A Generalized Accelerated Failure Time Model to Predict Restoration Time from Power Outages

Abstract

Major disasters such as wildfire, tornado, hurricane, tropical storm, and flooding cause disruptions in infrastructure systems such as power and water supply, wastewater management, telecommunication, and transportation facilities. Disruptions in electricity infrastructure have negative impacts on sectors throughout a region, including education, medical services, financial services, and recreation. In this study, we introduced a novel approach to investigate the factors that can be associated with longer restoration time of power service after a hurricane. Considering restoration time as the dependent variable and using a comprehensive set of county-level data, we estimated a generalized accelerated failure time (GAFT) model that accounts for spatial dependence among observations for time to event data. The model fit improved by 12% after considering the effects of spatial correlation in time to event data. Using the GAFT model and Hurricane Irma’s impact on Florida as a case study, we examined: (1) differences in electric power outages and restoration rates among different types of power companies—investor-owned power companies, rural and municipal cooperatives; (2) the relationship between the duration of power outage and power system variables; and (3) the relationship between the duration of power outage and socioeconomic attributes. The findings of this study indicate that counties with a higher percentage of customers served by investor-owned electric companies and lower median household income faced power outage for a longer time. This study identified the key factors to predict restoration time of hurricane-induced power outages, allowing disaster management agencies to adopt strategies required for restoration process.

Full text

Vol.:(0123456789)
International Journal of Disaster Risk Science
https://doi.org/10.1007/s13753-023-00529-3
ARTICLE
A Generalized Accelerated Failure Time Model toPredict Restoration
Time fromPower Outages
TasnubaBinteJamal1· SamiulHasan1
Accepted: 16 December 2023
© The Author(s) 2024
Abstract
Major disasters such as wildfire, tornado, hurricane, tropical storm, and flooding cause disruptions in infrastructure systems
such as power and water supply, wastewater management, telecommunication, and transportation facilities. Disruptions
in electricity infrastructure have negative impacts on sectors throughout a region, including education, medical services,
financial services, and recreation. In this study, we introduced a novel approach to investigate the factors that can be associ-
ated with longer restoration time of power service after a hurricane. Considering restoration time as the dependent variable
and using a comprehensive set of county-level data, we estimated a generalized accelerated failure time (GAFT) model that
accounts for spatial dependence among observations for time to event data. The model fit improved by 12% after consider-
ing the effects of spatial correlation in time to event data. Using the GAFT model and Hurricane Irma’s impact on Florida
as a case study, we examined: (1) differences in electric power outages and restoration rates among different types of power
companies—investor-owned power companies, rural and municipal cooperatives; (2) the relationship between the duration of
power outage and power system variables; and (3) the relationship between the duration of power outage and socioeconomic
attributes. The findings of this study indicate that counties with a higher percentage of customers served by investor-owned
electric companies and lower median household income faced power outage for a longer time. This study identified the key
factors to predict restoration time of hurricane-induced power outages, allowing disaster management agencies to adopt
strategies required for restoration process.
Keywords Generalized accelerated failure time model· Hurricanes· Investor-owned power companies· Median income·
Power outage· Restoration time
1 Introduction
Hurricanes have become more frequent and intense due to
global climate change. Hurricane induced damages have
significantly increased because of high wind intensities
when some major hurricanes made landfalls in recent years
(Grenier etal. 2020). For instance, Hurricane Irma caused
a damage of about USD 50 billion (Cangialosi etal. 2017)
and significant disruptions in infrastructure systems. After
Hurricane Irma, more than 6.2 million customers lost power
including 850,000 customers from Orange, Seminole, Lake,
and Osceola Counties in Florida (Gillespie etal. 2017).
Similarly, 8.5 million customers lost power during Hurri-
cane Sandy (Alemazkoor etal. 2020). Sustained winds and
excessive amount of precipitation/flooding during hurricanes
cause disruptions to infrastructure systems such as power
outage, disruptions in water supply and wastewater systems,
telecommunication failures, and transportation system dis-
ruptions. Local communities depend on these systems to
a great extent and failures in such infrastructure systems
highly affect their daily activities.
Infrastructure systems have become highly interconnected
and interdependent (Rinaldi etal. 2001; Grafius etal. 2020).
After Hurricane Sandy, damages in electricity stations sig-
nificantly affected the functions of transportation facilities
(Haraguchi and Kim 2016). Power outages hampered the res-
toration of subway services in New York City as trains could
not run without power restoration. Due to the interconnected
and interdependent relationships among infrastructure sys-
tems, the restoration process of a system is further delayed
www.ijdrs.com
www.springer.com/13753
* Tasnuba Binte Jamal
tasnubabinte.jamal@ucf.edu
1 Department ofCivil, Environmental, andConstruction
Engineering, University ofCentral Florida, Orlando,
FL32816, USA

Jamal and Hasan. Predict Restoration Time from Power Outages
when other systems are disrupted. As a result, infrastructure
services are unavailable, impacting the quality of life of the
population served. Among all types of infrastructure dis-
ruptions, a disruption in the electricity power infrastructure
is the most significant one. Power outages have significant
negative impacts on a region across different sectors such as
financial, business, education, medical services, and recrea-
tion (Koks etal. 2019; Kuntke etal. 2022).
To enhance community resilience against an extreme
event, faster restoration from power outages is necessary
for recovery efforts. For this reason, six steps are proposed
for power restoration process including restoration at power
plants, at transmission lines, at substations, for essential ser-
vices, in large service areas, and at individual home (Edison
Electric Institute 2019). To date, most of the works in infra-
structure disruption network analysis and modeling frame-
works have involved the first three steps (Ouyang and Wang
2015). However, a holistic approach for the restoration pro-
cess at the last two steps (restoration in large service areas
and at individual home) is essential to understand power
disruption patterns and durations at regional and household
levels.
Previous studies have investigated the durations of power
outages after hurricanes at a regional level. Liu etal. (2007)
applied an accelerated failure time (AFT) model to under-
stand restoration time of power outage. While this approach
provides useful insights for time to event data analysis, it
ignores spatial clustering of restoration time for power out-
age. To mitigate this problem, Mitsova etal. (2018) applied a
spatial autoregressive (SAR) model to understand restoration
time of power outage at a county level. While this model
considers the spatial autocorrelation among observations,
it does not provide useful insights for time to event data
analysis. The overall restoration time of power outages can
be explained by a set of readily accessible independent vari-
ables using a generalized accelerated failure time (GAFT)
model that allows both time to event data analysis and spa-
tial autocorrelation (non-independence of the observations)
(Zhou etal. 2020).
In this study, we investigated the spatial extent and cor-
relation of restoration time of electricity disruptions across
different counties after a hurricane by applying a spatial
clustering approach. We also developed a statistical model
(the GAFT model) considering a range of variables includ-
ing hazard, the built environment, and sociodemographic
characteristics to identify the factors associated with longer
restoration time for power outage at a county level after
a hurricane. If restoration time can be reliably predicted,
households may plan for alternatives of existing power ser-
vices during disruptions. At the same time, when policymak-
ers and stakeholders better understand the factors associ-
ated with longer disruptions, they can allocate resources to
manage restoration processes, reduce restoration time, and
mitigate the negative impacts of longer restoration times
from power outages. In areas that are likely to have power
outages for a long time, backup power plants and micro-
grids can be installed (Kwasinski 2010; Mishra etal. 2020).
Our study has the following contributions:
• We investigated the spatial distribution of restoration
time of electricity disruption of a region during a hur-
ricane using a statistical clustering approach.
• We developed generalized accelerated failure time
(GAFT)—a statistical model—to investigate the asso-
ciation between restoration time of power outage and a
wide range of variables including hazard, built environ-
ment factors, and sociodemographic characteristics of the
regions accounting for spatial dependence of observa-
tions. While power outage has been studied from the per-
spective of time to event data analysis (Liu etal. 2007)
and considering the spatial dependence of observations
(Mitsova etal. 2018), we add a new dimension by devel-
oping the GAFT model that can account both for time to
event data and spatial dependence of observations.
2 Literature Review
Infrastructure systems are highly interconnected and inter-
dependent and disruption in one system significantly affects
other systems (Rinaldi etal. 2001; Hasan and Foliente
2015; Grafius etal. 2020). In recent times, there has been
an increased interest in studying the impact of power out-
age on the performance of other infrastructure systems after
extreme events and how to enhance the resilience of such
interdependent systems.
For instance, previous studies focused on power–water
network disruptions for natural hazards and suggested pos-
sible solutions to ensure a resilient power–water distribution
system after a hurricane (Almoghathawi etal. 2019; Najafi
etal. 2019, Najafi etal. 2020; Kong etal. 2021). Disruptions
in electricity and petroleum infrastructures had negative
impacts on health care services and public transportation
systems in the New York metropolitan area after Hurri-
cane Sandy (Haraguchi and Kim 2016). Traffic congestion
increased three to four times due to the power outages after
Hurricane Isaac (Miles and Jagielo 2014). Previous studies
developed models to assess the resilience of interdepend-
ent traffic-power systems and to determine the parameters
to quantify the resilience of transportation systems against
hurricanes and other natural hazards and disasters (Kocatepe
etal. 2018; Ahmed and Dey 2020; Zou and Chen 2020). Zou
and Chen (2020) proposed strategies to improve the resil-
ience of traffic-power systems against a hurricane. Ouyang
and Wang (2015) modeled for the resilience optimization
of interdependent infrastructures. The consequences of the

