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Multi-fidelity optimization of horizontal axis wind turbines
Michael McWilliam
Danish Technical University
Introduction
Outline
• The Motivation
• The AMMF Algorithm
• Optimization of an Analytical Problems
• Structural Optimization
• Low Fidelity Tools• Optimization Results
• Aero-elastic Optimization
• Future work
• Closing statements
2 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
Introduction
Motivation
• Interested in applying design optimization toadvanced concepts:
• Swept blades• Flaps• Multi-rotor
• Typical optimization frameworks based onsimplified load cases
• Tuned to be overly conservative• Could miss potential opportunities
• Standard design tools and frameworks may notbe suitable
• Need higher fidelity analysis inoptimization
True Feasible Set
Simplified Feasible Set
Design Space
True Optimum
Simplified Optimum
Objective Contours
Improving
Design
Simplified
Constraint
True
Constraint
3 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
The AMMF Algorithm
4 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
The AMMF Algorithm
The AMMF Algorithm
Calculate fl, fh, ∇fl and ∇fh
Build/update correction model β(x)
Update trust region ∆:expand if |f̃ − fh| is smallconstrict if |f̃ − fh| is large
Calculate fl
Calculate f̃ = β(x)fl
Use optimization to find next design x
Calculate fl, fh, ∇fl and ∇fh
Initial design
Exit if converged
• High fidelity used for accuracy
• Low fidelity is used for speed
• Correction for first orderconsistency
f̃(x) = fl(x) + β(x)
β(x) = fh0 − fl0
+(∇fh0 −∇fl0)∆x
• Trust-region for robustness
5 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
The AMMF Algorithm
Constraints in the AMMF Algorithm
• Constraints are corrected in the same way
• The constraints are present in the low fidelity optimization
• Constraints receive special treatment in Approximation and Model ManagementFramework (AMMF)
• First an estimated Lagrangian is calculated
Φ = f + λ̃e · |c|+ λ̃i ·max(0,−ci)
• λ̃ are the Lagrange multipliers estimated from previous iterates.• λ̃ is specified for the first iteration
• New iterate only accepted when Φi < Φi−1
• Trust region is expanded or contracted based on M :
M =Φi−1 − Φi
Φi−1 − Φ̃i
• Trust region expanded if M is close to 1• Trust region contracts if M is far from 1
6 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
Preliminary investigation into
AMMF
7 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
Preliminary investigation into AMMF
Preliminary investigation into AMMF
• Objective
• Understand how different types of error affect AMMF convergence
• Methodology
• Used a simple 2D paraboloid optimization problem• Applied various offsets to simulate error in the low-fidelity model• Number of function evaluations used to assess computational cost
• Phase 1: Order of the error
• Constant offset, linear offset & quadratic offset
8 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
Preliminary investigation into AMMF
Effect of constant offset error on AMMF
0 2 4 6 8 10Major iteration of AMMF
0.1
1
10
Obj
ectiv
e di
ffer
ence
fro
m s
olut
ion
No errorError 0.1Error 0.2Error 0.5Error 1.0
9 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
Preliminary investigation into AMMF
Effect of linear offset error on AMMF
0 2 4 6 8 10Major iteration of AMMF
0.1
1
10
Obj
ectiv
e di
ffer
ence
fro
m s
olut
ion
No errorError 0.1Error 0.2Error 0.5Error 1.0
10 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
Preliminary investigation into AMMF
Effect of quadratic offset error on AMMF
0 2 4 6 8 10Major iteration of AMMF
0.1
1
10
Obj
ectiv
e di
ffer
ence
fro
m s
olut
ion
No errorError 0.1Error 0.2Error 0.5Error 1.0
11 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
Preliminary investigation into AMMF
AMMF investigation phase 2: Lateral offset errors
• Under or over-shooting low fidelity model
-2 -1 0 1 2 3 4Normalized distance
250
300
350
400
450
500O
bjec
tive
valu
eHigh fidelity functionWeak under-shootStrong under-shootWeak quadratic errorStrong quadratic errorWeak over-shootStrong over-shoot
