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A spatio-temporal analysis for power grid resilience to extreme weather

by   Shixiang Zhu, et al.

In recent years, extreme weather events frequently cause large-scale power outages, affecting millions of customers for extended duration. Resilience, the capability of withstanding, adapting to, and recovering from a large-scale disruption, has becomes a top priority for power sector, in addition to economics and sustainability. However, a good understanding on the power grid resilience is still lacking, as most approaches still either stay on the conceptual level, yielding no actionable results, or focus on a particular technical issue, revealing little insights on the system level. In this study, we take a quantitative approach to understanding power system resilience by directly exploring real power outage data. We first give a qualitative analysis on power system resilience and large-scale power outage process, identifying key elements and developing conceptual models to describe grid resilience. Then we propose a spatio-temporal random process model, with parameters representing the identified resilience capabilities and interdependence between service areas. We perform analyse using our model on a set of large-scale customer-level quarter-hourly historical power outage data and corresponding weather records from three major service territories on the east-coast of the United States under normal daily operations and three major extreme weather events. It has shown that weather only directly cause a small portion of power outages, and the peak of power outages usually lag the weather events. Planning vulnerability and excessively accumulation of weather effects play a key role in causing sustained local outages to the power system in a short time. The local outages caused by weather events will later propagate to broader regions through the power grid, which subsequently lead to a much larger number of non-local power outages.


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Resilience modeling

Figure 2: Weather intensity and power outage evolution during the first Nor’easter on March 2-3, 2018, in Massachusetts. Snapshots of outage and weather maps in every six hours during the first Nor’easter in March 2018, Massachusetts. Each bubble corresponds to a geographical unit. The size and colour depth of the bubbles represent number of reported customer power outages per 15 minutes. The depth of red cloud over the map represents the level of derived radar reflectivity (REFD) in the corresponding region, which reflects the intensity of the storm. The total number of customer power outages and the maximum REFD in the region of the map for each snapshot are shown in the lower left of the figure. The largest total number of outages and the maximum REFD over this period are highlighted in red.

Throughout the downtime, power grid resilience plays a key role in withstanding impacts from the extreme weather, slowing the propagation of blackouts, and bringing the blackout areas back to normal. The resilience can be specifically described from two aspects: infrastructural resistance and operational recoverability, which correspond to the responses of the system facing external impacts and restoration tasks, respectively [19], as shown in Fig. 1 (a). Infrastructural resistance refers to the inherent ability of power grid to anticipate, resist, and absorb the effects of hazardous events. Operational recoverability indicates the system’s recovery capability from severe existing damages induced by extreme weather through a series of operational measures adopted by system operators, utility companies and other support entities. It should be noted that outage occurrence and system restoration may occur concurrently, and the two opposite dynamics constitute the overall power outage evolution process.

We first demonstrate the existence of power grid resilience by presenting an illustrative example (Fig. 2). It shows a typical evolution process of customer power outage during the period of the first Nor’easter (winter storm) that mainly affected eastern Massachusetts in early March 2018. The storm began impacting Massachusetts in early morning of March 2nd and peaked at around noon. The peak wind gust reached hurricane level (about 97 miles per hour) and the storm brought strong precipitation. The derived radar reflectivity (REFD) is a metric directly reflecting the intensity of stormy precipitation and we use it as a measure of storm intensity. According to the HRRR data, the storm in Massachusetts reached peak (maximum REFD is 27.9 dB) at 12PM on March 2, but there were only a small number (6,217) of customer power outages reported in part of the Central and Eastern region of Massachusetts. However, after the storm passed it peak but wind and precipitation continued, a larger number and scale of customer power outages started to occur in most areas of Eastern Massachusetts. The power outages reached peak value of around 421,000 at 6AM on March 3, which is 18 hours after the storm peak. And by that time, the storm has greatly weakened and had been moving out from Massachusetts. According to a number of recent outage reports [9, 11, 14, 20], under the impact of extreme weather, these customer outages were majorly caused by damages to distribution systems. Much of the transmission and distribution networks, across the United States, and particularly in the Midwest and Eastern regions, are still above ground, leaving them vulnerable to the effects of extreme weather [20]. Continuous rainfalls and constant strong winds exert pressure on trees and power poles over time. Such accumulation of weather effects will eventually lead to a great number of structure failures; electric line poles may fall and downed trees could bring down power lines, causing sustained damages to the system with large-scale power outages.