International Journal of Disaster Risk Science
interdependencies in infrastructure failures starting from a
given outage were analyzed by considering severity, dura-
tion, spatial extent, and the number of people affected by a
disruption (Mcdaniels etal. 2007). Kong etal. (2021) cal-
culated the infrastructure efficiency by removing different
percentage of nodes in the system for both power and water
systems. Previous studies also investigated the societal, men-
tal, and economic impacts of power disruption (Dargin and
Mostafavi 2020; Stock etal. 2021) along with interdepend-
ency analysis among the infrastructure systems. Studies have
explored recovery strategies and efficiency (Ge etal. 2019;
Loggins etal. 2019) as well.
Most of the above studies considered infrastructure dis-
ruptions at an infrastructure facility level, analyzing how a
disruption in an electric power plant affects a water treat-
ment plant or water distribution systems, or a gas station
after a disaster. These studies focused on the restoration at
power plants, transmission lines, and substations. Also, these
studies focused on the impact of power outage. For exam-
ple, previous research mainly focused on how and to what
extent other facilities are disrupted when an extreme event
takes place causing power outages in a region. Few studies
investigated power service disruptions at a local (for exam-
ple, county) level and the time required for the restoration
process. Restoration time from power outages needs more
attention along with impact analysis at a local scale.
Researchers have developed models to identify the con-
tributing factors toward power outage following a disaster.
Liu etal. (2007) developed an accelerated failure time (AFT)
model for determining the time required for the restoration
of power outage after an extreme hazard, considering hur-
ricane and snowstorm. Nateghi etal. (2011) compared differ-
ent models such as accelerated failure time model, Cox pro-
portional hazard model, regression trees, Bayesian additive
regression trees (BART), and multivariate additive regres-
sion splines and found that BART performs the best. Models
based on various Geographic Information System databases
were developed to determine where outages are most likely
to occur by Liu etal. (2005). Han etal. (2009) considered
hurricane characteristics, land cover, and power system data
to analyze the number of outages and spatial distribution
of the power outages using negative binomial generalized
linear model for the Gulf Coast region of the United States.
Quiring etal. (2011) included soil characteristics and sug-
gested that these variables can implicitly inform about the
likelihood of trees being uprooted. McRoberts etal. (2018)
showed that the inclusion of elevation, land cover, soil,
precipitation, and vegetation characteristics improved the
accuracy of previously established statistical model by 17%.
However, sociodemographic characteristics and social vul-
nerability of population were not considered in these studies.
Dargin and Mostafavi (2020) considered sociodemographic
factors of a community and identified which group of people
were affected mostly from well-being perspectives due to
various infrastructure disruptions after Hurricane Harvey.
However, the spatial distribution of the recovery process for
a particular disruption, such as which group of people faced
longer disruption, was not considered.
Socioeconomic and sociodemographic characteris-
tics of the affected regions were considered in previous
research. Mitsova etal. (2019) considered characteristics
such as age, gender, race, housing tenure, education, and
income to identify whether households are recovered or not
from power outages after Hurricane Irma. They found that
while distributing federal financial assistance, low-income
households and minority groups were given less priority.
Duffey (2019) used a wide variety of extreme events, such
as hurricanes, wildfires, heavy snowstorms, and devastat-
ing cyclones, to calculate recovery times and probabilities
of failure to restore. He found that wildfires and hurricanes
may have different causes, but the non-restoration probabil-
ity patterns they produce are identical: a straightforward
exponential decline. Lee etal. (2019) studied the disparity
in getting social supports (for example, instrumental, emo-
tional, informational, and outside contact support) consider-
ing respondents’ sociodemographic characteristics such as
education, age, and religion. The results imply that older
and less educated people faced constraints in post-disaster
support. Previous studies also found that regions served by
rural municipalities faced longer disruption for electricity
disruption after Hurricane Maria and Irma (Mitsova etal.
2018; Román etal. 2019). Using satellite nighttime lights
data for Hurricane Maria, Román etal. (2019) found that
within the same urban area, poor residents possess higher
risk of power loss and longer disruption time. To determine
the relationship between physical and socioeconomic char-
acteristics and the power recovery effort, Azad and Ghan-
dehari (2021) developed a Quasi-Poisson regression model
and found that major challenges to the repair work were
poor road infrastructure and economically depressed com-
munities. These studies considered sociodemographic infor-
mation of the households and regions while analyzing the
effects of electricity disruption and time required for recov-
ery operation for power outage. However, these studies did
not focus on the distribution of restoration time to get back
the power over regions.
In addition, other considerations such as meteorologi-
cal, housing characteristics, and so on are equally impor-
tant to perceive the dynamics of restoration process for the
disaster response community. Watson etal. (2022) devel-
oped a machine learning model for predicting the effects of
extreme weather events on electrical distribution grids. They
found substantial diversity in the meteorological factors that
drive the predictions for the most severe events, suggest-
ing that weather hazards are more complex than they are
often treated in empirical analyses of their impacts. Wanik

