This is my paper presentation at the AIAA Fluid Dynamics conference, 2011 in Hawaii

1. Modeling the Uncertainty in Farm Performance
Introduced by the Ill-predictability of the Wind
Resource
Achille Messac#, Souma Chowdhury*, and Jie Zhang*
# Syracuse University, Department of Mechanical and Aerospace Engineering
* Rensselaer Polytechnic Institute, Department of Mechanical, Aerospace, and Nuclear Engineering
41st AIAA Fluid Dynamics Conference and Exhibit
June 27 – 30, 2011
Sheraton Waikiki and the Hawaii Convention Center
Honolulu, Hawaii

2. Wind Energy - Overview
 Currently wind contributes only 2.5% of the global electricity
consumption (WWEA report).
 The 2010 growth rate of wind energy has also been the slowest since
2004 (WWEA report)..
 Large areas of untapped wind potential exist worldwide and in the US.
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.
 Careful modeling of these uncertainties, together with their
propagation into the overall system, will allow for
1. More credible wind resource assessment, and
2. Development of wind farms that have a reliable performance.
3

4. Uncertainties in Wind Energy
Long Term
Uncertainties
Physical 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
4

5. Presentation Outline
• Research Objectives
• Illustrating the ill-predictability of the wind distribution
• Multivariate and Multimodal Wind Distribution model
• Modeling the WPD, the AEP and the COE
• Modeling the uncertainties in the wind distribution
• Illustrating the estimated uncertainties for onshore and
offshore wind sites
• Concluding Remarks
WPD: Wind Power Density; AEP: Annual energy Production; COE: Cost of Energy
5

6. Research Objectives
 Characterize the uncertainty in the predicted yearly variation
of wind conditions (wind speed, wind direction and air
density).
 Model the propagation of uncertainty from the wind
distribution to the WPD, the AEP, and the COE.
 Validate the uncertainty models for onshore and offshore
wind sites.
WPD: Wind Power Density; AEP: Annual energy Production; COE: Cost of Energy
6

7. Variability of 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 annual distribution of wind conditions also varies from year to
year, although the overall pattern remains somewhat similar.
 The long term variation of wind conditions is generally
represented using probability distribution models.
 These probability distribution models can be developed using
previous years’ recorded wind data at the site.
 In a practical scenario, the Measure-Correlate-Predict (MCP)
method is implemented to predict the long term distribution using
short term (1-year) onsite data, and co-occurring long term data at
nearby meteorological stations.
Zhang et al, 2011 7

8. Uncertainties in Wind Conditions
Uncertainty is introduced by:
 the assumption that, “The expected distribution of wind in the succeeding
years of operation of the wind farm is deterministically equivalent to the
wind distribution estimated from preceding years’ data, ”, and
 the inherent uncertainties in the MCP method.
Factors that are often not explicitly considered are:
 A single year data at the site may not be representative of the wind
pattern at the site; hence subsequent correlations may not be reliable.
 The wind data at the meteorological station is often recorded at lower
heights (approx. 3-5 m). Extrapolation of this data using standard wind
shear profiles that may be far from accurate, introduces further errors.
8

9. The Solution
 Economic and timeline constraints limit the feasibility of
recording detailed onsite wind data over a longer time period.
 Uncertainties in wind predictions thus remain unavoidable.
 Therefore, if these uncertainties can at least be accurately
quantified, a more credible farm resource assessment and a
reliable farm performance projection/economic evaluation can
be made.
9

10. 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.
• Case studies: An onshore site and an offshore sites
26NDSU, North Dakota Agricultural Weather Network, online, 2010. 10
27NOAA, National Data Buoy Center, online, 2011.

11. Year-to-Year Variations (Onshore Site)
Wind distributions estimated using the Multivariate and
Multimodal model for a site at Baker, ND
Zhang et al., 2011 11

12. Year-to-Year Variations (Offshore Site)
Estimated Wind May not be the right Predicted Long Term
Distribution way to account for
Deterministic assumption Variation of Wind
(preceding years’ data) wind variations (succeeding years)
Zhang et al., 2011 12

13. Wind Distribution in Wind Power Density
Wind Probability Distribution
• WPD of a potential site is given by:
• Using Monte Carlo integration, this integral equation can be numerically
expressed as:
Variability of Wind Uncertainty in Uncertainty in
Conditions the Predicted the predicted
Short time period of Yearly Wind Wind Power
recorded data Distribution Density
13

14. Wind Distribution in Annual Power Generation
Wind Probability Distribution
• Annual Energy Production of a farm is given by:
Wind Farm Power Generation
• This integral equation can be numerically expressed as:
Variability of Wind Uncertainty in Uncertainty in
Conditions the Predicted the Annual
Short time period of Yearly Wind Energy
recorded data Distribution Production
Kusiak and Zheng, 2010; Vega, 2008 14

15. Characterizing the Uncertainties
In this paper, two different models have been proposed.
 Parametric Wind Uncertainty (PWU) Model: We consider the
parameters of the wind distribution model to be stochastic - e.g. the k and
c parameters in the Weibull distribution.
 Non-Parametric Wind Uncertainty (NPWU) Model : We consider the
predicted yearly probability of a wind condition itself to be stochastic.
15

