Wind farm layout optimization (WFLO) is the process of optimizing the location of turbines in a wind farm site, with the possible objective of maximizing the energy production or minimizing the average cost of energy. Conventional WFLO methods not only limit themselves to prescribing the site boundaries, they are also generally applicable to designing only small-to-medium scale wind farms (<100 turbines). Large-scale wind farms entail greater wake-induced turbine interactions, thereby increasing the computa- tional complexity and expense by orders of magnitude. In this paper, we further advance the Unrestricted WFLO framework by designing the layout of large-scale wind farms with 500 turbines (where energy pro- duction is maximized). First, the high-dimensional layout optimization problem (involving 2N variables for a N turbine wind farm) is reduced to a 6-variable problem through a novel mapping strategy, which allows for both global siting (overall land configuration) and local exploration (turbine micrositing). Sec- ondly, a surrogate model is used to substitute the expensive analytical WF energy production model; the high computational expense of the latter is attributed to the factorial increase in the number of calls to the wake model for evaluating every candidate wind farm layout that involves a large number of turbines. The powerful Concurrent Surrogate Model Selection (COSMOS) framework is applied to identify the best surrogate model to represent the wind farm energy production as a function of the reduced variable vector. To accomplish a reliable optimum solution, the surrogate-based optimization (SBO) is performed by implementing the Adaptive Model Refinement (AMR) technique within Particle Swarm Optimization (PSO). In AMR, both local exploitation and global exploration aspects are considered within a single optimization run of PSO, unlike other SBO methods that often either require multiple (potentially mis- leading) optimizations or are model-dependent. By using the AMR approach in conjunction with PSO and COSMOS, the computational cost of designing very large wind farms is reduced by a remarkable factor of 26, while preserving the reliability of this WFLO within 0.05% of the WFLO performed using the original energy production model.

1. Surrogate-based Particle Swarm Optimization
for Large-scale Wind Farm Layout Design
Ali Mehmani*, Weiyang Tong*, Souma Chowdhury#, and Achille Messac#
* Syracuse University, Department of Mechanical and Aerospace Engineering
# Mississippi State University, Department of Aerospace Engineering
11th World Congress on Structural and Multidisciplinary Optimization
June 7 - 12, 2015, Sydney Australia
Research supported by the NSF Award: CMMI 1437746

2. Large-scale Wind Farm Layout Design – Overview
2
• Large utility-scale wind farms can involved more than 500 MW
installed capacity (consisting of hundreds of wind turbines)
• Such large utility-scale wind farms are central to the growth of the wind
energy industry as a energy source that can compete with conventional
energy resources (without financial incentives).
• Planning the layout of such a large scale wind farm however poses
substantial technical challenges – it entails a complex and extremely
time-consuming design optimization process.
It includes various mutually correlated factors and large-
scale effects, especially large number of turbine-wake
interactions, and energy losses due to the wake effects.

3. Research Motivation
3
• Wind farm layout optimization (WFLO) is the process of optimizing
the location of turbines in a wind farm site, with the objective of
minimizing the average cost of energy.
• WFLO methods in the literature limit themselves to majorly designing
small-to-medium scale farms (< 100 turbines), as their case studies.
• The wind farm layout optimization for large-scale wind farms is a very
high-dimensional and highly nonlinear optimization problem.
• Surrogate-based optimization (SBO) approaches can be applied to
alleviate the computational burden in large-scale WFLO.
• Direct surrogate modeling of the O(103)-dim problem is fraught with uncertainties.
• The need to maintain adequate accuracy of the surrogate model during the
optimization process (for a highly multi-modal problem) poses critical challenges.

4. Research Objectives
4
• Develop a design domain reduction strategy for reducing the very
high-dimensional (O(103)) WFLO process into a low-dimensional
design optimization process (O(101)) .
• Implement an adaptive model refinement technique in surrogate-
based optimization to achieve computational efficiency while
promoting high accuracy of the end/optimum results.

