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Seismic risk assessment of high-voltage transformers using Bayesian belief networks Siraj, Tammeen 2013

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SEISMIC RISK ASSESSMENT OF HIGH-VOLTAGE TRANSFORMERS USING BAYESIAN BELIEF NETWORKS  by Tammeen Siraj  B.Sc., Bangladesh University of Engineering & Technology, Bangladesh, 2009  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE COLLEGE OF GRADUATE STUDIES (Civil Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan)  April 2013 © Tammeen Siraj, 2013  Abstract  Past earthquake records showed that a large magnitude earthquake can cause severe damage to high-voltage substations, which may lead to power disruption for a significant amount of time. A high-voltage transformer is one of the key components of a substation. This thesis proposes a probabilistic framework using Bayesian belief network (BBN) model to predict the vulnerability of a high-voltage transformer for a seismic event. BBN has many capabilities that make it well suited for the proposed risk assessment method. This thesis considers past studies, expert knowledge and reported causes of failures to develop an initial integrated risk assessment framework that acknowledges multiple failure modes. Therefore, the framework incorporates major causes of transformer vulnerability due to seismicity, such as liquefaction, rocking response of transformer, or interaction between interconnected equipment. To demonstrate the application of this framework, this thesis elaborates each step of the framework. Finally, the sensitivity analysis was carried out to evaluate the effects of input variables on transformer damage. The paper also illustrates two predictive models using response surface method (RSM) and Markov chain. The proposed framework is particularly handy to perform, and the results can be useful to support decisions on mitigation measures and seismic risk prediction.  ii  Table of Contents Abstract .................................................................................................................................... ii Table of Contents ................................................................................................................... iii List of Tables ........................................................................................................................... v List of Figures ......................................................................................................................... vi List of Notations ................................................................................................................... viii Acknowledgements ................................................................................................................ ix Dedication.……………………………………………………………………………….….. x Chapter 1: Introduction ........................................................................................................ 1 1.1  Motivation………………… ............................................................................................ 2  1.2  Objective and Organization of the Thesis ........................................................................ 3  Chapter 2: Literature Review ............................................................................................... 5 2.1  Past earthquake performance of high-voltage substation components ............................ 5  2.2  Previous vulnerability studies on substation .................................................................... 8  2.3  Methods for risk assessment .......................................................................................... 11  2.4  Bayesian belief network ................................................................................................. 14  Chapter 3: Proposed Bayesian Belief Network ................................................................. 17 3.1  Ground motion intensity measure .................................................................................. 19  3.2  Soil instability ................................................................................................................ 23  3.3  Interaction coming from the conductors ........................................................................ 28  3.4  Rocking response of transformers ................................................................................. 33  3.5  Transformer vulnerability .............................................................................................. 38  3.6  Sensitivity analysis......................................................................................................... 42  Chapter 4: Predictive Model Development ....................................................................... 45 4.1  Model development using Markov chain ...................................................................... 45  4.2  Predictive model development using RSM.................................................................... 50 4.2.1  Screening of input variables................................................................................ 52  4.2.2  Prediction equation using Response Surface Method (RSM)............................. 56  4.2.3  Model validation ................................................................................................. 62  Chapter 5: Conclusion and Future Work.......................................................................... 65 iii  References .............................................................................................................................. 68 Appendices ............................................................................................................................. 77 Appendix A: NEHRP Site Class with Recommended Values of Average Shear-wave Velocity……………. .................................................................................................. ...77 Appendix B: Description of the CPT for Node Variable “Conductor Failure” .......... 78 Appendix C: Description of the CPT for Node Variable “Rocking Response of Transformer” …………………………………………………………………………..79  iv  List of Tables Table 1.1  Summary of power disruption and monetary loss due to substation damage during major earthquakes……………………………….……………………… 2  Table 2.1  Studies on vulnerability assessment of substation components……………….. 8  Table 3.1  Description of basic input parameters for “ground motion intensity measure”.. 22  Table 3.2  Snapshot of the CPT for node variable “PGA”…………………………………23  Table 3.3  Description of basic input parameters for “soil instability”…………………….25  Table 3.4  Snapshot of the CPT for node variable “liquefaction”……………………….... 27  Table 3.5  Typical equipment displacement (ASCE 1999)……………………………….. 30  Table 3.6  Calculated relative displacements between transformer and disconnect switch……………………………………………………………………………31  Table 3.7  Description of basic input parameters for IC: existing conductor length (ECL)/ required conductor length (RCL)…………………………………………….…32  Table 3.8  Snapshot of the CPT for node variable “conductor failure”……………………33  Table 3.9  Governing conditions for rest, slide, and rock modes…………………………. 34  Table 3.10  Description of basic input parameters for “Rocking response of transformer... 36  Table 3.11  Snapshot of the CPT for node variable “RT”…………………………………...37  Table 3.12  Causes and effects of failure of transformer components………………………39  Table 3.13  Description of CPT for node variable “foundation failure”…………………….40  Table 3.14  Description of CPT for node variable “failure of component class 1”…………40  Table 3.15  Description of CPT for node variable “failure of component class 2”…………41  Table 3.16  Description of CPT for node variable “transformer damage”…………………. 41  Table 3.17  Sensitivity analysis for “transformer damage” using proposed BBN model...... 44  v  List of Figures Figure 1.1  Basic structure of power system (Adapted from United States Department of Energy (2004))…………………………………………………………………. 1  Figure 2.1  Overturned electrical equipment at Sylmar Converter Station during 1971 San Fernando earthquake (Magnitude 6.6) (Makris and Zhang 1999)……………... 6  Figure 2.2  66 kV broken transformer bushing at Bromley substation during 2011 Christchurch earthquake (Magnitude 6.3), (Eidinger and Tang forthcoming).... 7  Figure 2.3  A sample Bayesian probabilistic network……………………………………... 15  Figure 3.1  Conceptual BBN for high-voltage transformers………………………………. 18  Figure 3.2  Proposed BBN for seismic risk assessment of transformer…………………….19  Figure 3.3  Attenuation of PGA as a function of site to fault distance (for soil type: NEHRP class E)………………………………………………………………...21  Figure 3.4  PGA vs. liquefaction occurrence (based on the historical data provided in Timothy and Scott (1995))……………………………………………………...26  Figure 3.5  Liquefaction occurrence versus σvo, σ'vo, qc, and D50…………………………...27  Figure 3.6  Behaviour of cable connected equipments (power transformer and disconnect switch) during an earthquake event……………………………………………..29  Figure 3.7  Boundaries of rest, slide, and rock modes, for H/B=2 (based on Shenton (1996))…………………………………………………………………………..35  Figure 3.8  Boundaries of rest, slide, and rock modes to calculate CPT……………………38  Figure 4.1  Simulated data plotted for (a) Low, (b) Medium and (c) High damage states of anchored transformer…………………………………………………………... 46  Figure 4.2  Simulated data plotted for (a) Low, (b) Medium and (c) High damage states of unanchored transformer………………………………………………………... 47  Figure 4.3  Transition probabilities calculated using Markov chain for anchored transformer……………………………………………………………………...50  Figure 4.4  Transition probabilities calculated using Markov chain for unanchored transformer……………………………………………………………………...50  Figure 4.5  Screening of important variables for transformer (Anchored) damage probability calculation using a Pareto chart…………………………………….53 vi  Figure 4.6  The internally studentized residuals and normal % probability plot of tranformer (Anchored) damage probability for the Resolution IV design…….. 54  Figure 4.7  Screening of important variables for transformer (Unanchored) damage probability calculation using a Pareto chart…………………………………….55  Figure 4.8  The internally studentized residuals and normal % probability plot of tranformer (Unanchored) damage probability for the Resolution IV design…...56  Figure 4.9  The internally studentized residuals and normal % probability plot of tranformer (Anchored) damage probability for quadratic model……………….57  Figure 4.10 The actual and predicted plot of tranformer (Anchored) damage probability using quadratic model…………………………………………………………..58 Figure 4.11 The internally studentized residuals and normal % probability plot of tranformer (Unanchored) damage probability for quadratic model…………….59 Figure 4.12 The actual and predicted plot of tranformer (Unanchored) damage probability using quadratic model…………………………………………………………..60 Figure 4.13 3-D interaction plot for d (fault to site distance in km) and ST (soil type, defined by shear wave velocity of soil, in m/sec) in the quadratic model (for anchored transformer)…………………………………………………………..61 Figure 4.14 3-D interaction plot for d (fault to site distance in km) and ST (soil type, defined by shear wave velocity of soil, in m/sec) in the quadratic model (for unanchored transformer)………………………………………………………..62 Figure 4.15 Bar charts showing the range of values of input variables…………………….. 63 Figure 4.16 Model validation for anchored transformer……………………………………. 64 Figure 4.17 Model validation for unanchored transformer………………………………… 64  vii  List of Notations BBN  Bayesian belief network  PGA  Peak ground acceleration  IC  Interaction coming from the conductors  RT  Rocking response of transformer  Mw  Earthquake magnitude  d  Site to fault distance  ST  Soil type  FM  Fault mechanism  EL  Extremely low  VL  Very low  L  Low  M  Medium  H  High  VH  Very high  EH  Extremely high  EEH  Extremely extremely high  D50  Average grain size  qc  CPT tip resistance  σvo  Effective vertical overburden pressure  σ’vo  Effective vertical overburden pressure  Drel  Maximum  horizontal  relative  displacement  between  adjacent  equipment  μs  Static coefficients of friction  B/H  Width to height ratio of transformer  RSM  Response surface method  viii  Acknowledgements  In the name of Allah, the Most Gracious and the Most Merciful All praises go to the almighty Allah to give me the strength and his blessing in completing this thesis. I express my sincere gratitude to my supervisor, Dr. Solomon Tesfamariam, who has inspired and guided me all through my MASc program. I would like to thank him for his believe in my potential. His constructive comments and suggestions throughout the thesis works have made a significant contribution to the success of this research. I offer my enduring gratitude to the faculty, and staff at the UBCO. I would also like to acknowledge Natural Sciences and Engineering Research Council of Canada (NSERC)’s support in this research work. I owe particular thanks to Dr. Leonardo Dueñas-Osorio and Dr. Goutam Mondal, whose invaluable suggestions taught me to question more deeply. Special thanks are owed to my parents for all their love, patience, and encouragement. They are the key source of inspiration for all my achievements. I am also thankful to those who indirectly contributed in this research, your kindness means a lot to me.  ix  DEDICATED TO MY PARENTS  x  Chapter 1: Introduction An electrical power transmission system is comprised of power generating stations,  substations, transmission lines, and supervisory control and data acquisition  (SCADA) facilities. Substations connect all other components (Figure 1.1) and transfer powers in different voltage levels. They perform many functions such as: change AC voltages from one level to another; switch generators, equipment, and circuits or lines in and out of a system, etc. Substations are comprised of different components, e.g., disconnect switch, circuit breaker, transformer, lightning arrester, control house, protection and control equipment (Blume 2007).  