# Meta-heuristic multidisciplinary design optimization of wind turbine blades obtained from circular pipes

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## Abstract

Aim of this paper is to present a methodology useful to optimize the geometry of the blades of a small-size wind turbine which are obtained from a circular pipe: an optimal chord distribution and airfoil sweep can be obtained with a proper cutting path. A strong reduction in manufacturing costs and time can be achieved for blades which are a critical element in wind turbine systems, especially in case of renewable plants in developing countries. An algorithm has been developed to obtain the shape of the blades and wind turbine performances are computed by the Blade-Element Method, due to its low computational simplicity; the XFoil tool has been used to compute the aerodynamic of the blades. Heuristic algorithms have been applied to obtain a feasible design solution assuring the best efficiency of the wind turbine. Also structural considerations are kept into account to provide a feasible configuration able to withstand the forces acting on the rotating blades. Results obtained suggest that an optimal design of such a kind of blades can be obtained thanks to this methodology. The mathematical framework developed for the optimization is efficient and the heuristics algorithms allow the convergence to feasible configurations. The computing time is compatible with a practical application of the method also in industries.

## Keywords

Multidisciplinary optimization Particle swarm algorithm Wind turbine Design CAD## Abbreviations

- a
Axial induction factor (–)

*a*_{c}Critical axial induction factor (–)

*a*′Angular induction factor (–)

*A*_{1},*A*_{2}Coefficients for the lift Viterna model (–)

- AR
Blade aspect ratio (–)

*B*_{1},*B*_{2}Coefficients for the drag Viterna model (–)

*C*_{D}Local drag coefficient (–)

*C*_{Dmax}Maximum lift coefficient (–)

*C*_{L}Local lift coefficient (–)

*C*_{Lstall}Local stall lift coefficient (–)

- ch
Chord of the airfoil (m)

- ch_
*a* Chord of the tip section (m)

- ch_
*b* Chord of the intermediate section (m)

- ch_
*c* Chord of the root section (m)

*D*Pipe diameter (m)

- dis_
*d* Position of the tip section chord (m)

- dis_
*e* Position of the intermediate section chord (m)

- dis_
*f* Position of the root section chord (m)

*F*_{x}Axial force (N)

- FS
Airfoil scale factor (–)

*K*Correction factor in Glauert equation (–)

- len_
*l* Blade span (m)

- len_
*g* Distance tip/intermediate section (m)

- len_
*h* Distance tip/rotation centre (m)

*N*Number of blades (–)

*r*Radius measured from wind turbine hub (m)

*R*Radius of the wind turbine from rotation axis (m)

*Q*Factor for tip loss (–)

*s*Pipe thickness (m)

*T*Torque (Nm)

*t*Parameter along Bezier curve (0 ÷ 1)

*V*Wind speed (m/s)

*W*Speed of the flow impacting the airfoil (m/s)

*X*_{LE},*Z*_{LE}Coordinates of the airfoil leading edge (m)

*X*_{TE},*Z*_{TE}Coordinates of the airfoil trailing edge (m)

*x*_{i},*y*_{i}Chord points’ coordinates in the plane (m)

*X*_{i},*Y*_{i},*Z*_{i}Chord points’ coordinates respect to blade longitudinal axis (m)

*X*_{i_up},*Y*_{i_up},*Z*_{i_up}Coordinates of the airfoil upper surface (m)

*X*_{i_down},*Y*_{i_down},*Z*_{i_down}Coordinates of the airfoil lower surface (m)

*x*_{i_rot},*y*_{i_rot}Rotated chord points’ coordinates in the plane (m)

*x*_{i_lean},*y*_{i_lean},*z*_{i_lean}Coordinates of the projection chord points on the pipe (m)

*α*Blade angle of attack (rad)

*α*_{stall}Blade stall angle of attack (rad)

*β*Relative flow angle on the blades (rad)

*λ*_{r}Tip–speed ratio at radius

*r*(–)*γ*_{0}Airfoil settling angle (rad)

*γ*_{rot}Blade geometry rotation angle (rad)

*Ω*Blade rotational speed (rad/s)

*ω*Wake rotational speed (rad/s)

*ρ*Air density (kg/m

^{3})*σ′*Local solidity (–)

*τ*Airfoil inclination angle on the pipe (rad)

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