5. Presentation Outline
5
• Layout Optimization of Large Scale Wind Farms
• Surrogate-based PSO for Large Scale WFLO
 Domain Reduction through Novel Layout Mapping
 Surrogate Model Selection
 Adaptive Model Refinement
• Numerical Experiments: Results and Discussion
• Concluding remarks
PSO: Particle Swarm Optimization

6. Layout Optimization of Large Scale Wind Farms: Review
6
The current approaches on solving large-scale layout optimization
problem is mostly limited to quantifying the layout using the
streamwise and the spanwise spacings between turbines (assuming a
specified number of turbines are uniformly distributed in pre-defined boundaries).
 Fuglsang et al. [1] defined the large scale wind farm layout as a function of the
spacing between rows and columns.
 Perez et al. [2] used the numbers of rows and columns, the streamwise and the
spanwise spacing between neighboring turbines, the turbine rotor diameter, and a
specified rectangular boundary to determine the large scale wind farm layout.
 Wagner et al. [3] developed a framework for the large scale wind farm layout in
which the initial location of turbines is restricted to an array-like layout, and a
radial displacement around each turbine is allowed.
[1] P. Fuglsang and T. Kenneth, Technical report, Risoe National Lab, Roskilde (Denmark), 1998.
[2] G. Perez et al., Wind Energy Association Offshore, 2013.
[3] M. Wagner et al., European Wind Energy Association Annual Event, 2011.

7. Surrogate-based PSO of Large Scale WFLO
7
 The proposed approach here is capable of optimizing the location of turbines
for large wind farms, (i.e., 500-turbine scale wind farms) without
prescribing the farm boundaries.
Mapping of the Layout
Surrogate model selection
Step 1:
• The high-dimensional layout optimization
problem (involving 2N variables for a N turbine
wind farm) is reduced to a 6-variable problem
through a novel mapping strategy.
Step 2:
• A surrogate model is used to substitute the
expensive analytical WF energy production
model.
• The powerful Concurrent Surrogate Model
Selection (COSMOS) framework is applied to
identify the best surrogate model to represent
the wind farm energy production as a function
of the reduced variable vector.
Step 3:
• To accomplish a reliable optimum solution,
the surrogate-based optimization (SBO) is
performed by implementing the Adaptive
Model Refinement (AMR) technique within
Particle Swarm Optimization (PSO).
Surrogate-based optimization

8. 8
Mapping of the Layout for a Large Scale Wind Farm
Design factors Lower bound Upper bound
rmax 5D 15D
smax 5D 15D
A − 20 20
B − 20 20
σ 0 1
Mapping
Wind Farm Layout
Wind Farm Layout
(X,Y)
rmax
smax
A
B
σ
φ
Input:
Output:
nput and out put st ruct ure of t he W ind Farm Layout M apping
Product ion M odel
on, first, the wind farm power generation model is adopted from the Unre-
arm Layout Optimization (UWFLO) framework [129] to estimate the total
 The developed mapping strategy allows for both global siting (overall land
configuration) and local exploration (turbine micro siting).
Rmax : maximum allowable streamwise spacing
Smax : maximum allowable spanwise spacing
A, B : control parameters for defining the spacing of rows and columns
σ : normalized local radial displacement which controls turbine micro-siting
Φ : farm site orientation

9. 9
Surrogate model selection using COSMOS
 Concurrent Surrogate Model Selection (COSMOS) framework is applied to select the
best surrogate model to represent the average annual energy production of a large-scale
wind farm as a function of the mapping factors.
Training data
average annual
energy production
probability of wind speed
and direction.power generation[1]
COSMOS
Best Surrogate model combination
(Model type-Kernel function-Hyperparameter)
[1] Chowdhury and Messac et al. (2013)

10. Surrogate-based optimization
10
 To reach a reliable optimum solution at a reasonable cost, surrogate-
based optimization is performed with Adaptive Model Refinement
(AMR).
 AMR is a novel model-independent approach to refine the surrogate model
during optimization.
Decisions regarding when to refine the surrogate model is guided by
the Adaptive Model Switching (AMS) technique.
Decisions regarding the batch size for the samples to be added is
guided by the Predictive Estimation of Model Fidelity (PEMF).

