Enhancing Power Distribution Grid Resilience Against Massive Wildfires
by Fei Teng
B.S. in Automation, June 2018, North China Electric Power University
A Thesis submitted to
The Faculty ofThe School of Engineering and Applied Science
of The George Washington Universityin partial satisfaction of the requirements
for the degree of Master of Science
August 31, 2020
Thesis directed by
Payman DehghanianAssistant Professor of Electrical and Computer Engineering
c© Copyright 2020 by Fei TengAll rights reserved
ii
Dedication
This MS Thesis is lovingly dedicated to my parents. They are the ones always nursing
me with affections and love. Their support and encouragement have sustained me throughout
my university life.
iii
Acknowledgments
I would like to express my sincere gratitude to my advisor, Prof. Payman Dehghanian,
for his guide, understanding, wisdom, patience, encouragements, and for supporting me
reach this point. Thanks also to my committee members, for their time and patience.
Appreciation goes to my friends and the member of the Smart Grid Laboratory, specifi-
cally Mostafa Nazemi, Jinshun Su, Dingwei Wang, Yifu Li, Naihao Shi, Mohannad Alhazmi,
Shiyuan Wang, and Bo Wang, for making my time at the George Washington University a
wonderful experience.
Most of all, I am fully indebted to my parents for their support, without whom, the
pursuit of this advanced degree would never have been started and accomplished.
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Abstract
Enhancing Power Distribution Grid Resilience Against Massive Wildfires
In the past decades, wildfire hazards occurred more and more frequently. During summer
seasons in regions with high temperature, power distribution systems especially those located
near the forests are prone to wildfires. The temperature of conductor lines exposed to the fire
increases rapidly, the shape and strength of which are, therefore, permanently reduced. The
conventional reliability view is insufficient to cope with these challenges in modern power
systems since such hazards cause prolonged and extensive outages, much more severe than
those previously accounted for in system reliability assessments. Improving the resilience of
the power grid, hence, becomes increasingly important and urgent. To enhance the resilience
of the system against wildfires, the characteristics of wildfires need to be first studied so that
effective mitigation strategies, e.g., dynamic line rating of the overhead power lines, can be
proposed taking into account the impact of fires and the existence of uncertainties.
This thesis mainly focuses on designing an optimal operation strategy to minimize load
shedding when distribution lines are affected by wildfires. Taking the uncertainties related
to solar and wind – that influence the spread of wildfire and the renewable generators – into
account, a stochastic mixed-integer nonlinear programming model is proposed and applied
to a modified 33-bus power distribution system. The formulation aims to minimize the social
cost which associates with the status of each wind turbine and the energy storage systems.
By this means, the most effective operation strategy against wildfires is found, thereby
enhancing the grid resilience and reducing the load outages during and following wildfire
event. A sensitivity analysis is conducted on the overhead lines affected and the number of
active components in the system to further investigate the best mitigation approach in the
power distribution system when exposed to massive wildfires.
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List of Figures
1.1 Elements of Resilience by Electric Power Research Institute (EPRI) . . . . . 71.2 The conceptual resilience curve related to a HILP event . . . . . . . . . . . . 8
2.1 Wildfire causes from 2013 to 2017 . . . . . . . . . . . . . . . . . . . . . . . 112.2 Number of arcs burned by wildfires in the world during 1871–2020 . . . . . 122.3 Flow of events, causes, and preventive solutions . . . . . . . . . . . . . . . . 17
3.1 Linear relationship for convection heat loss in case of non-zero wind speed(T a=273k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Piece-wise linearization of the radiation heat loss (T a=273k) . . . . . . . . 273.3 Weibull distribution of the wind speed . . . . . . . . . . . . . . . . . . . . . 283.4 Wind speed mean value and stochastic values . . . . . . . . . . . . . . . . . 283.5 Wind direction mean value during a 24-hour period . . . . . . . . . . . . . . 293.6 Solar radiation mean value and stochastic values . . . . . . . . . . . . . . . 293.7 Distance between the power line conductor and fire in different scenarios . . 303.8 Conductor heat gain rate from the fire in different scenarios . . . . . . . . . 31
4.1 Single-line diagram of the modified 33-node power distribution system . . . 334.2 Relationship between the output power of a wind turbine and the wind speed 354.3 Relationship between the output power of a PV and the solar radiation . . . . 364.4 Expected Load Shedding for 10, 50 and 100 Generated Scenarios . . . . . . 414.5 Expected energy exchange with the upstream system . . . . . . . . . . . . . 424.6 Power exchange price with the upstream system . . . . . . . . . . . . . . . . 434.7 Expected generated power and status of MTs . . . . . . . . . . . . . . . . . 434.8 Expected discharging power and SOC of ESS . . . . . . . . . . . . . . . . . 44
5.1 Objective function value: sensitivity analysis on every single line affected bywildfire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2 Expected load shedding cost: sensitivity analysis on every single line affectedby wildfire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.3 Expected load shedding when line 25 effected . . . . . . . . . . . . . . . . . 475.4 Objective function value when 2 lines are affected by wildfire . . . . . . . . 475.5 Expected load shedding when 2 lines are affected by wildfire . . . . . . . . . 485.6 Expected power exchange when 2 lines are affected by wildfire . . . . . . . 485.7 Objective function value when 3 lines are affected by wildfire . . . . . . . . 495.8 Expected load shedding when 3 lines are affected by wildfire . . . . . . . . . 505.9 Expected power exchange when 3 lines are affected by wildfire . . . . . . . 505.10 Objective function value when adding one distributed resource . . . . . . . . 515.11 Expected load shedding when adding one distributed resource . . . . . . . . 525.12 Expected power exchange when adding one distributed resource . . . . . . . 52
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List of Tables
1.1 Statistics of Outage Events in the U.S. Between 1984-2006 . . . . . . . . . . 21.2 The Conceptual Contrast Between Reliability and Resilience . . . . . . . . . 3
4.1 Location And Capacity of Distribution System Components . . . . . . . . . 344.2 Location and Capacity of Distribution System Components . . . . . . . . . 384.3 Simulation Results Concerning Different Number of scenarios . . . . . . . . 424.4 Revenues and costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
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Nomenclature
A. Sets and Indices
i, j ∈ NB Indices/set of nodes.
i j Indices/set of distribution lines between nodes i and j.
t ∈ NT Indices/set of time periods.
(i, j) ∈ L Indices/set of branches.
NB,NT,NL Number of all nodes, time periods, and branches.
ω Indices/set of scenarios.
MT ∈M Set of all Micro Turbines (MTs).
ESS ∈M Set of all energy storage systems (ESSs).
S ∈M Set of all photo-voltaic panels (PV).
B. Parameters and Constants
B.1. Fire Model
T f Flame zone temperature (k).
L f Fire front length (m).
α f Fire tilt angle (rad).
ρb The bulk density of the fuel (kg/m3).
ε f Flame zone emissivity
B.2. Environmental Conditions
τ Dimensionless atmospheric transmissivity.
B Stefan-Boltzman constant (W/m2K4)
V wind Wind speed (m/s).
σwind Angle between the fire and power line conductors (rad).
T a Ambient temperature (k).
ka Thermal conductivity of air (W/mK).
µα Dynamic viscosity of air (kg/ms).
viii
ρα Density of air (kg/m3).
kindex Shape index of the Weibull distribution.
C Scale index of the Weibull distribution.
B.3. Conductor Specifications
mCp Total heat capacity of conductor (J/mK).
D Conductor diameter (m).
K Solar absorptivity.
φ sun Solar radiation rate (W/m2).
Ri j,a Ambient line resistance.
T max Conductor maximum temperature permitted (k).
B.4. Price and Costs
VoLL Value of lost load ($/MWh).
cD Price for selling electricity ($/MWh).
cMT Price for micro turbine generation ($/MW).
csu/sd Micro turbine switching cost ($).
B.5. Power Distribution System Components
Pdemandi,ω,t Real power demand at node i at time t (MW).
Qdemandi,ω,t Reactive power demand at node i at time t (MVar).
nST Conversion efficiency of energy storage systems.
EST Energy capacity of energy storage systems.
C. Functions and Variables
C.1. Fire Model
θf
i j,t View angle between fire and conductor line i j at time t (rad).
d fi j,t Distance between wildfire and the conductor line i j at time t (m).
V ft Fire spread rate (m/s) at time t.
Ti j,t Conductor temperature of overhead line i j at time t (k).
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φf
t Radiative heat flux at time t (W/m2).
C.2. Heat Gain and Loss
qlinei j,t Resistance heat gain rate of line i j at time t (W/m).
qsuni j,t Solar heat gain rate of line i j at time t (W/m).
q f irei j,t Fire heat gain rate of line i j at time t (W/m).
qconi j,t Convection heat loss rate of line i j at time t (W/m).
qradi j,t ) Radiation heat loss rate of line i j at time t (W/m).
C.3. Power System Model
pDi,t ,qDi,t Real and reactive demand supplied at node i at time t (MW, MVar).
Pp fi j,t ,Q
p fi j,t Real and reactive power flow on branch (i, j) at time t (MW, MVar).
SOCSTi,t SOC of ESS at time t.
pChi,ω,t , pDC
i,ω,t Charging and discharging power of ESS at node i at time t (MW).
pMTi,t ,qMT
i,t Real and reactive power output of MT at node i (MW, MVar).