Figure 1.1 Basic structure of power system (Adapted from United States Department of Energy (2004))  Seismic shaking may cause damage to the substations in a power system. Failure of an electrical substation can cause power outages of varied amounts of time (Table 1.1). The consequence of those substation damages can lead to several hours of power outage and monetary loss. Table 1.1 presents a summary of power disruption and monetary loss during major earthquakes. 1  Table 1.1 Summary of power disruption and monetary loss due to substation damage during major earthquakes Earthquakes 1988, Saguenay earthquake 1989, Loma Prieta earthquake  1.1  Magnitude 5.9 6.9  1994, Northridge earthquake  6.7  2010, Christchurch earthquake  7.1  1971, San Fernando Earthquake  6.6  Power Disruption and Monetary Loss Power outages in Quebec City; complete power restoration occurred 9 hours after the earthquake (Mitchell et al. 1990). Electricity was lost to about 1,400,000 customers due that earthquake (Schiff 1998). Power outages in Los Angeles area lasted from a few seconds to several days; power restoration was done to all major substations and to about 95% of the customers within 24 hours (Schiff 1997). It took a day to achieve 90% restoration during that earthquake (Eidinger and Tang forthcoming). Estimated loss $22,000,000 due to component damage of Sylmar Converter station (500 kV DC), or 40 percent of the total value of loss (Eidinger and Ostrom 1994).  Motivation Experiences from past earthquake records showed that large magnitude earthquakes  could cause severe damage to substations and result in major service disruption of a power system. High-voltage substation components that have been designed without full consideration of the site seismicity and/or designed before the introduction of modern seismic design codes are the most vulnerable (Anagnos 1999). The commonly observed failure modes could be listed as failures in transformer anchorage, oil leakage from bushings and different parts of the transformers, shattering of porcelain columns, etc. Given diverse classes of assets exposed to seismic hazard, consistency in the process of risk assessment and management becomes a challenging task (Nuti et al. 2007). A key component of a substation is the transformer which is one of the single largest capital investments (60% of the total investment). Hence, replacing a damaged transformer with a new one or keeping a spare transformer in a substation is expensive. Also, according to the past earthquake observation, 2  the damage rate of high-voltage transformer is low, but their failure has high consequences (Abdelmoumene and Bentarzi 2012; Gaunt and Coetzee 2007); therefore seismic risk assessment of a transformer is of vital importance. Since, Bayesian belief network (BBN) takes into account the uncertainty in the modeling and aggregation of the different knowledge base, this thesis demonstrates a risk assessment framework using Bayesian belief network for high-voltage transformers in a substation. This thesis uses seismic vulnerability and risk in an alternative way as the definition of the vulnerability follows the same logic as that of risk (Aven 2008). Risk is related to future events (i.e., a seismic event) and their consequences (i.e., failure of substation components), and vulnerability is related to the combination of consequences (i.e., failure of substation components) and associated uncertainty (i.e., uncertainty of the consequences). Therefore, vulnerability is a feature of risk (Aven 2008). Risk assessments are crucial as they reduce the risks of unwanted events, which could be very costly both physically and financially. It is essential for Governments and decision makers to evaluate the seismic risk of substations. It directly contributes to the improvement of safety and security in a power system against seismic hazard of varying magnitudes. 1.2  Objective and Organization of the Thesis The objective of this research is to develop a risk assessment framework for high-  voltage transformers by combining current knowledge in the areas of seismic hazard. This thesis includes the use of conventional methodology to quantify site seismicity and the other effects that may induce from site seismicity (i.e., liquefaction, rocking response, conductor interaction of the interconnected equipment). Finally, a risk assessment framework is 3  developed to quantify the probability of failure of a high-voltage transformer using Bayesian belief network in the event of an earthquake. The thesis is divided into five chapters. Chapter 1 introduced the study by describing the objectives and motivation. Chapter 2 presents a state of the art review, explaining the past earthquake performances of substations. The chapter also discusses previous vulnerability studies on substations and its components. Later in this chapter, basics of risk assessment and a brief idea of Bayesian belief network are discussed. Chapter 3 elaborates on the proposed framework using Bayesian belief network and the sensitivity analysis using the proposed framework. Chapter 4 presents the predictive model development using response surface method and Markov chain separately. Finally, Chapter 5 presents a summary of the research and the conclusions. In addition, limitations of the current study and suggestions for future improvement are also presented in this chapter.  4  Chapter 2: Literature Review High-voltage equipment (e.g., porcelain members, transformers, bushing) are the most vulnerable parts of the substation during an earthquake (Anagnos 1999). Performance evaluation of high-voltage substation equipment during a seismic event has become a crucial issue. For substations and its components, this chapter reviews past earthquake induced damage and current state of practice on vulnerability assessment. Finally, a brief review of risk assessment, with more emphasis on Bayesian belief network, is provided. 2.1  Past earthquake performance of high-voltage substation components Typically, voltage rating of components of a substation is one of the factors that are  related to its seismic vulnerability (Stewart et al. 2003). Equipment with 115kV or below showed satisfactory performance during past earthquakes, if seismic installation practices of anchorage and conductor interconnection flexibility were present (Anagnos 1999). Different types of damages in substations observed from the past earthquake records are (ASCE 1999; Schiff 2003): • • • • • •  •  Leaking or breaking of bushing, Falling of inadequately anchored rail-supported transformers from the elevated platforms, Damage of bushings and post insulators, Failure of cast-aluminum hardware, Failure of porcelain insulator, Tilting of lighting arresters, and Tilting of dead end transmission tower.  Past earthquake records showed that damage to substation components may lead to severe consequences. For example, the 1994 Northridge earthquake caused severe damage to the electric power facilities of Los Angeles. The damage to electric power could have far 5  reaching consequences, as well. British Columbia, Montana, Wyoming, Idaho, Oregon and Washington experienced power outages due to the damage to substations in the Los Angeles area (Anagnos 1999; Schiff 1997). The Los Angeles Department of Water and Power’s (LADWP) Sylmer Converter Station suffered severe damage (Figure 2.1) that included transformer bushings, lightning arresters, disconnect switches, circuit switchers, bus supports, and potential measuring devices (Eidinger and Ostrom 1994). Similar damages were found in the 1988 Saguenay earthquake, the 1989 Loma Prieta earthquake, the 1995 Kobe earthquake, the 2010 Christchurch earthquake, etc.  Figure 2.1 Overturned electrical equipment at Sylmar Converter Station during 1971 San Fernando earthquake (Magnitude 6.6) (Makris and Zhang 1999)  Several components of transformers were also found damaged from different earthquakes (ASCE 1999). These include damage to anchorage, radiators, bushings, conservators, lightning arresters, sudden pressure and protective relays, tertiary bushings, etc. Figure 2.2 shows a photograph of a broken transformer bushing during the 2010 Christchurch earthquake. Also damages due to liquefaction were reported (Eidinger and Tang 6  forthcoming). Among 300 substations in Christchurch, two of the substations failed to operate due to liquefaction.  Figure 2.2 66 kV broken transformer bushing at Bromley substation during 2011 Christchurch earthquake (Magnitude 6.3), (Eidinger and Tang forthcoming)1  Based on the past performance of electrical substation components, it is evident that, mitigation measure has to be performed to improve performance during an earthquake. Recently, many substations are in the process of implementing mitigation measures which have performed relatively well during an earthquake event. For example, the electrical power companies in New Zealand implemented mitigation measures to some of the substations and  1  With permission from American Society of Civil Engineers (ASCE)  7  as a result, the substations performed relatively well during the Christchurch earthquake (2010) (Eidinger and Tang forthcoming). During the 1994 Northridge earthquake, it was also observed that equipment designed after the 1971 San Fernando earthquake suffered less damage comparing to older equipment (Jaigirdar 2005). 2.2  Previous vulnerability studies on substation Several experimental and analytical studies have been conducted for evaluating the  seismic performance of different substation components. A list of different substation vulnerable studies is shown in Table 2.1. Table 2.1 Studies on vulnerability assessment of substation components Substation components Transformer and bushing system  Rocking response and overturning of equipment Rigid electrical equipment anchored to a base foundation (e.g., transformer) Electrical substation equipment connected by nonlinear rigid bus conductors Cable-connected equipment items High-voltage disconnect switch 550 kV porcelain transformer bearings 230-kV porcelain transformer bushings High-voltage substation disconnect switches High-voltage transformer-bushing systems  Reference Ersoy (2002); Ashrafi (2003);Ersoy and Saadeghvaziri (2004); Saadeghvaziri et al. (2010); Ersoy (2001); Filiatrault and Matt (2006) Makris and Roussos (1998); Makris and Zhang (1999) Makris and Black (2001, 2002) Song et al. (2007) Hong et al. (2001) Paolacci and Giannini (2009) Gilani et al. (1999a) Gilani et al. (1999b) Whittaker et al. (2007) Whittaker et al. (2004)  The previous vulnerability studies showed that the analytical and experimental works are undertaken for a particular failure mode. Song et al. (2007) and Hong et al. (2001) worked on the response of cable connected equipment; Ersoy and Saadeghvaziri (2004) 8  worked on seismic interaction of transformer and bushing system; Makris and Roussos (1998) worked on rocking response and uplifting of equipment, etc. Combining all the failure modes together will be a challenging task, but it will help to start integrating such diversity of failure modes to give a better understanding on component vulnerability. Other studies that have been performed related to substation and its components vulnerability are discussed in the following paragraphs. Anagnos (1999) developed fragility curves for different equipment of a substation for specific types of failure modes. Data from twelve earthquakes which occurred between 1971 and 1994 was used to develop these fragility curves. Anagnos (1999) did not consider the statistical dependence among the observations while estimating the equipment fragilities. The dependency mainly comes when any equipment experiences multiple earthquakes. Straub and Der Kiureghian (2008) showed that the conventional approach of fragility estimation using the past earthquake damage data differs significantly if they introduce effects of statistical uncertainty and component statistical dependence on the system fragility. Developing fragility curves based entirely on the available damage data is not always sufficient as damage data is not enough to adequately define a fragility curve (e.g., there could be lack of data or incomplete data or missing failure modes) (Anagnos 1999). On the other hand, Hwang and Chou (1998) used risk assessment tools (i.e., fault tree and event tree) to evaluate the performance of an electrical substation for a seismic event. In their study, fault tree and event tree demonstrate the interrelationship between the substation components. Fragility of a substation was determined by using the fragility of individual components in event tree and fault tree. The fragilities of individual components were determined by analytical or experimental analysis by Huo and Hwang (1995). 