The document explores uncertainty propagation from wind conditions to optimal wind farm performance, emphasizing the importance of accurately characterizing and modeling these uncertainties for effective project planning, particularly for offshore wind farms. It presents two models for uncertainty in wind conditions and describes a robust wind farm optimization framework that aims to minimize costs and uncertainty while accounting for complex variable factors. The findings highlight the trade-offs between improving farm performance and managing uncertainty in energy costs.

1. Characterizing the Uncertainty Propagation from the
Wind Conditions to the Optimal Farm Performance
Souma Chowdhury*, Jie Zhang*, Achille Messac#, and Luciano Castillo*
# Syracuse University, Department of Mechanical and Aerospace Engineering
* Rensselaer Polytechnic Institute, Department of Mechanical, Aerospace, and Nuclear Engineering
52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and
Materials Conference
7th AIAA Multidisciplinary Design Optimization Specialist Conference
April 4 – 7, 2011
Sheraton Denver
Denver, Colorado

2. Wind Energy - Overview
 Currently wind contributes only 2% of the total electricity
consumption (worldwide and in the US).
 Wind energy is planned to account for 20% of the U.S. electricity
consumption by 2030.
 Steady improvement of wind energy technologies, particularly
for offshore wind farms would help accomplish this target.
www.prairieroots.org
NREL, 2011 2

3. Motivation
 One of the key factors restraining the development of wind energy is
the ill-predictability of the actual power that will be generated.
 The power generated by a wind farm is a variable quantity that is a
function of a series of highly uncertain parameters.
 A majority of these uncertainties are not well understood.
 Quantification of these uncertainties become all the more important
for offshore wind farms that generally require a higher investment
upfront.
 Careful modeling of these uncertainties, together with their
propagation into the overall system, will allow for more credible
planning of wind projects.
3

4. Presentation Outline
• Research Objectives
• Illustrating the Uncertainties in Wind Conditions
• Modeling the Uncertainties in Wind Conditions
• Multivariate and Multimodal Wind Distribution
• Robust Wind Farm Optimization
• Concluding Remarks
4

5. Uncertainties in Wind Energy
Long Term
Uncertainties
Uncertainties in Wind Energy may be broadly classified into:
Wind• Long
Environmental Turbine Operational
Term Uncertainties: Introduced by (i) theInterruptions Economic
long term variation of
Conditions Topography
Factors Performance Factors
wind conditions, (ii) turbine design, and (iii) other environmental,
operational and financial factors Turbine Changes in
Terrain/Surface Component
Wind Speed Rain/Snow Component Utility Price
Roughness Depreciation
Breakdown ($/kWh)
• Short Term Uncertainties: Introduced by boundary layer turbulence and
other flow variations that occur in a small time scale (order of minutes) in
Component Power Grid Changes
Wind Direction Storms Vegetation
Replacement Repair O&M Cost
Installation of
Man-made Changes is Govt.
Air Density Additional
Structures Policies
Turbines
Changes in
Interest Rates &
Insurance Rates
5

6. Research Objectives
 Characterize the uncertainty in the predicted long term
variation of wind conditions.
 Model the propagation of uncertainty from the wind
conditions to the power generated by the farm.
 Develop and apply a robust wind farm optimization
framework using the new uncertainty model and the
Unrestricted Wind Farm Layout Optimization (UWFLO)
method.
6

7. Uncertainties in Wind Conditions
• The wind speed, the wind direction, and the air density at a given site
vary significantly over the course of a year.
• The long term variation of wind conditions is generally represented using
probability distribution models.
• These probability distribution models are developed using previous
years’ recorded wind data (e.g. wind data from 2000 – 2009).
• Uncertainty is mainly introduced by the assumption, “the annual
distribution of wind conditions, estimated from preceding years’ data,
will directly apply to the succeeding years of operation of the wind farm
(being designed).”
• More often than not, the number of years for which credible wind data is
available is (for a particular site) significantly less than the designed
lifetime of the wind farm (15 – 20 years or more).
Zhang et al, 2011 7

8. Wind Distribution in Annual Power Generation
Wind Probability Distribution
• Annual Energy Production of a farm is given by:
• In farm power generation models this integral equation is expressed as:
Uncertainty in Uncertainty in
Uncertainty in
the Predicted the Annual
Wind
Yearly Wind Energy
Conditions
Distribution Production
Kusiak and Zheng, 2010; Vega, 2008 8

9. Year-to-Year Variations
Estimated Wind May not be the right Predicted Long Term
Distribution way to account for
Deterministic assumption Variation of Wind
Wind distributions estimated using the Multivariate and years)
(preceding years’ data) wind variations (succeeding
Multimodal model for a site at Baker, ND
Zhang et al., 2011 9

