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Reliability Assessment of the Floridian Electric Power Network System against Hurricanes

Permanent Link: http://ufdc.ufl.edu/UFE0043670/00001

Material Information

Title: Reliability Assessment of the Floridian Electric Power Network System against Hurricanes
Physical Description: 1 online resource (104 p.)
Language: english
Creator: Park, Young Jun
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: electricity -- florida -- hurricane -- regression -- reliability -- stochastic
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Civil Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This study aims to describe the regional performance on the electric power distribution systems (EPDS) against hurricane hazards with theories based on the uncertainty. The uniqueness of this study is the damage assessment models were proposed along with each device and customer minute interrupted was able to be estimated. A tree was revealed as the most significant factor to cause power outages. Wind momentums and rainfall were also verified as harmful elements. As damage extents, the number of each device break as well as customer minute interrupted could be assessed with statistical and stochastic models. The propose model using two estimable models was examined and accepted validly for describing power outages. Although this study aimed to describe the power disturbance with obtained weather, tree, and power outage data for a certain company’s service area, the methodologies as well as proposed damage assessment models could be more widely used to predict the other area serviced by the other companies. Furthermore, this ways can be used for indentifying vulnerable or safe zones relatively against to hurricanes with a hurricane simulation model. Ultimately, the outcomes of this study can be used to calculate the damage and restoration costs with predicted values of CMI and NB respectively, so that it helps to find out the cost analysis in order to determine which line is better between overhead and underground.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Young Jun Park.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Glagola, Charles R.
Local: Co-adviser: Gurley, Kurtis R.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0043670:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043670/00001

Material Information

Title: Reliability Assessment of the Floridian Electric Power Network System against Hurricanes
Physical Description: 1 online resource (104 p.)
Language: english
Creator: Park, Young Jun
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: electricity -- florida -- hurricane -- regression -- reliability -- stochastic
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Civil Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This study aims to describe the regional performance on the electric power distribution systems (EPDS) against hurricane hazards with theories based on the uncertainty. The uniqueness of this study is the damage assessment models were proposed along with each device and customer minute interrupted was able to be estimated. A tree was revealed as the most significant factor to cause power outages. Wind momentums and rainfall were also verified as harmful elements. As damage extents, the number of each device break as well as customer minute interrupted could be assessed with statistical and stochastic models. The propose model using two estimable models was examined and accepted validly for describing power outages. Although this study aimed to describe the power disturbance with obtained weather, tree, and power outage data for a certain company’s service area, the methodologies as well as proposed damage assessment models could be more widely used to predict the other area serviced by the other companies. Furthermore, this ways can be used for indentifying vulnerable or safe zones relatively against to hurricanes with a hurricane simulation model. Ultimately, the outcomes of this study can be used to calculate the damage and restoration costs with predicted values of CMI and NB respectively, so that it helps to find out the cost analysis in order to determine which line is better between overhead and underground.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Young Jun Park.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Glagola, Charles R.
Local: Co-adviser: Gurley, Kurtis R.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0043670:00001


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1 RELIABILITY ASSESSMENT OF THE FLORIDA ELECTRIC POWER NETWORK SYSTEM AGAINST HURRICANES By YOU NG JUN PARK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 201 2

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2 201 2 Young Jun Park

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3 To my family

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4 ACKNOWLEDGMENTS I would like to thank my committee chairs, Dr. Charles R. Glagola and Dr. Kurtis R. Gurley, and my committee members, Mr. Theodore J. Kury and Dr. Richard H. Schneider, for their guidance and support throughout the course of this research. Thanks also go t o my friends and colleagues and the department faculty and staff for making my time at the University of Florida a great experience. I also want to extend my gratitude to the Republic of Korea Army. Finally, thanks to my mother and parents in law for thei r encouragement and to my wife for her patience and love. To my three sons, when you cannot do anything anymore, make sure God is with you.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF ABBREVIATIONS ................................ ................................ ........................... 12 ABSTRACT ................................ ................................ ................................ ................... 13 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 15 Problem Statement ................................ ................................ ................................ 15 Literature Review ................................ ................................ ................................ .... 15 The EPDS Vulnerability ................................ ................................ .................... 15 The Most Hurricane Prone Region, Florida ................................ ...................... 16 The Cause of a Power System Failure ................................ ............................. 18 2 DATA COLLECTION AND MANAGEMENT ................................ ........................... 22 Sectionalized Units ................................ ................................ ................................ 22 Explanatory Variables ................................ ................................ ............................. 22 3 DAMAGE DESCRIPTION MODULE ................................ ................................ ...... 25 Uni Variate: Descriptive Statistics ................................ ................................ ........... 25 Bi Variate: Correlation Analysis ................................ ................................ .............. 36 Multi Variate: Multiple Linear Regression Analysis ................................ ................. 39 Normal Transformation ................................ ................................ ..................... 39 Model Stat ements ................................ ................................ ............................ 40 Results ................................ ................................ ................................ ............. 41 Outliers ................................ ................................ ................................ ............. 43 Assumptions ................................ ................................ ................................ ..... 43 Linearity ................................ ................................ ................................ ..... 44 Independence ................................ ................................ ............................ 49 Homoscedasticity ................................ ................................ ....................... 50 Normality ................................ ................................ ................................ .... 50 Multicollinearity ................................ ................................ ................................ 59 Summary ................................ ................................ ................................ ................ 60 4 RESTORATION ASSESSMENT MODULE ................................ ............................ 62 Variable Definition ................................ ................................ ................................ ... 62 Probability Distribution Fitting ................................ ................................ ................. 62 Correlation ................................ ................................ ................................ .............. 80 Random Number Generation ................................ ................................ .................. 82

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6 Generation of Uniformly Distributed Random Numbers ................................ ... 82 Generation of Standard Normal Random Numbers ................................ .......... 82 P rocedure to Generat e Random Numbers for Any Type of Distribution ........... 83 Simulation of Correlated Normal Random Variables ................................ ........ 83 Polynomial Regression ................................ ................................ ..................... 86 Procedure to Generate Correlated Non Gaussian Random Numbers ............. 87 Sensitivity Analysis ................................ ................................ ................................ 90 5 ASSESSMENT MODELS AND VALIDATION ................................ ........................ 92 6 CONCLUSION ................................ ................................ ................................ ........ 96 APPENDIX A OPERATION CENTERS OF PROGRESS ENERGY ................................ ............. 97 B WEATHER STATIONS IN FLORIDA ................................ ................................ ...... 98 C LETTER OF SOLICITATION ................................ ................................ ................ 100 LIST O F REFERENCES ................................ ................................ ............................. 101 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 103

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7 LIST OF TABLES Table page 1 1 The hurricane strikes on the U.S. mainland from 1851 2010. ........................... 16 1 2 Saffir/Simpson hurricane wind scale (SSHWS) ................................ ................. 17 1 3 Factors causing power system failures ................................ .............................. 18 3 1 Statistics for NB ................................ ................................ ................................ 27 3 2 Standard errors for NB ................................ ................................ ...................... 27 3 3 Correlation coefficient, significance, and data number ................................ ...... 38 3 4 for the normal transformation ................................ ................................ .......... 40 3 5 Coefficients and R 2 for the normal transformed linear regression ..................... 42 3 6 Durbin Watson statistics ................................ ................................ .................... 49 3 7 VIF ................................ ................................ ................................ ..................... 60 4 1 Goodness of fit of CI probability distribution for TF failure ................................ 65 4 2 Goodness of fit of M I probability distribution for TF failure ................................ 66 4 3 Goodness of fit of CI probability distribution for FS failure ................................ 67 4 4 Goodness of fit of M I probability distribution for FS failure ................................ 68 4 5 Goodness of fit of CI probability distribution for RC failure ................................ 69 4 6 Goodness of fit of M I probability distribution for RC failure ................................ 70 4 7 Goodness of fit of CI probability distribution for SR failure ................................ 71 4 8 Goodness of fit of CI probability distribution for FD failure ................................ 72 4 9 Goodness of fit of M I probability distribution for FD failure ................................ 73 4 10 Goodness of fit of CI prob ability distribution for SC failure ................................ 74 4 11 Goodness of fit of M I probability distribution for SC failure ................................ 75 4 12 Goodness of fit of CI probability distribution for TP failure ................................ 76 4 13 Goodness of fit of M I probability distribution for TP failure ................................ 77

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8 4 14 Goodness of fit of CI probability distribution for SG failure ................................ 78 4 15 Goodness of fit of M I probability distribution for SG failure ................................ 79 4 16 Probabilistic distribution of CI and MI ................................ ................................ 80 4 17 Correlation effects ................................ ................................ ............................. 89 5 1 E rror rates of the proposed model for the entire network ................................ .. 93

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9 LIST OF FIGURES Figure page 1 1 The hurricane strikes on the U.S. mainland from 1851 2010. .......................... 17 1 2 The distribution of factors causing power interruptions ................................ ..... 19 1 3 The distribution of factors causing device breaks ................................ ............. 19 3 1 Histogram for NB (TF) ................................ ................................ ...................... 28 3 2 Histogram for NB (FS) ................................ ................................ ...................... 28 3 3 Histogram for NB (RC) ................................ ................................ ..................... 29 3 4 Histogram for NB (SR) ................................ ................................ ...................... 29 3 5 Histogram for NB (FD) ................................ ................................ ...................... 30 3 6 Histogram for NB (SC) ................................ ................................ ...................... 30 3 7 Histogram for NB (TP) ................................ ................................ ...................... 31 3 8 Histogram for NB (SG) ................................ ................................ ..................... 31 3 9 Box plot for NB (TF) ................................ ................................ ......................... 32 3 10 Box plot for NB (FS) ................................ ................................ ......................... 32 3 11 Box plot for NB (RC) ................................ ................................ ......................... 33 3 12 Box plot for NB (SR) ................................ ................................ ......................... 33 3 13 Box plot for NB (FD) ................................ ................................ ......................... 34 3 14 Box plot for NB (SC) ................................ ................................ ......................... 34 3 15 Box plot for NB (TP) ................................ ................................ ......................... 35 3 16 Box plot for NB (SG) ................................ ................................ ......................... 35 3 17 Plot of residuals vs. predicted values (TF) ................................ ........................ 45 3 18 Plot of residuals vs. predicted values (FS) ................................ ....................... 45 3 19 Plot of residuals vs. predicted values (RC) ................................ ....................... 46 3 20 Plot of residuals vs. predicted values (SR) ................................ ....................... 46

