Seismic Vibration Control and Multi-Objective Optimization of Transmission Tower with Tuned Mass Damper Under Near-Fault Pulse-like Ground Motions

Abstract

Although the wind load is usually adopted as the governing lateral load in the design of transmission towers, many tall transmission towers may be damaged or even collapse in high seismic intensity areas, especially under near-fault pulse-like ground motions. To study the seismic vibration control effect of a tuned mass damper (TMD) attached to transmission tower, parametric analyses are conducted in SAP2000 through CSI OAPI programming, including TMD parameters such as the mass ratio μ from 0.5% to 10%, the frequency ratio f from 0.7 to 1.2, and the damping ratio ξ from 0.01 to 0.2. Based on the obtained analysis results, artificial neural network (ANN) is trained to predict the vibration reduction ratios of peak responses and the corresponding vibration reduction cost. Finally, the NSGA-III algorithm is adopted to perform the multi-objective optimization of a transmission tower equipped with TMD. Results show that the vibration reduction ratios first increase and then decrease with the increase of frequency ratio, but first increase and then remain stable with the increase of mass ratio and damping ratio. In addition, ANN fitting can accurately predict the nonlinear relationship between TMD parameters and objective functions. Through multi-objective optimization with the NSGA-III algorithm, TMD can simultaneously and significantly reduce different peak responses of transmission towers under near-fault pulse-like ground motions in a cost-effective manner.

1. Introduction

As significant lifeline structures, transmission towers are important for power supply, and failure of these towers may lead to electricity interruption and a series of socio-economic losses [1]. Due to its long natural period of vibration and small damping ratio, the transmission tower system is very sensitive to wind loads, rather than seismic loads [2]. As a result, the wind load is usually adopted as the governing lateral load in the design of transmission towers, and the corresponding mechanical behaviors under wind loads have been widely and deeply investigated [3,4,5,6,7]. However, many tall transmission towers may be damaged or even collapse in high seismic intensity areas [8,9], so it is of great importance to evaluate the seismic performance of transmission towers under strong seismic loads.

Vibration control of transmission towers can be mainly classified as active control, semi-active control, and passive control. Due to its many advantages, such as simple structure, low cost, ease of maintenance, and requiring no external energy, passive control has been widely applied in practical engineering by means of seismic base isolation and energy dissipation devices. In terms of the vibration reduction of transmission towers under wind and seismic loads, different kinds of dampers have been developed and applied in real-world projects, such as friction dampers [10,11,12], shape memory alloy (SMA) dampers [13,14,15,16,17], tuned mass dampers (TMD), fluid viscous dampers [18], buckling restrained braces (BRB) [19,20], and magnetorheological dampers [21].

As a dynamic absorber consisting of mass, spring, and damping, TMDs have attracted attention since they were first proposed in 1900s [22]. TMDs can dissipate the energy transferred from the main structure by the auxiliary mass through vibrating out of phase with the motion of the structure [23], and various types of TMD have been developed from conventional TMDs to improve vibration control performance. Tian et al. [24,25] established finite element models of a three-dimensional tower-line system and investigated the vibration reduction effect of TMDs. Analysis results indicated that the TMDs could effectively reduce the vibration response, and the mass ratio significantly influenced the vibration control effects. In addition, the analysis revealed that there exists an optimal mass ratio. Subsequently, they investigated the vibration control effect of a pounding TMD on a transmission tower under multi-component seismic excitations [26] and developed a bidirectional pounding tuned mass damper for transmission tower-line systems under seismic excitations [27]. Zhang et al. [28] developed a pounding TMD and verified its superiority over the traditional TMD under harmonic and seismic loads. Based on the developed eddy-current TMD, Yang et al. [29] conducted a vibration control test on an actual-scale transmission tower and found that the TMD device increased the damping ratio of transmission tower by 3%. Zhao et al. [30] performed shaking table tests on a 1/8-scaled test model equipped with and without a TMD and found that the vibration control effect was related to excitation type and intensity. In addition, the results indicated that the vibration control performance tended to be stable after the damping ratio of TMD reached a specific value. Lei et al. [31] proposed an eddy current damping-based tuned mass damper (ECD-TMD) and applied it to a 50-m-tall transmission tower. The results demonstrated that the proposed ECD-TMD can reduce the first-order bending vibrations of transmission tower along and across the direction of transmission line. For the mass ratio of approximately 2%, vibration reduction ratios of acceleration and displacement are 18–27% and 10–25%, respectively.

