Review on Energy Resilience

ARTICLES PUBLISHED: 29 APRIL 2016 | ARTICLE NUMBER: 16052 | DOI: 10.1038/NENERGY.2016.52

Large-scale data analysis of power grid resilience across multiple US service regions Chuanyi Ji1*, Yun Wei1, Henry Mei1, Jorge Calzada2, Matthew Carey3, Steve Church4, Timothy Hayes5, Brian Nugent6, Gregory Stella3, Matthew Wallace3, Joe White6 and Robert Wilcox2

Severe weather events frequently result in large-scale power failures, a�ecting millions of people for extended durations. However, the lack of comprehensive, detailed failure and recovery data has impeded large-scale resilience studies. Here, we analyse data from four major service regions representing Upstate New York during Super Storm Sandy and daily operations. Using non-stationary spatiotemporal random processes that relate infrastructural failures to recoveries and cost, our data analysis shows that local power failures have a disproportionally large non-local impact on people (that is, the top 20% of failures interrupted 84% of services to customers). A large number (89%) of small failures, represented by the bottom 34% of customers and commonplace devices, resulted in 56% of the total cost of 28 million customer interruption hours. Our study shows that extreme weather does not cause, but rather exacerbates, existing vulnerabilities, which are obscured in daily operations.

Severe weather disruptions, such as hurricanes and winterstorms, have become an important initiating cause of large-scale failures to the electric power grid. During Super Storm Sandy in 2012, power failures occurred across distribution system operators (DSOs) service regions in 21 states, affecting more than 8.5 million customers1. Power distribution (that is, the final stage of energy infrastructure) is particularly vulnerable, representing∼90% of failures2,3. Each disruption has left millions of people without electricity for days, motivating the need to characterize resilience and ultimately re-design energy infrastructure2,4,5.

Resilience is defined as the ability of the power grid to resist failures induced by external hazards, to reduce their impacts on people as much as possible, and to rapidly restore services to utility customers after disruptions2,4,6,7. Although the frequency and severity of extreme weather events vary, each event provides a unique opportunity to examine the otherwise hidden vulnerability (that is, lack of resilience) of power grids. However, the lack of comprehensive and sufficiently detailed failure and recovery data, as well as an attendant mathematical model, has impeded such analyses. In addition to data granularity, previous studies have been hampered by a lack of geographical scale spanning multiple service territories.

The necessity of obtaining both high-resolution and large-scale failure data poses a scientific challenge to studying resilience8,9. Data on failure and recovery have traditionally been collected mainly for reporting purposes1. Such data are stored within individual service territories, each managed by a DSO, and are not generally shared beyond service regions. For example, US federal and state governments require only aggregated information such as the total customer service interruption duration from investor-owned DSOs during major storms2. These data are often aggregated into hourly or daily statistics over townships, too crude to study the resilience of the infrastructure and services. As a result, there are only a

small number of studies that make use of failure data from even one service region10–14. A recent study examines power failures at a national scale across the United States for daily operations, where the data are aggregated annually and over service territories15. At a regional scale, existing studies examine resilience in terms of economic impacts through what-if scenarios (for example, potential blackout due to failures of power transmission in Los Angeles16). However, a resilience study that combines detailed failure data with a large scale across multiple service territories has not hitherto existed.

In addition to real data, a resilience study requires the analytical modelling of failure, recovery and cost with respect to time, geolocation, system location, customers, and their interdependencies. Models capturing the economic impacts of these events are also needed16.

