Objective Functions

Two conicting tness functions have been considered in order to take account of both the aerodynamic and the structural performances of the blades. Two further formulations of a cost function are also derived from the structural denition of the airfoils.

6.4.1 Aerodynamic Fitness Function

The aerodynamic performance of the blade is calculated through a BEM code, as described in Chapter 4 and more specically in Section 4.4. The code has been validated against the experimental data, reported in [65].

The input of the BEM code is represented by the polar curves of the three considered airfoils:

in order to calculate the CL and CD values for dierent angels of attack, both XFOIL and RFOIL panel codes have been tested. The best agreement with the experimental data has been found using RFOIL: it better predicts the experimental power curve at high wind speed, when stall phenomena occurs, as shown in Figure 6.5. In both the XFOIL and RFOIL settings, the performances of the three dierent airfoils of the blade have been calculated using the mean Reynolds Number Re of the baseline sections of the blade (1.0M, 1.5M and 1.5M respectively). The viscous analysis have been run using a total number of 200 panels along the airfoils and an angle of attack α in the range [0^◦− 40^◦]. The aerodynamic codes calculate the performances of the proles in terms of lift coecient CL and drag coecient CD until the stall occurs. In order to evaluate the performances in the entire range of angles of attack α, the polars have been extended using both the Viterna Method ([87]) and the Flat Plate Theory: the second (using a maximum CD limit of 1.8) has proved to be the most relevant. Figure 6.6 shows the comparison between the extension of the RFOIL polars using the two theories.

6 8 10 12 14 16 18 20 22

Figure 6.5: AOC 15/50 power curve: experimental data versus BEM results using XFOIL and RFOIL polars

After the evaluation of the power curve using the BEM code, the Annual Energy Production (AEP ) can be calculated as specied in Section 4.6, by multiplying the BEM power curve, the Weibull wind speed distribution curve and the total number of hours in a year (8760). Two dierent wind speed distributions, reported in Figures 6.7 and 6.8, have been examined in the present analysis: the adopted Weibull curves use a dierent velocity mean values (7.0 m/s and 7.5 m/s) and the same Rayleigh distri-bution (k = 2). Since the cut-in wind speed of the AOC 15/50 wind turbine is 4.9 m/s the contribute to the power prediction are studied in the range from 5 m/s to 22 m/s (cut-out wind speed).

6.4  Objective Functions

Figure 6.6: Experimental data versus Flat plate theory and Viterna method

0 5 10 15 20 25

Figure 6.7: Weibull curve n.1 of the considered wind speed distribution (Vave = 7.0 m/s)

0 5 10 15 20 25

Figure 6.8: Weibull curve n.2 of the considered wind speed distribution (Vave = 7.5 m/s)

Since the number of turbines that can be installed in an area is inversely proportional to the square of turbine radius, the AEP value is normalized with the turbine rotor radius R. Therefore, the Annual Energy Production density is dened as:

AEP d = AEP

R² [kW h/y/m²] (6.1)

The aerodynamic tness function fAEP, to be minimized, is therefore represented by ratio between the AEP dbas of the baseline conguration and the AEP d of the analysed individual.

f_AEP = AEP d_bas

AEP d (6.2)

6.4.2 Structural Fitness Function

In the previous work on AOC 15/50 of Dal Monte et al. [83], the mechanical behaviour of the wind turbine blade was the main topic of the analysis: a FEM model, using the software ANSYS Mechanical, has been set and validated, comparing the results to the structural test on the AOC 15/50 composite blade carried by SANDIA [66]. Two glass bre reinforced plastic (GFRP), A130 and the DB120, represent the composite material adopted in the blade layout composition for 0^◦ and ±45^◦ layers respectively, as specied in [66]. In the trailing edge zone (Layup n. 9) also balsa wood is used.

For reasons of clarity, the materials properties and the AOC 15/50 composite layout visualization, reported in Section 5.1, are also described in the present Section. The structural properties of the adopted materials are described in Table 6.6. Table 6.7 reports the layup schedule and the thickness of the dierent blade sections in both the span-wise and chord-wise directions. A graphic illustration of the layout zones of the blade is presented in Figure 6.9.

A130 DB120 Balsa Wood

Table 6.6: Structural properties of the materials adopted in the AOC 15/50 blade

6.4  Objective Functions

Component Location Layup schedule Thickness ρρρ_eqeqeq Layup n.

[mm] [mm] [kg/m²] [-]

5563 to 7493 [±45/0/±45/0/+45]S 4.3 7.40 6

Leading edge 1092 to 2311 [±45/02/±45]S 3.9 6.70 7

2311 to 7493 [±45/0/±45]S 2.8 4.75 8

Trailing edge 1092 to 6604 [±45/0/balsa/0/±45] 11.5 3.35 9

6004 to 7493 [±45/0]S 2.0 3.35 10

Spar ange 1092 to 7493 [±45/02/±45]S 3.9 6.70 11

Spar web 1092 to 7061 [±45/02/±45]S 3.9 6.70 12

Table 6.7: Layup schedule of the analysed rotor blade

In the proposed analysis, the layout and the total length of the blade are not being modied, hence the mass of the blade has been evaluated thought a simplied Matlab function. The 12 layup zones of Figure 6.9 are characterized by the value of ρeq, the Equivalent Surface Density in kg/m², as reported in Table 6.7. The ρeqparameter has been calculated by summing the product of the thickness and the layer density of every section of the blade:

ρ_eq=

N

X

i=1

t_i∗ ρ_i [kg/m²] (6.3)

In order to evaluate the total weight of the blade, the equivalent surface densities are later multiplied by the area of the sections (determined by the prole lengths, depending on the chosen control points) and summed. The mass of the considered baseline AOC 15/50 blade is 80.79kg.

The structural tness function fm, to be minimized, is therefore represented by ratio between the mass m of the considered individual and the mass m0 of the baseline blade individual.

fm = m

m₀ (6.4)

Figure 6.9: Exploded drawing of the analysed rotor blade geometry, showing the subdivision in 9 areas spanwise and 8 zones chordwise

6.4.3 Cost Fitness Function

An initial estimation of the cost function is done with the expression given by Giguere et al. [86] and used in Benini et al. [52]. The Cost Of Energy can be expressed as:

COE = (T C + BOS)

AEP ∗ F CR + O&M (6.5)

The turbine cost T C is proportional to the blades weight, assuming that the blades, made in E-glass material, represent the 20% of the total cost of the turbine. The BOS cost (200$/kW ) represents the balance of the station, proportional to the turbine rate power. The considered xed charge rate (F CR) is 11%/y and the costs for Operation and Maintenance O&M are estimated in 0.01$/kW h.

The cost model is based on the assumption that total turbine cost can be reconstructed on the basis of the wind turbine blade alone.

The cost tness function fCOE, to be minimized, is represented by the ratio between the COE of the analysed individual and the COE0 of the baseline conguration, for the Weibull wind speed distribution considered. The COE of the AOC 15/50 baseline conguration results 0.0301 $/kW h.

f_COE = COE

COE0 (6.6)