12 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
Preliminary investigation into AMMF
AMMF convergence rate vs. lateral offset error
0 2 4 6 8 10Major iteration of AMMF
1e-03
1e-02
1e-01
1e+00
1e+01
Obj
ectiv
e di
ffer
ence
fro
m s
olut
ion Weak under-shoot
Strong under-shootWeak quadratic errorStrong quadratic errorWeak over-shootStrong over-shoot
13 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
Preliminary investigation into AMMF
AMMF function evaluations vs. lateral offset error
0 2 4 6 8 10Major iteration of AMMF
0.0
5.0
10.0
15.0
20.0
25.0
30.0
Num
ber
of H
F fu
nctio
n ev
alua
tions Weak under-shoot
Strong under-shootWeak quadratic errorStrong quadratic errorWeak over-shootStrong over-shootDirect Optimization
14 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
Preliminary investigation into AMMF
Preliminary investigation summary
• Only affected by quadratic and higher order error
• Trust region is used to correct lateral offset error
• Extreme error requires more high fidelity function evaluations
• Best-case:Only 2-3 high fidelity function evaluations are required for convergence
• Worst-case:Convergence is the same as pure high fidelity optimization
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Multi-fidelity Structural Design
Optimization
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Low Fidelity Tool Development
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Multi-fidelity Structural Design Optimization
Summary of Low Fidelity Tools
Position EA EIx EIy GJ
0.05 0.0 2.6 -4.9 -5.40.15 0.5 1.1 -3.0 -0.80.25 -0.4 -1.8 2.1 -1.40.35 -0.7 -2.6 1.7 -3.10.45 -0.7 -3.1 1.0 -5.50.55 -0.9 -3.1 -0.3 -7.70.65 -0.8 -2.9 -1.7 -9.30.75 -0.6 -2.2 -2.2 -9.20.85 -0.6 -1.7 -3.5 -5.90.95 -0.1 -1.2 -2.0 -2.0
Table : Percent Error with BECAS
• Low fidelity cross section tool
• Thin-walled cross sectionassumption
• Rigid cross section(Euler-Bernoulli)
• Classic laminate theory• Written in C++• Python bindings with Swig• Will have analytic gradients• Within 10% compared to BECAS
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Multi-fidelity Structural Design Optimization
Summary of Low Fidelity Tools
Operation Calculation time [s]
Linear Beam Model 0.0035LF cross section model 0.0074BECAS 200.1866
Table : Speed Comparison of Low Fidelity Tools
• Linear Beam Model
• C++ code from my PhD• Analytic gradients wrt.
• Positions
• Orientation
• Cross section properties
• Applied forces
• Solves equivalent forces for givendeflection
• Speed comparison:
• With python bindings• Calculation for whole blade• 19 elements• DTU 10MW
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AMMF for equivalent static beam
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Multi-fidelity Structural Design Optimization
Problem Description
• Minimize DTU 10MW Blade Mass
• Varying spar cap thickness
• Subject to:
• Tip deflection constraint
• Analysis based on the equivalent static problem (i.e. Frozen loads)
• Compared pure BECAS, pure CLT and AMMF
• Looked at various AMMF configurations:
• Additive vs. Multiplicative corrections• Trust region size• Initial Lagrange multiplier (i.e. Penalty parameter)
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Multi-fidelity Structural Design Optimization
Optimization Results
• Low fidelity model is notconservative
• Will produce infeasiblesolutions
• AMMF reproduced the BECASsolution
• AMMF had betterconstraint resolution
• AMMF gives accuratecorrections
• Additive vs multiplicativecorrections:
• Gives similar solutions• Similar performance
0 0.2 0.4 0.6 0.8 1r/R
0
0.01
0.02
0.03
0.04
0.05
Thi
ckne
ss [
m]
InitialBECASCLTAMMF
0
0.01
0.02
0.03
0.04
Rel
ativ
e D
iffe
renc
e
AMMF Relative Difference
22 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
Multi-fidelity Structural Design Optimization
Optimization Convergence
0.0 5.0×104
1.0×105
1.5×105
2.0×105
Time [s]
5200
5300
5400
5500
5600
Obj
ectiv
e
BECAS ObjectiveAMMF Objective
0.01
0.1
1
Con
stra
int V
iola
tion
BECAS ViolationAMMF Violation
• AMMF converges 12 timesfaster
