# Seismic Vulnerability Assessment of Thermal Power Plants Based on Beta Function

Authors: Tian Deyuan; Liu Rushan; Xia Xuyi; Wang Zhenhui
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The study on seismic vulnerability of thermal power plants is of great significance for seismic damage prevention of the electrical power system and risk analysis of seismic economic loss. This paper presented a study method for obtaining the seismic vulnerability matrix and vulnerability curve of thermal power plants based on beta function. Firstly, according to the study results of thermal power plants in American ATC-25 report, the expected values of seismic damage indexes of thermal power plants under different seismic intensities were calculated. It was assumed that the expected values of seismic damage indexes of substation and thermal power plant were the same, and the dispersion of distribution of seismic damage indexes was the same. Then, based on the actual seismic vulnerability matrix of substations obtained from statistics in Wenchuan earthquake, the seismic vulnerability matrix of thermal power plants under different seismic intensities was fitted by using the probability density function of beta distribution. Finally, the seismic vulnerability curve of the thermal power plant based on peak ground acceleration (PGA) was obtained by using the cumulative function curve of standard lognormal distribution. The calculation results show that the seismic damage ratio calculated from the seismic vulnerability matrix of thermal power plants is basically consistent with the seismic damage ratio in ATC-25 report.


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## Body

## 1. Introduction

The power plant is a basic unit of the electrical power system. If it is severely damaged in the earthquake, it will not only cause direct economic loss, but also cause a series of chain hazards due to the loss of power generation function and the termination of power supply, which will seriously affect the postearthquake emergency rescue, people’s production and life, and postdisaster reconstruction. Electrical energy includes multiple types such as thermal power, hydropower, nuclear power, wind power, and solar power. Among them, the thermal power generation is the most widely used power generation mode in the world, even reaching more than 80% in China. Therefore, there are a large number of thermal power plants, and it is of great significance to study the seismic vulnerability of thermal power plants.The composition of a thermal power plant is very complex, including various civil construction facilities such as steam engine room, control room, cooling tower and chimney, various power generation equipment items such as generator set and boiler, and various substation electrical equipment items. The historical seismic damage showed that the seismic damage suffered by thermal power plants was quite complex. In addition to the direct damage of various facilities and equipment under the action of earthquake, enormous damage was also caused by interaction between various facilities and equipment.More past research results on thermal power plants were about the research on structural design, aseismic measures, and aseismic capacities of civil construction facilities, power generation, and transformation equipment [1]. For example, Wang et al. [2] studied the effect of structural irregularity of the thermal power plant on its aseismic performance; Dai et al. [3] designed PMI system of the thermal power plant through single objective and multiobjective optimization program; Mohammadi and Tehrani [4] studied the aseismic performance analysis of three interconnected high-voltage substations by using the incremental dynamic analysis method. The seismic vulnerability research is often aimed at the vulnerability of individual civil construction facilities or single equipment in thermal power plants. For example, Wang et al. [5] studied the seismic performance of a steel–concrete hybrid structure consisting of reinforced concrete (RC) tubular columns and steel braced truss with A-shaped steel frames and investigated the damage characteristics of the structure; Liu et al. [6] studied the relationship between damage rate of high-voltage electrical equipment and instrumental seismic intensity by using the normal distribution function. However, for seismic disaster prediction, economic loss assessment, or seismic insurance, it is often necessary to take the whole thermal power plant as a whole unit and calculate the seismic loss of the whole power plant according to the damage probabilities of different damage levels under different seismic intensities in the seismic vulnerability matrix. It is not necessary to study the specific damage of each facility or equipment item in the thermal power plant in detail. Therefore, it is very important to obtain the seismic vulnerability matrix of the thermal power plant as a whole unit and determine the damage probabilities of different damage levels under different seismic intensities for formulating corresponding disaster prevention policies and loss assessment and calculation [7, 8].Because there are many components in a thermal power plant, and the causes and characteristics of damage of various components are different, it is relatively difficult to analyze the seismic vulnerability of the thermal power plant as a whole unit. At the same time, compared with house buildings, there are little data and few statistics on actual seismic damage and overall damage or economic loss of the thermal power plant. At present, no relevant achievement is reached in the study of seismic vulnerability matrix of the thermal power plant as a whole unit. ATC-25 report [9] of American Applied Technology Council provides a calculation formula for the seismic damage ratio curve of the thermal power plant under different seismic intensities, but this calculation formula still cannot be used to obtain the seismic vulnerability matrix of overall damage of the thermal power plant and accurately calculate the probabilities of various damage levels of the thermal power plant under different seismic intensities. In the current related research results, it is a common method to study the seismic vulnerability of building structures by means of function fitting. Qiao and Wang [10] studied the effect of viscous damper on seismic vulnerability of high-rise steel structures by using the beta function and obtained the seismic vulnerability matrix of the structure; Zhang et al. [11] derived the seismic vulnerability curve of multistory masonry structures by using the lognormal distribution function and studied the impact of factors such as the number of stories, the seismic measure, the strength of mortar, and the area ratio of seismic wall on the seismic vulnerability of the multilayer masonry structure. Although the method of using functions to study the seismic vulnerability of buildings is widely used, there is still no relevant research on obtaining the seismic vulnerability matrix of overall damage of the thermal power plant by using function fitting.In view of this, this paper proposed a method for calculating the seismic vulnerability matrix of the thermal power plant based on beta function. According to the damage rates of the thermal power plant under different seismic intensities in ATC-25 report and the actual seismic damage data of substations obtained in Wenchuan earthquake, the seismic vulnerability matrix of thermal power plants based on seismic intensities was fitted by beta function. Then, the seismic vulnerability curve of the thermal power plant was calculated by lognormal distribution, and the seismic vulnerability matrix of the thermal power plant based on PGA was obtained, which provided a reference for seismic risk assessment and loss calculation of the thermal power plant.