11. Adaptive Model Refinement – Model Switching
11
 The switching criteria is based on whether the predicted model uncertainty
dominates the uncertainty associated with the improvement of the fitness func.
over the population.
pcr is the indicator of conservativeness
(user controlled)
Model Switching: Hypothesis Testing
Distribution of FF improvement (KDE)Distribution of Model Error (LogN)
Rejection of the test;
Don’t REFINE surrogate
Acceptance of the test;
REFINE surrogate

12. 12
 The inputs and outputs of PEMF in the AMR method are
• The desired fidelity is determined using the history of the fitness function improvement in the
optimization process
• The desired batch size is estimated using the inverse of regression functions used to represent the
variation of error with sample density in PEMF
Adaptive Model Refinement – Batch size estimation
PEMF[1]
[1] Mehmani and Messac, SMO (2015)

13. 13
Numerical Experiments
 Maximizing energy production of large-scale 500-turbine wind farm
This constraint is defined based on the average land usage of
US commercial wind farms in 2009
Assumptions:
1. The GE-1.5MW-XLE turbine is chosen as the specified turbine-type in this problem,
2. The minimum streamwise (smin) and spanwise (rmin) are set to the same value: 4D,
3. The wind data in this problem is obtained from the North Dakota Agricultural Weather Network
(NDAWN),
4. Initial sample size: N({Xin}) = 200 .The model refinement will be performed if the size of data
set is less than N({X}) = 500

14. 14
Numerical Experiments: results and discussion
improvement of the model fidelity through the
sequential model refinement process using the AMR
method.
 The farm layout optimization is started using the best surrogate model selected using
COSMOS (Kriging model with Linear correlation function).
COSMOS
Training
Data
Kriging-Linear
Computational cost of the energy production
model is reduced by a factor of 30.
 To reach a reliable optimum solution at a reasonable cost, surrogate-based optimization is
performed with Adaptive Model Refinement (AMR).

15. 15
Numerical Experiments: results and discussion
Convergence history of the optimization using AMR
Size of data set used to refine (update) the active
surrogate model in the AMR approach
Avg.AnnualEnergy
Production
While retaining an accuracy of within 0.05%, AMR improved the efficiency of the optimization process
by a remarkable factor of 26, when compared to optimization using the standard energy production model.

16. 16
Concluding remarks
 This paper presented a new approach to optimizing large-scale (500-turbine) wind
farms at an reasonable computational efficiency while reaching reliable optimum
results (i.e., attractive cost-accuracy tradeoffs).
 A novel stochastic mapping strategy allowed the reduction of the 1000-dim layout
problem into a 6-variable layout problem, which allows both global exploration
and local micro-siting flexibility.
 The COSMOS framework was then applied to select the globally-best surrogate
model to represent the energy production of the wind farm as a fast function of the
reduced set of layout variables.
 Surrogate-based optimization was then preformed using the Adaptive Model
Refinement approach, implemented through Particle Swarm Optimization.
 The 500-turbine WFLO results indicated that “AMR+PSO” improved
the efficiency of the optimization process by a factor of 26, while
retaining an accuracy of within 0.05% (compared to the results of
WFLO that uses the original energy production model).