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10 3 21 Plot of residuals vs. predicted values (FD) ................................ ....................... 47 3 22 Plot of residuals vs. predicted values (SC) ................................ ....................... 47 3 23 Plot of residuals vs. predicted values (TP) ................................ ....................... 48 3 24 Plot of residuals vs. predicted values (SG) ................................ ....................... 48 3 25 Q Q plot for the normal transformed linear regression residual (TF) ................ 51 3 26 Q Q plot for the normal transformed linear regression residual (FS) ................ 52 3 27 Q Q plot for the normal transformed linear regression residual (RC) ............... 52 3 28 Q Q plot for the normal transformed linear regression residual (SR) ................ 53 3 29 Q Q plot for the normal transformed linear regressi on residual (FD) ................ 53 3 30 Q Q plot for the normal transformed linear regression residual (SC) ................ 54 3 31 Q Q plot for the normal transformed linear regression residual (TP) ................ 54 3 32 Q Q plot for the normal transformed linear regression residual (SG) ............... 55 3 33 Histogram for the normal transformed linear regression residual (TF) ............. 55 3 34 Histogram for the normal transformed linear regression residual (FS) ............. 56 3 35 Histogram for the normal transformed linear regression residual (RC) ............. 56 3 36 Histogram for the normal transformed linear regression residual (SR) ............. 57 3 37 Histogram for the normal transformed linear regress ion residual (FD) ............. 57 3 38 Histogram for the normal transformed linear regression residual (SC) ............. 58 3 39 Histogram for the normal transformed linear regression residual (TP) ............. 58 3 40 Histogram for the normal transformed linear regression residual (SG) ............. 59 4 1 CI probability distribution for TF failure ................................ ............................. 65 4 2 MI probability distribution for TF failure ................................ ............................. 66 4 3 CI probability distribution for FS failure ................................ ............................. 67 4 4 MI probability distribution for FS failure ................................ ............................ 68 4 5 CI probability distribution for RC failure ................................ ............................ 69

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11 4 6 MI probability distribution for RC failure ................................ ............................ 70 4 7 MI probability distribution for SR failure ................................ ............................ 71 4 8 CI probability distribution for FD failure ................................ ............................. 72 4 9 MI probability distribution for FD failure ................................ ............................ 73 4 10 CI probability distribution for SC failure ................................ ............................ 74 4 11 MI probability distribution for SC failure ................................ ............................ 75 4 12 CI probability distribution for TP failure ................................ ............................. 76 4 13 MI probability distribution for TP failure ................................ ............................ 77 4 14 CI probability distribution for SG failure ................................ ............................ 78 4 15 MI probability distribution for SG failure ................................ ............................ 79 4 16 Scatter plot for MI (X axis) and CI (Y axis) ................................ ....................... 81 4 17 Gaussian (X axis) and non Gaussian (Y axis) distributions correlation. ........... 88 4 18 Sensitivity analysis. ................................ ................................ .......................... 91 5 1 Flowchart to predict NBs and CMIs ................................ ................................ .. 92 5 2 Errors of NB prediction ................................ ................................ ..................... 94 5 3 Observed and predicted values of NB ................................ .............................. 94 5 4 Errors of CMI prediction ................................ ................................ .................... 95 5 5 Observed and predicted values of CMI ................................ ............................ 95

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12 LIST OF ABBREVIATION S CI the number of C ustomers I nterrupted due to a system failure CMI CI MI EPDS E lectric Power D istribution System EPNS Electric Power Network System FD Feeder FS Fuse HURDAT Atlantic Hurricane Database MI the M inute s I nterrupted due to a system failure NB Number of Breaks NHC National Hurricane Center RC Recloser SC Sectionalizer SG Switchgear SR Service SSHWS Saffir/Simpson Hurricane Wind Scale TF Transformer TP Terminal Pole VIF Variation Inflation Factors

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13 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy RELIABILITY ASSESSM ENT OF THE FLORIDA ELECTRIC POWER NETWORK SYSTEM AGAINST HURRICANES By You ng Jun Park May 2012 Chair: Charles R. Glagola Cochair: Kurtis R. Gurley Major: Civil Engineering This study aims to describe the regional performance of the electric power distribution systems (EPDS) against hurricane hazards with theories based on the uncertainty. The uniqueness of this study is that the damage assessment models were proposed along w ith eight cr itical devices, i.e. transformers, fuses, reclosers, service lines, feeders, sectionalizers, terminal poles, and switchgears and the C ustomers M inutes I nterrupted (CMI) were able to be estimated. Trees were revealed as the most significant fact or to cause power outages. Wind momentum and rainfall were also verified as major contributing factors In terms of the extent of damage, the number of device breaks (NB) as well as the CMI could be assessed with statistical and stochastic models. The prop osed model using two estimable models was examined and determined to be valid for describing power outages. Although this study aimed to describe the power disturbance with measured/recorded weather, tree, and power outage data for a participating power di stribution methodologies as well as proposed damage assessment models could be more widely used to predict the areas serviced by other companies. Furthermore, these ways can be

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14 used for indentifying relatively vulnerable or safe zones against hurricanes with a hurricane simulation model. Ultimately, the outcomes of this study can be used to calculate the damage and restoration costs with predicted values of CMI and NB, respectively, so that they help in finding out the cost analy sis in order to determine whether an overhead or an underground line is better.

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15 CHAPTER 1 INTRODUCTION Problem Statement This study aims to describe the regional performance of the Electric Power Distribution Syst ems (EPDS) against hurricane hazards with theories based on uncertainty. For this study it is necessary to identify factors which cause EPDS failures. In particular, it should be verified whether a tree could be a major contributing factor or not to the power disturbances. Further, it needs to be identified how much the power interruption can be described with the adapted factors. Finally, in terms of structural integrity statistical models were developed to describe the number of device breaks as well as, in terms of stable operation, non Gaussian probability distributions to assess the interrupted electric power amount need to be calculated. The former is related to the damage assessment and the latter is intended to describe restoration of the damaged system. Once these uncertainty models are developed and tested it will be possible to assess the reliability of the EPDS in a region. Literature Review The EPDS Vulnerability Lifelines, as a unique network infrastructure, are so interconnected that their interdependency can make them vulnerable to not only a series of malfunctions but also widespread disruptiveness. Considering these social, economic, and corporeal damages, it is imperative to guarantee the structural safety as well as the stable operation of lifelines during hurricanes. Among the lifelines, the Electric Power Network

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16 System (EPNS) is the most vulnerable to hurricanes if it is not underground (Quanta Technology, 2008; Reed, 2008; Zhou, Pahwa, & Das, 2006; Winkler, Duenas Osorio, Stein, & Subramanian, 2010) The EPNS is classified into three sub systems: generation, transmission, and distribution. The genera ting plants and transmission phases are built and designed to resist winds, floods, etc. caused by the hurricanes with fairly high redundancy On the other hand, the EPDS are the most vulnerable and suffer from significant hurricane damages (Gonen, 1986) The Most Hurricane Prone Region, Florida Among the main states damaged by hurricanes, it is well known that Florida is one of the most hurricane prone regions. Table 1 1 and Figure 1 1 show the hurricane strikes on the U.S. mainland from 1851 2010 (Blake & Gibney, 2011) "Ma jor" means the number of strikes of categories 3, 4, and 5. Table 1 1 The hurricane s trikes on the U.S. m ainland from 1851 2010. Area Category n umber 1 2 3 4 5 All Major U.S. 113 75 75 18 3 284 96 Texas 27 18 12 7 0 64 19 Louisiana 21 16 16 3 1 57 20 Mississippi 4 6 8 0 1 19 9 Alabama 17 5 5 0 0 27 5 Florida 43 34 29 6 2 114 37 Georgia 15 5 2 1 0 23 3 South Carolina 17 7 4 2 30 6 North Carolina 25 14 11 1 0 51 12 Virginia 7 2 1 0 0 10 1 New York 6 1 5 0 0 12 5 Connecticut 5 3 3 0 0 11 3 Rhode Island 3 2 4 0 0 9 4 Massachusetts 6 2 3 0 0 11 3 Maine 5 1 0 0 0 6 0

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17 Figure 1 1 The h urricane strikes on the U.S. m ainland from 1851 2010. From 1851 1930 and 1990 2010, a strike has been designated to be the counts of sustained hurricane force winds on the coastline or inland. From 1931 1989, the case a hurricane center passed was defined as a strike. Table 1 1 is assigned as follows. The National Hurricane Center (NHC) assigns a hurricane scale according to the Saffir/Simpson Hurricane Wind Scale (SSHWS) as shown in Table 1 2 (Blake & Gibney, 2011) Table 1 2 Saffir/Simpson h urricane w ind s cale (SSHWS) Category Minimum central pressure (mb) Max s ustained w ind s peed (mph) 5 < 920 > 1 55 0 20 40 60 80 100 120 TEXAS LOUISIANA MISSISSIPPI ALABAMA FLORIDA GEORGIA SOUTH CAROLINA NORTH CAROLINA VIRGINIA MARYLAND DELAWARE NEW JERSEY PENNSYLVANIA NEW YORK CONNECTICUT RHODE ISLAND MASSACHUSETTS NEW HAMPSHIRE MAINE STRIKES MAJOR TOTAL

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18 4 920 944 130 155 3 945 964 110 130 2 965 979 94 110 1 > 9 80 74 94

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19 For reference, the original SSHWS was grounded on wind, central pressure, storm surge values, and their combinations (Herbert & Taylor, 1975) However, the NHC has categorized the SSHWS category just on the basis of the maximum s ustained wind speed for a minute since about 1990, i.e. during this term they used a single variable, the maximum sustained wind speed instead of using wind, central pressure, storm surge values, and their combinations. Thus, it is inevitable to be a cert ain amount of inconsistency with the categorization used in the Atlantic hurricane database (HURDAT) is to be expected (Landsea, et al., 2004) The original SSHWS was grounded on wind, central pressure, storm surge values, and their combinations (Herbert & Taylor, 1975) The Cause of a Power System Failure Table 1 3 Figure 1 2 and Figure 1 3 show the factors which caused a break in the EPDS during hurricanes and their percent ages. The time for this data is from August 1 to October 15, 2004; a nd from September 1 to 20, 2008 and reflects data from a major electric power company in Flor ida. For reference, it should be noted that this data include the break under normal conditions The target area is summarized as shown in A ppendix A. Table 1 3 Factors causing power system failures Power i nterruption Device b reak Cause CMI (minute) Portion NB Portion Normal 4763095141 31.03% 82687 4.15% Animal 5158625202 33.61% 264187 13.25% Birds 81057 0.00% 82455 4.14% Lightning 13478861 0.09% 546349 27.41% Storm 4570307900 29.78% 237402 11.91% Tree 461044120 3.00% 14059 0.71% Wind 62498808 0.41% 33102 1.66% Unknown 318765140 2.08% 732987 36.77% Sum 15347896229 100.00% 1993228 100.00%

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20 Figure 1 2 The distribution of factors causing power interruptions Figure 1 3 The distribution of factors causing device breaks Normal 31% Animal 34% Birds 0% Lightning 0% Storm 30% Tree 3% Wind 0% Unknown 2% Normal 4% Animal 13% Birds 4% Lightning 27% Storm 12% Tree 1% Wind 2% Unknown 37%

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21 The failure portions caused by storms, i.e., storms, wind, and trees, are about 33% and 15% in power interruptions and device breaks, respectively. Considering the failure portions of 31% and 4% caused under normal operations, it is concluded that the dama ges caused by storms cannot be overlooked. Furthermore the damages during hurricanes tend to be concentrated. This means that the restoration capability cannot follow the hurricane damages, so that it is hard to expect immediate power supplies because the delayed restoration should be inevitable. Even though falling trees (vegetation) have been recognized as the main factor concerning power disruption in the distribution system, identification of the statistical segmentation failed because it was historica lly difficult to collect and manage data concerning trees. In addition to falling trees, other factors such as rainfall, wind, terrain, and soil drainage were also considered as explanatory factors which affect electric outages or system failures. However, they need to be reexamined statistically or re clarified from previous studies (Han, Guikema, Quiring, Lee, Rosowsky, & Davidson, 2009; Liu, Davidson, David, & Stedinger, 2005; Davidson, Liu, Sarpong, Sparks, & Rosowsky, 2003) Besides damage sources, it could be assumed that each device has intrinsic resistance characteristics. Most studies just focused on a sole device which is relatively more fragile in hurricanes such as feeders, service lines, or poles (Quanta Technology, 2008) Otherwise, the system was analyzed entirely without identifying the failure of each device (Winkler, Duenas Osorio, Stein, & Subramanian, 2010; Han, Guikema Quiring, Lee, Rosowsky, & Davidson, 2009; Liu, Davidson, David, & Stedinger, 2005) However, it should be true that each device has its own strength to

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22 withstand the hurricane hazards, so that the damag e extents should be assessed or described along with the strength of each device. The Outline of This Study The core of this study is the EPNS damage assessment protocol against hurricanes using the statistical models to describe NB and the non Gaussian probability distribution models to generate CMI. For developing these two models, firstly, data concerning power outages, weather and tree density were collected and managed. The used data was related to the following hurricanes: Chaley, Frances, and Ivan in 2004 And then, two models, i.e. the multiple linear regression model for describing NB and non Gaussian probabilistic distribution for evaluating CMI were developed as well as proposed. In addition to managing data and establishing th e assessment models, the proposed damage assessment procedure and model was verified through the data concerning the hurricane Katrina in 2008.

