A synthesis of a blade pitch control for large wind turbines, taking into account the functional dependence of the torque generated in the blades from the azimuth angle

In multi-MW (1MW and up) wind generation systems a variable speed variable pitch (VS-VP) control technology is usually adopted to improve speed and torque characteristics and to obtain the optimal energy, which means to obtain the maximum power at low wind speed and the rated power at high wind speed. The power generated by an individual blade of the wind turbine changes considerably during rotation due for the change of wind speed in vertical direction, which decreases the angles of attack in the low blade positions and which leads to a reducing the generated energy. In the paper basing on BEM theory modification for none-homogenously fluid field and an optimization procedure is obtained an individual variable pitch control law as function of wind speed values and blades positions.


Introduction
The vertical speed gradient leads to significant changes of the torque generated from one blade during rotation. This is due to the fact that in the lower blade positions the wind speed decreases considerable, this leads to reducing the angles of attack and as a result of this, the torque diminishes. Therefore, if an individual blade pitch control is applied, it is possible to increase power to a certain extent, especially for wind speeds below the nominal. The determination of this control is complicated because the blades are building from sections with different profile and respectively different lift and drag coefficients. Thus if the angle of attack exceeded a certain value the aerodynamic interaction may get worse. Moreover due to the continuity of the fluid flow, the induction coefficients are in dependence with aerodynamic state of all blades, which is a prerequisite for additional complexity in determination of the optimum angle of attack in the given position of the blade. For a solution of this problem, basing on the adaptation of the classical blade element momentum (BEM) theory to the accounting of the vertical gradient of the wind speed, in this research an optimization approach for the determination of the individual pitch control law is proposed. The approach is realised for a set of discrete positions of the turbine and then the control dependence is obtained of by interpolation between them. The obtained results for a known prototype of wind turbine show simultaneously the validity of the BEM theory modification and of the propose approach for synthesis of an individual blade pitch control.

Pitch control synthesis
In [1][2][3][4][5][6][7][8][9] is investigated a pitch control synthesis that provided efficiency and high quality of produced electricity as well as maintaining the lowest possible dynamic loads in some elements of the wind generator's system.
In present paper this synthesis is treated in an aspect of obtaining both the maximum power at lower than the nominal wind speeds and the rated power at higher then nominal wind speeds, taking in account the fluid speed gradient in vertical direction. For the first purpose is investigated an individual form of pitch angle control and for the second is using collective pitch control form.

Wind speed distribution
The proposed algorithm is presented for the prototype wind turbine of high class -NREL5MW [10]. For the vertical wind speed distribution is preferred the logarithmic model [11]. For the parameters values defined in [12], the wind distribution as a function of the speed in the hub axis at a height of 90 m, and height z has the form:

BEM theory adaptation with taking in account the vertical wind speed gradient
The BEM theory is presented in many sources [13][14][15][16][17][18][19][20][21][22][23]. In the most of them the fluid field is considered as a homogeneous i.e. the differences of the wind speed in a vertical direction are neglected. Thus for all blades, the dynamic interaction with the wind flow is an identical and does not depend on the azimuth angle. For the turbines with standard size this is acceptable but for the high class it is observes significant different of the vertical speed distribution in the turbine disk. This causes that, for the given position each of the blades have a considerably different aerodynamic interaction with the wind. Respectively the aerodynamic forces that act on the blade become a function of the blade azimuth angle and varying even if the longitudinal turbulence of the wind velocity is not considered. The common used methodology for calculation of induction coefficients is needs to be revision. In [14, 16-19, 21, 23] some semi-analytical or numerical investigations in this or similar sense are made.
In the next text are presented the results obtained by the author in adaptation of BEM theory for the case of nonhomogeneous vertical wind speed distribution.
On Fig. 2 are shown the elements of the aerodynamic excitations in the plane of the blade cross-section.
The designations shown in the figure are as follows: r -distance from the hub axes; -blade azimuth angle (zero in vertical upward direction); -angular velocity of the turbine rotation; , -inflow speed; , -flow speed in turbine disk; , 1 , -tangential component of the relative flow velocity for the section; / -axial induction factor; 0,5 / -tangential induction factor; -angular velocity of the flow in the disk plane; , -section pitch angle; -construction pitch angle for the section; -blade pitch angle set by the system of the active control; , arctg , / , -inflow angle for the section; , , , -angle of an attack; , -torque force, F Thrust , -thrust forces, M wt ,torque , , , -torque, thrust, lift and drag coefficients of the section. The following dependencies are in effect: where: where: , ℓ ,

Obtaining of pitch control law
The relation between torque and angular velocity of the turbine is shown of Fig. 3. The results, from the numerical modelling, for the angular turbine velocities , average thrust force -, torque -and power -values, without using pitch control, obtained for values between the minimal and nominal wind velocity, related to the hub axis level -, are shown in Table 1. The proposed control law is individual pitch for wind velocity under the nominal value and collective for over the nominal.
The pitch angle has two components: , where: is the construction pitch angle for the blade section; blade pitch angle set by the system for active pitch control. The torque of the blade elements as function of the azimuth angle and the element position by the length of the blade for the different wind speed from 3 to 25 m/s is shown of Fig. 4. In this way the optimization can be described mathematically as:   Table 1). In this case the optimized parameter is the collective pitch angle, which is not treated as function of the azimuth angle and it is only a function of the wind speed. The mathematical presentation of this optimization task is: , 0 ,..

(4)
D= ∈ 0 ∈  ∈ 12,13, . .25 m/s. The optimization is realised by the Quasi-Monte Carlo method based on the using of the Sobol sequences [24]. It is generated cover of 150 points in the definition area, for each position. This number of points proves to be sufficient for the localization of the optimum.
The obtained results are given in Table 2. The individual pitch control as function of the blade azimuth angle, for different wind speeds is shown on Fig. 5. The collective pitch control as function of the wind speeds on the level of the hub axis is shown on Fig. 6.    This is not very large, but is a tangible improvement of the power characteristics.

Conclusion
The paper presents an optimal synthesis of a new type pitch control of the wind turbines of high class that takes into account the effects due to the vertical gradient of the wind velocity. It is based on proposed modification of BEM theory. The obtained from the simulation results show that they are some better in comparison of these without of such type control. The theory may to be applied ease for the synthesis of the blade pitch control when the longitudinal turbulence of the wind velocity is also included in the model.