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Presently, there exist few numerical methods which treat the inverse problem for the determination of the geometry of wind turbine blades. In this work, authors intend to solve the inverse optimum project for horizontal axis wind turbine in which the selection of the circulation distribution is obtained by resolving two variational problems: the first consists in sorting the circulation distribution on the lifting line, which, for a given power extracted by the wind turbine, minimizes the loses due to the induced velocity. In the second, the optimal circulation distribution is selected such that the kinetic energy of the wind downstream of the rotor disc is minimum, when the energy extracted by the wind turbine for one rotating period is imposed. A code has been developed which incorporates the real pitch of the helicoidal vortex wake. Very promising results have been obtained: the circulation distribution for a given extracted power and the chord lengths distribution law along the blade span.

The blade element momentum (BEM) model [

There are some fundamental shortcomings of the BEM model which can be ascribed to the assumptions introduced in its derivation. The equation relating the induction at the disc with the axial thrust is derived for a stream tube enclosing the whole rotor disc, but in the BEM model the same relation is used for differential stream tubes at different radial positions (strip theory). Comparison of the BEM model with more accurate induction models, such as the numerical actuator disc model with the flow solved with a CFD model, indicates that this assumption is not accurate in regions of the blade with strong radial variation of the loading, as, for example, towards the tip and at the root [

Owing to the assumptions in the BEM model, more physically based aerodynamic models such as vortex line and panel methods have been developed for some problems (rotor shapes or flow conditions). These models reflect detailed information about the induced flow field at the rotor disc and have been successfully applied, for example, to studies of dynamic inflow situations [

Though numerous studies have been effectuated on the resolution of the inverse problem on rotating machines, there exist few numerical methods which treat the inverse problem for the determination of the geometry of a wind turbine blade.

If the wake of the blade and its slipstream were to be known exactly, then the field of induced velocities and other relevant quantities could be computed more readily. This can be done, in fact, by appropriate manipulation of Biot-Savart equation for the inviscid vortex line. The classical expression of a vortex-induced velocity is the Biot-Savart law that is the fundamental relationship between a vortex, its shape, and the velocity that it induces. This method proves extremely valuable, because it can track the main pattern of the vortex system.

The main idea of the present article consists in determining the circulation distribution which corresponds to a given extracted power. It is evident that the power coefficient should be chosen under Betz limit moreover under the limit curve for 3-blade Betz rotor [

The selection of the circulation distribution for the inverse problem is obtained by resolving two variational problems: the first consists in sorting the circulation distribution on the lifting line, which, for a given power extracted by the wind turbine, minimizes the loses due to the induced velocity “OPHWT” [

It is shown that the optimal circulation distribution corresponding to the best energetic efficiency of the wind turbine induces a velocity which satisfies the slip condition. This actually occurs in the case where the helicoidally vortex sheet becomes rigid and rotates with the same speed as the wind turbine, but in the opposite sense.

The method used in calculating the induced velocities is publicized in [

The resolution is achieved by the singularities method based on discrete distributions of circulation: it consists in dividing the blade span-length

In this inverse “OPHWT” model, we intend in the reality to find the efficacy rotor. In fact, for a given wind site characterized by the wind speed and for a given number of the blades, we aim to use the machine with the minimum losses of power. These wishes consist in imposing some law for the induced velocities on the lifting line, and we determinate the optimum circulation distribution which induce these velocities.

The selection of the circulation distribution is based on the hypothesis of limiting the span length of the blade. An optimal circulation distribution is sorted by minimizing the power losses due to the effect of the induced velocity downstream of the turbine disc. It is a variational problem which consists in finding the circulation distribution

It should be noted that the wake shed along the blades is assumed to form a helicoidal vortex sheet originating from the trailing edge. The system of free vortices emanating from the trailing edge is specified to persist without disintegration up to a relatively far distance downstream of the rotor disc, by following a local stream line [

The method used in calculating the induced velocities by the vortex sheet is derived directly from the general lifting lines and lifting surfaces theories without any simplifications made. It was elaborated by Goldstein [

The integral equation for determining the optimum spanwise distribution of circulation

A semi-infinite helicoidal vortex is made up of a vortex filament which is moving from the rotor disc plane to downstream infinity while carrying the vortex intensity

Helicoidal vortex sheet originating from the lifting line.