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23 CHAPTER 2 DATA COLLECTION AND MANAGEMENT Sectionalized Units A sectionalized unit area was used for sp atial a nalysis because all data need to be collected and managed based upon a specified region. A grid system (mostly 1km 1 km) was first tried but eventually outcomes along with zip codes were used because the larger area could absorb more uncertainties, so tha t the errors between observed and predicted values tended to be decreased (Han, Guikema, Quiring, Lee, Rosowsky, & Davidson, 2009; Liu, Davidson, David, & Stedinger, 2005; Davidson, Liu, Sarpong, Sparks, & Rosowsk y, 2003) Thus, in this study, an observational center was used as a sectionalized unit because obtained power outage data were collected along with this quarter of observational center, and it should return such reasonable outcomes with decreasing uncertainties. Explanatory Variables Even though a falling tree has been acknowledged as a predominant factor to describe causes of power outages, (maximum) wind speed was typically used as an independent variable in statistical models because it was assu med that a falling tree is also caused by the wind. Meanwhile, the characteristics of the weather, trees, and soil were concurrently considered as independent variables to increase R 2 of regression models. As weather characteristics, rainfalls and wind sp eed were mostly chosen and indentified as significant factors. A tree type was also considered with a categorical variable, but unfortunately it was hard to be assigned as a significant variable due to the resolution limitation and the uncertainty it cause s. Soil characteristics such as a soil type

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24 and drainage were determined as non significant variables (Han, Guikema, Quiring, Lee, Rosowsky, & Davidson, 2009; Liu, Davidson, David, & Stedinger, 2005; Davidson, Liu, Sarpong, Sparks, & Rosowsky, 2003) In addition to these natural features, data concerning land use and cover, and population could also be used thanks to the GIS development, but these elements should be revised or eliminated because of their highe r Variation Inflation Factors (VIF), which mean these variables are strongly correlated with the others. Besides these studies using these multi variables, there were others which used slightly different approaches applying network topologies (Winkler, Duenas Osorio, Stein, & Subramanian, 2010) At this point, however, it is meaningless to consider network topologies in statistical and probabilistic models as variables because this system used the grid system instead of the noda l link system. Therefore, it is concluded that weather characteristics should be used as independent variables and the ones concerning trees need to be revised. The Florida Automated Weather Network keeps all historical weather information. Rainfalls (R F mm), average wind speeds (W A m/s), and maximum wind speeds (W M m/s) were used as independent variables. All weather data were recorded every 15 minutes and measured at a 10 meter height above the ground. Time is the pin point to connect two kinds of data: weather data and outage data. Power outage data was provided by Progress Energy in Florida. Eighteen observational centers collected power outages in their districts with disrupted and restored time. Subsequently, these recorded data were sent to headquarters, synthesized, stored in the database which holds data concerning blackout minutes and numbers of customers interrupted disrupted time, broken devices, and so on. The records of power outages in each observational center are

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25 joined with the w eather data of the closest weather station. If the recorded time difference between weather and outage is more than five minutes, those power outage records were eliminated in this study. While the transmission phase usual ly installs a setback to block certain failure s caused by a falling tree, the overhead distribution system is typically constructed at a height of 30 to 40 ft for making enough space to avoid physical damages incurred by a falling tree or its debris (Davidson Liu, Sarpong, Sparks, & Rosowsky, 2003) Thus, trees with a height of less than 50 ft could be neglected. On the other hand, if their heights are higher than 50 ft, debris, whether their branches or themselves, could threaten the overhead transmission system. Besides the tree height and density, there also may be other significant factors, which are related with power outages. Therefore, tree density (T D # of trees with heights more than 50 ft /Km 2 ) is used as an independent variable. However, in order to reduce the uncertainty caused by resolution, T D was treated as an ordinal variable.

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26 CHAPTER 3 DAMAGE DESCRIPTION M ODULE Power outages were associated with eight electrical distribution devices: transformers (TF), fuses (FS), reclosers (RC), service li nes (SR), feeders (FD), sectionalizers (SC), terminal poles (TP), and switchgears (SG). The dependent variables are the numbers of each device break (NB) per 15 minutes per 1,000 units failure is independent from the others. Transformers make power energy be transferred from a certain circuit from another inductively. Fuses are a kind of the resistor that act s as the over current protection to prevent overheating or fire. Reclosers are mounted just on the overhead EPD S to track down and break in momentary faults. It is cooperated with down stream protective devices, sectionalizers which do not interrupt fault current, but observes fault current and circuit interruption by reclosers. Swi tchgears are used to clear downstream faults for guaranteeing the reliability of the power supply. As a conductor, feeders carry power from upper level equipment to the panels at the lower level. These devices are used as a protective device, i.e. the equi pment to detect the system failure. Therefore, the power company can detect device breaks and power cut as well as can keep system failures database along with these devices (Gonen, 1986) Uni Variate: Descriptive Statistics Table 3 1 and Table 3 2 show descriptive stat istics and their standard errors for the Number of Breaks (NB) recorded for each device during hurricane Chaley, Frances, and Ivan in 2004. This enables one to determine the central tendency dispersion, and

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27 distribution of NB numerically. Figure 3 1 to Figure 3 16 show these distributi on s graphically. According to the descriptive statistics, the NB distributions for each device should be defined as positively skewed or a skewed to the right. As shown in Table 3 1 the means of NB for each device are bigger than the medians of NB, so that all distributions in Figure 3 1 to Figure 3 8 tend to have longer tails in the positive direction. Furthermore, the skewness values in Table 3 1 are also bigger than zero, i.e., the mass of the NB distribution for each device is concentrated on the left of the histograms as shown in Figure 3 1 to Figure 3 8 This tendency is stronger in TF, FS, and RC. In addition to skewness, the distribution of NBs for TF, FS, and RC are leptokurtic, i.e., these distributions have positive excess kurtosis as shown in Table 3 1 In addition, the kurtosis for NB distributions of SR, FD, SC, TP, and SG are also far from 3, i.e., NBs for these devices are not normally distributed. Lastly, considering variances, standard deviations, and interquartile ranges, the ranges seem to be anomalistic in TF, FS, RC, and SR, i.e., the ranges are extremely wide as shown in Table 3 1 These tendencies are caused by the extreme values as shown in Figure 3 9 and Figure 3 16 Therefore, to obtain the r obust analysis, consideration for these outliers or leverages is required. In sum, it is concluded that data are not normally distributed, so that it is indispensible to use non linear models in a regression analysis. Otherwise, the multiple linear models may be valid with normal transformations. Moreover, for scrutinizing relations between variables as preliminary tests the way to minimize the effect of

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28 outliers is needed. That is, Spearman correlations which are more robust to outliers and not affected by transformation should be utilized. Table 3 1 Statistics for NB TF FS RC SR FD SC TP SG Mean 1.83 1.73 1.40 2.41 1.38 1.24 0.87 1.41 Median 1.33 1.33 1.31 1.80 1.33 1.23 0.81 1.31 Mode 1.33 1.33 1.33 1.33 1.33 1.08 0.63 1.31 Variance 2.21 3.03 0.55 2.85 0.09 0.08 0.08 0.37 Std Dev 1.49 1.74 0.74 1.69 0.30 0.29 0.28 0.61 Min 0.60 0.63 0.60 0.63 1.05 0.67 0.63 0.67 Max 33.20 45.65 8.30 12.92 2.15 2.15 1.33 3.56 Range 32.59 45.02 7.69 12.29 1.10 1.49 0.70 2.89 IQR 0.94 0.92 0.36 1.92 0.20 0.36 0.51 0.73 Skewness 8.07 15.55 4.91 2.02 1.55 1.22 0.95 1.83 Kurtosis 144.19 362.99 39.49 5.60 2.28 3.65 0.70 4.96 Table 3 2 Standard e rrors for NB TF FS RC SR FD SC TP SG Mean 0.04 0.05 0.05 0.04 0.05 0.05 0.07 0.11 Skewness 0.07 0.07 0.18 0.05 0.39 0.38 0.60 0.44 Kurtosis 0.13 0.14 0.35 0.10 0.77 0.74 1.15 0.86

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29 Figure 3 1 Histogram for NB (TF) Figure 3 2 Histogram for NB (FS)

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30 Figure 3 3 Histogram for NB (RC) Figure 3 4 Histogram for NB (SR)

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31 Figure 3 5 Histogram for NB (FD) Figure 3 6 Histogram for NB (SC)

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32 Figure 3 7 Histogram for NB (TP) Figure 3 8 Histogram for NB (SG)

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33 Figure 3 9 Box p lot for NB (TF) Figure 3 10 Box p lot for NB (FS)

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34 Figure 3 11 Box p lot for NB (RC) Figure 3 12 Box p lot for NB (SR)

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35 Figure 3 13 Box p lot for NB (FD) Figure 3 14 Box p lot for NB (SC)

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36 Figure 3 15 Box p lot for NB (TP) Figure 3 16 Box p lot for NB (SG)

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37 Bi Variate: Correlation Analysis In a statistic analysis, relationship means the dependences between random variables. Correlation analysis refers to the ways to investigate this relationship between variables prior to the regression analysis which is the fundamental statistical way to calculate the dependencies among random variables. Table 3 3 s hows both Spearman and Pearson correlation coefficients, the significance of all independent variables (R F W A W M and T D ) with a dependent variable (NB), and the data count for each device. The Pearson correlation coefficient measures how well two random variables are linearly related. With covariance between two variables and their standard deviations, i t is defined as follows : However, in this study, it is impossible to u se this tool to look into the relationship between variables because the collected data cannot satisfy normality. Furthe rmore, it is hard to generalize the normal distribution of the damage data for TP and SC because of their insufficient data counts (whic h have fewer than 35 records) as shown in Table 3 1 The Spearman correlation coefficient measures how well two variables are monotonically related bec ause it is defined as the Pearson correlation coefficient between the raked v ariables. Therefore, thanks to using ranked variables, Spearman correlation coefficient s tend to be more robust to outliers and it is less affected by transformation than Pearson correlation coefficient s (Ott, 2008)

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38 As shown in Table 3 3 the results between both correlation results a re a little different. According to the Spearman correlation analysis, tree density is the most significant as well as predominant variable to be related with NB in all devices. In particular, NB has dependence just with tree density and less significant r elations with the other independent variables in SC, TP, and SG. These outcomes should be reflected in the regression analysis. For reference, the significance level is .05.