A satisfactory vibration control effect can be obtained by optimizing the TMD parameters, mainly including mass ratio, frequency ratio, and damping ratio. Den Hartog [32] developed closed-form expressions of optimal parameters for an undamped single-degree-of-freedom (SDOF) structure. Sadek et al. [33] proposed a method for estimating the TMD parameters of single and multiple degree-of-freedom structures under seismic excitations. Leung and Zhang [34] obtained the optimal TMD parameters of a viscously damped SDOF system under different excitations through the particle swarm optimization and presented their explicit expressions. Hoang et al. [35] investigated the optimal design of a TMD attached to a SDOF structure under seismic loads and found that the optimal TMD has lower tuning frequency and higher damping ratio as the mass ratio increases. Marano et al. [36] proposed a more complete approach for optimizing the parameters of TMD attached to a SDOF system, including mass ratio, frequency ratio, and damping ratio, which was then used to develop two different optimizations criteria minimizing the main system displacement or the inertial acceleration. Bekdas and Nigdeli [37] employed a metaheuristic algorithm called Harmony Search (HS) to find optimal TMD parameters. The comparisons between proposed method and simple expressions showed that the optimal parameters are more economical and feasible for HS approach. Erdogan and Ada [38] provided a computationally efficient approach to determine the optimal parameters of single-sided PTMDs used in vibration control of structural systems. Results showed that the effect of the damping ratio of the primary structure on the optimal coefficient of restitution value is not considerable, while it has significant influence on the optimal frequency ratio. Bian et al. [39] carried out the optimal design and performance evaluation of a tuned mass damper inerter (TMDI) designed for the vibration control of circular section members in transmission towers. Results showed that the TMDI not only effectively reduced the additional mass but also had better vibration control performance and robustness than the TMD.

To sum up, few studies investigate the influence and multi-objective optimization of all TMD parameters on seismic vibration control simultaneously, and the vibration reduction cost is rarely selected as an optimization objective. Moreover, characterized by intense velocity and long-period displacement pulses [40], near-fault pulse-like ground motions may cause extensive damage to transmission towers [41,42,43], but seismic vibration reduction of transmission towers under near-fault pulse-like ground motions is less mentioned. Therefore, to systematically study the seismic vibration control effect of TMDs attached to transmission towers, parametric analyses, ANN fitting, and multi-objective optimization are carried out simultaneously under ten near-fault pulse-like ground motions. The flowchart of this study is presented in Figure 1. Section 2 presents the finite element model of a transmission tower with a TMD and the selected near-fault pulse-like ground motions; Section 3 performs parametric analyses of a transmission tower with a TMD by using the self-compiled program; Section 4 establishes the ANN to predict the vibration reduction ratios of peak responses and the vibration reduction cost; Section 5 carries out the multi-objective optimization of a transmission tower with a TMD by using NSGA-III algorithm; and the conclusions are drawn in Section 6.

2. Finite Element Model of Transmission Tower with TMD

A finite element model of a transmission tower is established in SAP2000 (V20), without considering the tower-line coupling interaction, as shown in Figure 2. The total height of the transmission tower is 28 m and the space between the fixed feet is 6.24 m. The middle cross arm is 1.447 m × 9.558 m and the upper cross arm is 1.4 m × 8.45 m. The leg members are all steel angles with dimensions of L160 × 12, L125 × 10, and L100 × 8 from bottom to top. The diagonal members are all steel angles with dimensions of L90 × 6 and L70 × 5 from bottom to top. The redundant members are all steel angles with smaller dimensions. All the L-shaped steel members are simulated with beam and link elements according to their constraint conditions. The elastic modulus, Poisson’s ratio, and density of steel are 2.1 × 10¹¹ Pa, 0.3, and 7850 kg/m³, respectively, and the base points of the transmission tower are assumed to be fixed. The TMD is placed at the top center of the transmission tower and is regarded as a mass-spring-dashpot system, as shown in Figure 3. In this study, the TMD is assumed to only work in the Y direction since the first-order vibration mode of an uncontrolled transmission tower is the Y-direction translation with a vibration frequency of 5.2075 Hz. In practical applications, TMDs mainly consist of a mass block, dash-pot, springs, a baseplate, and a track. The track is fixed on the baseplate, and the mass block could move freely along the track.

