Wind-Resistant Capacity Modeling for Electric Transmission Line Towers Using Kriging Surrogates and Its Application to Structural Fragility

Featured Application: By applying the wind-resistant capacity model of transmission towers proposed in the article, it is capable to generate structural fragility curves for the tower, in which geo-spatial information of transmission line and wind (i.e., the line direction, the line span, and the wind direction) is considered simultaneously. Using the generated structural fragility model, the wind fragility assessment on transmission towers and line can be accomplished. Abstract: Wind loading on a transmission tower structure is jointly inﬂuenced by the wind ﬁeld, structural parameters, and the geo-spatial conﬁguration of the transmission line. Considering the multi-parametric effect, this paper aims at developing a limit capacity model for transmission towers under strong winds. To this end, the limit capacity of the tower is expressed via two equivalent means: one is the limit wind speed as a function of the wind angle of attack and the span of transmission line; the other is a limit capacity surface with three fundamental wind load components as the principal axes. An adaptive kriging surrogate modeling is constructed to approximate the function/surface with structural uncertainties considered. The performance of the surrogate model is improved by adding support points and then evaluated by the overall accuracy validation and local error check. A numerical example demonstrating the feasibility of the surrogate modeling for the limit capacity of the transmission tower under winds is presented. Finally, a fragility assessment concerning a practical transmission line and towers subjected to typhoons is accomplished using the established limit capacity model of the tower.


Introduction
Vulnerability/reliability analysis on power systems under extreme weather is widely of concern for ensuring the power supply in modern societies. Most research focuses on the overall performance of the power supply [1,2], the resilience of the power network, and the enhancement of the power system [3,4]. However, these studies mainly concentrate on electrical issues and the structural problems are usually expressed in terms of an oversimplified model and sometimes even with only one parameter. The overhead transmission line-tower system (a subsystem of the power network) is sensitive to winds, especially extreme ones (e.g., typhoon [5,6] and downburst [7]). The structural safety of transmission line-tower systems under extreme winds ought to be carefully concerned due to the limited mitigation strategies and time-consuming restoration. Transmission towers, as the support structure of transmission line, are the primary component to assure the safety and function of the system. Therefore, studies on the wind-resistant performance and reliability/fragility

Tower Capacity
The steel lattice tower considered in the paper comprises a series of ste which can be classified into main members, diagonal members, and auxiliar ( Figure 1). The main members and diagonal members are the primary force-be bers. The main members determine the global stability of the tower and t members provide the main members with constraints. There are two types members: the diaphragm and secondary bracing, which improve the overall s tower by offering redundant constraints.
where λ is the strength factor and R is the nominal strength, γ0 is an impor taken to be 1.0, γGi is the dead load factor taken to be 1.0 for tower gravity an equivalent gravity transferred from transmission wires (including the cond ground wires), SGik is the effect of nominal dead loads on tower members, ε is coefficient of variable loads taken to be 1.0, γQj is the variable load factor tak and SQjk is the effect of nominal variable loads on tower members. As the supporting structure, the total design load of transmission towers sified into two parts: one is the vertical load, which contains the tower grav equivalent gravity transferred from transmission wires; the other is the hori which contains the wind load acting on towers directly and the equivalen transferred from transmission wires. Wind loads are the only variable load c this paper and all the tower members are in the elastic state under the action o design wind speed.

Wind Loading
Wind loads on structures is generally calculated by: where ρair is the density of air, taken to be 1.235 kg/m 3 , V is the wind speed, C coefficient, and A is the effective area of the wind loading. Here, we set the w as the 10-min averaged wind speed U at 10 m above the ground. The drag co should jointly take into account the vertical variance of wind speed, the combin of the wind fluctuation and structural vibration, and the shape factor of struc sidering the effect of the wind angle of attack φ (i.e., the intersection angle b wind direction and the direction transverse to the transmission line), the w The design function of the steel members is given by DL/T 5154-2012 [17]: where λ is the strength factor and R is the nominal strength, γ 0 is an importance factor taken to be 1.0, γ Gi is the dead load factor taken to be 1.0 for tower gravity and 1.2 for the equivalent gravity transferred from transmission wires (including the conductors and ground wires), S Gik is the effect of nominal dead loads on tower members, ε is the weight coefficient of variable loads taken to be 1.0, γ Qj is the variable load factor taken to be 1.4, and S Qjk is the effect of nominal variable loads on tower members. As the supporting structure, the total design load of transmission towers can be classified into two parts: one is the vertical load, which contains the tower gravity and the equivalent gravity transferred from transmission wires; the other is the horizontal load, which contains the wind load acting on towers directly and the equivalent wind load transferred from transmission wires. Wind loads are the only variable load concerned in this paper and all the tower members are in the elastic state under the action of maximum design wind speed.

Wind Loading
Wind loads on structures is generally calculated by: where ρ air is the density of air, taken to be 1.235 kg/m 3 , V is the wind speed, C D is the drag coefficient, and A is the effective area of the wind loading. Here, we set the wind speed V as the 10-min averaged wind speed U at 10 m above the ground. The drag coefficient C D should jointly take into account the vertical variance of wind speed, the combination effect of the wind fluctuation and structural vibration, and the shape factor of structures. Considering the effect of the wind angle of attack ϕ (i.e., the intersection angle between the wind direction and the direction transverse to the transmission line), the wind load directly acting on transmission towers is given by [20,21]: ρ air U 2 · ∑ i µ i z β i z u i s,t A i s,t cos 2 φ + u i s,l A i s,l sin 2 φ · 1 + 0.2 sin 2 (2φ) (3) where i is the section number of the tower, µ z is a combined wind factor accounting for the altitude and terrain effects, β z is a gust response factor accounting for the wind fluctuation and structural vibration effects, and A s,t and A s,l and µ s,t and µ s,l are the projected areas and shape factors of the transverse and longitudinal faces of the tower, respectively. The term [1 + 0.2sin 2 (2φ)] is an amplification factor that related to the wind angle of attack. The wind load H T is commonly decomposed into the transverse and longitudinal directions of the transmission line: ρ air U 2 · ∑ i µ i z β i z u i s,t A i s,t cos 2 φ + u i s,l A i s,l sin 2 φ · 1 + 0.2 sin 2 (2φ) · cos φ (4) ρ air U 2 · ∑ i µ i z β i z u i s,t A i s,t cos 2 φ + u i s,l A i s,l sin 2 φ · 1 + 0.2 sin 2 (2φ) · sin φ (5) where H t and H l are the wind load components transverse and longitudinal to the transmission line, respectively. The wind load transferred form transmission wires to the tower structure is given by [20,21]: where j is the number of transmission wires, β z,w is the gust response factor of wires, µ s,w is the shape factor of wires, d is the outer diameter of the wires, L h is the horizontal span of the transmission line, and α is a span factor accounting for the span-wise non-uniform distribution of wind speed and determined by the wind speed U. The calculations for the above interaction parameters between the wind and tower, and between the wind and transmission wires, are given in the Table A1 in Appendix A.
The wind loading to a transmission tower is composed of three parts: H t , H l , and H w . For a given tower and transmission line, with the terrain type known, there are three independent variables that affect the wind loading to the tower structure, i.e., the wind speed U, the wind angle of attack ϕ, and the horizontal span of transmission line L h .