9  Matsuda et al. (1991) developed a method to assess substation damage and system impacts during an earthquake event. The method includes, 1) identifying scenario earthquake (high probability, large magnitude earthquake), 2) selecting and ranking the substations based on their exposure to scenario earthquakes and their importance to the system, 3) performing additional assessment on highest ranked substations, 4) assessing damage on the system due to scenario earthquakes, and 5) finally, developing seismic hazard mitigations. Their methodology was tested and verified for the Loma Prieta earthquake. Their assumptions were that 1) soil condition has a significant effect on substation components, 2) dead tank and bulk oil circuit breakers would not be damaged during the earthquake, 3) most of the damage would occur to the 500kV substation equipment, and 4) substation equipment below 115kV voltage level was unlikely to be damaged. Those assumptions matched with the actual scenario during the Loma Prieta earthquake. The previous studies reflected that work has been done to assess vulnerability of substation as a whole and its individual components. To evaluate the whole substation vulnerability, it is essential to get an idea of the vulnerability of its components. Vulnerability assessment of substation components could also be achieved by using risk assessment tools, other than developing analytical model or experimental analysis. When it comes to measuring the seismic risk of substations on a vast scale, analytical and experimental work may not be suitable. As analytical model is time consuming and computationally intensive, on the other hand, experimental work is expensive. The following section discusses the basics of risk assessment.  10  2.3  Methods for risk assessment In a very broader sense, risk can be defined as the combination of the probability, or  frequency, of occurrence and the consequence of a specific hazardous event (Sušnik et al. 2010). While doing risk assessment, it is necessary to integrate knowledge with diverse nature (i.e., qualitative and quantitative). Risk assessment can be done by specially focusing on qualitative knowledge. For example, organization and human analyses are more naturally modeled with qualitative knowledge and some of the qualitative techniques of risk assessment are Hazard and Operability Analysis (HAZOP), what-if analysis, preliminary risk analysis, Hazard Assessment Critical Control Points (HACCP), etc (Aven 2008). On the other hand, technical level risk assessment is usually modeled with quantitative information. There are several quantitative risk assessment techniques such as, Fault Tree Analysis (FTA), Event Tree Analysis (ETA), Failure Mode and Effects Analysis (FMEA), barriers and bowtie diagrams, etc. (Aven 2008). For seismic risk assessment of critical substation component (i.e., transformer in this thesis), the main characteristics to be modeled are stochastic behavior of earthquake ground motion (Song et al. 2004), induced effects from ground motion (i.e., liquefaction, rocking response) (ASCE 1999), behavior and performance of interconnected equipment (Der Kiureghian et al. 1999a),  the complexity of the equipment itself, the integration of  qualitative information with quantitative knowledge, the multi-state nature of equipment, the dependencies between events such as failures, and uncertainties on the parameter estimation (Weber et al. 2010), etc. Thus, classical dependability (i.e., availability, safety, reliability, maintainability, integrity) analysis methods, such as failure mode and effect analysis (Abdelgawad and Fayek 2012), fault trees (Bobbio 1999; Porter et al. 2006), Markov chains 11  (Yi et al. 2011), Petri nets (Wakefield and Sears 1997), and Bayesian belief networks (BBN) (Bensi et al. 2011; Liu et al. 2011), can be used. The BBN provides a robust probabilistic method of reasoning under uncertainty (Bobbio et al. 2001). Recent literature shows a growing interest on BBN, because of its usefulness in risk assessment of complex systems (Boudali and Dugan 2005; Langseth and Portinale 2007; Mahadevan et al. 2001). A brief discussion of each method with relation to BBN is provided in the following paragraphs. The discussions highlights that BBN has many capabilities that make them well suited for the risk assessment of substation components. This thesis uses the BBN as a risk assessment tool to perform vulnerability assessment of high-voltage transformers. It has been successfully applied in a variety of real-world tasks, and their suitability for risk assessment is considered by several researchers (e.g., Bensi et al. 2011; Liu et al. 2011; Tesfamariam and Liu 2013; Tesfamariam et al. 2011). Failure mode and effect analysis (FMEA) techniques are widely used to identify potential failure modes, to assess the risk associated with those failure modes, to rank the issues in terms of importance and to identify and carry out corrective actions to address the most serious concerns (Lee 2001). FMEA is an effective fault analysis technique but the ability of inference is not very enough and FMEA method is not suitable to use the fault related symptoms to do some posterior inference (Yang et al. 2009). In addition to that, this method is difficult to use in decision making and reuse, on contrary BBN is excellent at them (Chen et al. 2010). BBN can be constructed from FMEA and the events that cannot be done in FMEA, BBN takes care of them (Shi and Wang 2004; Yang et al. 2009).  12  Fault tree (FT) analysis is one of the most popular techniques for dependability analysis. The FT is a graphical model of various combinations of faults that may present as parallel and sequential way, which will result in the occurrence of the predefined undesired event. FT can easily be mapped into a BBN. By using BBN in comparison with FT, it is possible to obtain some additional power both at modeling and at the analytical level. Bobbio (1999) explained the different issues of FT and BBN. FT analysis could give exactly the probability of failure of a system or equipment. The problem arises when it deals with multiple failure of components that lead to several different consequences on the system (Weber et al. 2010). That is a common case in risk and dependability analysis, and those models need to be represented by multiple state variables. In that context, FT is not a preferable option. Another problem with FT analysis is that, it is limited to one top event. On the other hand, BBN has the similar capabilities as FT; in addition, BBN allows us to do multistate variable modeling and enables us to assess several output variables in the same model. Therefore, it is possible to represent FT as BBN, but the reciprocity is not always true. Markov chain (MC) is another suitable method for reliability (i.e., continuity of correct service of the system), and availability (i.e., readiness for correct service of the system) studies of systems. The analysis evaluates exact failure probability, even if there is presence of dependencies among components. It is possible to integrate diverse knowledge and to represent multistate variables with the help of MC. One of the main disadvantages of MC is the analysis becomes complex when a large number of variables are considered (De Souza and Ochoa 1992). On the other hand, BBN takes care of that issue since the parameters in conditional probability table becomes considerably low as compared to MC 13  (Weber et al. 2010). Later in this thesis, MC has been used to develop the simplified predictive model. Stochastic Petri Networks (SPNs) are considered as a traditional method to model reliability, availability etc.(Balakrishnan and Trivedi 1996).This is a powerful modeling process except that the reliability analysis method includes a rigorous simulation procedure. Therefore, SPN using simulation methods has two disadvantages: inefficient consideration of low frequency events and the simulation time (Weber et al. 2010). It is essential to consider low frequency events in risk analysis process as accidents are rare events with high consequences. In addition, SPNs do not allow integrating evidence easily, whereas BBN takes into account these events. As in the case with most computational methods, the limitation with BBN is that the calculations can be highly demanding when the BBN is densely connected (Bensi et al. 2011). A basic idea of BBN is given in the following section. 2.4  Bayesian belief network Bayesian belief network (BBN) is a graphical model that allows a probabilistic  relationship within a set of variables (Pearl 1988). A BBN consists of a directed acyclic graph (DAG). Here, the nodes represent variables of interest with several possible states, and the links between them indicate probabilistic cause-effect relationship among the variables. The uncertainties in a BBN model are reflected through subjective probability (Pearl 1988; Liu et al. 2011; Tesfamariam et al. 2011). The relations between the variables in a BBN are expressed in terms of family relationships, wherein a variable  and  is said to be  14  the parent of , and  the child of  and  if the link goes from  to  and  to  (as shown  in Figure 2.3). Based on Figure 2.3, a BBN may be formulated by the following steps; 1. Variables necessary and sufficient to model the problem framework of interest are identified (e.g., variable , , and ). 2. Causal interrelations existing between the nodes  ,  , and  are formulated,  graphically shown by arrows. 3. A number of discrete mutually exclusive states are assigned to each variable (e.g.,  and  has two states, state I and state II, and  has three states state I,  state II and state III). 4. The network supports the computation of the probabilities of any subset of variables (e.g., variable variable  ) given evidence about any other subset (e.g.,  and variable ). These dependencies are quantified through a set of  conditional probability tables (CPTs); each variable is assigned a CPT of the variable given its parents.  Variable A  Probability  State I  P1  State II  P2  Variable A  Variable B  Variable C  Variable A  Probability  Variable B  Probability  State I  P3  State II  P4  Conditional probability of variable C State I  State II  State III  State I  State I  P5  P9  P13  State I  State II  P6  P10  P14  State II  State I  P7  P11  P15  State II  State II  P8  P12  P16  Figure 2.3 A sample Bayesian probabilistic network  The conditional probability structure reduces to the unconditional probability (UP) if the variable does not have a parent node. These probabilities (i.e., unconditional or 15  conditional probabilities) are evaluated from historical data, expert judgment, or their combination (Abogrean and Latif 2012; Cockburn and Tesfamariam 2012; Liu et al. 2011). When it comes to computation of probabilities of rare events in complex structural and infrastructure systems, Straub and Der Kiureghian (2010a; b) proposed a new method which combines BBN and structural reliability methods (SRMs). In a BBN analysis, for n number of mutually exclusive hypotheses  (  ,......, ) ,  and a given evidence  , the updated  probability is computed by, =  ∑  × ) ( | )× ( )  2.1  which states that the belief for a hypothesis  (  computed by multiplying our previous belief  ( ) by the likelihood  materialize if  is true.  ,..., )  upon obtaining evidence ( | ) that  can be will  ( | ) is sometimes called the posterior probability (or simply  posterior), and ( ) is called the prior probability (or prior). The advantages of use of BBN are, it can be established according to historical data, expert judgment, or their combination, and can be used to update probabilities when new information is available (Liu et al. 2011; McCann et al. 2006; Uusitalo 2007). New information includes evidence on one or more variables, or observed states that can be entered into the BBN. The new updated information propagates throughout the network to provide up-to-date probabilistic values.  