The equation of motion of a transmission tower with a TMD under seismic load can be expressed as:

$$\left(M+m_d\right)\ddot{x}+C\dot{x}+Kx-c_d{\dot{x}}_d-k_d{x}_d=-\left(M+m_d\right){\ddot{x}}_g$$

(1)

where, M, C, and K are the mass, damping, and stiffness of the transmission tower, respectively; m_d, c_d, and k_d are the mass, damping, and stiffness of the TMD, respectively; x, $\dot{x}$, and $\ddot{x}$ are the displacement, velocity, and acceleration of the transmission tower with TMD, respectively; x_d and ${\dot{x}}_d$ are the displacement and velocity of the mass block of TMD, respectively; and ${\ddot{x}}_g$ is the base acceleration excitation.

The mass, damping, and stiffness of the TMD can be calculated as:

$$\left\{\begin{array}{l}{m}_d=\mu M_{1\mathrm{st}}\\ k_d=m_d{\omega }_d^2\\ c_d=2m_d{\omega }_d\xi \end{array}\right.$$

(2)

where, μ is the mass ratio, M_1st is the effective mass of the first-order vibration mode of uncontrolled transmission tower (M_1st = 0.47 × M in this study), ω_d is the frequency of the TMD (the frequency ratio between TMD and transmission tower is defined as f = ω_d/ω₀), and ξ is the damping ratio of the TMD.

To study the energy dissipation effect of the TMD, the energy equation of transmission tower without and with the TMD can be obtained through integrating the equation of motion, namely:

Energy equation of transmission tower without the TMD:

$$E_{KE}+E_{DE}+E_{SE}=E_T$$

(3)

Energy equation of transmission tower with the TMD:

$$E_{KE}+E_{DE}+E_{SE}+E_{TMD}=E_T$$

(4)

where, E_T is the total input energy, E_KE is the kinetic energy, E_DE is the structural damping energy, E_SE is the strain energy, and E_TMD is the energy dissipated by the TMD. E_KE and E_SE only participate in the process of energy transformation and cannot be dissipated. Therefore, the total input energy of the transmission tower without the TMD is dissipated only by the inherent structural damping, while that of the transmission tower with the TMD can be dissipated by both inherent structural damping and TMD damping.

In order to consider a wide frequency range of seismic loads, ten near-fault pulse-like ground motions are selected from Pacific Earthquake Engineering Research Center (PEER) [45] and scaled to fit the design response spectrum of GB B50011-2010 [44] in a statistical sense. The design response spectrum is determined according to the following parameters: seismic precautionary intensity of 8 degree, site of category III, second group of seismic design, design characteristic period of 0.6 s, the maximum seismic influence coefficient of 0.9, and the structural damping ratio of 0.05. All properties of the selected near-fault ground motions are presented in

Table 1. Properties of the selected near-fault ground motions.

RSN	Event Name	Year	Station Name	Mw	TP	Scale Factor
159	Imperial Valley	1979	Agrarias, 3	6.53	2.338	1.9300
178	Imperial Valley	1979	El Centro Array #3, 140	6.53	4.501	1.4594
179	Imperial Valley	1979	El Centro Array #4, 140	6.53	4.788	1.0797
180	Imperial Valley	1979	El Centro Array #5, 140	6.53	4.130	0.9323
185	Imperial Valley	1979	Holtville Post Office, 225	6.53	4.823	1.7900
802	Loma Prieta	1989	Saratoga-Aloha Ave, 0	6.93	4.571	1.2163
1085	Northridge	1994	Sylmar-Converter Sta East, 11	6.69	3.528	0.6424
1086	Northridge	1994	Sylmar-Olive View Med FF, 90	6.69	2.436	0.9091
1161	Kocaeli Turkey	1999	Gebze, 0	7.51	5.992	2.1491
6962	Darfield New Zealand	2010	ROLC, S29E	7.00	7.140	1.4152

, and the acceleration time histories and response spectra of the selected strong ground motions are shown in Figure 4 and Figure 5, respectively.