Jamal and Hasan. Predict Restoration Time from Power Outages
etal. (2018) simulated Hurricane Sandy like scenarios in
the future to determine the severity of tree-caused outages
in Connecticut, with each showing increased winds and
higher rain accumulation over the study area as a result of
large-scale thermodynamic changes in the atmosphere and
track modifications in 2100. Using an ensemble of Weather
Research and Forecasting simulations coupled with three
machine learning-based outage prediction models, they
found that future Sandy will lead to a 42–64% rise in out-
ages. Mukherjee etal. (2018) characterized the key factors
of severe weather-induced power outages and found that
power outage risk is a function of the type of natural haz-
ard, and investments in operations/maintenance activities
(for example, tree-trimming, replacing old equipment, and
so on). These studies found weather impacts on power grids
and density of power outages with simulations and machine
learning algorithms.
Previous studies used various statistical modeling
approaches for estimating power service restoration time.
Mitsova etal. (2018) developed a spatial autoregressive
(SAR) model at a county level for Hurricane Irma to deter-
mine the attributes associated with restoration time from
power outages. However, like the AFT model, the SAR
model does not provide useful insights for time to event data
analysis. We used the GAFT model for two reasons: (1) as
our dependent variable is restoration time, we considered
this to be a time to event data analysis; (2) as our dependent
variable is likely to have a spatial dependency, the GAFT
model can be used for modeling both spatial and non-spatial
data. Similarly, using a spatial autoregressive model, Ulak
etal. (2018) included wind speed, infrastructure and trans-
portation, demographic, and socioeconomic characteristics
to predict the number of power outages in the City of Tal-
lahassee of Florida for Hurricane Hermine. Rachunok and
Nateghi (2020) considered the spatial distribution of dis-
ruptions by demonstrating the network-performance of the
power distribution grid’s sensitivity to spatial characteristics.
However, we should give more emphasis on restoration time
rather than outage density. If many customers face power
outage after a hurricane but they get back their power ser-
vices within a short period, it may not hamper much to their
business, social, and other daily activities. Besides the SAR
model, a random forest model was used to predict hurricane-
induced power outage durations (Nateghi etal. 2014) and
outages (Guikema etal. 2014), which does not provide use-
ful insights for time to event data analysis and spatial cluster-
ing of restoration time.
In summary, the following observations can be made.
First, two types of dependent variables have been consid-
ered in the existing literature: the number of outages and
duration of outages. Second, duration of outages was exam-
ined at different geographical levels ranging from grid sizes
to county subdivisions and county levels. Lastly, different
types of factors such as meteorological, physical, and soci-
odemographic attributes were considered to explain the
outage durations by developing different machine learning
and statistical models. No models have been developed,
however, that account for time as a dependent variable and
spatial autocorrelation. This article claims a methodological
contribution for modeling power outage restoration time by
considering spatial dependence and time to event aspects
present in the data. This can provide more accurate and reli-
able predictions of restoration time with significant impli-
cations on policymaking related to infrastructure planning
and management. More specifically, the following research
questions need to be answered: (1) can we implement a sta-
tistical model that can account both for time to event data
and spatial dependence of the data to predict restoration
time from hurricane-induced power outages from a set of
common key factors that are publicly available? (2) Can we
explain the spatial distribution of restoration time of electric-
ity disruption due to a hurricane using a statistical cluster-
ing approach? As such, the objectives of this study were to
understand the spatial clustering patterns of the restoration
time of power outages due to a hurricane and to determine
the factors associated with prolonged restoration time con-
sidering a wide range of variables including hazard, the built
environment, and sociodemographic characteristics.
3 Data Collection andProcessing
We considered three types of factors: hazard characteristics,
built environment characteristics, and sociodemographic
factors that might be associated with longer restoration times
of power outages during a hurricane.
3.1 Restoration Time
We collected the data for the restoration time of power out-
ages during Hurricane Irma from Florida Today1 for each
county of Florida. The plots in Florida Today were drawn
using the data from the Florida Division of Emergency Man-
agement (FDEM). We used the duration between the time
when 20% customers or more of a particular county first lost
their electricity services and the time when 20% customers
or less were yet to restore their power services (Fig.1). In
other words, duration from when 20% of customers lost their
power to the time by which 80% of customers in a county
had their power restored. We chose 80% of customers’ res-
toration time from a sensitivity analysis.
While each customer’s power may be restored at differ-
ent times, companies typically make broad announcements
1 data.floridatoday.com/storm-power-outages/.

International Journal of Disaster Risk Science
that provide a single approximate restoration time for each
county or region (Liu etal. 2007). Liu etal. (2007) estimated
restoration times as the time by which an arbitrary Z% (say
90%) of the customers of the county will have their power
restored. They collected power outage data from a utility
company, allowing them to obtain such a threshold directly
from power companies. Since we did not collect datasets
from power companies, it was not possible for us to know the
company-specific threshold (Z%) values. In this study, we
chose a threshold Z% to quantify restoration time based on a
sensitivity analysis. For sensitivity analysis, we chose three
different values of the threshold, Z = 80%, 85%, and 90%.
We did not use a threshold of the time by which Z > 90%
of the customers in a county will have their power restored
because of model generalizability. A choice of Z > 90% may
overestimate the restoration time. We ran the GAFT model
for the three values of Z%, and selected the threshold based
on average log pseudo marginal likelihood (LPML). The
average LPML values were found to be −1.276, −1.279,
and −1.517 for 80%, 85%, and 90% of customers restora-
tion time, respectively. We used average LPML instead of
total LPML to compare the models (Iraganaboina and Eluru
2021) because the number of observations changes with the
selection of the threshold. The total LPML value is likely
to decrease with the increase in the number of observations.
So, a higher value of total LPML can result from a smaller
number of observations in the model rather than indicating
good model performance. The average LPML value is higher
for the model with 80% selected as a threshold to calculate
the restoration time. Thus, we chose a threshold of Z% =
80% to quantify the restoration time.
Although Florida consists of 67 counties, we considered
58 counties in this study. We did not consider nine coun-
ties because either there was no power outage (that is, no
customer lost power, as in counties such as Escambia, Hol-
mes, Oskaloosa, Santa Rosa, and Walton) or the percentage
of customers that lost power was less than 20%. Thus, for
these counties, the restoration time would be zero. Also, we
were interested in estimating power outage due to hurricane
impacts. From the wind speed values, it was evident that
those counties had very low (close to zero) sustained wind
speed, indicating that hurricane did not impact these coun-
ties. In most of the counties (19 counties), it took 3 days to
restore the power service for at least 80% of their customers
(Fig.2).
3.2 Hazard Characteristics
Under hazard characteristics we considered four types of
covariates: maximum sustained wind speed, the percentage
of power outage in each county, the percentage of census
tracts prone to flash flood in each county, and the percent-
age of census tracts prone to sea level rise in each county.
The wind speed for Hurricane Irma was estimated from
Fig. 1 Power outage for Bro-
ward County after Hurricane
Irma and considered restoration
time in this study
Fig. 2 Histogram for restoration times of in Florida after Hurricane
Irma

Jamal and Hasan. Predict Restoration Time from Power Outages
the HAZUS-MH wind model (Vickery etal. 2000; Vick-
ery etal. 2006). This model creates the wind speed profile
probabilistically due to a hurricane event. Using this model,
we obtained maximum sustained wind speed at census tract
level based on their distance to the center of the hurricane.
For a given county, the highest wind speed among all census
tracts was considered as the maximum sustained wind speed
for that county.
We considered the maximum percentage of customers
faced power outage from 9 to 28 September 2017. We col-
lected this information from the Florida Division of Emer-
gency Management.
3.3 Built Environment Characteristics
For built environment characteristics, we considered the per-
centage of customers served by investor-owned company,
and power system variables. According to the Florida Pub-
lic Service Commission (FPSC), there are three types of
electric service providers in Florida: investor-owned electric
utilities, rural electric cooperatives, and municipal electric
utilities. Investor-owned electric utilities include Florida
Power and Light Company, Duke Energy, Tampa Electric
Company, Gulf Power Company, and Florida Public Utilities
cooperation. Florida also has 34 municipally owned electric
utilities and 18 rural electric cooperatives. Investor-owned
electric companies are private companies not associated
with any government agency. We considered the percent-
age of customers served by investor-owned electric utilities
in each county. We added this variable for two purposes:
(1) to understand how these companies responded during
the restoration process and (2) if there is any discrepancy in
restoration across various electric companies. We collected
the number of total customers under each type of electric
companies along with total number of customers for each
county from FPSC,2 publicly available in their database.
We also considered the number of substations, power
plants, and total length of overhead line in each county.
They provide a measure of the extent of power system. We
collected power system data from U.S. Energy Information
Administration, EIA.3
3.4 Sociodemographic Characteristics
For sociodemographic characteristics, we included the
median income of the households, and the percentage of
non-White American population in each county from the
2013 to 2017 American Community Survey (ACS) 5-Year
Data Profile.4 We standardized median income before add-
ing to the model.
The descriptive statistics of the data are given in Table1.
To understand the presence of correlations among the pre-
dictors, Pearson’s correlation was calculated (Fig.3). Cor-
relation values among the number of power plants, the num-
ber of substations, and total length of overhead power lines
are high (0.72 and 0.77) (Raithel 2008). For the number of
substations and total length of overhead lines, variance infla-
tion factors were 8.1 and 11.8, respectively, indicating the
presence of multicollinearity issue. So, we considered only
the number of power plants among these three variables to
simplify the statistical model. The multicollinearity condi-
tion number with our considered six variables was 8.189
(which was below 30), indicating that collinearity should
not be an issue.
Table 1 Descriptive statistics of continuous variables
Variable Mean Std. Min Median Max
Dependent Variable Restoration time (days) 3.83 1.93 1 3 9
Hazard Characteristics Maximum sustained wind speed (mph) 60.90 27.83 0 64 114
% of customers who faced power outage 77.09 17.28 39 78.50 100
Built Environment Characteristics % of customers served by investor-owned
power company
55.03 33.90 0 53 100
Number of power plants 3.87 4.37 0 2 25
Number of substations 38.82 42.15 2 23.50 218
Length of overhead line (km) 1065.62 736.96 144.52 833.43 2937.25
Sociodemographic Characteristics % of non-White population 33.22 15.70 11.40 28.50 86.30
Median income (USD) 46,242 9029 31,816 45,424 73,640
2 psc.state.fl.us/Home/HurricaneReport.
3 eia.gov/maps/layer_info-m.php.
4census.gov/acs/www/data/data-tables-and-tools/data-profiles/2017/.

