Multi-fidelity optimization of horizontal axis wind turbines

Transcription

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

2 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,

3 Introduction Motivation Interested in applying design optimization to advanced concepts: Swept blades Flaps Multi-rotor Typical optimization frameworks based on simplified load cases Tuned to be overly conservative Could miss potential opportunities Standard design tools and frameworks may not be suitable Need higher fidelity analysis in optimization True Constraint Simplified Feasible Set Simplified Optimum Improving Design Objective Contours True Feasible Set Simplified Constraint Design Space True Optimum 3 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

4 The AMMF Algorithm 4 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

5 The AMMF Algorithm The AMMF Algorithm Initial design Calculate fl, fh, fl and fh Build/update correction model β(x) Use optimization to find next design x High fidelity used for accuracy Low fidelity is used for speed Correction for first order consistency f(x) = f l (x)+β(x) Update trust region : expand if f fh is small constrict if f fh is large Calculate fl Calculate f = β(x)fl Calculate fl, fh, fl and fh Exit if converged β(x) = f h0 f l0 +( f h0 f l0 ) x Trust-region for robustness 5 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

6 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 Management Framework (AMMF) First an estimated Lagrangian is calculated Φ = f + λ e c + λ i max(0, c i ) λ are the Lagrange multipliers estimated from previous iterates. λ is specified for the first iteration New iterate only accepted when Φ i < Φ i 1 Trust region is expanded or contracted based on M: M = Φ i 1 Φ i Φ i 1 Φ i 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,

7 Preliminary investigation into AMMF 7 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

8 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,

9 Preliminary investigation into AMMF Effect of constant offset error on AMMF Objective difference from solution No error Error 0.1 Error 0.2 Error 0.5 Error Major iteration of AMMF 9 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

10 Preliminary investigation into AMMF Effect of linear offset error on AMMF Objective difference from solution No error Error 0.1 Error 0.2 Error 0.5 Error Major iteration of AMMF 10 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

11 Preliminary investigation into AMMF Effect of quadratic offset error on AMMF Objective difference from solution No error Error 0.1 Error 0.2 Error 0.5 Error Major iteration of AMMF 11 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

12 Preliminary investigation into AMMF AMMF investigation phase 2: Lateral offset errors Under or over-shooting low fidelity model Objective value High fidelity function Weak under-shoot Strong under-shoot Weak quadratic error Strong quadratic error Weak over-shoot Strong over-shoot Normalized distance 12 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

13 Preliminary investigation into AMMF AMMF convergence rate vs. lateral offset error Objective difference from solution 1e+01 1e+00 1e-01 1e-02 Weak under-shoot Strong under-shoot Weak quadratic error Strong quadratic error Weak over-shoot Strong over-shoot 1e Major iteration of AMMF 13 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

14 Preliminary investigation into AMMF AMMF function evaluations vs. lateral offset error Major iteration of AMMF Number of HF function evaluations30.0 Weak under-shoot Strong under-shoot Weak quadratic error Strong quadratic error Weak over-shoot Strong over-shoot Direct Optimization 14 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

15 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 15 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

16 Multi-fidelity Structural Design Optimization 16 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

17 Low Fidelity Tool Development 17 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

18 Multi-fidelity Structural Design Optimization Summary of Low Fidelity Tools Position EA EIx EIy GJ Table : Percent Error with BECAS Low fidelity cross section tool Thin-walled cross section assumption Rigid cross section (Euler-Bernoulli) Classic laminate theory Written in C++ Python bindings with Swig Will have analytic gradients Within 10% compared to BECAS 18 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

19 Multi-fidelity Structural Design Optimization Summary of Low Fidelity Tools Operation Calculation time [s] Linear Beam Model LF cross section model BECAS 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 given deflection Speed comparison: With python bindings Calculation for whole blade 19 elements DTU 10MW 19 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

20 AMMF for equivalent static beam 20 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

21 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) 21 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

22 Multi-fidelity Structural Design Optimization Optimization Results Low fidelity model is not conservative Will produce infeasible solutions AMMF reproduced the BECAS solution AMMF had better constraint resolution AMMF gives accurate corrections Additive vs multiplicative corrections: Gives similar solutions Similar performance Thickness [m] AMMF Relative Difference Initial BECAS CLT AMMF r/r Relative Difference 22 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

23 Multi-fidelity Structural Design Optimization Optimization Convergence Objective BECAS Objective AMMF Objective BECAS Violation AMMF Violation Time [s] 1 Constraint Violation AMMF converges 12 times faster Just 2 major iterations AMMF had smoother convergence Only 1 iteration with constraint violation BECAS optimization ended due to maximum iterations Low fidelity models more suitable for optimization 23 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

24 Multi-fidelity Structural Design Optimization AMMF Robustness AMMF guards against poor approximations Unconstrained has all protections disabled Large violations Fails to converge Trust region is most robust Same progress as ideal configuration Large penalties work without trust region No large violations More searching Objective Ideal Violation Unconstrained Violation Trust Region Violation Large Penalty Violation Ideal 0.1 Unconstrained 4800 Trust Region Large Penalty Time [s] 10 1 Constraint Violation 24 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

25 AMMF for aero-elastic blade design 25 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

26 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,

27 AMMF for aero-elastic blade design Low-Fidelity Work-flow M dyn = M static A(r)σ dm dv Model for the dynamic loads M dyn based on: HAWCStab2 moment loads M static and dm dv Turbulence σ Correction A(r) Matches full DLB Used Dakota to tune A based on minr 2 No HAWC2 but still needs 75% of the time 27 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

28 AMMF for aero-elastic blade design AMMF Optimization Results AMMF ran in MPI Only 1 iteration achieved within cluster time limit Similar run time between low & high fidelity AMMF moving in the right direction Increase in AEP Direct 6.17% AMMF 4.61% 74.7% progress Blade failure index 0.79 < 0.9 AMMF was conservative Original AMMF Direct Optimization Figure : Normalized Chord vs. Blade Radius 28 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

29 Future Work 29 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

30 Future Work Ongoing Aero Elastic Design Optimization Figure : The full IEC Design Load Cases High fidelity based on HAWC2, the full set of International Electrotechnical Commission (IEC) design load cases with turbulence Low fidelity based on Classical Laminate Theory (CLT) and a reduced set of load cases without turbulence 30 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

31 Future Work Swept Blade Design Optimization x/l z/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,

32 Closing Statements 32 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

33 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 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,

34 Closing Statements Acknowledgments This work was supported by Natural Sciences and Engineering Research Council of Canada 34 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,

35 Closing Statements Thank-you for your interest Comments or Questions? 35 DTU Wind Energy Multi-fidelity HAWT Optimization January 12,