PWTi,t ,PS
i,t Real power output of WT and PV at node i at time t (MW).
V sqri,t Squared voltage magnitude at node i at time t (kV2).
pshedi,t ,qshed
i,t Real and reactive load shedding at node i at time t (MW, MVar).
pUPt Active power exchange with the upstream system at time t (MW).
D. Binary Variables
αi j,t Connection status of branch (i, j) at time t
1 (1 if the branch is connected, 0 otherwise).
usoci,t Charging and discharging status of ESS at node i at time t
1 (1 if charging, 0 otherwise).
ui,t Status of MT at node i at time t
1 (1 if the MT is generating, 0 otherwise).
ϕUPt Buying or selling energy to the up stream network at time t
1 (1 if buying, 0 otherwise).
x
Chapter 1: Introduction
1.1 Problem Statement
Power grids, as the most complex man-made cyber-physical system to date, have been
traditionally designed and planned to operate reliably under normal operating conditions and
withstand potential credible outages. In the last decade, it has become more apparent that
further considerations beyond the traditional system reliability view are needed for keeping
the lights on at all times [1]. Table 1.1 shows the statistics of 933 power outage events,
reported by the North American Electric Reliability Corporation (NERC), between 1984
to 2006 [2]. Extreme weather events and natural disasters have relatively low frequencies,
but a greater impact on the electric power supply and a larger size of affected electricity
customers, among these outage cause categories. According to the statistics provided in [3],
a total of 178 weather disasters occurred from 1980’s to 2014 in the US alone with the
overall damages exceeding the US $1 trillion.
Due to the growing demand to ensure higher quality electricity to end customers and
particularly critical services, and intensified public focus and regulatory oversights, safe-
guarding the nation’s electric power grid resilience and ensuring a continuous, reliable, and
affordable supply of energy in the face of the high-impact low-probability (HILP) events
are among the top priorities for the electric power industry and has become more and more
critical to people’s well-being and every aspect of our increasingly-electrified economy. The
HILP events include two categories: (i) natural hazards, such as hurricanes, earthquakes
tornadoes, windstorms, wildfires, ice storms, etc.; (ii) man-made disasters, such as cyber or
physical attacks on the power system infrastructure. Here in this thesis, the focus will be on
the power grid resilience to wildfires [4].
Wildfire, similar to other natural disasters, is the one for which everyone pauses to listen
each time it appears on the news. Sadly, for some families wildfires represent the loss of
1
% of events Mean size in MW Mean size in customers
Earthquake 0.8 1,408 375,900Tornado 2.8 367 115,439Hurricane/Tropical Storm 4.2 1309 782,695Ice Storm 5 1,152 343,448Lightning 11.3 270 70,944Wind/Rain 14.8 793 185,199Other cold weather 5.5 542 150,255Fire 5.2 431 111,244Intentional attack 1.6 340 24,572Supply shortage 5.3 341 138,957Other external cause 4.8 710 246,071Equipment Failure 29.7 379 57,140Operator Error 10.1 489 105,322Voltage reduction 7.7 153 212,900Volunteer reduction 5.9 190 134,543
Table 1.1: Statistics of Outage Events in the U.S. Between 1984-2006
their property, savings, and even life. Besides the excessive cost of physical damages from
fires to different properties, there are many other consequences such as costs for evacuation,
revenue loss for the businesses, costs for rehabilitation, etc. These are just some of the few
example consequences of the thousands of wildfires that affect thousands of homes and
burn millions of acres of land and business properties annually throughout the world [5]. In
October 2007, 17 people lost their lives in a single Southern California wildfire; 10 were
killed by the fire outright, 3 were killed while evacuating, 4 died from other fire related
causes and more than $1.5 billion in property damages was reported [6]. In February 2009,
a similar disaster happened in the state of Victoria, Australia. The Victorian Brush fires
Royal Commission estimates the cost of this “mega-blaze” to be more than $4 billion and
to have resulted in the death of 173 people [7]. On May 2016, a wildfire was initiated in
Alberta, Canada. The direct financial loss to insurance providers from the great Alberta
fire was estimated at about $3.7 billion [8]. In October 2017, a series of wildfires started to
burn across the wine country of Northern California. These wildfires caused at least $9.4
billion in insured damages and the death of 44 people [9]. In fiscal year 2017, the cost
2
of battling blazes topped $2.4 billion [10]. For the first time in its 110-year history, the
Forest Service is spending more than 50 percent of its budget fighting wildfires [11]. In
November 2018, California experienced one of the most destructive and deadliest seasons
in its wildfire history. Two major fires, Woolsey Fire near Los Angeles and Camp Fire at
Northern California, killed at least 86 and 3 people, respectively. The Woolsey Fire cost
about $4 billion while the Camp Fire destroyed more than 18,800 structures and caused
more than $11 billion in damages [12–14]. During 2018, more than 8,500 fires burned
across nearly 1.9 million acres in the state of California and resulted in more than $16.5
billion in the total damage. Cumulatively, the wildfires were the costliest natural disaster
of 2018, as well as one of the deadliest. In Northern California’s Camp Fire alone, more
than 85 people lost their lives [15]. Safeguarding the nation’s electric power grid resilience
against wildfire and ensuring a continuous, reliable, and affordable supply of energy during
such devastating events are among the top priorities for the electric power industry.
1.2 The Concept of Resilience
Unlike the widely adopted terminology "reliability" in many traditional principles, power
system resilience is an emerging concept and its definition and quantification measures are
unclear thus far; nonetheless, the definition has a common comprehension. "Resilience"
and "Reliability" seem to have a similar but essentially distinct meanings [16, 17]. The key
characteristic difference between resilience and reliability is presented in Table 1.2 [18].
Table 1.2: The Conceptual Contrast Between Reliability and Resilience [18]
3
The existing reliability metrics do not concentrate on the consequences of individual
HILP events. Beyond minimizing the probability of extensive and prolonged outages,
resilience also takes the following into account: acknowledgment of the occurrence of
such outages, smarter operations of the grid and harnessing its built-in flexibilities [19–27],
preparation to cope with them, minimization of the outage effect, rapid restoration of the
service, and learning from the experience to enhance the future performance [28–32].
National Academies of Sciences, Engineering, and Medicine [28] provides a definition
in 2017 for resilience as follows: "Resilience is not just about lessening the likelihood that
these outages will occur. It is also about limiting the scope and impact of outages when they
do occur, restoring power rapidly afterwards, and learning from these experiences to better
deal with events in the future." PJM Interconnection provides a definition in March 2017 as
follows [33]: "Resilience, in the context of the bulk electric system, relates to preparing for,
operating through and recovering from a high-impact, low-frequency event. Resilience is
remaining reliable even during these events". This definition is more specific to the HILP
events. In President Barack Obama’s Presidential Policy Directive [34], the term "resilience"
refers to "the ability to prepare for and adapt to changing conditions and withstand and
recover rapidly from disruptions. Resilience includes the ability to withstand and recover
from deliberate attacks, accidents, or naturally occurring threats or incidents". A definition
of resilience for energy system is provided by the UK Energy Research Center [35] as
follows:"Resilience is the capacity of an energy system to tolerate disturbance and to
continue to deliver affordable energy services to consumers. A resilient energy system can
speedily recover from shocks and can provide alternative means of satisfying energy service
needs in the event of changed external circumstances."
Among the existing definitions of resilience, four aspects of the system resilience are
summarized in [27, 36, 37] as follows:
• The state of electricity services in a power grid can be described by resilience when
confronting an interruption or outage. The description of resilience contains the extent
4
of the service degradation, the rapidity of service recovery, and the recovery extent of
the service. As can be seen, resilience does not only reveal a discrete state of whether
a disturbance has happened, but also demonstrates and accounts for the level and
extent of the disturbance.
• The system resilience is determined by its design and its operation. These affect
the degradation degree during a disturbance, the swiftness of the recovery and the
completion of the restoration. For instance, a more redundant system that considers
recovery strategies and additional contingency operation modes might undergo fewer
and shorter interruptions. On the other hand, such a redundant system is more
strenuous to reconstruct.
• Different resilience levels of the system can be resulted from different response at
different costs. For instance, the system rebuilt with additional resources and a more
efficient set of equipment can provide higher quality of service when exposed to
extremes than the original configuration.
• The system state of resilience changes over time. The quality and continuity of service
in a system could be enhanced with regular maintenance and upgrades but at a cost.
On the other hand, the service in a system without regular maintenance and upgrades
has a lower operating cost but it can be anticipated that the quality and reliability of
service will lessen in the future [38–44].
The National Infrastructure Advisory Council (NIAC), USA provided four main features
of resilience in [45]:
• Robustness: The capability to maintain operation or withstand disturbances when
disaster occurs, especially HILP events. Besides the system structure or design, it
also relates to the system redundancy in case of some important components damages,
along with the investment and maintenance of the critical infrastructure.
5
• Resourcefulness: The capability to expertly handle the occurred disaster. It incorpo-
rates determining the strategies and priority of the actions that should be taken to both
control and diminish the hazardous emergency, and convey the decisions to the crews
to execute. This feature mainly relates to the human, and not the adopted technology.
• Rapid Recovery: The capability to restore the system to its normal operating con-
dition as soon as possible following the extremes. It relates to elaborately prepared
contingency plans, capable emergency operations and strategic resources distribution
and crews dispatch.