International Journal of Disaster Risk Science
4 Methodological Approach
The methodological approach in this study has two main
parts. First, we determined the spatial distribution of the
restoration time based on the disruption of electricity ser-
vices. Second, we adopted a statistical modeling approach
to determine the factors associated with restoration time
from power outages.
4.1 Spatial Distribution forRestoration Time
ofPower Outage
To identify if there is a clustering pattern between restora-
tion times of electricity disruption in the affected areas,
we used global Moran’s I (Eq.1) (Ord and Getis 1995),
which is typically used to estimate spatial autocorrelation.
Moran’s I was used by Jackson etal. (2021) to understand
the spatial trends in county-level COVID-19 cases and
fatalities in the United States during the first year of the
pandemic.
where
wij
is the spatial weight, having a value of 1 if county
i
has a shared boundary with another county
j
or having a
value of 0 if otherwise;
Xi
is the restoration time; and
X
is
the average restoration time of all counties considered in
the analysis.
Global Moran’s I does not tell anything about the places
where the patterns are located. The concept of a local indica-
tor of spatial association was suggested to remedy this situ-
ation (Anselin 1995). We applied local Moran’s I (Eq.2) to
understand where the clustering patterns are located.
where
zj
is the deviation from the mean and the summation
over
j
such that only neighboring values are included. In
addition to local Moran’s I, we plotted a choropleth map to
visualize the spatial distribution of restoration times. We
(1)
I
=N

i

jwij

i

jwij(Xi−X)(Xj−X)

i

Xi−X

2
,
(2)
I
=zi
∑
j
wijzj
,
Fig. 3 Pearson’s correlations
between variables

Jamal and Hasan. Predict Restoration Time from Power Outages
used ESDA and PySAL packages in Python 3.9 to calculate
the global and local Moran’s I.
4.2 Statistical Modeling Approach
To determine the effects of the factors (described in Sect.3)
on restoration times from power outages, we developed a
generalized accelerated failure time model (GAFT). To
account for the spatial dependence, a random effect (frailty)
is introduced into the linear predictor of survival model (a
survival model is a statistical approach used to analyze the
time until an event of interest occurs). Both georeferenced
(that is, latitude and longitude are recorded) and areal ref-
erenced (that is, county of residence recorded) spatial data
are handled via random effects (frailties) (Zhou etal. 2020).
The GAFT model is given by the following equations (Zhou
etal. 2020).
Or equivalently,
where

Xij
is the matrix of covariates with an intercept term,
XT
ij
means the transpose matrix of
Xij
,

𝛽
is the vector of cor-
responding coefficients,
tij
is the time,
𝜖ij
is a heteroscedastic
error term independent of
vi
, and
S0(t)
is the baseline sur-
vival function. In the GAFT model,
S0(t)
may depend on
certain covariates,
zij
, where
zij
is a subset of
Xij
; in this
study, we considered
zij =Xij
. In the AFT model,
S0(t)
is
assumed to be a static parametric survival function, free of
covariates. That is, the resulting survival curves are not
allowed to vary for different covariates. In practical applica-
tion, this assumption does not always seem to be true (Hen-
sher and Mannering 1994). In the generalized AFT model,
S0(t)
is allowed to flexibly vary with covariates, which has
increased the flexibility of the model. Finally,
vi
is an unob-
served frailty term associated with a county;
i
indicates the
index of an observation (that is, county) and
j
indicates the
index of a predictor variable.
We estimated this model in R using the spBayesSurv
package and the frailtyGAFT function. The detailed descrip-
tion of this package and model can be found in Zhou etal.
(2020) and Hsu etal. (2015). As this is a Bayesian mod-
eling approach, it requires to set the prior distributions of the
parameters based on domain knowledge. However, this prior
knowledge is usually not available (Ulak etal. 2018). In this
study, we set most of the prior information according to the
default values of frailtyGAFT function under spBayesSurv
package in R due to the unavailability of the prior informa-
tion about the actual parameter distributions and validated
it using the trace plots obtained from the model.
(3)
S
xij (t)=S0,zij
(
e−X
T
ij β−vit
).
(4)
y
ij =𝑙𝑜𝑔
(
tij
)
=
X
T
ij

β+vi+ϵ
ij
,
The Bayesian specification for prior distribution of the
model used in this study is given below (Hsu etal. 2015;
Zhou etal. 2020):
For the coefficients

(𝛽)
, a normally distributed prior is
considered. For the frailty terms, in the GAFT model, a
conditional auto-regressive (CAR) prior is chosen for areal
data (indicating that the spatial data are included based over
a geographic area) and a GRF prior is chosen for georefer-
enced data (indicating that the data are included based on
coordinates). We chose CAR prior to model the frailty as
this study is county-level analysis. Since we included spatial
data at a county level, we can assume it as areal referenced
data instead of georeferenced data. For areal data, the intrin-
sic conditional auto-regressive (ICAR) prior smooths neigh-
boring geographic-unit frailties
v
=
(
v
1
,…., v
m)T
. Details
on ICAR
(
𝜏2
)
prior (Eq.6) is given by the set of conditional
distributions in Eq.9. Adjacency matrix, E = [
eij
] of
m×m
dimension for the
m
regions is used to calculate the frailties,
vi
. In Eq.9,
eij
is 1 if counties
i
and
j
share a common bound-
ary, 0 otherwise and
eii
= 0. While calculating
v
for a region
i
, the other regions under consideration are
j
.
e
i+=
∑m
j=1
e
ij
,
is the number of neighbors for region
i
(Zhou etal. 2017,
Zhou etal. 2020).
In GAFT, for spatial analysis, the error term
(𝜖ij)
is not
independent. For this reason, a heteroscedastic error term
is introduced over a probability measure
Gz
, defined on ℝ
for every
z∈X
and a linear dependent tailfree processes
(LDTFP) prior is considered for
Gz
. An LDTFP centered at
a normal distribution
𝜙𝜎
is focused with mean 0 and vari-
ance
𝜎2
, that is,
E(
G
z)
=N(0, 𝜎
2)
for every
z∈X
(details
are described in Jara and Hanson 2011 and Zhou etal. 2017).
Since the posterior distribution for coefficients of the
covariates are unknown, we ran Markov chain Monte Carlo
(MCMC) simulation. For MCMC simulation, we ran 4
chains, where 16,000 scans were thinned after a burn-in
period of 30,000 based upon examination of trace plots for
model parameters (Fig.6). A trace plot is a diagnostic tool
for assessing the mixing of a chain. It shows the iteration
(5)

β∼
N
(
m
0
,S
0),
(6)
(
v
1
,…,v
m)T|
τ∼ICAR
(
τ2
)
,τ−2=Γ
(
a
τ,
b
τ),
(7)
ϵ
ij
|
Gz
ij

indGz
ij ,
(8)
Gzij|
α,σ
2
∼LDTFP
L(
α,σ
2)
;α∼Γ
(
a
0,
b
0)
,σ
−2
∼Γ
(
a
σ,
b
σ).
(9)
v
i
|{
vj
}
j≠i∼N
(e
ij
v
j
e
i+
,τ2
e
i+)
,i=1, …
, m.