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39 Table 3 3 Correlation c oefficient, s ignificance, an d d ata n umber R F W A W M T D Spearman TF Coeff. .275 .214 .217 .476 Sig. .000 .000 .000 .000 N 1402 1402 1402 1402 FS Coeff. .182 .112 .164 .542 Sig. .000 .000 .000 .000 N 1156 1156 1156 1156 RC Coeff. .028 .173 .064 .693 Sig. .698 .016 .380 .000 N 191 191 191 191 SR Coeff. .155 .226 .297 .423 Sig. .000 .000 .000 .000 N 2275 2275 2275 2275 FD Coeff. .428 .545 .152 .593 Sig. .009 .001 .377 .000 N 36 36 36 36 SC Coeff. .177 .079 .233 .830 Sig. .280 .634 .154 .000 N 39 39 39 39 TP Coeff. .176 .365 .186 .967 Sig. .547 .200 .525 .000 N 14 14 14 14 Pearson SG Coeff. .041 .124 .162 .265 Sig. .836 .529 .411 .173 TF Coeff. .303 .305 .287 .313 Sig. .000 .000 .000 .000 FS Coeff. .310 .189 .203 .268 Sig. .000 .000 .000 .000 RC Coeff. .015 .008 .074 .466 Sig. .838 .911 .307 .000 SR Coeff. .073 .217 .289 .409 Sig. .000 .000 .000 .000 FD Coeff. .265 .550 .296 .375 Sig. .118 .001 .080 .024 SC Coeff. .076 .006 .178 .662 Sig. .648 .969 .278 .000 TP Coeff. .209 .325 .283 .993 Sig. .473 .257 .327 .000 SG Coeff. .112 .273 .176 .248 Sig. .571 .160 .370 .204

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40 Multi Variate: Multiple Linear Regression Analysis Normal Transformation Most statistical theories as well as tools are developed under the assumption of normality, i.e., if any test is not satisfied with this assumption, the results could not be accepted reasonably. Furthermore, this assumption makes most statistical tests sim ple, powerful, and convenient more than the ones which are not based on the normality assumption. Unfortunately, large amounts of data cannot satisfy this assumption. Therefore, it is indispensible to transform data normally because this normal transformat ion may yield tactical l results that follow normal distribution approximately. This normally transformed data make statistical analyses more applicable as well as useful because they can satisfy the normality assumption more than the original one. The Box Cox transformation is one of the most popularly used tactical tool for the normal transformation. It is defined as where particular transformation such as the Box Cox transformation defined above, it is helpful to define a measure of the normality of the resulting transformation. In a multiple linear regression, a data set also has to be distributed normally. However, the collected and managed data in this study is not normally distributed.

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41 Therefore, in this study the original dependent data, NBs, for each device should be as shown in Table 3 4 Table 3 4 for the n ormal t ransformation TF FS RC SR FD SC TP SG .4 .5 .5 .3 .3 .5 .3 .5 Model Statements in Table 3 4 the multiple linear regression models for each device break can be state d as follows: where are coefficients of multiple linear regression model and is an error. Therefore, the final equation to estimate NBs for each device can be provided as follows:

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42 Results Table 3 5 shows the results of the multiple linear regression analysis. The dependent variable is normally transformed Table 3 4 and the independent variables are T D W A R F and W M Table 3 5 hard to conclude that the variable is significant. The results of the correla tion analysis show that T D is also revealed as the most significant factor to be related with hurricane damage of power networks. Most researchers and engineers agree that trees are the predominant cause of the failure of power systems. However, in spite o f a few trials to verify the significance of the falling tree s there was not any clear evidence to support this claim in previous stud ies As verified in other studies, wind momentum and rainfall are also significant factors.

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43 Table 3 5 Coefficients and R 2 for the n ormal transformed linear r egression Device Mode l Variable e ntered Coefficients R 2 T D W A R F W M C TF 1 O 1.066 .066 .261 2 O O 1.147 .069 .019 .376 3 O O O 1.143 .073 .014 .036 .406 4 O O O O 1.150 .071 .007 .038 .002 .415 FS 1 O 1.121 .086 .324 2 O O 1.151 .090 .056 .412 3 O O O 1.183 .090 .047 .002 .432 4 O X O O 1.184 .090 .001 .046 .002 .432 RC 1 O 1.198 .092 .457 2 O O 1.192 .091 .000 .457 3 O X X 1.203 .093 .003 .001 .460 4 O X X X 1.205 .094 .001 .005 .001 .461 SR 1 O 1.001 .054 .225 2 O O 1.054 .054 .004 .318 3 O X O 1.053 .053 .002 .005 .318 4 O X X O 1.053 .053 .003 .005 .005 .319 FD 1 O .087 .007 .373 2 O O .917 .011 .006 .427 3 O O X .923 .013 .007 .007 .437 4 O O X X .927 .013 .008 .008 .000 .443 SC 1 O 1.168 .073 .567 2 O O 1.188 .78 .127 .593 3 O X X 1.185 .082 .005 .232 .609 4 O X X X 1.185 .081 .005 .226 .000 .609 TP 1 O .803 .071 .992 2 O O .803 .071 .010 .992 3 O O O .801 .071 .003 .010 .995 4 O O O O .802 .071 .002 .003 .008 .995 SG 1 O 1.089 .068 .166 2 O O 1.131 .068 .015 .195 3 O O O 1.144 .070 .020 .106 .223 4 O O O X 1.146 .070 .014 .111 .001 .223

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44 In addition, models for most failure have figures greater than 40% as their R 2 values. These values are relatively lower than the ones of other studies because they used nonlinear regression models. In particular, SR and SG have noticeable low R 2 It could be guessed that SR and SG have lower R 2 because their failure mainly occurred in a distribution phase. In other words, it may require more variables than the ones used in this study for describing the number of these failures in a distribution phase. In particular, TP have almost perfect R 2 values as shown in Table 3 6 Outlier Outliers cause violations of the main assumption in a regression analysis as well as potentially inducing invalid outcomes. Therefore, it is important to investigate outliers. However, it is hard to check whether outliers data are measured correctly at this point. Thus, the only way is to reduce the number of outliers is with normal transformation. Assumptions Quantitative analysis should satisfy some assumptions about the way the world works. In addition, a linear regression analy sis always rests on four main assumptions for justifying its model as a prediction tool. The four principal assumptions of a linear regression are linearity, independence, homoscedasticity, and normality. Linearity means the dependent variable has linear r elationships with independent variables. Independence means the errors in a regression analysis should not have auto correlation. Homoscedasticity means errors should have constant variance. Lastly, the error in a regression analysis should be distributed normally. If any of these assumptions is not satisfied, the analysis cannot guarantee reasonable prediction,

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45 standard error, and so on, so that it could be concluded that the obtained model is inaccurate Linearity Nonlinearity is extremely serious because it causes predictions that are likely to be substantially in error. Nonlinearity is usually most evident in the examination of residual plots, i.e., plots of the standardized residuals as a function of standardized predicted values which are a part of standard regression. Ideally, t he points should be symmetrically distributed along a line. For reference, the way to fix nonlinearity is to transform dependent and/or independent variables normally as was done previous ly. Figure 3 17 to Figure 3 24 show plots of the standardized residuals as a f unction of standardized predicted values. According to these plots, it could be concluded that the performed regression analysis for each device satisfies the linearity assumption.

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46 Figure 3 17 Plot of residuals vs. p redic ted v alues (TF) Figure 3 18 Plot of residuals vs. p redicted v alues (FS)

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47 Figure 3 19 Plot of residuals vs. p redicted v alues (RC) Figure 3 20 Plot of residuals vs. p redicted v alues (SR)

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48 Figure 3 21 Plot of residuals vs. p redicted v alues (FD) Figure 3 22 Plot of residuals vs. p redicted v alues (SC)

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49 Figure 3 23 Plot of residuals vs. p redicted v alues (TP) Figure 3 24 Plot of residuals vs. p redicted v alues (SG)

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50 Independence Autocorrelations are also very serious in time series regression models. The Durbin Watson (DW) statistic provides a te st for significant residual autocorrelation If is the residual associated with the observation at time t, then DW statistic is defined as follows: where T is the number of observations. Since d is approximately equal to 2(1 r ), where r is the sample autocorrelation of the residuals. Therefore, the DW statistic typically has a value between 0 and 4. If the DW statistic is close to 2, the analysis could be accepted as reasonabl e because it could be concluded that there is no autocorrelation in the residuals. If the DW statistic is close to zero or 4, there may be positive or negative autocorrelation, respectively (Ott, 2008) Howe ver, this statistic n eeds to be considered with the sample size. Ideally, if the DW statistic is between 1.4 and 2.6 for a sample size of 50, it could be concluded that there is no autocorrelation. According to Table 3 7 most models satisfy the assumption of independence without TF and SR. However, when considering both sample sizes as well as the DW statistics, it is possible to conclude that the residuals for these two failures are indepe ndent. Table 3 7 DW statistics TF FS RC SR FD SC TP SG Statistics 1.243 2.121 1.457 1.116 2.204 2.308 1.872 1.455 N 1402 1156 191 2275 36 39 14 28

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51 Homoscedasticity If the variance is not equal, i.e., homoscedasticity occurs in the regression analysis, it may be hard to gauge the acceptable standard errors, i.e. it results in confidence intervals that are too wide or too narrow. This violation can be detected with plots of residuals versus predicted values as shown in Figure 3 17 to Figure 3 24 Heteroscedasticity may also be related with nonlin earity or autocorrelation. Some combination of logging and/or deflating will often stabilize the variance because heteroscedasticity often arises due to the effects of inflation. In this study, normal transformation was used as a deflating tool. Therefore, although it was hard to conclude that variances are equal in a preliminary analysis, i.e., analysis before normal transformation, there is not any noticeable evidence which can reflect homoscedasticity as shown in Figure 3 17 to Figure 3 24 thanks to normal transformation. Normality Coefficients in a linear regression model are estimated with a least square method. Therefore, sometimes the distribution of residuals could be skewed due to extreme outliers or leverages. A linear regression analysis needs to be from the violation of norm ality because a few extreme observations can exert a disproportionate influence on parameter estimates. The best way to check whether distributed residuals are normal or not is just to use its histogram. The other way is a Q Q plot. A Q Q plot is a plot ba sed on estimates of the quantiles. The pattern of points in the plot is used to compare the two distributions If the distribution is normal, the points on this plot should fall close to the diagonal line in the Q Q plot (Ott, 2008)

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52 Figure 3 25 to Figure 3 32 show Q Q plot s. In these plots, it is hard to find a noticeable bow shaped pattern of deviations from the diagonal. For reference, a bow shaped pattern indicates that the residuals have excessive skewness. On the other hand, most plots seem to be an S shaped pattern of deviations which means that the residuals have excessive kurtosis. Furthermore, the residual histogram as shown in Figure 3 33 to Figure 3 40 seems not to be normal. Thus, it could be problematic that the residual is normally distributed in spite of normal transformation. However, the sample sizes in most models are larger than 35, so that it is possible to assume normality thanks to the C entral L imit T heorem because it states that if the original distribution has a mean and a standard deviation the distribution of the sample means will be normal with a mean and a standard deviation given by where n is the sample size (Ott, 2008) Figure 3 25 Q Q plot for the n ormal t ransformed linear regression r esidual (TF)

