**6. Results and discussions**

Both objectives are subject to the constraints:

214 Advances in Wind Power

Mass constraint: *M*

which is also too sensitive to the selected target frequency, is very slow.

**5. Optimization techniques**

Side constraints:X¯

^ <sup>=</sup>*<sup>∫</sup>* 0

where *XL* and *XU* are the lower and upper limiting values imposed on the design variables vec‐ tor *X* in order not to obtain unrealistic odd-shaped designs in the final optimum solutions. Ap‐ proximate values of the target frequencies are usually chosen to be within close ranges; sometimes called frequency – windows; of those corresponding to an initial baseline design, which are adjusted to be far away from the critical exciting pitching frequencies. Several com‐ puter program packages are available now for solving the above design optimization model, which can be coded to interact with structural and eigenvalue analyses software. Extensive computer implementation of the models described by Eqs. (18-21) have revealed the fact that maximization of the fundamental frequency is a much better design criterion. If it happened that the maximum frequency violates frequency windows, which was found to be a rare situation, another value of the frequency can be chosen near the global optima, and the frequency equa‐ tions (15-17) can be solved for any one of the unknown design variables instead. Considering the frequency-placement criterion, it was found that convergence towards the optimum solution,

The above optimization problem described by Eqs.(18-21) may be thought of as a search in an 3-dimensional space for a point corresponding to the minimum value of the objective function and such that it lie within the region bounded by the subspaces representing the constraint functions. Iterative techniques are usually used for solving such optimization problems in which a series of directed design changes (moves) are made between successive

points in the design space. The new design *Xi+1* is obtained from the old one *Xi*

where the vector *Si* defines the direction of the move and the scalar quantity *α<sup>i</sup>*

techniques are classified according to the way of selecting the search direction *Si*

there are two distinct formulations (Vanderplaats, 1999): the constrained formulation and the unconstrained formulation. In the former, the constraints are considered as a limiting

length such that *Xi+1* does not violate the imposed constraints, *Gj*

Such that *F*(*Xi*+1) < *F* (*Xi*

*X* ¯ *<sup>i</sup>*+1 = *X* ¯ *<sup>i</sup>* + *ai S* ¯

1 *C* ^ *h*

<sup>L</sup> ≤X ¯ ≤X ¯

^ *d x*^ =1 (20)

<sup>u</sup> (21)

as follows:

gives the step

. In general,

*(X)*. Several optimization

*<sup>i</sup>* (22)

) (23)

The developed mathematical model has been implemented for the proper placement of the fre‐ quencies of typical blade structure in free pitching motion. Optimum solutions are obtained by invoking the *MATLAB* routine "*fmincon"* which interacts with the eigenvalue calculation rou‐ tines. The target frequencies, at which the pitching frequencies needed to be close to, depend on the specific configuration and operating conditions of the wind machine. Various cases of study are examined including, blades with both locked and unlocked pitching conditions. The main features and trends in each case are presented and discussed in the following sections.

#### **6.1. Unlocked pitching mechanism condition**

Considering first the case of active pitching, figure 5 shows the variation of the first three resonant frequencies with the tapering ratio. It is seen that the frequencies decrease with in‐ creasing taper. Blades having complete triangular planforms shall have the maximum fre‐ quencies which is favorable from structural design point of view. However, such configurations violate the requirement of having an efficient aerodynamic surface produc‐ ing the needed mechanical power. Now, in order to place any frequency at its desired value *ω* ^ *i* \* , *i=1,2,3,* the first step is to calculate the dimensionless frequency *ω* ^ *i* , *i=1,2,3,* for known properties of the blade material and airfoil section, and then obtain the corresponding value of the taper ratio from the curves presented in figure 5. The next step is to choose appropri‐ ate value for the dimensionless thickness *h* ^ *<sup>o</sup>* at the blade root and find the corresponding chord length *C* ^ *<sup>o</sup>* at the determined taper ratio (see figure 6), which should satisfy the equali‐ ty mass constraint expressed by Eq. (20). It is to be noticed here that the dimensionless wall thickness *h* ^ *<sup>o</sup>* at root shall be constrained to be greater than a preassigned lower bound, which can either be determined from the minimum available sheet thicknesses or from con‐ siderations of wall instability that might happen by local buckling.

**Figure 5.** Normalized frequencies of free pitching motion (Unlocked blade)

**Figure 6.** Optimized tapered blades with constant mass (*C* ^ *<sup>o</sup>*- Level curves, *M* ^ =1)

#### **6.2. Condition of locked pitching mechanism**

which can either be determined from the minimum available sheet thicknesses or from con‐

siderations of wall instability that might happen by local buckling.