Limit Capacity
Tower structures exhibit nonlinear characteristics when the deformation grows to some extent [14,15]. The capacity curve relating the displacement at the tower top to the shear force at the tower bottom is commonly used to describe the wind-resistant performance of the tower, by which the linear and nonlinear stages can be readily identified [16,22]. The point of transition between the two stages is usually taken as the damage threshold of the tower because the nonlinear behavior is unexpected for the tower structure from the perspective of design and safety. However, a capacity curve can only show the performance of the tower under a single loading case. In reality, the wind loading to a tower is determined by the tower itself as well as transmission wires; it is jointly related to the wind intensity and the wind direction and it is influenced both by the wind field and by the spatial configuration of the transmission line. Hence, this paper uses the capacity surface instead of the capacity curves to describe the wind-resistance capacity of the tower.

Capacity Surface
According to Equations (4)-(6), the wind load on a transmission tower can be decomposed into three components, the transverse wind load acting on the tower (i.e., H t ), the longitudinal wind load acting on the tower (i.e., H l ) and the wind load transferred from transmission wires (i.e., H w ). Taking the three components as the principal axes, the limit capacity of the tower under winds can be described by the surface shown in Figure 2a. Correspondingly, the limit capacity of the tower can be expressed by a function in the form of Equation (7). Connecting the origin to the capacity point on the surface, we get the vector H C , whose direction and magnitude represent one load combination/ratio of three components and the load-carrying capacity of the tower under the combination, respectively. It is noteworthy that the three wind load components are not fully independent and they are influenced by some mutual parameters. In the real case, the capacity surface is usually not a complete surface, as Figure 2a shows, but a partial region. If the information of the tower structure and the line is given, there are three independent parameters determining the wind loading to the tower, i.e., the wind speed U, the wind angle of attack ϕ (here, we ignore the correlation between the wind speed and wind direction), and the horizontal span of the line L h . Therefore, the limit capacity of the tower can also be described by the capacity surface with the three independent parameters as the principal axes ( Figure 2b) and the corresponding limit capacity function is written as Equation (8). One point on the surface means the limit wind speed that the tower can sustain under the certain wind angle of attack and horizontal span. Here, the two capacity surfaces are named surface θ H , which is expressed by three wind load components, and the surface θ L , which is expressed by three independent external parameters. It is noteworthy that as the principal axes of the surface θ L are independent of each other, the region for the surface is explicit and determined by the ranges of U, ϕ, and L h .
According to Equations (4)-(6), the wind load on a transmission tower can be decom posed into three components, the transverse wind load acting on the tower (i.e., Ht), th longitudinal wind load acting on the tower (i.e., Hl) and the wind load transferred from transmission wires (i.e., Hw). Taking the three components as the principal axes, the lim capacity of the tower under winds can be described by the surface shown in Figure 2a Correspondingly, the limit capacity of the tower can be expressed by a function in th form of Equation (7). Connecting the origin to the capacity point on the surface, we ge the vector HC, whose direction and magnitude represent one load combination/ratio o three components and the load-carrying capacity of the tower under the combination, re spectively. It is noteworthy that the three wind load components are not fully independ ent and they are influenced by some mutual parameters. In the real case, the capacit surface is usually not a complete surface, as Figure 2a shows, but a partial region. If th information of the tower structure and the line is given, there are three independent pa rameters determining the wind loading to the tower, i.e., the wind speed U, the wind an gle of attack φ (here, we ignore the correlation between the wind speed and wind direc tion), and the horizontal span of the line Lh. Therefore, the limit capacity of the tower ca also be described by the capacity surface with the three independent parameters as th principal axes (Figure 2b) and the corresponding limit capacity function is written a Equation (8). One point on the surface means the limit wind speed that the tower ca sustain under the certain wind angle of attack and horizontal span. Here, the two capacit surfaces are named surface θH, which is expressed by three wind load components, an the surface θL, which is expressed by three independent external parameters. It is note worthy that as the principal axes of the surface θL are independent of each other, the re gion for the surface is explicit and determined by the ranges of U, φ, and Lh.
Here, we take a practical transmission line and the typical transmission tower in th line as the example, and generate the capacity surfaces for the typical transmission towe by numerical simulations. There are 109 towers in the example line with total length o 31.12 km. The typical transmission tower named tower ZY, as the suspension tower [19 accounts for more than 80% of the towers in the line. Table 1 shows the design informatio of tower ZY, transmission wires, and the line.