16  Chapter 3: Proposed Bayesian Belief Network This chapter demonstrates a seismic risk assessment framework using the Bayesian belief network (BBN) for high-voltage transformers in a substation. Although there are other reasons which are responsible for transformer vulnerability (e.g., deterioration of structural anchors, aging of transformer parts, etc.), this thesis only considers the vulnerability of highvoltage transformers due to seismic vibration. The beginning section gives a general ideal about the framework and subsequently, the framework is elaborated upon in the following sections. In the end, a sensitivity analysis is performed to investigate the significance of input parameters to the output result (i.e., transformer damage probability). A Bayesian belief network (BBN) was used to evaluate the performance of the transformer in the event of an earthquake. In this thesis the nodes (i.e., variables) in BBN represent the performance of different components of a transformer and consequences induced by earthquake shaking. The links between them indicate informational or causal dependencies among the variables. To develop the risk assessment framework using BBN, the seismic performance of substation components was reviewed from past earthquake records (ASCE 1999; Eidinger and Tang forthcoming; Mitchell et al. 1990; Schiff 1997). Cause and effect analysis of the transformer vulnerability based on the past earthquake records showed that transformer damage is initiated due to soil instability, rocking response and interaction coming from the conductors, due to relative movement of the substation components. Based on the cause and effect analysis of transformer vulnerability, a conceptual framework for transformer vulnerability was developed as shown in Figure 3.1. Figure 3.1 17  shows interaction coming from the conductors (IC), soil instability, and rocking response of transformer (RT) as a function of ground motion intensity measure. Finally, transformer vulnerability is related to earthquake induced objects (e.g., IC, soil instability, RT). This thesis denoted ground motion intensity measure, IC, soil instability, RC and transformer vulnerability as the objects of BBN. An object in a BBN framework is a distinct node behind which another BBN resides. When several variables are introduced to a BBN network, use of these objects reduces graphical clutter (Bensi et al. 2009). Thus, objects are introduced to proposed BBN framework to perform risk assessment of high-voltage transformer (Figure 3.1). The complete BBN framework with each of the BBN of these objects is shown in Figure 3.2 and elaborated in the following subsections and each of the BBN of the objects is described in the following subsections.  Ground motion intensity measure  Interaction coming from conductors (IC)  Soil instability  Rocking response of transformer (RT)  Vulnerability of critical substation component Figure 3.1 Conceptual BBN for high-voltage transformers  18  Ground motion intensity measure  d  ST FM  MW PGA  Interaction coming from the conductors (IC) Existing conductor Length  σvo D50  Required conductor length  Rocking response of transformer (RT)  qc  µs  σ'vo  Anchorage  B/H Liquefaction Conductor failure  RT Liquefaction  Failure of component class 1  Foundation failure  Failure of component class 2  Transformer damage Vulnerability of transformer  Figure 3.2 Proposed BBN for seismic risk assessment of transformer  3.1  Ground motion intensity measure Ground motion is characterized by an intensity measure, which could be any one of a  number of ground motion parameters (e.g., peak ground acceleration (PGA), spectral horizontal acceleration at a particular period) (Baker 2008). Several attenuation relationships are available to quantify ground motion intensity for several tectonic regions, such as western North America (Abrahamson and Silva 1997; Boore et al. 1997; Campbell 1997; Sadigh et al. 1997), eastern and central North America (Atkinson and Boore 1995; Toro et al. 1997), Cascadia subduction zone (Atkinson and Boore 1997; Youngs et al. 1997), etc. These models anticipate the probability distribution of ground motion intensity, as a function of different 19  variables such as the earthquake’s magnitude, distance, faulting mechanism, the near-surface site conditions, the potential presence of directivity effects, etc (Baker 2008). Kuehn et al. (2009) used BBN for modeling dependencies between ground motion intensity and source/site characteristics. In their study, the BBN was developed based on the data available in PEER NGA database (PEER 2005). Later on Bensi et al. (2011) proposed another BBN model for modeling seismic demands, addressing all the shortcomings found in the previous studies of Bayraktarli et al. (2005, 2006) and Kuehn et al. (2009). For this thesis, PGA was used as a ground motion intensity measure. To calculate PGA values, the attenuation equation provided by Boore et al. (1997) was used. Other studies on ground motion intensity measure could be used here, as well. Depending on the researcher’s preference, an appropriate method should be selected. In their study ground motion intensity (i.e., spectral acceleration, horizontal peak ground acceleration) was given as a function of moment magnitude (  ), site to fault distance ( ), fault type, and soil type.  Their model is valid if site to fault distance is in between 0-80km only and the moment magnitude range is in between 5.5 to 7.5. In their study, the faulting mechanism is defined into three types: strike-slip faulting, reverse-slip faulting and unknown faulting. For an unknown fault type the attenuation equation can be represented as follows, ln  = −0.242 + 0.527(  where, ( = √  2  − 6) − 0.778 ln ( − 0.371 ln(+,-. ⁄1396)  3.1  + 31.025 and +,-. is shear wave velocity of soil. Therefore, Figure 3.2  shows the ground motion intensity (i.e., horizontal PGA) as a function of moment magnitude (  ), site to fault distance ( ), soil type (34 ), and fault mechanism (56 ).  20  Site to fault distance (km)  80 70  Mw 5.5  60  Mw 6  50  Mw 6.5  40  Mw 7  30  Mw 7.5 20 10 0 0  0.2  0.4  0.6  0.8  1  1.2  PGA (g)  Figure 3.3 Attenuation of PGA as a function of site to fault distance (for soil type: NEHRP class E)  In this thesis, eight node states were used to describe PGA, namely, extremely extremely low ( (  89: ),  high (  789: ), extremely low ( 789: ), very low (+789: ), low (789: ), medium  89: ),  very high (+  89: ),  and extremely high (  89: );  corresponding PGA  range of each node states are 0≤ PGA <0.10, 0.10≤ PGA <0.15, 0.15≤ PGA <0.20, 0.20≤ PGA <0.30, 0.30≤ PGA <0.40, 0.40≤ PGA <0.60, 0.60≤ PGA <0.80, and 0.80≤ PGA <1.0, respectively. Five node states, with different range of <<;=. )  (  .;< ,  <; < ,  <;2< ,  2<;<< ,  and  were also defined to describe site to fault distance ( ). Selection of range of values,  corresponding to each node state of PGA ranges from 0.8 to 1g,  was done following Figure 3.3. For example, when  varies from 0 to 5km (which is representing very low value).  Table 3.1 shows the node states and corresponding values of input variable.  21  Table 3.1 Description of basic input parameters for “ground motion intensity measure” Variable (Unit)  MW  d (km)  ST (Vs30 in m/sec)  States M5.5 M6.0 M6.5 M7.0 M7.5 d0-5 d5-15 d15-25 d25-55 d55-80 Class A Class B Class C Class D Class E  Furthermore, moment magnitude ( corresponding states were named as  <.< ,  Value MW = 5.5 MW = 6 MW = 6.5 MW = 7 MW = 7.5 0≤ d <5 5≤ d <15 15≤ d <25 25≤ d <55 55≤ d <80 Vs30=1890 Vs30=1070 Vs30=520 Vs30=250 Vs30=150 >)  UP 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20  was discretized into 5.5, 6, 6.5, 7, and 7.5; the  ?.. ,  ?.< ,  @,  and  @.< ,  respectively. Soil type  (34 ) was discretized into five states based on the shear wave velocity of soil (+,-. ), similar to NEHRP site classes2 (i.e., class A, class B, class C, class D, and class E). The classification of soil type is described more elaborately in Appendix A. Finally, fault mechanism (56 ) was considered to be fixed (i.e., unknown faulting). The unconditional probability (UP) table corresponds to each states of input parameters are also shown in Table 3.1. The UPs of  ,  and 34 were defined as 1BA = 1B5 = 0.20 (Cockburn and Tesfamariam 2012). Table 3.2 shows the conditional probabilities for node variable PGA. To develop the CPT table, 124 pairs of  2  >  (  <.< ,  ?.. ,  ?.< ,  @,  @.< ),  (  .;< ,  <; < ,  <;2< ,  2<;<< ,  Weblink: http://www.seis.utah.edu/urban/nehrp.shtml (Last visited February 11, 2013)  22  <<;=. ),  and ST (class A, class B, class C, class D, class E) were selected and the median  PGA values were calculated using the Equation 3.1. Here, the subscript in the states denotes the range of values or discrete values to the corresponding state. Finally, the calculated values of PGA along with the UP of input variables were synthesized into Netica software (Norsys Software Corp 2006) to develop the CPT. A snapshot of these probabilities is shown in Table 3.2. Table 3.2 Snapshot of the CPT for node variable “PGA” (MW, d, ST) (M5.5, d0-5, Class A) (M6.0, d0-5, Class E) ……… ……… (M5.5, d15-25, Class A) (M6.0, d15-25, Class E) ……… ……… (M5.5, d55-80, Class A) (M7.5, d55-80, Class A)  3.2  PGA (ELPGA,VLPGA, LPGA, MPGA, HPGA, VHPGA, EHPGA, EEHPGA) (0.144, 99.017, 0.137, 0.136, 0.144, 0.132, 0.149, 0.141) (0.174, 0.161, 0.169, 0.177, 0.158, 98.836, 0.158, 0.165) ……… ……… (98.749, 0.169, 0.175, 0.184, 0.179, 0.188, 0.188, 0.169) (0.183, 0.181, 98.716, 0.190, 0.181, 0.188, 0.179, 0.182) ……… ……… (98.756, 0.187, 0.169, 0.164, 0.177, 0.186, 0.182, 0.178) (99.005, 0.147, 0.141, 0.141, 0.137, 0.148, 0.135, 0.145)  Soil instability Soil instability is one of the factors that may trigger by ground vibration and may  cause the failure of substation components (ASCE 1999; Eidinger and Tang forthcoming). This thesis considered only soil instability due to liquefaction. Several studies have been performed so far to predict the liquefaction initiation in the event of an earthquake. These studies quantified the soil liquefaction using deterministic and probabilistic techniques, which is either based on laboratory tests or empirical correlations. The deterministic empirical correlation proposed by Seed and Idriss (1971) using Standard Penetration Test 23  (SPT) is a widely used method in practice. Later on their method was revised by Youd et al. (2001). On the other hand, Bayraktarli (2006), Cetin et al. (2002), Tesfamariam and Liu (2013), etc. used a Bayesian framework for probabilistic assessment of the initiation of seismic soil liquefaction. For this thesis, BBN framework for predicting liquefaction initiation was adapted from Tesfamariam and Liu (2013) (Figure 3.2). Other studies on predicting liquefaction initiation could be used here, as well. Depending on the researcher’s preference, an appropriate method should be selected. In this thesis, liquefaction was conditioned on five factors, PGA, average grain size (D50), CPT tip resistance (qc), total vertical over-burden pressure (σvo) and effective vertical overburden pressure (σ’vo). Effective vertical overburden pressure (σ’vo) was conditioned on total vertical over-burden pressure (σvo). This thesis used the historical data available in the literature Timothy and Scott (1995) to predict the liquefaction initiation. The highest value of PGA in the database was found 0.6g. To capture the effect of PGA from 0.6g to 1g, a very high liquefaction initiation probability was assumed with a PGA of 0.6g to 1g. As liquefaction initiation requires local ground acceleration greater than 0.10g (Morales and Morales 2003), it was assumed that liquefaction would not occur between a PGA range of 0 to 0.1g. Table 3.3 shows the node states of all the input variables used in the liquefaction prediction.  24  Table 3.3 Description of basic input parameters for “soil instability” Variable (Unit)  PGA (g)  D50 (mm)  qc (MPa)  σvo (KPa)  σ'vo (KPa)  States Extremely Low (EL) Very Low (VL) Low (L) Medium (M) High (H) Very High (VH) Extremely High (EH) Extremely Extremely High (EEH) Very Low (VL0-0.05)  Value 0≤ PGA <0.10 0.10≤ PGA <0.15 0.15≤ PGA <0.20 0.20≤ PGA <0.30 0.30≤ PGA <0.40 0.40≤ PGA <0.60 0.60≤ PGA <0.80 0.80≤ PGA <1.0 0≤ D50 <0.05  Low (L0.05-0.10) Medium (M0.10-0.15)  0.05≤ D50 <0.10 0.10≤ D50 <0.15  High (H0.15-0.30) Very High (VH0.30-0.60)  0.15≤ D50 <0.30 0.30≤ D50 <0.60  Very Low (VL0-1.5)  0≤ qc <1.5  Low (L1.5-3) Medium (M3-6)  1.5≤ qc <3 3≤ qc <6  High (H6-12)  6≤ qc <12  Very High (VH12-30) Very Low (VL0-50)  12≤ qc <30 0≤ σvo <50  Low (L50-70)  50≤ σvo <70  Medium (M70-100) High (H100-150)  70≤ σvo <100 100≤ σvo <150  Very High (VH150-300) Very Low (VL0-40) Low (L40-60) Medium (M60-100) High (H100-150) Very High (VH150-250)  150≤ σvo <300 0≤ σ'vo <40 40≤ σ'vo <60 60≤ σ'vo <100 100≤ σ'vo <150 150≤ σ'vo <250  UP -  0.591 0.283 0.157 0.300 0.201 0.140 0.251 0.251 0.209 0.149 0.135 0.203 0.228 0.243 0.190 -  *Note: The subscript denotes the range of values for each state In this thesis, PGA was discretized into eight states (as discussed in the previous section). The range of values of PGA (see Table 3.3) for each node states was selected according to Figure 3.4. Figure 3.4 shows that liquefaction initiation was observed when PGA values were more than 0.1g. PGA ranges from 0.1 to 0.3g shows a significant amount of data points for liquefaction initiation; therefore, more discretization (i.e., very low, low, 25  medium) was made for this range of PGA. After 0.3g, PGA values were discretized uniformly.  