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# Modeling Power Outage Risk From Natural Hazards

## Summary and Keywords

Natural disasters can have significant widespread impacts on society, and they often lead to loss of electric power for a large number of customers in the most heavily impacted areas. In the United States, severe weather and climate events have been the leading cause of major outages (i.e., more than 50,000 customers affected), leading to significant socioeconomic losses. Natural disaster impacts can be modeled and probabilistically predicted prior to the occurrence of the extreme event, although the accuracy of the predictive models will vary across different types of disasters. These predictions can help utilities plan for and respond to extreme weather and climate events, helping them better balance the costs of disaster responses with the need to restore power quickly. This, in turn, helps society recover from natural disasters such as storms, hurricanes, and earthquakes more efficiently. Modern Bayesian methods may provide an avenue to further improve the prediction of extreme event impacts by allowing first-principles structural reliability models to be integrated with field-observed failure data.

Climate change and climate nonstationarity pose challenges for natural hazards risk assessment, especially for hydrometeorological hazards such as tropical cyclones and floods, although the link between these types of hazards and climate change remains highly uncertain and the topic of many research efforts. A sensitivity-based approach can be taken to understand the potential impacts of climate change-induced alterations in natural hazards such as hurricanes. This approach gives an estimate of the impacts of different potential changes in hazard characteristics, such as hurricane frequency, intensity, and landfall location, on the power system, should they occur. Further research is needed to better understand and probabilistically characterize the relationship between climate change and hurricane intensity, frequency, and landfall location, and to extend the framework to other types of hydroclimatological events.

Underlying the reliability of power systems in the United States is a diverse set of regulations, policies, and rules governing electric power system reliability. An overview of these regulations and the challenges associated with current U.S. regulatory structure is provided. Specifically, high-impact, low-frequency events such as hurricanes are handled differently in the regulatory structure; there is a lack of consistency between bulk power and the distribution system in terms of how their reliability is regulated. Moreover, the definition of reliability used by the North American Reliability Corporation (NERC) is at odds with generally accepted definitions of reliability in the broader reliability engineering community. Improvements in the regulatory structure may have substantial benefit to power system customers, though changes are difficult to realize.

Overall, broader implications are raised for modeling other types of natural hazards. Some of the key takeaway messages are the following: (1) the impacts natural hazard on infrastructure can be modeled with reasonable accuracy given sufficient data and modern risk analysis methods; (2) there are substantial data on the impacts of some types of natural hazards on infrastructure; and (3) appropriate regulatory frameworks are needed to help translate modeling advances and insights into decreased impacts of natural hazards on infrastructure systems.

# Introduction

Each year, energy infrastructure throughout the world is impacted by a range of extreme weather and climate events, leading to prolonged and widespread power outages and large economic and societal losses. In the United States, severe weather has been the primary contributor to widespread and sustained power outages, and the number of severe weather-related outages has increased substantially over the past decade (Executive Office of the President Report, 2013; Kenward & Raja, 2014). Hurricanes in particular have caused extensive power outages in the United States. For example, Hurricane Irene left more than 6.5 million people without power in 2011, and Hurricane Sandy caused power losses for more than 8.5 million customers in 2012 (Executive Office of the President Report, 2013). Many recent studies have highlighted the increasing trend in severity of extreme weather and climate events, leading to increased power outages and associated higher levels of economic losses (Nateghi et al., 2016; Larsen & Sweeney, 2015; Kenward & Raja, 2014; Campbell, 2012). There is also a rich body of literature indicating climate change will likely increase the severity or intensity of extreme events, which could further stress the reliability and resiliency of electric power systems (Coumou & Robinson, 2013; Peterson, Hoerling, Stott, & Herring, 2012; Webster et al., 2005).

Power outages due to natural hazards have long been a problem, even with modern power grids. Academically, much of the interest in research on power outages due to hazards initially focused on earthquakes, particularly through the National Science Foundation earthquake engineering research centers in the mid- to late 1990s. Researchers at the Multidisciplinary Center for Earthquake Engineering Research (MCEER) and the Mid-America Earthquake center led some of the early efforts in this area (Shinozuka et al., 1998, 1999, 2004). This work focused on improving postevent restoration of electric power and topology-based pre-event assessment of vulnerability of power systems (Çağnan et al., 2006; Xu et al., 2007). Data limitations and the fact that earthquakes could not be reliably pre-event forecasted prohibited a focus on pre-event impact forecasting. However, in the mid-1990s, many electric power utilities began installing and using an Outage Management System (OMS), and these systems generally included a database that recorded outages. This development created the potential for pre-event forecasting of hazard impacts for hydrometeorological hazards, such as hurricanes and thunderstorms, because, for the first time, an archive of outages for past events was available via the OMS. Early work focused largely on hurricanes, in part because they are relatively easy to forecast, with long lead times, at least relative to more spatially variable thunderstorms, tornadoes, and similar weather events. This work on hurricanes began in earnest in the early to mid-2000s and continues, with recent work increasingly focusing on other, more difficult to forecast weather and climate events.

Hurricanes cause power outages through three main mechanisms that directly or indirectly damage power system components and substations: (1) strong winds, (2) coastal flooding (i.e., storm surges), and (3) inland flooding. Much, but not all, of the research on hurricane power outage risk modeling has focused on wind-induced damage, which is the focus of this article. This is not to imply that flooding, either coastal or inland, does not cause damage to power systems; it does, as evidenced by the power outages in Manhattan due to coastal flooding during Hurricane Sandy. However, most research has focused on wind-induced outages, as these are much more widespread, and wind leads to the majority of outages. High winds can directly damage power system components through, for example, pushing poles over and causing overhead power lines to fail. These direct effects require very strong winds, weak foundations for poles, weak poles, or some combination of all three. Such direct pole failures have occurred, for example, along the beach in Galveston, Texas when poles were pushed to a nonvertical position during Hurricane Ike, leading to failure of the local power system. Most failures induced by high winds, however, are indirect, often through the impacts of trees, tree branches, or blowing debris impacting power poles, lines, and substations. Flooding-induced damage, either coastal or inland, occurs due to direct inundation of power system components.