16. Parametric Wind Uncertainty (PWU) Model
 The uncertainty in the parameters of the wind distribution model is
represented by their variance (in this paper).
 For a mp-parameter wind distribution model, the corresponding
uncertainties in the predicted yearly probabilities of the sample wind
conditions can be expressed in terms of a covariance matrix p as
 qk: kth parameter; q: Covariance of the distribution parameters;
 pi: frequency of the ith sample wind condition;
16

17. PWU Model continued…
 The uncertainty propagating into the AEP is modeled as a function of the
uncertainty in the wind distribution.
Uncertainty in Wind Distribution
 Subsequently, the Parameters
uncertainty in the COE can be expressed as
Uncertainty in the Predicted
Yearly Probability of Sample
Wind Conditions
where
Uncertainty in the Annual Energy
Production Lindberg, 1999 17

18. Jacobian of Popular Univariate Wind Distribution
Models
18

19. Non-Parametric Wind Uncertainty (PWU) Model :
Concept
Stochastic models of the wind distribution probabilities
5
Estimated probability of wind distribution, log(p(Ui, i))
 The variability in the predicted yearly probabilities MMWD is directly
3 10-yr pi
2000 MMWD
1
represented by a stochastic model. 2001 MMWD
2002 MMWD
-1 2003 MMWD
 Let us-3consider an example of the following five sample wind conditions 2004 MMWD
2005 MMWD
2006 MMWD
-5 2007 MMWD
2008 MMWD
-7 2009 MMWD
sample-1 DPSWC
-9 sample-2 DPSWC
sample-3 DPSWC
-11 Sample # Wind Speed (m/s) Wind Direction (deg) Air Density (kg/m3)
sample-4 DPSWC
sample-5 DPSWC
-13 1 6.50 180 1.245
-15
2 9.75 90 1.323
3 3.25 270 1.168
-17
1 2 3 4 5 6
4 4.88 Sample number, i 135 1.284
5 11.38 315 1.129
DPSWC: Distribution of the yearly probability of the sample wind condition
19

20. NPWU Model: Formulation
 The probability of a given wind condition was observed to vary in orders
of magnitude from year to year.
 To model this variability, a multivariate normal distribution of the
logarithms of the predicted yearly wind probabilities is used.
 The uncertainty in the predicted yearly wind probabilities is then given
by
 The uncertainty in the AEP and the COE can be determined as in PWU.
20

21. NPWU Model: Alternative
 The number of wind condition samples used (np) is significantly higher
than the number of years for which wind data is available.
 The estimation of the probability pp thus requires fitting a high
dimensional data with a significantly small number of data points.
 Alternatively, we can neglect the cross-covariance terms, thereby
assuming the sample wind conditions to be independent random variables.
 The uncertainty in the AEP is then given by:
ith diagonal element of the cov matrix
Lindberg, 1999 21

22. Comparing the Two Uncertainty Models
22

23. Illustration of the Estimated Uncertainty
Uncertainty in the univariate distribution of wind speed: Using NPWU
model without cross-covariance terms
 For a major portion of the wind distributions, there is approximately 20%
uncertainty.
23

24. Illustration of the Estimated Uncertainty
Uncertainty in the bivariate distribution of wind speed and direction: Using
NPWU model without cross-covariance terms
24

25. Uncertainty in the WPD: Validation
 The uncertainty in the annual WPD can also be readily evaluated by its
standard deviation over the ten years.
WPD
Uncertainty in the predicted WPD
Reasonably accurate Underestimation Overestimation
25

26. Uncertainty in the Farm Performance
• We consider a wind farm comprising 25 GE 1.5MW xle turbines at the
onshore site.
• Uncertainty is evaluated for the optimized farm layout, adopted from a
recent publication*.
• The AEP of the optimized wind farm was reported to be 4.4% higher
than that of a reference wind farm having a 5x5 array layout.
• The relative uncertainties in the AEP and in the COE, estimated
using the NPUW model without cross-covariance, are each
approximately 4%.
*Chowdhury et al. 2011 26

27. Concluding Remarks
 This paper presents a methodology 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 assessment of the wind resource (WPD) and a
more reliable prediction of the farm performance (AEP and COE) is unique
in the literature.
 Two uncertainty models are developed: (i) Parametric Wind Uncertainty
model (PWU), and (ii) Non-Parametric Wind Uncertainty model (NPWU).
 The relative uncertainty in the predicted yearly wind distribution was found
to be as high as 20% (approx.) for the onshore and the offshore sites.
27

28. Concluding Remarks
 The parametric model provides a reasonably accurate estimation of the
uncertainty in the WPD.
 Further advancement of the non-parametric model is necessary in order to
provide accurate uncertainty quantification.
 Significant uncertainties were also observed in the AEP and the COE of a
wind farm with an optimized layout.
 Therefore, an exploration of the trade-offs between optimal and reliable
wind farm design is crucial in wind project planning.
 Future research should also investigate the interaction of “the uncertainties
occurring due to year-to-year variations” with “the uncertainties
introduced by the MCP method”.
28

29. Thank you
Questions
and
Comments
29

30. 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.
30

31. Prediction of 5-year wind distribution
31

32. 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
32

33. Wake Model
 We implement Frandsen’s velocity deficit model
Wake growth Wake velocity
– 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 33