Figure 3: Spatially aggregated power outage and weather effect. (a) Spatially aggregated total number of customer outages per minute and corresponding two leading weather factors with respect to their maximum normal levels. The left vertical coordinate corresponds to the total number of customer outages per minute reported in the entire service territory, and the right vertical coordinate corresponds to the relative percentage of the weather data with respect to its Three-Sigma limits in the daily operation. (b) Total number of customer outage and corresponding decoupling according to our model estimation. Blue and red regions represent estimated number of outages induced by cascading outages and number of outages directly induced by extreme weather, respectively. Black dash lines represent the true total number of customer outages.

In comparing customer power outage records and corresponding weather data, we find a common behaviour from the three weather events (Fig. 3 (a) and Supplementary Fig. 10). The occurrence of large-scale severe customer power outages are normally 12-36 hours behind the peak of the extreme weather. The time lag between extreme weather and large-scale power outages demonstrates the power grid resilience to weather impacts mainly in two aspects: First, power grid has a certain risk-carrying capacity within which the power grid can withstand external impacts and does not cause power outages. Such principles are quite common in power grids. Transmission system is operated under N-1 requirements [29], which means the system is guaranteed secure after losing any one component. The power grid facilities are also designed with some safety margins to endure the external hazards up to a certain extent. The accumulation of weather impacts, including rainfall, wind, or snow, normally takes hours or even days to exceed the risk-carrying capacity of a power system, after which large-scale outages begin to occur. Second, utility companies and system operators are obliged to plan timely restoration for critical power facilities, and impose preemptive measures in response to potential damages. These operational restoration measures conducted throughout the downtime can slow the process of large-scale power outage.

To quantitatively assess infrastructural resistance and operational recoverability of a power system, we use volume data to analyse the spatio-temporal dependence of customer power outages under the impact of extreme weather. The change in the number of power outages is a result of the mutual interaction between disruption and restoration processes, and separately modeling these two dynamics is actually not practical. Thus, rather than modeling the physical processes, we directly build a spatio-temporal model from customer power outages and weather data, where the number of outages in each geographical unit (town, county or zip code) is regarded as a non-homogeneous Poisson process, and these processes interact with each other in an underlying topological space determined by distance and possible grid connectivity. The distribution of customer power outages in a unit at any given time is specified by an occurrence rate of customer power outage (we simply refer to it by outage occurrence rate in the rest of the paper). To simulate the dynamics of the power system under hazardous weather conditions, we assume that (1) the outage occurrence rate of a unit affected by extreme weather will grow as the accumulation of weather effects builds up until it reaches the system’s minimal functionality; (2) a unit with larger number of customer power outages have higher chance to raise the outage occurrence rate in its neighboring units (in the topological space) due to the potential damage of large transmission network among these units; (3) the outage occurrence rate of a unit will also decay exponentially over time as restoration plan being conducted throughout the region. Detailed modelling can be found in the Methods section. Here, infrastructural resistance can be described by the relationship between the accumulation of weather effects and the power system functionality (number of remaining customer with power supply), and operational recoverabilty can be regarded as the speed of unit recovering from power outages, described by the decay rate of outage occurrence rate in (3). Fig. 3 (b) shows that the prediction results (blue solid lines) using our model for the three service territories under different weather events, which have achieved promising results comparing to the ground truth indicated by black dash lines. As we assume the increase of outage occurrence rate can be possibly caused by either the accumulated weather effects, or the cascading failures in connected units, our results confirm that extreme weather triggered the occurrence of a large-scale power outages (red regions in Fig. 3 (b)), while a significant portion of these outages (blue regions in Fig. 3 (b)) are resulted by the cascading failures in a few affected units. More results on unit-level prediction can be found in the Supplementary Fig. 12.

Figure 4: System response Accumulation of weather effects versus outage ratio during degradation stage. Scatter plots of outage ratio during degradation stage given the accumulation of three different types of weather effects. The vertical and left horizontal coordinates are the outage ratio and unit ID, respectively. Each data point corresponds to the outage ratio (total number of customer outages/total number of customers) of a certain unit at a certain time. The color depth of data points also indicate the outage ratio. The unit IDs have been sorted by their historical maximal outage ratios in descending order. The right horizontal coordinate is the corresponding accumulation of weather effect in the past three days. The blue line on the left vertical plane is a sigmoid curve fitted by all the data points with outage ratio larger than 0. The green dotted line on the right vertical plane is the maximum outage ratios of all units.