• Just 2 major iterations
• AMMF had smootherconvergence
• Only 1 iteration withconstraint violation
• BECAS optimizationended due to maximumiterations
• Low fidelity models moresuitable for optimization
23 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
Multi-fidelity Structural Design Optimization
AMMF Robustness
AMMF guards against poor approximations
• Unconstrained has allprotections disabled
• Large violations• Fails to converge
• Trust region is most robust
• Same progress as idealconfiguration
• Large penalties work withouttrust region
• No large violations• More searching
0.0 2.5×104
5.0×104
7.5×104
1.0×105
Time [s]
4600
4800
5000
5200
5400
5600
5800
Obj
ectiv
eIdealUnconstrainedTrust RegionLarge Penalty
0.01
0.1
1
10
Con
stra
int V
iola
tion
Ideal ViolationUnconstrained ViolationTrust Region ViolationLarge Penalty Violation
24 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
AMMF for aero-elastic blade design
25 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
AMMF for aero-elastic blade design
Problem Description
• Maximize DTU 10MW AEP
• Varying all blade design parameters
• Subject to:
• Tip deflection constraint• Stress constraints• Geometric constraints
• Analysis based on BECAS, HAWCStab2, HAWC2
• Used a reduced DLB
• Preliminary optimization to see if it runs• Future work will use a full DLB
• Low-fidelity model based on corrected HAWCStab2 results
26 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
AMMF for aero-elastic blade design
Low-Fidelity Work-flow
Mdyn = MstaticA(r)σdM
dV
• Model for the dynamic loadsMdyn based on:
• HAWCStab2 momentloads Mstatic and dM
dV
• Turbulence σ
• Correction A(r)
• Matches full DLB
• Used Dakota to tune A
based on minR2
• No HAWC2 but still needs 75%of the time
27 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
AMMF for aero-elastic blade design
AMMF Optimization Results
• AMMF ran in MPI
• Only 1 iteration achievedwithin cluster time limit
• Similar run time betweenlow & high fidelity
• AMMF moving in the rightdirection
• Increase in AEP
• Direct 6.17%• AMMF 4.61%• 74.7% progress
• Blade failure index 0.79 < 0.9
• AMMF was conservative
0.0 0.2 0.4 0.6 0.8 1.0 1.20.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
OriginalAMMFDirect Optimization
Figure : Normalized Chord vs. Blade Radius
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Future Work
29 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
Future Work
Ongoing Aero Elastic Design Optimization
Figure : The full IEC 61400 Design Load Cases
• High fidelity based on HAWC2, the full set of International ElectrotechnicalCommission (IEC) 61400 design load cases with turbulence
• Low fidelity based on Classical Laminate Theory (CLT) and a reduced set of loadcases without turbulence
30 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
Future Work
Swept Blade Design Optimization
0 0.2 0.4 0.6 0.8 1z/L
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25x/
L
Figure : Swept Blade Shape
• High fidelity based on HAWC2 and Omnivor (time marching vortex code)
• Low fidelity based on HAWCStab2
• Aerodynamic only design optimization
31 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
Closing Statements
32 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
Closing Statements
Conclusions
• Promising results for Multi-fidelity optimization
• With the right low-fidelity model AMMF is 12 times faster• AMMF is robust against model and correction errors
• AMMF can perform aero-elastic blade optimization
• AMMF work-flow needs more refinement for aero-elastic optimization• Trying to include the full IEC 61400 DLC for high fidelity analysis
• Working on applying AMMF on a swept blade aerodynamic optimization
33 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
Closing Statements
Acknowledgments
This work was supported byNatural Sciences and Engineering Research Council of Canada
34 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017
Closing Statements
Thank-you for your interest
Comments or Questions?
35 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,2017