• Adaptability: The ability to learn from a hazard. It relates to the enhancement of the
robustness, resourcefulness and recovery abilities of the system for the future hazards
via new tools and technologies.
The Cabinet Office, U.K. also provides four main characteristics of infrastructure resilience
in [46] as follows:
• Resistance: provides the strength or protection to withstand the disaster and its main
effects to further mitigate the damage or disturbance.
• Reliability: the infrastructure components are ensured to be inherently designed to
maintain operation under certain conditions.
• Redundancy: the availability of the backup equipment or spare capacity to allow the
operation to be switched or redirected to alternative routes.
• Response and Recovery: rapid and effective response to and recovery from the
hazards.
Additionally, Electric Power Research Institute (EPRI) in the U.S. also determines the three
elements of resilience which are prevention, survivability, recovery for power distribution
systems in [47] as demonstrated in Figure 1.1:
6
Figure 1.1: Elements of Resilience by Electric Power Research Institute (EPRI) [47]
Notice that these elements and aspects of the system resilience can also lead to its
measurement (or metrics). Most of the existing definitions of resilience refer to the capability
of the system to withstand and rapidly recover from HILP events. A conceptual resilience
curve is proposed in [18] to describe the variations in the system resilience level over time
with regard to a HILP event, as illustrated in Figure 1.2, where R represents the resilience
level of the system. With respect to a HILP event, the power system experiences the
following operating states [18, 48]:
• Resilient State t0 ∼ te: before the HILP event happens at te, the power system should
be robust and resistant to withstand the first strike of the HILP event by sufficiently
predicting the time and location of the external disturbance and preventive actions (e.g.
preventive generation rescheduling) taken by the system operator, aiming to enhance
the disturbance resilience of the infrastructure.
• Event Progress te ∼ tpe: during the HILP event progress, the system is degraded to
post-event degradation state, where the system resilience decrease to Rpe. Emergency
or corrective actions (e.g. generation re-dispatch alone or generation re-dispatch
7
Figure 1.2: The conceptual resilience curve related to a HILP event [18]
coordinating with dynamic-boundary microgrid operation) can be taken to reduce the
effect of the external disturbance.
• Post-Event Degraded State tpe ∼ tr: after the event strike, the system enters the post-
event degraded state. At this stage, the key resilience features are the resourcefulness,
redundancy, adaptive self-organization, where they offer the necessary corrective
operational flexibility to accommodate and cope with the changing situation. This
assists in minimizing the consequence of the event and the degradation in the system
resilience level (e.g. R0−Rpe) while appropriate and effective coordination and
preparation enable rapid start of the restoration state.
• Restorative State tr ∼ tpr: the system should manifest fast response and recovery
ability to recover the system resilience level from Rpe to Rpr. Rpr may be the pre-event
resilience level R0 or a desired resilience level that is not as high as R0.
• Post-Restoration State tpr ∼ tir and Infrastructure Recovery tir ∼ tpir: Following
the restorative state, the event consequence on the system resilience and its perfor-
8
mance during the event need to be evaluated and analyzed to enhance the infrastructure
resilience for future similar or unpredictable events. Depending on the severity of the
event, the system may need longer time to recover the infrastructure in tir ∼ tpir.
1.3 Thesis Outline
The rest of this thesis is organized as follows:
Chapter 2 provides a background and reviews the existing literature on cause factors
of fires, their impacts on the power system operation, and several strategies to enhance the
power grid resilience to wildfires.
Chapter 3 explores the implements wildfire hazard modelling and formulations. The
heat balance equation is taken into account to help measure the thermal heat delivered to
the overhead lines in power distribution systems. The Dynamic Line Rating is proposed to
model and correlate this heat to the temperature increase of the power conductors. Then the
uncertainties of the surroundings is estimated and several sets of scenarios are generated
using the proper probability distribution functions.
Chapter 4 proposes a mixed-integer nonlinear programming (MINLP) optimization
model on a 33-bus power distribution system exposed to wildfires. The same scenarios
and models f or wildfires developed in Chapter 3 are employed serving as the inputs to the
optimization engine. The formulation is further linearized, hence reducing the computational
complexity. A base case is studied to find the best number of scenarios taken, and the efficacy
of the resilience measures and mitigation actions on the system operation.
In Chapter 5, the numbers and positions of lines out of service due to the progressive
wildfires is changed and additional power sources are added. The same test system and
damage scenarios are applied in the case studies with different components affected to
explore the sensitivity of each element when facing the wildfire extremes.
Chapter 6 presents the research conclusions and summarizes the main findings of this
thesis. Future work is also provided in this chapter.
9
Chapter 2: Literature Review
2.1 Introduction
This chapter reviews research in different fields of science and industrial projects that attempt
to address wildfire issues and their interactions with the power system. First, the causing
factors of wildfires are listed and the various scenarios of faulty electrical networks that may
lead to progressive wildfires are highlighted. Then the damages and negative effects that a
wildfire can cause to the electric grid are summarized. Finally, prediction and prevention
means and measures are discussed.
2.2 Root Causes of Wildfires
Wildfires are typically initiated from natural events such as lightning, spontaneous combus-
tion, or volcanic eruptions or as a result of day-to-day human activities or the operation of
indispensable equipment. A detailed list of wildfire triggers from 2013 to 2017 is illustrated
in Figure 2.1.
Wildfires and Climate Change. Meteorological data from many national agencies have
shown that one prime factor for recent wildfires is global climate change. In the western
parts of the North America, during the wet periods, generous rain creates substantial fuel
sources that can create a suitable condition for fire throughout the drought and warming
periods. During these seasons, the land is flammable due to its carbon-rich vegetation,
and oxygen-rich atmosphere [49, 50]. Weather-related effects on a fire consist of many
aspects such as behavior (e.g., wind speed and conditions), fuels (e.g., combustible material
created), and ignitions (e.g., lightning). Wildfires are also influenced by moisture availability,
snow-pack, temperature, precipitation, and other meteorological factors of the environment
that are all affected by climate change drivers. Further climate change would only exacerbate
10
Figure 2.1: Wildfire causes from 2013 to 2017
the problem, as increased temperatures, a reduced snow-pack, and altered precipitation
would lead to increased flammability of fuel for longer periods, which could affect the size,
frequency, and severity of wildfires in the future. These changing conditions may pose
an increasing threat to the energy infrastructure along the coast, including power plants,
transmission and distribution lines, gas storage facilities, and pipelines [51]. The 2012
and 2013 studies by Sathaye et al. [51] assessed the possible impacts that the increased
air temperature may have on the thermal performance of natural gas fired generation and
substations. Several studies have shown that climate change is likely to increase the size and
frequency of wildfires in California, a state that already leads U.S. wildfire-related economic
losses. Of the ten largest wildfires in California’s history, eight have occurred since 2001.
This is evidenced by an increase in the number of arcs burned by wildfires in the world as
shown in Figure 2.2.
Wildfires and Human Activities/Interventions. Wildfires can also start from day-
to-day human activities and the operation of essential equipment. According to official
11
Figure 2.2: Number of arcs burned by wildfires in the world during 1871–2020
reports, about 85-90% of wildfires are caused by human mistakes to deal with materials and
malfunction of equipment. The fire sources include equipment uses and faults, campfires
out of control, negligently discarded cigarettes, burning of debris, and intentional acts
of arson [52]. Some specific ignition sources are preventable, such as arson, discarded
cigarettes, electrical faults in power lines, failure of aged electrical equipment, oil-filled
transformers explosion, and sparks from vehicles and mechanical equipment.
Wildfire-Triggering Events in Power System. The chances of fire initiation by elec-
trical infrastructure are low (normally about 1.5% of all ignitions [?]), but in periods of
drought and when the heat tide comes, the percentage of fires linked to electrical assets rises
dramatically up to 30% of total ignitions [?]. For instance, the deadliest of the October 2017
wildfires in California’s wine country and the 2018 Camp Fire were started by electrical
equipment owned by Pacific Gas and Electric Company (PG&E) or by equipment that
was owned, installed and maintained by a third party [53]. Power system faults could be
12
caused by many different reasons and one prior one is the tree/vegetation/bush-related faults
(non-metallic short-circuit faults). It has been estimated that 80% of all vegetation-related
problems with power systems are the result of falling trees or branches, often from trees
that are off the electric utility’s right-of-way [54]. This may lead to the falling and breakage
of the line conductor. Therefore, current may flow for a long duration with high-energy,
causing the vegetation to dry, leading to high-temperature arcing and eventually start a fire.
This is one of the most common and critical cases for wildfires studies.
Power lines have caused more than 4,000 wildfires in Texas in the past 3 to 4 years, where
30% of cases are reported to be downed-line faults [55]. Vegetation can also cause wildfires
through non-mechanical mechanisms. For example, a branch could simultaneously contact
a phase conductor and neutral. As a result, a very low current flow can occur (e.g. tens of
milliamps) for a long period of time. This can result in charring of the vegetation, with the
end result of an arcing fault [56]. The phenomenon could also happen on a phase-to-phase or
phase-to-phase-to-ground basis. Non-mechanical vegetation faults are relatively uncommon
on single-phase basis and more common for phase- to-phase fault conditions, because of
higher voltage gradients. Line failure is another reason for power-system-related wildfires.
Short circuits in the power system lead to electromagnetic forces between conductors
carrying current. The conductors, even at non-faulted locations, begin to swing as the result.
