*5.2. The BDCDO of the Horizontal Axis Wind Turbine (HAWT)*

The design optimization problem of the HAWT system [4,15,20] is a complex co-design BDOP involving structural parameters and control variables, which can be simulated and estimated by the Advanced Wind Turbine program blade 27 (AWT27) in the Open FAST project. As shown in Figure 20, the structural parameters **x***<sup>p</sup>* in this paper mainly include the hub radius *Rh*, blade length *Lb*, and tower height *Ht*. The control variable **u** is the generator torque *Gt*. The fore-aft tower-top displacement *ξ*1, the side-to-side tower-top displacement *ξ*2, the fore-aft tower-top velocity *v*1, the side-to-side tower-top velocity *v*2, and rotor speed *ω* are regard as the state variables ξ. The co-design formulation of the HAWT system is expressed as follows:

$$\begin{array}{ll}\min\_{\mathbf{x}\_p, \mathbf{u}(t)} & I = w\_1 m\_s(\mathbf{x}\_p) + w\_2 \int\_0^{t\_f} (\lambda(t) - \lambda\_\*(t))^2 dt \\ \text{s.t.} & \dot{\xi}(t) = f(\xi(t), \mathbf{x}\_p, \mathbf{u}(t), t) \\ & \|\|\boldsymbol{\xi}\_1(t)\|\|\_\infty - \xi\_{1\text{max}} \le 0 \\ & \|\|\boldsymbol{\xi}\_2(t)\|\|\_\infty - \xi\_{2\text{max}} \le 0 \\ & P(V\_\ell) - P\_{\text{errin}} \ge 0 \\ & \mathbf{x}\_p \in [\mathbf{x}\_L, \mathbf{x}\_\ell] \end{array} \tag{26}$$

where *ms*(**x***p*) is the mass of the wind turbine, *tf* <sup>0</sup> (*λ*(*t*) <sup>−</sup> *<sup>λ</sup>*∗(*t*))<sup>2</sup> *dt* is the sum of the deviations of the wind blade tip tangential velocity ratio *λ*(*t*) and the optimal velocity ratio *λ*∗(*t*) over a period of time, *λ* is the ratio of leaf tip tangential velocity *ω* · *Lb* to wind speed *v*, *λ*<sup>∗</sup> can be calculated by the power coefficient function, and *w*<sup>1</sup> and *w*<sup>2</sup> are the weights of the two terms, respectively. Therefore, the optimization objective of the system is to minimize the sum of the mass and the deviations of speed ratios. What is more, the HAWT system needs to satisfy the structural deflection constraints and ξ2(*t*)∞. When the wind speed reaches the rated wind speed *Ve*, the system has to reach the minimum rated power *Pe*min. Since the simulation of the HAWT system involves several disciplines, the RHS function **.** ξ(*t*) = *f*(ξ(*t*),**X***p*, **u**(*t*), *t*) of the system is highly nonlinear, and AWT27 takes several seconds to execute a simulation valuation.

**Figure 20.** Schematic diagram of the HAWT.

The co-design and optimization problem of HAWT is optimized based on the finite difference technique without using the surrogate model, and the standard optimal objective 805.4801, hub radius *Rh* = 1.2000, blade length *Lb* = 13.7330, and tower height *Ht* = 32.3944 are obtained. Qiao et al. [20] adopted the HS-MASRI, EFDC-MASRI, and TEI-MASRI methods to solve this problem, while the SRIRMD method is used to build the surrogate model of a derivative function in the HAWT system. In the SRIRMD-STOR method, the number of initial points *N*<sup>0</sup> = 100, and maximum number of new samples per iteration Δ*<sup>N</sup>* = 10. The optimization outcomes of those different methods are listed in Table 6. Obviously, the original system-based optimization solution method is costly and necessitates extensive simulation evaluations of AWT27. In contrast, the surrogate model-based optimization methods significantly reduce the number of running valuations of AWT27 and decrease the computational costs. In these model-based optimization solutions, the plant parameters converge to the standard solution [1.2000, 13.7330, 32.3944]. Due to the accuracy of the surrogate models, the objective values obtained by various methods are different. Nevertheless, the errors with the standard solution are within the allowed range. More importantly, it is clear that the SRIRMD-STOR method uses the least number of samples in addressing the co-design problem of the HAWT system while it has the higher solution accuracy, improving the solution efficiency and conserving the computational resources.

**Table 6.** The computational cost, optimal plant parameters, and optimal objective values in the HAWT system.


The wind speed curve input to the HAWT system for a certain time period is displayed in Figure 21. The evolution of the generator torque *Gt*, the fore-aft tower-top displacement *ξ*1, the side-to-side tower-top displacement *ξ*2, and the rotor speed *ω* with this wind speed curve are shown in Figure 22. As can be observed from the figure, in order to optimize the HAWT system, the trend of the control and state variables obtained from the optimization solution is highly consistent with the trend of the wind speed.

**Figure 21.** The input of wind speed.

**Figure 22.** The trends of the state and control variables.

Figure 23 records the convergence processes of the state component trajectory overlap ratio and the state trajectory overlap ratio in the SRIRMD-STOR method, and *α*1, *α*2, *α*3, *α*4, *α*5, and A are the trajectory overlap ratios of the state components *ξ*1, *ξ*2, *v*1, *v*2, *ω*, and state ξ, respectively. According to Figure 23, the state trajectory overlap ratio A converges to 1 with the convergences of all *α<sup>i</sup>* in the 11*th* iteration. *α<sup>i</sup>* ≥ A, which verifies Theorems 1 and 2.

**Figure 23.** The convergence processes of A and all *α<sup>i</sup>* in the HAWT system.
