The discipline of SO has matured mostly within the aerospace industry, where advances in computational resources have opened up new ways to designing innovative solutions leveraging the progress made in CAE and FEA tools. In the wind industry, component optimization and, especially, system automatic optimization are still in their infancy. Several challenges exist, such as the non-linear dynamics of WTGs, the complex and stochastic loading scenarios, the importance of vibrational response and fatigue, and the consequent need for very specialized simulation tools. As far as SO within fatigue FLS-loading scenarios, not much literature support is available; in contrast, the majority of the work has been done within static (or ULS) scenarios, which is more readily achieved, requiring limited knowledge of the system response. In addition, the tight coupling among various components (see Section 10.7.1) demands multidisciplinary design optimization, which, to be accurate, should also be performed in the time domain. Hence, the analysis is quite complex and time-consuming. Frequency domain approaches, however, promise [120] to drastically reduce analysis times. An excellent review of methods and challenges of SO for OWTs is given by Ref. [121]. To try to overcome these challenges, new efforts are underway by turbine OEMs and by research laboratories [52,122–126].

In general, SO may encompass three different sets of problem [127]:

The basic constraint within the SO is that the response of the structure should be acceptable as per the various specifications, ie, it should at least be a feasible design per the applicable standards (eg, those of Section 10.3). Since there can be many feasible designs, it is desirable to choose the optimal one in terms of either minimum cost, minimum weight, maximum performance. or a combination of these. For example, in the case of a tubular tower for OWTs, the SO reduces to a sizing optimization with discrete variables (eg, wall thickness and outer diameter of the tower segments) within both ULS and FLS loading scenarios. The ultimate objective should be that of minimizing the system LCOE.

At the base of SO lies the mathematical formulation of a structural problem, which allows for finding optimal solutions via computer algorithms [128]. The mathematical nature of the problem is that of a non-linear programming problem, where approximations and successive linearizations are utilized to arrive at an iterative solution of the problem. The general SO can be written in mathematical terms as:

Different methods have been developed to solve Eq. [10.49] depending on whether the problem can be defined as either linear, convex, non-linear, non-convex, or with discrete vs. continuous parameters [127,129–132]. Derivatives and Jacobians of the functions f_obj and g_cnt with respect to the array x_d are usually needed to solve Eq. [10.49] (via the so-called gradient based optimization (GBO)). If analytical versions of the derivatives can be attained, the optimization will proceed much faster than the alternative, where numerical derivatives are employed. However, rarely one can confirm the convexity of the structural problem functions; therefore a GBO solution to Eq. [10.49] may not necessarily yield the global optimum, and multiple iterations may be necessary to determine the best solution. To overcome this drawback, genetic algorithms (GAs) can be employed, which are methods suitable for complex optimization problems with either continuous or discrete variables. GAs are robust at locating the global optimum, but at the cost of a large number of function evaluations; therefore they become advantageous when the number of variables is large.

As an example of optimization, a 10-MW turbine tower shell was optimized for mass via a GBO algorithm. The tubular tower was envisioned as mounted on top of a jacket, at a base elevation of approximately 21 m MSL. The steel density was augmented to account for flanges, hardware, secondary steel, and coatings. For sake of simplicity, only two ULS DLCs were considered, and only the tower-base and tower-top cross-sectional properties, and the length (Htwr₂) of a constant cross-section segment (tower waist) were selected as design variables. A more refined model could include all of the shell-segment elevations and associated D_sh and t_s as optimization variables, as well as more load cases and constraints. Here, this example aims to demonstrate the power of an optimization algorithm during preliminary design, and thus first-order approximations are acceptable. Further verifications would demand higher fidelity, and AHSE could then be used to assess fatigue damage to be included as extra g_cnt functions in the SO problem. The main turbine and environmental parameters for this SO example are given in Table 10.8, together with the imposed bounds for manufacturability (eg, minimum and maximum D_b, minimum and maximum DTRs for weldability). The main structural checks performed, besides targeting modal performance in terms of desired f₀ (see Table 10.8), are those described by Eqs. [10.29]–[10.31].

Results of the optimization process can be seen in Fig. 10.39 and Table 10.9. Note that, for this example, DLC 1.6 (a maximum thrust, operational condition) tends to drive the design of the tower shell (see Fig. 10.39(b)). This is in agreement with what was mentioned earlier regarding the overall importance of the RNA loads with the exception of very deep-water sites where FLS may be driven by hydrodynamics. In order to further optimize the shell, one could include more shell stations as design variables, and this effect will be shown later.

As visible from the results in Table 10.9, the tower mass is substantially reduced from an initial guess at the layout. Moreover, the target eigenfrequency is better captured, and the ULS maximum local-buckling utilization reaches unity. This is a good starting point for the more detailed design, and it was achieved in a matter of a few minutes on a personal computer. A verification of the results was also conducted through the ANSYS^® FEA package (FEA) (see Fig. 10.39(c)–(d)).

The design space could be further investigated for solution improvement, or to assess mass penalties associated with changes in design variables, as for instance due to manufacturing or transportation constraints. In order to address these sensitivities, a sweep over some 4480 cases was carried out by varying the design variables in their allowed ranges, and some of the results are shown in Fig. 10.40. The graph shows contours of tower mass, eigenfrequencies, and structural utilization (ie, global buckling utilization ratio (GLUtil) and shell buckling utilization ratio (EUUtil) that tend to drive the design). Because of the discretization of the various design variables in the sweep analysis, the graph represents only an approximate cross-section of the problem hyperspace in proximity of the attained optimal solution (ie, D_t = 3.5 m, rather than 3.65 m; DTR_t = 170, rather than 156; and Htwr₂ = 23.95 m, rather than 24 m). Yet, it can be seen how the obtained layout (denoted by the green cross) is indeed a global optimum, and that moving away from that would imply an increase in mass. For example, assuming a hard limit of 7.5 m on D_b, a feasible solution (one that satisfies the constraints) would likely result in a penalty of 100 t (the red cross in the graph of Fig. 10.40), at least as long as the Htwr₂ parameter is not significantly changed. Similar analyses to that in Fig. 10.40 could very well help in and speed up design decisions.

Other graphs such as that of Fig. 10.41 can be used to further explore the sensitivity of mass and other parameters to the design variables and objectives. In Fig. 10.41(a), the general trend in terms of tower mass and GLUtil is shown together with obtained eigenfrequencies and average DTRs along the tower span. It can be observed how a stiffer tower (higher f₀) demands larger steel mass as expected, but also how the wall thickness can be reduced (DTR increased) to save mass for a given target frequency. The two trends show how the DTR-versus-mass surface is less steep than the f₀-versus-mass one, illustrting how the wall thickness is not as effective at reducing mass as tower-base diameter, for example, as is shown in the analogous graph in Fig. 10.41(b). Fig. 10.41(c) shows a two-dimensional (2D) view of the same calculated solution space, with emphasis on the EUUtil ratio. This graph demonstrates that if the frequency target is pushed above the current 0.25 Hz, the buckling constraint would no longer be the major driver, being replaced by the modal performance requirement. This, of course, assumes no changes in the boundary conditions, as for example in the stiffness of the SbS. It can also be observed that a change of some 12% in calculated f₀ would bring forth a 20% mass penalty.

These results were achieved with just a few design variables, ie, with just two tower segments, and keeping a constant cross-section within the bottom segment. By relaxing the latter condition, and by adding one more intermediate station, and thus allowing for three tapered segments, the tower mass can be further reduced as shown in the last column of Table 10.9. A plot of the obtained tower profile and utilization distribution along the span is given in Fig. 10.42.