10. The Uncertainty Modeling Concept
Stochastic models of the wind distribution probabilities
Estimated probability of wind distribution, log(p(Ui,i)) 5
3 10-yr MMWD
2000 MMWD
1 2001 MMWD
2002 MMWD
-1 2003 MMWD
2004 MMWD
-3 2005 MMWD
2006 MMWD
-5 2007 MMWD
2008 MMWD
-7 2009 MMWD
sample-1 DPSWC
-9 sample-2 DPSWC
sample-3 DPSWC
-11 sample-4 DPSWC
sample-5 DPSWC
-13
-15
-17
1 2 3 4 5 6
Sample number, i
Sample # Wind Speed (m/s) Wind Direction (deg) Air Density (kg/m3)
1 6.50 180 1.245
2 9.75 90 1.323
3 3.25 270 1.168
4 4.88 135 1.284
10
5 11.38 315 1.129

11. Characterizing the Uncertainties
 In this paper, we model the uncertainty in the wind conditions as a
function of the uncertainty in the predicted wind distribution.
 Two different models have been proposed.
 Model 1: We consider the parameters of the predicted wind distribution
to be stochastic (e.g. the k and c parameters in the popularly used Weibull
distribution).
 Model 2: We consider the predicted yearly wind probability value itself
to be stochastic.
11

12. Uncertainty Model 1
 The uncertainty in the parameters of the wind distribution model is
represented by their variance (in this paper).
 The variability in the predicted yearly probability pi of wind coming at
speed Ui, direction i, and air density ri can thus be represented as a
function of the variance of the wind distribution parameters.
 qk: kth parameter; Sq: Covariance of the distribution parameters;
12

13. Uncertainty Propagation Model 1
 The uncertainty propagating into the annual energy production is
modeled as a function of the variance of the wind distribution.
Uncertainty in the predicted yearly
probabilitiesin Wind Distribution
Uncertainty of the sample wind conditions
Parameters
Uncertainty in the Predicted
Yearly Probability of Sample
Wind Conditions
Uncertainty in the Annual Energy
Production
Lindberg, 1999
13

14. Uncertainty Model 2
 The variability in the predicted yearly probabilities pi is directly
represented by a stochastic model.
 This approach has three key advantages:
Stochastic models of the wind distribution probabilities
5
Estimated probability of wind distribution, log(p(Ui,i))
1. Can be applied to both parametric and non-parametric10-yr MMWD distribution
3 wind
2000 MMWD
models
1 2001 MMWD
2002 MMWD
-1 2003 MMWD
2. Can -3 readily applied to univariate, bivariate and multivariate wind
be 2004 MMWD
2005 MMWD
distributions
-5
2006 MMWD
2007 MMWD
2008 MMWD
3. Avoids the bias that might be introduced by the estimation of the
-7 2009 MMWD
sample-1 DPSWC
nonlinear uncertainty propagation
-9 sample-2 DPSWC
sample-3 DPSWC
-11 sample-4 DPSWC
sample-5 DPSWC
 Robust wind farm optimization in this paper has been performed using
-13
-15
this model.
-17
1 2 3 4 5 6
Sample number, i
14

15. Uncertainty Model 2: Formulation
 The probability of a given wind condition was observed to vary in orders
of magnitude from year to year.
 We used lognormal distribution to represent the variation in the yearly
probabilities (over years) of a given sample wind condition.
 We call this model the Distribution of the Probability of a Sample Wind
Condition (DPSWC).
 The DPSWC model and the uncertainty in the predicted yearly wind
probability is given by
15

16. Uncertainty Propagation Model 2
 The uncertainty in the annual energy production can be determined by
where
 Subsequently, the uncertainty in the Cost of Energy (COE) can be
expressed as
where
Lindberg, 1999 16

17. Wind Distribution Model
• In this paper, we use the non-parametric model called the Multivariate
and Multimodal Wind Distribution (MMWD).
• This model is developed using the multivariate Kernel Density
Estimation (KDE) method.
• In this paper, we use 10-year wind data for a site at Baker, ND.
17

18. Robust Wind Farm Optimization
18

19. Motivation for Wind Farm Optimization
 The net power generated by a wind farm is reduced by the wake effects, which
can be offset by optimizing the farm layout.
 Optimally selecting the turbine-type to be installed may further improve the
power generation capacity and the economy of a wind farm.
Turbine
Rated Rotor Hub Power
Power Diameter Height Curve
www.wind-watch.org
19