17. 17
Thank you

18. Backup Slides
“The following Slides are not part of the normal presentation”

19. Backup Slides in PEMF

20. Predictive Estimation of Model Fidelity (PEMF)
20
PEMF - Error Measure: (1) Model Independent, (2) Predictive, and
(3) Minimally sensitive to outlier samples
en by:
eRAE(Xi) =
|
F(Xi) − ˆF(Xi)
F(Xi)
| if F(Xi) ̸= 0
|F(Xi) − ˆF(Xi)| if F(Xi) = 0
(8)
ere F is the actual function value at Xi, given by high fi-
ty simulation or experimental data, and ˆF is the function
ue estimated by the surrogate model.
In the original PEMF method, the distribution functions
be fitted over the median and the maximum errors at each
ation were selected using the chi-square goodness-of-fit
erion [38]. The following distributions were considered:
normal, Gamma, Weibull, logistic, log logistic, t-location
le, inverse gaussian, and generalized extreme value distri-
ion. However, in order to control the computational ex-
se of PEMF within model selection, only the lognormal
ribution is used. This distribution has been previously ob-
ved (from numerical experiments) to be effective in gen-
. The PDFs of the median and the maximum errors, pmed
pmax, can thus be expressed as
pmed =
1
Emedsmed
√
2p
exp(
(ln(Emed − µmed))2
2s2
med
)
pmax =
1
√ exp(
(ln(Emax − µmax))2
)
(9)
• The PDFs of the median and the maximum errors:
• The modal values of the median/max. error at any iteration
• The inputs and outputs of the PEMF method
for
istributions of the median error over all Mt combi-
ons
rmine the mode of the median and maximum error
ibutions; Emo,t
med and Emo,t
max
r
uct a final surrogate using all N sample points
e estimated Emo,t
med and Emo,t
max ∀t, to quantify their
on with # training points (nt) using regression func-
RN: The modal values of the median and the max-
errors in the final surrogate; emed and emax
e PEMF method, for a set of N sample points, inter-
urrogates are constructed at each iteration, t, using
tic subsets of nt training points (called intermedi-
ng points). These intermediate surrogates are then
r the corresponding remaining N − nt points (called
ate test points). The median error is then estimated
of the Mt intermediate surrogates at that iteration,
ametric probability distribution is fitted to yield the
ue, Emo,t
med . The smart use of the modal value of the
rror promotes a monotonic variation of error with
oint density, unlike mean or root mean squared error
highly susceptible to outliers [31]. This approach
MF an important advantage over conventionalcross-
n-based error measures, as illustrated by Mehmani
max
pmed =
1
Emedsmed
√
2p
exp(
(ln(Emed − µmed))2
2s2
med
)
pmax =
1
Emaxsmax
√
2p
exp(
(ln(Emax − µmax))2
2s2
max
)
(9)
In the above equations, Emed and Emax respectively rep-
resent the median and the maximum relative absolute errors
estimated over a heuristic subset of training points at any
given iteration in PEMF. The parameters, (µmed,smed) and
(µmax,smax) represent the generic parameters of the lognor-
mal distribution. The modal values of the median and the
maximum error at any iteration, t, can then be expressed as
Emo
med|t = exp(µmed − s2
med)|t
Emo
max|t = exp(µmax − s2
max)|t
where nt− 1 < nt ≤ N
(10)
Once we have the history of median and maximum er-
rors at different sample size (< N), the variation of the
modal values of the errors with sample density is then mod-
eled using the multiplicative (E = a0na1 ) or the exponen-
tial (E = a0ea1n) regression functions. The choice of these
regression functions leverage the monotonically decreasing

21. Predicted Median Error
MedianofRAEs
Number of Training Points
t1 t2 t3 t4
It. 3It. 1 It. 2
Momed
It. 4
Momax Mode of maximum
error distribution at
each iterationPredicted Maximum Error
PEMF: Variation of Error with Sample Density (VESD)
21

22. Regional & Global Error Prediction :
comparison of PEMF with cross-validation
Kriging RBF ERBF
0
50
100
150
200
250
300
RelativeError[%]
R
PEMF
R
CV
Kriging RBF ERBF
0
20
40
60
80
100
RelativeError[%]
R
PEMF
max
R
CV
max
Kriging RBF ERBF
0
100
200
300
400
500
600
RelativeError[%]
R
PEMF
max
R
CV
max
Kriging RBF ERBF
0
100
200
300
400
500
RelativeError[%]
R
PEMF
R
CV
Regional
Error
Prediction:
 Branin-Hoo Function
Global
Error
Prediction:
Mean or Median Error Maximum Error
Mean or Median Error Maximum Error
6.1 %
263.3 %
488.2 %
56.5 %
528.3 %
19.7 %
22
R[%]R[%]
R[%]R[%]
The PEMF method is up to two orders of magnitude more
accurate than the popular leave-one-out cross-validation

23. Prediction Estimation of Model Fidelity: Summary
PEMF vs. Other Measures
PEMF CV RMSE AIC BIC RMSEKriging
Model-independent ✓ ✓ ✗ ✗ ✗ ✗
Global Error Measure ✓ ✓ ✓ ✓ ✓ ✗
Local Error Measure ✓ ✗ ✗ ✗ ✗ ✓
Model Uncertainty Quantification ✓ ✓ ✓ ✗ ✗ ✗
Providing Maximum Error ✓ ✓ ✗ ✗ ✗ ✗
Providing Variance Error ✓ ✗ ✗ ✗ ✗ ✓
Expected Accuracy (if more resource available) ✓ ✗ ✗ ✗ ✗ ✓
Function Behavior with Sample Density ✓ ✗ ✗ ✗ ✗ ✗
Accuracy
Robustness
23