Prestorm forecasting of power outage risk and estimation of changes in risk over the longer term are critical for both short-term emergency response planning and longer-term system management decision making. In the short term, forecasts of the likelihood and projected number of outages due to an approaching storm in the days before landfall can significantly aid emergency response planning by power utilities, other utilities dependent on electric power, such as water and transportation systems, and local, state, and national level emergency response managers. For a single power utility facing a natural hazard, the decisions can be costly and of significant importance to society. Utilities must decide how many external crews and how much extra material (poles, lines, and transformers) to request to prepare for the storm. If they request too few, power outages take longer to restore; too many, and they spend more than they needed to, so this request is an important, expensive decision. As an example, for a large storm, a major utility in the United States may spend tens of millions of dollars per day on restoring power. Other power-dependent organizations such as water systems and local, state, and federal emergency response systems also have important, high-impact decisions to make, decisions that hinge at least partially on knowledge of where power outages are most likely to occur. For example, local emergency response managers must decide how many shelters to open and where to locate them. The number needed depends, in part, on the fraction of the population that will lose power, and they would prefer to not locate the shelters in areas likely to lose power. In the longer run, estimates of how power outage risk may change over time in response to changes in the hydrometeorological natural hazard environment in a changing climate can enable more informed long-term asset management decision making.

In this article, we summarize recent developments in modeling natural hazards-induced power outage risk, focusing specifically on hurricanes because of their substantial large-scale impacts and also because they are the type of natural hazard for which power outage risk models are most well developed. We provide an overview of both short-term and longer-term natural hazard power outage risk modeling work. Two main approaches are used for outage forecasting: one based on statistical learning theory, and the other on engineering fragility curves. We start in “Statistical Learning Methods” and “Fragility-Based Methods” by giving an overview of statistical methods and fragility curves, respectively. Then we show, in “Modeling Power Outage Restoration” and “Long-Term Hurricane-Induced Power Outage Risks,” how these methods can be leveraged to examine the potential for long-term changes in power outage risk due to disasters such as hurricanes in a changing climate. We discuss in “The Challenges of the U.S. Regulatory Environment” the challenges imposed by the current U.S. regulatory environment, and close with a brief summary. Our hope is that the discussion in this article prompts further research on the impacts of not only hurricanes, but other severe weather events on power systems.

# Statistical Learning Methods

In this section, we provide a brief discussion of the main types of statistical learning methods followed by examples of the uses of these approaches in developing natural hazard-induced power outage risk models. We conclude the section by discussing the most accurate statistical power outage forecast models to date.

## Overview of Statistical Modeling

With the use of statistical tools, a considerable body of literature exists on characterizing the risk and reliability of electric power systems impacted by severe weather. The main statistical approaches can be categorized into parametric, semiparametric, and nonparametric supervised learning methods. Generally speaking, the objective of supervised learning methods is estimating a target variable of interest (e.g., number of weather-induced outages) as a function of a set of input variables (e.g., the characteristics of the weather event, population density, and tree-trimming frequency). In supervised statistical learning methods, the outcome variable guides the learning process (Hastie et al., 2011). Denote the outcome variable as y and the matrix of input variables as X. The objective of supervised learning is to find the function that relates the input variables X to our outcome measure of interest y with some degree of inherent, irreducible variability ε‎. This relationship can be summarized as Equation 1:

$Display mathematics$
(1)

### Parametric Models

In parametric regression models, the response variable is assumed to have a particular type of probability distribution. Statistical tools can be utilized to determine the function that relates the input variables to response and to make inferences about the parameters of the function. Linear models and generalized linear models are among this class of statistical learning methods. Parametric models are easy to interpret and are widely used. However, if incorrect assumptions (e.g., linearity, normality, iid) are made in the analysis, the resulting inferences could be very misleading (Hastie et al., 2011).

Parametric regression models have been widely used in characterizing the impact of extreme weather events on the reliability of power systems. Here we summarize some of the past work that used several different parametric models to study the impact of storms on power system performance. Balijepalli et al. (2005) used a bootstrap method to estimate lightning storm parameters. The authors implemented Monte Carlo simulations based on the derived storm parameters and fault rates to estimate system reliability indices (i.e., both momentary and sustained outages). Li et al. (2010) predicted severe weather-induced outages using a Poisson regression model for spatial data in a Bayesian hierarchical framework.

Zhu, Cheng, Broadwater, and Scirbona. (2007) developed a two-stage prediction method to forecast power outages due to different types of storms (classified into six categories based on their temperatures and speed). The first stage involved fitting an exponential distribution to the historical data for each type of storm. In the second stage, they combined their error compensation methodology to their empirical model to get real-time forecasts. A correction factor was then applied based on the deviation of the predictions from the real-time storm data at the end of each hour into the storm. Their methodology can be implemented before and during the storms, and the prediction errors are sensitive to the error correction factor used.