## 2. Research Methods and Ideas of Seismic Vulnerability of Thermal Power Plants

The research methods and ideas of seismic vulnerability of thermal power plants based on beta distribution function were as follows: (1) The expected values of seismic damage indexes of the thermal power plants under different seismic intensities were determined according to the seismic damage ratio function of thermal power plants proposed by ATC25 report. (2) Since the substations and thermal power plants have many similarities in component composition and seismic damage characteristics, which are composed of buildings, electrical equipment, and indoor monitoring equipment, it was assumed that the thermal power plant and substation had the same dispersion of seismic damage index distribution when their expected values of seismic damage indexes were the same. According to this assumption, based on the expected value and the distribution variance of substation seismic damage index, the distribution variance corresponding to different expected values of seismic damage index of the thermal power plant was obtained by linear interpolation method. (3) The expected value and distribution variance of seismic damage index of the thermal power plant were taken as parameters to fit the seismic vulnerability matrix of the thermal power plant based on seismic intensity by using the beta distribution function. (4) Based on the seismic vulnerability matrix of the thermal power plant and the corresponding relationship between seismic intensity and PGA, the lognormal distribution was used to fit the seismic vulnerability curve of the thermal power plant, and the seismic vulnerability matrix of overall damage of thermal power plant based on PGA was determined. The calculation process is shown in Figure1.Figure 1
Flowchart of calculation method for seismic vulnerability matrix of thermal power plant.


## 3. Determination of Fitting Parameters

### 3.1. Expected Value of Seismic Damage Index of the Thermal Power Plant

The thermal power plant is a huge and complex plant producing electric energy and thermal energy, which consists of the following five systems: (a) fuel system; (b) combustion system; (c) steam-water system; (d) electrical system; (e) control system. Although there are many types of equipment and complex components in the thermal power plant, the equipment and facilities can be mainly divided into three categories: civil construction facilities, electromechanical equipment, and electrical equipment.(a)
Civil construction facilities: main power house, control building, power distribution unit building, office building, etc.(b)
Electromechanical equipment: boiler, generator, steam turbine, condenser, etc.(c)
Electrical equipment: various high-voltage electrical equipment items, output conductor, etc.In the ATC-25 report of the American Applied Technology Council, the seismic vulnerability of thermal power plants was studied. When the overall seismic damage ratio for thermal power plants was calculated, the damage characteristics of civil construction facilities, electromechanical equipment, and electrical equipment were comprehensively considered. Due to the different functions and economic values of different equipment, the damage of different equipment accounts for different proportions in the evaluation of the overall damage of the thermal power plant. Based on the seismic damage data of thermal power plants in historical earthquakes in the United States, the ATC-25 report proposes the calculation formula for the seismic damage ratio of these three types of facilities and equipment and determines the weight proportions and regression coefficientsα and β of various facilities in the overall damage of the power plant which can be found in the Facility Class table in Appendix G (ATC-13).The calculation formula of seismic damage ratio of each equipment type is shown as the following formula (1) [9, 12]:(1)DMG=expα.jβ,In (1), DMG is the seismic damage ratio, j is seismic intensity, and α and β are the regression coefficients.The regression coefficients and weight of various equipment types are shown in Table1.Table 1
Weights and regression coefficients of various equipment types.
Equipment typeWeightαβCivil construction facilities0.2−12.024.526Electrical equipment0.3−14.055.61Electromechanical equipment0.5−14.815.778According to the weight and regression coefficients of various equipment types in Table1, combined with the calculation formula for seismic damage ratio of the thermal power plant equipment, the overall seismic damage ratio function of thermal power plant was calculated as follows:(2)Rj=0.2.e−12.02.j4.526+0.3.e−14.05.j5.61+0.5.e−14.81.j5.778.In (2), R(j) is the overall damage ratio of the thermal power plant under j seismic intensity and j is the seismic intensity.The expected value of the seismic damage index under a certain ground motion intensity is the sum of the product of the damage proportion of each damage level and the seismic damage index corresponding damage level. The expected value of the seismic damage index can represent the overall damage level of the thermal power plant and has the same practical significance as the overall damage ratio. Therefore, the overall damage ratio calculated by (2) was used as the expected value of seismic damage index of the thermal power plant [13]. The expected values of seismic damage indexes of the thermal power plant under different seismic intensities are shown in Table 2.Table 2
Expected values of seismic damage indexes of the thermal power plant under different seismic intensities.
Seismic intensity67891011Expected value0.0150.0350.0730.1390.2480.420