Figure 3.4 PGA vs. liquefaction occurrence (based on the historical data provided in Timothy and Scott (1995))  Five node states very low, low, medium, high, very high were also defined to describe C<. , DE , FGH , and F′GH . Finally, liquefaction was considered to have two states, liquefaction occurrence (“Yes”) and non-occurrence (“No”). The range of values of each node state for all the variables is given in Table 3.3. Based on the available historical data (Timothy and Scott 1995) liquefaction occurrence versus, FGH , F′GH , C<. , and DE bar charts were plotted in Figure 3.5.The graphs show the relation between liquefaction occurrence and the variables. Finally, all the data were synthesized into Netica software (Norsys Software Corp 2006) to develop the CPT. A snapshot of these probabilities are shown in Table 3.3.  26  90  Liequfaction occurence  75  90  no yes  75  60  60  45  45  30  30  15  15  0  0 0-50  70-100 100-150 150-300  σ Vo (kPa)  90  Liequfaction occurence  50-70  0-40  75  60  60  45  45  30  30  15  15  0  60-100 100-150 150-250  σ 'vo (kPa)  90  75  40-60  0 0-1.5  1.5-3  3-6  6-12  12-30  0-0.05  qc (MPa)  0.05-0.1 0.1-0.15 0.15-0.3  0.3-0.6  D50 (mm)  Figure 3.5 Liquefaction occurrence versus σvo, σ'vo, qc, and D50 Table 3.4 Snapshot of the CPT for node variable “liquefaction” (PGA, D50, qc, σvo, σ'vo) (Extremely Low, VL0-0.05, VL0-1.5, VL0-50, VL0-40) (Medium, VL0-0.05, VH12-30, VL0-50, L40-60 ) ……… ……… (Very High, L0.05-0.10, VL0-1.5, H100-150, M60-100)  Liquefaction (Yes, No)  (High, H0.15-0.30, H6-12, L50-70, VL0-40) ……… ……… (Extremely High, VH0.30-0.60, M3-6, M70-100, VH150-250) (Extremely High, H0.15-0.30, H6-12, L50-70, L40-60)  (48.465, 51.535) ……… ………  (0.002, 99.998) (44.023, 55.977) ……… ……… (53.650, 46.350)  (99.998, 0.002) (99.998, 0.002)  27  3.3  Interaction coming from the conductors The conductor connection flexibility between substation equipment affects  earthquake equipment damage (ASCE 1999). Substation equipment may experience significant displacement at interconnected points (as shown in Figure 3.6), that may lead to conductor failure between the interconnected components. Der Kiureghian et al. (1999a) investigated the effect of interaction between the substation equipment interconnected by different type of conductors (i.e., rigid conductor and flexible conductor). They found that the seismic responses of cable connected equipment were significantly higher if compared with stand-alone equipment. Later on in Der Kiureghian et al. (1999b), they extended their previous work by doing further investigation on different conductor types. In addition to that Der Kiureghian et al. (2001) investigated the behavior of equipment connected with springdashpot-mass elements representing a conductor bus, Song et al. (2007) investigated the behavior of equipment connected with non-linear rigid bus conductor, and Hong et al. (2001) investigated the behavior of equipment connected with cable conductor and the influence of different parameters of cable conductors on the response of connected equipment.  28  Figure 3.6 Behaviour of cable connected equipments (power transformer and disconnect switch) during an earthquake event  Dastous et al. (2004) proposed a simple method to estimate expected displacement of the substation equipment under seismic excitation. Methodology proposed by Dastous et al. (2004) to calculate the displacement of the substation components used generalized SDOF method. The process to estimate the required slack between the equipment is reported in detail in IEEE-1527 (2006) and IEEE-693 (2005). The conductor failure will occur if sufficient slack has not been provided in the conductor between the moving components (Der Kiureghian et al. 1999a). The first step to estimate the displacement between interconnected equipment would be an estimation of individual displacement of the equipment. Once the displacement is known, it is possible to compare the required slack (based on calculated displacements) with the provided slack, to find out whether there would be any conductor failure or not. Following IEEE-693 (2005), the expected deflections for different voltage equipment (hence different natural frequencies), are provided in ASCE (1999) and the values are shown in Table 3.5. 29  Table 3.5 Typical equipment displacement (ASCE 1999) Operating Voltage Frequency 138kV  230kV  500kV  High : 8 Hz and greater: transformer, tank reactor, dead tank circuit breakers  25-50mm  25-75mm  100-300mm  Medium : 2.5-8Hz: disconnect switches, livetank circuit breakers, capacitor banks  50-150mm  200-1000mm  300-1500mm  Low : under 2.5Hz: capacitor voltage transformers, current transformers, wave traps, suspended components  150-500mm  200-1000mm  300-1500mm  The conductor length between two adjacent items of equipment should be enough to accommodate the relative displacement between the equipment. This relative displacement can be evaluated using the square root of the sum of the squares (SRSS), absolute sum method and combine displacements using the complete quadratic combination (CQC) (for more detail calculation see IEEE-693 (2005)). Equation 3.2 shows the calculation of relative displacement using SRSS method (IEEE-1527 2006). = 1.25 × where, ,  and  ,  +  3.2  ,  is the maximum horizontal relative displacement between equipment 1 and 2, is the maximum standalone displacement of equipment 1 in the conductor direction, ,  is the maximum standalone displacement of equipment 2 in the  conductor  direction. Using Equation 3.2, the relative displacement between transformer and disconnect switch (the adjacent equipment to power transformer) can be calculated. The calculated values are shown in Table 3.6.  30  Table 3.6 Calculated relative displacements between transformer and disconnect switch Operating Voltage Description Relative displacement between power transformer and disconnect switch  138kV  230kV  500kV  70-198mm  93-267mm  280-839mm  The required conductor length between interconnected components can be estimated using the following equation. Lo = L1 + L2 + Drel where, the 3.6) and  3.3  is the length of the straight line distance between attachment points (see Figure is an additional provision for the conductor configuration under consideration, as  to not transfer unnecessary additional loads to the equipment when fully stretched. Here, sum of  and  will give a constant value, hence,  in the equation will basically determine  whether the conductor length is adequate to withstand a seismic event or not. The conductor failure can be determined by comparing the existing conductor length and the required conductor length. Figure 3.2 shows that the conductor failure is a function of existing conductor length (  ) and required conductor length (  ), and the required  conductor length is conditioned on seismic hazard (i.e., PGA). When PGA is high, the required conductor length is high and vice versa. In this thesis, conductor failure was discretized into three states, namely, unlikely (  ), likely (  Furthermore, five node states corresponds to very low ( (!  "  ), high (#"  $  existing conductor length (  ), and very high (  $  ) and very likely ( ), low (  ).  ), medium  ) were also defined to describe  ) and required conductor length (  ). Here, the subscript in  31  each of the state denotes the range of values for the corresponding states, which is provided in Table 3.7. Table 3.7 Description of basic input parameters for IC: existing conductor length (ECL)/ required conductor length (RCL) Operating voltage of equipment  138kV  230kV  500kV  States Very low Low Medium High Very high Very low Low Medium High Very high Very low Low Medium High Very high  Value of conductor length (CL), (mm) 1≤ CL <50 50≤ CL <100 100≤ CL <150 150≤ CL <200 200≤ CL <250 1≤ CL <75 75≤ CL <150 150≤ CL <225 225≤ CL <300 300≤ CL <375 1≤ CL <150 150≤ CL <300 300≤ CL <450 450≤ CL <700 700≤ CL <1000  UP for ECL 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20  The range of values corresponds to each states were assumed based on Table 3.6. As mentioned before, the length of the conductor is a sum of a constant value (i.e., relative displacement,  ) and  ; the range of values for each state was assumed based on  (i.e., ignoring the constant part). parameter (i.e.,  +  and  The corresponding unconditional probability of each  ) was defined as 1&% (Cockburn and Tesfamariam 2012).  Finally, with the consideration of two input parameters (i.e.,  and  ), the CPT for the  “conductor failure” was generated. A snapshot of that CPT is shown in Table 3.8. The complete table is given in Appendix B. It is a knowledge based conditional probability table. For example, when (  ,  ) is (  ,!  "  ), the probability of conductor failure  will be high; corresponding conditional probabilities of conductor failure (ULCF, LCF, VLCF) 32  were assumed (5, 30, 65) (Table 3.8). Conversely, when the (  ,  ) is (!  "  ,  ), the probability of conductor failure will be low; corresponding conditional probabilities of conductor failure (ULCF, LCF, VLCF) were assumed (80, 15, 5) (Table 3.8). The basis of this assumption is, when existing conductor length is lower than the required, the conductor failure is very likely and vice versa. Table 3.8 Snapshot of the CPT for node variable “conductor failure” (ECL, RCL) (VL1-150, VL1-150) (L150-300, M300-450) ……… ……… (M300-450, L150-300) (M300-450, H450-700) ……… ……… (VH700-1000, H450-700) (VH700-1000, VH700-1000)  3.4  Conductor failure (ULCF, LCF, VLCF) (80, 20, 0) (5, 30, 65) ……… ……… (80, 15, 5) (25, 30, 45) ……… ……… (75, 20, 5) (60, 30, 10)  Rocking response of transformers Several cases have been found, by searching the past earthquake records, that the  transformer moved back and forth. Some of the records showed that due to back and forth movement of the transformer, the foundation tilted and remained tilted after the earthquake. In some cases, the foundation remained in its position, but damage to bus work indicated that there was a back and forth movement during the earthquake (ASCE 1999). Makris and Roussos (1998, 2000) investigated the rocking response of free standing block and Makris and Zhang (1999, 2001) investigated the rocking response of anchored blocks. The free standing transformer will experience the rocking response without experiencing any restoring 33  force coming from the anchorage. On the other hand, the anchored transformer will not rock until the strength of the restrainer has exceeded its fracture. Makris and Black (2002) investigated the uplifting and overturning of equipment anchored to a base foundation. Their study looked into the performance of high strength restrainer for a wide range of earthquake motion. This thesis considered both the rocking response of free standing transformers and anchored transformers. Strong ground motion may initiate slide or set up a rocking motion to electrical transformers that may result in substantial damage. Shenton (1996) derived the governing criteria of the initiation of the slide and rock modes from rest for a rigid, rectangular block (i.e., transformer) subjected to horizontal foundation acceleration. The governing equations for rest, slide, and rock modes for a rectangular free standing rigid block are presented in Table 3.9. Table 3.9 Governing conditions for rest, slide, and rock modes Modes Rest Slide Rock  Conditions '() ≤ +, ; and '() ≤ -/# 	+, ≤ '(); and -/# ≤ +, - ⁄# < '(); and -/# < +,  *Note: Ag = PGA, peak ground acceleration, B/H= width to height ratio of a rigid block, µ s=Static coefficients of friction  These criteria were plotted in the	+, , versus PGA parameter space for a block with a width to height ratio (e.g., -/#) of 0.50 as shown in Figure 3.7. The parameter space is divided into three regions by the lines corresponding to	+, = '(),	+, = -⁄# = 0.5, and '() = - ⁄# = 0.5. These three regions correspond to rest, slide, and rock.  34  Static coefficient of friction, µs  1  PGA=B/H  0.8 Rest  0.6  Rock  µs=B/H  0.4  µs= PGA  0.2  Slide  0 0  0.2  0.4  0.6  0.8  1  Peak ground acceleration, PGA (g) Figure 3.7 Boundaries of rest, slide, and rock modes, for H/B=2 (based on Shenton (1996))  Therefore, the rocking response of a transformer depends on four factors, anchorage,	PGA, static coefficients of friction (+, ), and width to height ratio (B/H) (Figure 3.2). This thesis considered that, if the transformer were unanchored, the rocking response would follow the criteria provided by Shenton (1996) and that if the transformer were anchored, it would not rock until the anchorage failed. Hence, the variable “anchorage” has two states, anchored and unanchored. In this thesis, RT assumed to have three states, rest, slide, and rock. Furthermore, B/H ratio had eight states, ) (  .4  and #  .42 .  . ",  -  . 2,  ."  ,  ."  ,  .  ,3. ,  Here, the subscripts denote the value of -⁄#. Each of the state denotes width  to height ratio (-/#) of transformers given in Makris and Roussos (1998). +, had only one state, representing the coefficient of friction between the concrete (the base at which the transformer is resting) and steel (material of the transformer bottom). The value of the coefficient of friction between concrete and steel is 0.45. The range of values that  35  corresponds to each state of the input parameters is shown in Table 3.10. The corresponding UP of each parameter was calculated as 1&% (as previous). Table 3.10 Description of basic input parameters for “Rocking response of transformer” Variable  PGA  B/H  Anchorage  States Extremely Low Very Low Low Medium High Very High Extremely High Extremely Extremely High A0.34 B0.39 C0.42 D0.43 E0.50 F0.51 G0.65 H0.69 Anchored Unanchored  Value 0≤ PGA <0.10 0.10≤ PGA <0.15 0.15≤ PGA <0.20 0.20≤ PGA <0.30 0.30≤ PGA <0.40 0.40≤ PGA <0.60 0.60≤ PGA <0.80 0.80≤ PGA <1.0 0.34 0.39 0.42 0.43 0.50 0.51 0.65 0.69 -  UP 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.50 0.50  Table 3.11 shows the snapshot of CPT for the node variable RT. The complete table is provided in Appendix C for the anchored transformer. The probabilities for an unanchored transformer were calculated based on the governing conditions provided by Shenton (1996) (see Table 3.9). While calculating the probabilities, it was assumed that 100% probability of being at rest, slide and rock mode would be considered when the combination of PGA and B/H values fell in the ‘ash colored zone’ (marked in the Figure 3.8). When the points fell outside the marked zone, the assumption was that the probability of being in that specific zone would be decreased linearly. For example, if a point was just outside the line connecting points 1 and 2, the probability of being at rest would not be 100%; there would be some 36  probabilities of being at slide mode. If the point moved towards the line	+, = '(), probability of being at rest would be decreased linearly; if the point was on the line, the probability of being at rest and slide would be equal. If the transformer was anchored, the probabilities were assigned based on some assumptions, such as the transformer would be at rest for low PGA values and the probability of transformer being at rocking and sliding mode would increase with the increment of PGA (as high PGA may fracture the anchorage). For example, Table 3.11 shows, when the input (PGA, -/#, Anchorage, μ, ) is (Extremely High, )  . ",  Unanchored, 0.45), the probabilities of being at rest, slide and rock mode are (0, 17.25,  82.75). On the other hand, when the input is (Extremely High, )  . ",  Anchored, 0.45,), the  probability of fracturing of anchorage is high (as PGA is high), as the output shows (Rest, Slide, Rock) = (43, 12, 45). Table 3.11 Snapshot of the CPT for node variable “RT” (PGA, B/H, Anchorage, µs) (Extremly Low, A0.34, Anchored, 0.45) (Low, F0.51, Unanchored, 0.45) ……… ……… (Very High, A0.34, Anchored, 0.45) (Very High, G0.65, Unanchored, 0.45) ……… ……… (Extremly High, A0.34, Anchored, 0.45) (Extremly High, A0.34, Unanchored, 0.45)  RT (Rest, Slide, Rock) (100, 0, 0) (100, 0, 0) ……… ……… (48, 10, 42) (46.02, 49.09, 4.88) ……… ……… (43, 12, 45) (0, 17.25, 82.75)  37  Figure 3.8 Boundaries of rest, slide, and rock modes to calculate CPT  3.5  Transformer vulnerability Based on past earthquake records, ASCE (1999) elaborated on the earthquake  performance of power transformers. The causes of failure of transformer components during an earthquake are listed in Table 3.12. Table 3.12 shows that causes of failure of transformer components during an earthquake are considered to be due to conductor interaction between interconnected equipment, liquefaction and rocking response.  38  Table 3.12 Causes and effects of failure of transformer components Component Foundation failure  Radiator failure (less probable to fail)  Internal parts  Cause Soil liquefaction/ differential settlement Rocking effect  Relative movement of the transformer and the radiator (for self supported radiators).  Earthquake induced vibration  Effect/Observed failure Tilt of a foundation Soil may remain level, but the due to back and forth movement other parts of the transformer may fail (e.g., low voltage bus, bushing, lightning arrester, radiators, conductor failure due to relative movement) Oil leakage at the coupling For non-manifold radiators, the observed failure was found for a PGA 0.15g during Northridge EQ Deformation in the manifold support is found Internal transformer fault, which may trigger the circuit breaker to open  Conservator failure  Relative movement of the conservator and the transformer case.  The structural support system failed The pipe connection between the conservator and transformer tank failed A piping failure may result in an oil spill  Lightning arrester and tertiary bushing failure  The flexibility of bus support structures imposed loads on the bushings  Failure of lightning arrester and tertiary bushing  Porcelain bushing failure  Inertial loads on bushings due to ground vibrations Rocking response of transformer Interaction loads from the conductors Effect of failed lightning arrester  Failure of porcelain bushing  Based on the studied causes and effects of transformer component failure, a BBN for transformer vulnerability was developed (shown in Figure 3.2). In this thesis, the components were classified into two groups, component class 1 (i.e., radiator, internal parts, and conservator) and component class 2 (i.e., lightning arrester and tertiary bushing failure, porcelain bushing failure). The classification was done depending on the causes of failure. Components that fail due to the transformer rocking response and foundation failure were 39  named as component class 1 and components that fail due to conductor failure and transformer rocking response, and foundation failure were named as component class 2 (Table 3.12). Hence, the foundation failure is conditioned on RT and liquefaction, component class 1 is conditioned on RT and foundation failure, and finally, component class 2 is conditioned on IC, RT and foundation failure. In the end, transformer damage is a function of foundation failure and component failure (classes 1 and 2). The foundation failure, the failure of component class 1, and the failure of component class 2 were considered to have two states, likely to fail and unlikely to fail. The variable transformer damage had three states, low, medium and high probability of damage. Table 3.13, Table 3.14, Table 3.15, and Table 3.16 show the CPT for foundation failure, failure of component class 1, failure of component class 2, and transformer damage. These tables are knowledge based conditional probability table. Table 3.13 Description of CPT for node variable “foundation failure” (RT, Liquefaction) (Rest, Yes) (Rest, No) (Slide, Yes) (Slide, No) (Rock, Yes) (Rock, No)  Foundation failure (Unikely, Likely) (30, 70) (99, 1) (5, 95) (70, 30) (1, 99) (45, 55)  Table 3.14 Description of CPT for node variable “failure of component class 1” (RT, Foundation failure) (Rest, Unlikely) (Rest, Likely) (Slide, Unlikely) (Slide, Likely) (Rock, Unlikely) (Rock, Likely)  Failure of component class 1 (Unlikely, Likely) (99, 1) (30, 70) (40, 60) (20, 80) (30, 70) (1, 99)  40  Table 3.15 Description of CPT for node variable “failure of component class 2” (Conductor failure, RT, Foundation failure) (Unlikely, Rest, Unlikely) (Unlikely, Rest, Likely) (Unlikely, Slide, Unlikely) (Unlikely, Slide, Likely) (Unlikely, Rock, Unlikely) (Unlikely, Rock, Likely) (Likely, Rest, Unlikely) (Likely, Rest, Likely) (Likely, Slide, Unlikely) (Likely, Slide, Likely) (Likely, Rock, Unlikely) (Likely, Rock, Likely) (Very Likely, Rest, Unlikely) (Very Likely, Rest, Likely) (Very Likely, Slide, Unlikely) (Very Likely, Slide, Likely) (Very Likely, Rock, Unlikely) (Very Likely, Rock, Likely)  Failure of component class 2 (Likely, Unlikely) (99, 1) (40, 60) (45, 55) (30, 70) (40, 60) (20, 80) (45, 55) (25,75) (35, 65) (15, 85) (20, 80) (10, 90) (35, 65) (15, 85) (30, 70) (10, 90) (10, 90) (1, 99)  Table 3.16 Description of CPT for node variable “transformer damage” (Foundation failure, Failure of component class 1, Failure of component class 1) (Unlikely, Unlikely, Unlikely) (Unlikely, Unlikely, Likely) (Unlikely, Likely, Unlikely) (Unlikely, Likely, Likely) (Likely, Unlikely, Unlikely) (Likely, Unlikely, Likely) (Likely, Likely, Unlikely) (Likely, Likely, Likely)  Tranformer damage (Low, Medium, High) (99, 1, 0) (20, 60, 20) (5, 30, 65) (3, 20, 77) (5, 25, 70) (2, 15, 83) (0, 10, 90) (0, 1, 99)  41  The assumption behind generating Table 3.13 was that foundation failure is more likely when transformer rocking and liquefaction occurrence probability are high. Table 3.14 shows that when the transformer is in rocking mode and there is a high chance of liquefaction occurrence, the failure of component class 1 is likely. When it is in rest mode and there is no probability of liquefaction occurrence, failure probability is very low, and when it is in slide mode it is considered to have low probability of failure. Table 3.15 was generated based on the assumption that there will be a higher probability of failure of the component class 2 when conductor failure, foundation failure, and transformer rocking occur, and vice versa. Finally, Table 3.16 was generated based on the assumption that failure of foundation and component class 1 cause more damage to the transformer, hence the damage probability is higher. The failure of component class 2 will not cause severe damage to the transformer, as it is possible to replace such components more easily. In fact, some substations keep spare porcelain bushings, lightning arresters and tertiary bushings. 3.6  Sensitivity analysis Since BBN employs prior conditional probabilities; it is essential to carry out the  sensitivity analysis to understand the significance of input parameters to the output results. In literature, different methods have been proposed to perform sensitivity analysis in a BBN (Jensen 1996; Pearl 1988). The variance reduction method was used in this thesis to determine the sensitivity of output parameter regarding the variation of a particular input parameter. This method was chosen as the input parameters needed to assess the transformer damage had discrete and continuous values (Norsys Software Corp 2006; Pearl 1988) and was applied for similar application (Cockburn and Tesfamariam 2012; Ismail et al. 2011; Tesfamariam and Martín-Pérez 2009). The method implies computation of variance 42  reduction of the expected real value of a query node 6 (e.g., transformer damage) due to varying variance node 3 (e.g., PGA, liquefaction). Therefore, the variance of the real value of 6	given evidence 3, (8|:), is computed as (Norsys Software Corp 2006; Pearl 1988), V (q | f ) = ∑ p (q | f )[ X q − E (Q | f )]2 q  3.4  where 8	is the state of the query node 6, : is the state of the varying variable node 3, <(8|:) is the conditional probability of 8 given	:, => is the numeric value corresponding to state 8, and (6|:) is the expected real value of 6 after the new finding :	for node 3. Results of the sensitivity analysis for the model illustrated in Figure 3.2 are summarized in Table 3.17. Table 3.17 shows the variance reduction and normalized percent contribution of the main input variables. The percent contribution values are normalized with respect to the sum of all the percent contribution of the input parameters.	?, !@ , and AB showed the highest normalized percent contribution towards the variance reduction, 67%, 16.12%, and 15.12%, respectively. To a lesser degree, the rest of the input variables showed normalized percent contribution less than 1%. Results of this analysis showed that the seismic hazard has the most contribution to the results and the sensitivity of the parent node (i.e.,	?, !@ , AB ,  ) significantly depends on the variability of the dependent children (i.e.,  PGA, conductor failure, liquefaction,  ). This indeed highlights the need for rigorous  method of quantifying site seismicity (e.g. see Bensi et al. 2011) and incorporation of mitigation techniques to delineate the seismic demand (e.g. use of base isolation).  43  Table 3.17 Sensitivity analysis for “transformer damage” using proposed BBN model Node D Mw ST ECL σvo qc  Anchorage B/H D50  Variance reduction 3.62E-02 8.73E-03 8.19E-03 4.11E-04 2.41E-04 1.32E-04 1.18E-04 6.01E-05 3.67E-06  Normalized percent contribution 67.00% 16.12% 15.12% 0.76% 0.44% 0.24% 0.22% 0.11% 0.007%  44  Chapter 4: Predictive Model Development This chapter consists of two main sections, development of predictive model using Markov chain, and response surface method (RSM). By creating different scenarios in the BBN framework (as shown in Chapter 3), these analyses were performed. 