Infrastructural resistance and recoverability

First, we focus on expounding the concept of infrastructural resistance. One of the key challenge is to define the accumulation of weather effects and quantify induced damages to the power system. As indicated by [20], power systems show vulnerability to the impact of various weather effects, including high winds, heavy rain, ice, and snow. In this paper, we focus on 34 weather variables defined in HRRR data set (Supplementary Table 1) that characterize the near-surface atmospheric activities, which are directly linked to the weather impacts on power grids. Under extreme weather conditions, sustained customer power outages can build up rapidly as damages to tree limbs or critical power facilities are excessively accumulated. However, in daily operations, regional and temporal weather variations rarely cause long-lasting and severe damages to the power system (Supplementary Fig. 11). Thus, we define the accumulation of each weather variable at time as a convolution of the time-series signal of this variable before and a exponential kernel function defined by , where is the decay rate and is the distance to . In other words, the impact of a weather feature is defined as the integral of the feature with decays over time. In our model, the decay rates of different weather features may be different. We further define the gross weather impacts as a collection of such convolutions for selected weather variables. To model the intricate dynamics of the growth of outages induced by extreme weather, we adopt a deep neural network, where the input of the network is the accumulation of weather effects in every unit, and the output is the corresponding increase in outage occurrence rate that are directly induced by weather damages.

Now we investigate the infrastructural resistance by analytically evaluating the relation with the outage ratio for the accumulated weather effects in each unit. Here we select three representative weather variables to demonstrate such kind of relations, including composite radar reflectivity (REFC), derived radar reflectivity (REFD), and wind speed (WIND). These features describe the intensity of precipitation and wind, which are among the most relevant factors of weather-induced outages. We calculate the accumulations of these features for last three days at every time point during three extreme weather incidents. Fig. 4

shows that the outage ratio stays close to zero until the accumulation of the weather variable reaches certain “threshold” where a sharp jump occurs (indicated by blue lines). Such a trait can be well captured by a sigmoid function of accumulated weather effects. The “threshold” can be interpreted as the risk-carrying capacity for the corresponding weather variable, which is jointly controlled by the decay rate

in the exponential kernel function and the parameters in the neural network. The result reveals that power grids have risk-carrying capacity within which the system can well withstand the external impacts without substantial outages. Systems with larger capacity are more resilient to weather effect (more accumulated weather effect is needed to initiate large-scale power outages). The prevalence of this phenomenon in three major service territories confirms the existence of the infrastructural resistance. With sufficient data, we are able to quantify the infrastructural resistance and the corresponding risk-carrying capacity of a power system. For example, in Fig. 4, we can observe that the risk-carrying capacity for wind speed in Massachusetts, North Carolina and South Carolina (around 10 m/s) is much smaller than that of Georgia (around 30 m/s). This indicates that the power systems in Massachusetts, North Carolina and South Carolina are more resilient to strong wind comparing to the system in Georgia. We can also find that there is a considerable difference in distribution of maximum outage ratios (indicated by green shaded area) among three service territories, and the maximum outage ratio is negatively related to its risk-carrying capacity. For example, the extreme weather is devastating to nearly half of the units (296) in Georgia as the maximum outage ratio for these units reaches close to 1 (), meanwhile the other half of the units are barely affected by the natural hazardous events (the maximum outage ratio stays around 0). As opposed to the polarization phenomenon in Georgia, the outage ratio for majority of the units in North Carolina and South Carolina remains remarkably low throughout the weather events. We note that the analyzed outages in North Carolina and South Carolina are most likely due to the damaged distribution network infrastructures, which are spread out and have lower mutual impacts. In other words, the transmission network infrastructures in these territories perform better in terms of withstanding extreme weather events compared with those in Georgia. This could be partly due to that the transmission line structures in North Carolina and South Carolina are required to withstand a gust wind with a speed of 45 m/s or greater as being installed increasingly closer to the coast per National Electric Safety Code.

Figure 5: Planning vulnerability and operational recoverability. Bubble sizes in each row from top to bottom represent planning vulnerability (), recovery rate (), and outage ratio of units during studied events, respectively. Value of planning vulnerability represents the average number of outages occurred in the given unit induced by a unit of accumulated weather effects. The number of outages in each unit decay exponentially, where the exponential rate is specified by the recovery rate. Colour depth of background indicates average weather intensity (REFD) during corresponding events.
Figure 6: Decay rate in accumulating weather effects. Bars represent the learned decay rate for each weather variables in the corresponding service territories, where colours indicate their categories. The weather variables have been sorted by their values.
Figure 7: Power outages propagation map. The map shows the spatial propagation of power outages among geographical units during extreme weather. An edge between two units indicates the power outages occurred in one unit (light red) are resulted by the other (dark red). Edge width and colour depth represent the number of customer power outages at the target attributed to the source. The size of dots represents the total number of customers of the corresponding geographical unit.