Reed (2008) used a combined statistical–Geographic Information System (GIS) methodology to study the performance of an urban distribution system in the Pacific Northwest of the United States impacted by four winter storms. The logs of the repair crews were plotted in GIS to study outage duration, fragilities, and restoration. To predict power distribution reliability indices such as SAIDI (System Average Interruption Index) and SAIFI (System Average Frequency Index) for a selected storm, they fitted linear regression models to the storm data and reported the R2 values as a measure of goodness of fit. They concluded that gust wind speeds are the best predictor. They then formulated a fragility model, making this a mixed statistical–fragility method, based on the ratio of damaged feeder length to the total length of the feeders (using the geocoded data) as a function of wind speed. They also fitted a Gamma distribution to fit the outage duration patterns of the storms with the shape parameter being correlated to the square of 5-s gust wind speed. Liu, Davidson, Rosowsky, and Stedinger (2005) used a negative binomial regression model to estimate the geographical distribution of hurricane-induced power outages using data from two major utility companies in North Carolina and South Carolina. Their input data included the following: number of power outages, power system inventories, hurricane wind speed, hurricane rainfall characteristics, types of trees, land cover, and soil drainage levels per geographical area. Their models also included hurricane and company indicator variables, limiting their model’s applicability to different systems and also future hurricanes. The spatial correlations associated with the data were not addressed. In later work, Liu et al. (2005) applied a generalized linear mixed model to account for spatial correlations in outages but found that this did not improve the model’s performance.

Han et al. (2009a) further advanced the models developed by Liu et al. (2005) and Liu, Davidson, and Apanasovich (2007) by removing the indicator variables and replacing them with a more extensive set of covariates such as the time between hurricanes, the central pressure difference of the hurricane, and a more extensive range of geographical and climatic variables. They used a negative binomial generalized linear model, achieving similar goodness of fit as Liu’s models. Moreover, their predictive accuracy was reported to be reasonable for the entire system. However, the model generally overestimated the number of outages in the urban areas and underestimated the number in rural regions.

### Nonparametric Models

Nonparametric models relax many of the assumptions used in the parametric models. No probability distribution is assumed for the response variable. Also, unlike parametric models, the functional relationship between the response and exploratory variables is not predetermined but can be adjusted to account for the nonlinear and complex patterns present in the data. The flexibility of this class of models comes at the cost of interpretability. Classification and regression trees and ensemble learning methods are among this class of statistical learning tools. In this section, we summarize some of the literature that involved leveraging nonparametric techniques to assess the impacts of severe weather on power distribution systems’ performance.

Quiring et al. (2011) used data similar to Han et al. (2009a, 2009b) and a method based on classification and regression trees (CART) to study the role of soil moisture and geographical effects on outages in power distribution systems. They also explored whether soil and topographic measures could be used as proxies for power systems data (i.e., number of poles, number of customers). They found that while soil and topographic variables did not significantly improve the predictive accuracy of their models, some of the land cover variables were reasonable proxies for power system data. They concluded that land cover variables could help with generalizing the power outage models and with extending their application to other service areas where detailed power system data are not available.

Nateghi, Guikema, and Quiring (2014) used the method of random forest and data from a power distribution system serving two states in the Gulf Coast region of the United States to predict the number of outages in power distribution systems. Their model showed enhanced predictive accuracy over all the previous models in the literature. They also demonstrated that modeling the reliability of power distribution systems as a function of the hurricane wind speed alone was not sufficient. They proposed using a multivariate approach to adequately characterize the reliability of power distribution systems.

Guikema et al. (2014) developed a spatially generalized model, based on the method of random forest, that could estimate the number of outgas and customer meters without power for an approaching hurricane using publicly available data. They also demonstrated how their modeling framework could be used in scenario analysis, assessing the potential impacts of hypothetical storms in different service areas.

Kankanala et al. (2014) used an ensemble learning approach based on a boosting algorithm to estimate lightning and weather-related outages in overhead distribution systems. The historical data used for their model training were from 2005 to 2011 for four cities in Kansas. Their model focused on understanding the variance of the past data as opposed to achieving good predictive power.

Nateghi (2018) used a multivariate boosting ensemble-of-trees approach to simultaneously estimate the joint distribution of the number of power outages, outage durations, and the number of customers affected due to Hurricane Katrina’s impact on a coastal community in the U.S. Gulf region. The study also illustrated how the multivariate modeling approach could be leveraged to assess the effectiveness of various resilience investment decisions.

### Semiparametric Models

Semiparametric models are the fuzzy category between the parametric and nonparametric models. While the response may be assumed to follow a particular probability distribution, the functional relationship between the inputs and response may be nonparameterized and flexible. Generalized additive regression models and multivariate adaptive regression splines fall into this category.

Han et al. (2009a, 2009b) further improved their outage predictions (described in “Parametric Models”) by using a Poisson generalized additive model (P-GAM). The P-GAM model captured the nonlinearity in the data, resulting in substantially better predictive accuracy. Guikema and Quiring (2011) used a hybrid classification tree–GAM approach to model power outages based on the same data set as Han (2009a, 2009b). Their model effectively handled the zero inflation issue that typically exists in outage data while capturing the complexities existing in outage data.

## Most Accurate Models to Date

Nateghi et al. (2014) used data from a power company in the Gulf Coast region of the U.S. to predict the number and spatial distribution of outages in power distribution systems (see “Nonparametric Models”). Their model is the most accurate model to date and has been in operational use by a major investor-owned utility.

Nateghi et al. (2014) started with an exploratory analysis of a large set of data on details of power system inventory and geographical and climatic characteristics of the region together with data on Hurricanes Danny, Dennis, Georges, Ivan, and Katrina. The special scale of their analysis was grid cells of 12,000 feet (3.66 km) × 8,000 feet (2.44 km).

They selected the “best subset variables” of their final model through rigorous cross validations and concluded their approach can capture the complex structure of outage data and yield accurate prediction estimates, even with a substantially reduced set of input variables relative to prior approaches. The schematic in Figure 1 shows the high correlation between predicted versus observed outages for each grid cell across all storms for one of the states in their analysis.