We study resilience characteristics by examining detailed data from both Super Storm Sandy and daily operations across four major service regions representing Upstate New York, an area of 50,590 square miles (Table 1). We develop a model that integrates failure, recovery and impact variables from the bottom up17,18. The resulting model is multi-scale in nature, starting from individual components, incorporating structures of power distribution and support to customers, then aggregating to DSO service regions (Methods). Failures, recoveries, and their customer impacts form non-stationary random processes that are dependent on time, geolocation, and system location (Methods). Therefore, resilience is a network property derived from the physical infrastructure to services and customers. We characterize these interdependent processes with network-wide metrics, such as disruption rates, and time-varying failure/recovery probability distributions. These metrics serve as guidelines for identifying vulnerabilities and quantifying time-varying cost using large-scale real data (Methods). In particular, we demonstrate that extreme weather does not cause, but rather exacerbates, certain existing

1 School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332–0250, USA. 2 National Grid, Waltham, Massachusetts 02451, USA. 3 New York State Public Service Commission, Albany, New York 12223-1350, USA. 4 New York State Electric and Gas Corporation, Binghamton, New York 13904, USA. 5 Central Hudson Gas & Electric Corp., Poughkeepsie, New York 12601, USA. 6 Orange & Rockland Utilities, Pearl River, New York 10965, USA. *e-mail: [email protected]

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ARTICLES NATURE ENERGY DOI: 10.1038/NENERGY.2016.52

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infrastructural vulnerabilities; traditional restoration approaches prioritizing disruptions that affect the highest number of customers are inadequate for minimizing total customer outage times.

Infrastructural vulnerability Failures that occurred with respect to power distribution during a severe weather event are largely due to local causes (for example, fallen debris, damaged poles and area flooding). Unlike transmission systems19–21, such failures typically do not cascade in radial power distribution designs. This is problematic because the majority of power distribution for our four DSOs has an overhead radial topology and corresponding protection6. Failures induced by severe weather can result in outages to downstream devices, which lose power without being damaged (see Supplementary Note 1). Nevertheless, disruptions (that is, either failures or outages)

resulting from exogenous events are expected to remain largely local (at individual devices and geolocations6,22).

Here, a fundamental question arises: is the impact of a local failure also local, affecting only a small number of customers? Disruption rates, which correspond to the marginal temporal costs per unit time, provide an answer (Methods). In evaluating customers and system-level disruptions, we find that a common behaviour emerges for the four DSOs shown in Fig. 1. During Super Storm Sandy, customer disruption rates were 93 to 193 times larger than system disruption rates (Supplementary Note 2). These higher rates demonstrate the existence of infrastructural vulnerability, in which local failures have a distinctly non-local impact on customers (Fig. 1). In fact, the top 20% of system disruptions affected 79% ∼89% of customers for the four DSOs, and each such disruption interrupted services for more than 74∼107 customers (Fig. 2a).

To quantify the prevalence of non-local impacts on customers, we analyse the dependence between infrastructural failures and affected customers. Because operational power distribution systems are based on engineering design and exhibit irregularities, the scaling behaviour we observe (for example, 20 ∼ 80) cannot be represented using a simple power-law distribution23. Thus, we obtain a generalized scaling law drawn from the data: a mapping between the probability W(x) for a customer affected by a disruption that interrupts electricity service for more than x users and the probability P(x) of such a disruption. Taking DSO 1 as an example, the generalized scaling law demonstrates that the top 0.1% ∼ 76% of system disruptions affected 2% ∼99.6% of all customers at the 90% confidence interval (Supplementary Note 3 and Supplementary Table 3). A similar generalized scaling law applies to the other three DSOs, with small variations (Fig. 2a and Supplementary Table 3).

Do exogenous events, such as Super Storm Sandy, create the infrastructural vulnerability or instead exacerbate an underlying lack of resilience? We apply the methodology outlined above to daily operations to show that vulnerability is inherent to the power distribution infrastructure. Using data on failures in daily operations from the year prior to the hurricane, a similar non-local impact on customers holds (Fig. 2b and Supplementary Fig. 2 and Supplementary Table 3). The mapping between the two probabilities differs by less than 2% ∼9% for daily operations and Super Storm Sandy at the 95% confidence level. Thus, customers interrupted

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Figure 2 | Scaling law for customer disruptions. a, Generalized scaling law for the four DSOs during Super Storm Sandy: empirical probability W̃(x) of the number of a�ected customers, where each of the disruptions a�ected more than x customers, and empirical probability P̃(x) that a disruption a�ected more than x customers. Shaded areas illustrate the estimation error at the 95% confidence level. The dotted horizontal lines point to the number of customers a�ected by the top 20% disruptions. The vertical lines on the right represent the number of customers a�ected per top x percentage disruptions. b, Generalized scaling law for DSO 1 during Super Storm Sandy (orange line) and daily operations (blue line) as a case study. Types of disrupted devices are shown on the right with respect to the number of interrupted customers. The histogram shows the percentage of customers from Super Storm Sandy storm (red) and the daily operations (blue). The generalized scaling laws for the other DSOs during daily operations are given in Supplementary Fig. 2.