International Journal of Disaster Risk Science
number against the value of the draw of the parameter at
each iteration. It also shows whether a chain gets stuck in
certain areas of the parameter space, indicating bad mixing.
5 Results
This section first presents the spatial distribution of power
outage restoration time. Second, it presents the result from
the statistical model.
5.1 Result forSpatial Distribution
We mapped the restoration time with associated county
over Florida (Fig.4). Figure 4 shows that the southern
parts of Florida (Monroe, Lee, Collier, Charlotte, Broward,
Miami-Dade, Palm-Beach, and Hendry Counties) needed
priority during restoration process after Hurricane Irma.
In addition, it shows that counties in the middle of Florida
(Seminole, Orange, St. Johns, Putnam, and Marion) and
some in the North (Hamilton, Suwannee, and Lafayette)
faced moderate (4–7 days) duration of disruption and needed
attention for fast recovery.
The obtained global Moran’s I value is 0.58 (p-value =
0.001), indicating the presence of spatial autocorrelation.
Figure5 shows locations of the clustering patterns for the
restoration time from power outages. The global Moran’s I
test within the entire study area shows significant (p < 0.05)
spatial autocorrelation for our target attribute.
Local Moran’s I plot (Fig. 5) shows the clusters of
longer restoration time (hot spots) and the clusters of short
Fig. 4 Restoration time in
Florida after Hurricane Irma
along with the hurricane path
(in blue color)

Jamal and Hasan. Predict Restoration Time from Power Outages
restoration time. The local Moran’s I test shows consider-
able spatial clustering for 17 counties (local clusters are
significant, p < 0.05). The grey areas in Fig.5 are the
locations where no significant spatial patterns were found;
the red areas are the counties where people had longer res-
toration time living closely to other counties with longer
restoration time. The low with low (L-L) are all the blue
areas, those are locations where people had shorter resto-
ration time living closely to other counties with shorter
time of restoration process. For Hurricane Irma in Florida,
we could not find any HL or LH clustering pattern.
5.2 Result fromStatistical Analysis
For statistical analysis, we considered spatial models
because of the obtained global Moran’s I statistics found
in Sect.5.1. A high Moran’s I value of 0.58 (p-value =
0.001) clearly indicates the presence of spatial correlations
among observations. As such, a non-spatial model assum-
ing independent and identically distributed (IID) observa-
tions, ignoring spatial correlations, will not be appropriate.
Spatial survival analysis is used to analyze clustered time to
event data when the clustering issue arises from geographi-
cal regions (Banerjee 2016).
Table2 presents the results of the generalized acceler-
ated failure time (GAFT) model. Two separate models were
fitted with and without considering spatial correlation. The
proposed GAFT model with CAR frailties has the larger log-
pseudo marginal likelihood (LPML) (−74) compared to the
non-frailty GAFT model (−83), indicating that considering
spatial correlation improves the model fit by 12%.
Bayes factors is a Bayesian alternative to classical hypoth-
esis testing. The Bayes factors for testing all the covariates’
effects on baseline survival were found to be greater than
100, indicating that the baseline survival function (Eq.3)
under the AFT model depends on these variables, and thus
the GAFT model should be considered (Zhou etal. 2020).
The mean posterior inference of conditional CAR frailty
variable was found to be 0.212, representing the amount
of spatial variation across counties. The trace plots of the
regression coefficients (Fig.6) have even and stationary pat-
tern, indicating that MCMC simulations converged (Zhou
etal. 2020).
Fig. 5 Local Moran’s I plot for restoration time of power outage
Table 2 Posterior inference of regression coefficients
90% HPD is reported, variables with ** were also significant at 95% HPD
*Significant at the 90% highest posterior density (HPD) interval
**Significant at the 95% HPD interval
Variable Model with conditional auto-regressive (CAR) frail-
ties
Model without CAR frailties
Mean (standard deviation) 90% HPD Mean (standard deviation) 90% HPD
Intercept −0.746 (0.347)** [−1.323, −0.192] −0.759 (0.224)** [−1.121, −0.389]
Maximum sustained wind speed 0.013 (0.003)** [0.007, 0.019] 0.011 (0.0016)** [0.009, 0.014]
% of customers faced power outage 0.0117 (0.0029)** [0.007, 0.017] 0.012 (0.0029)** [0.007, 0.017]
% of customers served by investor-
owned company
0.0062 (0.002)** [0.003, 0.009] 0.006 (0.0012)** [0.004,0.007]
Number of power plants −0.0134 (0.008)* [−0.027, −0.0002] −0.023 (0.009)** [−0.041, −0.008]
Median income −0.064 (0.047) [−0.14, 0.013] −0.062 (0.042)* [−0.12, −0.0009]
% of non-White population −0.001 (0.003) [−0.006, 0.004] 0.003 (0.002) [−0.001, 0.007]
Log pseudo marginal likelihood −74 −83

International Journal of Disaster Risk Science
Standard deviations of the maximum sustained wind
speed, the percentage of customers served by investor-owned
power companies, the percentage of customers faced power
outages, the number of power plants, and median income
are small compared to the mean (Table2). Moreover, 90%
high posterior density interval of the regression coefficients
do not contain zero, indicating that these variables have sig-
nificant influence on restoration time.
Among hazard characteristics, maximum sustained wind
speed and the percentage of customers faced power out-
ages were found to be significant and positively associated
with power service restoration time. A positive association
means that an increase in a predictor variable will increase
restoration time and a negative association indicates the
opposite. The exponentiated coefficient of maximum sus-
tained wind speed (
e0.013 =1.013
) is the factor by which
the mean restoration time increases by 1.3% with 1 mph
increase in maximum sustained wind speed. One percent
increase in % of customers without power (
e0.0117 =1.0117
)
increases the mean restoration time by 1.17%. Among built
environment characteristics, percentage of customers served
by investor-owned power companies and the number of
power plants were found to be significant. Among soci-
odemographic variables, median income was found to be
Fig. 6 Trace plots of regression
coefficients
Fig. 7 Survival curves for Hurricane Michael