24. Backup Slides in COSMOS

25. Surrogate Model Selection
Surrogate
Model
KRIGING
E-RBF
SVR
RBF
QRS
Gaussian
…
Exponential
Spherical
…
Gaussian
Sigmoid
…
σ
Shape
parameter
Correlation
parameter
𝜽
Kernel
parameters
c
Linear
Model Type Kernel Type Hyper-parameter
25

26. Concurrent Surrogate Model Selection (COSMOS)
 We developed a novel 3-level model selection framework called Concurrent
Surrogate Model Selection (COSMOS).
 This framework enables the designers to identify a globally best surrogate
model for any given application
26
 In COSMOS, the selection criteria depend on the type of application and the
user preference. These criteria are predicted using PEMF
PEMF

27. COSMOS
 COSMOS is uniquely formulated using a mixed integer nonlinear programming
(MINLP) problem.
To escape the potentially high computational cost of
theCascaded technique, thethree-level automated model
selection could also be performed by solving a single
(uniquely formulated) mixed integer nonlinear program-
ming (MINLP) problem. The major components and
the flow of information in the One-Step technique is il-
lustrated in Fig. 6. The general form of this MINLP
problem can be expressed as
Min
m,k,u
{ Em o
m ed, Em o
m ax , Eσ2
m ed, Eσ2
m ax , Em o
m ed,α }
subject to (5)
m ≤ NM , m ∈ Z> 0
k ≤ NK (m), k ∈ Z> 0
u = [u11
u12
... u21
u22
... um k
... uN M N K
]
um i n
m k ≤ um k ≤ um ax
m k
Min
z,u
{ Em o
m ed, Em o
m ax , Eσ2
m ed,
subject to
z ≤ N(Φp ), z ∈ Z> 0
0 ≤ u ≤ 1
In Eq. 6, z is the intege
combined model-kernel typ
uous variables that represen
ues; and N (Φp) represents
which is the total number
typesavailableunder thept h
It should be noted that a
used for each hyper-parame
are scaled based on the use
bounds. The upper and lo
Integer design variable that denotes the model type
Number of available Model type
Integer design variable that denotes the Basis (or Kernel) function
Number of available basis function for mth model type
Continuous variables that represent the hyper-parameter
values for the kth kernel of the mth candidate surrogate
27

28. 28
Concurrent Surrogate Model Selection (COSMOS)
 A new model selection approach, which simultaneously selects the
best model type, kernel function, and hyper-parameter.
Types of model Types of basis/kernel Hyper-parameter(s)
• RBF,
• Kriging,
• E-RBF,
• SVR,
• QRS,
• …
• Linear
• Gaussian
• Multiquadric
• Inverse multiquadric
• …
• Shape parameter in RBF,
• Smoothness and width
parameters in Kriging,
• Kernel parameter in SVM,
• …
• Searching for Globally-competitive surrogate models
• Necessitates a model-independent surrogate model selection Technique.
A complex MINLP
problem is formulated
and solved

29. COSMOS
 To solve this optimization problem; the global pool of model-
kernel candidates is divided into P smaller pool of model-
kernel candidates based on the number of constituent hyper-
parameters in them.
 Optimal model selection is performed
separately (separate MINLPs are run in
parallel) for each class.
he
he
ch
ves
n-
F.
ec-
nt.
ex-
es
≫
of
odel
gle
m-
nd
il-
LP
5)
binations which include p hyper-parameter(s). Subse-
quently, optimal model selection isperformed separately
(in parallel) for each candidatepool. Each model-kernel
combination/ candidate within a particular candidate
pool (Φp) is then assigned a single unique integer code,
as opposed to two separate integer codes, as given by
Eq. 5. The candidate model-kernels considered in this
paper are listed in Table 1, where the integer code as-
signed to each candidate is shown under their respec-
tive hyper-parameter class (Φp). For the Φ0 class of
model-kernel combinations, PEMF is applied to all the
candidates, followed by theapplication of a Pareto filter
to determine the final set of non-dominated or Pareto
optimal surrogate models. For all Φp with p > 0, the
mixed integer non-linear programming (MINLP) prob-
lem (Eq. 5) is reformulated as described in Eq. 6
Min
z,u
{ Em o
m ed, Em o
m ax , Eσ2
m ed, Eσ2
m ax , Em o
m ed,α }
subject to (6)
z ≤ N(Φp), z ∈ Z> 0
0 ≤ u ≤ 1
In Eq. 6, z is the integer variable that denotes the
combined model-kernel type; u is the vector of contin-
uous variables that represent the hyper-parameter val-
ues; and N (Φp) represents the size of the set of Φp,
which is the total number of candidate model-kernel
typesavailableunder thept h
hyper-parameter class(Φp).
It should be noted that a consistent range of (0,1) is
used for each hyper-parameter wherethehyper-parameters
Integer design variable that
denotes the model-kernel type
Continuous variables
that represent the
hyper-parameter
 Once the Pareto optimal surrogate models
for each p-class have been obtained, a Pareto
filter is applied to determine the globally
optimal set of surrogate models
29