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by disruptions in daily operations and the storm follow a similar scaling law. Compared to Super Storm Sandy, each disruption in daily operations affected 29, 19, 48 and 21 fewer customers for the four respective DSOs. The similarity between the scaling laws for the storm and daily operations demonstrates that Super Storm Sandy did not create but only exacerbated the non-local impacts on customers.

Although the overhead power distribution infrastructure is a complex combination of topology, protection schemes, and supporting mechanism for customers, the four service regions have a common radial and hierarchical structure. The infrastructure and disruptions can be represented by three commonly used system locations22: primary distribution, secondary distribution, and customer property. A disruption at the top level can disconnect customers away from a power source in the existing overhead power distribution7 owing to a lack of reconfiguration and distributed generation. Demonstrated by the data (Fig. 3a), among the top 20% of disruptions from Super Storm Sandy, approximately 77%

occurred at the primary distribution level; each such disruption affected ∼524 customers on average. The remaining 23% of disruptions occurred at the secondary distribution level, each of which affected∼218 customers on average.

The impact of the hierarchical design on the number of affected customers is further characterized by the five major types of activated protective devices at the corresponding system locations and through the generalized scaling law (Figs 2b, 3b, and Supplementary Fig. 5). For example, non-functional substations, open reclosers, and blown fuses at the primary distribution level represent the majority of the top 20% of disruptions that occurred during both Super Storm Sandy and daily operations; the same type of devices had a non-local impact on customers regardless of weather. Thus, the hierarchical structure is an important underlying cause of the infrastructural vulnerability for overhead power distribution with a radial topology.

The degree to which Super Storm Sandy exacerbated the infrastructural vulnerability is characterized by the non-stationary

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Figure 6 | Geographical distribution of the cost in Upstate New York during Super Storm Sandy. a, Number of disrupted customers. b, Customer interruption hours (CMI). Colours in a represent the top 20% versus the remaining disruptions. Colours in b represent Category 1 and Category 2 disruptions. Each marker represents a system disruption. The size of a marker represents the number of interrupted customers for a and customer interruption hours for b. Map reproduced using OpenStreetMap and arcGIS software.

probability distribution P1(t) for at least one disruption occurring in a service region at time t measured in a minute. Using empirical data from both Super Storm Sandy and daily operations (Supplementary Note 4), we find that P1(t) was less than 0.02 per minute in daily operations, but increased to 1.0 at the peak impact of the storm. The increase in the probability was particularly pronounced for the top

(for example, 20%) disruptions, in which P1(t) was less than 0.003 in daily operations, but increased to the maximum values of 0.6∼1.0 during the storm for the four DSOs (Fig. 3 and Supplementary Fig. 3 and Supplementary Table 4). Further relating to the hierarchy, the probability of a disruption at the primary distribution level shows a disproportionally large increase of 28 times for the top

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Table 1 | Data sets on the number of disruptions and the number of a�ected customers.

Daily operation Super Storm Sandy Disruptions (occurrence time) A�ected customers Disruptions (occurrence time) A�ected customers

DSO 1 5,643 (10/28/2011 ∼ 10/28/2012) 302,688 1,334 (7:43 10/28/2012 ∼ 12:59 10/31/2012) 109,392 DSO 2 9,992 (10/1/2011 ∼ 10/28/2012) 764,988 1,184 (1:53 10/28/2012 ∼ 12:51 10/31/2012) 113,488 DSO 3 13,063 (1/1/2012 ∼ 10/28/2012) 961,114 1,866 (1:58 10/28/2012 ∼ 12:57 10/31/2012) 249,339 DSO 4 3,023 (1/1/2012 ∼ 10/28/2012) 225,997 1,882 (2:39 10/28/2012 ∼ 12:56 10/31/2012) 174,549 Total 31,721 2,254,787 6,266 646,768 Occurrence time: time durations in which the disruptions occurred. Time accuracy of disruption and recovery: one minute. Location accuracy: latitude and longitude for each disruption. The data sets are preprocessed to remove the disruptions from intentional or prearranged disruptions, and small storms for normal daily operations.