4.1  Model development using Markov chain To understand the pattern of change of low, medium and high transformer damage  probabilities with the change of PGA, 10,000 random values of the input parameters (PGA, -/#, CDE , 8F ,  ,  ) were generated. The simulations were done for anchored and  unanchored transformer separately. The results were plotted and fitted for low, medium and high transformer damage probability (see Figure 4.1 and Figure 4.2). The pattern of the graphs for states, low, medium and high, show that the probabilities decrease gradually for low damage state, on the other hand, the probabilities increase gradually for high damage state. The probabilities for medium damage states increase for a certain time and after reaching a peak the probabilities decrease. The trend of the probabilities for low, medium and high damage states in the graphs follows the Markov property. A stochastic process has the Markov property if the conditional probability distribution of the future state only depends on the present state (Feller 1971).  45  0.4  Transformer damage probability  Transformer damage probability  1 0.8  Low 0.6 0.4 0.2 0 0  0.2  0.4  0.6  0.8  0.3  Medium 0.2  0.1  0  1  0  0.2  0.4  PGA  0.6  0.8  1  PGA  (a)  (b) Transformer damage probability  1 0.8 0.6 0.4  High 0.2 0 0  0.2  0.4  0.6  0.8  1  PGA  (c) Figure 4.1 Simulated data plotted for (a) Low, (b) Medium and (c) High damage states of anchored transformer  46  0.4  Transformer damage probability  Transformer damage probability  1 0.8  Low 0.6 0.4 0.2 0  0.3  Medium 0.2  0.1  0 0  0.2  0.4  0.6  0.8  1  0  0.2  0.4  PGA  0.6  0.8  1  PGA  (a)  (b)  Transformer damage probability  1  0.8  0.6  0.4  High 0.2  0 0  0.2  0.4  0.6  0.8  1  PGA  (c) Figure 4.2 Simulated data plotted for (a) Low, (b) Medium and (c) High damage states of unanchored transformer  The Markov chain model was developed using the simulation data. This model is widely known in developing a deterioration model of civil infrastructure (Baik et al. 2006; Enright and Frangopol 1999; Morcous and Lounis 2007). The Markov chain is a stochastic process whose development can be treated as a series of transitions between certain states (Lounis et al. 1998; Morcous 2006). The process will be considered as a first order Markov 47  process if the probability of being in a future state only depends on a present state and not on how it is achieved (Parzen 1962). This property can be given as follows, for a discrete parameter stochastic process ( ) with a discrete state space, = where  | = , = = | =  ,…,  =  ,  =  =  4.1  = state of the process at time ; and = conditional probability of any future event  given the present and past event. In this thesis, Markov chain was used to predict the transformer damage probability for different discrete damage states (i.e., low, medium, and high) and accumulating the probability of transition from one state to another over multiple discrete PGA interval. Transition probabilities are usually represented by a matrix of order ( × ) and called the transition probability matrix is represented by . Here,  is the number  = 3). The transition probability matrix can be  of condition states (e.g., for this thesis, represented as follows,  . . .  =  where each element,  ,  ,  ,  . . .  ,  ,  ,  ,  … … … … … …  . . .  ,  ,  4.2 ,  in the matrix represents the probability that a condition of the  transformer damage will go from state to state during a certain PGA interval. If the initial condition vector  0 of the transformer is known, the future condition vector  number of PGA intervals  at any  can be presented as follows (Morcous 2006), =  0 ×  4.3  48  The transition probability matrix of transformer damage for anchored and unanchored condition was developed using the simulated data. It was calculated based on the condition that it minimizes the root mean square estimation of distance between the Markovian expected value of the state and the damage probabilities for that state as predicted by the BBN framework. The transition probability matrix for anchored and unanchored transformer for estimating transformer damage probabilities are given in Equation 4.4 and 4.5.  " #$%&'(  0.989 0.011 = )0.000 0.956 0.000 0.000  1 2 #$%&'(  0.980 = )0.000 0.000  0.020 0.920 0.000  0.000 0.0440 1.000 0.000 0.0800 1.000  4.4  4.5  The transition probability graphs with the change of PGA values are also given in Figure 4.3 and Figure 4.4. The graphs show that the low state transition probabilities for unanchored transformer decreases more quickly than the anchored transformer and the high state transition probabilities reach more quickly to 100% for unanchored transformer in comparison to anchored transformer. That means the Markov model shows that with the use of anchorage, as expected, the transformer damage probability is reduced.  49  Transformer damage probability  1 0.8 Low Medium High  0.6 0.4 0.2 0 0  0.5  1  1.5  2  PGA Figure 4.3 Transition probabilities calculated using Markov chain for anchored transformer  Transformer damage probability  1 0.8 Low Medium High  0.6 0.4 0.2 0 0  0.5  1  1.5  2  PGA Figure 4.4 Transition probabilities calculated using Markov chain for unanchored transformer  4.2  Predictive model development using RSM This thesis aims at the screening of important variables followed by developing an  adequate functional relationship between the response of interest (i.e., transformer damage probability) and the associated input variables (i.e., 45 , 6, 78 , 9:;, </>, etc.) using a response surface method (RSM) approach. Several researchers have used RSM in different engineering application(Buratti et al. 2010; Möller et al. 2009; Rajeev and Tesfamariam 50  2012; Zhou et al. 2013). If the response of interest is denoted by ?, and the associated input  variables are denoted as @ , @ , @A , … . . @B , the unknown relationship between them can be approximated by response surface methodology which consists of a group of mathematical and statistical techniques (Khuri and Mukhopadhyay 2010). Two main approximations are commonly used in RSM, first order model or polynomial of higher degree (such as second degree model)(Rajeev and Tesfamariam 2012). The relationship can be represented by a low degree polynomial model of the form, ? = CD @ E + G  4.6  where, @ = @ , @ , @A … … … , @H D , C @ is a vector function of p elements that consists of  powers and cross products of powers of @ , @ , @A , … … , @H up to a certain degree denoted by 6 ≥ 1 , E is a vector of p unknown constant coefficient referred to as parameters and G is a random experimental error assumed to have a zero mean. Design Expert V8 (2009) software was used to perform the screening and RSM approach. A fractional factorial design (Resolution IV design) consisting of 18 runs was used to screen the most important input variables (i.e., 4J , 6, 78 , 9:;, </>, KL% , M# , and NO ) for anchored and unanchored transformer. The response of the input variables is denoted as transformer damage probability. For the study of the prediction model, mean transformer damage probability was used (instead of using different states). Resolution IV designs were used because they are useful for screening of main effects and it allows to determine if interactions exists (User manual of Design Expert V8 (2009)). The analysis results provide half normal plot and Pareto chart to screen the important variables. After selecting the important variables based on t-values and Bonferroni limits, 51  analysis of variance (ANOVA) along with its significance level was generated. After that the model significance was tested and the model was further used to select the important variables. The most significant factors or variables were studied later on using RSM approach. Design Expert V8 (2009) has an option named ‘optimal RSM approach’ which is useful to create a good design for fitting a linear, quadratic, cubic model and higher (up to 6th) order models. Optimal RSM approach was further used to generate an approximate relationship between the response and significant variables. 4.2.1  Screening of input variables Instead of 256 experimental runs as per full factorial design (28), just 18 runs were  performed as per Resolution IV design. It reduced the run without much compromise on errors. Each of the parameters varied with two levels, i.e., 4J (5.5 and 7.5), 6 (0 and 80  km), 78 (150 and 1890 m/sec), KL% (0 and 300 KPa), M# (0 and 30 MPa), NO (0 and 0.6 mm)  and 9:; (1 and 1000 mm). The Pareto chart gave that the 4J , 6, 78 , </>, KL% , M# , and  9:;	are the most significant factors for both anchored and unanchored transformer (Figure 4.5 and Figure 4.7). Thus, these variables were taken as model variables and the ANOVA model was constructed. Both of the models were found significant (for anchored p<0.0001 and unanchored p=0.0002) and the models passed the normal distribution test too (Figure 4.6 and Figure 4.8).  52  Design-Expert® Software T ransformer_damage  Pareto Chart B 29.08  C A  t-Value of |Effect|  A: Mw B: d C: ST (soil type) D: B/H E: sigma_vo F: qc G: D50 H: ECL Positive Effects Negative Effects  21.81  14.54  AD EH AG E  F  BE  7.27  D AC H  Bonferroni Lim it 6.46932  t-Value Lim it 2.77645  G  0.00  1  2  3  4  5  6  7  8  9  10 11 12 13 14 15 16 17  Rank Figure 4.5 Screening of important variables for transformer (Anchored) damage probability calculation using a Pareto chart  53  Normal Plot of Residuals  Normal % Probability  99 95 90 80 70 50 30 20 10 5 1  -2.00  -1.00  0.00  1.00  2.00  Internally Studentized Residuals Figure 4.6 The internally studentized residuals and normal % probability plot of tranformer (Anchored) damage probability for the Resolution IV design  54  Design-Expert® Software T ransformer_damage  Pareto Chart B 22.82  C 19.02  t-Value of |Effect|  A: Mw B: r_jb C: ST (soil type) D: B/H E: sigma_vo F: qc G: D50 H: ECL Positive Effects Negative Effects  A  15.22  11.41  AD EH AG  7.61  F  Bonferroni Lim it 6.46932  E BE AC D  3.80  H t-Value Lim it 2.77645  G 0.00  1  2  3  4  5  6  7  8  9  10 11 12 13 14 15 16 17  Rank Figure 4.7 Screening of important variables for transformer (Unanchored) damage probability calculation using a Pareto chart  55  Normal Plot of Residuals  Normal % Probability  99 95 90 80 70 50 30 20 10 5 1  -2.00  -1.00  0.00  1.00  2.00  Internally Studentized Residuals Figure 4.8 The internally studentized residuals and normal % probability plot of tranformer (Unanchored) damage probability for the Resolution IV design  4.2.2  Prediction equation using Response Surface Method (RSM) The significant variables found in screening (4J , 6, 78 , </>, KL% , M# , and 9:;) were  selected for further analysis using RSM approach. A total of 315 runs were conducted using optimal RSM analysis for both anchored and unanchored transformer. In this thesis, three of the variables were discrete (4J , 78 , and </>) and rest of them were continuous (6, KL% , M# , and 9:;). Five discrete values of 45 = {5.5, 6, 6.5, 7, 7.5} and 78 = {150 m/sec, 250 m/sec,  520 m/sec, 1070 m/sec, 1890 m/sec}, and three discrete values of </> = {0.34, 0.5, 0.69}  were varied in the model. On the other hand, two levels (low to high) were varied for continuous variables in the model, i.e., 6	= [0, 80 km], KL% = [0, 300 kPa], M# = [0, 30 MPa] and 9:; = [1, 1000 mm]. 56  For anchored transformer, the model summary showed quadratic as the most significant model (p<0.0001) with non-significant lack of fit. Analysis of ANOVA showed that 45 , 6, 78 , KL% , 9:;, 45 × 6, 45 × 78 , 6 × 78 , 6 × KL% , 6 , 78 , and M# were the most significant. Rest of the main variables and their interactions had p-values greater than 5% significance level. Thus, the model was again generated with the significant variables and their interactions. As qc did not have any significance, the higher level interaction of M# (i.e., M# ) was ignored in further analysis. The reduced quadratic model was again found  significant with p-value <0.001. The normal plot of residuals (Figure 4.9) and the actual vs. predicted plot (Figure 4.10) satisfied the normal distribution and prediction capability.  Normal Plot of Residuals  Normal % Probability  99 95 90 80 70 50 30 20 10 5 1  -3.00  -2.00  -1.00  0.00  1.00  2.00  3.00  Internally Studentized Residuals Figure 4.9 The internally studentized residuals and normal % probability plot of tranformer (Anchored) damage probability for quadratic model  57  Predicted vs. Actual 1.00  0.80 2  Predicted  0.60  0.40  0.20 2  2  2  2  0.00  -0.20  0.00  0.20  0.40  0.60  0.80  1.00  Actual Figure 4.10 The actual and predicted plot of tranformer (Anchored) damage probability using quadratic model  For unanchored transformer, the model summary also showed quadratic as the most significant model with non-significant lack of fit. Analysis of ANOVA showed that 45 , 6, 78 , KL% , 9:;, 45 × 6, 45 × 78 , 6 × 78 , 6 × KL% , 78 × </>, 6 , 78 , M# and 9:; were the most significant. The rest of the main variables and their interactions had p-values greater than 5% significance level. Thus, the model was again generated with the significant variables and their interactions. As </> and M# did not have any significance, the higher  level interactions of </> and M# (i.e., 78 × </> and M# ) were ignored in further analysis.  