**1. Introduction**

With the growing demand for cost-effective wind energy, optimization of wind turbine components has been gaining increasing attention for its acknowledged contributions made to design enhancement, especially in early stages of product development. One of the major design goals is the accurate determination of structural dynamics and control, which is di‐ rectly related to fatigue life and cost of energy production: a major design goal in exploiting wind energy. Modern wind turbines are designed with pitch-regulated rotor blades, which have to be able to turn around their longitudinal axis several times per second in order to face the rapidly changing wind direction. This fact emphasizes the need to improve the de‐ sign of pitch mechanisms using optimization techniques in order to increase availability of the turbines and reduce their maintenance overheads. (Florin et al., 2004; Jason et al. 2005) demonstrated the different tools for performing the analysis of the interaction between the mechanical system of the wind turbine and the electrical grid as well as the calculation of the dynamic loads on the turbine structure. In case of stronger winds it is necessary to waste part of the excess energy of the wind in order to avoid damaging the wind turbine. All wind turbines are therefore designed with some sort of power control. There are different ways of doing this safely on modern wind turbines: pitch, active stall and passive stall controlled wind turbines.

On a pitch controlled wind turbine (Hansen et al., 2005) the turbine's electronic controller checks the power output of the turbine several times per second. When the power output becomes too high, it sends an order to the blade pitch mechanism which immediately pitches (turns) the rotor blades slightly out of the wind. Conversely, the blades are turned back into the wind whenever the wind drops again. The rotor blades thus have to be able to turn around their longitudinal axis (to pitch) as shown in Fig. 1. The pitch mechanism

© 2012 Maalawi; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Maalawi; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

is usually operated using hydraulics or electric stepper motors. Fig. 2 shows the optimal operational conditions of a pitch-controlled 2 MW wind turbine. During normal operation the blades will pitch a fraction of a degree at a time, and the rotor will be turning at the same time. The computer will generally pitch the blades a few degrees every time the wind changes in order to keep the rotor blades at the optimum angle to maximize output power for all wind speeds.

**Figure 1.** Limiting power output using pitch control.

**Figure 2.** Operational conditions of a pitch-controlled, 2.0 MW wind turbine (Hansen et al., 2005)

On the other hand, passive stall controlled wind turbines (Leithed & Conner, 2002; Hoffmann, 2002) have the rotor blades bolted onto the hub at a fixed angle. The geome‐ try of the rotor blade profile however has been aerodynamically designed to ensure that the moment the wind speed becomes too high; it creates turbulence on the side of the rotor blade which is not facing the wind. This stall prevents the lifting force of the ro‐ tor blade from acting on the rotor. The rotor blade of a stall controlled wind turbine is twisted slightly along its longitudinal axis. This is partly done in order to ensure that the rotor blade stalls gradually rather than abruptly when the wind speed reaches its critical value. The basic advantage of stall control is that one avoids moving parts in the rotor itself, and a complex control system. On the other hand, stall control repre‐ sents a very complex aerodynamic design problem, and related design challenges in the structural dynamics of the whole wind turbine, e.g. to avoid stall-induced vibrations. Around two thirds of the wind turbines currently being installed in the world are stall controlled machines.

is usually operated using hydraulics or electric stepper motors. Fig. 2 shows the optimal operational conditions of a pitch-controlled 2 MW wind turbine. During normal operation the blades will pitch a fraction of a degree at a time, and the rotor will be turning at the same time. The computer will generally pitch the blades a few degrees every time the wind changes in order to keep the rotor blades at the optimum angle to maximize output

(a) (b)

On the other hand, passive stall controlled wind turbines (Leithed & Conner, 2002; Hoffmann, 2002) have the rotor blades bolted onto the hub at a fixed angle. The geome‐ try of the rotor blade profile however has been aerodynamically designed to ensure that

**Figure 2.** Operational conditions of a pitch-controlled, 2.0 MW wind turbine (Hansen et al., 2005)

power for all wind speeds.