20%, compared with 15 times at the two remaining levels (Fig. 3a and Supplementary Fig. 4). Thus, our study demonstrates that the infrastructural vulnerability was present only at low levels during daily operations until magnified by severe weather events such as Super Storm Sandy.

Impact to service and cost Multiple failures (that is, 6,266 system disruptions occurred during Super Storm Sandy) challenge the state of the art in recovery6. In daily operations, 90% of the interrupted customers recovered within 6 h, and all services were restored in one day (Fig. 4). However, recoveries from Super Storm Sandy lasted 8, 4, 10 and 15 days for the four DSOs.

We quantify recovery through the interdependence between failure and recovery processes. The probability density functions of downtime duration given the top percentage of disruptions exhibit a common characteristic in Fig. 5 for the four service regions during Super Storm Sandy. Disruptions that affected the highest number of customers were usually restored at an early stage of recovery (that is, the first 24%, 8%, 15% and 34% of the total restoration time) with peak probability density (0.006, 0.04, 0.003 and 0.002 per minute, respectively). These disruptions that affected the largest number of customers and had early recovery constitute so-called Category 1 (‘large/early’), amounting to 9% of the total system disruptions; each affected more than 89 customers (Supplementary Note 5). Such a characteristic reflects a common practice of repairs: power sources, such as substations, are restored first, followed by other components downstream. Such a pattern of recovery persists through daily operations, except that all disruptions are restored rapidly in minutes or hours (Supplementary Fig. 7). The remaining disruptions are in so-called Category 2, which mainly consists of ‘small’ system disruptions, each of which either affects the bottom 34% of customers or is represented by a common place device such as a fuse. Category 2 disruptions dominate the late stage of recovery at the four service regions almost exclusively (Supplementary Fig. 6). Category 2 (‘large/late’) also consists of a small number of disruptions (2%) that affected a large number of customers but recovered late (Table 2 and Supplementary Note 5).

The total service cost quantifies the impact of prolonged disruptions on customers, which as a special case of equation (2), is measured as the total customer interruption time by the data set. Aggregating over only disruptions occurring during the first four days when Super Storm Sandy impacted Upstate New York, the four service territories together experienced approximately 28 million customer hours of interruptions. The ‘small’ disruptions in Category 2 together (89% of the total) contribute to approximately 56% of the total cost but only amount to 34% affected customers (Table 2). Thus, whereas ‘large’ disruptions (9% of the total) that affected the highest number of customers were restored early, the aggregation of Category 2 disruptions from low-customer and small-device disruptions resulted in a disproportionately larger portion of the total cost. Hence, the total service cost in Fig. 6b exhibits a behaviour different from that of the generalized scaling

law for disruptions in Fig. 6a. This suggests that recovery, which is compounded by restoration strategies, available resources, the infrastructural vulnerability and the cost of enhancement, requires consideration of the overall cost of restoration in addition to individual disruptions that affect the highest number of customers.

We further examine the service cost in terms of disrupted devices and the power distribution hierarchy. Data samples with clearly labelled device types are used for this purpose (Methods). The total cost for the reduced data sets from Super Storm Sandy is 20 million customer interruption hours (Supplementary Table 6). Approximately 4% of the total cost results from disruptions by ‘large devices’ in Category 1 that were restored early (Supplementary Table 6). These interrupted devices represent about 28% of the affected customers. In contrast, 57% of the 20 million customer disruption hours results from Category 2 ‘small’ disruptions that recovered ‘late’. These interrupted devices amounted to 34% interrupted customers.