20. Existing Wind Farm Optimization Methods
Array layout approach Grid based approach
Computationally less expensive. Allows the exploration of different farm
Restricts turbine locating and introduces configurations.
a source of sub-optimality Results might be undesirably sensitive
to the pre-defined grid size
• Do not simultaneously optimize the selection of wind turbines
• Majority of them do not consider an appropriate multivariate wind
distribution model
• Do not explicitly account for the uncertainties in wind conditions
Sorenson et al., 2006; Mikkelson et al., 2007; Grady et al., 2005; 20
Sisbot et al., 2009; Gonzleza et al., 2010

21. Unrestricted Wind Farm Layout Optimization (UWFLO)
 Develops and uses a computationally inexpensive analytical power
generation model.
 Uses a response surface based wind farm cost (RS-WFC) model.
 Simultaneously optimizes the farm layout and the selection of the
turbine-type to be installed.
 Accounts for the annual distribution of wind speed and direction.
 The robust wind farm optimization framework, which is an evolution
from the UWFLO method, accounts for the uncertainties in wind
conditions.
Chowdhury et al., 2011 21

22. UWFLO Power Generation Model
 Turbines locations are defined by a
Cartesian coordinate system
 Turbine-j is in the influence of the wake
of Turbine-i, if and only if
Avian Energy, UK
 Effectiveapproach allows us to consider turbines with differing rotor-
 This velocity of wind  Power generated by Turbine-j:
approaching Turbine-j:
diameters and hub-heights
22

23. Wake Model
 We implement Frandsen’s velocity deficit model
Wake growth Wake velocity
a – topography dependent wake-spreading constant
 Wake merging: Modeled using wake-superposition principle
developed by Katic et al.:
Frandsen et al., 2006; Katic et al.,1986 23

24. Problem Definition
We apply a 2-step optimization process:
 Step-1: Minimize the Cost of Energy (COE).
 Step-2: Minimize the uncertainty in the COE. The minimum COE
obtained in Step-1 is relaxed (increased) by 5%, and applied as an
additional constraint.
Farm Boundaries
Inter-Turbine Spacing
COE Constraint
24

25. Optimized Farm Layouts
 Minimizing the COE has produced a well spread out farm layout, thereby
seeking to minimize the wake losses for wind coming from all directions.
 Minimizing the uncertainty in COE has biased the layout towards a
North-South spread, which probably seeks to extract more power from
relatively less uncertain wind conditions.
25

26. Performance of the Optimized Farms Designs
Parameter Step 1: Minimizing COE Step 2: Minimizing
Uncertainty in COE
Overall Farm Efficiency 0.776 0.752
COE ($/kWh) 0.0185 0.0191
Uncertainty in COE 3.7% 3.6%
 As expected, the reduction in the uncertainty of COE comes at the
expense of farm performance.
 Expectedly, robust optimization of wind farms presents conflicting
objectives.
26

27. Concluding Remarks
 This paper presents a model to characterize the uncertainties introduced by
the ill-predictability of the long term variation in wind conditions.
 To the best of the authors’ knowledge, such an uncertainty model that
provides a more credible quantification of the distribution of wind
conditions compared to traditional wind distribution models is unique in the
literature.
 A generalized uncertainty model is developed by considering the predicted
yearly wind probabilities themselves to be stochastic parameters.
 This generalized model can be used in conjunction with parametric/non-
parametric and univariate/bivariate/multivariate wind distribution models.
27

28. Concluding Remarks
 Robust wind farm optimization was performed by (i) minimizing the
COE and (ii) minimizing the uncertainty in COE in series.
 Interestingly, when we are minimizing the uncertainty, layout
optimization seeks to reduce the sensitivity of the farm power generation
to the relatively more uncertain wind speeds and wind directions.
 Expectedly, it is observed that farm performance and the uncertainty in
the farm performance are conflicting objectives.
 A multiobjective optimization scenario should be investigated in the
future.
 Applicability of the two different uncertainty models would be explored
and compared using a parametric wind distribution model.
28

29. Acknowledgement
• I would like to acknowledge my research adviser
Prof. Achille Messac, and my co-adviser Prof.
Luciano Castillo for their immense help and
support in this research.
• I would also like to thank my friend and colleague
Jie Zhang for his valuable contributions to this
paper.
29

30. Thank you
Questions
and
Comments
30

31. Mixed-Discrete Particle Swarm Optimization (PSO)
 This algorithm has the ability to
deal with both discrete and
continuous design variables, and
 The mixed-discrete PSO presents
an explicit diversity preservation
capability to prevent premature
stagnation of particles.
 PSO can appropriately address the
non-linearity and the multi-
modality of the wind farm model.
31

32. UWFLO Cost Model
• A response surface based cost model is developed using radial basis
functions (RBFs).
• The cost in $/per kW installed is expressed as a function of (i) the
number of turbines (N) in the farm and (ii) the rated power (P) of those
turbines.
• Data is used from the DOE Wind and Hydropower Technologies
program to develop the cost model.
32