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Figure 1. Predicted versus observed outages for Hurricanes Danny (1997), Dennis (2005), Georges (1998), Ivan (2004), and Katrina (2005) for a given state. The inset at the top left of both plots zooms in on the lower ranges of outages to better view the variability of data.

Figure 2 illustrates the spatial variation of the predicted and observed power outages for each grid cell for Hurricane Ivan in one of the states in their study. It can be seen that the overall predicted outages follow the same general spatial pattern as the observed outages.

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Figure 2. Spatial distribution of observed (a) and predicted (b) outages for Hurricane Ivan in 2004 for a given state.

They also plotted the partial dependencies between their input variables and the response variable of their final predictive model. Figure 3 shows the marginal change in the number of power outages per grid cell as a function of their six input variables of gust wind speed (in ms−1), duration of wind speed above 20 ms−1, fractional soil moisture at depths of 0–10 cm and 40 cm to bedrock, the number of customers in a grid cell served by the utility, and length-weighted time since the last tree trimming. The plots show that the wind speed and the duration of wind speed above 20 ms−1 explain only a fraction of the variability in service disruptions, demonstrating that methods based on only gust wind speed, as traditional engineering fragility curves often are, are unlikely to accurately model failures in the system. Tree trimming practices and the number of customers served by the utility in each grid cell are also key elements that characterize the vulnerability of the system to hurricane impacts and should be accounted for in the reliability analysis of power systems during hurricanes.

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Figure 3. Partial dependencies of model’s input variables.

While the model developed by Nateghi et al. (2014) predicts outages accurately prior to hurricane landfall, it relies on information specific to the utility company in the analysis, which limits its use to support wider emergency response planning. Guikema et al. (2014) developed a power outage prediction model applicable along the full U.S. coastline using only publicly available data. Having accurate forecast models that do not rely on proprietary data is particularly beneficial for interutility coordinations at a wider scale and for government agencies’ emergency preparedness and response decisions.

Guikema et al. (2014) demonstrated the use of the model for Hurricane Sandy. Figure 4 shows the result of running their forecast model 84, 60, 36, 12, and 0 hours prior to landfall. At 84 hours prior to landfall, Sandy was forecast to take a more southerly track than was later realized, and this is reflected in the map of forecast outages. There were significant outages forecasted for the northern edges of Virginia and relatively few forecasted for Long Island. At 60 hours, Sandy intensified and began to move more quickly, leading to a track that would penetrate much further inland while maintaining high wind speeds. This was reflected in their model output, with some areas along the northern edge of New York State having forecasts of 25% or more of customers without power. At 36 hours, the forecast track began to converge toward the ultimately experienced track. By the time of Sandy’s landfall, the model forecasts suggested widespread impacts from Long Island through the Mid-Atlantic region and well inland in Pennsylvania. Their final best estimate as Hurricane Sandy transited the Mid-Atlantic region of the United States was that there would be 10 million customers without power. They compared their outage totals by state to those from the DOE Situation Reports. Their estimates were within 8% for New York, Pennsylvania, Massachusetts, Rhode Island, and Virginia. However, their model substantially overestimated outages in Maryland and Delaware (possibly due to Sandy’s asymmetric wind field) and underestimated outages in Connecticut (possibly due to not directly accounting for surge in the model).

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Figure 4. Power outage forecasts for Hurricane Sandy at 84, 60, 36, 12, and 0 hours before landfall. Outage forecasts are at the census tract levels.

They also demonstrated the applicability of their models for implementing scenario analysis for hypothetical storms that could help with planning decisions. For example, they used their model to estimate the potential impacts of a number of historic storms—including Typhoon Haiyan—on the current U.S. energy infrastructure.

# Fragility-Based Methods1

The second major type of method that has been used to model power outages during hurricanes and other high-wind events is bottom-up engineering methods based on component-level fragility curves. These methods decompose the system into its constituent parts, estimate the probability of failure for each part, and then roll up the component-level estimates into an overall estimate of the system function or reliability. Fragility curves characterize the failure probability of a single component as a function of an event parameter such as wind speed for a hurricane event. Simulations are used to estimate the probability density function of component failures and loss of power to customers as a function of the relevant storm parameter by incorporating estimates of the demand parameter and component failure rates.

## Overview and Example of Fragility-Based Approach

One of the main examples of this approach is Winkler, Duenas-Osorio, Stein, and Subramanian (2010). Winkler et al. (2010) implemented a fragility-based approach for estimating power outages in Houston during hurricanes. This approach uses stochastic simulation to generate sets of pole failure scenarios for a given hurricane wind field and then translates these pole failure scenarios into customers without power based on the topology of the power system. Each of these steps is described here.

The first step in the Winkler et al. (2010) approach is to simulate N replications of system damage state. For each replication, poles are assumed to be in one of two possible states: failed or not failed. A hurricane wind field model is used to estimate the wind speed at each of the pole locations. Then an engineering fragility curve is used to estimate the probability of failure for each pole. A Monte Carlo simulation is done to generate sets of system states, with one set consisting of the state—failed or not failed—for each pole in the system.

The second step is to estimate which customers do not have power for each of the simulated system state. First, they assume that power lines follow roads because a detailed map of the topology of the power system was not available; that is, they assumed that the topology of the road network and the power system were the same. They then assumed pole locations. Note that these assumptions are not inherent in the fragility-based approach. Rather, the assumptions were made to address data limitations. Indeed, the federal government has also implemented a fragility-based approach using more realistic data. However, this approach is not published or described in detail in public documents, limiting our ability to report on its details. Winkler et al. (2010) use a connectivity-based model to estimate which customers would lose power. If a pole is damaged in a given simulated system state, all downstream customers beyond that pole are assumed to have lost power. This approach does not account for the physics of power flow, something that may limit the accuracy in some cases (LaRocca, Johansson, Hassel, & Guikema, 2014). The result of the Winkler et al. (2010) approach consists of a probability density function for the number of customers who would lose power during the storm. Winkler et al. (2010) report an overall prediction error of approximately 15%, higher than errors reported from the statistical models. However, they do provide much more localized damage estimates than the statistical models are able to provide. There are several potential ways in which the fragility-based approach could be further developed and improved.