This implies that aggregation of the small devices represented by blown fuses and other components at a lower level of the system hierarchy plus the majority of the non-functional transformers at the primary distribution level, although each affected a moderate number of customers (Fig. 3b and Supplementary Table 6), resulted in a larger portion of the total customer downtime. This again suggests that the early recovery of the components that affected the highest number of customers is insufficient for reducing the total cost to end users.

Data from daily operations is also studied for disrupted devices as a comparison (Supplementary Table 6). Open substation breakers and reclosers as well as non-functioning transformers at the primary distribution level in Category 1 (‘large/early’) amounted to 11% of customer interruption hours and 30% of the interrupted customers (Supplementary Table 6). In comparison, blown fuses and other devices in Category 2 result in 56% of the total customer interruption time and 38% affected customers. Therefore, a similar disparity of the costs is observed that reflects a similar trend of recovery on prioritizing disruptions that affected the highest number of customers in daily operations.

Discussion Using large-scale data collected by DSOs in Upstate New York, we obtain critical insights revealing that severe weather events, such as Super Storm Sandy, do not fundamentally cause infrastructural vulnerabilities. Vulnerabilities are inherently a complement of infrastructure resilience, and exist regardless of exogenous factors. Although such infrastructural vulnerabilities may manifest themselves locally, together, they have a profoundly non-local impact on customers. We characterize this asymmetrical relationship between disruptions and affected customers using the generalized scaling law estimated empirically from the data. The similarities of the scaling law for different service territories, and for Super Storm Sandy and daily operations, suggest that the infrastructural vulnerability exists independently of the exogenous events themselves and their effects. Whereas the number of DSOs

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Table 2 | Cost.

Disruptions Customers Customer interruption hours

Category 1 Category 2 Category 2 Category 1 Category 2 Category 2 Category 1 Category 2 Category 2 (large/early) (small) (large/late) (large/early) (small) (large/late) (large/early) (small) (large/late)

DSO 1 102 1,228 4 61,461 44,080 3,851 645,268 2,060,193 198,880 DSO 2 109 1,055 16 68,848 30,426 14,040 113,404 284,584 90,927 DSO 3 154 1,635 77 93,897 79,108 76,334 623,126 5,569,573 5,342,695 DSO 4 186 1,618 5 99,350 63,161 4,960 4,297,389 7,701,248 812,104 Total 6,189 639,516 27,739,390 Percentage 8.9% 89.4% 1.6% 50.6% 33.9% 15.5% 20.5% 56.3% 23.2% Cost is estimated as the number of a�ected customers and customer interruption hours for the two categories during Super Storm Sandy. Category 1: system disruptions that a�ected relatively large numbers of customers and recovered early. Category 2 (small): disruptions that either a�ected a relatively small number of customers or on commonplace devices. Category 2 (large/late): disruptions that a�ected a large number of customers but recovered late.

analysed is small, their combined service regions span nearly the entirety of Upstate New York. A large-scale analysis of data from multiple DSOs is necessary to understand the prevalence of such vulnerabilities and as a control against exogenous events24. As industry increasingly realizes the importance of data collection and study beyond experiences24, the availability of more detailed data25, especially regarding restoration and the root causes of failures, may allow for further analysis in recovery dynamics and yield optimal cooperative strategies.

When the recovery process was compounded by multiple factors (for example, infrastructure vulnerability, cost of enhancement, impacts from severe weather, and available resources), the restoration was apt to prioritize the recovery of disruptions that affected the highest number of customers during both the storm and daily operations. Although such system disruptions affected the highest number of customers, the aggregation of smaller disruptions proved to be a significant factor that must be considered in recovery. Unfortunately, recovery is also compelled by constraints introduced by the existing power distribution infrastructure: major components, such as substations, must be repaired first due to a lack of distributed generation. Moreover, whereas some failures are easily repairable (resetting breakers or replacing fuses), others may be arduous to restore owing to damaged power components or external circumstances, such as flooding26. Therefore, multiple factors should be considered jointly when designing enhancement options and recovery strategies for resilience. DSOs must weigh the impact of infrastructural vulnerability against the economic and societal costs of such improvements16, in addition to factors not considered in this paper such as life cycle cost27.

