Figure 1 shows a 3D CAD model and an FE (Finite Element) model for a small vertical axis wind turbine for seismic qualification analysis. In this FE modelling, shell element was used for the wing turbine’s blades and poles, while solid element was used for the gearbox. This FE model comprised 20,001 nodal points, 6753 shell elements, and 5156 solid elements. The material for the turbine blades is ABS 1020 (Young’s modulus = 3 GPa, Poisson’s ratio = 0.35, density = 1070 kg/m³, yield strength = 55 MPa, and allowable stress = 38 MPa). The material for the pole and gearbox is steel (young’s modulus = 200 GPa, Poisson’s ratio = 0.3, density = 7850 kg/m³, yield strength = 250 MPa, and allowable stress = 175 MPa).

The FE model in Figure 1 shows a multi-DOF (Degree Of Freedom) structural vibration system that bears the seismic load. If a seismic load is converted to an equivalent inertia force imposed on the structure by the ground acceleration, the equation of motion for the n-DOF undamped structural vibration system bearing the seismic load is represented as follows.

Here, , and denote mass-, damping- and stiffness-coefficient ma- trices of order n, respectively. is the DOF vector. is the seismic acceleration vector. is the excitation force acting direction vector.

1) Seismic response spectrum analysis

The seismic response spectrum analysis was performed as per the RRS of SSE 5% (Required Response Spectrum of Safe Shutdown Earthquake with 5% damping) prescribed in KBC 2009 [6] and ASCE 7-10 [7]-[9]. According to the seismic design standards, the peak ground acceleration required to generate the seismic acceleration spectrum input is. When the soil profile type is soft rock ground, the seismic coefficients (or the load maintenance coefficients) are and. When the seismic risk coefficient (return period = 100 years), the peak

ground acceleration is. This corresponds to the seismic design category D. Figure 2(a) shows the RRS in horizontal and vertical directions.

The seismic spectrum response of a wind turbine structure represented in the Equation (1) is obtained with the spectrum-mode analysis [1] [10] [11], where the maximum

seismic responses per single mode with high participation factors are determined, and then combined using the SRSS (Square Root of the Sum of the Squares) method to find the total response. The number of modes considered in the seismic response analysis is determined in the order of participation factors with the total modal mass accounting for more than 90% of the total mass of the structure. The ANSYS [12] for structural analysis was used to derive the maximum seismic stress () and the maximum seismic deformation () from the results of the modal analysis and response spectrum analysis. The acceleration response gain (G) is calculated from the maximum response acceleration and the maximum input acceleration.

2) Static analysis to dead weight and wind load

The static analysis of the FE model in Figure 1 was carried out to solve the maximum stress () and maximum deformation () of the wind turbine caused by the dead weight. Likewise, the FEM static analysis of the FE wind turbine model under the wind load was performed to calculate the maximum stress () and maximum deformation (). The wind load imposed on the section of the wind turbine structure was applied to three segments defined by the height above the ground as in Figure 2(b).

3) Total stress and total deformation

The maximum stress and the maximum deformation determined by the foregoing seismic response analysis and static structural analysis were combined using the SRSS(Square Root of the Sum of the Squares) to yield the total stress () and total deformation () as follows.

For seismic qualification of the wind turbine, total stresses of the wind turbine components are compared with their allowable stresses.

The design optimization problem of the small wind turbine structure is to determine the best design variables that meet the seismic qualification while minimizing the weight of the wind turbine structure. As in Figure 3, the thickness of turbine blade () and that of the pole (, ,) were defined as the design variables. Different thickness settings were applied to the 3 segments of the pole defined by height. Table 1 outlines the ranges of design variables. The design variables are represented as discrete values, considering the standard dimensions of commercial products.

The design optimization problem of the wind turbine meeting the seismic qualification and minimizing the total weight of the structure was formulated as follows.

Find; Design variable set.

To minimize; Object function. (3)

Subject to; Constraints. (4)

where,; Acceleration response gain. (5)

Here, is the object function. is the weight of the structure. is the SRSS (Square Root of Sum of Squares) value of the acceleration response gain in the horizontal and vertical directions. and are the scaling constants respectively. is the weighting factor., and are the acceleration response gains in, and directions, respectively. and are the maximum total stress () and maximum total deformation (), respectively., and are the allowable acceleration responses in horizontal (and) and vertical () directions, respectively. and are the allowable total stress and total deformation, respectively. is the peak ground seismic input acceleration. is the peak response acceleration of the structure by seismic load. The foregoing values were found with the seismic analysis using the commercial program ANSYS APDL [12]. In this design optimization, the weighting factor was 0.5. The allowable acceleration response gain was set at. The allowable stress and the allowable displacement were set at 37 MPa and 30 mm, respectively.

The GA (Genetic Algorithm) based optimization search program [5] was used for the design optimization. The FEM structure analysis was used to calculate the values defined as the object functions and constraints of the design problem. To consider both the object functions and constraints in GA, the fitness function and penalty function were defined as follows.

Fitness function: (6)

Penalty function: (7)

where, is number of violation, and

is amount of violation with their weighting factors of, , , , ,.

Table 2 shows the input parameters for the GA.

To test the robustness of the design optimization based on the genetic algorithm, each of 4 different initial designs underwent the design optimization, and then the results were comparatively analyzed. The 4 initial designs were as follows., , ,.

Figure 4 shows comparison of the convergence history of fitness and object functions in the design optimization search for each of 4 initial designs for checking robustness of the GA used.

Table 3 summarizes the design optimization results of 4 different initial designs. Here, each object function shows consistent results and the maximum deviation is 2%. Also, the design variables are consistent and the maximum deviation falls within 12.5%. Thus, this design optimization was proved to be robust enough.