Ice Storm Modelling in Transmission System Reliability ...12178/FULLTEXT01.pdf · av ett givet...

Ice Storm Modelling in Transmission SystemReliability Calculations

ELIN BROSTROM

Licentiate ThesisRoyal Institute of Technology

School of Electrical EngineeringElectric Power Systems

Stockholm, 2007

TRITA-EE-2007:022ISSN-1650-674xISBN 978-91-7178-690-6

School of Electrical EngineeringElectric Power SystemsRoyal Institute of TechnologyS-100 44 StockholmSWEDEN

Akademisk avhandling som med tillstand av Kungl Tekniska hogskolan framlag-ges till offentlig granskning for avlaggande av teknologie licentiatexamentorsdagen den 14 juni 2007 kl 10.00 i sal H1, Teknikringen 33, Kungl TekniskaHogskolan, Stockholm.

c© Elin Brostrom, June 2007

Tryck: Universitetsservice US-AB

Abstract

In this thesis a new technique of modelling non-dimensioning severe weatherfor power system reliability calculations is developed. The model issuitable for both transmission and distribution networks and is based ongeographically moving winds and ice storms. The modelled weather hasseverity levels that vary with time and change continuously as the weatherpasses a region. Different weather situations are represented with scenarios.For each scenario the weather parameters, such as size, strength, speedand direction can vary. A stochastic method for choosing parameters isalso described. This method is based on probabilities for different weathersituations for Swedish conditions.

A stochastic vulnerability model for the components is required for eachscenario to connect the risk of failure to the weather situation. The modeldeveloped here connects the direct wind impact with the impact from theice storm which is given by an ice accretion model. It is assumed that theprobability for an individual segment to break down due to the impact of agiven weather depends on load functions for wind and ice together with thevulnerability model for components. It is possible to estimate the outagerisk as well as the time difference between mean times to failure in differentlines. Monte Carlo methods, where many scenarios are simulated, are usedin the case studies. Studies of the system vulnerability is a future work ofthis project but in one small case study the probability for outage in a loadpoint is estimated.

To be able to estimate repair times after a severe weather the reliabilitycalculations are extended with a restoration model which gives distributionsof down times for the broken components. The situations after the icestorm that are studied are so severe that gathering of all or almost allpossible restoration resources is required to restore the system. Restorationtimes for different components are not assumed to be independent; on the

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other hand they are assumed to be strongly correlated. The restorationprocess is dependent on staff situation, distance between location of spareparts and the breakdown, forecasts, availability of roads and distance toother breakdowns; this is included in the model. A method for simulationof non-Gaussian correlated random numbers is developed to include thecorrelations during the restoration process. The case studies show theimpact of the different weather situations on the components and thefollowing restoration times for the broken components.

Sammanfattning

Denna avhandling presenterar ett nytt satt att modellera ovader fortillforlitlighetsberakningar pa kraftnat. Modellen ar applicerbar badepa transmissionsnat och pa distributionsnat. Ovadret beskrivs med enmodell dar intensitetsnivaerna for vind och nederbord varierar kontinuerligtmed tiden, allt eftersom ovadret passerar. Olika vadersituationerrepresenteras av scenarier. For varje scenario kan vaderparametrarna,sasom ovadrets omfattning, intensitet, framfartshastighet och riktningvariera. Avhandlingen beskriver ocksa en stokastisk metod for att valjavaderparametrar. Metoden bygger pa uppskattningar av sannolikheter forolika vadersituationer under svenska forhallanden.

For att bestamma sannolikheten for haveri, givet en vadersituation,har en stokastisk kanslighetsmodell for komponenterna i kraftnatet tagitsfram. Kanslighetsmodellen tar hansyn bade till den direkta paverkan avvinden och till paverkan av isbildning pa komponenterna. Sannolikhetenfor att ett enskilt segment av kraftnatet ska haverera under paverkanav ett givet vader antas kunna beskrivas av en vind- och islastfunktionsamt kanslighetsmodellen for komponenterna. Det ar mojligt att beraknaavbrottsrisken, liksom tidsskillnaden mellan forvantad tid till haveri forolika ledningar. Monte Carlo-metoder anvands genom att manga scenariersimuleras i numeriska exempel. I avhandlingen studeras tillforlitligheten forett enklare system, i projektets forlangning ska dock aven tillforlitligheten istorre slingade kraftnat betraktas.

Reparationstider for komponenter som havererat, pga. vind ochis, slumpas fram fran en sannolikhetsfordelning. Endast konsekvenserav allvarliga ovader studeras, da en stor del av de tillgangligareparatorerna samt reparationsresurserna kravs for att aterstalla systemet.Reparationstiden for olika komponenter modelleras att vara beroende,dvs. deras korrelation antas ha en stor betydelse. Faktorer som antas

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paverka reparationstiden ar antalet tillgangliga reparatorer, avstandetmellan haveriet och reservlager for komponenter, eventuell vaderprognos,framkomlighet pa vagar och avstand till ovriga haverier. For att modellerakorrelationen mellan reparationstider har en metod for att simuleraicke-normalfordelade korrelerade slumptal utvecklats.

Numeriska exempel visar hur olika vadersituationer paverkarkomponenterna i systemet, samt vilken tid det tar att aterstalla desom har havererat.

Acknowledgment

First of all I would like to thank my supervisor Professor Lennart Soder forintroducing me to this interesting topic and for his valuable contributionsand encouragement.

I would like to thank the Swedish Emergency Management Agency andthe Swedish system operator Svenska Kraftnat for their financial support.A special thank goes to Lillemor Carlshem at Svenska Kraftnat and toEva Sundin, Jorgen Martinsson and Roger Jansson at Vattenfall Powerconsultants for their support and for sharing their knowledge. A great thankgoes to Jesper Ahlberg for his valuable contributions to the weather model.

Many thanks go to my colleges and friends and especially to myroommate Elin Lindgren for being a dear friend and my own computersupport and Karin Alvehag, also a dear friend, for her support during thework with this thesis.

Finally I would like to thank Joakim and Eskil for all your love.

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Contents

Contents ix

List of Figures xiii

List of Tables xvii

List of Symbols xix

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Main contributions . . . . . . . . . . . . . . . . . . . . . . . . 31.4 The outline of the thesis . . . . . . . . . . . . . . . . . . . . . 51.5 List of publications . . . . . . . . . . . . . . . . . . . . . . . . 6

2 About Ice Storm Weathers 72.1 Wind and freezing rain . . . . . . . . . . . . . . . . . . . . . . 72.2 Ice storms in the world . . . . . . . . . . . . . . . . . . . . . . 9

3 Modelling of Ice Storm Weather 133.1 Weather models . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 The first weather model . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Wind load . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.2 Ice load . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.3 Size of weather and moving speed . . . . . . . . . . . 19

3.3 The improved weather model . . . . . . . . . . . . . . . . . . 203.3.1 Wind load . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.2 Precipitation . . . . . . . . . . . . . . . . . . . . . . . 22

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x CONTENTS

3.4 Modelling ice accretion . . . . . . . . . . . . . . . . . . . . . . 243.4.1 The Simple model . . . . . . . . . . . . . . . . . . . . 26

4 Modelling Component Vulnerability 294.1 Reliability of transmission lines . . . . . . . . . . . . . . . . . 29

4.1.1 Galloping . . . . . . . . . . . . . . . . . . . . . . . . . 324.2 Segment vulnerability models . . . . . . . . . . . . . . . . . . 32

4.2.1 The first segment vulnerability model . . . . . . . . . 354.2.2 The second segment vulnerability model . . . . . . . . 36

5 Modelling Restoration 395.1 Restoration models . . . . . . . . . . . . . . . . . . . . . . . . 395.2 The restoration time model . . . . . . . . . . . . . . . . . . . 41

5.2.1 Localization of breakdown . . . . . . . . . . . . . . . . 445.2.2 Identification of fault . . . . . . . . . . . . . . . . . . . 445.2.3 Localization and transport of spare parts and staff . . 445.2.4 Repair time . . . . . . . . . . . . . . . . . . . . . . . . 455.2.5 The covariance matrix . . . . . . . . . . . . . . . . . . 45

6 Description of Simulation Methods 476.1 Weather properties and loads . . . . . . . . . . . . . . . . . . 496.2 Impact of load on segments . . . . . . . . . . . . . . . . . . . 506.3 Restoration time . . . . . . . . . . . . . . . . . . . . . . . . . 506.4 Correlated non-Gaussian distributed random numbers . . . . 51

6.4.1 Correlated Gaussian distributed random numbers . . . 546.4.2 From Gaussian to uniform distribution . . . . . . . . . 546.4.3 From uniform to Weibull distribution . . . . . . . . . 54

6.5 System vulnerability and outage time . . . . . . . . . . . . . 54

7 Case Studies 577.1 The first weather model . . . . . . . . . . . . . . . . . . . . . 57

7.1.1 Case 1.1: Disconnected lines . . . . . . . . . . . . . . 607.1.2 Case 1.2: Restoration times . . . . . . . . . . . . . . . 61

7.2 The improved weather model . . . . . . . . . . . . . . . . . . 637.2.1 Case 2.1: The 1999 storm on the His-Kil power line . 657.2.2 Case 2.4: Storm moving parallel to the Bor-Kil power

line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

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7.2.3 Case 2.5: Storm moving perpendicular to the Bor-Kilpower line . . . . . . . . . . . . . . . . . . . . . . . . . 73

7.3 Modelling distributions of severe weather parameters . . . . . 757.3.1 Possible weather scenarios for Swedish conditions . . . 767.3.2 Main direction of weather . . . . . . . . . . . . . . . . 767.3.3 Weather code and direction . . . . . . . . . . . . . . . 777.3.4 Size, intensity and moving speed . . . . . . . . . . . . 807.3.5 Case 3.1: Distributions of weather parameters using

the method for generating different scenarios . . . . . 817.3.6 Case 3.2: Power system reliably using the method for

generating different scenarios . . . . . . . . . . . . . . 82

8 Conclusions and Future Work 85

A The Weibull Distribution 89

Bibliography 91

List of Figures

2.1 The warm and cold front in the different stages of a low pressure. 82.2 An ice storm can cause extensive damage. . . . . . . . . . . . . . 102.3 Ice storm impact on Canadian towers. . . . . . . . . . . . . . . . 10

3.1 The circular shape of the weather and its severity levels thatdecrease from the center. . . . . . . . . . . . . . . . . . . . . . . 15

3.2 The angle β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 A wind load function for a particular segment or a cross-section

of wind. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Ice build up function for a particular segment or a cross-section

of ice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5 Ice load function for a particular segment. . . . . . . . . . . . . . 193.6 Wind blows anti-clockwise around a low pressure on the north

hemisphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.7 The shape of the wind part of the weather model. The color

scale represent the intensities of the wind which are largest 300km south-west of the center (AW = 38 m/s). The center is atthis moment located at (1000,1000). . . . . . . . . . . . . . . . . 21

3.8 The shape of the precipitation part of the weather. The intensityis decreasing from (xc, yc) and AI = 10 mm/h. . . . . . . . . . . 23

3.9 The precipitation part of 1999 storm as it passes the lines in thecase studies. The darkest red color corresponds to the maximumprecipitation (9.3 mm/h). . . . . . . . . . . . . . . . . . . . . . . 24

4.1 Critical loads for one of the studied power lines. . . . . . . . . . 37

5.1 The restoration time is divided into five time intervals. . . . . . . 42

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xiv LIST OF FIGURES

5.2 The restoration time for segment i, Tres(i), is defined as the timefrom the breakdown to the time when the segment is functioningagain, assuming notification time to be zero. . . . . . . . . . . . 42

6.1 Flow chart of the proposed method. . . . . . . . . . . . . . . . . 486.2 ρG is the correlation between two Gaussian random numbers and

ρW is the resulting correlation between Weibull random numbers. 526.3 From the Φ-function of correlated Gaussian distribution random

numbers, to uniformly distributed random numbers, to Weibulldistributed random numbers. . . . . . . . . . . . . . . . . . . . . 53

6.4 Gaussian-Gaussian versus Weibull-Weibull correlations from1000 simulations. The Weibull distribution has parameters a=1.6and c=2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.1 A scheme of the studied network. . . . . . . . . . . . . . . . . . . 587.2 The ice part of the weather has a radius of 130 km and moves in

direction of the arrow. . . . . . . . . . . . . . . . . . . . . . . . . 597.3 The Nordic power system. . . . . . . . . . . . . . . . . . . . . . . 637.4 The studied Swedish transmission power lines. . . . . . . . . . . 647.5 The precipitation at the His-Kil power line. . . . . . . . . . . . 667.6 Gust wind (solid) and its perpendicular component (solid with

dots) during the 1999 storm at the His-Kil power line. The windblows more perpendicular to the line after about 20 hours. . . . . 66

7.7 Ice accretion according to the Simple model on the phase line(solid) and on the top line (dotted) of the His-Kil power line inmm and kg/m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7.8 Loads on the His-Kil power line compared to the critical loads forthis line. The ice thickness increases with time and the wind/icefunction can therefore also be seen as a function of wind thatvary with time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7.9 Ice accretion on the phase line in mm (solid) and kg/m (dotted). 687.10 Loads on the His-Kil power line during the 1999 storm with 100%

increase of wind compared to the critical loads. . . . . . . . . . . 697.11 Loads on the five segments (load on segment 1 is most to the left

and load on segment 5 is most to the right) compared to criticalload for Bor-Kil, which is similar to the critical loads for His-Kilon which segment 1 is placed. . . . . . . . . . . . . . . . . . . . . 71

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7.12 The wind part of 1999 storm with real weather data when passingthe studied lines. Red color represents the highest wind speed(22 m/s mean wind) and blue the lowest. . . . . . . . . . . . . . 71

7.13 Wind part from the weather model passing the studied lines.AW = 38 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

7.14 Precipitation part from the weather model passing the studiedlines. AI = 10 mm/h. . . . . . . . . . . . . . . . . . . . . . . . . 72

7.15 The weather moves parallel to the power line and the wind hitsthe line perpendicular to the line. . . . . . . . . . . . . . . . . . . 73

7.16 Loads on segment 3 compared to the critical loads when theweather is moving parallel to the Bor-Kil line. . . . . . . . . . . . 74

7.17 Loads on segment 3 compared to the critical loads when theweather is moving perpendicular to the Bor-Kil line. In this casethe wind that hits the line reach its maximum after about halfof the simulated time. . . . . . . . . . . . . . . . . . . . . . . . . 74

7.18 Flow chart of method for choosing weather properties for MonteCarlo Simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7.19 The distribution of maximal wind gust speed, AW . . . . . . . . . 827.20 The distribution of maximal precipitation, AI , (7.20(a)) and the

following ice load (7.20(b)) for a particular segment. . . . . . . . 837.21 The studied network with generation point (G) and load points

D1 and D2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

A.1 A Weibull distribution with c = 4 and a = 37.6. . . . . . . . . . . 89

List of Tables

2.1 The impact of different mean wind speeds. . . . . . . . . . . . . 7

4.1 λW as a function of wind load and design load . . . . . . . . . . 354.2 λI as a function of ice load and design load. . . . . . . . . . . . . 364.3 Failure rates for the different areas of figure 4.1. . . . . . . . . . 37

5.1 Correlations for adjacent and not adjacent segments, segment iand segment j are close, segment i and segment l are not close. . 46

6.1 Weather and simulation parameters. . . . . . . . . . . . . . . . . 49

7.1 Broken segments and the time for breakdown in the first tenscenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.2 Probabilities for different connections/disconnections. . . . . . . 617.3 The first five restorations times for scenario 5, 1000 scenarios. . . 627.4 Means and variances from 1000 simulations of restoration time. . 627.5 Data for two Swedish transmission lines. . . . . . . . . . . . . . . 647.6 Times for breakdowns (hours) for 10 out of 1000 simulations.

Segment 1 is located on His-Kil and segment 2-5 are located onKil-Bor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7.7 Probabilities for different main directions of the severe weather. . 777.8 Definition of weather codes and their probabilities. . . . . . . . . 787.9 The distribution of directions for low pressures in Sweden, given

the main direction. The most common direction for Sweden issouth-west-west or −45◦ ≤ Θ < 45◦. . . . . . . . . . . . . . . . . 80

7.10 Distribution of weather parameters given weather code. . . . . . 807.11 Return periods for different wind gust speeds. . . . . . . . . . . . 81

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List of Symbols

V (x, y, t) Wind function in cartesian coordinatesW (r,Θ, t) Wind function in polar coordinatesVh Moving speed of the weather [m/s]Vmean Mean wind speed [m/s]Vmax Maximal wind speed or gust [m/s]LW Load function for windLI Load function for iceβ The angle by which the wind force hits a line [rad]Θ Direction of the weather [rad]wβ Wind factor for perpendicular component of windRwind Radius of wind part of a weather [km]Rice Radius of ice part of a weather [km]tstopi,j Time when scenario i has passed segment j [h]

g(x, y, t) Function for circular part of precipitation part of the weatherh(x, y, t) Function for front zone part of precipitation part of the weatherAW Maximal wind gust speed [m/s]AI Maximal precipitation rate of circular part of the weather [mm/h]AIfront

Maximal precipitation rate of front zone part of the weather [mm/h]P Precipitation rate [mm/h]

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∆R(x, y) Increase of ice thickness [mm/h]∆t The length of a time stepxstart, ystart Start position of the center of the weather(xi, yi) Coordinates for segment ivi Angel to x-axis for segment iλW Failure rate due to wind [number of breakdowns/(h, km)]λI Failure rate due to ice [number of breakdowns/(h, km)]Tres(i) Total restoration time for segment i [h]Λ Correlation matrixm Number of scenariosn Number of liness Number of segmentsk Number of broken segments

Chapter 1

Introduction

1.1 Background

The Swedish power network consists of power plants, a transmission networkwith nationwide coverage and a distribution network. The main part of thepower production in Sweden comes from hydropower and nuclear power.The hydropower is generated in the northern part of Sweden and most ofthe consumers are located in the southern part. This implies high demandson transmission capacity from north to south, since an extensive interruptionof transmission between the northern and the southern part of Sweden canimply difficulties in maintaining power supplies in the southern part.

The technical infrastructure is of crucial importance for a modernsociety and electric power supplies are of particular importance.Society’s dependence on electrical energy, communications and informationtechnology is increasing and power supply is also a basic condition forcontinued economic growth and national security. At the same timeusers, both industry and households, have an overconfidence in technicalinfrastructure functionality and little preparedness for outages in the powernetwork [1].

A relative new concept used in Swedish politics is severe crises whichcan be defined as

”A severe crisis is not an individual event, for example anaccident, a sabotage etc, but a state that may occur when one ormore events develop or escalate to include many parts of society.Severe crises can be considered to constitute of different kinds

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2 CHAPTER 1. INTRODUCTION

of extreme situations with low probability that are separatedi question. The state is of such extent that it leads to severeinterruptions of important functions and requires efforts fromseveral different authorities and coordination of organs to handlethe situation and limit the consequences.”

The new concept replaces the former full preparedness which was used incase of war [2].

One example of a severe crisis in the power system is extremeweather conditions that result in large-scale power failures and extensivedamages that require gathering all possible repair resources to restore thesystem. Extremely severe weather is generally very unusual, however theconsequences can be serious since the failure rates of components, such asoverhead transmissions lines, increase sharply.

An interruption in the power system can have many causes: adverseweather, technical faults, operational problems, vandalism. Comparingthe primary causes of outages; adverse weather outages are accounting forapproximately 33% of all outages. One scenario that may lead to difficultiesin maintaining power supplies for a long time is ice storms. An ice stormis an extreme situation, which occurs very infrequently in most parts ofthe world but causes extensive damage when it does; freezing rain coatseverything in ice, often in combination with heavy wind. The power networkcomponents break down because of the heavy ice and wind and large areascan be affected.

The ice storm that hit eastern Canada and north-eastern United Statesin January 1998 is considered to be the worst in modern time in Canada; itcaused a crisis where about 1.5 million households where without electricityand the system was not completely restored until October 1998 [3] [4].Another recent example is the ice storm that hit Germany in November2005. More than 70 transmission towers were broken and 200 000 peoplewere affected by the blackout [5].

There was an ice storm in Sweden 1921; since the society is much moredependent on the infrastructure now the consequences if this storm wouldhave happen today are not comparable. A summary of some of the icestorms that have occurred during the last 100 years around the world canbe found in section 2.2.

Another example that may lead to a severe crisis is vandalism, where anorganized attack against vital constructions results in severe interruptions

1.2. AIM 3

in the system, and requires extensive efforts to restore system functionality.

1.2 Aim

The overall aim of the project Reliability of the power system under severecrises in the boundary between market and essential infrastructure, inwhich this thesis is one part, is to study how to handle reliability in thepower system under severe crises, which are extreme situations with lowprobability. The power system as an important infrastructure is centralwithin the project and the fact that the dominating part of the system isfinanced by the market.

The different phases of the project are the following:

A Descriptions of scenarios that may lead to severe crises. For studies afictive network similar to the Swedish system can be used.

B Descriptions of possible measures to reduce consequences of severecrises.

C How can costs for severe crises be estimated? Many conditions will betaken into consideration and Monte Carlo methods will be used.

D Which use of preparedness resources that is the most efficient?

Weather events such as lightnings are not considered within this project,since a transmission network is dimensioned for lightnings and they normallydo not cause as large damages as ice storms. The network is dimensionedfor lightnings and heavy wind but not for long lasting ice storms. In thisthesis severe weather and ice storms in particular is the only scenario thatis considered.

1.3 Main contributions

In order to mitigate severe consequences of future ice storms in an efficientway it is essential to be able to estimate the consequences based onassumptions of the technical system and the severity of possible storms.

Although many papers consider reliability and repair times there arevery few that consider both and none of the papers that will be described inchapter 3.1, 4.1 or 5.1 considers the time dependent risk level on lines when


a severe weather passes a region or the correlation between restoration timesfor different broken components.

Based on knowledge of how a severe weather is created and how itbehaves during its lifetime a new technique of modelling adverse weathersuch as ice storms is developed in this thesis. The model uses one functionfor the wind part of the weather and another function for the precipitationpart and is based on geographically moving winds and precipitation. Themodel is suitable for both transmission and distribution networks. A knownice accretion model is used for estimating the load due to ice.

The approach is to estimate the reliability of components when exposedto severe weather conditions with a Monte Carlo technique where eachscenario represents a certain weather situation. For different scenarios thestochastic weather parameters, such as size, strength, speed and directioncan vary. For each scenario a vulnerability model for the components isalso required, where the risk of transmission outage is connected to theweather situation. This vulnerability model for the components gives whichof the components that broke down and at which time. One benefit ofthis method is that it is possible to estimate the time difference between theoutages in different lines, not only the outage risk. The times for breakdownsare interesting for estimation of the restoration time. The situations aftera severe weather that are studied are so severe that gathering of all oralmost all possible restoration resources is required to restore the system.Restoration times for components are not assumed to be independent; theyare instead assumed strongly correlated. Another contribution in this thesisis therefore a method for simulating correlated repair times that is notassumed Gaussian distributed.

In the case studies the weather model together with the vulnerabilitymodel for the components gives estimations of the impact on componentsdue to the different weather situations. The loads on two Swedish power linesare estimated during different weather situations and conclusions about theconsistency to ice storms of the studied lines are presented. The risks ofpower outages in connection to these weather situations are also analyzed.

The main contributions of the thesis are the following:

• An overview of methods for modelling severe weathers, their impacton power systems and different methods for modelling the restorationprocess.

1.4. THE OUTLINE OF THE THESIS 5

• A new model for ice storm weathers and their impact on transmissioncomponents.

• A new vulnerability model for components such as overhead lines andtowers.

• A new restoration model for a transmission network after an ice stormevent.

• A method for generation of non-Gaussian distributed randomnumbers.

• A new method for choosing weather parameters for scenarios. Themethod includes possible distributions of weather parameters and adescription of possible weathers in Sweden.

• The developed methods are used in different case studies both on apart of the Swedish transmission network and on fictive networks.

1.4 The outline of the thesis

• Chapter 2 is a description of ice storms and other weatherphenomenons and their consequences.

• Chapter 3 contains weather models.

• Chapter 4 contains vulnerability models for components.

• Chapter 5 is about the restoration process and the model forrestoration times.

• Chapter 6 describes the overall simulation method.

• Chapter 7 contains case studies and a stochastic method for choosingweather parameters for different scenarios of Monte Carlo Simulations.

Conference paper [7] is included in sections 3.2 and 4.2. Conferencepaper [8] is included in section 5.2. The improvements of the weather model,section 3.3, and the vulnerability model in section 4.2.2 is developed inconference paper [9], accepted for publication in July 2007. Section 7.3contain new material that is accepted for publication in October 2007 [10].A flow chart of the method is found in chapter 6. All papers contain casestudies.


1.5 List of publications

The following publications are included in the thesis:

[7] ”Modelling of Ice Storms for Power Transmission ReliabilityCalculations”, Proceedings of the 15th Power Systems ComputationConference PSCC2005, Liege, Belgium, August 22-26 2005. Thispaper is the foundation of the models described in chapter 3 and 4.

[8] ”On Transmission Restoration Evaluation after Ice Storms usingMonte Carlo Techniques”, Proceedings of the Third InternationalConference on Critical Infrastructures 2006, Alexandria, USA,September 23-27 2006. This paper is the foundation of the restorationmodel described in chapter 5.

[9] ”Modelling of Ice Storms and their Impact Applied to a Part ofthe Swedish Transmission Network”, Accepted for proceedings ofPowerTech 2007, Lausanne, Switzerland, July 2-5 2007. This paperincludes the vulnerability model for the components described insection 4.2.2. The case studies in section 7.2 are based on this paper.The ice accretion model is described in this paper and in section 3.4of this thesis.

[10] ”Ice Storm Impact on Power System Reliability”, Abstract acceptedfor proceedings of the 12th International Workshop on AtmosphericIcing on Structures (IWAIS2007), Yokohama, Japan, October 2007.The paper will include the method for probabilities for different severeweather scenarios and the case studies described in section 7.3.

Chapter 2

About Ice Storm Weathers

2.1 Wind and freezing rain

A storm is defined according to the Swedish Meteorological and HydrologicalInstitute (SMHI) as heavy wind, often in combination with rain or snow.Wind is measured in direction and speed. The direction is stated as thedirection where the wind comes from and wind speed is measured in m/s.The wind speed given in a forecast, Vmean, is defined as mean speed over aperiod of 10 minutes at a height of 10 m above the ground [11]. Both windspeed and wind direction differ for low terrain, mountains and seas. Theimpacts of different wind speeds are listed in table 2.1 [12] [13]. The impactof different wind speeds increases rapidly with the wind speed, V , since thepower in the wind is proportional to V 3 [14].

There are two fronts in a low pressure; one warm front and one coldfront. A cold front is created when cold air replaces a warmer air massand the air behind the cold front is colder than the air ahead of it. A coldfront is represented by a blue line with triangles along the front, pointing

Wind speed Impact20.8-24.4 m/s Little damage on houses.24.5-28.4 m/s Trees falling with roots and large damage on houses.28.5-32.6 m/s Large damage.

>32.6 m/s Very large damage.

Table 2.1: The impact of different mean wind speeds.

7

8 CHAPTER 2. ABOUT ICE STORM WEATHERS

(a) (b)

(c) (d)

Figure 2.1: The warm and cold front in the different stages of a low pressure.

towards the warmer air and in the direction of movement, see figure 2.1. Awarm front is created when a warm air mass replaces a cold air mass and isrepresented by a red solid line with semicircles pointing towards the colderair in the direction of movement. The fronts move along with the center ofthe low pressure (L in figure 2.1) and are in the beginning separated fromeach other. The cold front moves faster than the warm front and therewill be a fusion of the cold front and warm front, this process is shown infigure 2.1(a)-2.1(d) [15]. As the weather gets more severe the pressure inthe center falls and the precipitation area grows. On the north hemispherethe wind is blowing anti-clockwise around the center of a low pressure dueto the rotation of the earth and Coriolis forces. Weathers move in directionfrom high pressure towards low pressure.

The wind is often stronger south and west of the center (behind the lowpressure). This means that the strongest winds often blows from the westor from north-west. Atmospheric icing of a radio- and TV-mast in northern

2.2. ICE STORMS IN THE WORLD 9

Sweden was studied in [16], measurements was performed for seven years.The most common wind direction during the two to six icing events perseason was from south-west.

The most intensive and longest lasting precipitation is found close tothe center of the low pressure [17]. Ice storms occur when supercooled rainfreezes on contact with tree branches or overhead conductors and forms alayer of ice. Ice loading requires specific combinations of precipitation andtemperature. Freezing rain is mostly associated with the passage of thewarm front. The air is colder in the lower air layers of the atmospherethan in the higher layers. As a result the rain freezes as it falls when thetemperature in the lower air layers is below freezing.

The shape of the area of the precipitation, the green area in figure 2.1,is changing during the lifetime of a low pressure since the cold front movesfaster than the warm front. A low pressure is often followed by another lowpressure to the south of the first low pressure. The combination of the twoweather systems can be devastating for the transmission network, especiallyif the first one brings a lot of freezing rain and wet snow and the secondbrings strong winds.

It is difficult to forecast how fast and in which direction a low pressurewill move. Some low pressures move very fast and others move slowly andthey can even be stationary. There is no obvious correlation between howfast a low pressure moves and how severe it is [17]. The frequency andintensity of icing depends strongly on the geographical location as well ason the local topography [18].

2.2 Ice storms in the world

The largest wind force measured in Sweden is 40 m/s (a mean value over10 minutes) and largest gust is 80 m/s (only for seconds) [12]. Most ofthe precipitation in Scandinavia in the autumn and winter comes from lowpressures created over the North Atlantic. Some of these low pressures turninto severe storms [17]. In the south-west of Sweden the weathers from seanormally come in from south-west and continues north-east (does not holdfor the severe weather 1921 and 1968).

Ice builds up on lines and places a heavy physical load on the linesand other structures and increases the cross-sectional area that is exposedto wind. The added weight and surface exposed to the wind increase the


Figure 2.2: An ice storm can cause extensive damage.

Figure 2.3: Ice storm impact on Canadian towers.

risk for the towers to break [13]. Figure 2.2 and 2.3 show lines and towersexposed to icing.

Icing is traditionally a phenomenon in the northern regions of theearth, such as Canada, Japan, Russia, the Nordic countries and centralEurope. However; there are many reports of icing in southern France, United

2.2. ICE STORMS IN THE WORLD 11

Kingdom, Spain, Algeria, south Africa and Latin America and there arereports on that the temperature has changed more rapidly during the latest20-50 years, which affects which areas that will be most exposed to icing inthe future [18]. Some of the ice storms in the last 100 years around the worldare listed below. These ice storm may not be the worst but for example thestorm Gudrun has become very famous, at least in Sweden.

Sweden October 1921

This storm with ice accretion is a part of the motivation for this project. Thecombination of a lot of rain and large wind speeds would have devastatingconsequences if it had happened today. In [19] it is concluded that 20−50%of today’s towers would have broke down in the affected area under the samecircumstances as 1921.

US January 1972

An ice storm hit the Lower Mainland of British Columbia in January 1972.Two 500 kV transmission lines were severely damaged. The duration of thestorm was approximately 48 hours and the maximum icing measured on aline was 9 mm. The wind was not as important as the ice for the towerbreakdowns during this storm [20].

Canada and US January 1998

The ice storm that hit eastern Canada and north-eastern United Statesin January 1998 has been called the worst storm i modern time [21].Extreme ice formation on power lines caused a crisis where about 1.5million households where without electricity. The system was not completelyrestored until October 1998 but 90% of the affected customers had receivedpower within two weeks [3].

France December 1999

A three day storm caused heavy damage on the transmission grid iFrance in December 1999. The maximum number of transmission linesout of operation was 38 and 5000 MW were not delivered to customers.Approximately 3.5 million households were without electricity [22].


Sweden January 2005 - The storm Gudrun

Southern Sweden was hit by a storm that is known under the name Gudrun.Gudrun had gust wind speeds of up to 46 m/s. The electricity distribution,the telecommunication services, the railways and many roads were affectedfor a long time. During the night between January 8th and 9th 650 000persons had their power supply interrupted. The restoration of the powersupply took up to seven weeks [23]. It is interesting to notice that the firstwarning of the expected storm was issued at lunchtime on January 7th andmy son Eskil was born at lunchtime January 7th, but he is more like ahurricane.

Other examples of recent ice storms are Switzerland January 2005,Germany November 2005 and the middle US December 2006.

Chapter 3

Modelling of Ice StormWeather

3.1 Weather models

Methods for including the impact of weather on power system reliabilitycalculations have been studied earlier. The most widely used model isthe two-state (normal and adverse) weather model that uses constantfailure and repair rates for each state [24]. The two-state and many othermodels, especially Markov models, assume the entire network to be inthe same weather environment, a reasonable assumption for geographicallyconstrained distribution networks but not for large transmission networks.In [25] a model applicable to transmission networks is described. Differentweather severities are considered, but the distribution of severity levels isdiscrete and the exposed area has to be divided into regions that are equallyaffected. In [26] it is stated that wind, icing and lightning are the mostinfluential weather phenomena and the daily wind gust speed is the onlyvariable selected to study wind effects. Since an ice storm is a very rareevent it is regarded as a special event and the analysis of icing is separatedfrom the other weather factors. In [27] the weather environment is dividedinto three states: normal, adverse and major adverse. The different stateshave different failure rates and the transitions rates between the differentstates are specified.

None of the above described papers consider the time dependent risklevel for lines when a severe weather passes a region. The new technique of

13

14 CHAPTER 3. MODELLING OF ICE STORM WEATHER

modelling severe weather, such as ice storms, in this thesis uses one functionfor the wind part of the weather and another function for the precipitationpart and is based on geographically moving winds and precipitation. Themodel is applicable to storms that cover large areas and is therefore suitablefor transmission networks. The model has severity levels that are continuousand vary with time as the weather passes for example transmission linesegments. In [8], the first weather model is described. In [9] this model isdeveloped into the improved weather model. Based on more knowledge ofhow a severe weather is created and how it behaves during its lifetime thewind and precipitation part of the weather is more realistic. A known iceaccretion model, the Simple model [28], is used to estimate the ice load. Inthe case studies the loads on two Swedish power lines are estimated fordifferent weather situations. The loads given by the simulated weathers arecompared to loads given by real weather data from a storm that hit Sweden1999 using the same ice accretion model. The loads are also compared to thecritical loads for the studied lines. The improved model is mainly describedin [29], which is a master thesis produced within this project in cooperationwith Svenska Kraftnat.

For different scenarios the stochastic weather parameters, such as size,strength, speed and direction can change. A method for choosing weatherparameters to different scenarios is described in section 7.3.

The first weather model for severe weather in section 3.2 is more generalthan the improved weather model in section 3.3, but it is less accurate forSweden where the improved model is more suitable. The ice accretion partis particulary improved in the improved weather model.

3.2 The first weather model

Since the largest precipitation occurs in the center of storms, a circularmodel with the largest strength in the middle can be applied [30]. Thetwo-variable function below has suitable properties to serve as a basic modelfor describing wind and ice load.

f(x, y) = A exp(− 1

2

((x− xcenter

σx

)2+

(y − ycenter

σy

)2))(3.1)

A is the amplitude, that is the severity level in the center of the weather.The center of the weather has coordinates (xcenter, ycenter). Figure 3.1 showsthe circular shape and the decreasing severity levels from the center.

3.2. THE FIRST WEATHER MODEL 15

100 200 300

50

100

150

200

250

km

km

Figure 3.1: The circular shape of the weather and its severity levels thatdecrease from the center.

Each line segment, which is a representation of transmission components,is exposed to certain load functions that depend on which intensities ofthe weather that meet the segment, and for how long. A load functioncorresponding to point (xj , yj), when the center of the weather movesaccording to the functions (xcenter(t), ycenter(t)), can be calculated from

L(xj , yj , t) =

A exp(− 1

2

((xj−xcenter(t)

σx

)2+

(yj−ycenter(t)

σy

)2)). (3.2)

The strength, or severity level, of the weather depends on ice loadand wind load. These loads are modelled with a load function for wind,LW (xj , yj , t), and a load function for ice, LI(xj , yj , t), these load functionshave different parameters and shape.

3.2.1 Wind load

Wind speed is often treated as a mean value, for example the mean valueof measured wind speed during a typical ten-minute period. Wind speedor wind load in this model refers to the instantaneous wind speed inside a


β(µx,µ

y)

(x1 ,y

1 )

(x2 ,y

2 )

r

u

Line segment

Figure 3.2: The angle β.

severe weather, also called gust. The maximum wind speed corresponds tothe amplitude, A, in equation (3.2).

Because the wind speed is zero in the absolute center of a storm; equation(3.2) can not be used directly. By subtracting an extra function with lessamplitude and smaller σx and σy, a more realistic model is achieved [30]. Thewind load function for a line segment represented by a point with coordinates(xj , yj) is obtained from equation (3.3):

LW (xj , yj , t)[m/s] = wβ(t)[

A1 exp(− 1

2

((xj−xcenter(t)

σx1

)2+

(yj−ycenter(t)

σy1

)2))

−A2 exp(− 1

2

((xj−xcenter(t)

σx2

)2+

(yj−ycenter(t)

σy2

)2))],

where wβ(t) ∈ [0, 1].(3.3)

wβ(t) in equation (3.3) is the wind factor needed to consider the impact ofthe angle, β, see figure 3.2, by which the wind load hits the line. Windload perpendicular to the line is the worst case; to include this in the modelthe perpendicular component of the wind load is used. The perpendicularcomponent of the wind load is then achieved from multiplication by the wind


factor, as in equation (3.3). The wind factor is obtained by equation (3.4).

wβ(t) = sinβ(t). (3.4)

Let the line segment be represented by a point with coordinates (x1, y1),see figure 3.2, and the center of the weather has coordinates (xcenter, ycenter)at time t. Choose an arbitrary point on the line in the same direction as thestudied line segment with coordinates denoted (x2, y2) such that the angle αbetween the vector r = (x1, y1), (µx, µy) and u = (x1, y1), (x2, y2) is between0 and π. Then α is given from cosα = ru

|r||u| and β = π2 − α if α ≤ π

2 andβ = α − π

2 if α > π2 , since the wind load is perpendicular to r. β is always

between 0 and π2 and a function of time for each segment.

When the wind load is parallel to the line, i.e. when β = 0 the windfactor is zero. Wind parallel to a line does not contribute to a breakdown;this is realistic since wind parallel to a line even can reduce the ice thickness[31]. The wind factor is one when the line is hit by perpendicular wind load,i.e. β = π

2 .LW (t) for a particular line segment is shown in figure (3.3). This figure

with distance on the x-axis can represent the wind part of the weather seenfrom the side (a cross-section of the wind part of the weather).

3.2.2 Ice load

Let the speed by which the ice build up on components ([mm/h]) also bemodelled with a two-parameter function. Figure 3.4 shows the ice build upfunction for a particular segment, or a cross-section of the ice part of theweather with distance on the x-axis.

The ice load function for a particular line segment, LI(xj , yj , t),corresponds to the total ice on (xj , yj) at time t and is achieved byintegration of the ice build up function for this segment.

LI(xj , yj , t)[mm/h] =∫ t

0

A3 exp(− 1

2

((xj−xcenter(u)

σx

)2+

(yj−ycenter(u)

σy

)2))du. (3.5)

The ice builds up continuously as in figure 3.5. The ice part of the weatherhas passed segment j when LI(xj , yj , t) is about equal to LI(xj , yj , t + ε)for a small positive ε, i.e. the time when the ice layer no longer increases onsegment j, in figure 3.5 this coincide with tstop for this particular segment.


0 2 4 6 8 100

10

20

30

40

time [s]

win

d sp

eed

[m/s

]

Figure 3.3: A wind load function for a particular segment or a cross-sectionof wind.

0 2 4 6 8 100

2

4

6

8

time [s]

ice

load

ing

[mm

/h]

Figure 3.4: Ice build up function for a particular segment or a cross-sectionof ice.


0 2 4 6 8 100

2

4

6

8mm

t

t−stop

Figure 3.5: Ice load function for a particular segment.

Wind load is one of the factors that affect the ice loading but theconnection between wind load and ice loading is not considered in the abovemodel for ice loading. In chapter 3.4 a more advanced ice accretion model isdescribed, where the impact of the wind load for ice accretion is considered.

3.2.3 Size of weather and moving speed

The spreading of a two parameter function as equation (3.1) is infinite.Normal weather is here defined as weather with severity level less than A

k ,for some k. The size of the severe weather is optional.

The weather moving speed, Vh, describes how fast the weather ismoving through the exposed area. The center of the weather located in(xcenter(t), ycenter(t)) at time t moves according to equation (3.6).

xcenter(t) = xstart + Vh cos(Θ)tycenter(t) = ystart + Vh sin(Θ)t, (3.6)

where Θ is as in figure 3.6 and Vh = VhW for the wind part of the weather andVh = VhI for the ice part. (xstart, ystart) is the start position of the weather.


m½¼

¾»

&%

'$

666Vh

LLLLq

Θ

moving direction

¡¡µ

Figure 3.6: Wind blows anti-clockwise around a low pressure on the northhemisphere.

Both moving speed and angle for the wind weather may be different fromthe speed and angle for the ice weather.

3.3 The improved weather model

The improved weather model is based on Swedish conditions but is probablymore accurate than the first weather model in other countries also. It isbased on the first weather model but the geometric shape of the wind andprecipitation areas have been more carefully studied. The model consistsof two parts: one function that describes the wind part of the weather andanother function that describes the precipitation part of the weather. Theprecipitation is most intense close to the center of the low pressure and thestrongest winds are blowing south-west of the center. The weather movesaccording to functions for how the centers of the wind part and the ice partmove in the improved model as well as in the first weather model.

3.3.1 Wind load

The wind is often stronger south and west of the center of the low pressureand the wind can be assumed to have its maximum 300 km south-west fromthe center [32], at least in Sweden.

The wind function in polar coordinates, W [m/s], is:

W (r, θ, t) =

AW e−1k(B∗(r(t)−300)2+C(min(θ(t)− 4π

3+2π,2π−(θ(t)− 4π

3+2π))2 , (3.7)

where r < Rwind,

0 < θ < 2π.

3.3. THE IMPROVED WEATHER MODEL 21

Rwind is the radius of the wind area. θ is the angle from the x-axis. Theamplitude AW [m/s] refers to the maximum wind 300 km away from thecenter with θ = 240◦. r [km] is the distance to the center. B, C and kare constants, with B = 0.08, C = 30 and k = 10000 the wind part of theweather has the shape shown in figure 3.7.

Figure 3.7: The shape of the wind part of the weather model. The color scalerepresent the intensities of the wind which are largest 300 km south-westof the center (AW = 38 m/s). The center is at this moment located at(1000,1000).

The wind factor, wβ(t), see equation (3.4), is needed also in this modelto consider the impact of the angle, β, by which the wind load hits the linesegment. The load function for wind, LW (x, y, t), is achieved according toequation 3.8, since the wind impact is direct.

LW (x, y, t) = wβ(t)V (x, y, t), wβ(t) ∈ [0, 1], (3.8)

where V (x, y, t) = W (r,Θ, t).The wind force on a line is also affected by the cross-section area of

the line, which is larger when there is ice on the line. In both the firstand the improved weather model the wind load is simply equivalent to theperpendicular component of the gust and independent of the cross sectionarea. A way to consider the increased area when there is ice on the line isdescribed in [13].


3.3.2 Precipitation

The precipitation area is modelled in two parts. Close to the center of thelow pressure the precipitation area can be assumed circular [15]. This ismodelled with one function that gives the largest values in the center anddecreases with the radius, Rice, as equation (3.9).

g(x, y, t) = AIe(− 1

30000∗(x−xc(t))2+(y−yc(t))2) (3.9)

if (x− xc)2 + (y − yc)2 < R2ice

else g(x, y, t) = 0

xc(t) and yc(t) are the x- and y-coordinates for the center of the circularpart of the precipitation part of the weather at time t. The constant AI isthe precipitation rate in this center.

A large precipitation area around the center of a low pressure often isfollowed by a large front zone. The front zone precipitation is here modelledwith largest intensity close to the center; the intensity decreases with thedistance from the center of the circle. The width of the front is dependenton the radius of circular part of the low pressure. The front zone is modelledwith the following function:

h(x, y, t) = AIfronte−

140000

(y−yc(t))2 , (3.10)

where AIfrontis the precipitation rate in the front zone area nearest the

circular area. In order to get the front zone in the right position in relationto the circular area described in equation (3.9) the following restrictions areneeded:

√(x− xfc(t))2

0.8+ (y − yfc(t))2 > 1.1Rice, (3.11)

√(x− xfc(t))2

0.8+ (y − yfc(t)2 < 2.6Rice, (3.12)

x > xfc(t) + 20,

g(x, y, t) = 0

elsewhereh(x, y, t) = 0. (3.13)

xfc(t) and yfc(t) are the x- and y-coordinates for the center of the front zonecircular area at time t, see figure 3.8. The whole precipitation area and its


Figure 3.8: The shape of the precipitation part of the weather. The intensityis decreasing from (xc, yc) and AI = 10 mm/h.

intensity, the precipitation rate, is obtained by:

f(x, y, t) = g(x, y, t) + h(x, y, t). (3.14)

The shape of the precipitation area is shown in figure 3.8. The shape of theprecipitation part of the weather is similar to figure 2.1(d). The reason forthe choice of modelling the weather in this phase is that the low pressurenormally is most violent in this phase [17].

Figure 3.9 shows a storm that hit Sweden 1999 using weather data fromweather stations. The shape of the precipitation part of the weather can becompared with the shape given by the weather model in figure 3.8.

The load function for the ice or snow, LI(x, y, t), is given by the Simpleice accretion model [28] described in chapter 3.4. The ice accretion isdependent on the wind. The mean wind, Vmean, is used to estimate theamount of ice that is deposited on a line. The relation between Vmean andthe gust or maximal wind, Vmax, can be approximated by (3.15).

Vmean = kgVmax. (3.15)

The factor kg differs for different storms and for different types of terrain.kg = 0.7 is used in the investigation of the Swedish ice storm 1921 [19] and


Figure 3.9: The precipitation part of 1999 storm as it passes the linesin the case studies. The darkest red color corresponds to the maximumprecipitation (9.3 mm/h).

in the case studies of this thesis. In [11] Vmean = 0.73Vmax, 25 m above sealevel.

3.4 Modelling ice accretion

There are many models for deposition of freezing rain and wet snow onobjects. The ice accretion models can be divided into two different types.The first type uses physical parameters and determines the heat balance ofthe object and requires parameters that are difficult to model. The secondkind of model uses meteorological data to estimate the accreted ice. Themodel suggested and used in this thesis is the Simple model [28] and is ofthe second type.

3.4. MODELLING ICE ACCRETION 25

An alternative to ice accretion models for estimation of ice thickness issuggested in [18]. Since ice load on overhead lines depends on height aboveground, which differs from tower to tower, and accretion is often a mixtureof different ice types a statistical method is suggested. In [18] is for exampleWeibull, exponential, gamma or log-normal distributions suggested for thedistribution of the ice thickness.

The Simple model uses parameters that are given by the weather modeldeveloped in this thesis, however it is necessary to assume the size of thedroplets. It can be assumed that all the droplets that hit the surface of theline freezes. This means that no icicles are developed. This is not necessarilyan overestimation of the accreted ice on the line because the icicles that areignored make the area for collecting new droplets larger, this can even leadto an underestimation of the ice load [33].

The Simple model has been developed to model the effects of freezingrain. The Goodwin model [34] is similar to the Simple model but it uses thefall speed of the precipitation instead of droplet sizes and can be used tomodel wet snow accretion also. The only difference for using the Goodwinmodel for freezing rain or snow is the density of the fall speed of theprecipitation. For wet snow a density of 0.3 − 0.6 kg/dm3 is suitable [35].There is a comparison of the Simple and the Goodwin model in [29]. Whenthere is no wind, or very light wind, the differences between the two modelsare negligible. The Goodwin model is very sensitive to the choice of dropletfall speed, therefore the Simple method is preferred in the case studies ofthis thesis. A very simple model for ice accretion is also suggested in [13],but it requires an initial ice layer for new ice to build up.

There are also numerical models for ice accretion which take the iciclegrowth into account [33]. These models are probably the most accurate forestimation of ice accretion but they require more meteorological data, forexample the humidity.

Different models assumes different shapes of the ice accretion aroundthe power line. Among others the Simple model assumes a perfect circularshape; this is reasonable because of the power lines ability to rotate. Whenone side of the line has been covered with snow or ice it becomes heavierand rotates [35]. Other models assume an elliptic accretion shape [33].


3.4.1 The Simple model

The Simple model estimates the amount of precipitation that hits the linefrom a horizontal and a vertical direction. The massflux is the mass ofrain that hits an area during one time unit. The vertical massflux can becalculated as:

mv = Pδ (3.16)

where P is the precipitation rate in [mm/h] and δ the water density in[g/cm3]. Let the perpendicular component of the mean wind speed be Vmean

[m/s], which can be estimated to 0.7 times the gust wind [36], as in equation(3.17).

Vmean = 0.7wβ(t)Vmax. (3.17)

The horizontal massflux, mh is given by:

mh = 3.6Vmeanv, (3.18)

where v [g/cm3] is the liquid water content. The liquid water contentaccording to [37] is:

v = 0.072P 0.88. (3.19)

The total massflux hitting the line, m0 is

m0 =√

m2v + m2

h =

=√

P 2δ2 + 3.62V 2meanv2. (3.20)

The increase of ice thickness on the line, ∆R ([mm/h]), when assuming acircular shape, is given by:

∆R =m0

πδi,

∆R =1δπ

√(Pδ)2 + (3.6Vmeanv)2 (3.21)

where δi is the density of the ice in g/cm3.P is the precipitation rate, the intensity of precipitation of the severe

weather, and is given by f in equation (3.14) for the improved weathermodel. The ice load function becomes:

LI(x, y, t) = LI(x, y, t−∆t) + ∆R(x, y)∆t, (3.22)

3.4. MODELLING ICE ACCRETION 27

where ∆t represent the length of a time unit.The Simple model does not include the radius of the line and the accreted

ice thickness does not depend on the initial radius of the line. However; theweight of the accreted ice is larger when the line is thicker. The heaviest icethat can be developed has a density of approximately 0.9 kg/dm3 [35]. Suchheavy ice is only created under very specific circumstances. During the 1921ice storm in southern Sweden ice with a density of 0.6-0.7 kg/dm3 was found[19]. In the cases studies a density of 0.9 kg/dm3 is used, a lower ice densitywould increase the wind load because the ice layer becomes thicker giventhe same precipitation rate which makes the area exposed to wind larger.

Chapter 4

Modelling ComponentVulnerability

4.1 Reliability of transmission lines

To be able to connect the risk of transmission outage to the weathersituation, a vulnerability model for the components is required. Thereare many different methods for estimation of reliability parametersfor transmission components, both deterministic and stochastic. Adeterministic method for estimation of breakdown means that if the loaddoes not exceed the specified load level for the component the component willnot fail. The threshold load is often equal to the design strength. In realityboth loads and strengths of the component are stochastic and stochasticmethods are therefore preferable [38].

Overhead lines are the most vulnerable components in distributionsystems [26]. The important issues for line damage are ice load, wind loadand type and conditions of components [39]. The studied components couldbe insulators, lines or foundation. Without any knowledge of sequence offailure is it hard to say which component that is the most critical. Duringan ice storm a transmission tower can fail due to buckling of its legs as infigure 2.3. Another failure sequence is when the line breaks first and causefailure to its adjacent towers [40]. Another way of breakdown is galloping,which is described in section 4.1.1. In this thesis tower and line breakdownsare considered and tower and lines are represented by segments.

A method for estimating the risk to transmission system components

29

30 CHAPTER 4. MODELLING COMPONENT VULNERABILITY

due to ice storms is described in [41]. Icing from a 1 in 50 year storm, a 1 in100 year storm and a 1 in 500 year storm are considered. A fault tree withprobabilities depending on how much ice load that is above the ultimatespecified load or design load is used to obtain a stochastic method. Thepaper considers different parts of a tower but only breakdowns due to ice,the wind is not considered. Data for the critical load for failure is estimatesbased on experiments on two existing transmission lines.

The basic transmission line model in [42] allows different weatherconditions (normal and adverse) with different failure rates for different partsof a transmission line. The same authors have in [43] described their methodthat also recognizes weathers that differ for different regions. Furthermore;they do not assume exponential distribution of the duration of an outageand discuss the effect of failure bunching when components in an adverseweather region have high failure rates simultaneously.

In [44] it is stated that the failure rates of overhead transmission linesare continuous functions of the weather to which they are exposed. Thetwo-state weather model in [24] and [42], among others, is expanded to athree weather state model which also consider major adverse weather withsharply increased failure rates. [44] suggests that these extreme weathersshould be considered as independent events with overlapping failure ratesfor the exposed area.

Since Canada experienced its severe ice storm in 1998 many case studieshave been performed on test networks based on Canadian conditions. Forinstance is the sequence of failure of an experimental distribution lineexamined in [45]. Other examples are [46] where the status of existingtransmission line components are considered and the standardized riskestimation spreadsheet developed in [41] which is used to calculate the riskof three different severity levels of ice storms.

Peak over threshold methods, where data from peak events that exceedsa specified threshold during a time series, are suggested in [18] to be suitablefor irregular events such as severe ice storms.

In [47] are Monte Carlo simulations used to model adverse weatherfor transmission line outages. Failure rate data is based on statisticsfor an existing transmission line and failures due to lightning, wind andprecipitation are studied. The paper shows that the distribution of up anddown times due to adverse weather are skewed and suggests to not usee.g. exponential distributions in reliability calculations for adverse weather.

[48] describes an analytical method with non-constant failure rates for

4.1. RELIABILITY OF TRANSMISSION LINES 31

adverse weather. The failure rate is a function of time which is low for mostof the year, and very high for a short period of the year which representadverse weather. In [49] past failure data during adverse weather is usedtogether with the weather data to achieve failure rates for different adverseweathers.

The methods described above have in common that needed componentreliability data is important but difficult to achieve.

In all the above methods for impact due to severe weathers on powersystems the failure rates are constant and none of the papers considers thetime dependent risk level on lines when a severe weather passes a region.In this thesis it is assumed that the probability for a breakdown due to theimpact of a given weather depends on the load function together with thevulnerability model for components, see section 4.2. The vulnerability modelis stochastic and is based on the design of the components. The approachis to estimate the reliability of components using Monte Carlo simulations.One benefit of this approach compared to many other methods is that itis possible to estimate the time difference between the outages in differentlines, not only the outage risk since Monte Carlo methods allow simulationsof time sequences.

Since a broken component during a severe weather probably has brokedown due to the weather the failure rates for all components in the severeweather region are correlated. This is often considered with increased butindependent failure rates for the components in the severe weather region.In this thesis the failure rates vary with time and the severity of the weatherand failure rates for components in the same weather environment havehereby correlated or similar failure rates. As shown in [50] it is likely thatadjacent towers to the tower that breaks first also breaks due to maximumweight on the line. The tower construction breaks due to the extra forceon the adjacent towers which already are exposed to high ice and windloads. However; often only a few towers are involved in this domino effect.In the case studies a segment represents many towers and the effect ofbreakdown of adjacent segment is not as obvious as the impact of adjacenttower breakdowns. In the model for the restoration process described inchapter 5 the status of adjacent segments is considered.

The sheltering effect on wind speed by trees and terrain in transmissioncorridors is not considered in this thesis.

When discussing transmission failures due to icing an often suggestedsolution is heating of the line by introducing large transmission loads.


During specific conditions the line can be up to twenty degrees warmer thanthe air temperature [51]. In this thesis it is assumed that this heating effectdoes not prevent the ice from building up on the power line. In Sweden thelargest transmission load occurs when the temperature is much lower thanthe ideal temperature for ice storms and the existing heating devices arenot in common use in Sweden where mechanical deicing methods are usedinstead. [52]

4.1.1 Galloping

Galloping occurs when there are ice in combination with wind on the spanof the line. The wind creates motions as it blows across the line and if thefrequencies of these motions coincide with the natural frequency of the linean oscillation occurs. Galloping does not result in an immediate breakdown,but causes extreme forces on the line and towers and can ultimately resultin a breakdown by fatigue [13]. Galloping can be mitigated by increasingthe tension of the line, or installing dampers. There have probably not beenany breakdowns due to galloping in Sweden, but there are known cases inCanada and England [52].

4.2 Segment vulnerability models

The exponential distribution with constant failure rates is often used inreliability calculations for simplicity and for being robust. According to [53]this can be misleading for the design of the system and the reliability canbe overestimated for some periods and underestimated for others. Anothercommon method for treating the risk of failure is to use a type of processthat includes a ”memory” in order to include the possibility that the riskalso increases when the load is constant due to fatigue. In the first segmentvulnerability model proposed in this thesis a time dependent exponentialdistribution of the time to failure was chosen. This means that the processis assumed to have ”no memory”. To include the changed risk of failurebecause of changed amount of wind and ice load, the parameters of thedistribution are controlled in order to obtain a realistic behavior of theconnection between the load and the risk of failure. In this way can theprocess be controlled more directly. The second segment vulnerability modeldiffers from the first segment vulnerability model, which is more generalbut requires data that is difficult to achieve. The second vulnerability

4.2. SEGMENT VULNERABILITY MODELS 33

model is applicable when knowledge of the critical wind and ice loads ofthe components can be estimated.

Because of the complexity of modelling the influence of severe weather,the network components such as line segments and towers are divided intosegments. A segment can for example consist of the line between two towersand one of the towers, but it can also represent longer parts of the line. Thestudied lines are divided into n line segments with a length and an angel vto the x-axis. The segment i is represented by its midpoint with coordinates(xi, yi), and angle vi. The segments may break down under influence of thesevere weather, the wind load function and the ice load function. The extentto which a segment is affected depends on segment vulnerability and severityand direction of the severe weather. A single segment breakdown is enoughfor the whole line to become disconnected, but the time for each breakdownis registered. Assuming that the load functions for each segment are known,how probable is it that a segment breaks down? The way in which a segmentis affected is not only dependent on severity of the weather, it is also heavilydependent on the design of network components and their condition at thetime of the severe weather (e.g. damage due to corrosion) [39]. Since it wouldbe impossible to consider the status of every component in the network it ishere assumed that the probability of an individual segment breaking downwithin a certain time under the impact of a given weather depends on theload function together with the segment vulnerability model.

A given weather will give a certain load on the components. Dependingon the vulnerability of the components it will take different times until thecomponent breaks. Both the weather, loading vulnerability and time tobreak are in reality stochastic. It is possible to treat only the vulnerabilityas stochastic, and the weather, the load and the threshold breaking asdeterministic. A stochastic behavior of a given weather and a deterministicmodel for component vulnerability can be replaced by a given weather’sdeterministic properties and a stochastic impact on segments.

The segment vulnerability model includes a stochastic method fordeciding whether a failure occurs or not. The model connects the directwind impact with the integrating impact from the ice storm, given byfor example an ice accretion model, and is based on the design of thecomponents. Different stress levels correspond to different failure rates,λ [number of breakdowns/(h, km)]. λ is a continuous function of the loads,which in turn are functions of time since the weather is moving.

Let the stochastic variable be time to failure. The parameter m(t) is


expected time to failure and related to the failure rate as in equation (4.1).

m(t) =1

λ(t). (4.1)

The probability density function with parameter m(t) is in bothvulnerability models described here assumed to be:

fT (t) =

{1

m(t)e− t

m(t) if t ≥ 0,

0 otherwise.(4.2)

The probability of breakdown of a segment in the time interval [t, t+∆t]is analytically given by (4.3) and can be approximated by (4.4).

P (failure in interval [t, t + ∆t]) =∫ t+∆t

tfT (u)du (4.3)

≈ fT (t +∆t

2)∆t if ∆t → 0. (4.4)

Assume that λ(t) is known; then the probability of breakdown in thetime interval [t, t + ∆t] in each segment is known by equation (4.3) or (4.4).

The time for a possible breakdown can be simulated for each scenario bydeciding stochastically whether a breakdown occurs or not for each t untilthe first break down. Whether a breakdown occurs or not is checked foreach time step, but the probability for a breakdown of a segment shall notbe dependent on the size of the time step. In the methods described here asmaller time step does not give a more vulnerable segment.

The stress due to only ice will increase as long as the ice builds up;thereafter, this stress level will become constant since the melting process isneglected. The stress due to a gust wind in combination with the ice can stillcause a breakdown. The increased stress associated with ice accumulationis in itself a time dependent failure rate. Monte Carlo techniques can beused to calculate the distribution of time to failure for different segmentsand lines.

The risks of power outages in connection to weather situations can beanalyzed given the structure of transmission and distribution networks inthe area. System reliability is not considered in this thesis except for a casestudy of a simple network.


LW λW [ 1h,50km ]

LW ≤ 0.9 · dl 1 · 10−5

0.9 · dl < LW ≤ 1 · dl 8 · 10−4

1 · dl < LW ≤ 1.1 · dl 0.0051.1 · dl < LW ≤ 1.2 · dl 0.0061.2 · dl < LW ≤ 1.5 · dl 0.03

1.5 · dl < LW 0.04

Table 4.1: λW as a function of wind load and design load

4.2.1 The first segment vulnerability model

In [49] failure rates due to ice and wind and other weather phenomena areanalyzed. In both [49] and [54] it is suggested that the failure rates due todifferent phenomena should be added during a severe weather.

The total failure rate function is given by weighing the failure rates forbreakdown due to ice and wind together, with weights aW and aI .

λ = aW λW + aIλI , (4.5)

where

λW (t) = f(LW (t)), (4.6)λI(t) = f(LI(t)). (4.7)

The weights are an important but difficult issue. To get a clue of the ratiowe have been in contact with utilities and they have agreed on that aW = 0.1and aI = 0.9 is a good choice for this first approach. λ is also dependenton design criteria, or design load (dl), of the considered tower. λW and λI

are in some of the case studies defined as in tables 4.1 and 4.2, where dl isdesign load. The failure rates in tables 4.1 and 4.2 are chosen to make thecase studies in chapter 7.1 more illustrative by studying more vulnerablecomponents.

λ = 0.1 means that the mean time to failure is 0.1 hours, i.e. every tenthhour, or one of ten components with the same failure rate breaks withinthe next hour. In [49] the failure rate has been estimated during the stormGudrun in Sweden 2005. One suggestion was λ = 0.12 [ 1

h,100km ]. Thisfailure rate was calculated from failure data experienced during the storm.The largest λW in table 4.1 is 0.04 [ 1

h,50km ] which is 0.08 [ 1h,100km ]. There


LI λI [ 1h,50km ]

LI ≤ 0.3 · dl 00.3 · dl < LI ≤ 0.5 · dl 4.5 · 10−3

0.5 · dl < LI ≤ 0.9 · dl 0.010.9 · dl < LI ≤ 1 · dl 0.0151 · dl < LI ≤ 1.1 · dl 0.03

1.1 · dl < LI ≤ 1.2 · dl 0.051.2 · dl < LI ≤ 1.5 · dl 0.07

1.5 · dl < LI 0.1

Table 4.2: λI as a function of ice load and design load.

was no ice formation during Gudrun and the failure rates in 4.1 and 4.2 arechosen such that the impact of wind is approximately 10% and the impactof ice is approximately 90%.

The above described model is preferred when tower data can beestimated and the weights and the stochastic nature of the segmentvulnerability can be estimated for example by experience. The secondvulnerability model requires simulated experiments for transmissioncomponents, or norms for the design load, together with a safety factor.

4.2.2 The second segment vulnerability model

This model also connects the wind impact with ice load impact, see equation(4.8) and is also based on the design of the components. A graph of thecritical conditions with ice thickness on x-axis and gust wind on y-axis isshown in figure 4.1.

λ(t) = f(LW (t), LI(t)). (4.8)

The weights in equation (4.5) and tables 4.1 and 4.2 are not needed in thesecond model. Instead figure 4.1 in combination with table 4.3 is used.

The tower and line data used in the second segment vulnerabilitymodel are for Swedish components, from the Swedish transmission systemoperator Svenska Kraftnat. The data is used together with detailedsimulations including the mechanical structure performed by Vattenfallpower consultants [51] on when the first tower breaks down at differentice and wind loads. These calculations where performed for two lines withinthe master thesis project connected to the project described in this thesis


0 10 20 30 400

10

20

30

40

50

Ice thickness [mm]

Gust wind [m/s]

1 2

3 4

Figure 4.1: Critical loads for one of the studied power lines.

Area λ

1 02 0.23 0.54 1

Table 4.3: Failure rates for the different areas of figure 4.1.

[29]. The ice and wind loads were placed at the midpoint of the line andthe density of the ice was 900 kg/m3 in the calculations. The loads in area1, see figure 4.1, was not enough for causing breakdowns. [51] provided adeterministic model with the solid line in figure 4.1 as threshold. To achievea stochastic model the failure rates in table 4.3 can be used for the otherareas of figure 4.1. Table 4.3 shows examples of failure rates which are usedin some of the case studies of this thesis.

According to [55] are Swedish transmission structures built for 18 mm iceduring normal wind. The calculation performed by [51] showed that whenthe ice thickness exceeds 28 mm at a particular power line, the weakesttowers break due to the ice load alone. The critical tower data are almostthe same for towers in the transmission network and in the regional network


in Sweden, this means networks built for > 45 kV [52].In [29] it is shown that a gust wind of 30 m/s perpendicular to the

power line in combination with an ice load of approximately 4.0 kg/m (theice radius depends on the thickness of the line) are critical to a common typeof tower structure. Examples of a few weather situations that can cause thisload on power lines using the improved weather model is listed below [29].

• Gust winds of 25 m/s with the wind direction almost parallel to theline and 8 mm/h of freezing rain for 10 hours.

• Precipitation rate in average 5 mm/h during 20 hours.

• Three days of freezing precipitation of approximately 1.5 mm/h andwinds almost perpendicular to the power line most of the time (worstcase). In the end of the period, when the ice load is maximal, gustwinds up to 35 m/s are needed to cause the critical load.

Chapter 5

Modelling Restoration

5.1 Restoration models

After an ice storm the electric utilities want to repair their transmissionand distribution system as fast and economically as possible. To be able toefficiently plan and perform restoration of a network after a severe weather,that has caused damages, estimations of the restoration times for differentline segments is needed. The optimal location of repair resources, such asreserve parts and staff, becomes an important factor. The restoration timesare also needed to estimate the impact of a severe weather and thereby theneed for reserve power plants.

Restoration times for power network components have been studiedearlier. In [56] models for optimal location of repair units are developed. Thetravel time between two nodes (repair unit stations, broken segments etc) isassumed known. However; the travel time under extreme weather conditionscan be extended by a weather factor, which will account for additionaldelays. A ”tactical model” is also suggested in [56]. In this short-termmodel for restorations are locations for repair unit stations and the numberof repair units fixed. To be better prepared a forecast of a severe weatheris useful for the electric power utility which can place repair units from theoriginal repair unit stations to appropriate repair unit stations inside theforecast region. Two long-term strategic models are also suggested. Theleast long-term of these strategic models still assumes that the number ofrepair units are fixed, while the number of repair unit locations can vary.

Models that determines the optimal location of ambulances and fire

39

40 CHAPTER 5. MODELLING RESTORATION

departments, which are common, cannot be used for extreme weatherbecause of the difference in predicability [57]. A forecast for an extremeweather can reduce the restoration process. Without the forecast there willbe less repair units in the forecast area and therefore less repair units tomove to the broken segment. For hurricanes as Katrina, USA 2005, it hasshown that a forecast is of great value [57]. With a prognosis the utilitiesdo not have to wait until a storm hits before requesting resources, whichcan take several days to receive. Fully equipped trucks can make a largedifference in restoration times .

In [58] the phases of power system restoration is discussed and theplanning process is described in detail. The paper points out the unusualcircumstances for restoration after very rare and severe events; the lackof restoration staff and unavailability of many other resources due to forexample heavy snow on the roads. The phases of a restoration process arealso described in [59]. The correlation between the times for the differentphases is however not mentioned.

A stochastic model widely used for system reliability is the Markovmodel. In [60] different Markov models are described. In [61] a Markovmethod for including non-exponential repair times is presented. In [62]Markov methods are used for dependent components and both failure ratesand repair rates vary with time. The Markov model is often restricted to theexponential distribution and constant failure rates; this makes it less usefulfor estimating repair times. In [42] Monte Carlo methods are used insteadand unavailability and repair time of transmissions lines are estimated. Thearea is divided into smaller weather regions and different but constant failureand repair rates are used for different weather situations.

Restoration times of components are influenced by variable weatherconditions and available repair resources according to [63]. The restorationtime also varies with time of occurrence such as day, night, weekday andweekend. The time-varying restoration time used in [63] considers the effectof the weather by weather weight factors (larger during adverse weather).The effect of available restoration resources can be represented by a daily orweekly weight factor (larger during night time and weekends than daytimeand weekdays). The weather factor and the available restoration resourcesfactor are considered independent. The restoration time during a severeweather becomes considerably larger than during normal weather. Averagerepair time for a line segment and other components after severe weatherare studied in [27]. However; the data are from weathers that can occur

5.2. THE RESTORATION TIME MODEL 41

as often as once in a year. Average time required to restore the systemafter a major storm (once in a year and lasts for one hour) is in one studyestimated to 5.6 days. [64] reports utility responses to 44 major stormsbetween 1989 and 2003. The report once again points out that useful datais hard to achieve and outage time, peak number of restoration staff, peaknumber of customers out of power and equipment damage are defined. Itis concluded that the number of restoration staff peaks a day or two aftera major storm, due to the time it takes to mobilize extra staff. The peaknumber of restoration staff is closely correlated to the severity of the storm.

Weather events such as lightning are considered in [54], however not inthis thesis since a transmission network is dimensioned for lightning andthey normally do not cause as much damage as ice storms.

There are many papers about restoration and reliability of power systembut none of the papers described above considers both reliability and repairtimes. One example with a Monte Carlo approach is [6] but here failure ratesare constant and repair facilities are unrestricted. Only a few paper discussthe correlations between repair times for different components during severeweather.

In this thesis a model for estimating repair times considering therestoration process without assuming times for different components to beindependent is developed. The model assumes that locations for repairunits and the number of repair units are fixed. The down time and itsvariance after an ice storm for particular line segments or system are theresult. A Monte Carlo approach is used for estimation of restoration times.Restoration times for components are assumed strongly correlated. Amethod for generating non-Gaussian distributed correlated random numbersis presented in section 6.4.

5.2 The restoration time model

This restoration model is general except for the correlations due to snow orice on the road. It can be used for whole countries or smaller regions and isnot necessarily restricted to restoration processes of transmission networksduring ice storms.

Restoration time, Tres, can be defined in many ways. The weathermodel and the segment vulnerability model give the opportunity to simulatepossible values of the restoration time from the time when the segment broke


-- - - -t1 t2 max(t3, t4) t5

Tres¡

¡ª

Breakdownin segment i

AAK

Staff and spare partsat segment i

@@R

Segment iis functioning again

t

Figure 5.1: The restoration time is divided into five time intervals.

Repair ofsegment

t5

Localizationand transportof spare part

t3

Collectionand transportof a team

t4

Faultidentification

t2

Breakdownlocalization

t1

Figure 5.2: The restoration time for segment i, Tres(i), is defined as thetime from the breakdown to the time when the segment is functioning again,assuming notification time to be zero.

down to the time the segment is restored for each segment, since the timewhen a segment broke down is given from these models. However; it willbe necessary to make an assumption about the time from the breakdown tothe notification. Assuming notification time to be zero is the same as sayingthat the restoration begins when the problem is notified.

The restoration time, Tres, can be calculated for each broken segment inthe network and is divided into five time intervals according to figures 5.1and 5.2.


The first interval is the time for localization of the breakdown, t1, nextis the time for identification of fault, t2. After the maximum of the times fortransports of reserve parts, t3, and staff, t4, the reparation or replacementcan start; the actual repair time is denoted t5. Let the number of brokensegments in the studied area be denoted k. The restoration time for segmenti is estimated according to equation (5.1),

Tres(i) = t1(i) + t2(i) + max(t3(i), t4(i)) + t5(i). (5.1)

The five time intervals for segment i are assumed positive correlated to eachother but also positive correlated to all the times for the other segments,both adjacent and non-adjacent. For example is the time for localizationof the failed segment i, t1(i) correlated to t1(j) if segment i and segment jare adjacent, the correlation is weaker if they are non-adjacent. All thesecross-correlations are considered within the method that will follow. Sincethe time intervals are correlated, Tres(i) becomes correlated to Tres(j), j =1, . . . , k, the restoration time for the other segments that have broke down,both close and more distant.

In addition to the correlations between time intervals the restorationtimes depend on three independent source variables.

• The general weather situation gives availability of roads and thelocations of breakdowns. (t1, t3, t4)

• The distance to store with spare parts. (t3, t4)

• The existence and quality of a forecast and preparedness of staff.(t1, t2, t3, t4, t5)

t1, ...t5 are functions of the source variables. These functions are however notspecified here. Instead the structure with three independent source variablesis used to motivate which time intervals that are correlated to each other.

The availability of the roads is most likely limited because of ice, snowand fallen trees during an ice storm. The availability of roads is here assumedto be similar in the whole area and thus the times for transports tend tobe correlated. Many ongoing reparations far from the studied segment mayaffect the availability of staff and thereby lengthen the time to get staff tothe location. The total amount of staff and the number of breakdowns willalso affect the restoration times. Given a forecast of the storm gives theopportunity to prepare enough staff for the whole area.


The times t1(i), t2(i), . . . , t5(i), i = 1, . . . , k are all assumed to beWeibull-distributed. Different parameters can be chosen to affect the shapeof the distribution and a constant can be added to the Weibull-number toachieve a smallest possible time larger than zero. The Weibull distributionis a common distribution within reliability analysis and is often used fortime to failure and repair times for components, see appendix A. Themethod for generation of correlated non-Gaussian random number presentedin section 6.4 is valid also for other distributions, for example the log-normaldistribution.

5.2.1 Localization of breakdown

The time for localization of the breakdown, t1, depends on whether therewas a prognosis or not; localization time is shorter in the whole area iflocalization staff are prepared. The availability of roads is similar in thewhole area. The availability of roads affects the localization time whichthereby varies in a correlated way. t1(i) is assumed correlated to t1(j), t3(j)and t4(j), j = 1, . . . , k.

5.2.2 Identification of fault

Identification of fault, t2, is the time for deciding how many restorationworkers and which spare parts that are needed. The identification timefor segment i, t2(i), is here assumed correlated to t2(j), j = 1, . . . , k. Themotivation is that if one fault is easy to identify it is likely that the otherfaults are easy to identify too. However, the type of fault can be assumedsimilar for the different broken segments. If this assumption is made the timefor identification of all broken segments is short, except for the identificationtime of the first fault which can be long or short.

5.2.3 Localization and transport of spare parts and staff

The time it takes to get the spare parts, t3(i), and staff, t4(i), to the locationof the broken segment i, depends on availability of roads and distance andare therefore correlated to t1(j), t3(j) and t4(j), j = 1, . . . , k. In some casesit is possible to transport staff and spare parts together. The transports aredependent on availability of roads.


5.2.4 Repair time

Since other reparations close tend to speed up the repair time if it is possibleto work according to a production line the actual repair time for segmenti, t5(i), with staff and spare parts in place, is assumed correlated to t5(j),j = 1, . . . , k if segment j is close to segment i. The repair time is assumedWeibull distributed with the same parameters for transmission componentsof the same kind.

5.2.5 The covariance matrix

The covariance matrix for the restoration times, Λ, can be estimated giventhe locations of the breakdowns, availability of roads, location of stores andthe staff situation as a starting point. The method used here for deciding thevalues in the covariance matrix considers the locations of the breakdowns.The condition for two segments being close is that they are adjacent and onthe same line. The size of the matrix is (5 × k) × (5 × k), where k is thenumber of broken segments. The covariance matrix Λ should be positivesemi-definite since every covariance matrix is positive semi-definite. Amatrix is positive-definite if and only if it has non-negative eigenvalues [65].For logical choices of correlations this is automatically achieved. However,a bad choice of matrix due to non-logical choices when the correlationsare estimates or guesses can occur, especially when strong correlations areinvolved. Choosing smaller correlations is one way to provide a positive-semidefinite matrix. Table 5.1 shows an example of correlations between timeintervals and between broken segment. Two segments are defined as close ifthey are adjacent and on the same line.


t1(i) t2(i) t3(i) t4(i) t5(i)t1(i) 1 0 0.5 0.5 0t2(i) 0 1 0 0 0t3(i) 0.5 0 1 0.8 0t4(i) 0.5 0 0.8 1 0t5(i) 0 0 0 0 1t1(j) 0.7 0 0.4 0.4 0t2(j) 0 0.8 0 0 0t3(j) 0.4 0 0.7 0.5 0t4(j) 0.4 0 0.7 0.7 0t5(j) 0 0 0 0 0.6t1(l) 0.3 0 0.1 0.1 0t2(l) 0 0 0 0 0t3(l) 0.1 0 0.3 0.3 0t4(l) 0.1 0 0.3 0.3 0t5(l) 0 0 0 0 0

Table 5.1: Correlations for adjacent and not adjacent segments, segment iand segment j are close, segment i and segment l are not close.

Chapter 6

Description of SimulationMethods

In this thesis series of events during severe weather are studied for powersystems. Analytical methods offer a structured way to analyze systemsand decide possible system states, but here are instead stochastic MonteCarlo methods used, which give the advantage that arbitrary complicatedmodels can be applied. Another reason for using Monte Carlo methodsare that they do not require exponential distribution assumptions of theweather and parameter properties which are required when using for exampleMarkov processes [66]. If the variance of the resulting indexes is of interestanalytical methods are impossible to apply, while Monte Carlo methods arewell suitable. Especially when estimating restoration times, where thereare correlations between different input variables, Monte Carlo methods arevery useful. A disadvantage with the Monte Carlo approach is that it canbe time consuming, but with modern computers this is becoming less of aproblem.

The general modelling approach in this thesis can be described by theflow chart in figure 6.1. A scenario represents a certain weather situationwith parameters. A segment is a representation of power system networkcomponents. Denote the number of scenarios m and let the total number ofsegments be s.

47

48 CHAPTER 6. DESCRIPTION OF SIMULATION METHODS

START

Choose m

For scenario i, i = 1...mchoose weather parametersA

For segment j, j = 1...sB calculate load of scenario i

on segment j

For time t, t = 0...tstopij

Stochastic calculation of timeto possible breakdown

C

?¾

½

»

¼Result for scenario i, seg. jD

?¾

½

»

¼Result for scenario i,all segmentsE

?'

&

$

%

Calculate Tres for all mscenarios,all broken segments

F

Result for all m scenarios.Statistical analysis

Figure 6.1: Flow chart of the proposed method.

6.1. WEATHER PROPERTIES AND LOADS 49

∆t Size of time step(xstart, ystart) Start position of weather

Rwind Radius of wind part of the weatherRice Radius of ice part of the weatherΘ Direction of weatherVh Weather moving speedAW Maximal intensity of wind part of the weatherAI Maximal precipitation rate of circular part of the weather

Table 6.1: Weather and simulation parameters.

6.1 Weather properties and loads

The first and the improved weather models described in sections 3.2 and 3.3have severity levels for wind and ice that vary with time for each segment asthe weather passes the segment. The time when the whole severe weatheri has passed segment j, tstop

ij , is when the weather properties are defined asnormal, that is weather with severity level less than A

k for some k (A = AW

or A = AI).The different weather and simulation parameters in table 6.1 can be

chosen for each scenario, this is done in box A. The weather parameters canbe chosen from their distributions according to a method used in the casestudies in section 7.3. It is also possible to simulate the same weather withidentical parameters in all simulations as in the cases in sections 7.1 and 7.2.The impact on the network will vary between different scenarios because ofthe stochastic nature of the segment vulnerability model.

The center of the weather moves according to equations (6.1) and (6.2)as:

xc(t) = xstart + Vh∆t cos(Θ)t, (6.1)yc(t) = ystart + Vh∆t sin(Θ)t. (6.2)

If a time step represents 0.01 h =36 s, as in many of the case studies,and Vh is 25 m/s = 90 km/h, the length that the weather moves in a timestep is 90× 0.01 = 0.9 km or 900 m in 36 s.

In box B the wind load function, LW , and the ice load function, LI , foreach segment are calculated according to (3.3) and (3.5) for the first weathermodel. The improved weather model uses (3.8) and (3.14) combined with


the Simple model for ice accretion described in section 3.4.1 to estimate thewind and ice loads.

The number of segments can be increased easily but that will affect thesimulation time, a larger time step can be chosen when many segments arestudied to decrease simulation time.

6.2 Impact of load on segments

The ice and wind loads are used together with the segment vulnerabilitymodels described in section 4.2 to estimate which of the segments that breakdown and at which time, see box C in the flow chart.

Given the failure rates by the first (section 4.2.1) or the second segmentvulnerability model (section 4.2.2) the approximation in equation (6.3) isused to get the probability for breakdown in each segment at each timestep. The probability of breakdown of a segment at time t is given by:

P (breakdown at time t) ≈P (breakdown in interval [t, t + ∆t]) ≈ fT (t + ∆t

2 )∆t. (6.3)

This approximation is valid if ∆t → 0. For each simulation it is decidedwhether a break down occurs or not for each segment and time step, stoppingat the time when a breakdown occurs or when the weather has passed. If theprobability of breakdown at time t exceeds a random number from a uniformdistribution, U(0, 1), a breakdown is registered for the studied segment. Forexample if equation (6.3) gives P (breakdown of segment i at time t) = 0.6and the random number from U(0, 1) is 0.3 there is a breakdown, if therandom number is 0.9 there will be no breakdown according to this method.

The failure rate functions are chosen equal for all segments, it wouldthough be possible to have different failure rates for segments placed ondifferent grounds, for example in the forest or on an open field.

In box D the result of scenario i is given for segment j, the steps in boxB and C are done for all segments, this is concluded in box E.

6.3 Restoration time

When the broken segments are identified for each scenario the time forrestoration is estimated according to the restoration model described in

6.4. CORRELATED NON-GAUSSIAN DISTRIBUTED RANDOMNUMBERS 51

section 5.2, see box F . Since the restoration time for a particular segmentis correlated to restoration times for the other broken segments and thiscorrelation is stronger if two segments are close a method for identifying thedistance between the broken segments is needed. The approach used hereis to treat segments of the same line directly connected to each other as”close”. A distance matrix is introduced for this purpose. If segment i andj are defined as close, that is adjacent and on the same line, position i, jand j, i has value one, if i, j is not adjacent position i, j and j, i has valuezero. This symmetric distance matrix can be set up at the same time as thestudied network is defined. For example the network in figure 7.1 has thefollowing distance matrix, D:

D =

1 1 0 0 0 0 0 0 01 1 1 0 0 0 0 0 00 1 1 1 0 0 0 0 00 0 1 1 0 0 0 0 00 0 0 0 1 1 0 0 00 0 0 0 1 1 0 0 00 0 0 0 0 0 1 1 00 0 0 0 0 0 1 1 10 0 0 0 0 0 0 1 1

. (6.4)

The restoration time, Tres, is divided into five time intervals for everybroken segment, see section 5.2. The down time after a severe weather forthe broken segment is the result of this part of the simulation. The meansand variances of the restoration times can be used to draw conclusions aboutthe distribution of the restoration time.

Since the restoration model includes correlated non-Gaussian distributedrandom numbers a method for generating these numbers is required.

6.4 Correlated non-Gaussian distributed randomnumbers

It is not common to generate correlated random numbers of otherdistributions than the Gaussian distribution. The method described in thischapter is applicable for generating correlated random numbers of otherdistributions than the Gaussian distribution and is used in the case studies.A similar method was later found in [67] and in [68]. In [68] the described


?

?

?

?

-¾

-¾

W1

U1

G1

W2

U2

G2

ρW

ρG

Figure 6.2: ρG is the correlation between two Gaussian random numbersand ρW is the resulting correlation between Weibull random numbers.

method gives almost exactly the wanted correlation after a few iterations.The result of the method described in this thesis are also correlations closeto the wanted correlations (see chapter 6.4.3). The method is suitable forapplications where achieving correlated random numbers does not requiremathematical refinement. The correlations between all restoration times forall segments will be estimates and the accuracy of this estimated correlationis less than the differences between the achieved and the wanted covariances.

Correlated random numbers from, for example, Weibull distributionscan be generated in three steps starting with a set of independentN(0,1)-distributed random numbers, see figure 6.2. After introducingthe wanted correlation between the Gaussian distributed random numbers(equation (6.5)) these can be translated to correlated Weibull distributedrandom numbers. This is done by using the distribution functions of theGaussian distributions to get uniformly distributed numbers (equation (6.6))and thereafter use these to generate Weibull numbers (equation (6.7)), seealso figure 6.3.

6.4. CORRELATED NON-GAUSSIAN DISTRIBUTED RANDOMNUMBERS 53

−3 −2 −1 0 1 2 30

0.2

0.4

0.6

0.8

1

Correlation=0.8

−3 −2 −1 0 1 2 30

0.2

0.4

0.6

0.8

1

Weibull

Weibull

Correlation=?

Figure 6.3: From the Φ-function of correlated Gaussian distribution randomnumbers, to uniformly distributed random numbers, to Weibull distributedrandom numbers.


6.4.1 Correlated Gaussian distributed random numbers

Let Y be a random vector whose components Y1, Y2, . . . , Ym are correlated,N(0,1)-distributed random variables. Introducing X as in equation (6.5)gives E(X) = µ and Cov(X) = Λ [65].

X = Λ12 Y + µ. (6.5)

6.4.2 From Gaussian to uniform distribution

With µ = 0 X is a random vector whose components X1, X2, . . . , Xm aredependent, N(0,1)-distributed, random variables, with covariance matrix Λ.For every Gaussian random distribution it is possible to get a correspondinguniform distribution, U(0,1), through the Φ-function, the distributionfunction of the standardized Gaussian distribution as in equation (6.6).

Φ(x) =1√2π

∫ x

−∞e−

t2

2 dt = P (Φ ≤ x). (6.6)

For x < 0 values of Φ(x) can be obtained from Φ(−x) = 1 − Φ(x). EveryX ∈ X1, X2, . . . , Xm corresponds to a particular U ∈ U(0, 1) and theseU1, U2, . . . , Um become correlated.

6.4.3 From uniform to Weibull distribution

Weibull distributed random numbers, w with scale parameter, a, and shapeparameter, c, can be constructed from a uniformly distributed u by equation(6.7). More about the Weibull distribution is found in appendix A.

w = a(− log(u))1c (6.7)

The correlation between the generated Weibull numbers turn out to be veryclose to the correlation of the origin Gaussian distributions, see figure 6.4.The Gaussian-Gaussian correlation between two numbers, on the x-axis, isthe wanted correlation, and the Weibull-Weibull correlation between thesetwo numbers, on the y-axis, is the result of the method.

6.5 System vulnerability and outage time

When considering system vulnerability and possible outages due to severeweather for customers at a particular load point the redundancy of the

6.5. SYSTEM VULNERABILITY AND OUTAGE TIME 55

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Gaussian−Gaussian correlation

Wei

bull−

Wei

bull

corr

elat

ion

Figure 6.4: Gaussian-Gaussian versus Weibull-Weibull correlations from1000 simulations. The Weibull distribution has parameters a=1.6 and c=2.

network has to be considered. To estimate when a blackout occurs theminimum time for breakdowns leading to a blackout is required and this timeis achieved by the stochastic method used in this thesis. System vulnerabilityis not particulary studied in this thesis and the conclusion of this section isthat a breakdown of one of the segments on a line is enough for the wholeline to become disconnected. Therefore a longer line with more segmentsbecomes more vulnerable. However; the same line is not more vulnerable ifmore segments are considered, since the failure rates also consider the lengthof the segment. A simple application is shown in case study 3.2 in section7.3.6.

Chapter 7

Case Studies

In this chapter is the impact on components during severe weathers studied.The simulated weathers consist of both real weather data from a stormin Sweden 1999 and data from the two weather models described in thisthesis. When weather data is simulated is it the same weather parameters ineach scenario or the weather properties are chosen according to a stochasticmethod.

A transmission network typically covers large areas and meets differentweather conditions due to geographic differences. In the case studies insection 7.1 the studied network is a fictive network consisting of threetransmission lines. In the case studies in section 7.2 a part of the Swedishtransmission network is studied. The ice and wind loads are varied inthe simulations and knowledge of how an increased ice load influences thecritical gust wind is necessary to estimate the failure rates for differentloads. In section 7.3 a method for stochastic choices of different weatherparameters for each scenario is described. The method is used togetherwith the improved weather model on a fictive network.

7.1 The first weather model

In the following cases the first weather model is used and the results aremainly presented in [7] and [8]. The studied transmission network consistsof three lines and is exposed to an ice storm. Lengths of lines are 200 km,162 km and 100 km respectively and divided into segments of 50 km. A

57

58 CHAPTER 7. CASE STUDIES

0 100 200 3000

50

100

150

200

250

300

km

km

1

2

3

4 5

6

7 8

9 A

B

C

Figure 7.1: A scheme of the studied network.

scheme of the network is shown in figure 7.1; the connections and segmentsare numbered according to the figure.

The aim is to investigate the persistency to ice and wind of thedifferent connections and calculate restoration times after breakdowns. Thefailure rate function is an increasing function of wind force and ice loadingaccording to the first segment vulnerability model. In each scenario weatherparameters are chosen identically.

Weather data are chosen from ice storm events in Sweden and Canada.Choosing weather parameters corresponds to box A in the flow chart infigure 6.1. The maximum ice loading (AI) is 2 mm/h and maximum windspeed (AW ) is 36 m/s [19]. Since the first weather model is used here theice load is given by integration of the precipitation. The ice storm hits thenetwork with same moving speed and same angel as the wind, i.e. Vh and Θare the same for both wind and ice. Thus the ice and wind load as functionsof time are the same for a specific segment in all scenarios. However, theinfluence on the components will vary because of the stochastic nature of thesegment vulnerability model. The ice build up function is symmetric andthe wind load function is of the type in figure 3.3. The radius of the ice partof the weather (Rice) is smaller than the radius of wind part of the weather


0 100 200 3000

50

100

150

200

250

300

km

km

Figure 7.2: The ice part of the weather has a radius of 130 km and movesin direction of the arrow.

(Rwind). σx and σy in equations (3.3) and (3.5) are calculated according toequation (7.1), with Rwind = 200 km and Rice = 130 km.

σx = σy = 0.4R for R = Rwind and R = Rice (7.1)

Direction for both wind and ice are Θ = 0 which with xstart = 0 andystart = 140 gives

xcenter(t) = 90t,ycenter(t) = 140. (7.2)

according to equation (3.6). The weathers move 0.9 km in x-direction foreach time step of 0.01 and 0 km in y-direction, i.e Vh = 90 km/h. Thecenter is located in (90t, 140). Figure 7.2 shows the size and direction of theweather compared to the studied network. Canadian standard for only ice is50 mm [4]. This standard demands a large and long lasting severe weatherfor an extensive damage on the network. To make this example illustrativemore vulnerable components are studied using tables 4.1 and 4.2 for λW andλI . The total failure rate is calculated as in equation (4.5), with aW = 0.1and aI = 0.9.

The number of scenarios, m, is 1000. To achieve convergence insimulations m = 800 is enough, the result differs on the third decimal fromthe case when m = 1000.


scenario Broken segments and time Disconnected lines1 3 (5.6 h), 7 (2.8 h) A-B, A-C2 8 (5.3 h), 9 (2.2 h) A-C3 6 (4.6 h), 8 (3.4 h) B-C, A-C4 - -5 1 (1.8 h), 2 (3.0 h), 7 (4.3 h) A-B, A-C6 3 (2.5 h), 8 (4.4 h), 6 (2.9 h) A-B, A-C, B-C7 - -8 - -9 3 (4.9 h), 6 (4.4 h) A-B, B-C10 2 (2.7 h), 8 (4.3 h) A-B, A-C

Table 7.1: Broken segments and the time for breakdown in the first tenscenarios.

7.1.1 Case 1.1: Disconnected lines

To be able to draw conclusions of which customers that will be affected of asevere weather it is interesting to examine how many of the scenarios thatlead to a lost connection between A, B and C respectively (i.e. disconnectionof at least two lines). Following box B to E in the flow chart gives table7.1 which shows the result of the first ten scenarios. A breakdown at time5.6 means a breakdown 5.6 hours after the center of the weather has passed(0, 140) in figure 7.1. Connection A-B was disconnected in 51.2% of the1000 simulations, connection A-C in 56.0%, and connection B-C in 26.1%.It is clear that connection B-C is the most persistent to the simulated storm.Connection B-C is affected a shorter time period and is hit by almost parallelwind in larger extent than connection A-B and A-C. A long line with manysegments is more vulnerable, since breakdown in one of the segments isenough for a disconnection of the line.

The ice storm impact of the studied network can be divided into eightconnection/disconnection cases, listed in table 7.2, together with theirprobability. Note that if connections A-B and A-C are functioning there is aconnection between B and C, even though line B-C is disconnected. Thereis a connection between B and C in 16.3%+16.1%+20.0%+5.1%+21.5% =79.0% of the simulations. Connection between A-B in 64.9% and connectionbetween A-C in 64.0% of the simulations.


event probabilityno disconnection 16.3%

disconnection of A-B 16.1%disconnection of A-C 20.0%disconnection of B-C 5.1%

disconnection of A-B and A-C 21.5%disconnection of A-B and B-C 6.5%disconnection of A-C and B-C 7.4%

disconnection of all lines 7.1%

Table 7.2: Probabilities for different connections/disconnections.

7.1.2 Case 1.2: Restoration times

In this case are the restoration time after breakdowns on one of the scenariosin case study 1.1 calculated, this is box F in the flow chart in figure 6.1.In scenario 5 there were three broken segments, segment 1, 2 and 7, seetable 7.1. Equation (7.3) with Y ∈ (0, 1) gives Gaussian distributed timeintervals with µ = 0 and the 15 × 15 symmetric covariance matrix Λ fort1(i), t2(i) . . . t5(i), j = 1, . . . , 3. (j = 1 corresponds to segment 1, j = 2corresponds to segment 2 and j = 3 corresponds to segment 7).

t1(1)...

t5(1)t1(2)

...t5(3)

= Λ12 Y. (7.3)

Equations (6.6) and (6.7) give Weibull distributions of the times with almostthe same correlation as chosen in Λ. A constant is added to the Weibullnumbers assuming that there is a minimum time for each time interval ofthe restoration process that never is below this constant. The constantsused here are 1.5h, 1h, 1.5h, 1.5h, 4h for t1, t2, . . . , t5 respectively.

The matrix in equation (7.4) is the covariance matrix for this scenarioand is based on table 5.1, see section 5.2.5, and the locations of the brokensegments. Since segment 1 and 2 are close their time intervals are closely


Tres(1) Tres(2) Tres(7)16.7 16.9 14.814.9 14.7 12.6213.6 12.9 16.912.9 12.4 10.816.3 15.1 13.7

Table 7.3: The first five restorations times for scenario 5, 1000 scenarios.

Segment E[Tres] V ar[Tres]1 13.92 3.042 13.83 2.997 13.83 2.89

Table 7.4: Means and variances from 1000 simulations of restoration time.

correlated and not so closely correlated to segment 7.

Λ =

1 0 0.5 0.5 0 0.7 0 0.4 0.4 0 0.1 0 0.3 0.3 00 1 0 0 0 0 0.8 0 0 0 0 0 0 0 0

0.5 0 1 0.8 0 0.4 0 0.7 0.7 0 0.1 0 0.3 0.3 00.5 0 0.8 1 0 0.4 0 0.5 0.7 0 0.1 0 0.3 0.3 00 0 0 0 1 0 0 0 0 0.6 0 0 0 0 0

0.7 0 0.4 0.4 0 1 0 0.5 0.5 0 0.3 0 0.1 0.1 00 0.8 0 0 0 0 1 0 0 0.6 0 0 0 0 0

0.4 0 0.7 0.5 0 0.5 0 1 0.8 0 0.1 0 0.1 0.1 00.4 0 0.7 0.7 0 0.5 0 0.8 1 0 0.1 0 0.1 0.1 00 0 0 0 0.6 0 0 0 0 1 0 0 0 0 0

0.1 0 0.1 0.1 0 0.3 0 0.1 0.1 0 1 0 0.5 0.5 00 0 0 0 0 0 0 0 0 0 0 1 0 0 0

0.3 0 0.3 0.3 0 0.1 0 0.1 0.1 0 0.5 0 1 0.8 00.3 0 0.3 0.3 0 0.1 0 0.1 0.1 0 0.5 0 0.8 1 00 0 0 0 0 0 0 0 0 0 0 0 0 0 1

.

(7.4)The total restoration time for the three broken segment Tres(1), Tres(2)

and Tres(7) are calculated according to equation (5.1) using the generatedcorrelated Weibull distributions for deciding t1(i), t2(i),. . . , t5(i). Examplesof the results of restoration times for the studied scenario are collected intable (7.3).

The mean, E[Tres], and variance, V ar[Tres], of the three restorationtimes are estimated from 1000 simulations and collected in table 7.4.


Figure 7.3: The Nordic power system.

The mean correlation between Tres(1) and Tres(2) is 0.72, 0.20 betweenTres(1) and Tres(7) and 0.13 between Tres(2)and Tres(7). These results agreewith the assumption that the correlations are stronger between adjacentsegments.

7.2 The improved weather model

The whole Nordic power system is shown i figure 7.3. The Swedishtransmission network consists of eight 500-1000 km long 400 kV transmissionlines from the northern part of Sweden, where the largest part of the powergeneration is located, to the central and southern part where the main loadis located [14]. In the case studies two transmission lines in the south part


Hisingen

Borgvik

Kilanda

Figure 7.4: The studied Swedish transmission power lines.

His-Kil Kil-Bordominating type of line A2 or A2X A1 or A1X

length (number of towers) 29 km (91 towers) 177 km (513 towers)diameter of phase line 36.2 mm 31.7 mmdiameter of top line 20.1 mm 20.1 mm or 10.6 mm

direction northeast-southwest north-south

Table 7.5: Data for two Swedish transmission lines.

of Sweden, located according to figure 7.4, are studied. It is the 400 kVline between Hisingen and Kilanda (His-Kil) and the 400 kV line betweenKilanda and Borgvik (Kil-Bor). The lines are divided into segments ofapproximately 30 km. (His-Kil) consists of a phase line and a top line,while (Kil-Bor) consists of a phase line and two top lines. Table 7.5 containsdata for the lines. Both lines consist of different types of towers and thedominating type is used in the calculations, more than 65% of the towers areof this type. Since the His-Kil line goes in a southwest-northeast directionit is most vulnerable to winds that hits the line from the north-west andsouth-east. Kil-Bor is most vulnerable if exposed to winds from east orwest.

The His-Kil towers are designed for an ice load of 2.76 kg/m incombination with a gust wind of 28 m/s. The weakest Bor-Kil towers break


due to ice alone at an ice load of 27 mm [51]. In all simulations the accretedice on the power lines is supposed to have the density, δi = 900 kg/m3.

The weather properties are chosen with inspiration from data from astorm that hit Sweden in 1999. This was not an ice storm, however, if thetemperature would have been a few degrees lower it could have developedinto an ice storm [32]. In the following examples it is assumed that theprecipitation falls as freezing rain instead of rain or wet snow which wasthe case in 1999. To be able to compare the improved weather model to aweather situation that only occurs once in a period of one hundred or twohundred years the precipitation and wind data for the 1999 storm have beenmodified.

In the first two cases that will be described in this section, real weatherdata from the 1999 storm is used and the loads are compared to criticalloads according to the second segment vulnerability model.

7.2.1 Case 2.1: The 1999 storm on the His-Kil power line

In this case the weather data is used without any modification except thatthe precipitation is supposed to fall as freezing rain instead of wet snow andrain. The precipitation at the location of the segment on His-Kil is shownin figure 7.5. The totally accumulated precipitation is 26 mm. The gustwind and the perpendicular component of the gust is showed in figure 7.6.The wind direction is important for the thickness of the ice layer. After 20hours the wind has turned around, and is almost perpendicular to the linewhich results in a faster ice accretion. The ice layer builds almost as fastafter 20 hours as earlier although the precipitation intensity is only 25% ofthe earlier intensity.

The ice accretion on the phase and top line according to the Simplemodel is shown in figure 7.7. The ice accretion in kg/m is larger for thephase line than the top line since it has a larger radius and therefore alarger area for ice accretion. There is no difference in mm, this assumptionis also proposed in [33].

In figure 7.8 the loads on the His-Kil line are compared to the criticalloads. The loads from the 1999 storm are too small for causing anybreakdowns, even with the stochastic second vulnerability model.


0 10 20 30 40 500

2

4

6

8

t [hours from 00 am dec 3 1999]

Precipitation [mm/h]

Figure 7.5: The precipitation at the His-Kil power line.

0 10 20 30 40 500

5

10

15


[m/s]

Figure 7.6: Gust wind (solid) and its perpendicular component (solid withdots) during the 1999 storm at the His-Kil power line. The wind blows moreperpendicular to the line after about 20 hours.


0 10 20 30 40 500

0.5

1

1.5

2

2.5

0 10 20 30 40 500

5

10

15

20


Ice accretion [mm]


Ice accretion [kg/m]

Figure 7.7: Ice accretion according to the Simple model on the phase line(solid) and on the top line (dotted) of the His-Kil power line in mm andkg/m.

0 10 20 300

10

20

30

40

50

Ice thickness [mm]

Gust wind [m/s]

Critical load

Figure 7.8: Loads on the His-Kil power line compared to the critical loadsfor this line. The ice thickness increases with time and the wind/ice functioncan therefore also be seen as a function of wind that vary with time.


0 10 20 30 40 500

5

10

15

20

25

30


Ice accretion [mm] and [kg/m] (dotted)

Figure 7.9: Ice accretion on the phase line in mm (solid) and kg/m (dotted).

Case 2.2: 1999 storm on the His-Kil power line withincreased wind

The 1999 storm had its maximal wind gusts about 200 km south of thestudied line. What would have happened if these wind speeds where closerto the studied line? In this case the wind is increased by 100% compared tothe 1999 wind. The precipitation and wind direction are not changed. Thewind forces that hit the lines in this case agree with wind forces in the partof the 1999 storm which passed the most southern part of Sweden.

The ice accretion on the phase line according to the Simple model isshown in figure 7.9. In figure 7.10 the loads are compared to the loads thatare assumed to cause damage to the towers of the line. Both the wind loadand ice load are larger when the wind is increased since the ice accretionincreases with the wind. In figure 7.10 the wind and ice loads exceed thecritical loads for the line for some times. The conclusion is that it is possiblethat some towers would have broke down if this where the circumstancesduring the 1999 storm. It is not impossible that this storm can occur in


0 10 20 300

10

20

30

40

50

Ice thickness [mm]

Gust wind [m/s]

Critical load

Figure 7.10: Loads on the His-Kil power line during the 1999 storm with100% increase of wind compared to the critical loads.

Sweden with both increased precipitation and wind [32]. In a simulationin [29] the precipitation of the 1999-weather was increased by 70% and thewind was the same as in case 2.1. The ice accretion on the studied lineestimated by the Simple model then increased with 100%.

Case 2.3: A storm with the weather model

In this case the aim is to mimic the 1999 storm by the improved weathermodel. Since most low pressures in Sweden comes from the Atlantic andmove east or north-east this is the direction used in the following simulations,i.e Θ = 45◦. The speed Vh is 28 m/s. This value is chosen since a severeweather in the studied part of Sweden 1969 moved approximately withthat velocity [69]. A maximal wind gust, AW , of 38 m/s and a maximalprecipitation, AI , of 10 mm/h where used and the weather passed the area


seg. 1 seg. 2 seg. 3 seg. 4 seg. 51 - - 2.95 - -2 - - - - -3 - - - - -4 - - - - -5 - - - 3.26 2.726 - - - - -7 - - - - 2.738 - - - - -9 - - - - -10 - - - 2.79 -

Table 7.6: Times for breakdowns (hours) for 10 out of 1000 simulations.Segment 1 is located on His-Kil and segment 2-5 are located on Kil-Bor.

during 45 hours with ∆t = 0.01. These data are comparable to the realweather data from 1999 with 100% increased wind as in case 2.1. Rwind is600 km and Rice is 300 km.

In 35% of the 1000 simulations there was a breakdown in one or twosegments, see table 7.1. In figure 7.11 the wind and ice loads on all fivesegments are shown, segment 4 and 5 are the most exposed to the storm,this is also shown in table 7.6. When the weather has passed the wind iszero while the ice load remains, this explains the abrupt ends of the ice/windfunctions in figure 7.11.

Figure 7.12 shows the wind part of the 1999 storm in the studied areausing real weather data. The shapes of the different parts of the weathercan be compared with the shapes given by the weather model in the samearea in figures 7.13 and 7.14.


0 10 20 30 400

10

20

30

40

50

Ice thickness [mm]

Gust wind [m/s]

Figure 7.11: Loads on the five segments (load on segment 1 is most to theleft and load on segment 5 is most to the right) compared to critical load forBor-Kil, which is similar to the critical loads for His-Kil on which segment1 is placed.

Figure 7.12: The wind part of 1999 storm with real weather data whenpassing the studied lines. Red color represents the highest wind speed (22m/s mean wind) and blue the lowest.


Figure 7.13: Wind part from the weather model passing the studied lines.AW = 38 m/s.

Figure 7.14: Precipitation part from the weather model passing the studiedlines. AI = 10 mm/h.


m½¼

¾»

&%

'$

666II

¡¡

¡¡

¡¡

¡¡µ

¡¡µ

Figure 7.15: The weather moves parallel to the power line and the wind hitsthe line perpendicular to the line.

7.2.2 Case 2.4: Storm moving parallel to the Bor-Kil powerline

The aim of case 2.4 and 2.5 is to study the importance of the direction ofthe wind.

Let the storm move parallel to the power line with the center on theline. The wind becomes perpendicular to the power line, see figure 7.15.The power line is exposed to a lot of precipitation and the perpendicularcomponent of the wind force will be equal to the gust wind. A maximumprecipitation rate of 10 mm/h and a maximum gust wind of 38 m/s is used.In figure 7.16 the simulated loads are compared to the critical loads. Thereis a breakdown in segment 3 in 15% of the 1000 simulations.

7.2.3 Case 2.5: Storm moving perpendicular to the Bor-Kilpower line

To compare the result of the parallel simulation a simulation with the samestorm moving perpendicular to the power line is performed with the samemaximal precipitation and maximal gust wind. The wind force is nowparallel to the power line and the start position is changed. As a resultthe power line is exposed to a minimum of precipitation and wind force.The ice load is now only 25% of the load in the parallel simulation. In figure7.17 the loads are compared to the critical loads.

These simulations show the importance of the wind direction. Exactlythe same weather conditions are used in the two simulations with theexception of the direction and start position of the centers. In parallelsimulation the wind is perpendicular to the line at all times which leads toa larger perpendicular component and larger ice accretion. The loads arelarge enough for several breakdowns. In the perpendicular simulation the


0 10 20 30 400

10

20

30

40

50

Ice thickness [mm]

Gust wind [m/s]

Figure 7.16: Loads on segment 3 compared to the critical loads when theweather is moving parallel to the Bor-Kil line.

0 10 20 30 400

10

20

30

40

50

Ice thickness [mm]

Gust wind [m/s]

Figure 7.17: Loads on segment 3 compared to the critical loads when theweather is moving perpendicular to the Bor-Kil line. In this case the windthat hits the line reach its maximum after about half of the simulated time.

wind is blowing parallel to the line and the ice accretion is therefore muchsmaller. The loads are not close to causing any damage to the power line.

7.3. MODELLING DISTRIBUTIONS OF SEVERE WEATHERPARAMETERS 75

7.3 Modelling distributions of severe weatherparameters

To be able to make assumptions of the probability for large power outages inthe future due to severe weather it is important to estimate probabilities forpossible weathers. With the green-house effect in mind these probabilitiesare likely to change which may influence the dimensioning of futuretransmission components and/or organization of how to consider situationsthat may lead to a severe crisis. However, these data are difficult to find, ifeven possible to obtain.

To structure different types of weathers, different weather codes is usedin [26] for the considered weather situations. Two methods for modellingfailure rates for overhead distribution lines during the different weatherstates represented by the weather codes are also presented in. The first usesa Poisson model for failures to describe the probability of failures duringdifferent weather codes. The other method consists of a Bayesian networkwith nodes which represent wind speed, lightning and number of events andis used together with a conditional probability table to map the relationshipbetween the weathers and the failure rates. Wind gust speed is for exampleclassified into three discrete intervals [0, 15.6 m/s), [15.6 m/s, 20.1 m/s)and [20.1 m/s, Maximal wind speed).

Another approach is to study weather statistics for the studied area.However to achieve data from the most severe possible weathers the weatherstatistic for even a few hundred years is not enough, especially if the weatherconditions will become more extreme in the future. Instead well documentedweather situations can be modified based on estimations of frequency ofsituations with increased precipitation and wind, and estimations of theprobability for a change in the weather conditions that would have leadto the more severe scenario. In [29] several simulations based on wellknown Swedish weather events and modifications of these weather eventsare presented. The weather events are described and modified to becomemore severe and the probability for the circumstances is discussed.

A common way to handle probability for different weather states is to usereturn periods for different weather conditions, for example wind speeds. Itis then common to connect the weather states with a certain return period todifferent component failure rates. In [38] is the variation of wind speed withreturn periods of 5, 50 and 500 years discussed. The method developed here


does not directly connect a weather situations to a failure rate. Instead themethod connects a forecast of a severe weather to distributions of weatherparameters. This new method for randomly choosing weather parameterssuch as size, moving speed and direction of the possible weathers can beused to simulate different weather scenarios which in combination with theweather models in chapter 3 give the loads on components. Thereafter canthe loads be connected to different failure rates by a vulnerability model forcomponents. The method is based on distributions and probabilities thatare assumed to be valid for Swedish conditions. The parameters are easilychanged within the model for different weather regions.