Example
Here, we take a practical transmission line and the typical transmission tower in the line as the example, and generate the capacity surfaces for the typical transmission tower by numerical simulations. There are 109 towers in the example line with total length of 31.12 km. The typical transmission tower named tower ZY, as the suspension tower [19], accounts for more than 80% of the towers in the line. Table 1 shows the design information of tower ZY, transmission wires, and the line.  Figure 3a gives the configuration of the typical transmission tower ZY with four cross arms and the tower body is divided into seven sections. Figure 3b,c present a sketch of the wind loading on the towers of a transmission line.     For the limit capacity surface θL of tower ZY, the change range of horizontal span Lh is set to 100-400 m according to Table 1 and the wind angle of attack φ keeps changing in the range of 0°-90°. For a certain Lh and φ, the capacity curve of tower ZY can be achieved by the explicit algorithm-based non-linear static analysis [19] on the finite element model of the tower structure with the wind speed U increasing. The wind loads at each increment is calculated by Equations (4)- (6). The interaction parameters involved in wind loads are determined by the rules shown in Table A1 in Appendix A. Then, the transition point of the simulated capacity curve directs to one point of surface θL as well as one point of surface θH. Simulating capacity curves in the ranges of Lh and φ, we get a group of points for surface θL, shown in Figure 4a. Based on the simulation results of surface θL, the capacity surface θH can be generated depending on the wind loading calculated from Equations (4)-(6) (see Figure 4b,c). For the limit capacity surface θ L of tower ZY, the change range of horizontal span L h is set to 100-400 m according to Table 1 and the wind angle of attack ϕ keeps changing in the range of 0 • -90 • . For a certain L h and ϕ, the capacity curve of tower ZY can be achieved by the explicit algorithm-based non-linear static analysis [19] on the finite element model of the tower structure with the wind speed U increasing. The wind loads at each increment is calculated by Equations (4)- (6). The interaction parameters involved in wind loads are determined by the rules shown in Table A1 in Appendix A. Then, the transition point of the simulated capacity curve directs to one point of surface θ L as well as one point of surface θ H . Simulating capacity curves in the ranges of L h and ϕ, we get a group of points for surface θ L , shown in Figure 4a. Based on the simulation results of surface θ L , the capacity surface θ H can be generated depending on the wind loading calculated from Equations (4)-(6) (see Figure 4b,c).

Discussion
The surface θL, shown in Figure 4a, can be fitted effectively by a cubic polynomial: From Equation (9), we see that the limit of the wind speed that the tower structure can sustain has a second-power relation with the horizontal span and a third-power relation with the wind angle of attack. Compared with the horizontal span, the wind angle of attack has more intensive influence in the limit capacity of the tower under winds.
From Figure 4b,c, the region of surface θH is a gradual-changing partial band. The tendency of surface θH approaches to a linear plane, but the non-linear tendency becomes obvious with the proportion of the wind load transferred from transmission wires (i.e., Surface θL and surface θH show the limit of the wind-resistant capacity of tower from the perspectives of the wind speed and wind load, respectively. Accordingly, the limit state function of the transmission tower under winds can be written as: The limit state function, as the primary function for fragility analysis, consists of a capacity function and a demand function. Surface θL expresses a capacity function with the limit wind speed Ulim dependent on the wind angle of attack and horizontal span of transmission line. Surface θH gives a capacity function in terms of three wind load components. The two capacity surfaces are high-dimensional especially when the structural parameters (presented by the vector ) involve uncertainties. Concerning the structural fragility of towers in a transmission line subjected to winds, the following study focuses on solving the capacity function that expressed by surface θL (as well as by surface θH) using surrogate models instead of high-fidelity models (e.g., finite element simulations).

Discussion
The surface θ L , shown in Figure 4a, can be fitted effectively by a cubic polynomial: From Equation (9), we see that the limit of the wind speed that the tower structure can sustain has a second-power relation with the horizontal span and a third-power relation with the wind angle of attack. Compared with the horizontal span, the wind angle of attack has more intensive influence in the limit capacity of the tower under winds.
From Figure 4b,c, the region of surface θ H is a gradual-changing partial band. The tendency of surface θ H approaches to a linear plane, but the non-linear tendency becomes obvious with the proportion of the wind load transferred from transmission wires (i.e., Surface θ L and surface θ H show the limit of the wind-resistant capacity of tower from the perspectives of the wind speed and wind load, respectively. Accordingly, the limit state function of the transmission tower under winds can be written as: The limit state function, as the primary function for fragility analysis, consists of a capacity function and a demand function. Surface θ L expresses a capacity function with the limit wind speed U lim dependent on the wind angle of attack and horizontal span of transmission line. Surface θ H gives a capacity function in terms of three wind load components. The two capacity surfaces are high-dimensional especially when the structural parameters (presented by the vector θ S ) involve uncertainties. Concerning the structural fragility of towers in a transmission line subjected to winds, the following study focuses on solving the capacity function that expressed by surface θ L (as well as by surface θ H ) using surrogate models instead of high-fidelity models (e.g., finite element simulations).

Kriging-Based Adaptive Surrogate Modeling for Limit Capacity of the Tower
We use ψ to represent the capacity function; then, ψ can be taken as the function of limit wind speed and written as Equation (12) according to the surface θ L : The wind angle of attack and horizontal span are taken as the external variables, and partial structural parameters are taken as the uncertainties. Thus, the capacity function ψ is a multi-dimensional probabilistic problem. In order to clarify the capacity function, numerous loading cases and simulations are required. Instead, we adopt the kriging method [23] in the study and establish an adaptive model to approximate ψ in a more efficient way. Compared with the classical methods (e.g., Taylor-expansion-based methods), kriging is more flexible when it is applied to the surrogate modeling, which allows it to construct an empirical surrogate error measure that can be propagated to the final quantity of interest.

