Study on Seismic Performance of Porcelain Pillar Electrical Equipment Based on Nonlinear Dynamic Theory

In order to consider the influence of nonlinear characteristics of porcelain pillar electrical equipment on the dynamic response under seismic excitation, a theoretical analysis method of nonlinear dynamics was raised to define the nonlinear parameter of the flange connection and establish a dynamic model of porcelain pillar electrical equipment. The theoretical analysis and the test results have a good degree of fitting, which verifies the correctness of the dynamic model and reveals the nonlinear seismic response law of the porcelain pillar equipment. According to the results, both the nonlinear calculation results of the displacement at the top end of the porcelain pillar and the stress at the bottom end of the porcelain pillar are smaller than the results in the linear calculation. The difference between them increases gradually with the increase of seismic excitation. The differences in displacement and stress at a seismic excitation of 0.5 g are 14.58% and 23.25%, respectively. When the nonlinear parameter increases to a certain value, the impact of maximum stress on the bottom of the porcelain pillar is very small and the change is not obvious. The research provides a theoretical reference for the seismic design of pillar electrical equipment.


Introduction
e electric power system is an important part of the lifeline project, which is related to the normal life of the people and the development of industry and manufacturing. e earthquakes have caused serious damage to the electrical equipment. An electrical substation, as the node of an electrical grid, is extremely important for the reliable service of an electrical grid system [1]. A wide variety of porcelain pillar electrical equipment, such as post insulators and surge arresters, is a major component of an electrical substation [2]. Due to the brittle nature of porcelain, equipment made of porcelain is vulnerable to earthquake shaking. For porcelain pillar electrical equipment when it is subjected to earthquake, due to the small rigidity of the cemented part of the flange connection, it is prone to damage at the flange connection.
e post-earthquake field investigation and shaking table tests (Figures 1 and 2) showed failure characteristics of fracture during the earthquake damage [3][4][5]. Many scholars have conducted extensive researches on the seismic response mechanism of porcelain pillars. Chinese codes are given in [6][7][8][9] combined with experimental research. In the recommended practice in the relevant code, the calculation equation for the bending stiffness of the joint between the flange and the porcelain bushing is in which K c is the bending stiffness of the connection between the flange and the porcelain bushing (in N·m/rad); d c is the porcelain bushing outer diameter (in m); h c is the height of porcelain bushing at cemented part (in m); t e is the thickness of the cemented part between the flange and the porcelain bushing (in m) (refer to Figure 3(a)); and β is the calculation parameter of the bending stiffness of the joint between the porcelain bushing and the flange, which is recommended as β � 6.54 × 10 7 in the Chinese code. e physical concept of the above equation is relatively clear, which basically reflects the mechanical characteristics of the cement layer between the flange and the porcelain bushing. For the bolted connection, the bolts are usually pretensioned and there is no opening between the two faces when the equipment is subjected to bending moment. e stiffness of bolted connection can be assumed infinitely large. erefore, the stiffness between metal cap and porcelain pillar is weaker and the bending capacity is determined by stiffness between metal cap and porcelain pillar. However, its accuracy depends on the empirical parameter of the bending stiffness of the porcelain bushing and the flange.
Zhang et al. [10] obtained the bending rigidity calculation parameter for the connections between UHV porcelain bushing and flange by conducting the bending rigidity test and, accordingly, applied this calculation parameter to the finite element calculation of the simulated UHV porcelain bushings. Singh et al. [11] introduced the response spectrum analysis to evaluate linear control systems for seismic inputs defined by code-prescribed or site-specific ground response spectra. Gilani et al. [12] carried out a shaking table test for two 230 kV, 3000 A, Type-U transformer bushings and found that both bushings survived earthquake shaking compatible with the IEEE spectrum for high-level qualification when mounted on a rigid frame. When mounted on a flexible frame, one 230 kV bushing was unable to sustain the high-level qualification shaking without oil leakage and a slip of the porcelain units. Filiatrault and Matt [13] investigated numerically the dynamic response of porcelain bushings mounted on transformer tanks. e results of the numerical study show that large amplification occurs when the fundamental frequency of the porcelain bushing is tuned with the fundamental frequency of the transformer tank. Bender and Farid [14] added additional evidence that bushing amplifications greater than 2.0 are possible, and they demonstrated that the complexity of these systems makes generalizations of design. Villaverde et al. [15] undertook the study to assess the adequacy of the amplification factor of 2.0 specified by the Institute of Electrical and Electronics Engineers in Standard IEEE 693-1997 for the seismic qualification of transformer bushings. Schiff [16] proposed a new method to seismically qualify power transformers and their bushings to meet the requirements of IEEE Standard 693. He et al. [17] analyzed the influence of different types of connections on the seismic responses of the oil-side bushings and a simplified method to rapidly estimate the seismic responses of transformer bushings was implemented.
Cao [18] conducted the characteristic research on the recovery capacity of pillar insulators and discussed the impact of various factors such as the insulator cementing parts between insulator flange and porcelain body on the seismic performance of insulator pillars. Schiff [19] presented a method for evaluating the reliability of ceramic structural members subjected to earthquake-induced vibrations which used the Weibull distribution to represent the probability of failure of the brittle ceramic material. e authors of [20][21][22] introduce the experimental study of seismic performance of pillar electrical equipment of different voltage classes and different types. In order to improve the seismic capacity of pillar electrical equipment, composite materials are often used in the porcelain pillar equipment. IEC 61462 [23] and IEEE Std 693-2005 [24] introduce the research on the improvement of bearing capacity of porcelain pillar equipment by using new composite materials. Mohammadi et al. [25] proposed a Four Degrees-of-Freedom (4-DOF) system in order to assess the dynamic properties of substation support structures and verified through the finite element method (FEM).
In the above researches, during the seismic analysis of porcelain pillar electrical equipment, the beams with equivalent rigidity are often used directly to represent the flange connections.
is paper focuses on the nonlinear factors at the flange connections by introducing a nonlinear rigidity parameter. e nonlinear dynamic model is then established. After that, the seismic performance of multiple porcelain pillars is studied and compared with the shaking table test results, which proved the correctness of the nonlinear dynamic model. Finally, the maximum stresses at the bottom of porcelain pillars are analyzed when the   Advances in Civil Engineering nonlinear rigidity parameter changes. e maximum stress at the bottom of the porcelain pillar with a different seismic excitation can be predicted through this way.