216 Advances in Wind Power

**Figure 5.** Normalized frequencies of free pitching motion (Unlocked blade)

**Figure 6.** Optimized tapered blades with constant mass (*C*

^

*<sup>o</sup>*- Level curves, *M*

^ =1)

Extensive computer solutions for the frequency equation (17) have indicated the existence of the frequency level curves in the selected design space. Figures 7, 8 and 9 depicts, respec‐ tively, the developed frequency charts for the design cases of locked pitching mechanism with *K* ^ *<sup>s</sup>= 10, 100 and 1000* representing flexible, semi-rigid and rigid blade root. Any other specific case can be easily obtained by following and applying the same procedures outlined before in sections 3 and 4. It is seen from the figures that the frequency function is well be‐ haved and continuous in the selected design space *(h*^ *<sup>o</sup>*, *C* ^ *<sup>o</sup>).* Actually, these charts represent the fundamental pitching frequency augmented with the equality mass constraint. There‐ fore, they reveal very clearly how one can place the frequency at its target value without the penalty of increasing the total mass of the main blade structure. Such charts also can be uti‐ lized if one is seeking to maximize the frequency under equality mass constraint. Maximiza‐ tion of the natural frequencies has the benefit of improving the overall stiffness/mass ratio of the vibrating structure (Maalawi and Negm, 2002).

**Figure 7.** Augmented frequency-mass contours (ω ^ <sup>1</sup>) for a blade with flexible blade root: *K* ^ *<sup>s</sup>* = 10 (*M* ^ =1)

As seen, the developed contours depicted in figure 7 has a banana- shaped profile bounded by two curved lines; the one from above represents a triangular blade (*Δ=0*) and the other lower one represents a rectangular blade geometry (*Δ=1*). It is not allowed to penetrate these two borderlines in order not to violate the imposed mass equality constraints. Each point in‐ side the feasible domain in the middle corresponds to different mass and stiffness distribu‐ tions along the blade span, but the total structural mass is preserved at a constant value equals to that of the rectangular reference blade. The lower and upper empty regions repre‐ sent, respectively, infeasible blade designs with structural mass less or greater than that of the baseline design. The global optimal design is too close to the design point *{C*^ *<sup>o</sup>, h* ^ *o Δ }={1.202, 2.011, 0.207}* with *ω* ^ 1,max*=2.6472.* If it happened that such global optima violates frequency windows, another value of the frequency can be taken near the optimum point, and an inverse approach is utilized by solving the frequency equation for any one of the un‐ known design variables instead.

**Figure 8.** Level curves of ω ^ <sup>1</sup> for a semi-rigid blade root; *K* ^ *<sup>s</sup>=100, M* ^ *=1*.

Other cases for semi-rigid and rigid blade root are shown in figures 8 and 9. It is seen that the contour lines become more flatten and parallel to the two borderlines as the hub stiffness increases. The calculated maximum values of the fundamental pitching fre‐ quency are 4.2161 at the design point {1.5, 2, 0} for *K* ^ *<sup>s</sup>*=100 and 4.4825 at the same de‐ sign point for *K* ^ *<sup>s</sup>*=1000. Such optimal blade designs having triangular planform are favorable from structural point of view. However, such configurations violate the re‐ quirement of having an efficient aerodynamic surface producing the needed mechanical power. In all, it becomes now possible to choose the desired maximum frequency, which is far away from the excitation frequencies, and obtain the corresponding opti‐ mum variables directly from the developed frequency charts. Actually, the charts repre‐ sent the fundamental frequency function augmented with the imposed mass equality constraint so that the problem may be treated as if it were an unconstrained optimiza‐ tion problem. Table 2 summarizes the final optimum solutions showing that good blade patterns ought to have the lowest possible tapering ratio. This means that the optimum design point is always very close to the lower limiting value imposed on the blade ta‐ pering ratio, i.e. 0.25.

**Figure 9.** Level curves of ω ^ <sup>1</sup> for a rigid blade root; *K* ^ *<sup>s</sup>=1000, M* ^ *=1*.

lower one represents a rectangular blade geometry (*Δ=1*). It is not allowed to penetrate these two borderlines in order not to violate the imposed mass equality constraints. Each point in‐ side the feasible domain in the middle corresponds to different mass and stiffness distribu‐ tions along the blade span, but the total structural mass is preserved at a constant value equals to that of the rectangular reference blade. The lower and upper empty regions repre‐ sent, respectively, infeasible blade designs with structural mass less or greater than that of the baseline design. The global optimal design is too close to the design point *{C*

frequency windows, another value of the frequency can be taken near the optimum point, and an inverse approach is utilized by solving the frequency equation for any one of the un‐

1,max*=2.6472.* If it happened that such global optima violates

*Δ }={1.202, 2.011, 0.207}* with *ω*

218 Advances in Wind Power

known design variables instead.