206 Advances in Wind Power

**Figure 1.** Limiting power output using pitch control.

Larger wind turbines (1-MW and up) are being developed with an active stall power control mechanism (Hoffmann, 2002). Technically the active stall machines resemble pitch controlled machines, since they have pitchable blades. In order to get a reasonably large torque at low wind speeds, the machines will usually be programmed to pitch their blades much like a pitch controlled machine at low wind speeds. One of the advantages of active stall is that one can control the power output more accurately than with pas‐ sive stall, so as to avoid overshooting the rated power of the machine at the beginning of a gust of wind. Another advantage is that the machine can be run almost exactly at rat‐ ed power at all high wind speeds. A normal passive stall controlled wind turbine will usually have a drop in the electrical power output for higher wind speeds, as the rotor blades go into deeper stall. As with pitch control it is largely an economic question whether it is worthwhile to pay for the added complexity of the machine, when the blade pitch mechanism is added. One of the most cost-effective solutions in reducing the produced vibrations and avoiding pitch-control failures on wind turbines (see Fig.3) is to separate the natural frequencies of the blade structure from the critical exciting pitching frequencies (Bindner et al., 1997). This would avoid resonance where large amplitudes of torsional vibration could severely damage the whole structure. The frequency-placement technique (Pritchard & Adelman, 1990; Maalawi, 2007; Maalawi & Badr, 2010) is based on minimizing an objective function constructed from a weighted sum of the squares of the differences between each important frequency and its desired (target) value. Approxi‐ mate values of the target frequencies are usually chosen to be within close ranges; some‐ times called frequency-windows; of those corresponding to a reference baseline design, which are adjusted to be far away from the critical exciting frequencies. Direct maximiza‐ tion of the system natural frequencies (Shin et al., 1988; Maalawi & EL-Chazly, 2002) is also favorable for increasing the overall stiffness-to-mass ratio level of the blade structure being excited. This may further other design objectives such as higher stability and fati‐ gue life and lower cost and noise levels. (Maalawi & Negm 2002) considered the optimal frequency design of a wind turbine blade in flapping motion. They used an exact power series solution to determine the exact mode shapes and the aeroelastic stability bounda‐ ries, where conspicuous design trends were given for optimum blade configurations. Both primal and dual optimization problems were thoroughly examined.

**Figure 3.** Typical blade failure of a three-bladed, 2 MW wind turbine

The scope of this chapter is not just to apply optimization techniques and find an opti‐ mum solution for the problem under study. The main aim, however, is to first; perform the necessary exact dynamical analysis of a pitch-regulated wind turbine blade by solving the exact governing differential equation using analytical Bessel's functions. Secondly, the behavior of the pitching fundamental frequency augmented with the mass equality con‐ straint will be investigated in detail to see how it changes with the selected design varia‐ bles. The associated optimization problem is formulated by considering two forms of the objective function. The first one is represented by a direct maximization of the fundamen‐ tal frequency, while the second considers minimization of the square of the difference be‐ tween the fundamental frequency and its target or desired value. In both strategies, an equality constraint is imposed on the total structural mass in order not to violate other economic and performance requirements. Design variables encompass the tapering ratio, blade chord and skin thickness distributions, which are expressed in dimensionless form, making the formulation valid for a variety of blade configurations. The torsional stiffness simulating the flexibility of the inboard panel near the rotor hub is also included in the whole set of design variables. Case studies include the locked and unlocked conditions of the pitching mechanism, in which the functional behavior of the frequency has been thor‐ oughly examined. The developed exact mathematical model guarantees full separation of the frequency from the undesired range which resonates with the pitching frequencies. In fact, the mathematical procedure implemented, combined with exact Bessel's function sol‐ utions, can be beneficial tool, against which the efficiency of approximate methods, such as finite elements, may be judged. Finally, it is demonstrated that global optimality can be achieved from the proposed model and an accurate method for the exact placement of the system natural frequencies has been deduced.