Kriging Method
The kriging method assumes that the target function relating the input (vector x) to the output (vector y) is a sample path of a Gaussian process whose mean and auto-covariance functions are determined by observation points X = [x 1 . . . x p ] T and the corresponding observation results Y = [y 1 . . . y p ] T (p is the number of observations) [24]. The observation result Y is usually solved by high-fidelity simulations or experiments, and can be viewed as the true solutions for the input observation point set X. The observation set Y(X) is named as the design of experiment (DoE) and the set [X Y] is also called the experiment set. The kriging method constructs a Gaussian process based on the DoE, and then predicts the output y corresponding to an arbitrary input x. The Gaussian process for the approximation of a target function can be written as: where f(x) is a functional set used for regression, β is the regression coefficient vector, and z(x) is a zero-mean stationary Gaussian process with the variance σ 2 y and correlation function R. A kriging model involves three steps [25,26]: (i) generating the observation points X = [x 1 . . . x p ] T and results Y = [y 1 . . . y p ] T ; (ii) selecting the kriging functions, i.e., the regression functional set f(x) and the correlation function R; (iii) producing the kriging prediction model. Here, the Latin-hypercube sampling are used to generate the initial observation points X, and the observation results Y are solved based on finite element simulations. The regression functional set f(x) and correlation function R are selected as: where x i (i = 1, 2, . . . , n) is the element of the input vector x, n is the number of input variables that contained in the vector x, j, k = 1, . . . , p, and θ i is the correlation function parameter which is optimized using the principle of maximum likelihood estimation (MLE) [24]. Then, the mean and error variance of the kriging prediction in terms of a new input vector x are, respectively, expressed as [23]: indicates the correlation between the observation input points x and the new input x.
The above kriging prediction is fundamentally an interpolation process whose fitting quality depends on the experiment set (i.e., DoE). The interpolation process considers the tendency of the target function and the random correlations of points in the region of concern simultaneously. The kriging model yields a mean value for the predicted output as well as a prediction error variance, which is helpful to measure the accuracy of surrogate modeling.

An Adaptive Modeling Framework
Based on the kriging method, we propose a kriging-based adaptive surrogate modeling framework ( Figure 5) in which three main parts are involved, i.e., the kriging model, support points, and model validation. In the framework, the performance of the kriging surrogate is improved continuously by adding support points in loops. The common approach for selecting the support point is sequential, that is, just one support point (i.e., the point at which the prediction error is largest) is chosen in each loop [24]. To be more efficient, we try to identify a larger sample set of support points in each loop. In this strategy, the points at which the performance of the kriging surrogate is lower have a larger probability of being selected as support points. Concerning the overall accuracy of the surrogate model, an additional issue should be considered, namely, that the selected support points may be close to one another, which can lead to a redundant effect on surrogate performance improvements. To be more effective, the optimization of the support point set is necessary in each loop. At the end of each loop, the accuracy of the kriging model and the validation of the surrogate should be examined, based on which we can decide if we continue the next loop or not. Overall, in this framework, three issues are of major concern: (i) selecting support points (i.e., to generate the initial support point set in an efficient way); (ii) optimizing the support point set (i.e., to remove the inclusion of support points that are close to one another); (iii) validating the surrogate model (i.e., determining whether to terminate the loop or not). (16) and (17), the kriging prediction gives both the mean valueŷ(x) and the error variance σ 2 y . The error variance reveals the accuracy of the predicted mean value. Here, we take the prediction error variance as the measure for the selection of support points. That is, more support points should be selected from the regions where the error variance is larger. This means that the larger the prediction error variance is, the more likely the point is to be selected as the support point. Therefore, the original sample set of support points should satisfy a probability distribution whose density monotonically increases with the prediction error variance. To this end, the reject sampling technique is used [27] to generate the initial set of support points. Then, the clustering method is used to eliminate the inclusion of support points that close to one another. Considering that the k-means method is easy to trap in local optimum [28], the bisecting k-means method [29] combined with the iterative self-organizing data technique algorithm (ISODATA) is used to implement the optimization. The steps for generating the support point set in each loop are listed as:

From Equations
(i). Generate the original sample set of support points X * (with the scale of n * ) using the reject sampling technique.
(ii). Divide the original support point set into K clusters using the bisecting k-means method.
(iii). Set the lower limit of distance between two nearby clusters as d min ; set the lower limit of number of clusters as N s,min ; set the upper limit of standard deviation of each cluster as Sigma; set the largest number of iterations as N iter ; denote the number of clusters as N c (the initial value is K).
(iv). Calculate the scale of each cluster and the standard deviation of the samples in each cluster.
(v). Find the clusters whose standard deviations are larger than Sigma and whose scales are two times larger than n s,min ; split the cluster into two new clusters with their two center points located at ±Sigma to the original center point.
(vi). Calculate the distance between any two of the clusters; find the two clusters between which the distance is less than d min and merge them afterwards (only one merging is allowable for each cluster).
(vii). If N c < K/2, turn to step (iv); if N c > 2K, turn to step (vi); if the number of iterations reaches N iter or the iteration approaches to convergence, move to step (viii).
(viii) Gather the center points of the clusters and take these center points as the optimized support point set X**.
Taking the support point set X** as the input, we get the corresponding output Y** through high-fidelity simulations or experiments. Then, add Z** = [X** Y**] to the experiment set and the updated DoE can then be used to construct a new kriging model with the higher performance. From Equations (16) and (17), the kriging prediction gives both the mean value ˆ( yx and the error variance 2  y . The error variance reveals the accuracy of the predicted mea value. Here, we take the prediction error variance as the measure for the selection of sup port points. That is, more support points should be selected from the regions where th error variance is larger. This means that the larger the prediction error variance is, th more likely the point is to be selected as the support point. Therefore, the original sampl set of support points should satisfy a probability distribution whose density monoton There are two tasks in the model validation. One is to judge if the loop terminates or not. The other is to check the effectiveness of the kriging surrogate. To this end, three parts are included in the validation. First, considering the overall accuracy of the kriging model, a cross-validation method [30] is adopted. The cross validation divides the experiment set (i.e., DoE) into two groups. One group is used to construct the kriging model and the other is used as the validation set. Based on the error measures, the accuracy of model is evaluated depending on the two groups. Instead of establishing an independent validation set, the cross-validation technique is more efficient and feasible. Here, we utilize the random sampling for the cross validation: (i) sample n points from the experiment set (scale N) that are taken as the validation set, and the residual (N-n) points are used to construct the kriging model; (ii) predict the output for validation points using the constructed kriging model and measure the prediction errors; (iii) repeat the sampling, prediction, and error measuring; (iv) calculate the statistical error. To assure the reliability of cross validation, the sample scale n should be small enough and the number of repeats should be large enough. Therefore, we set n = 1 with N being the number of repeats. The coefficient of determination (RD, Equation (18)), accounting for the overall accuracy of the prediction, and the mean error percent (ME, Equation (19)), accounting for the overall deviation of the prediction, are used as the statistical error measures. The larger DR and smaller AME implies better overall performance of the kriging model.
where i means the i-th element of output vector y, y represents the observation result (i.e., true value),ŷ is the prediction of y (i.e., estimated value), x is the sampled validation point, the upper index p means the p-th repeat, and n is the total number of repeats. Second, considering the local error of the kriging model, we take the support point set generated in the loop as another validation set for local validation. The roots of the mean square error (RMSE, Equation (20)) and mean absolute error (MAE, Equation (21)) are used to measure the local performance of the kriging model: where m is the number of validation points (i.e., the scale of the support point set in each loop). The scale of the support point set is generally small and insufficient to check the overall accuracy of the kriging model. Most points in the support point set have a larger prediction error variance, which is helpful to validate the local performance of the kriging surrogate. Third, the kriging method, as a mathematical approach, does not reveal the nature of the engineering or physical issues. Therefore, it is necessary to validate the performance of the surrogate in engineering practice. In this study, the surrogate for the wind-resistant capacity surface of transmission towers is of concern. In the surrogate, a Gaussian process is used to approximate the points on the capacity surface, and these capacity points are characterized by probabilities. In other words, instead of a single surface, the surrogate model aims at a group of surfaces that characterized by probabilities. The details of the issue-based validation are demonstrated in the following example study.