Nonlinear Dynamic Model of Porcelain Pillar Equipment
In an electrical substation, some porcelain pillar equipment is directly installed on the foundations. Take a piece of typical flanged 4-section porcelain post insulator as an example, as illustrated in Figure 4. Figure 4 also shows the key structural and typical sections of a solid core post insulator with sheds. e equipment has structural characteristics (1): the main body of the equipment is made of porcelain core as annotated in Figure 4; (2) the porcelain component is connected through metal caps at both ends with bolted flange connections. e connections between porcelain pillars contain two parts, cemented connections between metal caps and flange, and connection between two flanges with bolts. Figure 5 is an analysis diagram of the porcelain pillar equipment structure, where x b is the seismic excitation, and the sequential numbers of porcelain pillars are 1, 2, 3, and 4, respectively, from bottom to top. e equipment is mainly in a bending state when it is subjected to seismic and the flange connection between two pieces of porcelain pillars rotates by a certain angle θ. e bottom end of the upper porcelain pillar rotates by θ i , and the upper end of the lower porcelain rotates by θ i − 1 . So, the relative angle θ can be expressed as θ � θ i − θ i − 1 . e angles of each flange connection are θ 1 , θ 2 , θ 3 , and θ 4 , respectively, as shown in Figure 5. e analysis diagram of flange connections is illustrated in Figure 3. Figure 3(a) shows the deformation of the cemented part. Figure 3(b) shows the mechanical simplified diagram of the flange connection. According to the deformation characteristics, the connection between porcelain and metal cap can be regarded as a spring with rigidity k id . Correspondingly, the connection between lower porcelain and flange can be regarded as a spring with rigidity k (i − 1)u . e cemented part between metal caps and flange can be equivalent to a series spring with the equivalent rigidity k i , and k i, can be expressed as follows:

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where k id and k (i − 1)u are calculated as follows [2]: in which d u and d d are the porcelain diameters at cemented parts of (i − 1) th and i th porcelain pillar, respectively; h u and h d are the cemented heights of (i − 1) th and i th porcelain pillar, respectively; t u and t d are the thicknesses of cement layer of (i − 1) th and i th porcelain pillar, respectively; β u and β d are bending rigidity parameters at the connections of (i − 1) th and i th porcelain pillar, respectively, the value of β is 6.54 × 10 7 when the porcelain diameter at the cemented part is smaller than 275 mm, the value of β is 5.0 × 10 7 when the porcelain diameter at the cemented part is larger than 375 mm, and the value of β can be obtained by linear interpolation when the porcelain diameter at the cemented part is between 275 mm to 375 mm. As can be seen from the above analysis, the rotational bending moment caused by the relative rotation of the up and down flanges can be expressed using Equation (4) expresses rotational bending moment under ideal conditions. e exact mechanics of the rotation stiffness k i is complex, which involves nonlinear mechanical properties of the cemented material and the mechanics at the contact interfaces. ese factors should be included in M i to make it more realistic. In most previous studies, the flange joint was equivalent to a beam with equivalent rotational stiffness [2,26,27]. Figure 6 shows a diagram of the force of the microelement. According to the elastic theory, when the beam subjected to large deformation, the strain can be expressed as in which u represents the horizontal displacement, θ represents the rotational angle, and ωrepresents the curvature. en, the stress can be calculated using in which E represents Young's modulus. Further, the bending moment on the cross section of the porcelain bushing is in which A represents the area of cross section. In addition, In equation (8), I represents the moment of inertia. Consequently, equation (7) is reduced to θ can be solved by methods of separating variable; the expression of θ is shown in where Φ(x) represents the vibration shape function of porcelain pillar, and θ(t)is the rotational angle of flange connection.
When the height of the cemented part is h, the bending moment provided by the flange is Referring to equation (12), equation (4) is rewritten as follows after considering the nonlinear factors: In equation (13), the parameter η is the nonlinear rigidity parameter combined with discontinuous deformation at the flange when subjected to large deformation. It is closely related to the vibration shape function of porcelain pillar assembly. It is difficult to get through a theoretical solution, so the value of η is based on an experimental solution. When η � 0, equation (13) is used to calculate the rotational bending moment under the ideal conditions represented by equation (4). e different values of η correspond to different degrees of the nonlinearity of the flange cemented part.
If there are N c porcelain pillars, there will be N c flange connections. e elastic potential energy at a flange connection is calculated as follows: Since the rotation of porcelain pillars causes the change of gravitational potential energy of porcelain pillars, the total gravitational potential energy is calculated as follows: in which L j represents the length of j th porcelain pillar. e kinetic energy of porcelain pillars consists of average kinetic energy and rotational kinetic energy: in which v c,i is the centroid velocity of the i th porcelain pillar, and m i is the mass of the i th porcelain pillar. For the four-section porcelain pillar system, the speed relationship of the i th porcelain pillar is illustrated in Figure 7: As illustrated in Figure 7, the velocity of 1 st porcelain pillar bottom is e centroid velocity of the 1 st porcelain pillar is calculated as follows: e centroid velocity of the 2 nd porcelain pillar is calculated as follows: e centroid velocity of the 3 rd porcelain pillar is calculated as follows: e centroid velocity of the 4 th porcelain pillar is calculated as follows: e Hamilton variational equation [28] is as follows: In equation (21), Figure 7: Speed relationship diagram of porcelain pillars.

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After equations (23) and (24) are substituted into equation (22) and further simplification, equation (22) is rewritten in matrix form as follows: where In addition, Table Test. In this section, the accuracy and effectiveness of the proposed modeling method will be validated and demonstrated by comparing the analysis results with the experimental results. e shaking table test set-up of a 1000 kV porcelain surge arrester is shown in Figure 8(a). is equipment contains 4 insulators with a similar configuration. e mass, dimension, and material parameters are listed in Table 1.

Shaking
In the shaking table tests, the strain gauges were placed in the X direction of the root of each section of the casing (two on the opposite side); the accelerometer is placed in the X direction of the top of the vibrating table and the top of each section of the casing, which is shown in Figure 8(b). e amplitudes of the input artificial waves are 0.1 g, 0.15 g, 0.2 g, and 0.3 g, and the direction of the input artificial wave is a single X direction (refer to Figure 8(b)). Figure 9 shows the input seismic wave of the artificial ground motions, a broad resonant period range from 0.1 s to 0.9 s, and the stress responses at the bottom with peak ground acceleration of 0.15 g. e vibration table test obtains the strain at the bottom end of the porcelain pillar. e stress can be obtained by multiplying the elastic modulus of the porcelain bushing (110 GPa) by the strain. Figure 9(b) shows the maximum stress at the bottom of porcelain.
As seen in Figure 9, the maximum acceleration of the ground is about 2.3 m/s 2 and the maximum stress at the bottom end of the porcelain pillar is 11.14 MPa. Figure 10 shows the results when the peak value of the input wave is 0.3 g. As seen in Figure 10, the maximum acceleration of the ground is about 4 m/s 2 and the maximum stress at the bottom end of the porcelain pillar is 22.02 MPa.