Storms typically cause simultaneous failure of the system components. In order to implement simultaneous fragility analysis for several components, it has usually been necessary to assume conditional independence, given the demand parameter value. However, the assumption of conditional independence is not necessarily valid during a hurricane. For example, poles in a given utility service area generally have similar maintenance histories and perhaps ages, inducing probabilistic dependencies, even conditional on the wind speeds. In addition, standard fragility curves usually do not incorporate a wide range of covariates that could help explain the spatial variance of outages. Instead, they are usually based on gust wind speed alone. The work reviewed in “Statistical Learning Methods” points out the difficulties with this approach; hurricane impacts depend on a host of factors, not just gust wind speed (Nateghi et al., 2014). The fragility curves used are often standard fragility curves from the U.S. HAZUS model, and these curves do not adequately incorporate data from recent storms, another area in which the methods could be improved.

The fragility-based approach has promise in that it provides localized probabilistic estimates of impacts, but to date it has not achieved as high a predictive accuracy as statistical approaches. Additional research may help improve the outage estimates from the fragility-based approach. This research could focus on improving the fragility curves themselves, making the fragility curves multidimensional in the sense of capturing more aspects of the hazard beyond gust wind speed, and better accounting for probabilistic dependencies. As an example of recent work in the first of these areas, we give a brief overview of Han et al. (2014).

## A Bayesian Approach to Updating Fragility Curves Based on Storm Data

Han et al. (2014) developed a Bayesian approach to combine reliability models and failure data from the impacts of hurricanes on utility poles in the central Gulf Coast region of the United States. This approach yields fragility curves updated with observed storm data and would allow the curves to improve over time as additional storms occur, leading to an increasing amount of damage data.

In evaluating the probability of failure of a component of a power system, one must account for the fact that the system can often fail due to more than one failure mechanism. In other words, the probability of failure of a component can be defined by individual failure mechanisms such as flexural or foundation failures due to severe wind, trees falling on lines or poles, and wind-born debris striking lines or poles. In Han et al. (2014), only two failure mechanisms were considered: (1) flexural failure of poles due to wind, and (2) foundation failure of poles due to wind. While the other failure modes (e.g., tree-induced failures) are likely to play a significant role in terms of overall system reliability, the focus of this study was on only direct wind-induced failures. Han et al. (2014) did not address other wind-induced failures such as trees and debris falling on or being blown into poles and lines due to high winds directly in our structural reliability models. They also did not directly include failures due to other hazards such as inland flooding or storm surge along the coast. These failure mechanisms are accounted for indirectly through the Bayesian updates based on real-world failure data where these failure mechanisms do have an influence.

The Han et al. (2014) approach starts with a structural reliability model of poles. This approach uses first-order reliability methods to model the probability of failure of an individual pole due to the pole breaking because of wind forces acting on the pole and connected wires, or foundational failure due to wind forces acting on the pole and connected wires. These models estimate the probability of failure of a pole as a function of the gust wind speed.

The structural reliability models provide the basis for developing a prior probability distribution for the future frequency of pole failure as a function of wind speed. With $P( E i)$ representing the probability of failure due to the ith failure mechanism, the probabilities of the individual failure modes can be defined by:

$P(E1)=P(flexuralfailure|V)$: conditional probability of a pole breaking due to a bending moment induced by wind speed V, and

$P( E 2)=P( foundationfailure|V)$: conditional probability that the pole fails because the soil that the pole is supported by loses bearing strength given wind speed V.

Assuming that the two failure events are statistically independent, a simplification that provides a starting point, the probability of failure of the power distribution system poles is

$Display mathematics$
(2)

and the cumulative density function is

$Display mathematics$
(3)

While the assumption of independence is not strictly correct given that the formulation of the limit state functions involves common random variables, it facilitates the analysis for the prior, which can be obtained simply. Furthermore, the assumption of independence is really one of conditional independence here: the two failure modes are assumed to be conditionally independent, given wind speed. While there may still be sources of dependence (e.g., span length appearing in both failure mode equations, inducing a dependency), the assumption of conditional independence may be a reasonable first approximation.

The limit state function for each of the individual failure modes for the reliability analysis is defined as

$Display mathematics$
(4)

where $(X1,X2,…,Xn)$ are random variables, R is the resistance capacity for the individual failure mode, and W is the wind load. The resistance capacity and geometry information of the power distribution system are obtained from the ANSI standard O5.1 classification of pole structures.

For evaluating the probability of failure for the individual failure modes, Han et al. (2014) used first-order reliability methods (FORM) because the limit state function for each failure mechanism can be assumed to be linear and the random variables used in structural reliability models are uncorrelated. Specifically, the advanced first-order second-moment (AFOSM) method was used in order to include non-normally distributed random variables. Using the AFOSM approach, the probability of failure of a single pole as a function of wind speed was estimated.