Resilience has a profound impact at all scales, from DSOs to states and nations. Ideally, the inclusion of additional weather data would enable a more comprehensive analysis of power distribution under the effect of extreme meteorological events and, more broadly, climate change28–30. It is challenging for multiple organizations to participate in a large-scale study with detailed organizational data owing to a variety of issues, ranging from proprietary information and confidentiality among DSOs to governmental regulations. Using data from a portion of the geographic area affected by Super Storm Sandy, this work demonstrates the feasibility and potential for multiple DSOs and policy makers to collaborate with a common goal of resilience and to make meaningful advances.

As the power industry traditionally learns from severe weather through experience24,31, this work demonstrates how the knowledge of resilience can be learnt from large-scale data. We have created a theoretical framework that provides a system perspective, starting with detailed information on infrastructural vulnerabilities, service, and cost to customers by employing spatiotemporal non-stationary random processes. Such a model connects interdependent networks from the infrastructure and DSOs to customers. Moreover, we

demonstrate the potential and feasibility of large-scale data analysis guided by modelling at the final stage of the energy grid. Advancing data collection and analysis will benefit DSOs, customers and policy makers alike, moving all of us towards a resilient energy infrastructure. We hope that our study stimulates more organizations to actively participate in resilience studies.

Methods Data. Real data used in this work are both sufficiently detailed and at a large scale. Data are rarely available that provide detailed information at the power distribution level (for example, the occurrence and restoration time as well as the geolocation on each failure and recovery). Data in this work are provided by four DSOs in Upstate New York. DSO 1 and DSO 4 are medium-size distribution system operators (that is, each serving <500,000 customers), whereas DSO 2 and DSO 3 are large system operators (that is, each serving >500,000 customers)32. The four DSOs have disjointed service territories, spanning more than 50,950 square miles. The data set consists of 6,266 system disruptions that occurred during the four-day period (10/28/2012 ∼ 10/31/2012) when Super Storm Sandy affected Upstate New York (Table 1). As many as 646,768 customers lost electricity service. The data sets also consist of 31,721 disruptions during daily operations in the same year.

The data contain detailed information regarding failures and recoveries at the final stage of the electric grid (Table 1). Each data sample consists of the following common information across the four service territories: time when the first customer reported loss of power, time when power was restored, the number of customers affected, the type(s) of activated protective device(s), geolocation and system location of activated protective device. Activated protective devices include blown fuses at feeders and transformers, open substation breakers and reclosers. The accuracy of each sample is a minute in time, latitude and longitude in geolocations (Supplementary Note 6). System locations of the activated protective devices are provided in terms of the hierarchy (that is, primary distribution, secondary distribution and customer properties). From the data sets, the cost can be obtained as customer interruption time but not the actual economic loss used in ref. 16. Simulated data (without geolocations) can be obtained from figshare.com (https://dx.doi.org/10.6084/m9.figshare.3119893.v1).

The data set from DSO 4 has the following exceptions on protective devices. The information on system locations is provided only for activated protective devices at primary distribution. Two-thirds of the samples do not have the recovery time specified uniquely on the types of disrupted devices. Hence, when the cost is examined with respect to different types of devices (Supplementary Table 6), data samples used are only those disruptions that are uniquely identified by their device types. This results in a reduced data set of 5,171 samples used (Supplementary Table 6).

Overall, it is challenging to obtain sufficiently accurate data during a severe weather event, where data accuracy can be compromised when the impact is severe. The data from Upstate New York are sufficiently accurate, and thus chosen for this study. Although the time of a disruption was taken from the customer calls, the occurrence time is expected to be reasonably accurate (that is, in minutes) at Upstate New York. This is because there were often multiple calls from customers reporting loss of electricity, especially during a severe weather event.