### 3.2. Variance of Distribution of Seismic Damage Index of the Thermal Power Plant

The substation is an important node in the electrical power system. It has many similarities with the thermal power plant in equipment composition and seismic damage characteristics: (1) The substation is also the main body composed of multiple types of equipment such as transformer and control equipment, which can be divided into three types of equipment such as civil construction facilities, outdoor high-voltage electrical equipment, and indoor equipment. (2) In historical earthquakes, there are two main reasons for the damage of substations. One reason is the direct damage caused by earthquakes, and the other reason is the indirect damage of internal facilities caused by the damage of civil construction facilities, which is basically the same as the overall damage of the thermal power plants. (3) Based on the actual earthquake damage of 110 KV and above substations and the asset value of various facilities in the Wenchuan earthquake, Liu et al. [14] determined the weight coefficients of the outdoor high-voltage electrical equipment, indoor equipment, and civil construction facilities of the substation. The weights of outdoor high-voltage electrical equipment, indoor equipment, and civil construction facilities of 110 KV and above substations are shown in Table 3, which are 0.671, 0.201, and 0.128, respectively, similar to the economic value weights of various equipment types of the thermal power plant in Table 1.Table 3
Weights of various equipment types of 110 KV and above substations.
Equipment typeHigh-voltage electrical equipmentIndoor equipmentCivil construction facilitiesWeight0.6710.2010.128Due to the similarity between substation and thermal power plant in component composition, damage type, and weight coefficient, this paper assumed that the dispersion of seismic vulnerability matrix distribution of thermal power plant and substation was relatively close. When their expected values of seismic damage index were the same, their variances of seismic damage index distribution were the same. Since there is no research on seismic vulnerability matrix of the thermal power plant, the variance of seismic damage index distributionσi2 is unknown. Therefore, based on the assumption, this paper adopted the variance of seismic damage index distribution in seismic vulnerability matrix of existing substations to obtain the variance of seismic damage index distribution of the thermal power plant under different seismic intensities by interpolation calculation.After the Wenchuan earthquake in 2008, Liu et al. conducted statistical analysis on 121 110 KV and above substations in Sichuan power grid and obtained the seismic vulnerability matrix of overall damage of substations [15]. The expected values and variances of seismic damage distribution indexes of substations under different seismic intensities are shown in Table 4.Table 4
Expected value and variance of seismic damage index of substation.
Seismic intensity67891011Expected value0.0250.0960.2260.5690.820.875Variance0.000210.013090.028930.078050.017770.005Based on the expected values and variances of seismic damage index distribution of existing substations and the expected value of seismic damage index of the thermal power plant, this paper used (3) to calculate the variance of seismic damage index distribution of the thermal power plant by piecewise linear interpolation:(3)δj2=δA2+Drj−DrADrB−DrAδB2−δA2.In (3), Drj is the expected value of seismic damage index of the thermal power plant under j intensity, δj2 is the variance of seismic damage index distribution of the thermal power plant under j intensity, DrA and DrB are the expected values of seismic damage index of the substation, and δ2A and δ2B are the variances of seismic damage index distribution of the substation.The variance of seismic damage index distribution of the thermal power plant is shown in Table5.Table 5
Expected value and variance of seismic damage index of the thermal power plant.
Seismic intensity67891011Expected value0.0150.0350.0730.1390.2480.420Variance0.000130.002060.008860.018310.032130.05665


## 3.1. Expected Value of Seismic Damage Index of the Thermal Power Plant

The thermal power plant is a huge and complex plant producing electric energy and thermal energy, which consists of the following five systems: (a) fuel system; (b) combustion system; (c) steam-water system; (d) electrical system; (e) control system. Although there are many types of equipment and complex components in the thermal power plant, the equipment and facilities can be mainly divided into three categories: civil construction facilities, electromechanical equipment, and electrical equipment.(a)
Civil construction facilities: main power house, control building, power distribution unit building, office building, etc.(b)
Electromechanical equipment: boiler, generator, steam turbine, condenser, etc.(c)
Electrical equipment: various high-voltage electrical equipment items, output conductor, etc.In the ATC-25 report of the American Applied Technology Council, the seismic vulnerability of thermal power plants was studied. When the overall seismic damage ratio for thermal power plants was calculated, the damage characteristics of civil construction facilities, electromechanical equipment, and electrical equipment were comprehensively considered. Due to the different functions and economic values of different equipment, the damage of different equipment accounts for different proportions in the evaluation of the overall damage of the thermal power plant. Based on the seismic damage data of thermal power plants in historical earthquakes in the United States, the ATC-25 report proposes the calculation formula for the seismic damage ratio of these three types of facilities and equipment and determines the weight proportions and regression coefficientsα and β of various facilities in the overall damage of the power plant which can be found in the Facility Class table in Appendix G (ATC-13).The calculation formula of seismic damage ratio of each equipment type is shown as the following formula (1) [9, 12]:(1)DMG=expα.jβ,In (1), DMG is the seismic damage ratio, j is seismic intensity, and α and β are the regression coefficients.The regression coefficients and weight of various equipment types are shown in Table1.Table 1
Weights and regression coefficients of various equipment types.
Equipment typeWeightαβCivil construction facilities0.2−12.024.526Electrical equipment0.3−14.055.61Electromechanical equipment0.5−14.815.778According to the weight and regression coefficients of various equipment types in Table1, combined with the calculation formula for seismic damage ratio of the thermal power plant equipment, the overall seismic damage ratio function of thermal power plant was calculated as follows:(2)Rj=0.2.e−12.02.j4.526+0.3.e−14.05.j5.61+0.5.e−14.81.j5.778.In (2), R(j) is the overall damage ratio of the thermal power plant under j seismic intensity and j is the seismic intensity.The expected value of the seismic damage index under a certain ground motion intensity is the sum of the product of the damage proportion of each damage level and the seismic damage index corresponding damage level. The expected value of the seismic damage index can represent the overall damage level of the thermal power plant and has the same practical significance as the overall damage ratio. Therefore, the overall damage ratio calculated by (2) was used as the expected value of seismic damage index of the thermal power plant [13]. The expected values of seismic damage indexes of the thermal power plant under different seismic intensities are shown in Table 2.Table 2
Expected values of seismic damage indexes of the thermal power plant under different seismic intensities.
Seismic intensity67891011Expected value0.0150.0350.0730.1390.2480.420