30. Surrogate model candidates
Concurrent Surrogate Model Selection (COSMOS): Optimizing Model Type, Kernel Function, and Hyper-parameters 7
Table 1 Candidate model-kernel combinations and their integer-codes
Surrogate Kernel Φ0 Φ1 Φ2 Hyper-Parameter(s)
Radial Basis Function
Linear 1 - - -
Cubic 2 - - -
Gaussian - 1 - Shape parameter, σ
Multiquadric - 2 - Shape parameter, σ
Kriging
Linear - 3 - Correlation parameter, θ
Exponential - 4 - Correlation parameter, θ
Gaussian - 5 - Correlation parameter, θ
Spherical - 6 - Correlation parameter, θ
Support Vector Regression
Linear - 7 - Penalty parameter, C
Gaussian - - 1 Kernel parameter, γ and Penalty parameter, C
Sigmoid - - 2 Kernel parameter, γ and Penalty parameter, C
Table 2 Range of hyper-parameters
Surrogate Hyper-parameter Lower
bound
Upper
bound
RBF Shape parameter, σ 0.1 3.0
Kriging Correlation parameter, θ 0.1 20
SVR Kernel width parameter, γ 0.1 10
SVR Penalty parameter, C 0.1 100
ons and their integer-codes
nel Φ0 Φ1 Φ2 Hyper-Parameter(s)
ar 1 - - -
c 2 - - -
ssian - 1 - Shape parameter, σ
iquadric - 2 - Shape parameter, σ
ar - 3 - Correlation parameter, θ
onential - 4 - Correlation parameter, θ
ssian - 5 - Correlation parameter, θ
erical - 6 - Correlation parameter, θ
ar - 7 - Penalty parameter, C
ssian - - 1 Kernel parameter, γ and Penalty parameter, C
moid - - 2 Kernel parameter, γ and Penalty parameter, C
Table 2 Range of hyper-parameters
Surrogate Hyper-parameter Lower
bound
Upper
bound
RBF Shape parameter, σ 0.1 3.0
Kriging Correlation parameter, θ 0.1 20
SVR Kernel width parameter, γ 0.1 10
SVR Penalty parameter, C 0.1 100
PEMF method, as given by 30

31. Backup Slides in
Model Switching and Model Refinement

32. Adaptive Model Switching (AMS)
32
 The AMS metric is a hypothesis testing that is defined by a comparison between
(I) the distribution of the relative fitness function improvement, and
(II) the distribution of the error associated with the model.
pcr regulates the trade-off between
reliability and computational cost
Rejection of the test; Don’t Refine a model Acceptance of the test; Refine a model
Fitness Func. Improvement (KDE)
Distribution of Model Error (LogN)

33. Number of Training Points
t1 t2 t3 t4
Adaptive Model Refinement (AMR)
33
MedianofRAEs
Fina
l
Momed
FF improvement (KDE)PEMF Error (LogN)
Rejection of the test;
Don’t REFINE surrogate
Acceptance of the test;
REFINE surrogate
MODEL REFINEMENT BASED ON PEMF
In the previous sub-section, the formulations and c
of the AMR metric is defined. Based on the AMR me
model refinement will beperformedat thet∗ -th iteration
under thecondition that
QP
SMCURR
≥ Qt= t∗
Θ
whereSMCURR representsthecurrent surrogatemodel in
timization process. Model refinement isperformed to ef
improve the fidelity of the current surrogate model to
the“desired fidelity” for theupcoming iterationsof SBO
paper, the desired fidelity, ε∗
mod, is determined using the
of thefitness function improvement in the optimization
which isgiven by:
ε∗
mod = |1−
Qt= t∗
Θ − Qt= t∗ − τ
Θ
Qt= t∗
Θ
| × εCURR
mod
In Equation11, εCURR
mod isthepredictedmodal error v
sociated with the current surrogate model; and τ (∈ Z
user-defined parameter which regulates the occurrence
“surrogate model refinement” in the proposed SBO ap
Numerical experiments exploring different values for t
dicated that the3 ≤ τ ≤ 5 can bethesuitablechoice.

