (the Unrestricted Wind Farm Layout Optimization (UWFLO)) to maximize the farm efficiency. Response surface based cost models are used to estimate the cost of the wind farm as a function of the the turbine rotor diameters and number of tur- bines. Optimization is performed using a Particle Swarm Optimization (PSO) algorithm. A robust mixed-discrete version of the PSO algorithm is implemented to appropriately account for the discrete choice of feasible rotor diameters. The use of an optimal combi- nation of turbines with differing rotor diameters was observed to significantly improve the net power generation. Exploration of the influences of (i) the number of turbines, and (ii) the farm size, on the cost per KW of power produced, provided interesting observations.

1. Exploring Key Factors Influencing Optimal Farm Design
Using Mixed-Discrete Particle Swarm Optimization
Souma Chowdhury*, Jie Zhang*, Achille Messac#, and Luciano Castillo*
* Rensselaer Polytechnic Institute, Department of Mechanical, Aerospace, and Nuclear Engineering
# Syracuse University, Department of Mechanical and Aerospace Engineering
13th AIAA/ISSMO Multidisciplinary Analysis Optimization (MAO) Conference
September 13-15, 2010
Fort Worth, Texas

2. Presentation Outline
 Technical background and Motivation
 Objectives of this paper
 UnrestrictedWind Farm Layout Optimization (UWFLO) framework
 Mixed-Discrete Particle Swarm Optimization
 Optimal use of a combination of available non-identical turbines
 Exploring the influence of number of turbine and farm size
 Concluding Remarks
2

3. Wind Farm Optimization
3
• Currently wind energy contributes 2% of worldwide electricity consumption.
• Planned increase in USA by 2030 – 10 fold.
• Advancing wind energy would require optimal wind farm design strategies.
Critical aspects in wind farm
design include (not limited to)
 Farm layout
 Number and types of
turbines to be installed
 Farm size
www.prairieroots.org

4. Mixed-Discrete Non-Linear Optimization (MDNLO)
MDNLO
Criterion
Functions
Non-linear
Objectives
Non-linear
constraints
Wind farm layout optimization involving
optimal selection of turbines:
MDNLO with non-uniformly distributed
discrete variables
Design
Variables
Continuous
Variables
Discrete
Variables
Uniformly
Distributed
e.g. Integers
Non-uniformly
Distributed
4

5. Motivation
 The net power generated by a wind farm is reduced by the wake effects, which
can be offset by optimizing the farm layout.
 A combination of different types of turbines is expected to further improve the
power generation capacity and the economy of a wind farm. Commercially
available turbines provide a set of discrete choices.
 Exploration of the influence of key
farm planning factors such as the farm
size and the number of turbines,
within the context of layout
optimization would be uniquely
helpful.
www.wind-watch.org 5

6. Existing Wind Farm Optimization Methods
6
Grid based approach
Yields a computationally expensive
mixed-integer problem for large
number of turbines
Array layout approach
Restricts turbine locating and
introduces a source of sub-optimality
• Do not simultaneously optimize the selection of wind turbines
• Assume a constant induction factor

7. Research Objectives
• Develop and use an analytical wind farm model that avoids conventional
restrictions in layout planning.
• Implement a generalized Mixed-discrete Particle Swarm Optimization to
simultaneously optimize (i) the selection of turbine rotor diameters, and
(ii) the layout of the wind farm.
• Explore the influences of the farm size and the number of turbines on the
net performance of the optimized wind farm
7

8. Basic Components of the UWFLO Framework
Power Generation Model
 Develops a turbine influence matrix based on the wake effects
 Considers a variable induction factor and partial wake-rotor overlap
 Determines the net power generated by the wind farm
Optimization Framework
 Implements a wind farm cost model
 Simultaneously optimizes the selection of differing types of turbines
 Maximizes the net power generation using the PSO algorithm
8

9. 9
Component 1 • Wind Farm Model
• Wind Farm Optimization
Framework
Component 2

10. UWFLO Power Generation Model
• The flow pattern inside a wind farm is complex, primarily due to the wake
effects and the highly turbulent flow.
• Rotor averaged velocity is determined from the flow profile*
• Step 1
Transformed co-ordinates are evaluated
based on wind direction
10
x X
y Y
   cos   sin

  
      
   sin  cos

  
i i
i i
* Cal et al., 2010

11. Mutual Influence of Turbines
• Step 2
An influence matrix is defined as
where Turbine-i influences Turbine-j if
• Step 3
 

j wake ij
The turbines are ranked in the increasing order of their x-coordinate. Power
generated by turbines is calculated in the increasing order of their rank.
11
1 if Turbine- influences Turbine-
1 if Turbine- influences Turbine-
0 if there is no mutual influence
ij
i j
M j i
 


, 0 &
2 2
ij ij
D D
x  y  

12. • Step 4
Power Generated by the Wind Farm
Effective velocity of wind approaching Turbine-j:*
The power generated by turbine-j:
• Step 5
Coefficient of power
Power generated by the farm: Farm Efficiency:
Power generated by
a standalone turbine
* Katic et al., 1986 12

13. Wake Model
UWFLO uses Frandsen’s wake model*, which calculates the diameter of the
growing wake and the wake velocity as:
Wake spreading constant
However, UWFLO has the flexibility to use any standard wake model.
13
* Frandsen et al., 2006

14. 14
Component 1 • Wind Farm Model
• Wind Farm Optimization
Framework
Component 2

15. UWFLO – Problem Definition
• An unidirectional uniform wind at 7.09 m/s and at 0o to X-axis is considered.
15
Cost Constraint: Applied when optimizing the
selection of wind turbines