## 3.2. Variance of Distribution of Seismic Damage Index of the Thermal Power Plant

The substation is an important node in the electrical power system. It has many similarities with the thermal power plant in equipment composition and seismic damage characteristics: (1) The substation is also the main body composed of multiple types of equipment such as transformer and control equipment, which can be divided into three types of equipment such as civil construction facilities, outdoor high-voltage electrical equipment, and indoor equipment. (2) In historical earthquakes, there are two main reasons for the damage of substations. One reason is the direct damage caused by earthquakes, and the other reason is the indirect damage of internal facilities caused by the damage of civil construction facilities, which is basically the same as the overall damage of the thermal power plants. (3) Based on the actual earthquake damage of 110 KV and above substations and the asset value of various facilities in the Wenchuan earthquake, Liu et al. [14] determined the weight coefficients of the outdoor high-voltage electrical equipment, indoor equipment, and civil construction facilities of the substation. The weights of outdoor high-voltage electrical equipment, indoor equipment, and civil construction facilities of 110 KV and above substations are shown in Table 3, which are 0.671, 0.201, and 0.128, respectively, similar to the economic value weights of various equipment types of the thermal power plant in Table 1.Table 3
Weights of various equipment types of 110 KV and above substations.
Equipment typeHigh-voltage electrical equipmentIndoor equipmentCivil construction facilitiesWeight0.6710.2010.128Due to the similarity between substation and thermal power plant in component composition, damage type, and weight coefficient, this paper assumed that the dispersion of seismic vulnerability matrix distribution of thermal power plant and substation was relatively close. When their expected values of seismic damage index were the same, their variances of seismic damage index distribution were the same. Since there is no research on seismic vulnerability matrix of the thermal power plant, the variance of seismic damage index distributionσi2 is unknown. Therefore, based on the assumption, this paper adopted the variance of seismic damage index distribution in seismic vulnerability matrix of existing substations to obtain the variance of seismic damage index distribution of the thermal power plant under different seismic intensities by interpolation calculation.After the Wenchuan earthquake in 2008, Liu et al. conducted statistical analysis on 121 110 KV and above substations in Sichuan power grid and obtained the seismic vulnerability matrix of overall damage of substations [15]. The expected values and variances of seismic damage distribution indexes of substations under different seismic intensities are shown in Table 4.Table 4
Expected value and variance of seismic damage index of substation.
Seismic intensity67891011Expected value0.0250.0960.2260.5690.820.875Variance0.000210.013090.028930.078050.017770.005Based on the expected values and variances of seismic damage index distribution of existing substations and the expected value of seismic damage index of the thermal power plant, this paper used (3) to calculate the variance of seismic damage index distribution of the thermal power plant by piecewise linear interpolation:(3)δj2=δA2+Drj−DrADrB−DrAδB2−δA2.In (3), Drj is the expected value of seismic damage index of the thermal power plant under j intensity, δj2 is the variance of seismic damage index distribution of the thermal power plant under j intensity, DrA and DrB are the expected values of seismic damage index of the substation, and δ2A and δ2B are the variances of seismic damage index distribution of the substation.The variance of seismic damage index distribution of the thermal power plant is shown in Table5.Table 5
Expected value and variance of seismic damage index of the thermal power plant.
Seismic intensity67891011Expected value0.0150.0350.0730.1390.2480.420Variance0.000130.002060.008860.018310.032130.05665


## 4. Seismic Vulnerability Matrix Based on Seismic Intensity Fitted by Beta Distribution Function

### 4.1. Properties of Beta Function

The probability distribution range of most random variables is unbounded at one or both ends, but in some engineering applications, the values of random variables may have lower and upper limit values. In this case, it is more appropriate to use the probability distribution function with upper and lower limit values. If the random variable is bounded and has upper and lower limit values, the beta distribution function is one of the few appropriate distributions [16].The random variable of beta function has upper and lower limit values. When the upper and lower limit values of an independent variablex are [0,1], it can be called standard beta distribution.The probability density function of standard beta distribution is shown in the following formula:(4)BEx,a,b=xa−11−xb−1Ba,b,0≤x≤1,0,x<0orx>1.In (4), a ＞ 0, b ＞ 0, and the expression of B(a,b) is shown in the following formula:(5)Ba,b=∫01ta−11−tb−1dt.The expected value of random variable of standard beta distribution function is shown in the following formula:(6)Ex=aa+b.The variance calculation formula is as follows:(7)Vaxx=aba+b2a+b+1.Obviously, the probability density function of beta distribution conforms to the property of total probability distribution in the interval ofx ∈ [0,1]; that is,(8)∫01BEx,a,bdx=1.This has a similar property as the sum of the probability distribution of each damage level of the structure under a certain intensity in the seismic vulnerability matrix is 1. When the parametersa and b are not less than 1 simultaneously, the probability density function of the standard beta distribution has the property of single peak. Therefore, if the seismic damage index Dr of thermal power plant is taken as a variable, the beta distribution probability density function can be used to fit the seismic vulnerability matrix of the thermal power plant. Here, the seismic damage index Dr is a continuous variable.