34. 34
 AMR is a novel model-independent approach to refine the surrogate model during optimization,
with the objective to maintain a desired level of fidelity and robustness “where” and “when”
needed.
 Reconstruction of the model is performed by sequentially adding a batch of new samples at any
given iteration (of SBO[1-3]) when a refinement metric is met.
[1] Forrester et al. (2008)
[2] Rai et al. (2006)
[3] Romero et al. (2011)
Adaptive Model Refinement (AMR)

35. AMR: Location of Infill Points
35
 The location of the new infill points in the input space is determined based on a
hypercube enclosing promising current candidate designs in the optimization process.
infill points :
Lower and Upper bounds
of the jth dimension
lower and upper bounds of
the entire set of the current
candidate solutions
 The distance-based criterion is then applied to select the optimum setting for the new
infill points
Euclidean Distance

36. Backup Slides in
Energy Production Model
& Case Study

37. Large-scale Wind Farm Layout Design:
Energy production model
37
average annual
energy production
probability of wind speed and direction, estimated by
Multivariate and Multimodal Wind Distribution model.
power generation[1]
number of turbines
Power generated by Turbine- j.
For any given incoming wind speed and direction, the power generated by the
individual turbines is determined by the power generation model developed by [1]
[1] Chowdhury and Messac et al. (2013)

38. Large-scale Wind Farm Layout Design: Assumptions
38
1. The GE-1.5MW-XLE turbine is chosen as the specified turbine-type in this problem.
2. The minimum streamwise (smin) and spanwise (rmin) are set to the same value: 4D
3. The wind data in this problem is obtained from the North Dakota Agricultural Weather
Network (NDAWN)
model
Below are other assumptions applied to the numerical experiments:
1. The GE-1.5MW-XLE turbine is chosen as the specified turbine-type in this problem.
The features of this turbine are listed in Table 9.2.
Table 9.2: Feat ures of t he GE-1.5M W -X LE t urbine [130]
Turbine feature Value
Rated power (Pr 0) 1.5MW
Rated wind speed (Ur 0) 11.5m/ s
Cut-in wind speed (Uin0) 3.5m/ s
Cut-out wind speed (Uout0) 20.0m/ s
Rotor-diameter (D) 82.5m
Hub-height (H ) 80.0m
2. The minimum streamwise (smin ) and spanwise (rmin) are set to the same value: 4D;
and
3. The wind data this problem is obtained from the North Dakota Agricultural Weather
Network (NDAWN). The local wind distribution is shown in Fig. 9.5, and the onshore
192
rm scenario is assumed, and the ambient turbulence (10%) is constant over the
arm site.
nd data used in this problem is obtained from the North Dakota Agricultural
er Network (NDAWN) [113]. We use thedaily averaged data for wind speed and
n, measured at the Baker station between the years 2000 and 2009. Fig. 9.4
he Baker station, and further details are provided in Table 9.3.
Figure 9.4: Baker st at ion set up [113]
ble 9.3: Det ails of t he N DAW N st at ion at Baker, N D [113]
Parameter Value
Location Baker, ND
Period of Record 01/ 01/ 2000 to 12/ 31/ 2009
192
arm scenario is assumed, and the ambient turbulence (10%) is constant over the
farm site.
nd data used in this problem is obtained from the North Dakota Agricultural
er Network (NDAWN) [113]. We use thedaily averaged data for wind speed and
on, measured at the Baker station between the years 2000 and 2009. Fig. 9.4
the Baker station, and further details are provided in Table 9.3.
Figure 9.4: Baker st at ion set up [113]
able 9.3: Det ails of t he N DAW N st at ion at Baker, N D [113]
Parameter Value
Location Baker, ND
Period of Record 01/ 01/ 2000 to 12/ 31/ 2009
Latitude 48.167
Longitude -99.648
Elevation 512m
Measurement height 3m
Baker station setup
Wind rose diagram for the site at Baker