The power extracted by the wind turbine is written in the following form:

The power coefficient

The aim of this method is to minimize the kinetic energy of the wind down stream of the rotor disc. This leads to increasing the induced velocity downstream in order to attain the optimum efficiency which is under the limit curve for 3-blade Betz rotor [

A variational calculation is used to show that the optimal distribution of the circulation

Indeed, if we designate by

The power extracted by the wind turbine for one functioning period is given by

The target now is to look for the condition to be satisfied so that the kinetic energy

The corresponding variation of

Now the velocity

Let

The resulting variation of

The corresponding variation of

If

By substituting

By introducing the Lagrange constant:

By considering the expressions which give

We notice that this is a homogenous equation. If

But the component in the direction of

Hence, the normal component of the induced velocity corresponding to the optimal condition is given by

Now, the normal component of the induced velocity is linked to the axial component

It comes out that the optimal condition concerning the component

The optimal distribution of

The resolution of (

For a given extracted power, we present in Figures

Circulation Coefficient on the lifting line for

Circulation Coefficient on the lifting line for

This result is very important in designing the blade of a wind turbine. In effect, by using this code we can determine the optimum circulation for every given power coefficient; later on we can determine the chord lengths distribution along the blade span leading to the optimum shape of the blade. This is the main the advantage of the OPHWT model.

A comparison of the next two figures reveals some discrepancy between the two models. This is presented in Figure

Comparison of the circulation coefficient distribution on the lifting line between the OPHWT model and the kinetic energy model for

For the same coefficient power

Figures

Spanwise variation of the chord in the inverse OPHWT model for

Span wise variation of the chord in the kinetic energy model for

Figure

However for the same functioning parameter

The OPHWT is confirmed very efficient to calculate the optimal performances and geometry of a wind turbine. In effect, it is applied in the case of reel functioning conditions. Figure ^{−1} and for a coefficient power

Comparison of the circulation ^{2}/s) between the inverse OPHWT model and the near wake model for the 40 m rotating blade at a free wind speed of 7 ms^{−1} and for

Spanwise variation of the chord in the inverse OPHWT model and the Near wake model for

As can be observed in Figure

The Near wake model was tested on a 40 m rotating blade with an angular velocity of 2.0 rad s^{−1} corresponding to a tip speed of 80 m s^{−1} and with a plan-form as shown in Figure

In this paper, an inverse problem has been resolved by two methods: the first consists in the minimization of the power losses due to the induced velocity (inverse OPHWT model) and in the second we have minimized the kinetic energy imparted to the fluid downstream of the rotor disc (prop fan).

Confrontation of the results obtained with that of Helge Aaggard and Flemming (A near wake model for Beddoes) has revealed the potential of the optimum project in predicting improved and higher rotor performances even at relatively high wind speeds.

In effect, for a given wind site characterized by a wind-rose, that is a given functioning parameter

As a follow-up of this work we intend to construct a complete blade of a wind turbine in 3D by coupling the inverse OPHWT model with an inverse code which solves the inverse problem of the flow over a 2D airfoil [

The aim of the project is to visualize the actual level of wind turbine technology in Tunisia (the Sidi Daoud wind site, Tunisia): essentially the blades and the aerodynamic efficiency in order to set up a local wind turbine industry. Tunisia intends to install other wind sites in order to reach an energy production of about 200 MW by 2012. This paper can help the decision makers in the choice of the blades suitable for a given wind site, which can be purchased, and also to design new forms of blades for the small installations of wind turbines for isolated sites.