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53 Figure 3 26 Q Q plot for the n ormal t ransformed linear regression r esidual (FS) Figure 3 27 Q Q plot for the n ormal t ransfor med linear regression r esidual (RC)

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54 Figure 3 28 Q Q plot for the n ormal t ransformed linear regression r esidual (SR) Figure 3 29 Q Q plot for the n ormal t ransformed linear regression r esidual ( FD)

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55 Figure 3 30 Q Q plot for the n ormal t ransformed linear regression r esidual (SC) Figure 3 31 Q Q plot for the n ormal t ransformed linear regression r esidual (TP)

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56 Figure 3 32 Q Q plot for the n ormal t ransformed linear regression r esidual (SG) Figure 3 33 Histogram for the normal transformed l inear r egression r esidual (TF)

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57 Figure 3 34 Histogram for the normal transformed l inear r egression r esidual (FS) Figure 3 35 Histogram for the normal transformed l inear r egression r esidual (RC)

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58 Figure 3 36 Histogram for the normal transformed l inear r egression r esidual (SR) Figure 3 37 Histogram for the normal transformed l inear r egression r esidual (FD)

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59 Figure 3 38 Histogram for the normal transformed l inear r egression r esidual (SC) Figure 3 39 Histogram for the normal transformed l inear r egression r esidual (TP)

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60 Figure 3 40 Histogram for the normal transformed l inear r egression r esidual (SG) Multicollineari ty If a linear regression analysis aims to check how well independent variables predict dependent variables, multicollinearity can be ignorable. However, if this analysis aims to find out how much independent variables impact dependent variables, it is ess ential to check this problem. Multicollinearity usually occurs when there are relationships between independent variables. The way to detect multicollinearity is with the Variation Inflation Factor (VIF). If the VIF is larger than 10, there are correlations between the independent variables. According to the VIF results as shown in Table 3 8 it could be concluded that WA and WM are highly related as expected in TP. In addition to TP, W A is a little related with W M in TF. This tendency occurred in models which used both WA and WM

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61 as significant independent variables, the wind momentum. There are several ways to reduce the correlation between variables such as making interaction effects. However, this could be somewhat anticipated because both variables came from the same root. Thus, this muliticollinearity was ignored. Table 3 8 VIF Device Model Variable e ntered VIF TD WA RF WM TD WA RF WM TF 4 O O O O 1.015 3.189 1.285 2.792 FS 3 O O O 1.010 1.133 1.122 RC 2 O O 1.101 1.101 SR 2 O O 1.000 1.000 FD 3 O O 1.086 1.086 SC 2 O O 1.069 1.069 TP 4 O O O O 1.216 83.503 7.303 113.94 SG 3 O O O 1.007 1.108 1.115 Summary Consequently, it could be concluded that average wind speed, maximum wind speed, and rainfall are significant factors to impact electric equipment failure like the outcomes in previous studies. The most important of these is tree density. Tree density was revealed as the most significant factor to affect electric equipment failure. Most researchers tried to find out the ways to pro ve this fact previously but failed due to lack of data However, numerous engineers as well as technicians have brought forth the same opinions as ever: trees are the most significant factors which cause power outages. In addition to the significance of th ese factors, the NB for each device can be predicted with the following equations.

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62

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63 CHAPTER 4 RESTORATION ASSESSME NT MODULE Most studies had focused primarily on identifying the number of power outages caused by storms in a region. It is true that power outages have to do with the entire network damages directly, so that it is possible to represent the power outages in describing the entire power network damages t o a certain extent. Meanwhile in terms of electric end users, it is important to consider whether power is lost or not during a storm, but the more crucial consideration should be the extent of power outages i.e., blackout durations. Once any e quipment ha s failed it is unavoidable power will be lost for a while or long time. This outage is definitely damage as well as directly related to economic loss for both the electric suppliers and demanders. Therefore, it is also critical to estimate how many unit s (customers) are interrupted and how long (minutes) the power cut lasts. V ariable D efinition First of all, it is required to define a critical variable. CI means the number of customers interrupted due to a system failure, and MI means the minutes inte rrupted due to a system failure. Lastly, CMI means CI times MI. Probability Distribution Fitting Once a device is broken, CI and MI, which are caused by a device failure, need to be assessed. Unfortunately, these CI and MI cannot be estimated deterministi cally but stochastically due to uncertainty. Statistical models were presented to estimate a

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64 wrecked number of a certain electric device. Likewise, each electric device has its own probabilistic distributions of CI and MI. Distribution fitting enables one s to find tenable models of random processes because Non Gaussian Probability distributions could be a useful way to consider the uncertainty of the extent of CI and MI. Ordinarily, the n o rmal distribution, which is the oldest and most frequently used dist ribution, is used. However, this distribution has the following demerits: symmetric, unbounded and shape constant. The probability density function of the normal distribution is symmetric, i.e., this distribution could not be appropriate for the skewed d ata. Furthermore, this distribution could not guarantee a competent goodness of fit if the dataset is bounded or non negative. Finally, the normal distribution shape does not depend on its parameters. Even if the dataset is symmetric by nature, it is possible that it is best described by one of the heavy tailed models. The first procedu re is to choose the best Non Gaussian probability distributions which would be able to describe the dataset of CI and MI. With the most suitable probability distribution, it is possible to calculate required answers as well as make reasonable decisions. Ot herwise, if an inadequate probability distribution is used, it is hard to expect to have fair results. Therefore, after the distributions are fitted, it is required to check how well the candidate distributions fit to the dataset as the second step. This c an be done both numerically or visually by comparing the obtained empirical dataset and theoretically fitted distribution graphs. As a numerical method, goodness of fit tests could be available. Table 4 1 to Table 4 15 show goodness of fits for the appropriate Non Gaussian probability distributions, which would be able to represent the distribution of CI and MI. As a

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65 measurement of a goodness of fit, Maximum Likelihood (MLE), Normalized Area (NMA), Mean Square Error (MSE), and Mean Percent Difference (MPD) were used. In this study, ten Non Gaussian distributions we re used as candidates: Extreme Value distribution (EV), Exponential distribution (EXP), Gamma distribution (GAM) Generalized Extreme Value distribution (GEV), Generalized Pareto distribution (GP), Lognormal distribution (LOGN), Negative Binomial distribu tion (NB), Poisson distribution (POISS), Rayleigh distribution (RAYL), and Weibull distribution (WBL). In addition to a numerical method, Figure 4 1 to Figure 4 15 were drawn in a visual way. The left side shows a histogram and possible probabilistic distributions of each device for CI and MI, respectively. On the right side, the u pper one is the histogram for the entire range of the CI and MI dataset, the middle is a curve of the probability distribution for the most suitable distribution, and the lower one is a curve of the cumulative density for the most suitable distribution.

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66 Table 4 1 Goodness of f it of CI p robability distribution for TF f ailure Goodness of f it EXP GAM GEV GP LOGN WBL Maximum l ikelihood 4.0E+04 3.8E+04 3.8E+04 8.1E+04 3.8E+04 3.9E+04 (RANK) 6 3 2 7 1 4 Normalized a rea 8.4E 01 9.5E 01 9.8E 01 4.0E 01 9.7E 01 9.1E 01 (RANK) 9 5 2 10 4 8 Mean s quare e rror 2.2E 04 9.4E 05 7.4E 05 6.4E 04 8.5E 05 1.4E 04 (RANK) 6 3 1 7 2 5 Mean p ercent d ifference 1.75E+02 1.89E+02 1.77E+02 1.99E+02 1.74E+02 1.89E+02 (RANK) 2 5 3 6 1 4 Figure 4 1 CI probability d istribution for TF f ailure

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67 Table 4 2 Goodness of f it of M I p robability distribution for TF f ailure Goodness of f it EXP GAM GEV LOGN RAYL WBL Maximum l ikelihood 1.3E+05 1.3E+05 1.4E+05 1.4E+05 1.4E+05 1.3E+05 (RANK) 4 1 5 6 8 3 Normalized a rea 9.9E 01 9.9E 01 9.4E 01 9.3E 01 1.0E+00 9.9E 01 (RANK) 3 5 7 8 2 4 Mean s quare e rror 4.1E 09 3.5E 09 6.7E 09 4.7E 09 1.0E 08 4.0E 09 (RANK) 3 1 5 4 7 2 Mean p ercent d ifference 7.6E+01 7.8E+01 7.1E+01 9.2E+01 8.2E+01 7.6E+01 (RANK) 2 4 1 6 5 3 Figure 4 2 MI probability d istribution for TF f ailure

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68 Table 4 3 Goodness of f it of CI p robability distribution for FS f ailure Goodness of f it EXP GAM GEV GP LOGN WBL Maximum l ikelihood 5.7E+04 5.7E+04 5.7E+04 5.2E+05 5.6E+04 5.7E+04 (RANK) 6 4 3 9 1 2 Normalized a rea 9.8E 01 9.5E 01 9.7E 01 4.0E 01 9.8E 01 9.5E 01 (RANK) 4 6 5 10 3 7 Mean s quare e rror 4.4E 06 4.4E 06 8.3E 07 3.1E 05 1.2E 06 3.7E 06 (RANK) 5 4 1 8 2 3 Mean p ercent d ifference 1.34E+02 1.16E+02 5.52E+01 2.00E+02 4.79E+01 9.79E+01 (RANK) 5 4 2 8 1 3 Figure 4 3 CI probability d istribution for FS f ailure

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69 Table 4 4 Goodness of f it of M I p robability distribution for FS f ailure Goodness of f it EXP GAM GEV LOGN RAYL WBL Maximum l ikelihood 1.1E+05 1.1E+05 1.1E+05 1.1E+05 1.1E+05 1.1E+05 (RANK) 4 1 6 5 8 3 Normalized a rea 1.0E+00 1.0E+00 9.3E 01 9.4E 01 1.0E+00 1.0E+00 (RANK) 3 5 8 7 2 4 Mean s quare e rror 4.5E 09 3.4E 09 7.5E 09 3.7E 09 1.1E 08 3.8E 09 (RANK) 4 1 5 2 7 3 Mean p ercent d ifference 8.3E+01 8.6E+01 8.4E+01 1.0E+02 1.1E+02 8.6E+01 (RANK) 1 4 2 5 6 3 Figure 4 4 MI probability d istribution for FS f ailure

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70 Table 4 5 Goodness of f it of CI p robability distribution for RC f ailure Goodness of f it EXP GAM GEV LOGN RAYL WBL Maximum l ikelihood 1.0E+04 1.0E+04 1.0E+04 1.0E+04 1.1E+04 1.0E+04 (RANK) 3 2 5 6 7 1 Normalized a rea 1.0E+00 1.0E+00 9.7E 01 9.8E 01 1.0E+00 1.0E+00 (RANK) 3 4 8 7 1 5 Mean s quare e rror 5.3E 08 5.3E 08 9.7E 09 2.7E 08 5.0E 07 5.5E 08 (RANK) 3 4 1 2 7 5 Mean p ercent d ifference 6.94E+01 6.90E+01 5.71E+01 5.40E+01 1.32E+02 6.52E+01 (RANK) 5 4 2 1 7 3 Figure 4 5 CI probability d istribution for RC f ailure

