1. An Estimation Theory Approach to Decision Under Uncertainty
with Application to Wind Farm Siting
Motivation Demonstration Results
Conclusions
References/Acknowledgments
Define objectives and requirements.
Identify and characterize the sources of uncertainties.
Develop analysis models that map factor information to objectives and
requirements.
Identify key contributors to uncertainties in objectives and requirements.
Allocate resources to update the parameters, models to reduce the risk.
Fatma Demet Ulker, Douglas Allaire, John Deyst, and Karen Willcox
SIAM Conference on Computational Science and Engineering
Bayesian estimation framework permits us to tract the risk of not realizing
the required power, which is used to support decision-making in
 Resource allocation for risk mitigation via refining our estimate of
quantity of interest.
 Redesigning the site, or
 Abandoning the site.
To minimize the risk of not meeting
the requirements for complex system
developments and operations:
1. Quantitative and systematic risk
assessment methodologies
2. Efficient management of resources
Source: National Renewable Energy Laboratory
Classification of Uncertainties1
Basis of a rigorous approach for
1. Treating appropriately the different kinds of uncertainties
2. Achieving efficient allocation of resources to mitigate risk.
Parameter Variability: not always possible to model certain inputs.
Residual Variability: not always an outcome of a process is the same
even when the conditions are fully specified.
Observation Error: not all actual observations are error free.
Model Discrepancy: not all models are perfect.
Parameter Uncertainty: not all model inputs are certain.
Code Uncertainty: not possible to execute the code at every possible
input configurations when a code is so complex.
1. Wind Speed Estimation
Parameter Uncertainty
𝐴 = 𝑈 10,12 and 𝑘 = 𝑈 2.0,2.3
2. Turbulence Intensity
Parameter Variability
𝑇𝐼 = 𝑈 5%, 20% ,
PowerT𝐼 = 𝐻𝐺𝑃~(𝜇, Σ 𝑢ℎ𝑢𝑏 )
3. Blade Twist Angle
Parameter Uncertainty.
𝑖 = 𝑈 𝑖 ± 10% 𝑖 = 1 … 𝑁𝑠𝑝𝑎𝑛
𝑃𝑤 𝑢 = 𝑃𝑤
∞
0
𝑢 𝜋 𝑈 𝑢 𝑑𝑢
𝐶𝐹 =
𝑃𝑤
𝑃𝑅
=
1
𝑀
𝐶𝐹 𝑇,1 + 𝐶𝐹 𝑇,2 + ⋯
Quantity of Interest:
Average Capacity Factor, 𝐶𝐹 of 𝑀 turbines
with Rated Power, 𝑃𝑅
[1] Kennedy, M. and O'Hagan, A., “Bayesian calibration of computer models," Journal of Royal
Statistical Society, Vol. 63, 2001, pp. 425-464.
[2] Renkema, D., “Validation of wind turbine wake models using wind farm data and wind tunnel
measurements”, Master's Thesis, Delft University of Technology, 2007.
[3] Rozenn, W., Michael, C., Torben, L., and Uwe, P., “Simulation of shear and turbulence
impact on wind turbine performance“, Technical Report, Riso National Laboratory for
Sustainable Energy, 2010.
This work was supported in part by the BP-MIT Research Program
Approach
Modeling
Turbulence Simulation
Data3
Stochastic Models
Variations in the power curve due to turbulence in the flow are added to
the nominal power obtained using blade element momentum theory.
Heteroscedastic Gaussian Process Model
Analytical Models
Blade Element Momentum Theory
Kinematic Wake Model2
for Wake Deficit
Sensitivity Analysis
Main Effect Sensitivity Indices
Weibull Scale Factor (A)
Weibull Shape Factor (k)
Turbulence Intensity
Wind Shear < 1%
69%
26%
3%
Y= 𝑓(𝑋1 , 𝑋2,..., 𝑋 𝑁)
Inputs (r.v.)Quantity of Interest Sobol Main Effect Indices
𝑋1 , 𝑋2,..., 𝑋 𝑁 𝑆𝑖
var(Y) = var(E[Y|𝑋𝑖])+E[var(Y|𝑋𝑖)]
𝑆𝑖=
var(E[Y|Xi])
var(Y)
=
var(Y)−E[var(Y|Xi)]
var(Y)
4. Wake Turbulence Intensity
Parameter Variability
𝑇𝐼 𝑤 = 𝑈 5%, 20%
5. Wind Shear
Parameter Uncertainty
𝑢 𝑧 = 𝑢ℎ𝑢𝑏(
𝑧
𝑧ℎ𝑢𝑏
)α
α = 𝑈[0.1,0.3]
Power 𝑊𝑆 = 𝐻𝐺𝑃~ 𝜇, Σ 𝑢ℎ𝑢𝑏
6. Wake Modeling
Parameter Uncertainty &Model Discrepancy
Average Power:
Weibull distribution:
Uncertainties
Capacity Factor Bins
CapacityFactorFrequency
0.1 0.2 0.3 0.4 0.5 0.6
0
50
100
150
200
250
Downstream
Turbine
Upstream
Turbine
Mean Wind Speed (( uhub
) (m/s))
Power(W)
0 2 4 6 8 10 12
0
1000
2000
3000
4000
Laminar Flow (TI=0)
Turbulent Flow (TI %5- %20)
Mean Wind Speed (( uhub
) (m/s))
Power(W)
0 2 4 6 8 10 12
-600
-300
0
300
600 PowerTI (Simulation)
Mean ()
MLHGP 
MLHGP 
GP 
GP 
Mean Wind Speed (( uhub) (m/s))
Power(W)
0 5 10 15
0
50
100
150
200
250
Power fluctuations
due to turbulence
Nominal power