3. Parametric Analyses Based on CSI OAPI Programming

Based on the established finite element model, parametric analyses of transmission tower are carried out as below. The Newmark-β method is adopted in the numerical integration and the damping ratio of the transmission tower is assumed to be 0.02. The first vibration mode is controlled by unidirectional TMD installed at the top of transmission tower, namely the seismic excitation is applied in the Y direction of transmission tower. The effects of TMD parameters such as mass ratio μ, frequency ratio f, and damping ratio ξ are also investigated. When studying the effect of one parameter, the other two parameters are assumed to be specific values.

The vibration reduction ratio can be defined as:

$$VRR={\displaystyle \frac{R_{\mathrm{O}}-R_{\mathrm{C}}}{R_{\mathrm{O}}}}$$

(5)

where, R_O and R_C are the response of the transmission tower without and with TMD, respectively. Therefore, VRR_d, VRR_v, and VRR_a represent the vibration reduction ratios of peak displacement, peak velocity, and peak acceleration at the top of transmission tower, respectively.

In order to systematically and conveniently perform a large number of numerical simulations in SAP2000 (V20), CSI OAPI programming is applied through Visual Basic for Applications (VBA) in this study, as presented in Algorithm 1.

Algorithm 1 CSI OAPI programming with VBA

1: Input: mass ratio μ, frequency ratio f, damping ratio ξ, and .sdb file 2: Output: VRRd, VRRv, and VRRa 3: Define variables by type 4: Dim mySapObject As SAP2000v20.cOAPI 5: Dim mySapModel As cSapModel 6: mySapObject.ApplicationStart 7: Set mySapModel = mySapObject.SapModel 8: mySapModel.File.OpenFile(Filename) 9: for j = 1 to 11 10:  for k = 1 to 11 11:    for l = 1 to 11 12:    Update TMD parameters μj, fk, and ξl 13:    Define output result variables 14:    Start analysis ret = mySapModel.Analyze.RunAnalysis() 15:    Extract nodal displacement D, velocity V, and acceleration A 16:    Calculate vibration reduction ratios VRRd, VRRv, and VRRa 17:    end for 18:  end for 19: end for 20: mySapObject.ApplicationExit False 21: Set the object to Null

3.1. Effect of Mass Ratio

In order to study the effect of mass ratio, μ is assumed to be 0.5%, 1%, 2%, 3%, 4%, 5%, 6%, 7%, 8%, 9%, and 10%, while the frequency ratio f is set to be 1.0 and the damping ratio ξ is set to be 0.1. Figure 6 shows the effect of mass ratio on vibration reduction ratios VRR_d, VRR_v, and VRR_a of the transmission tower with a TMD.

It can be noted from Figure 6 that the vibration reduction ratios for peak displacement are less than those of peak velocity and peak acceleration at the top of transmission tower. TMD has a similar vibration reduction effect on the peak acceleration and peak velocity. All vibration reduction ratios first increase and then tend to be stable with the increase of mass ratio μ. The optimal mass ratio for VRR_d, VRR_v, and VRR_a is 3%, 5%, and 6%, respectively.

3.2. Effect of Frequency Ratio

In order to study the effect of frequency ratio, f is assumed to be 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, 1.0, 1.05, 1.1, 1.15, and 1.2, while the mass ratio μ is set to be 5% and the damping ratio ξ is set to be 0.1. Figure 7 shows the effect of frequency ratio on vibration reduction ratios VRR_d, VRR_v, and VRR_a of the transmission tower with a TMD.

It can be seen from Figure 7 that the vibration reduction ratios for peak responses have similar variation trends, first increasing and then decreasing with the increase of frequency ratio, and that the optimal frequency ratio is 0.9.