The structural reliability model provided the basis for the fragility curves, which were then updated with data from observed pole failures during hurricanes in the central Gulf Coast region. The standard Bayesian updating formula is defined as

$Display mathematics$
(5)

where the parameter p is the probability of failure of a power distribution system pole, given the observed data consisting of f failed poles out of t poles under a wind speed of V. Considering p as the frequency of pole failure, $f(p|V)$ represents the prior probability density function of the probability of failure for poles under wind loads. In estimating the posterior probability density function (PDF) for the number of damaged poles in a given grid cell, a beta distribution is an appropriate prior, provided that the pole failures are assumed to be conditionally independent, given wind speed. This distribution is constrained to the (0, 1) interval, as are probabilities. The beta distribution is also highly flexible in its ability to model differing degrees of information and accuracy in the prior. Pole failures are discrete, nonnegative counts, and, barring additional information that could be used as additional conditioning variables, the probability of failure can reasonably be assumed to be constant across all poles that experience the same wind speed. These lead to a representation of the likelihood as a series of independent and identically distributed Bernoulli events, making the binomial distribution a natural probability mass function (PMF) for the likelihood function. The model thus uses a conjugate beta-binomial pair, allowing the Bayesian updating to be carried out analytically. With a beta prior with parameters f and t obtained from the structural reliability model (see Han et al., 2014 for details) and a binomial likelihood for the observed data consisting of f' failures out of t' poles, the posterior distribution is also a beta distribution with parameters f', t', f, and t. The posterior is given by Equation 6:

$Display mathematics$
(6)

The mean and variance of the posterior beta distribution are given by Equations 7 and 8, respectively:

$Display mathematics$
(7)

$Display mathematics$
(8)

This process produces the mean frequency of pole failure, given a wind speed, as well as the full PDF for the future frequency of pole failure given a wind speed. Southern Pine is a common type of tree used for utility poles, and Figure 5 shows the results of updating the prior with observed failure data. The prior is the rightmost line in Figure 5, and the mean of the posterior is the heavy dark line. The dash-dot lines give the 95% credible intervals for the posterior. We see that the posterior has shifted considerably to the left, meaning that the structural reliability model was providing substantially higher estimates of pole reliability than a purely data-driven approach would have.

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Figure 5. Prior fragility curve, posterior fragility curve, and its confidence intervals for Southern Pine, 12.47 kV distribution line.

# Modeling Power Outage Restoration

Various approaches have been leveraged to model power system restoration after disasters. The approaches include engineering fragility curve fitting, deterministic resource constraint methods, simulation and optimization methods, network-based approaches, and regression. In the empirical curve fitting approach, data are gathered either from past records or by eliciting expert opinion to plot restoration curves that represent utility performance during extreme events (Nojima & Sugito, 2002). These restoration curves depict the percentage of customers with restored service or percentage of demand met as a function of time since the occurrence of the event. The restoration curves usually do not account for spatial variations in restoration times and are typically not validated.

Isumi and Shibuya (1985) used a deterministic resource constraint method to model a postearthquake restoration process. They used an optimization model to minimize the average time that a customer is without power, using decision variables such as damage assessments, inspections, and repair times and constraints such as the system’s characteristics and the total number of available crews. Sensitivity analysis was used to estimate the impact of various factors such as the number of crews available on restoration times. Discrete-transition Markov processes have also been used to model the performance of individual systems after a natural hazard impact such as an earthquake. However, it is a challenging approach as it is very data-intensive. Discrete event simulation is structurally similar to the discrete-transition Markov approach in that it is a data-intensive simulation-based representation of postdisaster restoration. In the network approach, the power system can be modeled as a series of supply and demand nodes connected to one another via links. Graph theory and optimization techniques can then be leveraged to minimize the mean restoration time, with recovery defined as customers being connected to supply nodes. This technique allows for the incorporating of spatiotemporal variations in the restoration procedures. For example, Xu et al. (2007) developed an approach for optimizing the order in which substations are repaired after a major disaster based on the detailed simulation model of Cagnan et al. (2006). Regression models have also been used to estimate power outage durations. This approach uses a set of covariates and outage duration data from similar past events to predict the outage durations during future events. Davidson et al. (2003) investigated the performance of the electric power distribution system in North Carolina and South Carolina using historical data from five hurricanes. Through a correlation-based study of the outage duration data, they established that there is a weak relationship between the percentage of prolonged outages and increasing hurricane intensity. Based on overlaying the maps of outage durations and maximum gust wind speed, rainfall, and population density, they observed that the cluster of outages were more substantial for densely populated areas. They also reported that maximum gust wind speed and rainfall amounts did not seem to be related to duration of outages.

Liu et al. (2007) used survival analysis to model power outage restoration times for ice storms and hurricanes. They used both Accelerated Failure Time (AFT) and Cox Proportional Hazard (CPH) regression techniques to predict outage duration times. They recommended fitting AFT models over CPH due to its better interpretability. They did not implement out-of-sample holdout tests to assess the predictive power of their proposed model. Nateghi et al. (2014) used an ensemble tree-based regression model (method of Random Forest) to estimate power outage durations in the Gulf Coast of the United States. Their model was rigorously validated and was shown to be substantially more accurate than the existing statistical regression models in the literature.

# Long-Term Hurricane-Induced Power Outage Risks

The link between climate change and hurricane activity has been researched extensively (Emanuel et al., 2008; Knutson et al., 2010; Mann & Emanuel, 2006). However, there are still substantial uncertainties involved (Staid et al., 2014; Berner et al., 2017). Staid et al. (2014) modeled the long-term hurricane risks to power systems through simulating plausible scenarios in which climate change may affect hurricane intensity, frequency, and intensity. In their sensitivity analysis-based approach, they investigated how changes in tropical cyclone activity could influence extreme wind speeds, probability of power outages, and proportion of people without power. Their simulation results demonstrate the impact under both the status quo of the baseline case and the climate change-induced scenarios that were evaluated. The scenarios exemplify the sensitivity of various areas of the country to potential changes in tropical cyclone behavior, and the results can be evaluated on a local level. Figures 6 and 7 summarize parts of their findings visually. Figure 6 shows the impact for simulated years with impact measures of (1) the 100-year wind speed (the wind speed with an annual probability of exceedance of 0.01), (2) the probability of at least 10% of customers losing power in a given year, and (3) the 100-year fraction of customers without power from a given storm, as assessed for each census tract individually. These results show the estimated conditions under the historic state of tropical cyclone activity.