16. Wind Farm Cost Model
Quadratic response surface based cost models* are developed to represent the
farm cost, as a function of the turbine rotor diameters and number of turbines.
To this end we used data for wind farms in the state of New York*
For wind farm with non-identical turbines
The cost per KW of power produced is given by
* Chowdhury et al., IDETC2010 16

17. Particle Swarm Optimization (PSO)
Swarm Motion*
t  1 t t

1
i i i
t t t t
i i l i i g g i
x x v
v  v  r p x  r p x
   
1
1 2

 
    
Solution Comparison
The constraint dominance principle**
is used.
PSO can appropriately address the
non-linearity and the multi-modality of
the wind farm model.
17
* Kennedy and Eberhart, 1985
** Deb et al., 2002 (NSGA-II)

18. Generalized Approach to MDNLO - Principles
• Divides the variable space into continuous and discrete variable spaces.
• Implements continuous optimization as the primary search strategy
• Approximates candidate solutions to nearby feasible discrete locations
based on certain criterion.
• Saves computational expense by evaluating criterion functions only at
feasible discrete locations.
• Implemented through non-gradient based optimization algorithms
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19. Vertex Approximation Techniques
In the discrete variable domain, the
location of a candidate solution can be
defined by a local hypercube
Nearest Vertex Approach (NVA)
Approximates to the nearest discrete
location based on Euclidean distance.
Shortest Normal Approach (SNA)
Approximates to the discrete location with
shortest normal to the connecting vector.
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20. Experimental Scale Wind Farm
The UWFLO model has been validated** against a
wind tunnel experiment on a scaled down farm.*
Mean rotor diameter of commercial turbines: 75m
Scaled down to experimental dimensions: 0.12m
Resulting feasible set of diameters at the
experimental scale:
0.03 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17
* Cal et al., 2010; ** Chowdhury et al., IDETC2010 20

21. Case 1 – Non-Identical Turbines
21
Using NVA Using SNA
Incoming
Wind Speed

22. 22
Case 2 – Identical Turbines
• Identical turbines, with rotor diameter, D = 0.12m, are considered
• Original continuous PSO was used in this case
• Approximated power curve: The power generated is assumed to remain
constant at the rated power (0.385W) for U > Rated speed (6.17m/s)
0.5
 To investigate the influence of the number of turbines, we optimize five
wind farms with 6, 0.4
9, 12, 15, and 18 turbines laid out in 14D x 6D wind farm
0.3
 To investigate the influence of the farm size, we optimize five wind farms
with a length to breadth ratio of 7/3.
0.2
0.1
0.0
3 4 5 6 7 8
Approaching Wind Velocity, U (m/s)
Power Generated, P (W)
U = 6.17 m/s
P = 0.385 W Approximated
Power curve

23. UWFLO – Influence of the Number of Turbines
23

24. UWFLO – Influence of the Farm Size
Cost information relating the farm size to the total cost was not readily
available.
24

25. Concluding Remarks
 The proposed UWFLO technique allows simultaneous optimization of (i) the
selection of turbine rotor diameters, and (ii) the layout of the wind farm.
 To this end the developed mixed-discrete PSO is found to be highly effective.
The nearest vertex approach performs better than the shortest normal approach.
 This wind farm optimization technique increases the power generation by 44%
compared to the array layout (at no additional cost).
 The determination of the appropriate number of turbines, and the farm size is
crucial to optimal wind farm design.
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26. Future Work
 In future research, each commercially available turbine, with a unique
combination of rotor diameter, hub height, and performance
characteristics, will be explicit considered.
 Future research will also consider the variability of the speed and
direction of wind, in the case of commercial wind farms.
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27. Selected References
1. World Wind Energy Report 2008. Bonn, Germany, February 2009.
2. Katic, I., Hojstrup, J., and Jensen, N. O. A Simple Model for Cluster Efficiency. In Proceedings of European
Wind Energy Conference and Exhibition (Rome, Italy 1986).
3. Frandsen, S., Barthelmie, R., Pryor, S, Rathmann, O, Larsen, S, Hojstrup, J, and Thogersen, M. Analytical
Modeling of Wind Speed Deficit in Large Offshore Wind Farms. Wind energy, 9, 1-2 (2006), 39-53.
4. Grady, S. A., Hussaini, M. Y., and Abdullah, M. M. Placement of Wind Turbines Using Genetic Algorithms.
Renewable Energy, 30, 2 (February 2005).
5. Sisbot, S., Turgut, O., Tunc, M., and Camdali, U. Optimal positioning of Wind Turbines on Gökçeada Using
Multi-objective Genetic Algorithm. Wind Energy (2009).
6. Mosetti, G., Poloni, C., and Diviacco, B. Optimization of Wind Turbine Positioning in Large Wind Farms by
Means of a Genetic Algorithm. Journal of Wind Engineering and Industrial Aerodynamics, 54, 1 (January
1994), 105-116.
7. Kennedy, J. and Eberhart, R. C. Particle Swarm Optimization. In Proceedings of the 1995 IEEE International
Conference on Neural Networks ( 1995), 1942-1948.
8. Cal, R. B., Lebron, J., Kang, H.S., Meneveau, C., and Castillo, L., “Experimental study of the horizontally
averaged flow structure in a model wind-turbine array boundary layer”, Journal of Renewable and
Sustainable Energy, 2, 1 (2010).
9. Lebron, J., Castillo, Cal, R. B., Kang, H. S., and Meneveau, C., 2010, “Interaction Between a Wind Turbine
Array and a Turbulent Boundary Layer,” Proceeding 49th AIAA Aerospace Sciences Meeting including the
New Horizons Forum and Aerospace Exposition, January 4-9.
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28. Acknowledgement
28
Thank you

29. Questions
or
Comments
29