Jamal and Hasan. Predict Restoration Time from Power Outages
statistically significant in the model without CAR frailties.
After adding counties as frailties, the model accounted for
spatial autocorrelation, reducing the apparent significance
of median income and number of power plants.
Since Hurricane Irma’s data were used to fit the GAFT
model, we generated survival curves for counties affected by
Hurricane Michael to ensure that the model is not overfitting.
The model captured restoration time with minimal deviation
for seven counties present in our study area. Among those
seven counties, Fig.7 shows Leon and Franklin Counties’
survival curves (median with 95% confidence interval) pre-
dicted by the model. Survival curves of all seven counties
indicated that Jefferson, Leon, Wakulla, Franklin, Gadsden,
Liberty, and Gulf Counties had median restoration times of
about 2, 4, 5, 6, 10, 12, and 12 days, respectively and these
counties had actual restoration times of 2, 4, 4, 6, 11, 12, and
13 days, respectively.
6 Discussion
In this study, we examined how hazard, built environment,
and socioeconomic characteristics of a region are associ-
ated with restoration time of power outages due to a hur-
ricane. Our results indicate that counties with higher wind
speed had longer restoration times. It is likely that high wind
speed during Hurricane Irma caused greater damages to the
electric infrastructure systems, causing a longer restoration
time. The positive coefficient for the percentage of custom-
ers faced power outage indicates that for regions where
higher percentage of customers were out of electricity, it
took longer time for the maintenance teams to restore power
service in such places.
The percentage of customers of a county served by an
investor-owned utility company is also positively associated
with restoration time. It indicates that counties with a higher
percentage of customers served by investor-owned electric
companies faced longer restoration time, adjusting for other
covariates and county of residence. This may have happened
because the regions where most of the households are served
by investor-owned utility companies also faced higher wind
speed, and had a large number of customers with power out-
ages. As a result, it took long time for the investor-owned
power companies to restore electricity disruption.
The number of power plants is negatively associated with
restoration time, adjusting for other covariates and county
of residence. That is, counties with more power plants were
able to restore their power services fast. A greater number
of power plants indicates a more extensive and better power
system of a region. In other words, these areas are prioritized
to get more systems up and running, resulting in a shorter
restoration time of power outages. Utility companies might
have prioritized restoration in regions with large number of
power plants since component-based restoration strategies
prioritize critical components in the following order: power
plants, substations, transmissions, and distributions (Esma-
lian etal. 2022). Moreover, we found that it took a longer
time for investor-owned power companies to restore electric-
ity disruption, perhaps because of a high number of outages
present in the regions served by investor-owned companies.
Hence, instead of a component-based restoration strategy, an
outage-based restoration strategy can be prioritized, focus-
ing the regions with a greater number of customers without
power. Population and vulnerability-based restoration strate-
gies were found to be better than a component based strategy
in the agent-based simulation by Esmalian etal. (2022).
Figure8 highlights counties with significant factors of
longer power service restoration time using county-level
data. For example, maximum sustained wind speeds in
southwest counties of Florida (Monroe, Collier, Lee, Hen-
dry, and Highlands) were greater than southeast counties
(Miami-Dade, Broward, Palm Beach, Martin, and St. Lucie)
and northwest counties (Taylor, Jefferson, Leon, Wakulla,
Gadsden, Gadsden, Liberty, and Franklin). As a result,
southwest counties on average (8 days) had longer time of
power outage, southeast counties faced on average 4.5 days,
and northwest counties on average 1.75 days of power dis-
ruption. Counties where 75% or more customers were served
by investor-owned power companies on average faced 4.75
days of electricity disruption. Collier and Highlands Coun-
ties faced 9 days of power disruption where about 87% of the
customers were served by investor-owned power companies.
In such counties, the mean percentage of customers who
faced power disruption was also higher (79%). In Collier
and Highlands, about 97% customers lost power services
due to Hurricane Irma. Counties with 4 or more number of
power plants (Polk, Leon, Hillsborough, Alachua, Orange,
and Osceola) on average faced 3.5 days of power disruption.
Previous studies on Hurricanes Irma (Mitsova etal. 2018)
and Hurricanes Bonnie, Isabell, Dennis, and Floyd (Liu etal.
2007) showed that maximum sustained wind speed is posi-
tively associated with power service restoration time. The
number of power plants is important to predict thunderstorm-
induced power outages (Kabir etal. 2019). Mitsova etal.
(2018) found longer disruption for municipal owned power
companies and rural cooperatives. Besides, they found the
percentage of Hispanic population to be significant, which
contradict with our results. One possible reason for these dis-
crepancies could be that Mitsova etal. (2018) considered wind
speed information as a dichotomous variable, which cannot
account for the differences of wind speeds across counties.
Thus, the effect of wind speed on restoration times is captured
by other variables (for example, % of customers served by dif-
ferent power companies and % of Hispanic population). On
the contrary, we have considered actual maximum sustained
wind speed for each county. It is often assumed that poor,

International Journal of Disaster Risk Science
minority communities are less prioritized, reflecting inequal-
ity in power service restoration activities. Previous studies also
found disparities in experienced hardship due to power outages
in Puerto Rico and Texas during Hurricane Maria and Harvey
(Coleman etal. 2020; Azad and Ghandehari 2021). Consist-
ent with these studies, we found disparity issue with respect
to median income for power restoration time in Florida during
Hurricane Irma. This necessitates accelerated recovery activi-
ties and better infrastructure systems in low-income communi-
ties to make them resilient to hurricane impacts.
Based on the significant factors (for example, maximum
sustained wind speed, % of customers faced power outage,
% of customers served by investor-owned power companies,
and the number of power plants) obtained from the GAFT
model with CAR frailties, areas likely to face a longer dis-
ruption time after a hurricane can be identified. For most
of the counties, these four variables could capture the pos-
sible critical regions for restoration process of power outages
(Fig.8).
7 Conclusion
In this study, spatial distribution of restoration time was
investigated at a county level to identify less resilient loca-
tion for electricity disruption. We presented a generalized
accelerated failure time (GAFT) model to determine the
factors that have impacts on electricity infrastructure sys-
tems. Considering spatial correlation in time to event data
analysis has improved the model fit by 12% compared to
the model without considering spatial correlation. The pro-
posed model holds potential for the analysis of power service
restoration time due to extreme events as it can consider
spatial clustering particularly for time as a dependent vari-
able. The findings of this study suggest that counties with
a higher percentage of customers served by investor-owned
electric companies, smaller number of power plants, and
lower median household income faced power outage for a
longer time. Hence, recovery strategies based on number of
outages and vulnerability (in terms of median income) may
improve power outage recovery time.
The described approaches and finding of the study can
aid policymakers and emergency management officials in
understanding factors that should be given importance dur-
ing the restoration process after a hurricane. This study will
also allow them to identify which critical counties or regions
need attention for restoration process and can ensure rapid
restoration and minimize losses in the affected regions. In
general, electricity companies have the knowledge about
power system variables (for example, the number of power
plants, substations, and total length of overhead lines) and
number of outages but do not have much knowledge about
Fig. 8 Counties in Florida
mapped by significant variables
for power service restoration
time (the color bar represents
the factors and dots represent
the restoration time in days)

