Anonymous MIT Students Introduction Problem Formulation Design Vector Analysis Methodology & Parameters Fidelity & Complexity Optimization Methods Single Objective Results Multi Objective Results Conclusions & Future Work Focus on renewable energy Disciplines involved◦ Aerodynamics◦ Control◦ Structures◦ Acoustics◦ Electrical engineering Interactions◦ Control (rotational speed affects aerodynamics)◦ Structures (blade deflection affects aerodynamics)Considered in this projectRenewable Energy ProjectionSource: http://www.paulchefurka.ca/WEAP/WEAP.htmlConsidered in this project Objective Function◦ Over a range of incoming wind speeds Penalty Function Model reduction by Piecewise Cubic Hermite Interpolating PolynomialsDecision variable pointsAssumed pointPCHIP SplineDecision variable pointsk = 4Weibull statisticsMean = 4.43 m/sStd. Dev. = 2.13 m/s N2Diagram Aero – Blade Element Momentum Theory◦ Relaxed iteration root-finding Expected Power – Simpson’s rule integration over Weibull distribution Structure – Equivalent beam theory Control – Line search convex optimization (fminbnd in MATLAB)Inputsx, paramx, paramx, paramx, paramx, paramOutputWrappercweibull,kweibullPE, Vblades, Max(σmax(v0))Max(σmax(v0))Expected Powerv0ω, v0ω, Ft/ΔR, Fa/ΔRωControlω, v0Q, P, Ft/ΔR, Fa/ΔRQ, P, Ft/ΔR, Fa/ΔRAeroσmax(v0)StructureInter-module optimizationDiscretization Parameters (affect fidelity) Model order reduction◦ PCHIP Analysis methodology low fidelity◦ Quick computation times Validation◦ Structures code equates with analytical classical beam theory for cantilevered beam◦ Ran out of time to compare with Qprop / VABS high-fidelity codes◦ Tell us if you know of a wind turbine design to benchmark against Design of Experiments (DOE)◦ Complex design space (initial idea)◦ Main effects◦ Space-filling starting points for Gradient-based methods◦ Good results, short running times Gradient-based methods◦ Continuous design variables with no discontinuities◦ Constraints imposed by square term penalty method◦ Implemented SQP with MATLAB’s ‘fmincon’◦ Re-scaling Hessian◦ Multi-start (non-convex objective function and feasible space)◦ Good results, long running times Heuristic methods◦ Multi-Objective Genetic Algorithm (MOGA)◦ Poor results, long running times-Varying convergence rates-Varying solution values  Non-convexityDesign Variables Values CommentsR 14.13Qmax 20000t 0.004k 4T(mid) 0.79 lower boundT(tip) 0.78F 0.25C(root) 1.91 upper boundC(mid) 0.79C(tip) 0.24beta 0.790.750.36162 Sensitivity Analysis (2ndorder central difference)◦ Decision variables that are tight on box bounds have directions of improvement without violating feasibility (Qmax increase  PE/Vbladesincrease, σmax decrease◦ Decision variables that are free on box bounds have no directions of improvement that do not violate constraints (R increase  σmax increase) Connection to Lagrange multipliers Slope of Pareto Front◦ initially benefit from going to higher expected power◦ later stages cost outweighs benefits of increasing expected power◦ optimum somewhere in between Utopia point – highest expected power for the lowest cost SQP outperforms MOGA◦ Not enough running time◦ Computational expenseUtopia pointBest result from SQP DOE is powerful and inexpensive SQP works great for local optimization Heuristic methods may be too expensive Higher resolution optimization◦ More decision variables for distributions Higher fidelity analysis◦ Increase blade discretization◦ Qprop◦ VABS Higher-powered optimization◦ DAKOTA implementation may be more powerful than the Matlab Optimization Toolbox Questions?Also for validation Thickness of structural spar compensates for light structure◦ Adds leewayttboxBlade Cross-sectionBest combination of factor levelsMIT OpenCourseWarehttp://ocw.mit.edu ESD.77 / 16.888 Multidisciplinary System Design OptimizationSpring 2010 For information about citing these materials or our Terms of Use, visit: