Ice Storm Modelling in Transmission System
Reliability Calculations
ELIN BROSTR ¨
OM
Licentiate Thesis
Royal Institute of Technology
School of Electrical Engineering
Electric Power Systems
Stockholm, 2007
TRITA-EE-2007:022 ISSN-1650-674x
ISBN 978-91-7178-690-6
School of Electrical Engineering Electric Power Systems
Royal Institute of Technology S-100 44 Stockholm
SWEDEN
Akademisk avhandling som med tillst˚and av Kungl Tekniska h¨ogskolan framl¨ag-ges till offentlig granskning f¨or avl¨aggande av teknologie licentiatexamen torsdagen den 14 juni 2007 kl 10.00 i sal H1, Teknikringen 33, Kungl Tekniska H¨ogskolan, Stockholm.
c
° Elin Brostr¨om, June 2007
Abstract
In this thesis a new technique of modelling non-dimensioning severe weather for power system reliability calculations is developed. The model is suitable for both transmission and distribution networks and is based on geographically moving winds and ice storms. The modelled weather has severity levels that vary with time and change continuously as the weather passes a region. Different weather situations are represented with scenarios. For each scenario the weather parameters, such as size, strength, speed and direction can vary. A stochastic method for choosing parameters is also described. This method is based on probabilities for different weather situations for Swedish conditions.
A stochastic vulnerability model for the components is required for each scenario to connect the risk of failure to the weather situation. The model developed here connects the direct wind impact with the impact from the ice storm which is given by an ice accretion model. It is assumed that the probability for an individual segment to break down due to the impact of a given weather depends on load functions for wind and ice together with the vulnerability model for components. It is possible to estimate the outage risk as well as the time difference between mean times to failure in different lines. Monte Carlo methods, where many scenarios are simulated, are used in the case studies. Studies of the system vulnerability is a future work of this project but in one small case study the probability for outage in a load point is estimated.
To be able to estimate repair times after a severe weather the reliability calculations are extended with a restoration model which gives distributions of down times for the broken components. The situations after the ice storm that are studied are so severe that gathering of all or almost all possible restoration resources is required to restore the system. Restoration times for different components are not assumed to be independent; on the
iv
other hand they are assumed to be strongly correlated. The restoration process is dependent on staff situation, distance between location of spare parts and the breakdown, forecasts, availability of roads and distance to other breakdowns; this is included in the model. A method for simulation of non-Gaussian correlated random numbers is developed to include the correlations during the restoration process. The case studies show the impact of the different weather situations on the components and the following restoration times for the broken components.
Sammanfattning
Denna avhandling presenterar ett nytt s¨att att modellera ov¨ader f¨or tillf¨orlitlighetsber¨akningar p˚a kraftn¨at. Modellen ¨ar applicerbar b˚ade p˚a transmissionsn¨at och p˚a distributionsn¨at. Ov¨adret beskrivs med en modell d¨ar intensitetsniv˚aerna f¨or vind och nederb¨ord varierar kontinuerligt med tiden, allt eftersom ov¨adret passerar. Olika v¨adersituationer representeras av scenarier. F¨or varje scenario kan v¨aderparametrarna, s˚asom ov¨adrets omfattning, intensitet, framfartshastighet och riktning variera. Avhandlingen beskriver ocks˚a en stokastisk metod f¨or att v¨alja v¨aderparametrar. Metoden bygger p˚a uppskattningar av sannolikheter f¨or olika v¨adersituationer under svenska f¨orh˚allanden.
F¨or att best¨amma sannolikheten f¨or haveri, givet en v¨adersituation, har en stokastisk k¨anslighetsmodell f¨or komponenterna i kraftn¨atet tagits fram. K¨anslighetsmodellen tar h¨ansyn b˚ade till den direkta p˚averkan av vinden och till p˚averkan av isbildning p˚a komponenterna. Sannolikheten f¨or att ett enskilt segment av kraftn¨atet ska haverera under p˚averkan av ett givet v¨ader antas kunna beskrivas av en vind- och islastfunktion samt k¨anslighetsmodellen f¨or komponenterna. Det ¨ar m¨ojligt att ber¨akna avbrottsrisken, liksom tidsskillnaden mellan f¨orv¨antad tid till haveri f¨or olika ledningar. Monte Carlo-metoder anv¨ands genom att m˚anga scenarier simuleras i numeriska exempel. I avhandlingen studeras tillf¨orlitligheten f¨or ett enklare system, i projektets f¨orl¨angning ska dock ¨aven tillf¨orlitligheten i st¨orre slingade kraftn¨at betraktas.
Reparationstider f¨or komponenter som havererat, pga. vind och is, slumpas fram fr˚an en sannolikhetsf¨ordelning. Endast konsekvenser av allvarliga ov¨ader studeras, d˚a en stor del av de tillg¨angliga reparat¨orerna samt reparationsresurserna kr¨avs f¨or att ˚aterst¨alla systemet. Reparationstiden f¨or olika komponenter modelleras att vara beroende, dvs. deras korrelation antas ha en stor betydelse. Faktorer som antas
vi
p˚averka reparationstiden ¨ar antalet tillg¨angliga reparat¨orer, avst˚andet mellan haveriet och reservlager f¨or komponenter, eventuell v¨aderprognos, framkomlighet p˚a v¨agar och avst˚and till ¨ovriga haverier. F¨or att modellera korrelationen mellan reparationstider har en metod f¨or att simulera icke-normalf¨ordelade korrelerade slumptal utvecklats.
Numeriska exempel visar hur olika v¨adersituationer p˚averkar komponenterna i systemet, samt vilken tid det tar att ˚aterst¨alla de som har havererat.
Acknowledgment
First of all I would like to thank my supervisor Professor Lennart S¨oder for introducing me to this interesting topic and for his valuable contributions and encouragement.
I would like to thank the Swedish Emergency Management Agency and the Swedish system operator Svenska Kraftn¨at for their financial support. A special thank goes to Lillemor Carlshem at Svenska Kraftn¨at and to Eva Sundin, J¨orgen Martinsson and Roger Jansson at Vattenfall Power consultants for their support and for sharing their knowledge. A great thank goes to Jesper Ahlberg for his valuable contributions to the weather model. Many thanks go to my colleges and friends and especially to my roommate Elin Lindgren for being a dear friend and my own computer support and Karin Alvehag, also a dear friend, for her support during the work with this thesis.
Finally I would like to thank Joakim and Eskil for all your love.
Contents
Contents ix
List of Figures xiii
List of Tables xvii
List of Symbols xix
1 Introduction 1
1.1 Background . . . 1
1.2 Aim . . . 3
1.3 Main contributions . . . 3
1.4 The outline of the thesis . . . 5
1.5 List of publications . . . 6
2 About Ice Storm Weathers 7 2.1 Wind and freezing rain . . . 7
2.2 Ice storms in the world . . . 9
3 Modelling of Ice Storm Weather 13 3.1 Weather models . . . 13
3.2 The first weather model . . . 14
3.2.1 Wind load . . . 15
3.2.2 Ice load . . . 17
3.2.3 Size of weather and moving speed . . . 19
3.3 The improved weather model . . . 20
3.3.1 Wind load . . . 20
3.3.2 Precipitation . . . 22 ix
x CONTENTS
3.4 Modelling ice accretion . . . 24
3.4.1 The Simple model . . . 26
4 Modelling Component Vulnerability 29 4.1 Reliability of transmission lines . . . 29
4.1.1 Galloping . . . 32
4.2 Segment vulnerability models . . . 32
4.2.1 The first segment vulnerability model . . . 35
4.2.2 The second segment vulnerability model . . . 36
5 Modelling Restoration 39 5.1 Restoration models . . . 39
5.2 The restoration time model . . . 41
5.2.1 Localization of breakdown . . . 44
5.2.2 Identification of fault . . . 44
5.2.3 Localization and transport of spare parts and staff . . 44
5.2.4 Repair time . . . 45
5.2.5 The covariance matrix . . . 45
6 Description of Simulation Methods 47 6.1 Weather properties and loads . . . 49
6.2 Impact of load on segments . . . 50
6.3 Restoration time . . . 50
6.4 Correlated non-Gaussian distributed random numbers . . . . 51
6.4.1 Correlated Gaussian distributed random numbers . . . 54
6.4.2 From Gaussian to uniform distribution . . . 54
6.4.3 From uniform to Weibull distribution . . . 54
6.5 System vulnerability and outage time . . . 54
7 Case Studies 57 7.1 The first weather model . . . 57
7.1.1 Case 1.1: Disconnected lines . . . 60
7.1.2 Case 1.2: Restoration times . . . 61
7.2 The improved weather model . . . 63
7.2.1 Case 2.1: The 1999 storm on the His-Kil power line . 65 7.2.2 Case 2.4: Storm moving parallel to the Bor-Kil power line . . . 73
xi 7.2.3 Case 2.5: Storm moving perpendicular to the Bor-Kil
power line . . . 73 7.3 Modelling distributions of severe weather parameters . . . 75 7.3.1 Possible weather scenarios for Swedish conditions . . . 76 7.3.2 Main direction of weather . . . 76 7.3.3 Weather code and direction . . . 77 7.3.4 Size, intensity and moving speed . . . 80 7.3.5 Case 3.1: Distributions of weather parameters using
the method for generating different scenarios . . . 81 7.3.6 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
List of Figures
2.1 The warm and cold front in the different stages of a low pressure. 8 2.2 An ice storm can cause extensive damage. . . 10 2.3 Ice storm impact on Canadian towers. . . 10 3.1 The circular shape of the weather and its severity levels that
decrease from the center. . . 15 3.2 The angle β. . . 16 3.3 A wind load function for a particular segment or a cross-section
of wind. . . 18 3.4 Ice build up function for a particular segment or a cross-section
of ice. . . 18 3.5 Ice load function for a particular segment. . . 19 3.6 Wind blows anti-clockwise around a low pressure on the north
hemisphere. . . 20 3.7 The shape of the wind part of the weather model. The color
scale represent the intensities of the wind which are largest 300 km south-west of the center (AW = 38 m/s). The center is at
this moment located at (1000,1000). . . 21 3.8 The shape of the precipitation part of the weather. The intensity
is decreasing from (xc, yc) and AI = 10 mm/h. . . 23
3.9 The precipitation part of 1999 storm as it passes the lines in the case studies. The darkest red color corresponds to the maximum precipitation (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
xiv LIST OF FIGURES
5.2 The restoration time for segment i, Tres(i), is defined as the time
from the breakdown to the time when the segment is functioning
again, assuming notification time to be zero. . . 42
6.1 Flow chart of the proposed method. . . 48
6.2 ρGis the correlation between two Gaussian random numbers and ρW is the resulting correlation between Weibull random numbers. 52 6.3 From the Φ-function of correlated Gaussian distribution random numbers, to uniformly distributed random numbers, to Weibull distributed random numbers. . . 53
6.4 Gaussian-Gaussian versus Weibull-Weibull correlations from 1000 simulations. The Weibull distribution has parameters a=1.6 and c=2. . . 55
7.1 A scheme of the studied network. . . 58
7.2 The ice part of the weather has a radius of 130 km and moves in direction of the arrow. . . 59
7.3 The Nordic power system. . . 63
7.4 The studied Swedish transmission power lines. . . 64
7.5 The precipitation at the His-Kil power line. . . 66
7.6 Gust wind (solid) and its perpendicular component (solid with dots) during the 1999 storm at the His-Kil power line. The wind blows 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 in mm and kg/m. . . 67
7.8 Loads on the His-Kil power line compared to the critical loads for this line. The ice thickness increases with time and the wind/ice function can therefore also be seen as a function of wind that vary with time. . . 67
7.9 Ice accretion on the phase line in mm (solid) and kg/m (dotted). 68 7.10 Loads on the His-Kil power line during the 1999 storm with 100% increase of wind compared to the critical loads. . . 69
7.11 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 critical load for Bor-Kil, which is similar to the critical loads for His-Kil on which segment 1 is placed. . . 71
xv 7.12 The wind part of 1999 storm with real weather data when passing
the 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 studied lines. AI = 10 mm/h. . . 72 7.15 The weather moves parallel to the power line and the wind hits
the line perpendicular to the line. . . 73 7.16 Loads on segment 3 compared to the critical loads when the
weather is moving parallel to the Bor-Kil line. . . 74 7.17 Loads on segment 3 compared to the critical loads when the
weather is moving perpendicular to the Bor-Kil line. In this case the wind that hits the line reach its maximum after about half of the simulated time. . . 74 7.18 Flow chart of method for choosing weather properties for Monte
Carlo Simulations. . . 79 7.19 The distribution of maximal wind gust speed, AW. . . 82
7.20 The distribution of maximal precipitation, AI, (7.20(a)) and the following ice load (7.20(b)) for a particular segment. . . 83 7.21 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 . . . 35
4.2 λI as a function of ice load and design load. . . 36
4.3 Failure rates for the different areas of figure 4.1. . . 37
5.1 Correlations for adjacent and not adjacent segments, segment i and 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 ten scenarios. . . 60
7.2 Probabilities for different connections/disconnections. . . 61
7.3 The first five restorations times for scenario 5, 1000 scenarios. . . 62
7.4 Means and variances from 1000 simulations of restoration time. . 62
7.5 Data for two Swedish transmission lines. . . 64
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. . . 70
7.7 Probabilities for different main directions of the severe weather. . 77
7.8 Definition of weather codes and their probabilities. . . 78
7.9 The distribution of directions for low pressures in Sweden, given the main direction. The most common direction for Sweden is south-west-west or −45◦≤ Θ < 45◦. . . 80
7.10 Distribution of weather parameters given weather code. . . 80
7.11 Return periods for different wind gust speeds. . . 81
List of Symbols
V (x, y, t) Wind function in cartesian coordinates
W (r, Θ, t) Wind function in polar coordinates
Vh 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 wind
LI Load function for ice
β The angle by which the wind force hits a line [rad] Θ Direction of the weather [rad]
wβ Wind factor for perpendicular component of wind
Rwind 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 weather
h(x, y, t) Function for front zone part of precipitation part of the weather
AW Maximal wind gust speed [m/s]
AI Maximal precipitation rate of circular part of the weather [mm/h]
AIf ront Maximal precipitation rate of front zone part of the weather [mm/h] P Precipitation rate [mm/h]
xx LIST OF SYMBOLS
∆R(x, y) Increase of ice thickness [mm/h] ∆t The length of a time step
xstart, ystart Start position of the center of the weather
(xi, yi) Coordinates for segment i
vi Angel to x-axis for segment i
λW 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 matrix
m Number of scenarios
n Number of lines
s Number of segments
Chapter 1
Introduction
1.1
Background
The Swedish power network consists of power plants, a transmission network with nationwide coverage and a distribution network. The main part of the power production in Sweden comes from hydropower and nuclear power. The hydropower is generated in the northern part of Sweden and most of the consumers are located in the southern part. This implies high demands on transmission capacity from north to south, since an extensive interruption of transmission between the northern and the southern part of Sweden can imply difficulties in maintaining power supplies in the southern part.
The technical infrastructure is of crucial importance for a modern society and electric power supplies are of particular importance. Society’s dependence on electrical energy, communications and information technology is increasing and power supply is also a basic condition for continued economic growth and national security. At the same time users, both industry and households, have an overconfidence in technical infrastructure functionality and little preparedness for outages in the power network [1].
A relative new concept used in Swedish politics is severe crises which can be defined as
”A severe crisis is not an individual event, for example an accident, a sabotage etc, but a state that may occur when one or more events develop or escalate to include many parts of society. Severe crises can be considered to constitute of different kinds
2 CHAPTER 1. INTRODUCTION
of extreme situations with low probability that are separated i question. The state is of such extent that it leads to severe interruptions of important functions and requires efforts from several different authorities and coordination of organs to handle the situation and limit the consequences.”
The new concept replaces the former full preparedness which was used in case of war [2].
One example of a severe crisis in the power system is extreme weather conditions that result in large-scale power failures and extensive damages that require gathering all possible repair resources to restore the system. Extremely severe weather is generally very unusual, however the consequences can be serious since the failure rates of components, such as overhead transmissions lines, increase sharply.
An interruption in the power system can have many causes: adverse weather, technical faults, operational problems, vandalism. Comparing the primary causes of outages; adverse weather outages are accounting for approximately 33% of all outages. One scenario that may lead to difficulties in maintaining power supplies for a long time is ice storms. An ice storm is an extreme situation, which occurs very infrequently in most parts of the world but causes extensive damage when it does; freezing rain coats everything in ice, often in combination with heavy wind. The power network components break down because of the heavy ice and wind and large areas can be affected.
The ice storm that hit eastern Canada and north-eastern United States in January 1998 is considered to be the worst in modern time in Canada; it caused a crisis where about 1.5 million households where without electricity and the system was not completely restored until October 1998 [3] [4]. Another recent example is the ice storm that hit Germany in November 2005. More than 70 transmission towers were broken and 200 000 people were affected by the blackout [5].
There was an ice storm in Sweden 1921; since the society is much more dependent on the infrastructure now the consequences if this storm would have happen today are not comparable. A summary of some of the ice storms that have occurred during the last 100 years around the world can be found in section 2.2.
Another example that may lead to a severe crisis is vandalism, where an organized 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 severe
crises in the boundary between market and essential infrastructure, in
which this thesis is one part, is to study how to handle reliability in the power system under severe crises, which are extreme situations with low probability. The power system as an important infrastructure is central within the project and the fact that the dominating part of the system is financed by the market.
The different phases of the project are the following:
A Descriptions of scenarios that may lead to severe crises. For studies a fictive network similar to the Swedish system can be used.
B Descriptions of possible measures to reduce consequences of severe crises.
C How can costs for severe crises be estimated? Many conditions will be taken 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 normally do not cause as large damages as ice storms. The network is dimensioned for lightnings and heavy wind but not for long lasting ice storms. In this thesis severe weather and ice storms in particular is the only scenario that is considered.
1.3
Main contributions
In order to mitigate severe consequences of future ice storms in an efficient way it is essential to be able to estimate the consequences based on assumptions of the technical system and the severity of possible storms.
Although many papers consider reliability and repair times there are very few that consider both and none of the papers that will be described in chapter 3.1, 4.1 or 5.1 considers the time dependent risk level on lines when
4 CHAPTER 1. INTRODUCTION
a severe weather passes a region or the correlation between restoration times for different broken components.
Based on knowledge of how a severe weather is created and how it behaves during its lifetime a new technique of modelling adverse weather such as ice storms is developed in this thesis. The model uses one function for the wind part of the weather and another function for the precipitation part and is based on geographically moving winds and precipitation. The model is suitable for both transmission and distribution networks. A known ice accretion model is used for estimating the load due to ice.
The approach is to estimate the reliability of components when exposed to severe weather conditions with a Monte Carlo technique where each scenario represents a certain weather situation. For different scenarios the stochastic weather parameters, such as size, strength, speed and direction can vary. For each scenario a vulnerability model for the components is also required, where the risk of transmission outage is connected to the weather situation. This vulnerability model for the components gives which of the components that broke down and at which time. One benefit of this method is that it is possible to estimate the time difference between the outages in different lines, not only the outage risk. The times for breakdowns are interesting for estimation of the restoration time. The situations after a severe weather that are studied are so severe that gathering of all or almost all possible restoration resources is required to restore the system. Restoration times for components are not assumed to be independent; they are instead assumed strongly correlated. Another contribution in this thesis is therefore a method for simulating correlated repair times that is not assumed Gaussian distributed.
In the case studies the weather model together with the vulnerability model for the components gives estimations of the impact on components due to the different weather situations. The loads on two Swedish power lines are estimated during different weather situations and conclusions about the consistency to ice storms of the studied lines are presented. The risks of power 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 impact
on power systems and different methods for modelling the restoration process.
1.4. THE OUTLINE OF THE THESIS 5
• A new model for ice storm weathers and their impact on transmission
components.
• A new vulnerability model for components such as overhead lines and
towers.
• A new restoration model for a transmission network after an ice storm
event.
• A method for generation of non-Gaussian distributed random
numbers.
• A new method for choosing weather parameters for scenarios. The
method includes possible distributions of weather parameters and a description of possible weathers in Sweden.
• The developed methods are used in different case studies both on a
part 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 weather
phenomenons 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 for
restoration times.
• Chapter 6 describes the overall simulation method.
• Chapter 7 contains case studies and a stochastic method for choosing
weather parameters for different scenarios of Monte Carlo Simulations. Conference paper [7] is included in sections 3.2 and 4.2. Conference paper [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 in conference paper [9], accepted for publication in July 2007. Section 7.3 contain 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 case studies.
6 CHAPTER 1. INTRODUCTION
1.5
List of publications
The following publications are included in the thesis:
[7] ”Modelling of Ice Storms for Power Transmission Reliability Calculations”, Proceedings of the 15th Power Systems Computation
Conference PSCC2005, Liege, Belgium, August 22-26 2005. This
paper is the foundation of the models described in chapter 3 and 4. [8] ”On Transmission Restoration Evaluation after Ice Storms using
Monte Carlo Techniques”, Proceedings of the Third International
Conference on Critical Infrastructures 2006, Alexandria, USA,
September 23-27 2006. This paper is the foundation of the restoration model described in chapter 5.
[9] ”Modelling of Ice Storms and their Impact Applied to a Part of the Swedish Transmission Network”, Accepted for proceedings of
PowerTech 2007, Lausanne, Switzerland, July 2-5 2007. This paper
includes the vulnerability model for the components described in section 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 3.4 of this thesis.
[10] ”Ice Storm Impact on Power System Reliability”, Abstract accepted
for proceedings of the 12th International Workshop on Atmospheric Icing on Structures (IWAIS2007), Yokohama, Japan, October 2007.
The paper will include the method for probabilities for different severe weather 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 Hydrological Institute (SMHI) as heavy wind, often in combination with rain or snow. Wind is measured in direction and speed. The direction is stated as the direction 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 a
period of 10 minutes at a height of 10 m above the ground [11]. Both wind speed and wind direction differ for low terrain, mountains and seas. The impacts of different wind speeds are listed in table 2.1 [12] [13]. The impact of different wind speeds increases rapidly with the wind speed, V , since the power in the wind is proportional to V3 [14].
There are two fronts in a low pressure; one warm front and one cold front. A cold front is created when cold air replaces a warmer air mass and the air behind the cold front is colder than the air ahead of it. A cold front is represented by a blue line with triangles along the front, pointing
Wind speed Impact
20.8-24.4 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. A warm front is created when a warm air mass replaces a cold air mass and is represented by a red solid line with semicircles pointing towards the colder air in the direction of movement. The fronts move along with the center of the low pressure (L in figure 2.1) and are in the beginning separated from each other. The cold front moves faster than the warm front and there will be a fusion of the cold front and warm front, this process is shown in figure 2.1(a)-2.1(d) [15]. As the weather gets more severe the pressure in the center falls and the precipitation area grows. On the north hemisphere the wind is blowing anti-clockwise around the center of a low pressure due to the rotation of the earth and Coriolis forces. Weathers move in direction from high pressure towards low pressure.
The wind is often stronger south and west of the center (behind the low pressure). This means that the strongest winds often blows from the west or 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 per season was from south-west.
The most intensive and longest lasting precipitation is found close to the center of the low pressure [17]. Ice storms occur when supercooled rain freezes on contact with tree branches or overhead conductors and forms a layer of ice. Ice loading requires specific combinations of precipitation and temperature. Freezing rain is mostly associated with the passage of the warm front. The air is colder in the lower air layers of the atmosphere than in the higher layers. As a result the rain freezes as it falls when the temperature 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 moves faster than the warm front. A low pressure is often followed by another low pressure to the south of the first low pressure. The combination of the two weather systems can be devastating for the transmission network, especially if the first one brings a lot of freezing rain and wet snow and the second brings strong winds.
It is difficult to forecast how fast and in which direction a low pressure will move. Some low pressures move very fast and others move slowly and they can even be stationary. There is no obvious correlation between how fast a low pressure moves and how severe it is [17]. The frequency and intensity of icing depends strongly on the geographical location as well as on 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 over 10 minutes) and largest gust is 80 m/s (only for seconds) [12]. Most of the precipitation in Scandinavia in the autumn and winter comes from low pressures created over the North Atlantic. Some of these low pressures turn into severe storms [17]. In the south-west of Sweden the weathers from sea normally come in from south-west and continues north-east (does not hold for the severe weather 1921 and 1968).
Ice builds up on lines and places a heavy physical load on the lines and other structures and increases the cross-sectional area that is exposed to wind. The added weight and surface exposed to the wind increase the
10 CHAPTER 2. ABOUT ICE STORM WEATHERS
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 towers exposed to icing.
Icing is traditionally a phenomenon in the northern regions of the earth, such as Canada, Japan, Russia, the Nordic countries and central Europe. 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 are reports on that the temperature has changed more rapidly during the latest 20-50 years, which affects which areas that will be most exposed to icing in the future [18]. Some of the ice storms in the last 100 years around the world are listed below. These ice storm may not be the worst but for example the storm 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. The combination of a lot of rain and large wind speeds would have devastating consequences 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 same circumstances 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 the storm was approximately 48 hours and the maximum icing measured on a line was 9 mm. The wind was not as important as the ice for the tower breakdowns during this storm [20].
Canada and US January 1998
The ice storm that hit eastern Canada and north-eastern United States in January 1998 has been called the worst storm i modern time [21]. Extreme ice formation on power lines caused a crisis where about 1.5 million households where without electricity. The system was not completely restored until October 1998 but 90% of the affected customers had received power within two weeks [3].
France December 1999
A three day storm caused heavy damage on the transmission grid i France in December 1999. The maximum number of transmission lines out of operation was 38 and 5000 M W were not delivered to customers. Approximately 3.5 million households were without electricity [22].
12 CHAPTER 2. ABOUT ICE STORM WEATHERS 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 affected for a long time. During the night between January 8th and 9th 650 000 persons had their power supply interrupted. The restoration of the power supply took up to seven weeks [23]. It is interesting to notice that the first warning of the expected storm was issued at lunchtime on January 7th and my son Eskil was born at lunchtime January 7th, but he is more like a hurricane.
Other examples of recent ice storms are Switzerland January 2005, Germany November 2005 and the middle US December 2006.
Chapter 3
Modelling of Ice Storm
Weather
3.1
Weather models
Methods for including the impact of weather on power system reliability calculations have been studied earlier. The most widely used model is the two-state (normal and adverse) weather model that uses constant failure and repair rates for each state [24]. The two-state and many other models, especially Markov models, assume the entire network to be in the same weather environment, a reasonable assumption for geographically constrained distribution networks but not for large transmission networks. In [25] a model applicable to transmission networks is described. Different weather severities are considered, but the distribution of severity levels is discrete and the exposed area has to be divided into regions that are equally affected. In [26] it is stated that wind, icing and lightning are the most influential weather phenomena and the daily wind gust speed is the only variable selected to study wind effects. Since an ice storm is a very rare event it is regarded as a special event and the analysis of icing is separated from the other weather factors. In [27] the weather environment is divided into three states: normal, adverse and major adverse. The different states have different failure rates and the transitions rates between the different states are specified.
None of the above described papers consider the time dependent risk level for lines when a severe weather passes a region. The new technique of
14 CHAPTER 3. MODELLING OF ICE STORM WEATHER
modelling severe weather, such as ice storms, in this thesis uses one function for the wind part of the weather and another function for the precipitation part and is based on geographically moving winds and precipitation. The model is applicable to storms that cover large areas and is therefore suitable for transmission networks. The model has severity levels that are continuous and vary with time as the weather passes for example transmission line segments. In [8], the first weather model is described. In [9] this model is developed into the improved weather model. Based on more knowledge of how a severe weather is created and how it behaves during its lifetime the wind and precipitation part of the weather is more realistic. A known ice accretion model, the Simple model [28], is used to estimate the ice load. In the case studies the loads on two Swedish power lines are estimated for different weather situations. The loads given by the simulated weathers are compared to loads given by real weather data from a storm that hit Sweden 1999 using the same ice accretion model. The loads are also compared to the critical loads for the studied lines. The improved model is mainly described in [29], which is a master thesis produced within this project in cooperation with Svenska Kraftn¨at.
For different scenarios the stochastic weather parameters, such as size, strength, speed and direction can change. A method for choosing weather parameters to different scenarios is described in section 7.3.
The first weather model for severe weather in section 3.2 is more general than the improved weather model in section 3.3, but it is less accurate for Sweden where the improved model is more suitable. The ice accretion part is particulary improved in the improved weather model.
3.2
The first weather model
Since the largest precipitation occurs in the center of storms, a circular model with the largest strength in the middle can be applied [30]. The two-variable function below has suitable properties to serve as a basic model for describing wind and ice load.
f (x, y) = A exp ³ −1 2 ³³ x − xcenter σx ´2 +³ 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 shows
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 that decrease from the center.
Each line segment, which is a representation of transmission components, is exposed to certain load functions that depend on which intensities of the weather that meet the segment, and for how long. A load function corresponding to point (xj, yj), when the center of the weather moves according to the functions (xcenter(t), ycenter(t)), can be calculated from
L(xj, yj, t) = A exp ³ −1 2 ³³ xj−xcenter(t) σx ´2 + ³ yj−ycenter(t) σy ´2´´ . (3.2) The strength, or severity level, of the weather depends on ice load and 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 functions
have different parameters and shape.
3.2.1 Wind load
Wind speed is often treated as a mean value, for example the mean value of measured wind speed during a typical ten-minute period. Wind speed or wind load in this model refers to the instantaneous wind speed inside a
16 CHAPTER 3. MODELLING OF ICE STORM WEATHER β (µ x,µy) (x 1 ,y1 ) (x 2 ,y2 ) r u Line segment
Figure 3.2: The angle β.
severe weather, also called gust. The maximum wind speed corresponds to the 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 less amplitude and smaller σxand σy, a more realistic model is achieved [30]. The
wind 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) h A1exp ³ − 12 ³³ xj−xcenter(t) σx1 ´2 + ³ yj−ycenter(t) σy1 ´2´´ −A2exp ³ −12 ³³ xj−xcenter(t) σx2 ´2 + ³ yj−ycenter(t) σy2 ´2´´i , where wβ(t) ∈ [0, 1]. (3.3)
wβ(t) in equation (3.3) is the wind factor needed to consider the impact of the angle, β, see figure 3.2, by which the wind load hits the line. Wind load perpendicular to the line is the worst case; to include this in the model the perpendicular component of the wind load is used. The perpendicular component of the wind load is then achieved from multiplication by the wind
3.2. THE FIRST WEATHER MODEL 17 factor, as in equation (3.3). The wind factor is obtained by equation (3.4).
wβ(t) = sin β(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 the studied line segment with coordinates denoted (x2, y2) such that the angle α
between the vector r = (x1, y1), (µx, µy) and u = (x1, y1), (x2, y2) is between
0 and π. Then α is given from cos α = |r||u|ru and β = π2 − α if α ≤ π2 and
β = 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 wind factor 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 seen from 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 be modelled with a two-parameter function. Figure 3.4 shows the ice build up function for a particular segment, or a cross-section of the ice part of the weather 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 by
integration of the ice build up function for this segment.
LI(xj, yj, t)[mm/h] = Z t 0 A3exp ³ −1 2 ³³ xj−xcenter(u) σx ´2 + ³
yj−ycenter(u)
σy
´2´´
du. (3.5) The ice builds up continuously as in figure 3.5. The ice part of the weather has passed segment j when LI(xj, yj, t) is about equal to LI(xj, yj, t + ²)
for a small positive ², i.e. the time when the ice layer no longer increases on segment j, in figure 3.5 this coincide with tstop for this particular segment.
18 CHAPTER 3. MODELLING OF ICE STORM WEATHER 0 2 4 6 8 10 0 10 20 30 40 time [s] wind speed [m/s]
Figure 3.3: A wind load function for a particular segment or a cross-section of wind. 0 2 4 6 8 10 0 2 4 6 8 time [s] ice loading [mm/h]
Figure 3.4: Ice build up function for a particular segment or a cross-section of ice.
3.2. THE FIRST WEATHER MODEL 19
0
2
4
6
8
10
0
2
4
6
8
mm
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 the connection between wind load and ice loading is not considered in the above model for ice loading. In chapter 3.4 a more advanced ice accretion model is described, 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 is moving 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+ Vhcos(Θ)t
ycenter(t) = ystart+ Vhsin(Θ)t, (3.6)
where Θ is as in figure 3.6 and Vh= VhW for the wind part of the weather and
20 CHAPTER 3. MODELLING OF ICE STORM WEATHER m ½¼ ¾» &% '$ 666 Vh L L L L q Θ moving direction ¡¡µ
Figure 3.6: Wind blows anti-clockwise around a low pressure on the north hemisphere.
Both moving speed and angle for the wind weather may be different from the speed and angle for the ice weather.
3.3
The improved weather model
The improved weather model is based on Swedish conditions but is probably more accurate than the first weather model in other countries also. It is based on the first weather model but the geometric shape of the wind and precipitation areas have been more carefully studied. The model consists of two parts: one function that describes the wind part of the weather and another function that describes the precipitation part of the weather. The precipitation is most intense close to the center of the low pressure and the strongest winds are blowing south-west of the center. The weather moves according to functions for how the centers of the wind part and the ice part move 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 pressure and the wind can be assumed to have its maximum 300 km south-west from the center [32], at least in Sweden.
The wind function in polar coordinates, W [m/s], is:
W (r, θ, t) = AWe− 1 k(B∗(r(t)−300)2+C(min(θ(t)−4π3 +2π,2π−(θ(t)−4π3 +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. The
amplitude AW [m/s] refers to the maximum wind 300 km away from the
center with θ = 240◦. r [km] is the distance to the center. B, C and k
are constants, with B = 0.08, C = 30 and k = 10000 the wind part of the weather has the shape shown in figure 3.7.
