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

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.

Appendix

As already introduced in the Sect. 3 of the paper, the basic methodology to optimize wind turbine blades comes from the integration of Momentum Theory and Blade-Element Model. The Momentum Theory finds the axial force and torque of a wind turbine evaluating the behaviour of the air stream through the area swept by the wind turbine blades.

In Momentum Theory [4], the kinetic energy of the air flow passing through the wind turbine and the quantity of motion are equated, and the axial force on the turbine disk area (Fig. 16) can be computed with:

Fig. 16

Turbine axial view

$${\text{d}}{F_x}=Q\rho V_{1}^{2}[4a(1 - a)]\pi r{\text{d}}r,$$

(38)

where V₁ is the velocity of the undisturbed air flow before the wind turbine (Fig. 17). In a similar manner, the torque generated by the air flow over the blades can be evaluated with:

Fig. 17

Wind turbine side view

$${\text{d}}T=Q4a^{\prime}(1 - a)\rho V\Omega {r^3}\pi {\text{d}}r.$$

(39)

The losses on the blades can be accounted with the following equation, function of the radial distance from the turbine hub:

$$Q=\frac{2}{\pi }\arccos \left[ {\exp \left\{ { - \left( {\frac{{(N/2)\left[ {1 - r/R} \right]}}{{(r/R)\cos \beta }}} \right)} \right\}} \right].$$

(40)

The Blade-Element Theory exploits the aerodynamics to compute the axial force and torque on the wind turbine: the integral of the lift and drag of the blade airfoils along the span is required in this approach. Once defined the rotational speed of the wind turbine blades (Ω) and the speed of the rotational wake (ω), the axial induction factor (a) and the angular induction factor (a′) can be defined as follows (see also Fig. 17):

$$a^{\prime}=\frac{\omega }{{2\Omega }};\,a=\frac{{{V_1} - {V_2}}}{{{V_1}}}.$$

(41)

Considering the triangle of the air relative speed over a rotating blade (see Fig. 18), the following formula applies:

Fig. 18

Wind turbine speed triangle

$$\Omega r+\frac{{\omega r}}{2}=\Omega r(1+a^{\prime}).$$

(42)

The settling angle (γ) depends on the blade geometry, while the angle between wind turbine hub axis and the relative air speed acting on the blade (β) can be found with:

$$\tan \beta =\frac{{\Omega r(1+a^{\prime})}}{{V(1 - a)}}.$$

(43)

The local tip ratio (λ_r) is an index of the ratio between wind turbine rotational speed and the speed of the undisturbed axial wind flow:

$${\lambda _r}=\frac{{\Omega r}}{V}.$$

(44)

Following the rules of the trigonometry, the relative speed impacting the blade sections can be found with:

$$W=\frac{{V(1 - a)}}{{\cos \beta }}.$$

(45)

The axial force acting on the infinitesimal area of the blade surface dS given by the local chord (c) multiplied by dr is:

$${\text{d}}{F_x}=\sigma ^{\prime}\pi \rho \frac{{{V^2}{{\left( {1 - a} \right)}^2}}}{{{{\cos }^2}\beta }}({c_{\text{L}}}\sin \beta +{c_{\text{D}}}\cos \beta )r{\text{d}}r,$$

(46)

where C_L and C_D are lift and drag airfoil non-dimensional coefficients. A similar formula can be derived for the torque on the infinitesimal surface:

$${\text{d}}T=\sigma ^{\prime}\pi \rho \frac{{{V^2}{{\left( {1 - a} \right)}^2}}}{{{{\cos }^2}\beta }}\left( {{c_{\text{L}}}\cos \beta - {c_{\text{D}}}\sin \beta } \right){r^2}{\text{d}}r.$$

(47)

In the previous formulas, the local solidity (σ′) has been introduced:

$$\sigma ^{\prime}=\frac{{{\text{Nc}}}}{{2\pi r}}.$$

(48)