The reduced quadratic model was again found significant with p-value <0.001. The normal plot of residuals (Figure 4.11) and the actual vs. predicted plot (Figure 4.12) satisfied the normal distribution and prediction capability.  58  Normal Plot of Residuals  Normal % Probability  99 95 90 80 70 50 30 20 10 5 1  -3.00  -2.00  -1.00  0.00  1.00  2.00  3.00  Internally Studentized Residuals Figure 4.11 The internally studentized residuals and normal % probability plot of tranformer (Unanchored) damage probability for quadratic model  59  Predicted vs. Actual 1.20  1.00  2  Predicted  0.80  0.60  2 2  0.40  2  2  2 2  2  2  0.20  2  2  2 2  2 4 2  22 2 2  2  3  0.00  2  2 7 2 3  4  -0.20  0.00  0.20  0.40  0.60  0.80  1.00  Actual Figure 4.12 The actual and predicted plot of tranformer (Unanchored) damage probability using quadratic model  The final predictive equations are given in Equations 4.7 and 4.8 for anchored and unanchored transformers, respectively. PQR SCTQUVQ	6RURWV	 QTXRX Y ? = −0.79436 + 0.24306 × 45 − 6.086419 A × 6 + 1.214079 \ × 78 − 4.241049 \ × KL% − 7.29979 O × 9:; − 1.62829 A × 45 × 6 − 6.69129 O × 45 × 78 + 1.716769 × 6 × 78 + 6.017689 ] × 6 × KL% + 1.197359 \ × 6 + 6.117529 ^ × 78  ]  PQR SCTQUVQ	6RURWV	 QTXRX Y ? = −0.87825 + 0.26834 × 45 − 5.779159 A × 6 + 1.216109 \ × 78 − 4.468559 \ × KL% − 1.951859 \ × 9:; − 1.946879 A × 45 × 6 − 6.88079 O × 45 × 78 + 2.141979 × 6 × 78 + 6.359419 ] × 6 × KL% + 1.28839 \ × 6 + 5.204279 ^ × 78 + 1.289139 _ × 9:;  4.7  ]  4.8  60  Figure 4.13 and Figure 4.14 show the interactions between the variables 6	and 78 has a positive effect on the response, for both anchored and unanchored transformer.  0.8  Transformer damage  0.6 0.4 0.2 0.0 -0.2  1890  0 1455  20 1020  40  d (km)  585  60  ST (m/sec)  80 150  Figure 4.13 3-D interaction plot for d (fault to site distance in km) and ST (soil type, defined by shear wave velocity of soil, in m/sec) in the quadratic model (for anchored transformer)  61  0.8  Transformer damage  0.6 0.4 0.2 0 -0.2  0 1890  20  1455 40  d (km)  1020 60  585 80 150  ST (m/sec)  Figure 4.14 3-D interaction plot for d (fault to site distance in km) and ST (soil type, defined by shear wave velocity of soil, in m/sec) in the quadratic model (for unanchored transformer)  4.2.3  Model validation To validate the predictive equations, again 10,000 random runs were generated by  randomly changing the eight input variables (i.e., 4J , 6, 78 , </>, KL% , M# , NO , and 9:;) for both anchored an unanchored transformer. The input values of the variables are shown in Figure 4.15 in bar charts. The results from the BBN framework were compared with the results found using the prediction equation. Figure 4.16 and Figure 4.17 show comparison between the actual and predicted values of transformer damage probability for anchored and unanchored conditions, respectively. The graphs show how the data points are scattered over the line of equality. For the anchored transformer the model predicted comparatively well in compared to unanchored transformer. Both of the graphs show that models predict well for the high values of transformer damage probability. On the other hand, while predicting the 62  low transformer damage probability the model predicted values less than zero in some cases. This is the weakness of the predictive model.  Earthquake magnitude, Mw  5.5  6.0  6.5  7.0  Earthquake distance, d  Soil type, ST  7.5  Width to height ratio, B/H  Total vertical overburden pressure, σvo  CPT tip resistance, qc  0.34 0.39 0.42 0.43 0.5 0.51 0.65 0.69  Existing conductor length, ECL  Average grain size, D50  Figure 4.15 Bar charts showing the range of values of input variables  63  1  0.8  Predicted  0.6  0.4  0.2  0  -0.2 0  0.2  0.4  0.6  0.8  1  Actual  Figure 4.16 Model validation for anchored transformer  1  0.8  Predicted  0.6  0.4  0.2  0  -0.2 0  0.2  0.4  0.6  0.8  1  Actual  Figure 4.17 Model validation for unanchored transformer  64  Chapter 5: Conclusion and Future Work Based on the observed failure from the past earthquakes, this thesis proposed a risk assessment framework using BBN, for high-voltage transformers. Cause and effect analysis of the transformer vulnerability showed that the transformer damage is initiated due to soil instability, rocking response and interaction coming from conductors, due to the relative movement of the substation components. This thesis used information from published literature to develop the conditional probability tables in BBN. The framework predicts the damage probability of a high-voltage transformer subjected to seismic events. Using the proposed framework a sensitivity analysis was performed. The sensitivity analysis showed that the input parameters related to seismic hazard had more contribution towards the transformer damage and average grain size (NO ) has lowest contribution towards the transformer damage. Hence, the best strategy is to reduce the PGA at the transformer base that could be confirmed by using the base isolation at the transformer base. After that, response surface method (RSM) was used to develop a predictive model. To perform the RMS an initial screening was performed as the number of input variables are more than five. The results from screening showed that average grain size (NO ) had insignificant contribution to the transformer damage probability, similar to the sensitivity analysis. Two quadratic equations were developed using RSM for anchored and unanchored transformers, to predict the mean probability of transformer damage. Finally, for different combinations of input variables, the transformer damage probability was calculated using the predictive model and compared with the actual one. Both of the results for anchored and unanchored transformer showed that models predict well for the high values of transformer 65  damage probability. On the other hand, for the low transformer damage probability, the model predicted values less than zero in some cases. This is the weakness of that predictive model. Another analysis was performed by doing 10,000 simulations with randomly generated eight input variables. The output data (transformer damage probabilities in different states, i.e., low, medium, and high) was then plotted to understand the pattern of the dataset. The result showed that the damage probabilities of different states followed the Markov property. Therefore, in the end of the case study, the transition probability matrices for anchored and unanchored transformer were developed, to predict the probability of transformer damage in different states (i.e., low, medium, high) with the change of PGA values. The results are particularly useful supporting decisions on mitigation measures and seismic risk prediction in a region. One of the major advantages with BBN frameworks is that when new information is available it can be updated very easily. This intuitive nature is also useful for different stakeholders. Note that BBNs could also benefit from analytical and experimental results, which are typically explored for single failure modes of substation components, but that can be systematically integrated through BBNs. This thesis combined most of the critical failure modes observed for high-voltage transformers to date. The main limitation of the current study is that several conditional probabilities in the transformer vulnerability assessment were knowledge based. This may change with the perspective of researchers and the stakeholders. However the methodology is ready to accept conditional probabilities that are developed based on other evidence data (i.e., historical data, 66  results from analytical and experimental work). For future study this part could be refined. Also, this thesis showed risk assessment of high voltage transformer without considering any retrofitting options. Further study can be carried out by including retrofitting options to the transformer components. 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(2013). “Response surface method based on radial basis functions for modeling large-scale structures in model updating.” Computer-Aided Civil and Infrastructure Engineering, 28(3), 210–226. 76  Appendices Appendix A: NEHRP Site Class with Recommended Values of Average Shear-wave Velocity  Site  Soil Profile Name  NEHRP class A NEHRP class B  Hard rock Rock Very dense soil and soft rock Stiff soil profile Soft soil profile  NEHRP class C NEHRP class D NEHRP class E  Vs30 Recommended (m/sec) 1890 1070  NEHRP Vs30 Range (m/sec) > 1500 760 to 1500  520  360 to 760  250 150  180 to 360 < 180  *Note: The provided information is available in: http://peer.berkeley.edu/course_modules/eqrd/index.htm?c227top.htm&227cont.htm&IntExmp/atten00.htm (Last visited February 11, 2013)  77  Appendix B: Description of the CPT for Node Variable “Conductor Failure”  (ECL, RCL)  Conductor failure (ULCF, LCF, VLCF)  (VL1-150, VL1-150) (VL1-150, L150-300) (VL1-150, M300-450) (VL1-150, H450-700) (VL1-150, VH700-1000) (L150-300, VL1-150) (L150-300, L150-300) (L150-300, M300-450) (L150-300, H450-700) (L150-300, VH700-1000) (M300-450, VL1-150) (M300-450, L150-300) (M300-450, M300-450) (M300-450, H450-700) (M300-450, VH700-1000) (H450-700, VL1-150) (H450-700, L150-300) (H450-700, M300-450) (H450-700, H450-700) (H450-700, VH700-1000) (VH700-1000, VL1-150) (VH700-1000, L150-300) (VH700-1000, M300-450) (VH700-1000, H450-700) (VH700-1000, VH700-1000)  (80, 20, 0) (40, 50, 10) (5, 20, 75) (0, 15, 85) (0, 5, 95) (90, 10, 0) (80, 10, 10) (5, 30, 65) (5, 20, 75) (0, 15, 85) (90, 10, 0) (80, 15, 5) (70, 20, 10) (25, 30, 45) (10, 15, 75) (95, 5, 0) (85, 10, 5) (80, 15, 5) (65, 25, 10) (25, 30, 45) (98, 2, 0) (95, 5, 0) (85, 10, 5) (75, 20, 5) (60, 30, 10)  78  Appendix C: Description of the CPT for Node Variable “Rocking Response of Transformer”  (PGA, B/H, µ s) (Extremely Low, A0.34, Anchored, 0.45) (Extremely Low, B0.39, Anchored, 0.45) (Extremely Low, C0.42, Anchored, 0.45) (Extremely Low, D0.43, Anchored, 0.45) (Extremely Low, E0.50, Anchored, 0.45) (Extremely Low, F0.51, Anchored, 0.45) (Extremely Low, G0.65, Anchored, 0.45) (Extremely Low, H0.69, Anchored, 0.45) (Very Low, A0.34, Anchored, 0.45) (Very Low, B0.39, Anchored, 0.45) (Very Low, C0.42, Anchored, 0.45) (Very Low, D0.43, Anchored, 0.45) (Very Low, E0.50, Anchored, 0.45) (Very Low, F0.51, Anchored, 0.45) (Very Low, G0.65, Anchored, 0.45) (Very Low, H0.69, Anchored, 0.45) (Low, A0.34, Anchored, 0.45) (Low, B0.39, Anchored, 0.45) (Low, C0.42, Anchored, 0.45) (Low, D0.43, Anchored, 0.45) (Low, E0.50, Anchored, 0.45) (Low, F0.51, Anchored, 0.45) (Low, G0.65, Anchored, 0.45) (Low, H0.69, Anchored, 0.45) (Medium, A0.34, Anchored, 0.45) (Medium, B0.39, Anchored, 0.45) (Medium, C0.42, Anchored, 0.45) (Medium, D0.43, Anchored, 0.45) (Medium, E0.50, Anchored, 0.45) (Medium, F0.51, Anchored, 0.45) (Medium, G0.65, Anchored, 0.45) (Medium, H0.69, Anchored, 0.45) (High, A0.34, Anchored, 0.45) (High, B0.39, Anchored, 0.45) (High, C0.42, Anchored, 0.45) (High, D0.43, Anchored, 0.45) (High, E0.50, Anchored, 0.45)  RT (Rest, Slide, Rock) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (100, 0, 0) (99, 1, 0) (99, 1, 0) (99, 1, 0) (99, 1, 0) (99, 1, 0) (100, 0, 0) (100, 0, 0) (99, 1, 0) (85, 7.5, 7.5) (85, 15, 0) (0, 44, 56) (95, 0, 5)  79  (PGA, B/H, µ s) (High, F0.51, Anchored, 0.45) (High, G0.65, Anchored, 0.45) (High, H0.69, Anchored, 0.45) (Very High, A0.34, Anchored, 0.45) (Very High, B0.39, Anchored, 0.45) (Very High, C0.42, Anchored, 0.45) (Very High, D0.43, Anchored, 0.45) (Very High, E0.50, Anchored, 0.45) (Very High, F0.51, Anchored, 0.45) (Very High, G0.65, Anchored, 0.45) (Very High, H0.69, Anchored, 0.45) (Extremely High, A0.34, Anchored, 0.45) (Extremely High, B0.39, Anchored, 0.45) (Extremely High, C0.42, Anchored, 0.45) (Extremely High, D0.43, Anchored, 0.45) (Extremely High, E0.50, Anchored, 0.45) (Extremely High, F0.51, Anchored, 0.45) (Extremely High, G0.65, Anchored, 0.45) (Extremely High, H0.69, Anchored, 0.45) (Extremely Extremely High, A0.34, Anchored, 0.45) (Extremely Extremely High, B0.39, Anchored, 0.45) (Extremely Extremely High, C0.42, Anchored, 0.45) (Extremely Extremely High, D0.43, Anchored, 0.45) (Extremely Extremely High, E0.50, Anchored, 0.45) (Extremely Extremely High, F0.51, Anchored, 0.45) (Extremely Extremely High, G0.65, Anchored, 0.45) (Extremely Extremely High, H0.69, Anchored, 0.45)  RT (Rest, Slide, Rock) (95, 0, 5) (99, 1, 0) (99, 1, 0) (48, 10, 42) (50, 15, 35) (54, 16, 30) (55, 16, 29) (60, 25, 15) (61, 28, 11) (69, 30, 1) (69, 31, 0) (43, 12, 45) (38, 22, 40) (40, 43, 57) (42, 26, 32) (40, 32, 28) (48, 34, 18) (50, 45, 5) (50, 49, 1) (25, 15, 60) (25, 25, 50) (30, 30, 40) (30, 32, 38) (35, 35, 30) (38, 42, 20) (38, 54, 8) (39, 59, 2)  80  

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