The higher the fault current, the larger the physical motion. This movement could lead to
conductor clash and create a consequent fault with the possibility of arcing [57]. The wire
slap may also happen under windy conditions. This phenomenon can ignite combustibles
directly if close to vegetation. Also, the ejection of hot metal particles can ignite dried
vegetation on the ground or any combustible such as transformers’ oil.
Other faults include back-fed faults which are ‘wire down’ faults caused by a fallen con-
ductor (on a radial system) that remains connected to electrical source from the ‘downstream’
side where the line is broken; pole-related failures such as corrosion of structures, bare
conductors, insulated/covered conductors, conductor swing and uplift into the structure, live
13
down-wire and a host of pole components, such as long rod insulator, spindle, post insulator,
cross arm, jumper, lug and crimp, conductor joint, pole twist, stay, pole-top fire and pole
foundation and soil mechanics failures; electrical apparatus failures which means the failure
of power system equipment such as transmission and distribution lines, an explosion of
transformers, failure of breakers, switches, clamps, bushings, surge arrestors or capacitors
might be a reason for initiation of a wildfire [58, 59].
2.3 Wildfire Impacts on Power Systems
Increases in the size and frequency of wildfires in California would affect the state’s major
electricity transmission lines. Interestingly, these transmission line-related impacts resulted
from wildfires are not limited to the actual destruction of the structures. In the case of
an intensive wildfire, in a forest for example, the wooden poles would likely catch fire
and the conductors would melt. So, the line will fall in ruins and must be reconstructed.
Nevertheless, there are many small or moderate wildfires having an extensive front length,
in places with combustibles of low height that cause thermal stress to the power lines. In
these situations, the transmission capacity of a line can be indirectly affected by the heat,
smoke, and particulate matter from a fire, even if there is no actual damage to the physical
structure [5]. For example, the combined impacts of temperature and ash will significantly
reduce the gap’s insulation strength, and the breakdown of the air gap results in outages of
the electrical transmission line. Soot can accumulate on the insulators that attach the lines to
the towers, creating a conductive path and causing leakage currents that may force the line
to shut down. Ionized air in smoke can act as a conductor, causing arcing, either between
lines or between lines and the ground, that results in a power line fault and potential outages
in the power grid [60].
Additionally, the surface temperature increase of the conductor can affect the conductor’s
rate of annealing and reduce its tensile strength. It is well known that the degradation in
strength of the conductor exposed to high temperatures is cumulative. This loss of strength
14
of the conductor is a result of cumulative annealing during its lifetime. When the wires of
a conductor are heated, two processes take place: recovery and re crystallization. During
recovery, some of the non-mechanical properties of the metal may change [61]. All these
effects finally lead to the violation of safety clearance between the conductor and the ground
due to excess conductor sag. Finally, even if the power lines are protected from fires, the
effects of firefighting can also negatively affect transmission operation either by aircraft
dumping loads of fire retardant that can damage the lines or through preventive shutdowns
for safety measures.
2.4 Wildfire Mitigation Strategies
Vegetation Management. One useful approach to stop the spread of the wildfire is to build
an isolation zone. The principle of this approach is to isolate combustibles from the source of
the fire. Since no combustibles are available in the area, the fire will automatically extinguish.
Fire isolation belts are mainly used to prevent a fire from spreading and expanding when
a fire occurs in a piece of building or a large area of forest with a further fire resistance
level because the fire cannot be extinguished quickly. Power distribution company Southern
California Edison (SCE) inspects approximately 900,000 trees annually and prunes nearly
700,000 of them per year, including 400,000 trees in high fire risk areas. The company also
frequently monitors trees outside SCE’s designated pruning zones that could potentially fall
into lines to determine whether they are dead, dying, diseased or hazardous [62].
Sensors and Situational Awareness Technologies. Usually, the wildfires start from
a small region and spread quickly with certain wind conditions. Many accidents can be
prevented if the information about the fire can be collected timely before the situation gets
serious. In [63], the authors proposed a distributed control framework designed for a team
of unmanned aerial vehicles to help closely monitor a wildfire in open space, and precisely
predict its development. This method is promising because the vehicles can replace humans
in hazardous fire tracking and significantly reduce operation costs. In 2013, SCE completed
15
a system-wide meteorological study and used the updated wind speed data to implement
new pole designs and construction standards appropriate for expected conditions. SCE then
launched a comprehensive pole replacement program in 2014, concentrating first on poles
located in areas that posed both high wind and high fire risks, and assessing those poles
against the updated wind standards. When extreme weather conditions are forecasted, the
company restricts certain types of work and does not automatically re-energize distribution
power lines in high fire risk areas after a circuit interruption. Under normal conditions, the
grid automatically tests the circuit and, if the fault condition no longer exists, the circuit is
quickly re-energized. During extreme fire conditions, affected circuits are not automatically
re-energized and SCE crews physically inspect the lines before they are re-energized.
Public Safety Power Shutoff. Another operational practice that can reduce fire risk
is Public Safety Power Shutoff (PSPS). When there is a high probability that an extreme
wildfire would happen, electrical companies might shut down power preemptively in limited,
high fire risk areas. While this method is planned only as a “last resort,” millions of
customers experienced blank-out in practice.
Resilience Zone and Microgrids. To mitigate the effects of PSPS, the electric utilities,
particularly the ones in California, have laid out a raft of proposals, from sectionalism
portions of the grid to reducing the scope of outages during PSPS events, to building
“resilience zones” which could extend emergency backup power to services such as grocery
stores and gas stations in areas suffering from a PSPS. The preparation work includes re-
configuring segments of the distribution system that can be quickly isolated from the broader
grid, and reinstalled interconnection hubs (a transformer and associated interconnection
equipment, ground grid, and grid isolation and protection devices) where Pacific Gas and
Electric Company (PG&E) can connect the “temporary mobile generation” to energize the
area. Some novel concepts, like working with non-utility partners to install or utilize on-site
generation or distributed energy resources for continuous power during safety outages and
exploring the potential for true microgrid systems to play a role are also proposed and under
16
investigation. The ability to island (to disconnect completely from the centralized grid) at
key times can allow for the sustained backup generation to critical facilities in communities
working to respond and recover from wildfires and other natural disasters.
Hardening Power Lines. The design of the infrastructure can be strengthened by
increasing the use of fire-resistant poles, composite cross arms, and covered conductors in
select high fire risk areas.
The following Figure 2.3 illustrates a general overview of the wildfire morphology,
highlighting its causes, and preventive solutions at each stage.
Figure 2.3: Flow of events, causes, and preventive solutions [5]
2.5 Pathways to Enhanced Resilience
2.5.1 Resilience in Modern Power Grids
The road to a resilient power grid is cluttered with a myriad of formidable challenges.
Resilience, as defined by the Cabinet office of the government of the United Kingdom,
is “the ability of assets, grids, and systems to anticipate, absorb, adapt to, and/or rapidly
recover from a disruptive event." This definition identifies four components of a resilient
electricity grid: fault-tolerance, fast response, maintenance and recovery, and reliability.
These aspects should be tailored into the power grid operation and planning paradigms
to ensure its resilience to natural disasters. Note that large and centralized power plants,
bulk transmission lines, substations, and transformers are potential points of vulnerability
in power systems since an uncontrollable minor incident could lead to the interruption of
17
megawatt flows [64, 65].
Both long-term and short-term strategies for enhancing grid resilience against extreme
conditions have been addressed in the literature. In the former, enhancing the grid structural
resilience is primarily the focus of concern, and suggestions are made toward “grid harden-
ing” plans through reinforcement, preventive maintenance of critical infrastructure [66–80],
vegetation management, and efficient allocation of flexible energy resources (e.g., storage
units). In the latter, improving the operational resilience is targeted through fast emergency
response and remedial actions, defensive islanding and micro-grids. Power system resilience
is quite a complicated concept with many driving factors. A number of past research has
identified that the high variability and intermittency of renewable energy have limited their
integration and storage largely to day-to-day normal operating conditions [81–100], where
some solutions in response are proposed. To efficiently deal with weather-driven HILP
incidents in a timely and effective manner, today’s operational planning tools should smartly
account for high levels of unpredictability. They should take advantage of environmentally-
friendly renewable as possible grid-support resources equipped with both analytical and
human insights in deriving resilience plans and emergency response strategies [64]. The GW
Laboratory researchers have studied the resileince challenges that the power grid faces to a
wide range of threats and have proposed solutions for detection, verification, and mitigation
in response [16, 17, 27, 29–32, 37, 101–117].
2.5.2 Techniques to Enhance Resilience against Wildfires
Several papers have done some research on enhancing the resilience of the power system
to extreme fire conditions. In [118], the thermal rating of the at-risk lines was dynamically
adjusted to reduce the loading of the line counteract the heat gained from the fire. S. Dian
et al. [60] have proposed a line outage model (LOM), based on wildfire prediction and
breakdown mechanisms of the air gap, to predict the breakdown probability varying with
time and the most vulnerable poles at the holistic line scale. Reference [119] proposed
18
a dynamic line rating of the overhead lines in order to model the impact of wildfires on
conductor temperature and flowing current. Transmission faults caused by recent wildfires
in California have induced the disconnection of utility-scale converters in power plants.