In this study, we categorize the factors that decide power systems’ infrastructural resistance into three general aspects: planning vulnerability, maintenance sufficiency, and criticality (Fig. 1 (b)). Our results demonstrate that resistance ability of a power grid is jointly determined by these factors. First, planning vulnerability determines the long-term risks of exposure to extreme weather hazards. In particular, power infrastructural planning, including site selection of power facilities, transmission or distribution route planning, model selection of power devices, needs to consider the likelihood of natural threats (e.g. high winds, flood, landslide, icing and vegetation growth). For example, power lines going through zones with high density of tall trees could put the system at a higher risk of outages caused by fallen tree limbs. In our model, we introduce a set of trainable coefficients in our model for each unit representing their overall planning vulnerabilities to extreme weather, where the coefficient value represents the number of outages occurred in this unit induced by a unit of accumulated weather effects. A larger coefficient indicates the unit is more vulnerable to extreme weather and more likely to suffer a massive blackout (Fig. 5). Second, power industry regularly makes inspection and maintenance for electrical components and facilities to sustain the health of infrastructure. Power grid maintenance refers not just to the maintenance of assets and equipment but also to the environment in the vicinity, e.g. vegetation inspection and trimming. Insufficient maintenance may leave the system vulnerable to the impact of extreme weather and exacerbate existing planning vulnerabilities [18]. This factor can be captured by the decay rates defined in the exponential kernel function; the smaller decay rate a unit has, the less maintenance has been completed, and the faster weather effects will be accumulated. Our results suggest that different service territories are vulnerable to different variables: the accumulation of precipitation is rapid in Massachusetts; the accumulation of wind is rapid in Georgia; North Carolina and South Carolina can absorb both quickly, however sensitive to the temperature-related variables. (Fig. 6) Last but not least, some of the components or facilities hold critical status in the resilience of a power system. Modern power grids are interdependent networked systems [23, 24, 31, 34, 8], and failure of some critical portions may not only affect the customers’ power supply within their own service territories, but also degrade the entire system’s resistance ability and lead to failure of other dependent nodes in the same networks. This is a common mechanism of cascading failures and blackouts. Thus, we consider each unit in the power network as a node, and model such connectivity between nodes using a directed graph [38], where directed edges between nodes represent the direction of power outages’ propagation and the weight of each edge indicates the average number of outages in the target unit resulted by occurrence of every outage in the source unit. We further assume the increase in number of power outages at a unit can lead to a proportional increase in the number of power outages at the unit it connects to. Thus, the criticality of a unit can be precisely evaluated by the total number of customer power outages in other directly connected units resulted by its failure. Our results show disruptions in only a few number of critical units can lead to a large-scale subsequent customer power outages in three service territories we studied (see the units with major outgoing links in Fig. 7). This implies that these units are critical in the evolution of power outages: a major portion of the power outages can be directly or indirectly attributed to these units. Such characteristic is important because it shows some bottlenecks of the resilience of the grid, and people need to concentrate resources on enhancing these critical units.

We next discuss the effect of operational recoverability among geographical units. In daily operations, 92% of customer power outages recovered within 6 hours, and all customers were back to normal within one day (Supplementary Fig. 11). However, recoveries from damages induced by extreme weather lasted 4, 4, and 2 days for the three service territories and units in different region may also carry out their restoration activities differently. The differences in restoration process may be incurred by various factors, including the infrastructure damage severity, restoration plan efficacy, resources (human, funding and supplies), and the coordination of restoration progress [27]. For example, six months after Hurricane Maria hit Puerto Rico, more than 10 percent of the island’s residents were still without power [3]; Similarly, it took the U.S. Virgin Islands more than four months to fully restore power after being hit by Hurricanes Irma and Maria. Meanwhile, the recovery in the mainland of US has been much quicker — in part because the resources are more abundant. In particular, after Irma hit the Southeast US, knocking out power to 7.8 million customers, 60,000 utility workers from across the country deployed and restored power to 97 percent of the population in about a week [26]. Therefore, we assume the recovery process in each unit reduces the number of outages in an exponential rate, which is determined by another set of parameters ; the parameters may be different in different units. Here we quantify recovery of each unit through the learned recovery rate from real outage data (Second row in Fig. 5). We find that areas with more population and various of easily accessible resources (e.g. Atlanta in GA and Charlotte in NC) have higher recovery rates. Instead, some remote areas that are away from the big cities and with significantly high outage ratios, such as Nantucket, Martha’s Vineyard in MA, Bainbridge in GA, and Wilmington in NC, have remarkably lower recovery rates.