Example Study
In the example study, the capacity function ψ in Equation (12) is the target function we need to approximate using kriging models. Function ψ is an implicit function dependent on the external variables (i.e., the wind angle of attack ϕ and horizontal span of line L h ) and structural parameters of transmission line towers (i.e., θ S ). The structural vector θ S contains the information of material, geometry, the initial state, and boundary conditions. Uncertainties in these structural parameters may lead to the uncertainty in the windresistant 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 ϕ 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 ϕ is 0 • -90 • and that of L h is set to 150-400 m. In the kriging modeling, the wind angle of attack ϕ, the horizontal span L h , and structural parameters θ S are the input of the model, i.e., x = [ϕ L h θ S ]. The structural vector θ S contains six random parameters, i.e., θ S = [f y,Q345 , f y,Q235 , E s , ν, t, l]. The logarithm of limit wind speed lnU lim was taken as the output of the model, i.e., y = ln(U 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 Gaussian'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.
for the support point set in each loop ranged from 50 to 200.
At the validation stage, we first calculated the overall accuracy measure Equations (18) and (19)) of the kriging model. Then, we took the optimized su set as the local validation set and calculate the local error measures (shown a (20) and (21)). Figure 6 gives the trends of the accuracy/error measures with th ber increasing. As can be seen in Figure 6a, the coefficient of determination (RD) grew value of 0.98 and then kept stable. The mean error percent (ME) continuousl with the scale of DoE increasing. When the value of ME reached 2%, we term adaptive modeling. Then, the modeling process ended when the number of to construct the kriging model was close to 3500. Concerning the local perform kriging surrogate, Figure 6c,d show the trends of the local error measures MAE. In this process, RMSE fluctuated around the value of 3.5 and the fluctu ened with the scale of DoE increasing. MAE shows a decreasing trend with the approximating to 2, and the fluctuation of MAE became less obvious in the lat Figure 7 further gives the comparison result between the kriging predict observation result in the last loop of the adaptive modeling. Most of the pred were located in the range of ±5% error, and the prediction yielded satisfac when the limit wind speed was less than 100 m/s. As can be seen in Figure 6a, the coefficient of determination (RD) grew quickly to a value of 0.98 and then kept stable. The mean error percent (ME) continuously decreased with the scale of DoE increasing. When the value of ME reached 2%, we terminated the adaptive modeling. Then, the modeling process ended when the number of points used to construct the kriging model was close to 3500. Concerning the local performance of the kriging surrogate, Figure 6c  The kriging model is expected to predict an assembly of capacity surfaces for transmission towers that subjected to winds, and these capacity surfaces are characterized by probabilities. The prediction on limit wind speed aims at a probability distribution rather than a single value. Hence, in the region of interest, the kriging prediction is the prediction of an assembly of probability distributions. At the issued-based validation step, we first generated 10,000 samples randomly, depending on probabilistic features of structural parameters (shown in Table 2). The 10,000 samples were taken as the set of prediction points, which was characterized by randomness. Then, we took 42 points of the capacity surface uniformly based on the ranges of the wind angle of attack and horizontal span, and the 42 surface points were taken as the set of validation points. In each loop, the constructed kriging model was used to predict the limit wind speed at each validation point of the capacity surface and 10,000 samples of the tower were involved in each validation point. Then, the mean value and standard deviation of the limit wind speed at each validation point were calculated, which further yielded the change ratio of the mean value and standard deviation (Equations (22) and (23)) with the loop number increasing: The kriging model is expected to predict an assembly of capacity surfaces for transmission towers that subjected to winds, and these capacity surfaces are characterized by probabilities. The prediction on limit wind speed aims at a probability distribution rather than a single value. Hence, in the region of interest, the kriging prediction is the prediction of an assembly of probability distributions. At the issued-based validation step, we first generated 10,000 samples randomly, depending on probabilistic features of structural parameters (shown in Table 2). The 10,000 samples were taken as the set of prediction points, which was characterized by randomness. Then, we took 42 points of the capacity surface uniformly based on the ranges of the wind angle of attack and horizontal span, and the 42 surface points were taken as the set of validation points. In each loop, the constructed kriging model was used to predict the limit wind speed at each validation point of the capacity surface and 10,000 samples of the tower were involved in each validation point. Then, the mean value and standard deviation of the limit wind speed at each validation point were calculated, which further yielded the change ratio of the mean value and standard deviation (Equations (22) and (23)) with the loop number increasing:  (Figure 8b,d).
According to the above validation results (Figures 6-8), the presented adaptiv kriging modeling for capacity surface θL of transmission towers is feasible and effective Figure 9a shows the surrogate result for capacity surface θL of tower ZY using the pro posed adaptive modeling framework. Correspondingly, the approximated capacity sur face θH is shown in Figure 9b according to the wind loading relations (i.e., Equations (4) (6)).  (Figure 8b,d).
According to the above validation results (Figures 6-8), the presented adaptive kriging modeling for capacity surface θ L of transmission towers is feasible and effective. Figure 9a shows the surrogate result for capacity surface θ L of tower ZY using the proposed adaptive modeling framework. Correspondingly, the approximated capacity surface θ H is shown in Figure 9b according to the wind loading relations (i.e., Equations (4)- (6)).
According to the above validation results (Figures 6-8), the presented adaptive kriging modeling for capacity surface θL of transmission towers is feasible and effective. Figure 9a shows the surrogate result for capacity surface θL of tower ZY using the proposed adaptive modeling framework. Correspondingly, the approximated capacity surface θH is shown in Figure 9b according to the wind loading relations (i.e., Equations (4)-(6)). In Figure 9a, most of the support points are in the two regions enclosed by green curves, which indicates that the limit capacity of the tower is more sensitive to the structural uncertainty in these two regions. Comparing Figure 9b with Figure 4b, it is shown that the trends of capacity surface θH are similar. With the load component Hw increasing, the surfaces exhibit a more significant nonlinear trend.

