## 3 Kriging-Based Adaptive Surrogate Modeling for Limit Capacity of the Tower
### Example Study  
In the example study, the capacity function \(\psi\) in Equation (12) is the target function we need to approximate using kriging models. Function \(\psi\) is an implicit function dependent on the external variables (i.e., the wind angle of attack \(\varphi\) and horizontal span of line \(L_{h}\)) and structural parameters of transmission line towers (i.e., \(\theta_{S}\)). The structural vector contains the information of material, geometry, the initial state, and boundary conditions. Uncertainties in these structural parameters may lead to the uncertainty in the wind-resistant capacity of the tower. In general, the boundary condition of tower legs was taken as rigid, and the initial state and damping ratio of the tower structure had neglectable influences on the limit capacity of the tower [31]. Hence, we did not take into account the uncertainties of the boundary condition, initial state, and structural damping ratio in this case. Thus, the material and geometry are the two primary uncertainties concerned in the study. Taking the practical transmission tower ZY in Figure 3 as an example, the probabilistic features of structural parameters are listed in Table 2.  
Two types of steel with different yield strengths (\(f_{y,Q345}\) and \(f_{y,Q235}\)) were used in the tower structure. The two external variables (i.e., the wind angle of attack \(\varphi\) and horizontal span of line \(L_{h}\) in Equation (12)) were taken as normal variables with their values distributing uniformly in their ranges. According to the transmission line information shown in Table 1, the range of \(\varphi\) is \(0^{\circ}\)-\(90^{\circ}\) and that of \(L_{h}\) is set to 150-400 m. In the kriging modeling, the wind angle of attack \(\varphi\), the horizontal span \(L_{h}\), and structural parameters \(\Theta_{S}\) are the input of the model, i.e., \(\mathbf{x}=[\varphi\ L_{h}\ \Theta_{S}]\). The structural vector \(\Theta_{S}\) contains six random parameters, i.e., \(\Theta_{S}=[f_{y,Q345},f_{y,Q235},E_{s},\,\
u,\,t,\,l]\). The logarithm of limit wind speed \(\ln U_{\text{lim}}\) was taken as the output of the model, i.e., \(\mathbf{y}=\ln(U_{\text{lim}})\). Following the above framework, we first generated the initial DoE [\(X_{0}\ Y_{0}\)] using the Latin hypercube sampling technique. The scale of the initial experiment set is taken to be \(n_{0}=100\). Based on the initial DoE, we constructed the initial kriging model by adopting the first-order polynomial regression function set (shown as Equation (14)) and [PERSON]'s correlation function (shown as Equation (15)). Then, we moved to the second stage, i.e., generating the support point set. The scale for the initial set of support points was taken to be \(n^{*}=1000\). During the optimization process, the expected scale for the support point set was taken to be \(100\) (i.e., the initial number of clusters \(K=100\)). The final scale for the support point set in each loop ranged from 50 to 200. At the validation stage, we first calculated the overall accuracy measures (shown as Equations (18) and (19)) of the kriging model. Then, we took the optimized support point set as the local validation set and calculate the local error measures (shown as Equations (20) and (21)). Figure 6 gives the trends of the accuracy/error measures with the loop number increasing.