### 4.2. Fitting of Seismic Vulnerability Matrix of Thermal Power Plant Based on Seismic Intensity

In the existing research results, the beta distribution function has been applied to fit the correlation function relationship in many fields, and a good fitting effect has been obtained. Liu et al. [17, 18] proposed to take the PGA as the fitting parameter, fit the probability distribution of each damage level of the structure under different seismic intensities with beta distribution function, and determine the seismic vulnerability matrix of the building structure. Since the sum of probability density of each damage level of seismic vulnerability matrix is 1, which has the same property as the probability density function of the standard beta distribution, the existing research results show that the result of fitting seismic vulnerability matrix using beta distribution function is good. Therefore, this paper selected the beta function to fit the seismic vulnerability matrix of the overall damage of the thermal power plant. Taking the seismic damage index as a continuous variable and the expected value and the variance of the seismic damage index under different seismic intensities as fitting parameters, the beta function was used to fit the seismic vulnerability matrix of the overall damage of the thermal power plant. The process was as follows:According to the relationship of the beta distribution function parametersa and b with the expected value and variance in (6) and (7), the values of the beta function parameters aj and bj under j intensity could be calculated by using the expected value and variance of the seismic damage index. The calculation formulas are as follows:(9)aj=Drj2−Drj3δj2−Drj,(10)bj=1−DrjDrj2−Drj3δj2−1.In (9) and (10), Drj is the expected value of seismic damage index under j intensity, and δj2 is the variance of seismic damage index distribution under j intensity.The parametersaj and bj were substituted into (4), and the probability density function of beta distribution of seismic damage index under j intensity is shown in the following formula:(11)fDr;aj;bj=Draj−11−Drbj−1∫01taj−11−tbj−1dt.The damage probability of each damage level of the thermal power plant under different seismic intensities was fitted and obtained by subsection integral calculation using (12), and finally the seismic vulnerability matrix of overall damage of the thermal power plant was obtained.(12)pij=∫Dri1Dri2fDr;aj;bjdDr.In (12), Dr is the continuous variable of seismic damage index, pij is the level i damage probability under j intensity, Dri1 is the lower limit value of seismic damage index interval of level i damage, and Dri2 is the upper limit value of seismic damage index interval of level i damage.The corresponding relationship between the range of value of seismic damage index and damage level of thermal power plant is shown in Table6.Table 6
Corresponding relationship between seismic damage index interval and seismic damage level of the thermal power plant.
Damage levelBasically intactSlightly damagedModerately damagedSeverely damagedDestructedSeismic damage index[0,0.04](0.04,0.08](0.08,0.35](0.35,0.70](0.70,1]As mentioned above, the expected value and the variance of seismic damage index of the thermal power plant in Tables2 and 4 were substituted into (9) and (10) to calculate and obtain the parameters aj and bj. The calculation results are shown in Table 7.Table 7
The fitting parametersaj and bj of probability density function of beta distribution of the thermal power plant.
Seismic intensity67891011a1.790.550.480.771.191.38b115.3714.986.154.773.611.91The fitting parametersaj and bj of the probability density function of beta distribution of thermal power plant in Table 7 were substituted into (11) to obtain the probability density distribution curve of beta distribution of the thermal power plant under different seismic intensities, as shown in Figure 2.Figure 2
Probability density distribution of seismic damage index of thermal power plant under different seismic intensities.It can be seen from Figure2 that the probability density value of seismic damage index of thermal power plants was relatively large in areas with a small seismic damage index when the seismic intensity was low. With the increase of seismic damage index, the probability density decreased rapidly, indicating that the damage of thermal power plants was mainly slight when the seismic intensity was low. With the increase of seismic intensity, the probability density value of seismic damage index of the thermal power plant gradually increased in areas with a larger seismic damage index, indicating that the thermal power plant began to have large damage level seismic damage when the seismic intensity was high.Based on the probability density function of beta distribution, the seismic vulnerability matrix of overall damage of the thermal power plant under different intensities was calculated by using (12), as shown in Table 8.Table 8
Seismic vulnerability matrix of the thermal power plant based on seismic intensity (%).
Seismic intensity67891011Basically intact96.3859.7842.2921.313.920.00Slightly damaged3.5728.0926.4824.7319.2310.59Moderately damaged0.0412.0828.7845.1052.6632.27Severely damaged0.000.052.438.6522.4442.25Destructed0.000.000.020.201.7414.89Based on the seismic vulnerability matrix of the thermal power plant fitted by the method in this paper, the expected values of seismic damage index of the thermal power plant under different seismic intensities were calculated. As shown in Table9, comparing the expected value of seismic damage index fitted by the beta function with the original expected value in American ATC-25 report, we find that all fitting errors of the expected values of seismic damage indexes under different seismic intensities were less than 0.03. This indicated that there was no systematic deviation in the fitted vulnerability matrix during back calculation of the expected value of seismic damage index. The beta function was used to fit the seismic vulnerability matrix of the thermal power plant, and the fitting effect was good.Table 9
Comparison between the fitted value and original value of the expected value of seismic damage index of the thermal power plant.
Seismic intensity67891011Fitted value0.0210.0550.0990.1630.2580.424Original value0.0150.0350.0730.1390.2480.420Error0.0060.0200.0260.0240.0100.004The seismic vulnerability matrix of thermal power plant fitted by the method in this paper showed the following: when the seismic intensity was 6 degrees, the thermal power plant was basically not damaged; when the seismic intensity was 7/8 degrees, the thermal power plant began to have slight and moderate damage, and the damage level was low; when the seismic intensity was 10 degrees, the probability of moderate seismic damage in thermal power plants was the highest, and some thermal power plants might be severely damaged; when the seismic intensity was 11 degrees, the damage of thermal power plant was relatively severe, the damage level was mainly severe damage, and the probability of seismic destruction was about 15%.


