As one of the pioneer works on FCs of electrical equipment Schiff and Newsom (1979) did a survey which shows that: 1) Most fragility data is based on engineering estimates rather than on the results of tests or analysis; 2) for many types of equipment, estimates are quite speculative; and 3) the range of estimates for a given type of equipment, for the few cases where there are more than one, often vary by a factor of three or more. Obviously, risk estimation based on such fragility data would have a very low level of reliability.
Years later, Huo and Hwang (1995) and also Hwang and Huo (1998) presented a seismic fragility analysis of equipment and structures in an electric substation in the eastern United States, using Substation 21 in Memphis as an example. Mentioning that seismic damage to electric facilities in the eastern United States has been rare and information on dynamic testing of electric equipment similar to those installed in the substation is not available; they used an analytical approach to perform the fragility analysis. They noted that the analytical method may not cover all the possible failure modes of substation structures and equipment. However, it should be mentioned that they have used spectral seismic response analysis, not THA.
Anagnos (1999) developed a database for an electrical substation equipment performance to evaluate the equipment fragilities. She considered performance of substation equipment in 12 California earthquakes. The majority of data were related to equipment operating at 220/230 kV and 500 kV. The purpose of the database was to provide a basis for developing or improving equipment vulnerability functions. The probabilities of failure were calculated by dividing the number of damaged items by the total number of items of that type at each site. Using peak ground acceleration (PGA) as the ground motion parameter, failure probabilities were compared with opinion-based FCs for a few selected equipment classes. Comparisons were somewhat crude as the calculated failure probabilities did not include information about the mode of failure. The comparisons indicated that some of the existing FCs provide reasonable matches to the data and others should be modified to better reflect the data.
Also Der Kiureghian (1999) presented fragility estimates for electric substation equipment by using Bayesian approach based on observed performance during past earthquakes. He presented point estimates of the fragility based on maximum likelihood, posterior mean and predictive analysis, as well as confidence intervals on fragility that reflect statistical uncertainties. He accounted for errors in the measurement of ground motion intensity and correlation between observations at the same substation. In that study it has been mentioned that the accuracy of the formulas proposed for the joint first-passage probability is highly dependent upon the accuracy of the marginal first-passage probability formulas. Therefore, it has been recommended to improve the accuracy of the formulas for marginal and joint first-passage probability, especially for the case of strongly narrowband response.
In 2003, Shinozuka and colleagues presented experimental curves as two-parameter normal distribution functions for which the data were obtained from Kobe earthquake of 1995. Apart from the vulnerability of transformers, the seismic vulnerability of other equipment, such as circuit breakers and disconnect switches, was integrated into the analysis by using corresponding FCs. Paolacci and Giannini (2005) evaluated the seismic fragility of electrical insulators, by using spectral acceleration as the IM. Jaigirdar (2005) analyzed the seismic risk index of electric power substations of Hydro-Quebec. Mentioning that vulnerability of substations largely depends on the performance of circuit breakers and control buildings during earthquakes, they identified the critical parameters responsible for vulnerability of substations by statistical analysis of the field data. They also studied the correlation of different parameters with vulnerability, sensitivity of the weighting factors of critical parameters and sensitivity of seismic exposure levels to seismic risk index by statistical analysis
Straub and Der Kiureghian (2008) worked on improved seismic fragility modeling from empirical data. Their improved empirical fragility model addresses statistical dependence among observation of seismic performances which arises from common but unknown factors influencing the observation. Their proposed model accounts for this dependence by explicitly including common variables in the formulation of the limit state for individual components. Additionally, the fact that observations of the same component during successive earthquakes are correlated is considered in the estimation of the model parameters
It should be notified that although the improved formulation proposed by Straub and Der Kiureghian (2008) can lead to significantly different fragility estimates than those obtained by using the conventional assumption of statistical independence among the empirical observation, still there is a remarkable difference between the developed curves and the observed data. They have expressed that, consideration of statistical dependence among observations leads to larger uncertainty on the estimated fragility. This is because dependence among observation reduces the information content of the data. It follows that neglecting these dependences, as in the conventional approach, may lead to a serious underestimation of the statistical uncertainty and overconfidence in the estimated fragility.
Bradley (2010) has discussed the epistemic uncertainties in component fragility functions used in performance-based earthquake engineering. He has expressed that there exist many uncertainties in the development of such fragility functions, and has presented and discussed the sources of epistemic uncertainty in fragility functions, their consideration, combination, and propagation. He has used two empirical fragility functions presented in literature to illustrate the epistemic uncertainty in the fragility function parameters due to the finite size of the datasets (see Fig. 1)
Bradley has claimed that those examples and the associated discussions illustrate that the magnitude of epistemic uncertainties are significant and there are clear benefits of the consideration of epistemic uncertainties pertaining to the documentation, quality assurance, implementation, and updating of fragility functions. He has concluded that epistemic uncertainties should therefore always be addressed in future fragility functions developed for use in seismic performance assessment. Porter and colleagues (2010) have provided fragility functions for 52 varieties of mechanical, electrical, and plumbing equipment of commercial and industrial buildings.
There are large dispersions in the data points used for developing the FCs, which shows their relatively low level of reliability. In a study by Cavalieri and colleagues (2011) on fragility functions of electric power stations, fragility values for PGA value of 0.3g varies from 0.05 in Metcalf 230 kV substation to 0.98 for San Mateo 230 kV substation, as shown in Fig. 2.
Another case of large variation in the fragility values can be found in the curves presented in a by Buriticá (2013), who conducted a study on seismic vulnerability assessment of power transmission networks based on system thinking approach, and presented a set of FCs for low, medium and high voltage substations. Fig. 3 shows the curves presented for high-voltage substations
It is observed in Fig. 3 that there are remarkable differences between the presented FCs of HAZUS and other curves. For example, the values of exceedance probability corresponding to 0.2g vary from around 5%, given by HAZUS, to almost 70%, presented by Hwang and Huo (1998). These variations and difference show the very low level of reliability in the FCs presented in almost all of the previous studies.
This low level of reliability has encouraged the designers of such systems in recent years to focus more on more reliable techniques such as the use of dampers for seismic response reduction (Yue et al. 2019). A few reasons can be mentioned for the aforementioned low level of reliability of FCs presented in previous works of different scholars. One is the age or the passed time of the operation lifetime of the equipment which may make it more vulnerable against earthquake. For example, the fatigue conditions resulted from Aeolian vibrations in the conductors of power transmission lines, (Zhao et al. 2020), can create near-failure conditions in the conductor just before the earthquake. Another reason is believed to be the factors that can have either increasing or decreasing effect on the seismic response of the electric equipment, such as the coupling effect with the adjacent equipment (Yang 2021). Finally, another reason, which is the main incentive for developing the present paper, is believed to be the high dispersion in the data used for this purpose. Therefore, it would be quite desired to decrease this dispersion in some logical ways. In the following sections of this paper first the studies on improvement of fragility functions are briefly reviewed, and then three means are proposed for reducing the dispersion of the data used for development of FCs.