**Figure 8.** Level curves of ω

sign point for *K*

^

^

<sup>1</sup> for a semi-rigid blade root; *K*

quency are 4.2161 at the design point {1.5, 2, 0} for *K*

^ *<sup>s</sup>=100, M* ^ *=1*.

Other cases for semi-rigid and rigid blade root are shown in figures 8 and 9. It is seen that the contour lines become more flatten and parallel to the two borderlines as the hub stiffness increases. The calculated maximum values of the fundamental pitching fre‐

^

*<sup>s</sup>*=1000. Such optimal blade designs having triangular planform are

*<sup>s</sup>*=100 and 4.4825 at the same de‐

^

^ *<sup>o</sup>, h* ^ *o*

> Figure 10 depicts the variation of the maximum fundamental frequency with the stiffness at blade root. It is seen that the frequency decreases sharply with increasing the stiffness coeffi‐ cient up to a value of 10, after which it increases in the interval between *K* ^ *<sup>s</sup>*=10 and 100 and then remain approximately constant at the principal values π/2 and π. The average attained optimization gain reached a value of about 86.95 % as measured from the reference design.

**Figure 10.** Variation of the constrained maximum fundamental frequency ω ^ 1,max with blade root stiffness *K* ^ *<sup>s</sup>*, (*M* ^ =1)


**Table 2.** Constrained optimal solutions for different blade root flexibility.

## **6.3. Model validation: Actual operation case**

As a part of the ministry of electricity plans for wind energy programs in Egypt, a study is currently performed concerning the design and manufacture of an upwind, two-bladed, pitch-controlled, horizontal-axis wind turbine producing *100 KW* electrical power output. The wind turbine will be erected for testing and experimental investigation in the western coast of the Gulf of Suez near Hurghada, which has the most favorable wind condition with average wind speeds between *7-12 m/s*. The followings are the relevant values of the refer‐ ence blade design parameters:


*ω* ^ *<sup>r</sup>= π* for unlocked pitch

*= π/2* for locked pitch

**Figure 10.** Variation of the constrained maximum fundamental frequency ω

*(C ^ <sup>o</sup>,h ^*

Reference rectangular

ω ^

<sup>1</sup> ω

(Unlocked pitch) 3.1416 (π) 4.4871 (1.4520, 1.5861, 0.2514)

0.01 2.9235 4.3891 (1.7289, 1.3122, 0.2522)

0.1 2.5987 4.1871 (1.5973, 1.4221, 0.2541)

1 1.9546 3.6542 (1.3794, 1.6583, 0.2527)

10 1.2322 2.6467 (1.1651, 1.9546, 0.2504)

100 1.59811 3.2741 (1.2533, 1.7982, 0.2532)

1000 1.5731 3.2435 (1.4523, 1.5822, 0.2531)

^ =1

*^ <sup>o</sup>* ≤ 2.0

(Perfect rigidity) 1.5708 (π/2) 3.2389 (1.4763, 1.5428, 0.2529)

Equality mass constraint : *M*

Inequality side constraints: 0.5 ≤ *C*

0.25 ≤ *h ^ <sup>o</sup>* ≤ 2.0 0.25 ≤ Δ ≤ 0.75

**Table 2.** Constrained optimal solutions for different blade root flexibility.

Stiffness coefficient (*K* ^ *s*)

220 Advances in Wind Power

0.0

∞

^

*<sup>o</sup>* , Δ )=(1, 1, 1) Optimized tapered blade

1,max *(C*

^

1,max with blade root stiffness *K*

*^ <sup>o</sup>,h ^* ^ *<sup>s</sup>*, (*M* ^ =1)

*<sup>o</sup>* , Δ )*optimum*

∴Dimensional circular frequency *ωr = 58.65 ω* ^ *<sup>r</sup> rad/sec*. (refer to Table 1).

Frequency in *HZ: fr=ωr*/2*π*


The final attained optimal design for the case of active pitch is (see Table 2 and Figure 5):

**•** The first three frequencies are *fi,max= 41.8846, 67.802, 95.548 HZ*, which corresponds to the optimal chord and thickness distributions:

*C( x* ^ *)= 1.452 (1-0.7486 <sup>x</sup>* ^ *) m h( x* ^ *)= 7.931x10-3(1-0.7486 <sup>x</sup>* ^ *) m, 0 ≤ <sup>x</sup>* ^ *≤ 1.*

*Δ=0.2514*.

Other cases with different blade root flexibilities can be obtained using the dimensionless optimal solutions given in Table 2.