## 4.1. Properties of Beta Function

The probability distribution range of most random variables is unbounded at one or both ends, but in some engineering applications, the values of random variables may have lower and upper limit values. In this case, it is more appropriate to use the probability distribution function with upper and lower limit values. If the random variable is bounded and has upper and lower limit values, the beta distribution function is one of the few appropriate distributions [16].The random variable of beta function has upper and lower limit values. When the upper and lower limit values of an independent variablex are [0,1], it can be called standard beta distribution.The probability density function of standard beta distribution is shown in the following formula:(4)BEx,a,b=xa−11−xb−1Ba,b,0≤x≤1,0,x<0orx>1.In (4), a ＞ 0, b ＞ 0, and the expression of B(a,b) is shown in the following formula:(5)Ba,b=∫01ta−11−tb−1dt.The expected value of random variable of standard beta distribution function is shown in the following formula:(6)Ex=aa+b.The variance calculation formula is as follows:(7)Vaxx=aba+b2a+b+1.Obviously, the probability density function of beta distribution conforms to the property of total probability distribution in the interval ofx ∈ [0,1]; that is,(8)∫01BEx,a,bdx=1.This has a similar property as the sum of the probability distribution of each damage level of the structure under a certain intensity in the seismic vulnerability matrix is 1. When the parametersa and b are not less than 1 simultaneously, the probability density function of the standard beta distribution has the property of single peak. Therefore, if the seismic damage index Dr of thermal power plant is taken as a variable, the beta distribution probability density function can be used to fit the seismic vulnerability matrix of the thermal power plant. Here, the seismic damage index Dr is a continuous variable.


## 4.2. Fitting of Seismic Vulnerability Matrix of Thermal Power Plant Based on Seismic Intensity

In the existing research results, the beta distribution function has been applied to fit the correlation function relationship in many fields, and a good fitting effect has been obtained. Liu et al. [17, 18] proposed to take the PGA as the fitting parameter, fit the probability distribution of each damage level of the structure under different seismic intensities with beta distribution function, and determine the seismic vulnerability matrix of the building structure. Since the sum of probability density of each damage level of seismic vulnerability matrix is 1, which has the same property as the probability density function of the standard beta distribution, the existing research results show that the result of fitting seismic vulnerability matrix using beta distribution function is good. Therefore, this paper selected the beta function to fit the seismic vulnerability matrix of the overall damage of the thermal power plant. Taking the seismic damage index as a continuous variable and the expected value and the variance of the seismic damage index under different seismic intensities as fitting parameters, the beta function was used to fit the seismic vulnerability matrix of the overall damage of the thermal power plant. The process was as follows:According to the relationship of the beta distribution function parametersa and b with the expected value and variance in (6) and (7), the values of the beta function parameters aj and bj under j intensity could be calculated by using the expected value and variance of the seismic damage index. The calculation formulas are as follows:(9)aj=Drj2−Drj3δj2−Drj,(10)bj=1−DrjDrj2−Drj3δj2−1.In (9) and (10), Drj is the expected value of seismic damage index under j intensity, and δj2 is the variance of seismic damage index distribution under j intensity.The parametersaj and bj were substituted into (4), and the probability density function of beta distribution of seismic damage index under j intensity is shown in the following formula:(11)fDr;aj;bj=Draj−11−Drbj−1∫01taj−11−tbj−1dt.The damage probability of each damage level of the thermal power plant under different seismic intensities was fitted and obtained by subsection integral calculation using (12), and finally the seismic vulnerability matrix of overall damage of the thermal power plant was obtained.(12)pij=∫Dri1Dri2fDr;aj;bjdDr.In (12), Dr is the continuous variable of seismic damage index, pij is the level i damage probability under j intensity, Dri1 is the lower limit value of seismic damage index interval of level i damage, and Dri2 is the upper limit value of seismic damage index interval of level i damage.The corresponding relationship between the range of value of seismic damage index and damage level of thermal power plant is shown in Table6.Table 6
Corresponding relationship between seismic damage index interval and seismic damage level of the thermal power plant.
Damage levelBasically intactSlightly damagedModerately damagedSeverely damagedDestructedSeismic damage index[0,0.04](0.04,0.08](0.08,0.35](0.35,0.70](0.70,1]As mentioned above, the expected value and the variance of seismic damage index of the thermal power plant in Tables2 and 4 were substituted into (9) and (10) to calculate and obtain the parameters aj and bj. The calculation results are shown in Table 7.Table 7
The fitting parametersaj and bj of probability density function of beta distribution of the thermal power plant.
Seismic intensity67891011a1.790.550.480.771.191.38b115.3714.986.154.773.611.91The fitting parametersaj and bj of the probability density function of beta distribution of thermal power plant in Table 7 were substituted into (11) to obtain the probability density distribution curve of beta distribution of the thermal power plant under different seismic intensities, as shown in Figure 2.Figure 2
Probability density distribution of seismic damage index of thermal power plant under different seismic intensities.It can be seen from Figure2 that the probability density value of seismic damage index of thermal power plants was relatively large in areas with a small seismic damage index when the seismic intensity was low. With the increase of seismic damage index, the probability density decreased rapidly, indicating that the damage of thermal power plants was mainly slight when the seismic intensity was low. With the increase of seismic intensity, the probability density value of seismic damage index of the thermal power plant gradually increased in areas with a larger seismic damage index, indicating that the thermal power plant began to have large damage level seismic damage when the seismic intensity was high.Based on the probability density function of beta distribution, the seismic vulnerability matrix of overall damage of the thermal power plant under different intensities was calculated by using (12), as shown in Table 8.Table 8
Seismic vulnerability matrix of the thermal power plant based on seismic intensity (%).
Seismic intensity67891011Basically intact96.3859.7842.2921.313.920.00Slightly damaged3.5728.0926.4824.7319.2310.59Moderately damaged0.0412.0828.7845.1052.6632.27Severely damaged0.000.052.438.6522.4442.25Destructed0.000.000.020.201.7414.89Based on the seismic vulnerability matrix of the thermal power plant fitted by the method in this paper, the expected values of seismic damage index of the thermal power plant under different seismic intensities were calculated. As shown in Table9, comparing the expected value of seismic damage index fitted by the beta function with the original expected value in American ATC-25 report, we find that all fitting errors of the expected values of seismic damage indexes under different seismic intensities were less than 0.03. This indicated that there was no systematic deviation in the fitted vulnerability matrix during back calculation of the expected value of seismic damage index. The beta function was used to fit the seismic vulnerability matrix of the thermal power plant, and the fitting effect was good.Table 9
Comparison between the fitted value and original value of the expected value of seismic damage index of the thermal power plant.
Seismic intensity67891011Fitted value0.0210.0550.0990.1630.2580.424Original value0.0150.0350.0730.1390.2480.420Error0.0060.0200.0260.0240.0100.004The seismic vulnerability matrix of thermal power plant fitted by the method in this paper showed the following: when the seismic intensity was 6 degrees, the thermal power plant was basically not damaged; when the seismic intensity was 7/8 degrees, the thermal power plant began to have slight and moderate damage, and the damage level was low; when the seismic intensity was 10 degrees, the probability of moderate seismic damage in thermal power plants was the highest, and some thermal power plants might be severely damaged; when the seismic intensity was 11 degrees, the damage of thermal power plant was relatively severe, the damage level was mainly severe damage, and the probability of seismic destruction was about 15%.