Application to the Structural Fragility Assessment on a Transmission Line
The above study constructed a capacity function for the limit state of transmission towers subjected to winds, and solved the capacity function by the proposed kriging- In Figure 9a, most of the support points are in the two regions enclosed by green curves, which indicates that the limit capacity of the tower is more sensitive to the structural uncertainty in these two regions. Comparing Figure 9b with Figure 4b, it is shown that the trends of capacity surface θ H are similar. With the load component H w increasing, the surfaces exhibit a more significant nonlinear trend.

Application to the Structural Fragility Assessment on a Transmission Line
The above study constructed a capacity function for the limit state of transmission towers subjected to winds, and solved the capacity function by the proposed krigingbased surrogate modeling. Considering the capacity surface θ L , the limit state function is written as: where ϕ and L h are the normal variables; θ S is a structural vector containing stochastic structural parameters. Based on the above limit state function, the fragility curves can be simulated through the workflow shown in Figure 10 (Θ S is the sample assembly of the stochastic structural vector θ S ; N s is the scale of Monte Carlo sampling; N(g < 0) means that the number of elements that less than zero in the limit state vector g).  (24) where φ and Lh are the normal variables; θS is a structural vector containing stochastic structural parameters. Based on the above limit state function, the fragility curves can be simulated through the workflow shown in Figure 10 (ΘS is the sample assembly of the stochastic structural vector θS; Ns is the scale of Monte Carlo sampling; N(g < 0) means that the number of elements that less than zero in the limit state vector g). Depending on the established kriging surrogate model of capacity surface θL, we can generate multiple groups of fragility curves for the transmission tower with its structural uncertainties considered and the external variables (i.e., the wind environment and line span) involved. Figure 11 further gives the fragility curves of tower ZY achieved by the Depending on the established kriging surrogate model of capacity surface θ L , we can generate multiple groups of fragility curves for the transmission tower with its structural uncertainties considered and the external variables (i.e., the wind environment and line span) involved. Figure 11 further gives the fragility curves of tower ZY achieved by the above flow, in which different wind angles of attack and three horizontal spans are of concern.
Appl. Sci. 2021, 11 Taking the practical transmission line (Table 1) as the example and considering the normal segment of the line (i.e., the blue section in Figure 12), we conducted the structural fragility assessment on the towers and the line segment under typhoons by implementing the workflow of the fragility calculation, shown in Figure 10. In the normal segment of the line, there were 86 transmission towers in total, including 71 suspension tangent towers, 14 tension towers, and 1 terminate tower. Suspension tangent towers, as the wind-sensitive structure, are concerned in the fragility assessment. Tension towers and terminate towers generally have a higher safety level under winds and are not considered here. Taking the wind and the line information into account simultaneously, the annual probability of failure of each tower can be estimated by convolving the structural fragility with the joint model of the annual extreme wind speed of typhoons (v) and the corresponding wind direction (β) as: (25) in which Pf,T is the annual probability of failure of the tower structure and FT represents the cumulative probability of failure of the tower with the horizontal span and line direction taken as Lh and α, respectively. This cumulative probability is a function of the annual Taking the practical transmission line (Table 1) as the example and considering the normal segment of the line (i.e., the blue section in Figure 12), we conducted the structural fragility assessment on the towers and the line segment under typhoons by implementing the workflow of the fragility calculation, shown in Figure 10.  (Table 1) as the example and c normal segment of the line (i.e., the blue section in Figure 12), we conducted fragility assessment on the towers and the line segment under typhoons by the workflow of the fragility calculation, shown in Figure 10. In the normal segment of the line, there were 86 transmission towers i ing 71 suspension tangent towers, 14 tension towers, and 1 terminate tow tangent towers, as the wind-sensitive structure, are concerned in the fragili Tension towers and terminate towers generally have a higher safety leve and are not considered here. Taking the wind and the line information int ultaneously, the annual probability of failure of each tower can be estimate In the normal segment of the line, there were 86 transmission towers in total, including 71 suspension tangent towers, 14 tension towers, and 1 terminate tower. Suspension tangent towers, as the wind-sensitive structure, are concerned in the fragility assessment. Tension towers and terminate towers generally have a higher safety level under winds and are not considered here. Taking the wind and the line information into account simultaneously, the annual probability of failure of each tower can be estimated by convolving the structural fragility with the joint model of the annual extreme wind speed of typhoons (v) and the corresponding wind direction (β) as: (25) in which P f,T is the annual probability of failure of the tower structure and F T represents the cumulative probability of failure of the tower with the horizontal span and line direction taken as L h and α, respectively. This cumulative probability is a function of the annual extreme wind speed of typhoons v and the corresponding wind angle of attack ϕ (ϕ is calculated according to the line direction α and the direction of annual extreme wind β), which can be calculated using the workflow shown in Figure 10. f v,θ is the joint probability density of the annual extreme wind speed (v) and its wind direction (β), in which the wind speed v and direction β range from 0-v max and 0 • -360 • , respectively. Here, we adopt the joint probability model of v and β that are generated based on the 60-year typhoon on-site data ( Figure 13) [34][35][36]. This joint model is established by combining the Monte Carlo simulation with the two-dimensional non-parametric kernel density estimation [37]: ppl. Sci. 2021, 11, x FOR PEER REVIEW which can be calculated using the workflow shown in Figure 10. fv,θ is the density of the annual extreme wind speed (v) and its wind direction (β), i speed v and direction β range from 0-vmax and 0°-360°, respectively. He joint