Super Storm Sandy. Super Storm Sandy was one of the most damaging natural disasters that occurred in 2012. During Super Storm Sandy, power distribution was disrupted for more than 29 DSOs in the eight most-impacted Northeastern states, affecting 8,511,251 customers at the peak of the storm1. As reported by the National Hurricane Center33, the storm started to impact the Northeastern coast

6

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NATURE ENERGY DOI: 10.1038/NENERGY.2016.52 ARTICLES of the United States on 28 October 2012, and the centre of the storm made landfall at about 23:30 UTC in New Jersey. After the landfall, the storm turned towards the west and then northeast, before gradually weakening. The centre of the storm became ill-defined and the hurricane was considered dissipated after 0:00 UTC 31 October33.

We identify when the service territories were under the impact of Super Storm Sandy based on the following information: the observed best-track of the storm34, the radius of the hurricane-force winds and the tropical-storm-force winds35. As the result, 28 October to 31 October 2012 is considered as the dominant period of the storm for the four DSOs in Upstate New York.

Joint disruption, recovery and cost random processes. Modelling here provides an analytical formulism for a larger number of interdependent variables at multiple spatiotemporal scales.

Variables and spatiotemporal scales. Infrastructural failures are the first type of variables, which correspond to either damaged power components or activated protective devices induced by severe weather. An example of component failure includes a down-wire, a damaged transformer or a non-functional distribution substation36. Protective devices, activated by failures or fault currents, include open switches and blown fuses that protect the remainder of the system10. Outages are further caused by failures within a distribution system (for example, with a radial topology and the corresponding protection), where devices downstream lose power but are not damaged (Supplementary Note 1). A (system) disruption refers to either a failure or an outage. Service is the second type of variable and is signified by recovery provided by DSOs. Cost to customers is another type of variable; it exhibits a general form in our model but is estimated here as the customer interruption time37 constrained by the available information from the data.

The temporal scale at which failures occur is in minutes or hours, consistent with that of evolving severe weather. Outages often occur in groups at a shorter timescale of sub-seconds7. Disruptions are usually restored at a longer timescale, that is, from minutes to days. Spatial scales in our model start at components and their connections to customers, and then aggregate from a small to large area, such as a township, service territory, region or nation.

Dependent spatiotemporal processes. Modelling starts at the finest level of individual failures in the infrastructure, incorporates recoveries as services, and finally evaluates costs to customers as impacts. We assume that a system disruption is already detected6,38. Because failures are randomly induced by individual local causes owing to severe weather, a system disruption I (d)i (t) can be represented by a random state transition of device i from normal operation to disruption. Such a state transition occurs during time duration (t −1t, t] and at a given location i for sufficiently small 1t > 0. Device index i includes its type, geolocation and system location.

Recovery characterizes resilience by how rapidly services to customers are restored after disruptions4. Recovery can be delayed by both random factors, such as conditions aftermath, and constraints (for example, availability of resources, recovery strategies and infrastructure)14. The cost to customers induced by delayed recovery is signified by the downtime duration Di(v) of disruption i occurred at time v, where the indicator function I[Di(v) > t − v] represents the recovery event, through which the system disruption is not yet recovered at t, 0 < v < t. Finally, the cost to customers evaluated at time t is represented by Gi(v, t) for disruption i, which occurs at time v for v < t.

Disruptions, recoveries and costs are dependent for a given severe weather event, evolving in time and locations. Considering the randomness in individual occurrences of disruptions, their durations and cost, the spatiotemporal non-stationary random processes model a collection of interdependent infrastructural failures, recoveries and costs as follows:

ˆ Disruption process: {I (d)i (v), i ∈ S(v), v > 0}, ˆ Recovery process: {I[Dk(v) >t−v],k∈S(t),0<v <t}, ˆ Joint disruption–recovery–cost process: {Gj(v, t), j ∈ S(v), 0 < v < t}.

Here, S(v) and S(t) consist of nodes in normal operations at time v and disruptions at time t, respectively. As disruptions are random and time-varying, so are S(v) and S(t).