Postmortem investigations reported that tripping commands were caused by phase-locked
loops and dc-side dynamics, which are typically unmodified in classical transient stability
studies. To address this shortcoming, [120] set forth a positive-sequence model for PV power
plants that are derived from physic’s and control’s first principles. Instances of the developed
model are integrated into illustrative power systems containing conventional generators.
Numerical simulations of the obtained multi-machine multi-converter power systems were
assessed via a suitable set of stability and performance metrics. Reference [121] proposed
a method for enhancing the resilience of distribution systems that are exposed frequently
to disastrous conditions. In case of a fault, the distribution system is sectionalized into
self-sufficient microgrids to prevent a possible spread of the fault with the minimum load
shedding of the islanded portion of the distribution system. A new ultra-fast circuit breaker
technology is introduced that enables a new approach to the optimal protection of rural lateral
lines. A novel protection scheme is presented and is shown to represent the optimal solution
for both reliability performance and prevention of electrical faults causing wildfire ignition
on extreme weather days. For fault levels typical on rural lateral lines, when switched off
in less than a half cycle, there is inadequate energy in the arc to cause fire ignition. This
alternate protection mode can be remotely commanded over SCADA to allow rapid network
response to extreme weather. Recent research efforts on wildfires and the technological
developments can be found in [122–129].
19
Chapter 3: Wildfire Hazard Model and Formulations
3.1 Introduction
In this chapter, a model for the fire flame is formulated to present its impacts on the power
distribution system in the later research. The rest of this chapter is organized as follows. First,
the propagation mode of the flame will be discussed, where the equations corresponding to
each model are presented and explained. Afterward, the dynamic line rating constraints will
be addressed to demonstrate the influences of wildfires on the power system in detail. At
last, some uncertain factors related to the fire will be stressed to make it a general model.
3.2 Wildfire Model
As a process, a wildfire can take many forms, all of which involve a chemical reaction
between combustible species and oxygen. In other words, fire is an oxidation reaction
releasing thermal energy to the surroundings. Heat transfer is a process that concerns the
exchange of thermal energy between physical systems. Heat transfer is classified into three
basic mechanisms, such as thermal conduction, thermal convection, and thermal radiation.
While these mechanisms have distinct characteristics, they often occur simultaneously in
the same system.
Thermal Conduction is the transfer of internal energy by microscopic collisions of
particles and the movement of electrons within a body. The colliding particles, which
include molecules, atoms, and electrons, transfer disorganized microscopic kinetic and
potential energy, jointly known as internal energy. Conduction takes place in all phases:
solid, liquid, and gas. The rate at which energy is conducted as the heat between two
bodies depends on the temperature difference (and hence temperature gradient) between
the two bodies and the properties of the conductive interface through which the heat is
20
transferred [130].
Heat Convection occurs when the bulk flow of a fluid (gas or liquid) carries heat along
with the flow of matter in the fluid. The flow of fluid may be forced by external processes, or
sometimes (in gravitational fields) by buoyancy forces caused when thermal energy expands
the fluid (for example in a fire plume), thus influencing its transfer. The latter process is
often called "natural convection". All convective processes move heat partly by diffusion, as
well. Another form of convection is forced convection. In this case, the fluid is forced to
flow by use of a pump, fan, or other mechanical means.
Thermal Radiation is the electromagnetic radiation generated by the thermal motion of
particles in matter. All matter with a temperature greater than absolute zero emits thermal
radiation. Particle motion results in charge-acceleration or dipole oscillation which produces
electromagnetic radiation.
In our case, the heat from wildfire is transferred through radiation and convection. But
since convection affects the conductors’ temperature in a short distance—i.e., nearly under
the power line, it is most likely that a power line will already be out of order when the
convection happens. So in this Section, only radiation transfer is considered. For a large
fire incident, the flame model is based on the radiant surface approach shown in [131]. This
approach does not account for the flow and the fire dynamics and only represents the flame
as a radiant surface (solid-flame assumption). The model relates to the flame front width and
the estimation of the flame front. The radiative heat flux φ f from the whole flame emitted to
a conductor is then determined considering the geometry of the flame and the properties of
the fire front according to:
φf
i j,ω,t =τ · ε f ·B ·T f 4
2· sin(θ f
i j,ω,t)(3.1)
where τ , ε f and B are all parameters related to the environment. T f is the temperature
of the wire front set as 1200k [132], and θ f is the view angle between the threatened line
21
and the fire front expressed in the following:
θf
i j,ω,t = tan−1(L f · cos(α f )
d fi j,ω,t− (L f · sin(α f ))
) (3.2)
In the equation above, L f is the length of the fire front. d f represents the distance
between wildfire and the conductor line affected and is computed in Equation (3.3).
d fi j,ω,t = d f
i j,ω,t−1 ·Vf
ω,t ·∆t · cos(σwindi j,ω,t) (3.3)
V fω,t =
k · (1+V windω,t )
ρb(3.4)
V f (m/s) is the specific rate of flame spread in wildland on a flat ground that depends on
the wind speed V wind(m/s). ρb is the bulk density of the fuel equal to 40 kg/m3 in the forest.
k is equal to 0.07 for wildland fire and 0.05 for wood crib [118].
3.3 Dynamic Line Rating (DLR): Heat Balance Equation
In practice, power line thermal ratings are calculated seasonally assuming the given or
forecasted weather conditions. It is worth mentioning before giving basic aspects of the
DLR that the current-temperature relationship of power line conductors can be computed by
the guidelines presented in the IEEE Std. 738 and the CIGRE SC.B2 WG 22.12. According
to the comparison in [133], one can demonstrate that they provide essentially the same
results. It is assumed that the conductor cross-sections are circular.
The power line conductor’s total heat is the multiplication of coefficients and heat loss
rates, which include the convective heat loss rate qcon and the radiative heat loss rate qrad ,
and heat gain rates, which include ohmic losses resistant of the power line qline, radiative heat
flux from fire q f ire and solar heat gain rate qsun. Therefore, any changes in the temperature
for any time interval is calculated using the following non-steady-state heat equation which
22
is a first-order nonlinear differential constraint:
(Ti j,ω,t+1−Ti j,ω,t) =∆t
mCp· (qline
i j,ω,t +qsuni j,ω,t +q f ire
i j,ω,t−qconi j,ω,t−qrad
i j,ω,t) (3.5)
Each of the terms in the introduced equation is explained in the following sections
3.3.1 Heat Gain
The heating terms in the given equation are the solar heat energy that can be absorbed by
the conductor, the resistive thermal energy generated due to currents flowing through the
power line conductor and the fire radiation heat calculated as follows:
qsuni j,ω,t = Di j ·Ki j ·φ sun
i j,ω,t (3.6)
qlinei j,ω,t = Rline(Ti j,ω,t) · (Ii j,ω,t)
2 (3.7)
q f irei j,ω,t = Di j ·φ f ire
i j,ω,t(3.8)
In Equation (3.6), Di j is the diameter of the conduction lines and φ suni j,ω,t is the sun
radiation rate. Ki j is the solar absorptivity that varies between 0.27 for the bright stranded
aluminum conductor and 0.95 for the weathered conductor in an industrial environment. It
is equal to the ratio of the solar heat absorbed by the conductor to the solar heat absorbed by
a perfectly black body of the same shape orientation and increases with age for the overhead
power lines. This value for a brand new power line is 0.2 to 0.3. A typical value for a
conductor that has been in use for more than 5 (in industrial environments) to 20 years (in
rural clean environment) is 0.9. A value of 0.5 is often used if nothing is known about the
conductor absorptivity [134].
23
In Equation (3.7), Rline(Ti j,ω,t is a function that describes the relationship between the
resistance of the power line conductor and its temperature. Ri j,a is the resistance of the line
at ambient temperature Ti j,a (298k). di j is the conductor thermal resistant coefficient.
Rline(Ti j,ω,t) = Ri j,a · (1+di j · (Ti j,ω,t−Ti j,a)) (3.9)
3.3.2 Heat Loss
The last two terms in Equation (3.5) accounts for the cooling down of the power line
conductor. The convection loss in this paper is forced convection meaning the conductor is
cooled down via a cylinder of moving air around the conductor. The convection heat loss
is the largest value between high-speed wind qconi j,ω,t,(high) and low-speed wind qcon
i j,ω,t,(low)
according to the IEEE standard [135]. Equation (3.10) and Equation (3.11) represent the
calculation of the convection loss.
qconi j,ω,t,(high) = Kangle ·0.754 ·N0.6
Re · ka · (T −T a) (3.10)
qconi j,ω,t,(low) = Kangle · [1.01+1.35 ·N0.52
Re ] · ka · (T −T a) (3.11)
The magnitude of the equation depends on NRe, the Reynolds number and wind direction
factor Kangle given by:
NRe =Di j ·ρα ·V wind
ω,t
µα(3.12)
Kangle = 1.194− cos(σwindi j,ω,t)+0.194cos(2σ
windi j,ω,t)+0.368sin(2σ
windi j,ω,t) (3.13)
24
Next, the cable radiated heat rate can be described by the following equation:
qradi j,ω,t = 17.8Di j · ε · [(
Ti j,ω,t
100)4− (
Ti j,a
100)4] (3.14)
Equation (3.7), Equation (3.10), Equation (3.11) and Equation (3.14) make the heat
balance equation non-linear, non-convex and, consequently, hard to solve. In the following
section, we propose a methodology to effectively solve this problem.