3.3. Effect of Damping Ratio

In order to study the effect of damping ratio, ξ is assumed to be 0.01, 0.02, 0.04, 0.06, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18, and 0.20, while the mass ratio μ is set to be 5% and the frequency ratio f is set to be 1.0. Figure 8 shows the effect of damping ratio on vibration reduction ratios VRR_d, VRR_v, and VRR_a of the transmission tower with a TMD.

As can be seen from Figure 8, the vibration reduction ratios VRR_d, VRR_v, and VRR_a increase with the damping ratio. However, when the damping ratio is greater than 0.08, the vibration reduction ratio cannot be improved efficiently, which agrees well with the results obtained by other scholars. Similar to Figure 6 and Figure 7, TMD has the better vibration reduction effect on peak acceleration and velocity.

4. ANN Fitting of Vibration Reduction Ratios and Cost

Although Section 3 presents the effects of mass ratio μ, frequency ratio f, and damping ratio ξ on vibration reduction ratios of transmission tower with TMD, single-variable analysis cannot precisely characterize the global distribution of vibration reduction ratios. Therefore, vibration reduction ratios in the entire parameter space are calculated through CSI OAPI programming and then used as training data. ANN fitting is further applied to establish the nonlinear mapping between TMD parameters (μ, f, and ξ) and vibration reduction ratios and cost.

4.1. Training Data

Considering TMD parameters such as mass ratio μ from 0.5% to 10%, frequency ratio f from 0.7 to 1.2, and damping ratio ξ from 0.01 to 0.2, the vibration reduction ratios VRR_d, VRR_v, and VRR_a are calculated by CSI OAPI programming. In addition to the vibration reduction effect, this study also considers the vibration reduction cost as an optimization objective. In the vibration reduction practices of a transmission tower with a TMD, the vibration reduction cost mainly comes from the additional mass, the increased stiffness, and the additional damping. Therefore, the vibration reduction cost is assumed to be the weighted sum of normalized mass ratio, frequency ratio, and damping ratio. From an economic standpoint, the cost of mass block installation and the additional damper is much greater than that of stiffness change, because the stiffness can be easily adjusted by the number of springs. The corresponding weight coefficient of vibration reduction cost related to normalized mass ratio, frequency ratio, and damping ratio is set to be 0.4, 0.1, and 0.5, respectively. Thus, the vibration reduction cost can be expressed as:

$$V{R}_{\mathrm{c}}=0.4·{\displaystyle \frac{\mu -{\mu }_{\min }}{{\mu }_{\max }-{\mu }_{\min }}}+0.1·{\displaystyle \frac{f-f_{\min }}{f_{\max }-f_{\min }}}+0.5·{\displaystyle \frac{\xi -{\xi }_{\min }}{{\xi }_{\max }-{\xi }_{\min }}}$$

(6)

where, μ_min and μ_max are the lower and upper bounds of mass ratio μ, respectively, f_min and f_max are the lower and upper bounds of frequency ratio f, respectively, and ξ_min and ξ_max are the lower and upper bounds of damping ratio ξ, respectively.

The vibration reduction ratios VRR_d, VRR_v, and VRR_a and the vibration reduction cost VR_c are calculated and presented in Figure 9. In the entire parameter space, vibration reduction ratios present a complicated relationship with the input variables, which cannot be simply characterized by single-factor analysis. Therefore, a more complex fitting method is required, and the ANN fitting is adopted in this study.

4.2. ANN Fitting

In this section, the feed-forward multilayer perceptron (MLP) structure is used for the ANN with two hidden layers and ten neurons based on calculated results (Figure 9). In addition, the ANNs using more than two hidden layers and ten neurons in each layer have greater complexity and require more processing time, while performance of the ANNs does not improve significantly. The initial weights of ANN have random values between −1 and 1 and are initialized using a symmetric random weight function. The network is trained using the Levenberg–Marquardt learning algorithm [46] and the sigmoidal tangent activation function [47]. Training data are split into training (70%), validation (15%), and test (15%) sets in this study. The learning rate of ANN training is set to 0.001 and the maximum number of epochs is set to 1000. The ANN training continues until generalization stops improving, and both MSE-observation and R-square are used to evaluate the performance of ANN fitting to obtain a performance measure that is independent of the scale of data. After training completion, the ANN fitting effects of vibration reduction ratios VRR_d, VRR_v, and VRR_a and vibration reduction cost VR_c are shown in Figure 10.