Jamal and Hasan. Predict Restoration Time from Power Outages
disaster conditions. Therefore, if utility companies can work
with emergency managers to understand the relationship
between disaster condition and electricity disruption, they
could take necessary steps that would account for disaster
conditions. Such efforts can improve electrical grid resil-
ience during extreme events and lead to improved recovery
outcomes.
Most previous studies (Liu etal. 2007; Kabir etal. 2019)
were based on proprietary data from utility companies. This
does not allow reproducibility of the research and prevents
implementation in actual crisis management. All the factors
included in this study were collected from publicly available
data. For example, projected hurricane path or wind speed
information can be obtained from the National Weather Ser-
vice (NWS) and National Hurricane Center (NHC) when
planning for power restoration before a hurricane strikes.
Similarly, socioeconomic characteristics of a community are
available in ACS. Thus, the variables used in this study can
be easily collected and used before the occurrence of a hur-
ricane to predict restoration time. Such predictions will help
policymakers and emergency management officials to accel-
erate the overall restoration process from power outages.
Our analysis has several limitations, which include: this
study is a county-level analysis for power service restoration
time. However, county is not the finest geographic unit. In
the future, focus can be given at smaller level of geographic
units (for example, county subdivision, zip code, or cen-
sus tracts) based on data availability. These limitations can
be overcome if relevant agencies such as utility companies
share outage data at a higher resolution.
Acknowledgments The authors are grateful to the U.S. National Sci-
ence Foundation for the Grant CMMI-1832578 to support the research
presented in this article. However, the authors are solely responsible
for the findings presented here.
Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article are
included in the article's Creative Commons licence, unless indicated
otherwise in a credit line to the material. If material is not included in
the article's Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will
need to obtain permission directly from the copyright holder. To view a
copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.
References
Ahmed, S., and K. Dey. 2020. Resilience modeling concepts in trans-
portation systems: A comprehensive review based on mode, and
modeling techniques. Journal of Infrastructure Preservation and
Resilience 1(1): Article 8.
Alemazkoor, N., B. Rachunok, D.R. Chavas, A. Staid, A. Louh-
ghalam, R. Nateghi, and M. Tootkaboni. 2020. Hurricane-
induced power outage risk under climate change is primarily
driven by the uncertainty in projections of future hurricane
frequency. Scientific Reports 10(1): Article 15270.
Almoghathawi, Y., K. Barker, and L.A. Albert. 2019. Resilience-
driven restoration model for interdependent infrastructure net-
works. Reliability Engineering and System Safety 185: 12–23.
Anselin, L. 1995. Local indicators of spatial association-LISA. Geo-
graphical Analysis 27(2): 93–115.
Azad, S., and M. Ghandehari. 2021. A study on the association of
socioeconomic and physical cofactors contributing to power res-
toration after Hurricane Maria. IEEE Access 9: 98654–98664.
Banerjee, S. 2016. Spatial data analysis. Annual Review of Public
Health 37: 47–60.
Cangialosi, J.P., A.S. Latto, and R. Berg. 2017. National Hurricane
Center cyclone report. Miami: National Hurricane Center.
Coleman, N., A. Esmalian, and A. Mostafavi. 2020. Equitable resil-
ience in infrastructure systems: Empirical assessment of dispar-
ities in hardship experiences of vulnerable populations during
service disruptions. Natural Hazards Review 21(4). https:// doi.
org/ 10. 1061/ (asce) nh. 1527- 6996. 00004 01.
Dargin, J.S., and A. Mostafavi. 2020. Human-centric infrastructure
resilience: Uncovering well-being risk disparity due to infra-
structure disruptions in disasters. PLoS ONE 15(6). https:// doi.
org/ 10. 1371/ journ al. pone. 02343 81.
Duffey, R.B. 2019. Power restoration prediction following extreme
events and disasters. International Journal of Disaster Risk Sci-
ence 10(1): 134–148.
Edison Electric Institute. 2019. Restoring power after a storm: A
step-by-step process. Edison Electric Institute. https:// www. eei.
org/-/ media/ Proje ct/ EEI/ Docum ents/ Issues- and- Policy/ Relia
bility- and- Emerg ency- Respo nse/ Resto ration_ Proce ss_ Step_
by_ Step. pdf? la= en& hash= 62A37 8F3A4 5FFDD 6619A A4F25
24D75 77AB0 51560. Accessed 20 Oct 2021.
Esmalian, A., W. Wang, and A. Mostafavi. 2022. Multi-agent mod-
eling of hazard–household–infrastructure nexus for equitable
resilience assessment. Computer-Aided Civil and Infrastructure
Engineering 37(12): 1491–1520.
Ge, Y., L. Du, and H. Ye. 2019. Co-optimization approach to post-
storm recovery for interdependent power and transportation
systems. Journal of Modern Power Systems and Clean Energy
7(4): 688–695.
Gillespie, R., J. Weiner, and L. Postal. 2017. Central Florida lights
could be out for days, weeks. Orlando Sentinel, 2017. https://
www. orlan dosen tinel. com/ weath er/ hurri cane/ os- hurr i cane-
irma- centr al- flori da- power- outag es- 20170 911- story. html.
Accessed 5 Feb 2022.
Grafius, D.R., L. Varga, and S. Jude. 2020. Infrastructure interde-
pendencies: Opportunities from complexity. Journal of Infra-
structure Systems 26(4). https:// doi. org/ 10. 1061/ (asce) is. 1943-
555x. 00005 75.
Grenier, R.R., P. Sousounis, and J.S.D. Raizman. 2020. Quantifying
the impact from climate change on U.S. hurricane risk. www.
air- world wide. com/ Legal/ Trade marks/. Accessed 5 Feb 2022.
Guikema, S.D., R. Nateghi, S.M. Quiring, A. Staid, A.C. Reilly, and
M. Gao. 2014. Predicting hurricane power outages to support
storm response planning. IEEE Access 2: 1364–1373.
Han, S.-R., S.D. Guikema, and S.M. Quiring. 2009. Improving the
predictive accuracy of hurricane power outage forecasts using
generalized additive models. Risk Analysis 29(10): 1443–1453.
Haraguchi, M., and S. Kim. 2016. Critical infrastructure interde-
pendence in New York City during Hurricane Sandy. Interna-
tional Journal of Disaster Resilience in the Built Environment
7(2): 133–143.