Seismic Wave Input in eoretical Analysis.
e seismic wave data are those for artificial waves jointly proposed by China Electric Power Research Institute and China Earthquake Disaster Prevention Center.
is artificial wave is fitted by the seismic acceleration response spectrum with a characteristic cycle of 0.9 s that can envelop the characteristic cycles of Category I∼Category III soil sites, and it is suitable for seismic design, test, and seismic performance evaluation of HV and UHV electrical equipment. e artificial wave time history and the fitting data for artificial wave response spectrum and target spectrum at a peak seismic acceleration of 1g are illustrated in Figure 11. e data X b can be fitted with a periodic function X b : in which When the value of N F is 500, the fitting result is illustrated in Figure 11. As seen from Figure 11, the blue fitting curve has a high matching degree with the black seismic wave curve.

Method Validation.
In order to verify the correctness of the theoretical model, the numerical method was used to solve equation (25) with the fitting seismic wave proposed in Section 3.2, and the results were compared with the curve obtained by the shaking table test in Section 3.1.
Based on the calculation by the kinetic model established according to equation (25), the first-order frequency of the four-section porcelain pillar equipment is 2.28 Hz, and the result obtained by shaking table test is 2.18 Hz, so there is a difference of 4.6% between the test value and the theoretical value. It indicates that the theoretical analysis is correct. Figure 12 is a comparison curve of stress time histories at the bottom end of the porcelain pillars obtained by theoretical analysis and vibration table test at a � 0.15 g, a � 0.3 g respectively, when the value of nonlinear rigidity parameter η is taken as 20,000. e maximum stresses at the bottom of the 1 st section of the equipment obtained by test and theoretical analysis are 11.14 MPa and 11.00 MPa, respectively, at a � 0.15 g, indicating that the difference between maximum stress values is 1.2%. e theoretical and test curves also show a similar change trend.
e maximum stresses at the bottom of the 1 st section of the equipment obtained by test and theoretical analysis are 22.02 MPa and 20.30 MPa, respectively, at a � 0.3 g, indicating that the difference between maximum stress values is 9.04%. Seismic excitation usually has nonlinear characteristics. With the increase of seismic excitation, the nonlinear characteristics of the porcelain pillar become more significant, which leads to the fact that the difference between test and analytical value becomes larger. Figure 13 shows a comparative diagram of maximum stresses at the bottom of the 2 nd section pillar. When the peak acceleration of the input seismic wave is 0.15 g, the maximum value of the nonlinear theoretical result is 9.41 MPa and the maximum value of the test is 9.6 MPa. e difference between them is 2.0%. When the peak acceleration of the input seismic wave is 0.3 g, the maximum value of the nonlinear theoretical result is 19.25 MPa and the maximum value of the test is 21.00 MPa. e difference between them is 8.3%.

Seismic Response of 1,000 kV Porcelain Pillar Arrester Equipment
Equation (25)          be obtained, as shown in Table 2. Figure 26 illustrates the dynamic response of the equipment under different seismic excitations with the linear and nonlinear values used. As seen from the analysis results in Table 2 and Figure 26, for the 1000 kV arrester equipment, when the flange connection is linear, the maximum displacement and stress grow linearly with the increase of peak ground acceleration; when the nonlinear factor is considered, the maximum displacement and stress increase nonlinearly, and the magnitude of the increase gradually decreases. It can be seen from Figure 25 that the nonlinear seismic response of the equipment is smaller than the linear seismic response under seismic excitation. e greater the ground motion excitation, the larger the difference between nonlinear response results and linear response results of the equipment. It indicates that equipment exhibits more obvious nonlinear characteristics as seismic excitation increases.
When the peak ground acceleration increases from 0.1 g to 0.5 g, the maximum displacement of the top of the porcelain pillar increases from 22.26 mm to 111.28 mm without consideration of nonlinear factor; when the nonlinear factor is considered, the range of this value is from 22 mm to 95.06 mm. Correspondingly, the variation range of the maximum stress at the bottom end of the porcelain pillar is from 7.68 MPa to 38.41 MPa and 7.48 MPa to 29.48 MPa, respectively. When the peak ground acceleration is 0.1 g, 0.2 g, 0.3 g, 0.4 g, and 0.5 g, the percentages of the difference between nonlinear analysis results and linear analysis results of displacement are 1.17%, 2.88%, 5.72%, 9.66%, and 14.58% respectively, and the percentages of the difference between nonlinear analysis results and linear analysis results of stress are 2.60%, 6.77%, 11.93%, 17.51%, and 23.25%, respectively.