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Figure 6. Baseline impacts of (a) 100-year wind speed, (b) annual probability of at least 10% of customers losing power, and (c) 100-year fraction of utility customers without power plotted for each census tract.

To study the potential impacts of climate change on hurricane intensity, Staid et al. (2014) ran the simulations with varied intensity levels to see the potential impacts on 100-year wind speeds and also on the fraction of customers losing power. When looking at wind speeds, the effects of varying intensity are felt primarily in coastal areas. This can be seen in Figure 7, where the changes are seen primarily along the coasts. The biggest changes are in those areas that receive the most frequent hurricanes, indicating that they are particularly sensitive to changes in hurricane intensity.

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Figure 7. Changes in 100-year wind speeds for varying storm intensity away from baseline.

The fraction of customers without power, however, depends both on wind speed and storm size. Stronger storms are larger, and the simulations show the effects of storm size as the “reach” of the stronger (or weaker) storms creates bands of increased (or decreased) impact. These are the areas on the margins of the impacted area and are areas of the country particularly sensitive to changes in hurricane intensity. Figure 8 shows this result, as, on average, the stronger storms impact areas further inland than in the baseline case. The areas that fall on the edge of the impacted area see the largest changes when the average storm intensity varies. This is of particular interest because many of these areas are farther inland and do not have a strong history of experience with hurricanes. Additional consideration of hurricane preparation planning may be appropriate in such areas.

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Figure 8. Changes in 100-year fraction of customers without power for varying storm intensity away from baseline.

In general, the results of Staid et al. (2014) demonstrate that inland areas tend to be more protected from the strongest impacts, but some areas may still see considerable changes in maximum wind speeds, power outage likelihood, and the number of customers losing power during a hurricane. Their analysis shows that a shift in landfall location could result in impacts to areas of the country with little experience in dealing with hurricanes. Also, they discuss how coastal areas are particularly sensitive to increases in storm intensity. According to their results, 100-year wind speeds are projected to increase by more than 50% in some areas with a 20% increase in storm intensity. The probability of customer power outages in a given area increases slightly, but the actual number of customers losing power would change more drastically as a result of stronger, and often larger, storms. A reduction in storm frequency, however, is projected to have a corresponding reduction in the likelihood of power outages. Future work is needed to better understand the link between climate change and the effects on hurricane frequency, intensity, size, and landfall location.

# The Challenges of the U.S. Regulatory Environment

Our modern society relies largely on electric power for its proper functioning. Electric consumption is an indicator of the economic health of every society. Unfortunately, the United States continues to fall behind developed nations in terms of electricity reliability. The Reliability Demand Survey (RDS) conducted in 2012 revealed that many customers’ levels of power reliability were less than their expectations (King, 2012). It has been argued that the costs associated with improved grid reliability are too high. However, research has shown that the expense of repetitive customer interruptions and system repairs is far greater (O’Neil, 2004).

Here we briefly describe the current landscape of electric reliability standards and regulations in the United States and discuss their challenges.

Three years following the large-scale Northeast blackout of 1965, the North American Electric Reliability Corporation (NERC) was established as a voluntary membership organization. After a series of blackouts following the establishment of NERC—including the 2003 Northeast blackout—the need for enforceable reliability standards became evident. The Federal Energy Regulatory Commission (FERC) certified NERC in 2007 as the Electric Reliability Organization (ERO), making compliance with reliability standards mandatory and enforceable (NERC, 2007). NERC is in charge of developing mandatory reliability standards for the U.S. bulk power system (BPS) and of assessing power system reliability annually through both long-term (10-year) and short-term (winter and summer) assessments. Along with monitoring the bulk power system, NERC educates, trains, and certifies industry personnel (NERC, 2007). While NERC oversees the reliability of the bulk power systems, which consist of generation and transmission, the reliability of power distribution systems falls under the jurisdiction of individual states.

Before further discussing the current landscape of reliability standards for power systems, it is imperative to explore what the term “reliability” encompasses. One of the earliest formal definitions of reliability was in the military reliability handbook where reliability was defined as “the probability of performing without failure, a specific function under given conditions for a specified period of time” (MIL-HDBK-338B). This basic definition of reliability is the same one used in most reliability engineering textbooks. (Modarres et al., 1999; Barlow, 1998).

The NERC definition of reliability differs from the standard engineering definition of reliability. NERC defines the reliability of a bulk power system as “an electricity service level or the degree of performance of the bulk power system defined by accepted standards and other public criteria.” According to NERC, there are two basic functional components of reliability: “operating reliability” and “adequacy.” According to NERC’s definition, operating reliability refers to the ability of the electric power system to withstand sudden disturbances (e.g., electric short circuits) or unexpected component failures. Adequacy refers to the ability of the system to supply the electrical demands of the customers at all times, taking into account scheduled and expected unscheduled outages. Comparing this definition to the earlier definition of reliability, it can be seen that NERC’s definition is more deterministic and the probabilistic aspect of performance is missing. Moreover, NERC’s definition does not specify many components of reliability beyond vague statements such as “accepted standards.”