probability model of v and β that are generated based on the 60-yea data ( Figure 13) [34][35][36]. This joint model is established by combining simulation with the two-dimensional non-parametric kernel density esti Then, we obtain the annual probability of failure of each suspens normal segment of the transmission line using Equation (25), and the fai of tension towers and terminate towers are taken to be zero. Clusterin towers based on the line direction and span, we obtain three clusters of their centers corresponding to different line directions (Figure 14a). In notes the number of the towers located in the corresponding cluster. F shows the failure class of each tower in three clusters. The failure class annual probability of failure calculated from the Equation (25) and five fied according to the calculation result. In Figure 14a,b, the red cluster majority of the towers and their failure probabilities scatter in the five cla the blue cluster and the black cluster, we find that most of the towers h probabilities. Then, we obtain the annual probability of failure of each suspension tower in the normal segment of the transmission line using Equation (25), and the failure probabilities of tension towers and terminate towers are taken to be zero. Clustering the suspension towers based on the line direction and span, we obtain three clusters of the towers, with their centers corresponding to different line directions (Figure 14a). In Figure 14a, n denotes the number of the towers located in the corresponding cluster. Figure 14b further shows the failure class of each tower in three clusters. The failure class is defined by the annual probability of failure calculated from the Equation (25) and five classes are classified according to the calculation result. In Figure 14a,b, the red cluster accounts for the majority of the towers and their failure probabilities scatter in the five classes. Considering the blue cluster and the black cluster, we find that most of the towers have larger failure probabilities.
Defining one strain section of a line as starting from one tension tower and ending with the adjacent tension/terminate tower, there are 14 strain sections in the normal segment of the transmission line. To evaluate each strain section, Figure 15a,b first give the basic information (including the length and the number of the towers) of each strain section, and we have the essential cognition that the longer the strain section is, the more towers are included in the section. Figure 15c,d further show the statistical result of the annual failure probability of the towers in each strain section. notes the number of the towers located in the corresponding cluster. Figure 14b further shows the failure class of each tower in three clusters. The failure class is defined by the annual probability of failure calculated from the Equation (25) and five classes are classified according to the calculation result. In Figure 14a,b, the red cluster accounts for the majority of the towers and their failure probabilities scatter in the five classes. Considering the blue cluster and the black cluster, we find that most of the towers have larger failure probabilities.  Comparing the two strain sections with the largest numbers of towers, i.e., section No. 5 and No. 9, we found that although the No. 5 section owns the largest number of towers, its annual failure risk is quite low in terms of both the average and deviation. No. 9 section has the highest failure risk from the perspective of average, while No. 12 and 14 sections have the larger deviations. Furthermore, we considered the failure of stain sections using the bound theory (shown as Equations (26) and (27) (27) where Pf,L is the annual probability of failure of one strain section or line segment, Pf,T (Tj) represents the annual probability of failure of the j-th tower, j = 1, 2, …, m, and m is the total number of towers in the strain section. Using the Equations (26) and (27), we obtained the failure bounds of each strain section, as shown in Figure 16.  Comparing the two strain sections with the largest numbers of towers, i.e., section No. 5 and No. 9, we found that although the No. 5 section owns the largest number of towers, its annual failure risk is quite low in terms of both the average and deviation. No. 9 section has the highest failure risk from the perspective of average, while No. 12 and 14 sections have the larger deviations. Furthermore, we considered the failure of stain sections using the bound theory (shown as Equations (26) and (27)): Upper bound P f , where P f,L is the annual probability of failure of one strain section or line segment, P f,T (T j ) represents the annual probability of failure of the j-th tower, j = 1, 2, . . . , m, and m is the total number of towers in the strain section. Using the Equations (26) and (27), we obtained the failure bounds of each strain section, as shown in Figure 16.  (27) where Pf,L is the annual probability of failure of one strain section or line segment, Pf,T (Tj) represents the annual probability of failure of the j-th tower, j = 1, 2, …, m, and m is the total number of towers in the strain section. Using the Equations (26) and (27), we obtained the failure bounds of each strain section, as shown in Figure 16. In Figure 16, the No. 9 section has the highest lower bound as well as the highest upper bound among all the strain sections, which indicates that the most dangerous tower is in the No. 9 strain section and most of the towers in this section have a higher risk. Moreover, the lower bounds of No. 10-No. 14 sections are also considerably high, which means that there are also dangerous towers located in these sections.
Based on the above assessment, we conclude the following (from the perspective of annual failure risk): In Figure 16, the No. 9 section has the highest lower bound as well as the highest upper bound among all the strain sections, which indicates that the most dangerous tower is in the No. 9 strain section and most of the towers in this section have a higher risk. Moreover, the lower bounds of No. 10-No. 14 sections are also considerably high, which means that there are also dangerous towers located in these sections.
Based on the above assessment, we conclude the following (from the perspective of annual failure risk): (i). The primary attention should be drawn on the No. 9 strain section due to its high risk of failure.
(ii). The No. 5 stain section, owning the largest number of the towers, shows a relatively low failure risk.
(iii). Considering the whole normal segment, the first part of the segment (i.e., No. 1-8 strain sections) exhibits a lower failure risk than the latter part of the segment (i.e., No. 9-10 strain sections), which indicates that the latter part of the normal segment is of greater significance for assuring the annual safety of the line that subjected to typhoons.