The R-square of vibration reduction ratios VRR_d, VRR_v, and VRR_a is 0.9853, 0.9907, and 0.9902, respectively. Since the vibration reduction cost VR_c is calculated by the weighted sum of normalized inputs, ANN is just suitable for the fitting of VR_c and the corresponding R-square is equal to 1.

5. Optimization and Verification of Transmission Tower with TMD

5.1. Multi-Objective Optimization

Compared with its predecessor, NSGA-II, used for dealing with many-objective optimization problems (MaOPs), the NSGA-III algorithm [48] replaces the crowding distance measure with reference points in the objective space to ensure diversity of the converged solutions along the pre-determined solutions in the environmental selection phase. Since the reference points are adaptive, the NSGA-III algorithm also performs well on MaOPs with differently scaled objective values.

(1)

Design variables and constraints

In this study, the TMD parameters are modeled by the design variables vector, including mass ratio μ, frequency ratio f, and damping ratio ξ, whose definitions can be found in Section 2.

Table 2. Lower and upper limit of design variables.

Name	Lower Limit	Upper Limit	Unit
μ	0.5	10	%
f	0.7	1.2	-
ξ	0.01	0.2	-

presents the names of the variables and values of lower and upper limits. During the optimization process, stress constraints are taken into account, namely principal tensile and compressive stresses of structural members at the integral points, and should not exceed their allowable values.

(2)

Objective functions

Four objective functions are considered, namely the vibration reduction ratios of peak displacement, peak velocity and peak acceleration at the top of transmission tower, and the corresponding vibration reduction cost.

The first three objective functions are safety criteria and optimal when maximized, while the fourth objective function is assumed to be the criterion of construction cost and optimal when minimized. These criteria are in conflict with each other. In other words, the vibration reduction cost is usually increased when trying to reduce the peak responses. The aforementioned objective functions are expressed as below:

$$\left\{\begin{array}{l}{J}_1=VR{R}_d\hfill \\ J_2=VR{R}_v\hfill \\ J_3=VR{R}_a\hfill \\ J_4=V{R}_c\hfill \end{array}\right.$$

(7)

(3)

Optimization strategy

The above four objectives can be obtained by the ANN fitting model trained in Section 4. A program is developed using the MATLAB (R2023b) code to invoke SAP2000 (V20) in batch mode and assess the stress constraints. The NSGA-III algorithm is then used for multi-objective optimization. The number of reference points is set to 10, the maximum number of iterations is set to 150, and the population size is set to 150. The crossover and mutation percentage is 0.5 and the mutation rate is set to 0.02.

(4)

Optimization results

The optimal variables are shown in Figure 11, and the corresponding Pareto front is shown in Figure 12. Specifically, the mean values of optimal mass ratio, frequency ratio, and damping ratio are 8.275%, 0.8707, and 0.1203, respectively. In addition, the mean values of the four objectives, VRR_d, VRR_v, VRR_a, and VR_c, are 0.3108, 0.4255, 0.4154, and 0.6523, respectively.

Multi-objective optimization aims to obtain nondominated solutions or a Pareto front. However, decision-makers tend to select the best solution in practical applications with appropriate methods [49]. In view of this, the shortest distance to the ideal point method is applied without obtaining more information from the decision-makers. The ideal point p_ideal takes all optimization objectives into account at the same time, and the closest point on the Pareto front to p_ideal is selected as the best solution. In order to calculate the distance between point p₁ ($J_1^{p_1}$, $J_2^{p_1}$, $J_3^{p_1}$, $J_4^{p_1}$) and p_ideal (0.3207, 0.4450, 0.4409, 0.3774) on the Pareto front, the Euclidean distance (L2 norm) [50,51] is applied:

$$‖p_1-p_{ideal}‖=\sqrt{{\displaystyle \sum _{i=1}^4{\omega }_i{\left(\frac{J_i^{p_1}-J_i^{p_{ideal}}}{J_i^{max}-J_i^{min}}\right)}^2}}$$

(8)

where ω_i is the weight of the i-th objective function, and ω_i = 0.25 (i = 1, 2, 3, 4) is set in this study; $J_i^{p_1}$ and $J_i^{p_{ideal}}$ are the i-th objective functions at point p₁ and ideal point p_ideal, respectively; $J_i^{max}$ and $J_i^{min}$ are the maximum and minimum values of the i-th objective function, respectively.