Disruption rate and probability distribution. The model is particularly characterized by the first moments as rates and probability distributions. The disruption rate λ(d)i (t) is the average increment of cost 1C

(d) i (t) from new

disruptions occurring around location i,

λ (d) i (t)= lim

1t,1Ai→0

E [ 1C(d)i (t)|S(t)

] 1t1Ai

(1)

where 1C(d)i (t) = ∑

kC(d)k (t), aggregating over cost C (d) k (t) from newly occurring

disruptions in (t −1t, t] and in the vicinity of location i represented by a small area 1Ai. E[] is the conditional expectation over randomly occurring disruptions given i ∈ S(v). λ(d)i (t) shows the impacts of failures induced by extreme weather and outages from the infrastructure of power distribution.

As a special case, the customer disruption rate is when the cost is measured with respect to the number of affected customers. The customer disruption rate further reduces to the system disruption rate, that is, the average number of new disruptions per unit time at t and in an area, when all disruptions incur a unit cost. Therefore, the disruption rate λ(d)i (t) relates failures, power distribution infrastructure, and corresponding impacts on customers.

Cost. Let C(t) be the combined cost from failures and recoveries with respect to impacts on customers at time t for a service area. The expected total cost is

E[C(t)]= ∫ t

0 ES(v){λ

(d) i (v|S(v))E[Gi(v,t)|S(v)]}dv (2)

where ES(v)[·] is the expectation over a random trajectory of disruptions occurring at location i and time v for i ∈ S(v). λ(d)i (v|S(v)) is the system disruption rate. Conditional expectation E[Gi(v, t)|S(v)] is related to random disruption durations that occur at time v but have not yet recovered at t > v. Intuitively, λ(d)i (v|S(v))dv is the number of failures or outages that occur in (v −dv, v]. Product λ(d)i (v|S(v))E[Gi(v, t)|S(v)]dv is the cost of interrupted customers induced by the system disruptions that occur in (v −dv, v] and do not recover at t. The integral adds up all such costs induced by disruptions occurring in [0, t]. The system disruption rate λ(d)i (v|S(v)) and expected cost E[Gi(v, t)|S(v)] vary over time, illustrating the non-stationarity of the random processes.

A special case of the cost is the customer minutes of interruption (CMI), a standard metric and a simple economic cost that can be measured using the available data37. Let ci be the number of affected customers. The cost is proportional to ci and the downtime duration of disruption i, where Gi(v,t)=ci ·min{t−v,Di(v)} for 0 < v < t (Supplementary Note 1). Aggregating such Gi(v, t) over all disruptions, the total cost evaluated at time t is the entire customer downtime experienced thus far in a given area.

The cost in equation (2) is motivated by and extends from non-stationary queueing networks and the previous works7,14,39. These previous works assume first-come-first-serve policy for restoration which does not apply to recovery strategies with priorities used by DSOs. Moreover, the formulation here extends the previous works to include costs on customers. As customers are supported by power distribution infrastructure, the formulation must be derived from bottom up—that is, starting at individual components, each of which supports various customers.

Received 9 September 2015; accepted 27 March 2016; published 29 April 2016

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Acknowledgements The authors thank J. Love, D. Mitra, M. Rodriguez, T. Spatz and M. Worden for their help and insightful discussions. In addition, the authors thank D. Mitra as an academic adviser for the project, M. Rodriguez for sharing observations relating to the non-local impact to customers from other hurricanes, and A. Afsharinejad for critiquing the manuscript. Support from the New York State Energy Research and Development Authority (NYSERDA) to Georgia Tech is gratefully acknowledged. The opinions in this paper are of the authors and do not represent those of the New York State Department of Public Service.

Author contributions C.J. formulated the problem; Y.W. and H.M. conducted the experiments and wrote codes with contributions from M.C.; J.C., S.C., T.H., B.N., J.W. and R.W. gathered granular data; G.S. and M.W. provided their related Data Envelopment Analysis (DEA); C.J. derived the analytical model with Y.W.; C.J., Y.W. and H.M. analysed the results with insightful suggestions from all the other authors; C.J., Y.W. and H.M. wrote the paper with contributions from M.W.

Additional information Supplementary information is available online. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to C.J.

Competing interests C.J. and Y.W. are co-authors of a pending patent application; C.J., Y.W. and H.M. are co-authors of a pending provisional patent application.

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