International Journal of Disaster Risk Science
Hasan, S., and G. Foliente. 2015. Modeling infrastructure system
interdependencies and socioeconomic impacts of failure in
extreme events: Emerging R&D challenges. Natural Hazards
78(3): 2143–2168.
Hensher, D.A., and F.L. Mannering. 1994. Hazard-based duration
models and their application to transport analysis: Foreign sum-
maries. Transport Reviews 14(1): 63–82.
Hsu, C.H., J.M.G. Taylor, and C. Hu. 2015. Analysis of acceler-
ated failure time data with dependent censoring using auxiliary
variables via nonparametric multiple imputation. Statistics in
Medicine 34(19): 2768–2780.
Iraganaboina, N.C., and N. Eluru. 2021. An examination of factors
affecting residential energy consumption using a multiple dis-
crete continuous approach. Energy and Buildings 240: Article
110934.
Jackson, S.L., S. Derakhshan, L. Blackwood, L. Lee, Q. Huang, M.
Habets, and S.L. Cutter. 2021. Spatial disparities of COVID-
19 cases and fatalities in United States counties. International
Journal of Environmental Research and Public Health 18(16):
Article 8259.
Jara, A., and T.E. Hanson. 2011. A class of mixtures of dependent
tail-free processes. Biometrika 98(3): 553–566.
Kabir, E., S.D. Guikema, and S.M. Quiring. 2019. Predicting thun-
derstorm-induced power outages to support utility restoration.
IEEE Transactions on Power Systems 34(6): 4370–4381.
Kocatepe, A., M.B. Ulak, L.M.K. Sriram, D. Pinzan, E.E. Ozguven,
and R. Arghandeh. 2018. Co-resilience assessment of hurricane-
induced power grid and roadway network disruptions: A case
study in Florida with a focus on critical facilities. In Proceed-
ings of the 21st International Conference on Intelligent Trans-
portation Systems (ITSC), 4–7 November 2018, Maui, HI, USA,
2759–2764.
Koks, E., R. Pant, S. Thacker, and J.W. Hall. 2019. Understanding
business disruption and economic losses due to electricity fail-
ures and flooding. International Journal of Disaster Risk Science
10(4): 421–438.
Kong, J., C. Zhang, and S.P. Simonovic. 2021. Optimizing the resil-
ience of interdependent infrastructures to regional natural hazards
with combined improvement measures. Reliability Engineering
and System Safety 210: Article 107538.
Kuntke, F., S. Linsner, E. Steinbrink, J. Franken, and C. Reuter. 2022.
Resilience in agriculture: Communication and energy infrastruc-
ture dependencies of German farmers. International Journal of
Disaster Risk Science 13(2): 214–229.
Kwasinski, A. 2010. Technology planning for electric power supply in
critical events considering a bulk grid, backup power plants, and
micro-grids. IEEE Systems Journal 4(2): 167–178.
Lee, S., A.M. Sadri, S.V. Ukkusuri, R.A. Clawson, and J. Seipel. 2019.
Network structure and substantive dimensions of improvised
social support ties surrounding households during post-disaster
recovery. Natural Hazards Review 20(4): Article 04019008.
Liu, H., R.A. Davidson, A.M. Asce, D.V. Rosowsky, M. Asce, and J.R.
Stedinger. 2005. Negative binomial regression of electric power
outages in hurricanes. Journal of Infrastructure Systems 11(4):
258–267.
Liu, H., R.A. Davidson, and T.V. Apanasovich. 2007. Statistical fore-
casting of electric power restoration times in hurricanes and ice
storms. IEEE Transactions on Power Systems 22(4): 2270–2279.
Loggins, R., R.G. Little, J. Mitchell, T. Sharkey, and W.A. Wallace.
2019. CRISIS: Modeling the restoration of interdependent civil
and social infrastructure systems following an extreme event.
Natural Hazards Review 20(3): Article 04019004.
Mcdaniels, T., S. Chang, K. Peterson, J. Mikawoz, D. Reed, and M.
Asce. 2007. Empirical framework for characterizing infrastructure
failure interdependencies. Journal of Infrastructure Systems 13(3).
https:// doi. org/ 10. 1061/ ASCE1 076- 03422 00713: 3175.
McRoberts, D.B., S.M. Quiring, and S.D. Guikema. 2018. Improv-
ing hurricane power outage prediction models through the
inclusion of local environmental factors. Risk Analysis 38(12):
2722–2737.
Miles, S.B., and N. Jagielo. 2014. Socio-technical impacts of Hur-
ricane Isaac power restoration. In Vulnerability, uncertainty and
risk: Quantification, mitigation and management, ed. M. Beer,
S.-K. Au, and J.W. Hall, 567–576. Reston: American Society of
Civil Engineers.
Mishra, S., K. Anderson, B. Miller, K. Boyer, and A. Warren. 2020.
Microgrid resilience: A holistic approach for assessing threats,
identifying vulnerabilities, and designing corresponding mitiga-
tion strategies. Applied Energy 264: Article 114726.
Mitsova, D., A.M. Esnard, A. Sapat, and B.S. Lai. 2018. Socioeco-
nomic vulnerability and electric power restoration timelines in
Florida: The case of Hurricane Irma. Natural Hazards 94(2):
689–709.
Mitsova, D., M. Escaleras, A. Sapat, A.M. Esnard, and A.J. Lamadrid.
2019. The effects of infrastructure service disruptions and socio-
economic vulnerability on hurricane recovery. Sustainability
11(2): Article 516.
Mukherjee, S., R. Nateghi, and M. Hastak. 2018. A multi-hazard
approach to assess severe weather-induced major power outage
risks in the U.S. Reliability Engineering and System Safety 175:
283–305.
Najafi, J., A. Peiravi, A. Anvari-Moghaddam, and J.M. Guerrero. 2019.
Resilience improvement planning of power-water distribution
systems with multiple microgrids against hurricanes using clean
strategies. Journal of Cleaner Production 223: 109–126.
Najafi, J., A. Peiravi, A. Anvari-Moghaddam, and J.M. Guerrero. 2020.
An efficient interactive framework for improving resilience of
power-water distribution systems with multiple privately-owned
microgrids. International Journal of Electrical Power and Energy
Systems 116: Article 105550.
Nateghi, R., S.D. Guikema, and S.M. Quiring. 2011. Comparison and
validation of statistical methods for predicting power outage dura-
tions in the event of hurricanes. Risk Analysis 31(12): 1897–1906.
Nateghi, R., S.D. Guikema, and S.M. Quiring. 2014. Forecasting hur-
ricane-induced power outage durations. Natural Hazards 74(3):
1795–1811.
Ord, J.K., and A. Getis. 1995. Local spatial autocorrelation statistics:
Distributional issues and an application. Geographical Analysis
27(7): 286–306.
Ouyang, M., and Z. Wang. 2015. Resilience assessment of interde-
pendent infrastructure systems: With a focus on joint restoration
modeling and analysis. Reliability Engineering and System Safety
141: 74–82.
Quiring, S.M., L. Zhu, and S.D. Guikema. 2011. Importance of soil and
elevation characteristics for modeling hurricane-induced power
outages. Natural Hazards 58(1): 365–390.
Rachunok, B., and R. Nateghi. 2020. The sensitivity of electric power
infrastructure resilience to the spatial distribution of disaster
impacts. Reliability Engineering and System Safety 193: Article
106658.
Raithel, J. 2008. Quantitative Forschung, 2., Durchges. Aufl, Wies-
baden: Verl. Für Sozialwiss, 207–213 (in German).
Rinaldi, S.M., J.P. Peerenboom, and T.K. Kelly. 2001. Identifying,
understanding, and analyzing critical infrastructure interdepend-
encies. IEEE Control Systems Magazine 21(6): 11–25.
Román, O., M.E.C. Stokes, R. Shrestha, Z. Wang, L. Schultz, E.A.
Sepúlveda Carlo, Q. Sun, and J. Bell etal. 2019. Satellite-based
assessment of electricity restoration efforts in Puerto Rico after
Hurricane Maria. PLoS ONE. https:// doi. org/ 10. 1371/ journ al.
pone. 02188 83.
Stock, A., R.A. Davidson, J. Kendra, V.N. Martins, B. Ewing, L.K.
Nozick, K. Starbird, and M. Leon-Corwin. 2021. Household

Jamal and Hasan. Predict Restoration Time from Power Outages
impacts of interruption to electric power and water services. Natu-
ral Hazards 115: 2279–2306.
Ulak, M.B., A. Kocatepe, L.M.K. Sriram, E.E. Ozguven, and R.
Arghandeh. 2018. Assessment of the hurricane-induced power
outages from a demographic, socioeconomic, and transportation
perspective. Natural Hazards 92(3): 1489–1508.
Vickery, P.J., P.F. Skerlj, A.C. Steckley, and L.A. Twisdale. 2000. Hur-
ricane wind field model for use in hurricane simulations. Journal
of Structural Engineering 126(10): 1203–1221.
Vickery, P.J., J. Lin, P.F. Skerlj, L.A. Twisdale, and K. Huang. 2006.
HAZUS-MH hurricane model methodology. I: Hurricane hazard,
terrain, and wind load modeling. Natural Hazards Review 7(2):
82–93.
Wanik, D.W., E.N. Anagnostou, M. Astitha, B.M. Hartman, G.M.
Lackmann, J. Yang, D. Cerrai, J. He, and M.E.B. Frediani. 2018.
A case study on power outage impacts from future Hurricane
Sandy scenarios. Journal of Applied Meteorology and Climatol-
ogy 57(1): 51–79.
Watson, P.L., A. Spaulding, M. Koukoula, and E. Anagnostou. 2022.
Improved quantitative prediction of power outages caused by
extreme weather events. Weather and Climate Extremes 37: Arti-
cle 100487.
Zhou, H., T. Hanson, and J. Zhang. 2017. Generalized accelerated
failure time spatial frailty model for arbitrarily censored data.
Lifetime Data Analysis 23(3): 495–515.
Zhou, H., T. Hanson, and J. Zhang. 2020. SpBayesSurv: Fitting Bayes-
ian spatial survival models using R. Journal of Statistical Soft-
ware 92(9): 1–33.
Zou, Q., and S. Chen. 2020. Resilience modeling of interdependent
traffic-electric power system subject to hurricanes. Journal of
Infrastructure Systems 26(1): Article 04019034.