3.3.3 Convexification of the Nonlinear Terms
The heat gain results from the ohmic losses, presented in Equation (3.7), is proportional
to the multiplication of the square of the current flow and conductor resistance. For an
ohmic conductor, as shown in Equation (3.9), the resistance can be calculated by a function
of conductor temperature. In order to convexify the heat produced by the current, we
considered that the resistance of the conductor is a constant value equal to its maximum at
the highest temperature Ti j,(max). Also, the voltage is considered close to 1 p.u.. Applying
this method, the current flow is equal to the apparent power flow and the equality constraint
Equation (3.7) is relaxed to an inequality one given by:
qlinei j,ω,t ≥ Rline(Ti j,(max) · (|P
f lowi j,ω,t |
2 + |Q f lowi j,ω,t |
2) (3.15)
This relaxation is accurate concerning the considered approximations when the temper-
ature is less than the maximum permissible one and this inequality constraint above will
be identical to the equality one. The error introduced by such an approach is progressively
reduced when the conductor line temperature comes closer to its maximum limit. Since we
are interested to better exploit the lines during the wildfire conditions when they reach their
maximum operating temperatures, the error in this approximation is acceptable.
The convection heat loss can be considered as a function of wind speed, the difference
between ambient temperature and internal conductor temperature, and altitude. As shown in
25
Figure 3.1, the relations between the convection heat at each generic wind speed and the
difference of the conductor temperature with the ambient temperate are linear. However,
since the equation is different for high and low wind speeds, as discussed before, the one
producing the largest heat losses should be used. In this study, we can use the maximum of
the slope calculated for high and low wind speeds as inputs to the optimization problem.
The radiation heat loss rate can be piece-wise linearized. The radiation heat loss depends
Figure 3.1: Linear relationship for convection heat loss in case of non-zero wind speed(T a=273k)
on the fourth power of the conductor temperature as shown in Equation (3.14). To piece-
wise linearize this term, it is written as a function of the conductor temperature with a
domain between the maximum conductor temperature and the ambient temperature. This
approximation is shown in Figure 3.2 where the term is evenly divided into three pieces.
The proposed piece-wise linearization approximations can be replaced with a linear
approximation to avoid binary variables and this simpler approximation brings a very small
error but effectively decreases the computational time. The radiation heat rate is finalized as:
26
Figure 3.2: Piece-wise linearization of the radiation heat loss (T a=273k)
qradi j,ω,t = a ·Ti j,ω,t +b (3.16)
3.4 Uncertainties
The parameters related to the environment such as wind speed, wind direction and solar
radiation are considered uncorrelated but with uncertainties.
3.4.1 Environment Parameters
A large number of experiments have summarized that the stochastic wind speed approxi-
mately follows the Weibull and von Mises distributions. Suppose that the stochastic wind
speed V wind is a stochastic quantity with the following probability density function:
f (V wind) =kindex
Ck ·Vkindex−1 · e(−V/C)kindex
(3.17)
where kindex and C are the shape index and the scale index of the Weibull distribution.
27
Equation (3.17) could be depicted as Figure 3.3.
Figure 3.3: Weibull distribution of the wind speed
In this thesis, a k-factor of 2 is assumed. The standard deviation is considered equal
to 15% of the mean value. With the above probability distribution, the wind speed and
direction can be used as inputs to the optimization engine. The mean value and the standard
deviation of the wind speed is shown below.
Figure 3.4: Wind speed mean value and stochastic values
10, 50, and 100 series of values are generated and considered in the test cases. All these
28
stochastic numbers representing uncertainties are generated in Matlab.
Figure 3.5: Wind direction mean value during a 24-hour period
The illumination intensity is considered the main factor affecting the output power of
the photovoltaic panels. Suppose that the stochastic illumination intensity also follows the
Weibull distribution. The mean value and normal distribution value of the solar radiation
data is exploited below in Figure 3.6.
Figure 3.6: Solar radiation mean value and stochastic values
29
3.4.2 Fire Parameters
The stochastic parameters of wind are substituted into the constraints in Section 3.2 and
Section 3.3. We can get the distance between the wildfire and the threatened conductor line
in Figure 3.7.
Figure 3.7: Distance between the power line conductor and fire in different scenarios
It can be concluded that the fire reaches the conductor line in the last hours of the
simulation. And the slopes of the lines are similar with different scenarios.
Figure 3.8 presents the fire radiation rate. Comparing the two figures, it is observed that
the heat is greater than zero when the distance between the fire and the threatened line is
less than 400m and increases sharply after the fire comes to a close distance. Also, one can
observe that the time when the heat flux reaches the peak point is different but is, in this
case, later than 18 hours.
3.5 Conclusion
Wildfire affects the power line conductors in different ways, but the most evident is the heat
exchange with the line. In this chapter, the thermal energy exchange between the wildfire
30
Figure 3.8: Conductor heat gain rate from the fire in different scenarios
and overhead lines is quantified using the heat balance equation. After the convexification
of the nonlinear terms in the equation, the optimization problem is transformed into a
mixed-integer problem that has quadratic constraints and needs to be solved using a CPLEX
solver. Due to the uncertain nature of the environmental factors, the typical distributions
for the stochastic parameters are used to generate different series of scenarios and account
for such uncertainties. In this way, the heat flux from the fire, the wind speed, the solar
radiation, and the parameters of convection loss is computed and will be used as the inputs
to the optimization engine developed in the following chapters.
31
Chapter 4: Power Distribution System Optimization Model for Enhanced
Wildfire Resilience
4.1 Introduction
This chapter investigates the optimization model for resilient operation of the distribution
system (DS) when the wildfire approaches the system. The chapter is organized as follows:
first, a mixed-integer linear programming (MILP) optimization model is developed for
the DS considering the wildfire model presented in Chapter 3. Afterward, the parameter
uncertainties are accounted for through 10, 50, and 100 scenarios generated using Beta
Distribution and Weibull Distribution and integrated into the optimization model for testing
and evaluations.
4.2 Formulation
In this thesis, we consider a 33-bus distribution system (DS) that consists of the energy
storage system (ESSs) and distributed generators (DGs). The wind turbines (WTs) and
solar photo-voltaic (PVs) are modeled as stochastic resources while micro turbines (MTs)
are considered fully controllable. In this section, we discuss only a base case condition
meaning that the wildfire affects only one power line (line 1-2) which connects the DS with
the upstream network as shown in Figure 4.1.
A stochastic optimization model is used for enhancing resilience when a wildfire ap-
proaches the DS. Although resilience relates directly with load shedding according to [136],
operating costs should also be considered to further provide the most economic solution.
Accordingly, the DS operates in the most efficient way at a minimum load shedding cost
during the emergency operating conditions. Thus, the objective function is designed to
32
Figure 4.1: Single-line diagram of the modified 33-node power distribution system
33
minimize the expected social cost as expressed below:
min(NT
∑t=1
NΩ
∑ω=1
πω ·NB
∑i=1
(VoLL · pshedi,ω,t− cD · pD
i,ω,t)
+NT
∑t=1
NΩ
∑ω=1
πω ·NB
∑i=1
(cMT · pMTi,ω,t)
+NT
∑t=1
NB
∑i=1
(suMTi,t + sdMT
i,t ))
+NT
∑t=1
NΩ
∑ω=1
πω · cUPt · (pUPB
ω,t − pUPSω,t )
(4.1)
In the first line, VoLL · pi,ω,t represents the load shedding cost and cD · pDi,ω,t represents
the revenue from providing energy to the end customers. The second and third terms
represent the generation, start-up, and shut down costs of MT. And the last term represents
the power exchange cost with the upstream network. To represent the operation of a DS,
multiple constraints should be considered. The optimization model includes components
constraints, power balance constraints, and linearization constraints.
4.2.1 Renewable Generation Constraints
In this thesis, multiple distributed generators are connected to the system as shown in
Table 4.1. In this section, the output power of renewable generators will be discussed.
Table 4.1: Location And Capacity of Renewable Resources
Type Bus Capacity (MW)
WTs 14/16/31 0.8/0.8/0.8PV 11 0.5MTs 8/13/16/25 3/2/2/3ESSs 19/26 0.5/0.5
Based on the known probability distribution function of the wind speed expressed in
Chapter 3, the relationship between the output power of a wind generating unit and the wind
34
speed can be formulated as follows:
PWTi,t = 0, 0≤ v≤ vciorvco ≤ v
PWTi,t = Pw
rated · (v− vci
vr− vci), vci ≤ v≤ vr
PWTi,t = Pw
rated, vr ≤ v≤ vco
(4.2)
Where v is the wind speed at the hub height of the wind unit; vci, vco, and vr are,
respectively, the cut-in wind speed, the cut-out wind speed, and the rated wind speed; and
Pwrated is the rated output power of the wind unit [137]. In this thesis, the cut-in, cut-out,
and rated wind speeds are assumed 4, 20, and 12 m/s, respectively. Thus, the relationship
between the output power of a wind turbine and the wind speed at the given height could be
shown as Figure 4.2.
Figure 4.2: Relationship between the output power of a wind turbine and the wind speed
As for the solar power, the illumination intensity is usually considered the dominant
factor affecting the output power of the solar panel. The relationship between the illumination
35
intensity and the output power of a solar generating source can be described as follows:
PSi,t = PS
rated · (SSr), 0≤ S≤ Sr
PSi,t = PS
rated, Sr ≤ S(4.3)
In the equations above, we have S as the illumination intensity, Sr the rated value, and
PSrated the rated output power of the solar cells.