Characteristics of Nonlinear Seismic
Response within Structural Parameter Domains Figure 27 illustrates the curves for the influence of changes of nonlinear rigidity parameter within a certain range on the maximum stress at the bottom end of porcelain pillar when the seismic excitations are 0.1 g, 0.2 g, 0.3 g, 0.4 g, and 0.5 g. Table 3 shows the corresponding maximum stress value at the bottom end of the porcelain pillar with the change of nonlinear rigidity η under different seismic excitations.
As seen from Figure 27, for different seismic excitations, as the nonlinear rigidity increases, the maximum stress at the bottom end of the porcelain pillar exhibits change characteristic of decreasing first and then becoming stable. When the seismic excitation is 0.1 g and the nonlinear rigidity is 12 × 10 4 , the maximum stress at the bottom end of porcelain is 6.67 MPa, which has a decrease of about 9.5% compared with that when the nonlinear rigidity is 0. e nonlinear rigidity change has a rather little impact on the maximum stress at the bottom end of the porcelain pillar. As seen from Table 3, the impact of nonlinear rigidity on the maximum stress at the bottom end of the porcelain pillar increases gradually when the seismic excitation increases. When the seismic excitation is 0.2 g, 0.3 g, 0.4 g, and 0.5 g, respectively, and the nonlinear rigidity is 12 × 10 4 , the maximum stress at the bottom end of the porcelain pillar decreases by 23.8%, 26.6%, 35.5%, and 42.8% compared with that when the nonlinear rigidity is 0.
According to Section 3.3, η � 20,000 can reflect the nonlinear factors, and the maximum stress at the bottom end of the porcelain pillar is analyzed with η changes in the range of [0, 4 × 10 4 ] in order to consider the influence of fluctuation of η on the maximum stress at the bottom end of porcelain pillar. When the seismic excitation is 0.1 g, 0.2 g, 0.3 g, 0.4 g, and 0.5 g, respectively, and η increases from 0 to 2 × 10 4 , the maximum stress at the bottom end of the porcelain pillar decreases by 1.9%, 6.7%, 13.43%, 21.2%, and 23.7%; when η increases from 2 × 10 4 to 4 × 10 4 , the maximum stress at the bottom end of porcelain pillar decreases by 1.8%, 5.9%, 11.33%, 3.7%, and 4.01%, respectively. As seen from the above analysis, the decrease rate of the maximum stress at the bottom end of the porcelain pillar when η increases from 2 × 10 4 to 4 × 10 4 is smaller than that when η increases from 0 to 2 × 10 4 .
As seen from Figure 27, as the seismic excitation increases, the nonlinear rigidity change ranges corresponding to rapid decrease stages of maximum stress at the bottom end of porcelain pillar are [0, 12 × 10 4 ], [0, 10 × 10 4 ], [0,8 × 10 4 ], [0, 4 × 10 4 ], and [0, 2 × 10 4 ], and the change ranges have a trend of gradual decrease. It indicates that the influence of nonlinear rigidity on the maximum stress at the bottom end of the porcelain pillar under high seismic excitation is larger and the maximum stress changes rapidly. When the nonlinear rigidity increases to a certain value, the impact of nonlinear rigidity on the maximum stress at the    14 Advances in Civil Engineering bottom end of the porcelain pillar is rather small, and the stress change of the porcelain pillar has entered stationary change.

Conclusions
(1) According to elastic mechanics, in the range of elasticity, it is assumed that the glued part of flange connection assembly before and after deformation follows the plain-section assumption; that is, the cross section after deformation at the glued part assembly is perpendicular to the neutral plane. A dynamic model of porcelain pillar electrical equipment was established for considering the nonlinear characteristics of flange connections. e theoretical analysis results and the test results have a rather good degree of the fitting, which verifies the correctness of the dynamic model. When the nonlinear rigidity parameter is 20,000, the influence of nonlinear factors at the cementing parts can be reflected (2) Both the nonlinear calculation results of the displacement at the top end of the porcelain pillar and the stress at the bottom end of the porcelain pillar are smaller than the linear calculation results. e difference between such two kinds of results increases gradually with the increase of seismic excitation. e differences in displacement and stress at a seismic excitation of 0.5 g are 14.58% and 23.25%, respectively (3) When the nonlinear rigidity increases to a certain value, the influence of nonlinear rigidity on the maximum stress at the bottom end of the porcelain pillar is rather small, the stress change of the porcelain pillar has entered stationary change. As the seismic excitation increases, the nonlinear rigidity change ranges corresponding to rapid decrease stages of maximum stress at the bottom end of the porcelain pillar decrease gradually Data Availability e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.