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71 Table 4 6 Goodness of f it of M I p robability distribution for RC f ailure Goodness of f it EV EXP GAM GEV LOGN WBL Maximum l ikelihood 1.4E+04 1.3E+04 1.3E+04 1.4E+04 1.4E+04 1.3E+04 (RANK) 7 4 1 6 5 3 Normalized a rea 7.7E 01 9.9E 01 9.7E 01 9.1E 01 9.1E 01 9.7E 01 (RANK) 9 3 4 8 7 6 Mean s quare e rror 2.1E 08 1.0E 08 6.4E 09 1.5E 08 5.7E 09 7.2E 09 (RANK) 6 4 2 5 1 3 Mean p ercent d ifference 7.6E+01 4.7E+01 4.7E+01 5.5E+01 6.6E+01 4.9E+01 (RANK) 6 1 2 4 5 3 Figure 4 6 MI probability distribution for RC f ailure

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72 Table 4 7 Goodness of f it of CI p robability distribution for SR f ailure Goodness of f it EXP GAM GEV LOGN RAYL WBL Maximum l ikelihood 2.7E+05 2.7E+05 2.7E+05 2.7E+05 2.8E+05 2.7E+05 (RANK) 3 2 5 6 7 1 Normalized a rea 1.0E+00 1.0E+00 9.5E 01 9.6E 01 1.0E+00 1.0E+00 (RANK) 4 5 8 7 1 3 Mean s quare e rror 1.0E 09 9.9E 10 3.1E 09 2.4E 09 8.7E 09 1.1E 09 (RANK) 2 1 5 4 7 3 Mean p ercent d ifference 7.1E+01 7.2E+01 7.8E+01 9.3E+01 1.2E+02 6.9E+01 (RANK) 2 3 4 5 7 1 Figure 4 7 MI probability d istribution for SR f ailure

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73 Table 4 8 Goodness of f it of CI p robability distribution for FD f ailure Goodness of f it EV EXP GAM GEV RAYL WBL Maximum l ikelihood 5.7E+03 5.6E+03 5.6E+03 5.6E+03 5.7E+03 5.6E+03 (RANK) 6 5 4 1 7 2 Normalized a rea 8.7E 01 9.5E 01 9.6E 01 9.6E 01 1.0E+00 9.8E 01 (RANK) 8 7 5 6 2 3 Mean s quare e rror 2.2E 08 2.6E 08 2.5E 08 1.8E 08 2.4E 08 2.2E 08 (RANK) 3 6 5 1 4 2 Mean p ercent d ifference 5.78E+01 5.32E+01 5.06E+01 4.05E+01 4.66E+01 4.67E+01 (RANK) 6 5 4 1 2 3 Figure 4 8 CI p robabi lity distribution for FD f ailure

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74 Table 4 9 Goodness of f it of M I p robability distribution for FD f ailure Goodness of f it EV EXP GAM GEV LOGN WBL Maximum l ikelihood 7.5E+03 5.4E+03 5.2E+03 5.3E+03 5.2E+03 5.2E+03 (RANK) 7 6 2 5 4 1 Normalized a rea 1.0E+00 1.0E+00 1.0E+00 1.1E+00 1.1E+00 1.0E+00 (RANK) 5 1 4 8 9 7 Mean s quare e rror 1.5E 08 4.7E 09 8.1E 09 1.1E 08 1.0E 08 8.4E 09 (RANK) 7 1 2 5 4 3 Mean p ercent d ifference 1.9E+02 1.9E+02 1.8E+02 1.8E+02 1.8E+02 1.8E+02 (RANK) 5 6 2 4 1 3 Figure 4 9 MI p robability d istribution for FD f ailure

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75 Table 4 10 Goodness of f it of CI p robability distribution for SC f ailure Goodness of f it EXP GAM GEV LOGN RAYL WBL Maximum l ikelihood 1.5E+03 1.5E+03 1.5E+03 1.5E+03 1.6E+03 1.5E+03 (RANK) 6 2 1 3 7 5 Normalized a rea 9.8E 01 9.9E 01 9.8E 01 9.9E 01 1.0E+00 9.9E 01 (RANK) 8 3 7 4 2 6 Mean s quare e rror 8.7E 06 6.1E 06 4.5E 06 6.1E 06 2.0E 05 6.9E 06 (RANK) 5 3 1 2 6 4 Mean p ercent d ifference 1.03E+02 1.04E+02 1.01E+02 1.00E+02 1.33E+02 1.05E+02 (RANK) 3 4 2 1 7 5 Figure 4 10 CI p robability d istribution for SC f ailure

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76 Table 4 11 Goodness of f it of M I p robability distribution for SC f ailure Goodness of f it EXP GAM GEV LOGN RAYL WBL Maximum l ikelihood 2.9E+03 2.9E+03 2.9E+03 2.9E+03 3.0E+03 2.9E+03 (RANK) 4 1 5 6 8 3 Normalized a rea 9.7E 01 9.6E 01 9.1E 01 9.0E 01 1.0E+00 9.7E 01 (RANK) 2 5 7 8 1 4 Mean s quare e rror 7.2E 09 6.0E 09 1.0E 08 7.4E 09 1.5E 08 6.7E 09 (RANK) 3 1 5 4 7 2 Mean p ercent d ifference 5.6E+01 5.6E+01 6.0E+01 7.0E+01 7.6E+01 5.7E+01 (RANK) 2 1 4 5 7 3 Figure 4 11 MI probability distribution for SC f ailure

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77 Table 4 12 Goodness of f it of CI p robability distribution for TP f ailure Goodness of f it EV EXP GAM GP LOGN RAYL Maximum l ikelihood 2.7E+02 2.3E+02 2.3E+02 1.3E+03 2.3E+02 2.8E+02 (RANK) 6 4 2 9 5 7 Normalized a rea 7.7E 01 9.9E 01 9.3E 01 4.4E 01 8.8E 01 1.1E+00 (RANK) 8 1 3 9 6 7 Mean s quare e rror 6.4E 04 4.9E 04 4.3E 04 2.3E 04 3.6E 04 6.6E 04 (RANK) 7 6 5 2 3 8 Mean p ercent d ifference 1.50E+02 1.53E+02 1.55E+02 1.93E+02 1.62E+02 1.47E+02 (RANK) 2 3 4 8 5 1 Figure 4 12 CI probability d istribution for TP f ailure

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78 Table 4 13 Goodness of f it of M I p robability distribution for TP f ailure Goodness of f it EV EXP GAM GEV LOGN WBL Maximum l ikelihood 5.1E+02 4.1E+02 3.9E+02 3.6E+02 3.7E+02 3.9E+02 (RANK) 7 6 4 1 2 3 Normalized a rea 5.8E 01 1.0E+00 8.6E 01 1.0E+00 9.5E 01 8.5E 01 (RANK) 8 2 5 1 3 7 Mean s quare e rror 1.1E 06 8.1E 07 7.7E 07 3.5E 07 4.9E 07 6.8E 07 (RANK) 6 5 4 1 2 3 Mean p ercent d ifference 1.8E+02 1.8E+02 1.8E+02 1.8E+02 1.8E+02 1.8E+02 (RANK) 1 6 2 4 5 3 Figure 4 13 MI probability d istribution for TP f ailure

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79 Table 4 14 Goodness of f it of CI p robability distribution for SG f ailure Goodness of f it EV EXP GAM GEV LOGN WBL Maximum l ikelihood 1.2E+03 1.1E+03 1.1E+03 1.1E+03 1.1E+03 1.1E+03 (RANK) 7 5 1 6 4 3 Normalized a rea 7.2E 01 9.9E 01 9.5E 01 8.7E 01 8.9E 01 9.5E 01 (RANK) 9 2 5 8 7 4 Mean s quare e rror 1.3E 06 6.1E 07 2.7E 07 1.8E 07 1.1E 07 2.3E 07 (RANK) 6 5 4 2 1 3 Mean p ercent d ifference 1.07E+02 1.53E+02 1.55E+02 1.78E+02 1.62E+02 1.89E+02 (RANK) 1 3 4 6 5 7 Figure 4 14 CI probability d istribution for SG f ailure

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80 Table 4 15 Goodness of f it of M I p robability distribution for SG f ailure Goodness of f it EXP GAM GEV LOGN RAYL WBL Maximum l ikelihood 1.4E+03 1.4E+03 1.4E+03 1.4E+03 1.6E+03 1.4E+03 (RANK) 6 1 5 4 8 3 Normalized a rea 9.9E 01 9.5E 01 8.7E 01 9.1E 01 1.0E+00 9.5E 01 (RANK) 2 3 7 6 1 5 Mean s quare e rror 6.1E 08 3.5E 08 1.4E 08 1.7E 08 1.0E 07 3.2E 08 (RANK) 5 4 1 2 7 3 Mean p ercent d ifference 9.6E+01 9.0E+01 1.2E+02 1.1E+02 1.2E+02 9.4E+01 (RANK) 3 1 6 5 7 2 Figure 4 15 MI probability distribution for SG f ailure

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81 Based on these visual and numeral results, the probability distribution for the CI and MI of each device are determined as shown in Table 4 16 Table 4 16 Probabilistic d istribution of CI and MI Device Interruption CI MI TF GEV(0.3407, 2.1631, 3.2874) GAM(0.8686, 3.3013) FS LOGN(2.9921, 1.3191) GAM(0.7999, 3.3286) RC GEV(0.6086, 132.4796, 133.9526) GAM(0.7183, 3.5886) SR 1 WBL(660.0172, 0.6041) FD RAYL(55.0803) WBL(2.5655, 0.9103) SC GEV(0.3786, 20.7168, 25.2679) GAM(0.8518, 3.6193) TP EXP(23.8909) GEV(1.1831, 70.4997, 73.0177) SG WBL(282.4281, 0.6413) GAM(0.5165, 4.1144) Correlation If there is a correlation between the CI and MI for each device, both values cannot be generated under statistically independent condition, i.e., the CI and MI need to be simulated with correlation coefficients between two factors. Figure 4 16 shows scatter plots for the MI (X axis) and CI (Y axis) for (A) TF, (B) FS, (C) RC, (D) SR, (E) FD, (F) SC, (G) TP, and (H) SG respectively.