The relationship between the output power of a PV unit and the illumination intensity
could be shown as in Figure 4.3.
Figure 4.3: Relationship between the output power of a PV and the solar radiation
The rated illumination intensity of the PVs is set at 1000 W/m2. The solar and wind data
and the corresponding uncertainties are tackled through the scenario approach described
earlier in Chapter 3.
4.2.2 Micro Turbine Constraints
As shown in Table 4.1, several MTs are connected to the DS. In this optimization model, the
active and reactive output power of MTs and their start-up and shut-down costs need to be
36
considered as follows to guarantee the power balance in the system at a minimum cost.
pMTi(min) ·u
MTi,t ≤ pMT
i,ω,t ≤ pMTi(max) ·u
MTi,t (4.4)
qMTi(min) ·u
MTi,t ≤ qMT
i,ω,t ≤ qMTi(max) ·u
MTi,t (4.5)
suMTi,t ≥ 0,suMT
i,t ≥ cSUi · (uMT
i,t −uMTi,t−1) (4.6)
sdMTi,t ≥ 0,sdMT
i,t ≥ cSDi · (uMT
i,t −uMTi,t−1) (4.7)
where Equation (4.4) and Equation (4.5) set the limits for MTs; pMTi(max) is the maximum
output power of the turbine as shown in Table 4.1, while pMTi(min) is set as 10 percent of the
capacity of the unit. Equation (4.7) and Equation (4.6) reflect the start up and shut down
costs of the turbines. The binary variable ui,t is used to determine the status of MTs, 1 for
start up and 0 for shut down. The corresponding prices cSDi and cSU
i are considered the same
and set to $300.
4.2.3 Energy Storage System (ESS) Constraints
The system also consists of two ESSs as shown in Figure 4.1, the operations and limits of
which can be expressed using the following constraints:
socSTi,ω,t = socST
i,ω,t−1 +(nST
i · pChi,ω,t · (
∆t3600)
ESTi
)− (pDC
i,ω,t · (∆t
3600)
nSTi ·EST
i)) (4.8)
socSTi,(min) ≤ soci,ω,t ≤ socST
i,(max)(4.9)
37
0≤ pChi,ω,t ≤ pCh
i,ω,t,(max) ·usoci,ω,t (4.10)
0≤ pDCi,ω,t ≤ nST
i · pDCi,ω,t,(max) · (1−usoc
i,ω,t) (4.11)
qESSi(min) ≤ qESS
i,ω,t ≤ qESSi(max)
(4.12)
socSTi,ω,tend
≥ socthres (4.13)
In the equations above, Equation (4.8) calculates the state of charge (SoC) of ESSs. The
limitation on the SoC of ESSs is set by Equation (4.9). Equation (4.10) and Equation (4.11)
guarantee that the active power charged or discharged by ESSs is within the limits consid-
ering their operation mode. Equation (4.12) represents the reactive power limits of ESSs.
Equation (4.13) is to ensure that the SOC of ESSs is above a certain threshold socthres at
the end of the simulation. nSTi is the conversion efficiency of the ESSs, EST
i represents the
energy capacity, pChi,ω,t and pDC
i,ω,t are respectively the charging and discharging active power
of the ESS, and ∆t is the duration of time intervals. The parameters related and the node
connected to the ESSs are shown inTable 4.2. The boundary socthres is set to 30 percent. A
time step of 30 minutes is chosen since it is a suitable time step for DLR models [138].
Table 4.2: Location And Capacity of Renewable Components
Energy Storage System EST (MWh) pCh/DCi,ω,t(max)(MW) qESS
max(MVAr) qESSmin (MVAr)
ESS(node 19) 1.5 0.5 0.3 -0.3ESS(node 26) 1.5 0.5 0.3 -0.3
38
4.2.4 Power Balance Constraints
Each bus should maintain a real and reactive power balance between the generated power and
demanded loads. With constraints listed in subsections above, the power balance constraints
of the system will be as in Equation (4.14) and Equation (4.15).
NB
∑i=1
P f lowi,ω,t = pMT
i,ω,t + pWTi,ω,t + ps
i,ω,t + pUPω,t + pCh
i,ω,t− pDCi,ω,t− pD
i,ω,t (4.14)
NB
∑i=1
Q f lowi,ω,t = qMT
i,ω,t +qChi,ω,t−qD
i,ω,t (4.15)
−M1 ∗ull,ω,t ≤ p f P
i,ω,t ≤M1 ∗uli j,ω,t (4.16)
−M1 ∗ull,ω,t ≤ p f Q
i,ω,t ≤M1 ∗uli j,ω,t (4.17)
Equation (4.16) and Equation (4.17) allow the power flow through each line only when
uli j,ω,t is equal to 1 meaning that the line is functional and online. The large enough
positive number M1 value is a relaxation parameter. The variables pDi,ω,t and qD
i,ω,t are the
demanded active and reactive power served to the customer. They are calculated by the load
shedding pshedi,ω,t and the original demand of each node Pi,ω, tDemand in Equation (4.18) and
Equation (4.19).
pDi,ω,t = Pdemand
ω,t − pshedω,t (4.18)
39
qDω,t = Qdemand
ω,t −qshedω,t (4.19)
0≤ pshedi,ω,t ≤ Pi,ω, tDemand (4.20)
qshedi,ω,t = pshed
i,ω,t ·Qi,ω, tDemand
Pi,ω, tDemand(4.21)
In Equation (4.14), the active power pUPω,t represents the power exchange with upstream
system during the optimization. It depends on the energy buying from or selling to the
main grid and needs to be limited as shown in Equation (4.22), Equation (4.23), and
Equation (4.24). The binary variable ϕUPω,t is used to determine buying (1) or selling (0)
energy during the studied time horizon. The maximum energy import from and export to
the main grid are set 12MWh and 4MWh.
pUPω,t = p
UPbuyω,t − pUPsell
ω,t (4.22)
0≤ pUPbuyω,t ≤ p
UPbuymax ·ϕUP
ω,t (4.23)
0≤ pUPsellω,t ≤ pUPsell
max · (1−ϕUPω,t ) (4.24)
4.2.5 linearization Constraints
Based on the DistFlow branch equations in [139], constraint (4.25) and (4.26) represent the
power flow equation. The large enough positive number M2 value is a relaxation parameter
to relax these two constraints for open branches. Constraint (4.27) states the boundary for
40
the nodal voltage magnitudes across the network.
V sqri,t−V sqr j,t ≤(1−αi j,t) ·M2 +2 · (ri j · p fi j,t + xi j ·q fi j,t), ∀(i, j) ∈ L,∀t ∈ T
(4.25)
V sqri,t−V sqr j,t ≥(αi j,t−1) ·M2 +2 · (ri j · p fi j,t + xi j ·q fi j,t), ∀(i, j) ∈ L,∀t ∈ T
(4.26)
Vsqri≤V sqri,t ≤ Vsqri,∀i ∈ B,∀t ∈ T (4.27)
4.3 Case Study
The wild model proposed in Chapter 3 is applied in order to optimize the DS operation
during an approaching wildfire in the next 24 hours. In this case, only line 1-2 which
connects the DS with the main grid is affected by the wildfire. The proposed model was
solved using GAMS and CPLeX solver considering different numbers of scenarios on wind
speed, wind direction, and solar radiation.
The results on the expected load shedding for each case are presented in Figure 4.4.
Between hours 19 and 24, the demand is not met due to the disconnection of line 1-2 and
Figure 4.4: Expected Load Shedding for 10, 50 and 100 Generated Scenarios
the low output power of DGs and ESSs. After hour 22, the wind is increasing so the load
shedding is not recorded for some cases. According to the uncertainty analysis, the line
41
affected is most likely to be out of service after 18 hours, and the load shedding is increasing
with more scenarios being considered. This is reasonable since the uncertainty makes it
harder for the system to go back to normal.
Table 4.3 shows the value of the objective function, the load shedding, and the total
Cplex time.
Table 4.3: Simulation Results Concerning Different Numbers of scenarios
# of Scenarios OF ($×103) Load Shedding (MW) Computation Time (s)
10 -52.31 2.52 21.550 -52.32 2.67 791.56100 -52.95 2.96 3802.5
10 scenarios seem to be the best choice since the computation time is the least while
the results are similar to other cases. In the following sections, we will consider only 10
scenarios.
The expected energy exchange with the upstream network is presented in Figure 4.5.
Figure 4.5: Expected energy exchange with the upstream system
It is observed that until time 16, the power is only bought from the upstream system.
Considering the power exchange price in Figure 4.6, the reason is that the price of MT
42
Figure 4.6: Power exchange price with the upstream system
generation (80$/MWh) is greater than the buying price from the upstream network before
time 16, and the system cannot balance with only renewable generators and ESSs. After
16 hours, the power exchange has a higher price; accordingly, to make most revenue the
power is generated by MTs and sold to the main grid as in Figure 4.7. At time 20, the energy
exchange cost is still higher than the cost of MTs, but the power is bought from the upstream
network to charge the ESSs to prepare for the outage of line 1-2.