Conclusions
The fragility/vulnerability assessment of transmission tower structures subjected to winds is a multi-parametric issue. To this end, the paper proposes the surface expression (i.e., the surfaces θ L and θ H ) for the wind-resistant capacity of the tower, based on the wind loading rule. The limit capacity of a certain tower is described by a function related to three external parameters, as well as by a function related to three wind load components, after which the corresponding limit sate functions of tower structures can be obtained. In order to acquire the capacity surfaces, the explicit algorithm-based non-linear static simulation is applied to the finite element model of the tower and a series of wind loading cases are involved. The simulation results yield the fitted/approximated capacity surfaces of an example tower by taking the transition point of capacity curves as the damage threshold.
Applying the capacity surface θ L to the limit state function of the tower structure under winds, the capacity function term is constructed. Considering the structural fragility, the capacity function is high-dimensional and characterized by the stochasticity. For solving the high-dimensional probabilistic relation, a kriging-based adaptive modeling framework is established to approximate the capacity function. The constructed kriging model is expected to predict the points on the capacity surface θ L , and these capacity points are characterized by stochasticity. Therefore, rather than a single surface, the kriging surrogate aims to approximate an assembly of surfaces.
In the proposed modeling framework, the performance of the kriging surrogate is improved by adding support points. Three steps (i.e., the generation of support points, the optimization of the support point set, and the model validation) are involved to assure the effectiveness of the adaptive modeling. In the example study, the validation results (including the overall accuracy validation, the local error check and the practical issue-based validation) demonstrate the feasibility of the framework with all of the accuracy/error measures approaching to convergence/stationarity after a sufficient number of loops.
By applying the established kriging surrogate of the capacity surface, a group of structural fragility curves for transmission towers are obtained, in which geo-spatial information of the line and wind (i.e., the line direction, the line span and the wind direction) is considered simultaneously. Depending on the fragility model, it is capable to do the fragility assessment on transmission towers and line under winds. Considering the safety of tower structures under typhoons, the annual failure probability of each tower in a practical transmission line is calculated by combining the capacity-surface-based fragility model with the joint probability model of the annual extreme wind speed and wind direction of typhoon. By the failure risk assessment on each strain section of the line, some suggestions are made for evaluating and improving the safety of the transmission line.
Taking the line direction and span, the wind direction, and intensity into account, this paper explores an effective approach to establish and solve the capacity function for transmission towers under winds, and then assess the structural fragility of tower structures and transmission line. To do the fragility analysis more precisely, much more factors are needed to be involved (e.g., the wind stochasticity, the variance of terrain, and the service time of the transmission line). Correspondingly, a higher dimensional model for the wind-resistant capacity of the tower is required.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. Interaction between the Wind and the Transmission Line Towers
Four types of interaction parameters are concerned, the combined wind factor (µ z and µ z,w ), accounting for the effect of the altitude and terrain on the wind field, the gust response factor (β z and β z,w ), accounting for the load effect due to the turbulence of the wind and the dynamic amplification of the tower and wires, the shape factor (µ s and µ s,w ), accounting for the effect of structural members (e.g., the member size, shape and solidity), and the span factor (α), accounting for the reduction effect due to the non-uniform distribution of the wind speed along the line. In this work, these interaction parameters are calculated according to the design codes/specifications (Table A1), referring to the IEC Standard 60826 [21], the ASCE-74 [38], the Chinese load code (GB 50009-2012) [39], and the Chinese code for design of 110 kV-750 kV overhead transmission line (GB 50545-2010) [40]. To determine the interaction parameters, four types of data are necessary: the wind intensity data (e.g., the 10-min averaged wind speed at 10 m over the ground); structural data, including the geometric and mechanical properties of the tower and wires; spatial data, including the direction and span of the line; and terrain data, which affects the altitude-dependent wind distribution and wind turbulence. Table A1. Wind-structure interaction parameters.

Parameters Tower Structure Transmission Wires Remarks
Combined wind factor [21] µ z = (z/z 0 ) α 0 µ z,w = (z/z 0 ) α 0 The power law is adopted here. z is the height of concern, z 0 is the reference height taken to be 10 m, and α 0 is the roughness exponent.
Gust response factor [37] Ls I z is the turbulence intensity of winds, B (including B t and B w ) is the background component of the structural response, L s is the integral scale of turbulence of winds, z is the height of the tower section, and S is the span of line.

Appendix B. Simulation Results of the Limit Capacity of Transmission Towers under Winds
Considering the ranges of the wind angle of attack (0 • -90 • ) and horizontal span (100-400 m), we conducted a series of simulations on the finite element model of tower ZY and the capacity curves of tower ZY under different wind loading cases generated by the explicit algorithm-based non-linear static analysis ( Figure A1). The capacity curve was drawn by relating the tower top displacement to the wind speed and to the bottom shear force, respectively (see Figure A1a,b). From Figure A1, the mechanical behavior of the tower structure can be divided into two stages: one is the linear/elastic stage and the other is the nonlinear/plastic stage. The transition of the two stages is defined as the damage threshold of the structure. The wind speed magnitude that corresponds to the transition point was taken as the limit wind speed that the tower can sustain. Figure A1 further notes the proportions of three wind load components (i.e., H t , H l , and H w ) at the limit/transition point of the capacity curves. From Figure A1, we have the plastic stage gets shorter with the proportion of the sum of H t and H l increasing. Figure A2 further shows the failure modes of tower ZY under different combinations/ratios of three wind load components.

Appendix B. Simulation Results of the Limit Capacity of Transmission Towers under Winds
Considering the ranges of the wind angle of attack (0°-90°) and horizontal span (100-400 m), we conducted a series of simulations on the finite element model of tower ZY and the capacity curves of tower ZY under different wind loading cases generated by the explicit algorithm-based non-linear static analysis ( Figure A1). The capacity curve was drawn by relating the tower top displacement to the wind speed and to the bottom shear force, respectively (see Figure A1a,b). From Figure A1, the mechanical behavior of the tower structure can be divided into two stages: one is the linear/elastic stage and the other is the nonlinear/plastic stage. The transition of the two stages is defined as the damage threshold of the structure. The wind speed magnitude that corresponds to the transition point was taken as the limit wind speed that the tower can sustain. Figure A1 further notes the proportions of three wind load components (i.e., Ht, Hl, and Hw) at the limit/transition point of the capacity curves. From Figure A1, we have the plastic stage gets shorter with the proportion of the sum of Ht and Hl increasing. Figure A2 further shows the failure modes of tower ZY under different combinations/ratios of three wind load components.