Multi-fidelity optimization of horizontal axis wind turbines · Multi-ﬁdelity optimization of horizontal axis wind turbines Michael McWilliam Danish Technical University

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


The AMMF Algorithm


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


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


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



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



Effect of linear offset error on AMMF


0.1

1

10

Obj

ectiv

e di

ffer

ence

fro

m s

olut

ion




Effect of quadratic offset error on AMMF


0.1

1

10

Obj

ectiv

e di

ffer

ence

fro

m s

olut

ion




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



AMMF convergence rate vs. lateral offset error


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



AMMF function evaluations vs. lateral offset error


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



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


Multi-fidelity Structural Design

Optimization


Low Fidelity Tool Development


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



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


AMMF for equivalent static beam



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)



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



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



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


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



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



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


Future Work


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


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


Closing Statements


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


Closing Statements

Acknowledgments

This work was supported byNatural Sciences and Engineering Research Council of Canada


Closing Statements

Thank-you for your interest

Comments or Questions?


Documents

Multi-fidelity optimization of horizontal axis wind turbines · Multi-ﬁdelity optimization of horizontal axis wind turbines Michael McWilliam Danish Technical University