Among 150 nondominated designs, the compromise solution on Pareto front is obtained by the shortest distance to the ideal point method. The corresponding compromise design variables are marked with the red ball in Figure 11 and the compromise solution is marked with the black ball in Figure 12.

5.2. Case Verification

To verify the optimization results obtained in the above section, design variables marked by the red ball in Figure 11 are imported into the finite element model, namely μ = 8.021%, f = 0.8752, and ξ = 0.1137. The vibration reduction ratios of peak responses of the transmission tower under different ground motions are presented in Figure 13. It can be seen that due to different seismic spectral characteristics, the vibration reduction ratios vary greatly among different ground motions. However, when the vibration reduction ratios are averaged, the mean vibration reduction ratios agree well with those predicted by the ANN model, as listed in

Table 3. Comparison between calculated and predicted mean objectives.

Objective	Calculated Value	Predicted Value	Relative Error
VRRd	0.316975	0.312899	1.286%
VRRv	0.441968	0.424429	3.969%
VRRa	0.420393	0.412964	1.767%
VRc	0.624608	0.625301	−0.1109%

. As can be noted from the table, the relative errors between calculated and predicted objectives are smaller than 4%, in which the vibration reduction ratio of peak velocity has the maximum relative error while the vibration reduction cost has the minimum relative error. Therefore, the ANN fitting model established in this study is proved to be reliable and the compromise solution obtained from the Pareto front can significantly reduce the peak response while considering the cost.

Figure 14 shows the energy response of the transmission tower without and with the TMD under the scaled RSN 6962 ground motion. After the TMD is installed, the total input energy increases from 665.28 J to 892.47 J and the proportion of structural damping energy E_DE in the total input energy decreases from 100% to 32.82%. In other words, more than two thirds of the total input energy is dissipated by TMD damping. In addition, both kinetic energy E_KE and strain energy E_SE are also suppressed by the TMD, which makes the total input energy of the controlled transmission tower smoother. Figure 15 shows the time histories of displacement, velocity, and acceleration at the top of transmission tower without and with the TMD. The seismic responses are greatly suppressed when a TMD with appropriate parameters is installed.

6. Conclusions

In this study, parametric analyses of a transmission tower with a TMD under near-fault pulse-like ground motions are conducted in SAP2000 (V20) through CSI OAPI programming, and ANN is then trained to predict the vibration reduction ratios of peak responses and the corresponding vibration reduction cost. Finally, the multi-objective optimization of a transmission tower with a TMD is carried out by using the NSGA-III algorithm. The following conclusions can be drawn:

• TMD has better vibration reduction effects on the peak velocity and acceleration. The vibration reduction ratios first increase and then decrease with an increase of frequency ratio, but first increase and then remain stable with the increase of mass ratio and damping ratio.
• ANN fitting can accurately predict the nonlinear relationship between TMD parameters (μ, f, and ξ) and vibration reduction ratios and cost; the minimum R-square of four objectives is 0.9853, with statistical significance. Since the vibration reduction cost is calculated by the weighted sum of normalized inputs, ANN is suitable for the fitting of vibration reduction cost and the corresponding R-square is equal to 1.
• According to the Pareto front, the mean value of optimal mass ratio, frequency ratio, and damping ratio is 8.275%, 0.8707, and 0.1203, respectively, and the mean values of vibration reduction ratios and cost are 0.3108, 0.4255, 0.4154, and 0.6523, respectively. Therefore, TMDs can simultaneously and significantly reduce different peak responses of transmission tower in a cost-effective manner after selecting appropriate TMD parameters.