Figure 3.7: The shape of the wind part of the weather model. The color scale represent the intensities of the wind which are largest 300 km south-west of 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 model to consider the impact of the angle, β, by which the wind load hits the line segment. The load function for wind, LW(x, y, t), is achieved according to
equation 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 first and the improved weather model the wind load is simply equivalent to the perpendicular component of the gust and independent of the cross section area. A way to consider the increased area when there is ice on the line is described in [13].
22 CHAPTER 3. MODELLING OF ICE STORM WEATHER 3.3.2 Precipitation
The precipitation area is modelled in two parts. Close to the center of the low pressure the precipitation area can be assumed circular [15]. This is modelled with one function that gives the largest values in the center and decreases 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 circular part of the precipitation part of the weather at time t. The constant AI is
the precipitation rate in this center.
A large precipitation area around the center of a low pressure often is followed by a large front zone. The front zone precipitation is here modelled with largest intensity close to the center; the intensity decreases with the distance from the center of the circle. The width of the front is dependent on the radius of circular part of the low pressure. The front zone is modelled with the following function:
h(x, y, t) = AIf ronte
−400001 (y−yc(t))2, (3.10)
where AIf ront is the precipitation rate in the front zone area nearest the
circular area. In order to get the front zone in the right position in relation to the circular area described in equation (3.9) the following restrictions are needed: r (x − xf c(t))2 0.8 + (y − yf c(t))2> 1.1Rice, (3.11) r (x − xf c(t))2 0.8 + (y − yf c(t)2< 2.6Rice, (3.12) x > xf c(t) + 20, g(x, y, t) = 0 elsewhere h(x, y, t) = 0. (3.13)
xf c(t) and yf c(t) are the x- and y-coordinates for the center of the front zone circular area at time t, see figure 3.8. The whole precipitation area and its
3.3. THE IMPROVED WEATHER MODEL 23
Figure 3.8: The shape of the precipitation part of the weather. The intensity is 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 the precipitation part of the weather is similar to figure 2.1(d). The reason for the choice of modelling the weather in this phase is that the low pressure normally is most violent in this phase [17].
Figure 3.9 shows a storm that hit Sweden 1999 using weather data from weather stations. The shape of the precipitation part of the weather can be compared 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 Simple
ice accretion model [28] described in chapter 3.4. The ice accretion is
dependent on the wind. The mean wind, Vmean, is used to estimate the
amount of ice that is deposited on a line. The relation between Vmean and
the 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.
24 CHAPTER 3. MODELLING OF ICE STORM WEATHER
Figure 3.9: The precipitation part of 1999 storm as it passes the lines in the case studies. The darkest red color corresponds to the maximum precipitation (9.3 mm/h).
in the case studies of this thesis. In [11] Vmean= 0.73Vmax, 25 m above sea
level.
3.4
Modelling ice accretion
There are many models for deposition of freezing rain and wet snow on objects. The ice accretion models can be divided into two different types. The first type uses physical parameters and determines the heat balance of the object and requires parameters that are difficult to model. The second kind of model uses meteorological data to estimate the accreted ice. The model suggested and used in this thesis is the Simple model [28] and is of the second type.
3.4. MODELLING ICE ACCRETION 25 An alternative to ice accretion models for estimation of ice thickness is suggested in [18]. Since ice load on overhead lines depends on height above ground, which differs from tower to tower, and accretion is often a mixture of different ice types a statistical method is suggested. In [18] is for example Weibull, exponential, gamma or log-normal distributions suggested for the distribution of the ice thickness.
The Simple model uses parameters that are given by the weather model developed in this thesis, however it is necessary to assume the size of the droplets. It can be assumed that all the droplets that hit the surface of the line freezes. This means that no icicles are developed. This is not necessarily an overestimation of the accreted ice on the line because the icicles that are ignored make the area for collecting new droplets larger, this can even lead to an underestimation of the ice load [33].
The Simple model has been developed to model the effects of freezing rain. The Goodwin model [34] is similar to the Simple model but it uses the fall speed of the precipitation instead of droplet sizes and can be used to model wet snow accretion also. The only difference for using the Goodwin model for freezing rain or snow is the density of the fall speed of the precipitation. 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]. When there is no wind, or very light wind, the differences between the two models are negligible. The Goodwin model is very sensitive to the choice of droplet fall speed, therefore the Simple method is preferred in the case studies of this 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 icicle growth into account [33]. These models are probably the most accurate for estimation of ice accretion but they require more meteorological data, for example the humidity.
Different models assumes different shapes of the ice accretion around the power line. Among others the Simple model assumes a perfect circular shape; this is reasonable because of the power lines ability to rotate. When one side of the line has been covered with snow or ice it becomes heavier and rotates [35]. Other models assume an elliptic accretion shape [33].
26 CHAPTER 3. MODELLING OF ICE STORM WEATHER 3.4.1 The Simple model
The Simple model estimates the amount of precipitation that hits the line from a horizontal and a vertical direction. The massflux is the mass of rain that hits an area during one time unit. The vertical massflux can be calculated as:
mv = P δ (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 content
according to [37] is:
v = 0.072P0.88. (3.19) The total massflux hitting the line, m0 is
m0 = q m2 v+ m2h= =pP2δ2+ 3.62V2 meanv2. (3.20)
The increase of ice thickness on the line, ∆R ([mm/h]), when assuming a circular shape, is given by:
∆R = m0 πδi , ∆R = 1 δπ p (P δ)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 weather model. The ice load function becomes:
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; the weight of the accreted ice is larger when the line is thicker. The heaviest ice that can be developed has a density of approximately 0.9 kg/dm3[35]. Such
heavy ice is only created under very specific circumstances. During the 1921 ice 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 density
would increase the wind load because the ice layer becomes thicker given the same precipitation rate which makes the area exposed to wind larger.