## 5. Fitting of Seismic Vulnerability Curve of the Thermal Power Plant Based on PGA

Because the seismic intensity is assessed by factors such as the damage of building structure, people’s perception, and ground surface damage, it is not a physical quantity, so there is some uncertainty in the assessment result of seismic intensity. With the development of earthquake prevention and disaster reduction research in the world, PGA is widely used to represent the ground motion intensity, whether it is observation of ground motion intensity, aseismic design, or seismic vulnerability assessment of buildings, PGA is widely used to represent the ground motion intensity. In this paper, according to the basic corresponding relationship between seismic intensity and PGA, the seismic vulnerability matrix based on seismic intensity was converted into the seismic vulnerability matrix based on PGA by using the lognormal distribution function.Fitting seismic vulnerability curve with lognormal distribution function has been widely used in previous studies [19, 20]. In order to obtain detailed damage and loss status of thermal power plant under different PGAs, based on the fitted seismic vulnerability matrix, this paper used the two-parameter lognormal cumulative distribution function to fit the seismic vulnerability curve of the thermal power plant under different damage levels, and the two-parameter lognormal distribution function as the vulnerability function expression is shown as follows:(13)Fa=ϕlna/cζ.In (13), F(a) is the exceedance probability of the thermal power plant reaching a certain damage level; a is PGA; Φ is standard normal distribution function; and c and ζ are the median value and logarithmic standard deviation of vulnerability function, respectively.The calculation formula for the exceedance probability of damage level of the thermal power plant is shown as follows:(14)fij=∑j5pij.In (14), fij is the exceedance probability of level i damage under intensity j.There were many corresponding relationships between seismic intensity and PGA. In different conversion relationships, the corresponding PGA values of seismic intensity are often different, as shown in Table10. In order to meet China’s standards for seismic design and damage assessment of thermal power plants, the input parameters of seismic vulnerability matrix of the thermal power plant were converted from seismic intensity to PGA based on the corresponding relationship between seismic intensity and PGA in China’s Code for Seismic Design of Buildings (GB 50011-2010) [21].Table 10
Correspondence between seismic intensity and PGA.
Seismic intensity6 (g)7 (g)8 (g)9 (g)10 (g)11 (g)China’s Code for Seismic Design of Buildings0.050.10.20.40.81.6Trifunac–Brady0.070.130.260.531.052.10MSK0.050.100.210.420.841.67Gutenberg–Richter0.030.070.150.320.701.50Neumann0.060.130.260.511.022.04Although this paper used the conversion relationship between seismic intensity and PGA in China’s Code for Seismic Design of Buildings (GB 50011-2010), according to other corresponding relationships between seismic intensity and PGA, the method can still be used to obtain the seismic vulnerability matrix of the thermal power plant.According to the seismic vulnerability matrix of the thermal power plant in Table8 and corresponding values of seismic intensity and PGA in Table 10, the exceedance probability matrix of the thermal power plant with the PGA as input parameter was calculated by using (14), as shown in Table 11.Table 11
Exceedance probability matrix of the thermal power plant (%).
PGA/g0.050.10.20.40.81.6Basically intact3.6240.2257.7178.6996.08100.00Slightly damaged0.0512.1331.2353.9676.8589.41Moderately damaged0.000.052.458.8624.1957.14Severely damaged0.000.000.020.201.7414.89According to the corresponding relationship between PGAs and the exceedance probabilities of different damage levels in Table11, the cumulative function of the standard normal distribution shown in (13) was fitted by the least square method, and the seismic vulnerability function curve of the thermal power plant (the exceedance probability curve of each damage level) was calculated, as shown in Figure 3. The fitted median parameter value (c) and logarithmic standard deviation parameter value (ζ) of the transcendence probability function curve of each damage level are shown in Table 12.Figure 3
Seismic vulnerability curve of the thermal power plant.Table 12
Median value and logarithmic standard deviation of seismic vulnerability function of the thermal power plant.