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82 (A) (B) N/A (C) (D) (E) (F) (G) (H) Figure 4 16 Scatter p lot for MI (X axis) and CI (Y axis)

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83 Random Number Generation The general process of generating random numbers from two correlated non Gaussian distributions is as described through the following steps. S TEP 1 (Ang & Tang) Generation of Uniformly Distributed Random Numbers The number, which is generated randomly among the ones that are distributed from 0 to 1 uniformly, is the essence of the Monte Carlo Simulation. Most computational programs have a built in subroutine to generate these random numbers using their own tables, i.e., a set of numbers generated randomly. S TEP 2 (Ang & Tang) Generation of Standard Normal Random Numbers In most simulations, Gaussian (standard normal) probability distribution is absolutely important Therefore, th is bas is is the competence to simulate standard normally distributed random variable For generat ing a set of Gaussian random numbers z 1 z 2 z n i t is required to generate a corresponding set of uniformly distributed random variables, u 1 u 2 n as explained in S TEP 1 Then, for each randomly generated u i a corresponding z i should be generated as follows: w here i s the inverse of the Gaussian cumulative distribution function. S TEP 3 (Ang & Tang)

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84 P rocedure to Generat e Random Numbers for Any Type of Distribution First, let s a ssume a random variable X with a cumulative distribution function F X (x). Once a sample value u i or z i for a uniform or Gaussian distribution is generated a corresponding sample value x i can be generated as follows : w here is the inverse of Fx. S TEP 4 (Ang & Tang) Simulation of Correlated Normal Random Variables In S TEP 3 implicitly it was assumed that generated random variables were statistically independent, i.e., uncorrelated. However, in practice, it is inevitable to consider a correlation between/among random variables. Therefore, as realistic as possible, these correlation effects should be reflected in a simulation. First o f all, it should be understood how to simulate correlated normal random variables prior to generating correlated non normal random numbers. A transformation technique for simulating correlated normal random variables is the key. Assume that X 1 X 2 N b e correlated normal random variables. The matrix for mean values and covariance are given as follows :

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85 To generate correlated random numbers for the random variable, X 1 X 2 N it is necessary to first generate a set of uncorrelated random numbers Y 1 Y 2 N using the techniques in STEP 2 Then X 1 X 2 N are calculated using the variable transform ation as follows : w here is a transformation matrix. To apply this approach, the matrix needs to be determined as well as the mean and variance values for the uncorrelated Y i variables need to be also calculated. To do this, some concepts from linear a lgebra is required, especially E igen values and E igen vectors as follows. Let to be a symmetric N by N matrix A diagonal matrix and a square matrix can be found as follows :

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86 The orthogonal matrix which means that its inverse is equal to its transpose is c omposed of the ortho gonal normal E igen vectors corresponding to the E igen values of of the original, correlated variable The diagonal matrix the covariance matrix of the uncorrelated variables has the E igen values of For reference, the first column of contains the ortho gonal normal E igenvec tor corresponding to the first E igen values, the second column of contains the ortho gonal normal E igen vect or corresponding to the second E igen values, and so on. The matrix is an orthogonal matrix, meaning that its inverse is equal to its transpose. In addition, t he diagonal elements of contain the variance of the uncorrelated variables needed to do the simulation. The mean values of the can be obtained as follows : Once simulated values of are obtained, s imulated values of can be obtained as follows : Although it is valid for normal random variables only, it can be used for other types of rando m variables as an approximation through STEP5 and STEP6 (Ang & Tang)

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87 STEP 5 Polynomial Regression As mentioned above, i f random variables are statistically independent each other, random numbers can be generated through STEP 1 to STEP 3 simply. On the other hand, if random variables are correlated each other, the procedure to generate random number needs to follow STEP 4 fir st. However, STEP 4 is just to generate random numbers for the correlated normal distributions. The way to calculate non Gaussian random numbers can be possible as an approximation, i.e. it is required to find the relationship between correlation coefficients of a pair of Gaussian dist ribution and a pair of non Gaussian distribution as shown in Figure 4 17 with polynomial regression models. In this model, the dependent variable is the correlation coefficient of Non Gaussian and the independent variable is the correlation coefficient of Gaussian di stribution as shown in equation: when P is equation to denote polynomial regression model, means correlation coefficient for a set of non Gaussian distributions, and means correlation coefficient for a set of Gaussian distribu tions (Ang & Tang)

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88 STEP 6 Procedure to G enerat e C orrelated Non Gaussian R andom N umbers Once regression models are calculated, inversely correlation coefficient for a set of Gaussian distributions which is corresponded with the one for a set of Non Gaussian distribution s for (A) TF, (B) FS, (C) RC, (D) SR, (E) FD, (F) SC, (G) TP, and (H) SG as shown in Figure 4 17 by the following equation : With the calculated correlation coefficient of a set of correlated Gaussian distributions, it is possible to generate the random numbers through STEP 4. Then a set of correlated non Gaussian random numbers are able to be c alculated with this set of generated Gaussian random numbers through STEP 3. For reference, in this study, the built subroutine in MATLAB was used for generating uniformly distributed random numbers between 0 and 1. In addition to uniformly distributed ra ndom numbers, MATLAB also has a standard normal random number generation built in as well as built in functions to generate non Gaussian random numbers using the above theory and procedures (Ang & Tang)

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89 (A) (B) N/A (C) (D) (E) (F) (G) (H) Figure 4 17 Gaussian ( X axis) and n on Gaussian (Y axis) d istributions c orrelation

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90 Table 4 17 Correlation e ffects Trial No TF FS FD SC SG Trial No. RC TP Non Gaussian N/A 0.0139 0.0270 0.1111 0.0352 0.0756 N/A 0.0072 0.0045 Gaussian 50,000 0.0210 0.0521 0.1525 0.0640 0.1287 200,000 0.0398 0.0340 100,000 0.0190 0.0519 0.1521 0.0644 0.1291 400,000 0.0258 0.0275 150,000 0.0189 0.0497 0.1521 0.0618 0.1293 600,000 0.0248 0.0293 200,000 0.0188 0.0510 0.1522 0.0607 0.1290 800,000 0.0289 0.0285 250,000 0.0205 0.0506 0.1524 0.0607 0.1286 1,000,000 0.0182 0.0282 300,000 0.0205 0.0518 0.1522 0.0608 0.1287 1,200,000 0.0216 0.0293 350,000 0.0200 0.0522 0.1525 0.0603 0.1287 1,400,000 0.0204 0.0294 400,000 0.0192 0.0528 0.1526 0.0592 0.1285 1,600,000 0.0221 0.0291 450,000 0.0196 0.0531 0.1526 0.0569 0.1285 1,800,000 0.0232 0.0289 500,000 0.0199 0.0537 0.1526 0.0577 0.1286 2,000,000 0.0215 0.0287 550,000 0.0183 0.0537 0.1526 0.0585 0.1284 2,200,000 0.0225 0.0291 600,000 0.0194 0.0534 0.1525 0.0582 0.1283 2,400,000 0.0265 0.0294 650,000 0.0194 0.0536 0.1524 0.0583 0.1283 2,600,000 0.0254 0.0295 700,000 0.0193 0.0540 0.1522 0.0584 0.1283 2,800,000 0.0244 0.0301 750,000 0.0194 0.0543 0.1522 0.0587 0.1281 3,000,000 0.0234 0.0297 800,000 0.0196 0.0538 0.1522 0.0587 0.1281 3,200,000 0.0241 0.0301 850,000 0.0195 0.0536 0.1522 0.0593 0.1281 3,400,000 0.0244 0.0302 900,000 0.0194 0.0536 0.1521 0.0592 0.1281 3,600,000 0.0247 0.0297 950,000 0.0195 0.0537 0.1521 0.0593 0.1281 3,800,000 0.0245 0.0293 1,000,000 0.0195 0.0536 0.1521 0.0593 0.1281 4,000,000 0.0246 0.0294

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91 Sensitivity Analysis In practice, most sets of random variables may have correlations. Thus, it is required to reflect these correlations in generating random numbers. However, it is hard to say that these correlations affect all simulating outputs predominantly, i.e., althoug h there is correlation between random variables, the impact of this correlation could be ignorable. One of the ways to determine whether to consider correlation effects in a model is sensitivity analysis for (A) TF, (B) FS, (C) RC, (D) SR, (E) FD, (F) SC, (G) TP, and (H) SG as shown in Figure 4 18 In each graph, the x axis means the number of trials and the y axis means the difference rate defined as According to these analyses, in FS and FD, the difference between CMI considering correlation and CMI simulated without correlation is a little higher than the others. However, in other devices, this effect s o minimal that it is possi ble to ignore the correlation.

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92 (A) (B) N/A (C) (D) (E) (F) (G) (H) Figure 4 18 Sensitivity a nalysis

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93 CHAPTER 5 ASSESSMENT MODELS AN D VALIDATION Figure 5 1 shows the general procedure to do this. Above all, it is required to obtain the data: R F W A W M and T D Then, the NB for each device per a specifically sectionalized unit can be calculated with the obtained regression model. As mentioned previously, the calculated NB is for 1,000 customer units, and the actual NB should be tuned along with the number of customers of a specific region. Next, w ith the CMI estimation models and the calculated NB, the CMI should be evaluated. In this study, the Monte Carlo simulation was just applied to the CMI estimation phase, but if using the Hurricane Simulation Engine, the Monte Carlo Simulation needs to be a pplied from the (weather) data input phase. Figure 5 1 Flowchart to p redict NBs and CMIs To investigate the validation of the proposed model, the data for storm damages to power networks as well as the weather from Aug 20 to 30, 2008, was used. The hurricane which ruined the power lines at this time was Katrina. As a validation assessment tool, error rates of the proposed model can be evaluated by

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94 where means o bserved values, means Predicted values. Figure 5 2 and Figure 5 4 shows the error rates of NB and Figure 5 3 and Figure 5 5 show the observed and predicted values of NB for each observation center. NBs and CMIs which have more than 15 % error rate were predicted in seven and four observational centers among 19, respectively, but the error for the entire power ne twork was so insignificant that it could be concluded that the proposed model is sound as shown in Table 5 1 The reason for the error rate for the wh ole network may be that the uncertainly tends to be reduced more in larger regions. However, at this point it is hard to identify this reason in detail. In addition, the error rates in both NB and CMI tends to be in a similar direction as well as amount. Considering the NB was predicted earlier Table 5 1 Error rates of the proposed model for the entire network Observed Predicted Error NB 3,860 3,707 3.96% CI 82,624,769 80,340,456 2.76% MI 237,459 228,442 3.80% CMI 1,802,574 1,726,759 4.21%

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95 Figure 5 2 Errors of NB p rediction Figure 5 3 Observed and predicted v alues of NB CWY LKW HIL DEL BNV APK JAM SEV LNG WGN INV OCA MON WAL EUS STP CLW CLR ZEP ERROR 11% 10% 7% 12% 14% 12% 1% 27% 19% 1% 12% 29% 1% 10% 16% 24% 5% 15% 13% 30% 20% 10% 0% 10% 20% 30% 40% CWY LKW HIL DEL BNV APK JAM SEV LNG WGN INV OCA MON WAL EUS STP CLW CLR ZEP OBSERVED 97 118 54 558 39 438 251 63 356 115 317 469 536 67 180 81 73 42 6 PREDICTED 108 106 50 490 33 384 247 46 423 116 353 331 540 61 209 101 69 36 5 0 100 200 300 400 500 600

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96 Figure 5 4 Errors of CMI p rediction Figure 5 5 Observed and p redicted v alues of CMI CWY LKW HIL DEL BNV APK JAM SEV LNG WGN INV OCA MON WAL EUS STP CLW CLR ZEP ERROR 19% 11% 7% 11% 15% 12% 2% 28% 28% 1% 13% 30% 2% 10% 20% 22% 5% 15% 16% 40% 30% 20% 10% 0% 10% 20% 30% 40% CWY LKW HIL DEL BNV APK JAM SEV LNG WGN INV OCA MON WAL EUS STP CLW CLR ZEP OBSERVED 487347 331840 122136 2E+07 260732 2E+07 6E+06 139899 1E+07 878049 5E+06 1E+07 8E+06 210561 8E+06 257409 310746 306474 10933 PREDICTED 581907 296698 113892 1E+07 222665 1E+07 6E+06 101357 1E+07 884905 5E+06 8E+06 7E+06 189884 9E+06 313562 294214 259216 9151.6 0.E+00 2.E+06 4.E+06 6.E+06 8.E+06 1.E+07 1.E+07 1.E+07 2.E+07 2.E+07

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97 CHAPTER 6 CONCLUSION Trees have been recognized as the most destructive factor which cause s power outages while previous studies have failed to clearly identify this fait because it was hard to collect appropriate data concerning tree properties directly related to power outages. Furthermore, it was also a demanding assignment due to the extent of sectionalized units, i.e. to solve the resolution difficulties regarding tree data special management. In this study, a tree, more specifically, densities for trees with heights taller than 50 ft, could be appropriately analyzed thro ugh the Spearman correlation analysis and multiple regression analysis. Besides, it was determined that the other factors concerning weather characteristics such as wind momentum and rainfall are also statistically significant. In the CMI estimations, it w as found that the CI and MI distributions for all devices were non Gaussian. Moreover, in a few devices, the correlations between the CI and MI were somewhat sensitive to the final results. Lastly, the proposed model could be accepted as a reasonable predi ctor Although there are a couple of quarters with errors between the observed and predicted values of more than 20%, the NB and CMI errors for the entire service region were only around 3% and 4%, respectively because the uncertainly could be eliminated m ore in larger regions. This study used linear regression models to describe the NB. Although biased aspects could be deflated using normal transformations, the R 2 for all devices was lower than expected excluding TP. To increase R 2 it may be better to use non linear regression models or additional factors.