In calculating risk indices to characterize the risks posed to power systems, NERC defines extreme events such as deliberate attacks and natural disasters as high-impact, low-frequency events (HILF) (NERC, 2011). HILF events can have widespread and catastrophic impacts on the electric power system. These events are categorized by NERC as “macro-prudential” risks, meaning that these are risks that are “not transferrable and cannot be fully insured against, diversified or hedged at the individual firm level” (NERC, 2011). Note that this definition by NERC ignores the existence of catastrophe insurance and bond markets. These financial instruments deal with precisely these types of large-scale, high-impact, low-probability events NERC is concerned with. Furthermore, this characterization assumes a very finance-centric view of risk rather than a view focused on a system’s ability to adequately meet customer expectations. Stating it simply, NERC’s justification for not addressing HILF events in their risk assessments is focused not on customers’ expectations of service levels but on financial issues at the utility firm level. NERC’s justification for treating these risks differently is based on the argument that they are rare and therefore cannot be modeled or quantified. Note, however, that there is a large body of research on modeling HILF events, much of it within the context of power system risk analysis modeling for high-impact, low-probability risk (Guikema & Aven, 2010; Guikema et al., 2014; Bell & Glade, 2004; Staid et al., 2014; Nateghi et al., 2011, 2014; Cagnan et al., 2006). Furthermore, many consulting firms and government agencies are also engaged in practical work to examine HILF events.

Categorizing HILF events as a different type of risk that is excluded from risk assessments based on the argument that they are rare and hard to model and prepare for is problematic. HILF events have substantial impacts on customers’ electric power service; they impact reliability from a customer’s point of view. By excluding HILF events, NERC discounts the customer’s expectation and utilizes an incomplete view of reliability. Is a power outage due to a HILF weather event any less disruptive to a customer than an outage due to a technical failure of a substation component? Not necessarily. From a customer’s perspective, a lost power event is disruptive, regardless of the trigger event. NERC’s risk indices provide a different picture than this. Extreme events risks should be dealt with adequately. Research shows that climate change will likely increase the frequency and intensity of extreme weather events. Therefore, utilities need to ensure that appropriate measures are taken to protect their systems and mitigate the impacts of disasters (Peterson et al., 2012). If the appropriate reliability standards for handling the impacts of extreme events are not put in place, we may face a more highly stressed grid in the future.

While there exists a national NERC-mandated, though problematic, definition of reliability for the bulk power system, there is no standard definition for reliability of distribution systems. The reliability of power distribution systems is regulated at the state level and varies greatly, ranging from states with no regulations to states with comprehensive return on equity performance-based regulations. According to a study by Lawrence Berkley National Laboratory (LBNL), only 37 utility commissions have some type of reliability accounting system where either the utility reports reliability indices or raw outage data. It should be emphasized that understanding and ensuring the reliability of the power distribution systems is very important, since about 90% of electricity outages occur in the distribution system rather than the bulk power system (Executive Office of the President Report, 2013). Ensuring the reliability of power distribution systems can be a challenging task depending on the network characteristics of the system (underground vs. overground, radial vs. interconnected), physical characteristics of the system, the age of the system, and the geographical characteristics of the service area (e.g., areas with frequent storms, heavy precipitation, or wildfires).

A wide range of techniques exist for measuring the reliability of power distribution systems. The most commonly used performance measures are indices defined by the Institute of Electrical and Electronics Engineers (IEEE) Standard 1366. These indices calculate the average frequency and duration of interruptions at the customer and utility level (IEEE Std 1366-2003). Nateghi et al. (2016) identifies several issues with the way power distribution reliability indices are calculated and reported. For example, utilities vary widely in the way they calculate their reliability indices, which makes it hard to compare the indices from different utilities in different regions (Nateghi et al., 2016). Moreover, in calculating reliability indices, many utilities tend to exclude major events, which does not give a comprehensive picture of the state of reliability of a utility company. Also, the reported indices are average values that could misrepresent the reliability of a service area. Pockets of poor reliability (meaning areas with a large frequency or duration of outages) will not have a large impact on system-wide reliability averages. In general, mean values are not a good representation of central tendency in the presence of outliers or when the data are skewed and thus reporting average values is not fully informative. To address this issue, some states have resorted to reporting a geographically disaggregated measurement (i.e., using a two-tiered reporting system to report the reliability of rural areas separately from areas with high population densities).

After extreme weather events that account for the majority of distribution-level outages, trees have been stated as the second most important predictor of outages (First Quantile Consulting Report, 2009). Research has also shown the importance of tree-trimming practices for reducing outages, particularly during high wind events (Guikema et al., 2006; Nateghi et al., 2014). Vegetation management is a big part of power companies’ operation and management (O&M) budgets. Schedules for tree trimming have come under scrutiny recently, as some utilities appear to be increasing the period between these operations in an effort to reduce maintenance costs. Some states already have vegetation management targets in place; however, many states do not regulate the vegetation management practices of power companies.

Lastly, it is worth mentioning that calculating reliability indices and metrics is most useful when there is a target level of performance to be met. Targets cannot be selected based on a “one-size-fits-all” strategy but should be tailored to the particular characteristics of each utility. Currently only 25 states have established reliability targets, and 20 have implemented some form of penalty. For example, the states of Massachusetts and New York already have imposed penalties for noncomplying utilities. However, in many cases the reliability targets are not stringent enough, for example, in selecting target values for average interruption frequency that are much larger than the national average or in imposing small penalties that could easily be ignored (First Quantile Consulting Report, 2009).

# Acknowledgments

The work underlying much of this article was funded by the U.S. National Science Foundation under grants 1621116, 1631409, and 1555582. This support is gratefully acknowledged. The opinions offered in this article are those of the authors and do not necessarily represent the opinions of the sponsors.

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## Notes:

(1.) This section draws heavily from and summarizes the paper by R. Nateghi and S. D. Guikema (2011). A comparison of top-down statistical models with bottom-up methods for power system reliability estimation in high wind events. Proceedings of ICVRAM, Vulnerability, Uncertainty, and Risk: Analysis, Modeling, and Management (pp. 594–601).