Damage levelSlightly damagedModerately damagedSeverely damagedDestructedMedian (c)0.1600.3591.3943.203Log standard deviation (ζ)0.9971.1050.8580.666According to the calculated exceedance probabilities of different damage levels of the thermal power plant, the seismic vulnerability matrix of the overall damage of thermal power plant under different PGAs can be calculated using the following formula:(15)pia=Fi+1a−Fia.In (15), a is the PGA, pia is the probability of level i damage under the PGA a, Fi(a) is the exceedance probability of level i damage under the PGA a, and Fi+1a is the exceedance probability of level i + 1 damage under the PGA a.The calculated seismic vulnerability matrix of the thermal power plant based on PGA was shown in Table13:Table 13
Seismic vulnerability matrix of the thermal power plant based on PGA (%).
PGABasically intactSlightly damagedModerately damagedSeverely damagedDestructed0.0587.838.453.720.010.000.168.1319.5012.260.110.000.241.1529.0328.641.180.000.326.4230.0439.883.650.020.417.9028.2046.627.190.090.512.6525.5650.1811.340.260.69.2522.8651.6015.700.600.76.9420.3451.6219.981.120.85.3218.0950.7124.011.860.94.1616.1249.2227.672.831.03.3014.3947.3730.914.021.12.6612.8945.3333.705.431.22.1611.5843.1936.057.021.31.7810.4341.0337.978.791.41.489.4338.9039.5010.701.51.248.5436.8140.6712.731.61.057.7734.8141.5114.87According to the fitted seismic vulnerability curve of thermal power plant, the damage ratio curve of thermal power plant under different PGAs could be calculated. According to the corresponding relationship between seismic intensity and PGA in Table10, the relationship between seismic damage ratio and seismic intensity of the thermal power plant in ATC-25 report was converted into the corresponding relationship between seismic damage ratio and PGA, and the corresponding seismic damage ratio curve of thermal power plant could also be obtained.The comparison between the damage ratio for thermal power plants calculated by the method in this paper and the damage ratio for thermal power plants in ATC report is shown in Figure4.Figure 4
Comparison of seismic damage ratio curve of the thermal power plant.As shown in Figure4, the seismic damage ratio for thermal power plant calculated by the method in this paper was larger than that in ATC report, but the difference between two ratios was small, and these two ratios tended to be the same with the increase of PGA. The result of this comparison could not prove that the vulnerability curve was correct, but it could well demonstrate that when the seismic vulnerability curve established in this paper was used to calculate the seismic damage ratio of the thermal power plant under a given PGA, the achieved results would not deviate from the original basic assumption, namely, the results reported by ATC-25.


## 6. Conclusion

In this paper, a calculation method of seismic vulnerability matrix for the overall damage of the thermal power plant based on PGA is proposed by studying the seismic damage characteristics and seismic loss of the thermal power plant. The expected value and variance of seismic damage index of overall damage of thermal power plant are obtained based on the research results in ATC-25 report and the actual seismic vulnerability matrix of substations in Wenchuan earthquake. Thereafter, the probability density function of beta distribution is used to fit the seismic vulnerability matrix of thermal power plant under different seismic intensities. Finally, the cumulative function formula of standard lognormal distribution is used to fit the seismic vulnerability curve reflecting the overall damage of thermal power plant and thus determine the seismic vulnerability matrix of thermal power plant based on PGA. The overall seismic damage ratio of the thermal power plant calculated by this method is close to the calculation result of seismic damage ratio in ATC-25 report, which meets the expected requirements.Compared with the previous relevant studies on thermal power plants, this method obtains the seismic vulnerability matrix of the overall damage of thermal power plants, with the PGA as the parameter, and comprehensively analyzes the overall loss and damage of the thermal power plant. This method is concise and easy to implement. Due to the lack of data on actual seismic damage of thermal power plants, the relationship curve between seismic damage ratio and seismic intensity and the discrete parameters used in the fitting of the seismic vulnerability matrix of the thermal power plant still need to be further studied.



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*Source: 5799241-2022-04-13.xml*