7.3.1 Possible weather scenarios for Swedish conditions

The Swedish Meteorological and Hydrological Institute (SMHI) has provideddata from two storms in the southern parts of Sweden for the master thesis[29]. The first occurred in November 1995 and the second in December 1999.The data contains information on precipitation, the mean wind, the maximalgust wind and wind direction every third hour for different coordinates. Datais from weather stations and estimated between the weather stations. The1995 storm was a storm with a lot of wet snow precipitation. The windswere up to 20 m/s. During the 1999 storm the gust wind speeds reached32 m/s. The temperatures were higher than in the 1995 storm and mostof the precipitation fell as rain. Small changes in temperature, wind speed,and moisture content can dramatically alter the intensity and duration offreezing rain [32].

In case studies an increase of the data with a factor of 70 %, and insome cases 150% (see [29]) has been made. This is not unrealistic since theprecipitation rates and wind speeds from 1995 and 1999 were low for a severeweather. There has not been enough information concerning how probablethe simulated weathers are.

7.3.2 Main direction of weather

During the two most severe ice storms in southern Sweden the last hundredyears (1921 and 1968) the weather has come from north-east [19]. Both the1995 and the 1999 storm came from the west and south-west and most stormsin Sweden comes from that direction. There is not enough data available tosay if it is possible that ice storm weathers are more likely than other storms


Main direction ProbabilityThe most common direction 0.8

All other directions 0.2

Table 7.7: Probabilities for different main directions of the severe weather.

to come from north-east. This is not further investigated within this thesis,but the most probably direction is supposed to be from south-west.

To estimate distributions of possible directions it is important to considerthe possible behavior of weathers in the region. Each studied region has a fewdirections that are more likely than others, because severe weather originatesin a particular area and usually follow one or two prevailing directions,for example from the ocean and in over land [70]. To include the smallprobability that a severe weather originates from another direction than themost common ones the main direction of the weather is chosen, for exampleaccording to table 7.7.

The flow chart in figure 7.18 shows the stochastic method for choosingweather parameters for one possible specific scenario within the frame oftable 7.8. This flow chart is within the box marked with A in the overallmethod flow chart in figure 6.1. In each scenario the main direction is chosenby generation of a uniform random number between 0 and 1. If the randomnumber is smaller than the probability for the most common direction, the”most common direction” is chosen for this particular scenario, else is ”allother directions” chosen. This is box a in figure 7.18.

7.3.3 Weather code and direction

Severe weather is here defined as wind forces above 25 m/s or that icebuilds up on the lines, or a combination, these cases are listed in table7.8 and represented by a weather code for simplicity. The approach is toassume that a forecast of a coming adverse weather is available includingassumptions of the probabilities for the different scenarios listed in in table7.8. These probabilities are difficult to estimate and the probabilities intable 7.8 is one possibility.

When the probabilities for the different weather codes have beenestimated a weather code can be chosen according to the probabilitiesin table 7.8. Weather code 1 may have precipitation but not freezingprecipitation. In each scenario the weather code is chosen by generation of a


Weather Code Condition Probabilty1 No ice but wind > 25 m/s 0.852 Ice and wind ≤ 10 m/s 0.13 Ice and wind > 10 m/s 0.05

Table 7.8: Definition of weather codes and their probabilities.

uniform random number between 0 and 1. If the random number is smallerthan the probability for weather code 1, weather code 1 is chosen for thisparticular scenario. If the random number is smaller than the probabilityfor weather code 1 added to the probability for weather code 2 weather code2 is chosen and if the random number is larger than the sum of probabilitiesfor weather code 1 and 2 weather code 3 is chosen. This gives the wanteddistribution of weather codes for the scenarios. This is the first part of boxb in figure 7.18.

After choosing whether the main direction is the ”most common” or ”allother directions” there is a forecast of a weather that will be severe accordingto one of the definitions in table 7.8, and its main direction is known. Herea forecast is what usually is known by meteorologists a few days before asevere weather hits the region [12], for example that a severe weather iscoming and from which main direction.

Since it is not unusual for the wind direction to change during a storm[17] the distribution of Θ is chosen after the forecast with warnings of severeweather and where it comes from is given. This is the second part of boxb. Θ is independent of weather code, but depended on main direction andis here assumed uniformly distributed according to for example table 7.9.Other distributions of the direction are possible, for example the Gaussiandistribution. The start position, (xstart, ystart), for the center of the weatherdepends on main direction. In the case studies is (xstart, ystart) = (50, 200)for the most common direction and (xstart, ystart) = (200, 200) for all otherdirections.

Most low pressures that reach Sweden are developed on the NorthAtlantic and moves towards the north-east in over Scandinavia. Thisdirection is used as the main direction in the simulations for Sweden. Θis assumed to be the same for the wind and ice parts of the weather in thecase studies.


Continue to calculate impacton all segments, time to possiblebreakdown and restoration timefor each broken segment.

¾

½

»

¼Properties of scenario i.

Choose weather codeaccording to table 7.8and directionaccording to table 7.9.

b.)

Choose parametersfor the choosenweather codeaccording to table 7.10.

c.)

Choose main directionfor scenario iaccording to table 7.7.

a.) -

A forecast ofadverse weatherand its maindirection.

@@

@R

Choose weather properties for the adverse weather scenario i.

?

?

?

@@

@R

Figure 7.18: Flow chart of method for choosing weather properties for MonteCarlo Simulations.


Main direction ΘThe most common direction for the region. U(−45◦, 45◦)

All other directions U(45◦,−135◦)

Table 7.9: The distribution of directions for low pressures in Sweden,given the main direction. The most common direction for Sweden issouth-west-west or −45◦ ≤ Θ < 45◦.

Weather code Rwind Rice

1 U(300,900) 02 U(400,800) U(50,500)3 U(300,900) U(50,500)

Weather code AW AI

1 N(35,15) Min 25, Max 60 02 N(5,5) Min 0, Max 10 N(8,5) Min 0, Max 153 N(38,20) Min 10, Max 60 N(8,5) Min 0, Max 15

Weather code Vh

1,2,3 N(16,15) Min 10, Max 50

Table 7.10: Distribution of weather parameters given weather code.

7.3.4 Size, intensity and moving speed

The other scenario parameters needed in the improved weather model, seesection 3.3, are: maximal wind force, AW ; radius of the wind area, Rwind;the radius of the circular part of the precipitation part of the weather Rice;the maximal intensity in the circular part AI ; the weather moving speed,Vh. The maximal intensity in the front zone AIfront

is always smaller thanAI by the weather model or max(h(x, y)) < max(g(x, y)), g and h are fromequation (3.14). For example AIfront

= 0.9AI is used in [29]. Choosing theseparameters corresponds to box c in figure 7.18.

Rwind and Rice are assumed uniformly distributed within a suitable rangefor each weather code, see table 7.10. The wind part and the ice part ofsevere weather may have different sizes and the ice part of the weather istypically smaller than the wind weather.

The distribution of the intensity of precipitation and the maximal windspeed have to be estimated for each weather code. This is done here forSwedish conditions by studying different return periods of wind speeds and


Wind gust speed Return period52 m/s 3 years68 m/s 50 years74 m/s 150 years80 m/s 500 years

Table 7.11: Return periods for different wind gust speeds.

ice load and the well documented weathers and their manipulations of theseweathers described in [29], see section 7.3.1. When Gaussian distributionsare used a minimum and a maximum value are specified to avoid for examplewind speeds below zero. In table 7.11 the return period of different maximalwind speeds is shown when conversion factors from [11] have been used. If awind gust speed and its return period is known there are conversion factorsto achieve the wind gust speeds of the other return periods in table 7.11. InSweden meteorological data have been collected during more than 200 years[12] and the largest wind gust measured in Sweden is 81 m/s. Assumingthat 80 m/s has a return period of 500 years in the studied area the windspeeds for the other return periods estimated by the conversion factors arelisted in [11]. Comparing the wind speed intervals in table 2.1 with the windspeeds in table 7.11 it is important to note that the winds in table 2.1 arethe mean wind speed during 10 minutes while the winds in table 7.11 arethe gust wind speed.

7.3.5 Case 3.1: Distributions of weather parameters usingthe method for generating different scenarios

Table 7.10 shows which distributions are used in this case study and thesecan easily be changed to be suitable for any other studied region. For 1000simulations the maximal wind speed is distributed according to figure 7.19using the method with weather codes described in the flow chart 7.18.

Weather code 2 and 3 include precipitation rates and the precipitation isin this case study supposed to fall as freezing rain. For weather code 1 thereis no icing and the maximal precipitation rate is zero. Other precipitationthan freezing precipitation is not considered in the weather model or thevulnerability model for components. The precipitation rates in table 7.10give precipitation and ice load distributed as in figure 7.20 for the 138 out


0 10 20 30 40 50 600

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Wind speed [m/s]

Den

sity

Figure 7.19: The distribution of maximal wind gust speed, AW .

of 1000 simulations that gave non-zero freezing precipitation and the 135 ofthese cases that gave non-zero icing on segment 4 in the network studiedin case 3.2. The distribution of the ice load requires a specified segmentsince ice formation only occurs on an object or surface. The distribution ofthe maximal wind gust and the maximal intensity of precipitation can beviewed for any coordinate of the studied area.

The moving speed of the wind and precipitation part of the weather areassumed to be the same in the case studies. The moving speed is independentof weather code since it is not connected to the strength of the weather, seetable 7.10.

7.3.6 Case 3.2: Power system reliably using the method forgenerating different scenarios

1000 simulations of different weather situations are performed on thenetwork in figure 7.21. A longer line is more vulnerable than a shorterline that consists of fewer segments. To be able to compare outage timesfor load points D1 and D2 the network is almost symmetric around thegeneration point.

The broken segments are identified and the times for breakdown areestimated as well as restoration times for these segments. To be able toestimate the system reliability, the distribution of outage times for the loadpoints D1 and D2 can be estimated. However the probabilities for outagesin the load points are low; 0.2% of the scenarios lead to an outage at loadpoint D1 and 0.3% lead to an outage at load point D2. There is an outage


0 2 4 6 8 10 12 14 160

0.02

0.04

0.06

0.08

0.1

Precipitation rate [mm/h]

Den

sity

(a)

0 50 100 1500

0.01

0.02

0.03

0.04

0.05

0.06

Ice thickness [mm]

Den

sity

(b)

Figure 7.20: The distribution of maximal precipitation, AI , (7.20(a)) andthe following ice load (7.20(b)) for a particular segment.


0 50 100 150 200 25050

100

150

200

250

300

km

km

1

2

3

4

G

6

5

7

8

D291011D1

Figure 7.21: The studied network with generation point (G) and load pointsD1 and D2.

in D1 if one or more of segments 1 to 4 and one or more of the segments 5to 11 are broken. There is an outage in D2 if one or more of segments 5 to8 and one or more of the segments 1-4, 9-11 are broken. To make it morelikely that an outage occurs the line between D1 and D2 can be removed,now a breakdown in any of segment 1 to 4 is enough for breakdown in D1and a breakdown in any of segment 5 to 8 is enough for breakdown in D2.

Chapter 8

Conclusions and FutureWork

Conclusions

The main contributions of the thesis are; a technique for modelling adverseweather, a model for the impact on towers and line segments and a model forrestoration times of broken components. The thesis also includes simulationmethods for applications of the models. The model for restoration timesassumes correlations between time intervals for different restoration phasesand between different failed line segments. Since the time intervals for thedifferent restoration phases are assumed to be non-Gaussian distributeda method for generation of non-Gaussian distributed correlated randomnumbers is presented in the thesis. Except which of the components thatbroke down and the restoration time for these components the method alsoprovides results concerning at which time the breakdown occurred and howextensive the loads were.

The many case studies performed illustrate the failure risk for particularline segments and the time of occurrence for possible breakdowns. The timeof occurrence for individual breakdowns is interesting when for exampleestimating the restoration times. Both fictive power networks and a part ofthe Swedish transmission network are studied. Different weather situationsare simulated; both using data from real weather situations that haveoccurred in Sweden, and weather scenarios generated by the weather models.The impact of the wind direction and the impact of the wind on the ice

85

86 CHAPTER 8. CONCLUSIONS AND FUTURE WORK

accretion are also investigated in the case studies. The Simple model for iceaccretion is used. Graphs of the critical conditions for ice thickness combinedwith wind gust, which is a part of the model for the impact on towers andline segments, are presented.

During the work with this thesis it has often been difficult to findsimulation data, both regarding the stochastic nature of the componentsand regarding distributions of possible weather parameters. Discussionswith experienced people from utilities have instead been very valuable andthe references are therefore partly of this kind. Weather data is especiallydifficult to obtain for future weathers. Many scientists agree on that extremeadverse weather is both becoming more frequent and more violent [44] andfuture decision makers may have to reconsider the design of the powernetworks to maintain the same reliability as today and/or organization ofhow to consider situations that may lead to a severe crisis..

Future work

System reliability or system vulnerability, i.e. the probability for outages indifferent load points, is studied briefly in this thesis for a simple meshedfictive power network. For a meshed network the probability of an outageis low and most of the simulated weathers will not lead to an outage for aparticular load point, using the methods described in this thesis. Luckilythis agrees with reality. To be able to draw conclusions about systemreliability for larger systems, more efficient methods for generating thescenarios are needed. This can be done with variance reduction techniques,which in future work will be applied to reduce the variance of the reliabilityparameters and thereby the number of simulations needed for convergence.Also, improved model efficiency is necessary to decrease the computationtimes.

By the end of this project the aim is to answer which use of preparednessresources that is the most efficient. To be able to answer this question,finding the most important components or parts of the network for avoidingblackouts is crucial. Both the costs for decreasing the risk for failures andthe costs for limiting the consequences of a scenario are necessary to studyin order to find the optimal improvements. Costs for blackouts will alsobe future work and will lead to estimations of costs for severe crises. Tocompare the costs for restoring the system the possible measures to reduce

87

consequences of severe crises should be described. The model for risks foroutages should be connected to customer costs in future work.

In combination with the above suggestions of future work improvementsof the existing models described in the licentiate thesis will be needed. Animproved and extended version of the model for restoration described inchapter 5 can for example include a detailed model for availability of staffand the value of a forecast. In the optimization of resource location the sizeand location of reserve towers should also be clear. More realistic estimationsfor restoration times should be achieved by for example discussions with theutilities.

Improved models for segment vulnerability and a sensitivity analysisof impact due to different weather parameters would be interesting.The weather model of the wind can easily be extended to consider thecross-section area of the ice, a larger area gives a larger wind force.

The master thesis in cooperation with Svenska Kraftnat [29] was usefulfor the project, especially for simulation of the real weathers in the casestudies and the improved weather model. This cooperation can be expandedand a study of distribution networks would improve the project. One ofthe most challenging part of the project is to consider all the correlationsduring a severe crisis. Mathematical considerations for improving the toolsand understanding of these problems will be useful for future work.

It would also be interesting for future work to study other scenarios thanice storms that may lead to a severe crisis. The restoration model can bedeveloped to be useful for other situation than outages due to ice storms,for example terrorist attacks.

Appendix A

The Weibull Distribution

The Weibull distribution is one of the most commonly used distributions inreliability analysis because of the many shapes it attains for various values ofthe parameters. Another advantage is that it only assumes positive numberwhich often is wanted when studying physical quantities. The 2-parameterWeibull probability density function (pdf) is given by

f(t) =ctc−1

ace−( t

a)c

,

where f(t) ≥ 0, t ≥ 0, c ≥ 0, a ≥ 0. (A.1)

a is a scale parameter and c is a shape or slope parameter. An example withc = 4 and a = 37.6 gives the distribution in figure A.1.

0 20 40 60 80 1000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Figure A.1: A Weibull distribution with c = 4 and a = 37.6.

89

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[70] C.D. Ahrens. Meteorology today: An introduction to weather, climateand the environment. Pacific Grove Brooks/Cole, 2000.

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