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98 APPENDIX A O PERATION CENTERS OF PROGRESS ENERGY VP OP CENTER NAME ABBREVIATION CUSTONERS SERVED STREET CITY NORTH CENTRAL APOPKA APK 92,836 275 W PONKAN RD APOPKA DELAND DEL 75,215 400 N SPRING GARDEN AVE DELAND JAMESTOWN JAM 123,293 2801 W SR 426 OVIEDO LONGWOOD LNG 81,380 150 PROGRESS ENERGY WAY LONGWOOD NORTH COASTAL INVERNESS INV 69,759 515 INDE PENDENCE HWY INVERNESS MONTICELLO MON 46,455 1295 ROCKY BRANCH RD MONTICELLO OCALA OCA 76,268 4361 SE MARICAMP RD OCALA SOUTH CENTRAL BUENA VISTA BNV 95,337 3250 BONNET CREEK RD LAKE BUENA VISTA CLERMONT CLR 28,071 401 CITRUS TOWER BLVD CLERMONT SE ORLANDO CWY 79,763 8407 BOGGY CREEK RD ORLANDO HIGHLANDS HIL 55,487 5020 KENILWORTH BLVD SEBRING LAKE WALES LKW 89,415 199 E MOUNTAIN LAKE CUTOFF RD BLDG A LAKE WALES WINTER GARDEN WGN 69,467 452 E CROWN POINT RD WINTER GARDEN SOUTH COASTAL CLEARWATER CLW 149,776 2166 PALMETTO ST BLDG F CLEARWATER SEVEN SPRINGS SEV 165,298 4121 SAINT LAWRENCE DR NEW PORT RICHEY SAINT PETERSBURGSTP 159,807 2501 25TH ST N SAINT PETERSBURG WALSINGHAM WAL 145,786 12600 WALSINGHAM RD LARGO ZEPHYRHILLS ZEP 24,098 36453 EILAND BLVD ZEPHYRHILLS

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99 APPENDIX B W EATHER STATIONS IN F LORIDA TOWER NAME TOWER NUMBER LATITUDE LONGITUDE ELEVATION START DATE FACILITY ALACHUA 260 29.80266 82.41081 160' 1999 APR 13 DEPT. OF AGRONOMY FORAGE RESEARCH UNIT APOPKA 320 28.63771 81.54675 107' 1997 DEC 19 MID FLORIDA REC ARCADIA 490 27.22621 81.83838 64' 2005 JUN 8 DESOTO COUNTY EXTENSION OFFICE AVALON 304 28.47485 81.6530 0 196' 1997 DEC 15 MID FLORIDA CITRUS FOUNDATION FARM BALM 350 27.75998 82.22410 129' 2003 DEC 19 GULF COAST REC HILLSBOROUGH BELLE GLADE 410 26.65678 80.63001 11' 2003 NOV 18 EVERGLADES REC PALM BEACH BRONSON 230 29.40038 82.58611 116' 2002 SEP 24 UF DEPT. OF ASTRONOMY ROSEMARY HILL OBSERVATORY BROOKSVILLE 310 28.63471 82.28511 107' 2000 MAR 27 USDA BROOKSVILLE SUBTROPICAL AGRICULTURAL STATION HERNANDO CARRABELLE 150 29.84240 84.69511 19' 2003 MAR 24 DIVISION OF FORESTRY WORK CAMP FRAN KLIN CITRA 250 29.41058 82.17021 60' 2000 OCT 10 PLANT SCIENCE REC CLEWISTON 405 26.73906 81.05278 19' 2007 NOV 21 AIRGLADES AIRPORT DOVER 360 28.01716 82.23378 69' 1998 MAY 5 GULF COAST REC HILLSBOROUGH FROST PROOF 390 27.76551 81.53735 164' 2005 NOV 16 BEN HIL L GRIFFIN ELEMENTARY SCHOOL FT. LAUDERDALE 420 26.08530 80.24050 5' 2001 JA N 24 FORT LAUDERDALE REC FT. PIERCE 430 27.42598 80.40196 19' 1998 APR 30 INDIAN RIVER REC HASTINGS 270 29.69332 81.44485 25' 1999 AUG 4 AGRICULTURA L EXPERIMENT STATION HOMESTEAD 440 25.50941 80.49924 8' 1997 DEC 29 TROPICAL REC IMMOKALEE 450 26.46225 81.44033 35' 1997 DEC 30 SOUTHWEST FLORIDA REC INDIAN RIVER 371 27.61916 80.57259 23' JAY 110 30.77516 87.14015 210' 2002 SEP 18 WFREC

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100 KENANSVILLE 340 27.96221 81.05123 69' 2003 JAN 28 THREE LAKES WILDLIFE MANAGEMENT AREA LAKE ALFRED 330 28.10185 81.71128 154' 1997 DEC 31 CITRUS REC LIVE OAK 170 30.30500 82.89876 165' 2002 SEP 24 NFREC MACCLENNY 180 30.28148 82.13798 126' 2002 SEP 17 BAK ER COUNTY EXTENSION OFFICE MARIANNA 130 30.85000 85.16516 115' 2002 SEP 24 NFREC MONTICELLO 160 30.53190 83.92031 163' 2003 APR 30 NFREC NORTH PORT 480 27.14340 82.33741 16' 2007 JUL 5 T. MABRY CARLTO N, JR. MEMORIAL RESERVE OCKL AWAHA 280 29.02033 81.96896 79' 1998 DEC 19 CARNEY ISLAND COUNTY PARK OKAHUMPKA 303 28.68165 81.88565 90' 1997 DEC 12 US DA WHITMORE FOUNDATION ONA 380 27.39750 81.93973 75' 199 8 MAR 10 RANGE CATTLE REC PALMDALE 460 26.92480 81.31455 38' 200 3 APR 2 DIV ISION OF FORESTRY STATION PIERSON 290 29.21717 81.46065 54' 199 8 APR 3 RICHARDSON FARMS PUTNAM HALL 240 29.69700 81.98600 148' 2001 JAN 24 ORDWAY S WISHER BIOLOGICAL STATION QUINCY 140 30.54581 84.59898 240' 2002 SEP 24 NFREC SEBRING 470 27.42108 81.40095 118' 2003 APR 2 DIVISI ON OF FORESTRY STATION UMATILLA 302 28.91913 81.63075 120' 1997 DEC 12 GOLDEN GEM FARM

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101 APPENDIX C L ETTER OF SOLICITATIO N

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102 LIST OF REFERENCES Ang, A. H. S., & Tang, W. H. Probability Concepts in Engineering Planning and Design. John Wiley & Sons. Blake, E. S., & Gibney, E. J. (2011). The Deadliest, costliest, and Most Intense United States Tropical Cyclones from 1851 to 2010 (and Other Frequent ly Required Hurricane Facts). National Hurricane Center Miami Brown, B. E., & Ochia, J. R. (1998). Distribution system reliability: Default data and model validation. IEEE Transactions on Power Systems 704 709. Davidson, R. A., Liu, H., Sarpong, I. K., Sparks, P., & Rosowsky, D. V. (2003). Ele c tric Power Distribution System Performance in Carolina Hurricanes. Natural Hazards Review 36 45. Filliben, J. J., Gurley, K., Pinelli, J. P., & Simiu, E. (2002). Fradility curves, damage matrics, and wind i nduced loss estimation National Institute of Standard and Technology. Gonen, T. (1986). Electric Power Distribution System Engineering. McGraw Hill, New York. Han, S. R., Guikema, S. D., Quiring, S. M., Lee, K. H., Rosowsky, D., & Davidson, R. A. (2009). Estimating the spatial distribution of power outages during hurricanes in the Gulf coast region. Reliability Engineering and System Safety 199 210. Herbert, P. J., & Taylor, J. G. (1975). Hurricane Experience Levels of Coastal County Populations Texas to Maine. NWS Community Prepareness Staff and SR. Landsea, C. W., Anderson, N. C., Clark, G., Dunion, J., Fernadez Partagas, J., Hungerford, P., et al. (2004). The Atlantic Hurricane Database Reanalysis Project. Documentation for 1851 1910 alterations and additions to HURDAT. Hurricanes and Typhoons: Past, Present and Future. Columbia University Press. Liu, H., Davidson, R. A., David, R. V., & Stedinger, J. R. (2005). Negative Binomial Regression of Electric Power Outages in Hurr icanes. Journal of Infrastructure Systems 258 267. Melchers, R. E. Structural Reliability Analysis and Prediction. John Wiley & Sons, Australia. Nowak, A. S., & Collins, K. R. Reliability of Structures. McGraw Hill. Ott, R. L. (2008). An Introduction t o Statistical Methods and Data Analysis. Brooks/Cole, Belmont.

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103 Quanta Technology. (2008). Undergrounding Assessment Phase 3 Report: Ex Ante Cost and Benefit Modeling. Florida Electric Utilities. Rawlings, J. O., Pantula, S. G., & Dickey, D. A. Applied Re gression Analysis: A Research Tool. Springer, New York. Reed, D. A. (2008). Electric utility distribution analysis for extreme winds. Journal of Wind Engineering and Industrial Aerodynamics 96 123 140. Scheaffer, R. L., & McClave, J. Probability and Statistics for Engineers. Duxbury Press. Winkler, J., Duenas Osorio, L., Stein, R., & Subramanian, D. (2010). Performance assessment of topologically diverse power systems subjected to hurricane events. Reliability Engineering and System Safety 98 323 336. Zhou, Y., Pahwa, A., & Das, S. (2006). Prediction of weather ralated failure of overhead distribution feeders. Probability in the Engineering and Informational Sciences 117 125.

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104 BIOGRAPHICAL SKETCH Young Jun Park received his Bachelor of Engineering degree from the Korea Military Academy in March 2000, Master of Science degree from Seoul National University in February 2004, and Doctor of Philosophy from Texas A&M University in December 2009 He entered the Civil Engineering program, sp ecializing in the College of Engineering, University of Florida in August 2009 and received his Doctor of Philosophy degree in May 201 2 His research interests include natural hazards and their mitigation using statistics and stochastic models. Mr. Park c an be reached at the Korea Military Academy, Department of Construction Engineering and Environmental Sciences, P.O. Box 77, Goneung dong, Nowon gu, Seoul, Korea, 139 799. His email is yjpark@kma.ac.kr.