Chapter 4
Modelling Component
Vulnerability
4.1
Reliability of transmission lines
To be able to connect the risk of transmission outage to the weather situation, a vulnerability model for the components is required. There are many different methods for estimation of reliability parameters for transmission components, both deterministic and stochastic. A deterministic method for estimation of breakdown means that if the load does not exceed the specified load level for the component the component will not fail. The threshold load is often equal to the design strength. In reality both loads and strengths of the component are stochastic and stochastic methods are therefore preferable [38].
Overhead lines are the most vulnerable components in distribution systems [26]. The important issues for line damage are ice load, wind load and type and conditions of components [39]. The studied components could be insulators, lines or foundation. Without any knowledge of sequence of failure is it hard to say which component that is the most critical. During an ice storm a transmission tower can fail due to buckling of its legs as in figure 2.3. Another failure sequence is when the line breaks first and cause failure 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 breakdowns are 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 in 100 year storm and a 1 in 500 year storm are considered. A fault tree with probabilities depending on how much ice load that is above the ultimate specified load or design load is used to obtain a stochastic method. The paper 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 estimates based on experiments on two existing transmission lines.
The basic transmission line model in [42] allows different weather conditions (normal and adverse) with different failure rates for different parts of a transmission line. The same authors have in [43] described their method that also recognizes weathers that differ for different regions. Furthermore; they do not assume exponential distribution of the duration of an outage and discuss the effect of failure bunching when components in an adverse weather region have high failure rates simultaneously.
In [44] it is stated that the failure rates of overhead transmission lines are continuous functions of the weather to which they are exposed. The two-state weather model in [24] and [42], among others, is expanded to a three weather state model which also consider major adverse weather with sharply increased failure rates. [44] suggests that these extreme weathers should be considered as independent events with overlapping failure rates for the exposed area.
Since Canada experienced its severe ice storm in 1998 many case studies have been performed on test networks based on Canadian conditions. For instance is the sequence of failure of an experimental distribution line examined in [45]. Other examples are [46] where the status of existing transmission line components are considered and the standardized risk estimation spreadsheet developed in [41] which is used to calculate the risk of three different severity levels of ice storms.
Peak over threshold methods, where data from peak events that exceeds a specified threshold during a time series, are suggested in [18] to be suitable for irregular events such as severe ice storms.
In [47] are Monte Carlo simulations used to model adverse weather for transmission line outages. Failure rate data is based on statistics for an existing transmission line and failures due to lightning, wind and precipitation are studied. The paper shows that the distribution of up and down times due to adverse weather are skewed and suggests to not use e.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 most of the year, and very high for a short period of the year which represent adverse weather. In [49] past failure data during adverse weather is used together with the weather data to achieve failure rates for different adverse weathers.
The methods described above have in common that needed component reliability data is important but difficult to achieve.
In all the above methods for impact due to severe weathers on power systems the failure rates are constant and none of the papers considers the time 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 the impact of a given weather depends on the load function together with the vulnerability model for components, see section 4.2. The vulnerability model is stochastic and is based on the design of the components. The approach is to estimate the reliability of components using Monte Carlo simulations. One benefit of this approach compared to many other methods is that it is possible to estimate the time difference between the outages in different lines, not only the outage risk since Monte Carlo methods allow simulations of time sequences.
Since a broken component during a severe weather probably has broke down due to the weather the failure rates for all components in the severe weather region are correlated. This is often considered with increased but independent failure rates for the components in the severe weather region. In this thesis the failure rates vary with time and the severity of the weather and failure rates for components in the same weather environment have hereby correlated or similar failure rates. As shown in [50] it is likely that adjacent towers to the tower that breaks first also breaks due to maximum weight on the line. The tower construction breaks due to the extra force on the adjacent towers which already are exposed to high ice and wind loads. However; often only a few towers are involved in this domino effect. In the case studies a segment represents many towers and the effect of breakdown of adjacent segment is not as obvious as the impact of adjacent tower breakdowns. In the model for the restoration process described in chapter 5 the status of adjacent segments is considered.
The sheltering effect on wind speed by trees and terrain in transmission corridors is not considered in this thesis.
When discussing transmission failures due to icing an often suggested solution is heating of the line by introducing large transmission loads.
32 CHAPTER 4. MODELLING COMPONENT VULNERABILITY
During specific conditions the line can be up to twenty degrees warmer than the air temperature [51]. In this thesis it is assumed that this heating effect does not prevent the ice from building up on the power line. In Sweden the largest transmission load occurs when the temperature is much lower than the ideal temperature for ice storms and the existing heating devices are not in common use in Sweden where mechanical deicing methods are used instead. [52]
4.1.1 Galloping
Galloping occurs when there are ice in combination with wind on the span of the line. The wind creates motions as it blows across the line and if the frequencies of these motions coincide with the natural frequency of the line an oscillation occurs. Galloping does not result in an immediate breakdown, but causes extreme forces on the line and towers and can ultimately result in a breakdown by fatigue [13]. Galloping can be mitigated by increasing the tension of the line, or installing dampers. There have probably not been any breakdowns due to galloping in Sweden, but there are known cases in Canada and England [52].
4.2
Segment vulnerability models
The exponential distribution with constant failure rates is often used in reliability calculations for simplicity and for being robust. According to [53] this can be misleading for the design of the system and the reliability can be overestimated for some periods and underestimated for others. Another common method for treating the risk of failure is to use a type of process that includes a ”memory” in order to include the possibility that the risk also increases when the load is constant due to fatigue. In the first segment
vulnerability model proposed in this thesis a time dependent exponential
distribution of the time to failure was chosen. This means that the process is assumed to have ”no memory”. To include the changed risk of failure because of changed amount of wind and ice load, the parameters of the distribution are controlled in order to obtain a realistic behavior of the connection between the load and the risk of failure. In this way can the process be controlled more directly. The second segment vulnerability model differs from the first segment vulnerability model, which is more general but requires data that is difficult to achieve. The second vulnerability