Figure 4.7: Expected generated power and status of MTs
As shown in Figure 4.8, at the beginning of the day, ESSs are charging until time 16 to
maximize the revenue from selling energy to the customers and the upstream network. After
time 18, the discharging power is lower to save for later use when line 1-2 is in outage due
43
to the progressive wildfire.
Figure 4.8: Expected discharging power and SOC of ESS
Table 4.4 presents the load shedding cost, the revenue from selling energy to customers,
the cost of DGs and the cost of exchanging power with the upstream network.
Table 4.4: revenues and costs
Load shedding cost Revenue from customers Generation cost Power exchange cost
3.56 69.16 8.04 4.302.547 67.505 8.099 4.54
4.4 Conclusion
In this chapter, a stochastic mixed-integer linear programming model is developed consider-
ing the wildfire model with different series of scenarios. The proposed model is applied to
an IEEE 33-bus system and the social cost was set as the objective function. The results
show that the 10 scenario model is the proper one for further analysis considering the com-
putational complexity.The proposed optimization model provides the mitigation strategies of
a resilient operation of the power distribution system when facing an approaching wildfire.
44
Chapter 5: Sensitivity Analysis
5.1 Introduction
In this chapter, the proposed model and analysis data in Chapter 4 will be further studied to
investigate the sensitivity of the solutions to changes in each element in the system. Four
cases are studied to get a better understanding of the impact of wildfires on the system
operation and how to minimize the wildfire consequences considering possible uncertainties
in the system elements and how the fire progresses.
5.2 Case Study: Wildfire Affecting Different Overhead Power Lines
5.2.1 Case 1: Wildfire Affecting Different Single Lines
In Chapter 4, we assume that only line 1 (from node 1 to node 2) was threatened by the
approaching fire. In order to show how a wildfire affects different power line and to quantify
the impacts accordingly, in this section all 32 lines are set as possible targets affected
separately by a wildfire. The objective function value and the load shedding cost are shown
below in Figure 5.1. The results demonstrate that when power line 25 (connecting node 6 to
node 26) is out of service, the consequence load shedding is the maximum. This is because
the branch isolated from the system comprises only one ESS and one WT, the capacity of
which can not satisfy the total demand in the system.
This observation can be further supported by the detailed load shedding chart at each
node when line 25 is affected by wildfire in Figure 5.3.
5.2.2 Case 2: Wildfire Affecting Different Double Lines
In this case, the same model in Chapter 4 is applied to cases where the wildfire is considered
to affect two lines at the same time. These two lines need to be connected lines (adjacent).
45
Figure 5.1: Objective function value: sensitivity analysis on every single line affected bywildfire
Figure 5.2: Expected load shedding cost: sensitivity analysis on every single line affectedby wildfire
46
Figure 5.3: Expected load shedding when line 25 effected
Based on the results observed in Case 1, lines 1&2, lines 8&9 and line 22&23 are chosen as
three pairs of lines being affected by a the studied wildfire.
Figure 5.4: Objective function value when 2 lines are affected by wildfire
The objective function values of the base case 1 and case 2 are shown in Figure 5.4.
When the wildfire affect line 1&2, the social cost is the maximum value; this is mostly due
to the load shedding cost depicted in Figure 5.5. When lines 1&2 are affected, the branch
from node 2 to node 22 is separated from the system, and the only ESS left cannot meet
47
the demand, while in other cases, only the node between these two lines is effected. When
Figure 5.5: Expected load shedding when 2 lines are affected by wildfire
line 1 and line 2 are affected simultaneously by the fire, since line 1 is the line connecting
the system to the main grid, there is no energy exchange when it is out of service. In other
situations, the generators and storage systems connected with the separated (isolated) parts
cannot meet their demand, so the energy is always imported from the main grid as can be
seen in Figure 5.6.
Figure 5.6: Expected power exchange when 2 lines are affected by wildfire
48
5.2.3 Case 3: Wildfire affecting different three lines
Case 3 focuses on the situation when 3 lines are disconnected when wildfire approaches
the system. As discussed in case 2, these 3 lines are also connected (adjacent) lines. Line
1&2&3 and line 4&6&25 are the two scenarios selected here for analysis. Figure 5.7 depicts
the social cost in each situation.
Figure 5.7: Objective function value when 3 lines are affected by wildfire
In the first case, node 2 is isolated, while the rest of the DS is divided into two parts: the
individual distribution system, and a single branch from node 19 to node 22 supplied by the
ESS connected to node 19. Since the main grid is unavailable for the system, DGs and ESSs
are critical for maintaining the demand-supply balance. At node 11, only a PV is connected,
and the its capacity is not large enough compared with the demand; also, the ESS in the
isolated branch is not able to satisfy the demand. So the load has to be shed in nodes 2, 11,
19, 21, 27, and 30 (see in Figure 5.8). In the second case, node 6 is isolated, while the rest
part is separated into 3 parts: the upper part of node 5 which connects with the upstream
system, a microgrid system consisting of six generators from node 7 to node 18, 3 and a
branch from node 26 to node 33. The first 2 parts are able to supply the load demand while
49
the third part has only one ESS and one MT, so the nodes of this branch had load shedding
in this situation.
Figure 5.8: Expected load shedding when 3 lines are affected by wildfire
When line 1 is effected by the wildfire, there is no energy exchange with the upstream
network since it is the only transfer line. When other lines are affected, part 1 of the divided
system which connects with the main grid buys the energy from it continuously since it
cannot supply its demand by itself.
Figure 5.9: Expected power exchange when 3 lines are affected by wildfire
50
5.3 Case Study: Modified IEEE 33-Node Distribution System with Different Num-
bers of DGs and ESSs
In this section, one more distributed resource is connected to the system. The capacity of
each element is the same as that in Chapter 4. The results are shown below compared with
those in the base case condition studied earlier. Only one element is added at each time, and
they are all connected to node 30.
Figure 5.10: Objective function value when adding one distributed resource
Considering the difference in the results obtained in each case, it is observed that much
less load shedding would be recorded in the system if a MT was connected. And the
generators and storage systems can decrease the load shedding in general.
The power exchange profile in each case is also shown in Figure 5.12.
When renewable generators are added, they can supply some demand so less power is
bought from the main grid before MT start up at hour 16 and more sold afterward. When
MT is added, it is not working until time 16 when the price of selling the electricity to the
upstream network is greater than the generation cost. After 16 hours, more energy is sold
making more revenue. When ESS is added to the system, more energy is bought at the
51
Figure 5.11: Expected load shedding when adding one distributed resource
Figure 5.12: Expected power exchange when adding one distributed resource
52
beginning for charging purposes while ESS discharging to the grid happens more toward the
end of the studied time horizon.
5.4 Conclusion
In this chapter, different numbers of lines are set out of service as a result of the progressive
wildfire, and DGs are added to the network to analyze their sensitivity and performance
when facing a wildfire. It is recognized that the spatial load shedding depends on the lines
affected by wildfire as well as the position, number, and type of generators and storage
systems in the network. The analysis of this chapter will help distribution system planners
and decision makers better decide on sizing and siting of distributed energy resources across
the network aiming to achieve an enhanced grid-support and resilience against wildfires
should they happen in the future. Future research could include the analysis of the system
in response to wildfires in the presence of different energy storage technologies and grid-
support resources [140–149].
53
Chapter 6: Conclusion
6.1 Conclusion
With the recent increase in the frequency and intensity of wildfires around the world, and the
projection for a higher trend in the years to come, maintaining and enhancing the ability of
the power system to be resilient against such disruptions is a challenge. When the wildfire
approaches the power system, the thermal burden and stress is added and the affected
lines could allow less current flowing through them or even become out of service. It is
significantly important to efficiently and smartly exploit the available resources to minimize
the consequences of a wildfire, e.g., load shedding in the system. Dynamic Line Rating
is considered to model the thermal impacts of wildfires on power lines and a stochastic
mixed-integer linear program (MILP) optimization model is established aiming to minimize
the social cost of the system when exposed to a wildfire.
In Chapter 3, the wildfire hazard model and the associated uncertainties were presented.
Different numbers of scenarios were generated employing the normal distribution approach
to account for the extreme event uncertainty. The non-steady state heat balance constraint
was used to model the impact of wildfires on the temperature of power line conductors. The
results were considered as inputs in the following chapters.
In Chapter 4, the optimization model aiming to mitigate the impacts of a progressive
wildfire on the power distribution system was presented and studied. To explore the ef-
fectiveness of the suggested model, a base case was considered with fire progression. It
is observed that the resilient operation of the system can be achieved with reduced load
shedding and a lower social cost.
In Chapter 5, the impact of various factors in the optimization model was studied. The
same test system with the same wildfire model was optimized with different lines being
affected and different availability of distributed energy resources across the network. The
54
results revealed that the load shedding depends on the combinations of the position (spatial
factor) of the resources and lines being affected as the wildfire progresses (temporal factor).
6.2 Future Research
Future work may include investigating the combination of the thermal and physical influ-
ences of wildfires on the shape of overhead lines since the high temperature can cause sag
to the conductors. Future research may also include the contribution of flexible loads and
the interdependent services in the resilience evaluation of the system [150] and the repair
strategies in facilitating the power system restoration during the post-disaster outage scenar-
ios. Also, the role of intelligent electronic devices (IEDs) [151–156] as well as the existing
and next-generation sensors [157–167] on detecting and preventing the wildfire-triggering
events in power systems should be studied.
55
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