Resilience enhancement analysis

A large body of previous efforts [1, 19, 22, 25, 27, 28, 36] have assessed a variety of techniques that can be employed before an event occurs in an effort to enhance system resilience. These include but not limited to (1) improving system architectures to further reduce the criticality of individual units in the power network, (2) enhancing the health and reliability of the individual units, (3) making restoration plan ahead of events to speed up help to those who may need it. (4) asset health monitoring and preventive- and reliability-centered maintenance. However, examining and evaluating the effectiveness of resilience enhancement strategies could be technically intractable. There are number of reasons: first, due to the low-frequency nature of extreme weather events, there is a lack of weather data and corresponding power outage records for such evaluation; second, the uniqueness of each extreme weather event and its ripple effects hinders the reproducibility of large-scale outages in resilience study; last but not least, accurate evaluation of these strategies requires a partnership across all levels of government and the private sectors, which is time- and cost-consuming to be carried out in practical terms.

We now aim to identify and evaluate strategies in four general aspects (planning vulnerability, recoverability, criticality, and maintenance) that, if adopted, could significantly increase the power grid resilience. Even though these strategies could not be reliably and accurately assessed in the absence of detailed failure and restoration records, our models still offers an opportunity to simulate the entire process of large-scale customer power outages under different circumstances, and explore potential opportunities to help alleviate the effects of extreme weather. To be specific, we “improve” certain aspects of the power grid resilience by adjusting the corresponding parameters of the fitted model, then simulate the outage process given the same weather condition, and finally calculate the outage reduction rate comparing to the original number of outages (Fig. 8). We observe that de-escalating the criticality of a small number of units with the largest amount of customer power outages can effectively mitigate the ultimate impact of extreme weather. For example, in Massachusetts, reweighting the largest 50 edges of top 10 units by their number of customer power outages, such as Boston and Cambridge, can reduce 47.8% customer power outages; similarly in Georgia, the same strategy can reduce 38.4% customer power outages. However, the same strategy may not achieve competitive performance in North Carolina and South Carolina; instead, by improving the planning vulnerability for the unit with the largest number of customer power outages, i.e., Wilmington, the total number of outages can be drastically reduced by 32%; it can be further improved to 49.8% if top 12 units are considered with the same strategy. We also notice that improving the rate of accumulating weather effects () is not rewarding comparing to other strategies. However, the system will degrade remarkably if the rate is lower than the current level. The phenomenon can be exaplained bacause the maintenance have been managed at a fairly good level by the local operators.

Figure 8: Resilience enhancement simulation results. (a) is the outage reduction by re-weighting the edges with the largest weights in the network to the average level. The left horizontal coordinate represents the number of top units by their maximum number of outages that we aim to improve during the extreme weather incidents; the right horizontal coordinate represents the number of edges per unit we need re-weight; The vertical coordinate represents the simulated corresponding percentage of outage reduction with respect to the real number. (b) is the outage reduction by re-weighting the units with the largest vulnerability and smallest recoverability coefficients, respectively, to their average levels. The horizontal coordinate represents the number of top units with the largest vulnerability and smallest recoverability coefficients, respectively, that we aim to improve during the extreme weather incidents; The vertical coordinate represents the corresponding percentage of outage reduction with respect to the real number.


From the quantitative analysis of power grid resilience and simulated resilience enhancement, some planning and operational measures to prevent and mitigate weather-induced power outages can be inspired. The first observation is that power grid shows some vulnerability of major outage propagation, which significantly exacerbates the initial impact of weather events. If such outage evolution and propagation can be prevented or mitigated, the weather impact would be much smaller and self-contained. A key to stem the outage propagation is to enable the interdependency reduction when facing outage risks. Interdependency of power grids are highly related to their interconnected nature and it is neither economical nor practical to simply break the power grids into local pieces. A desired way to reduce the risks of major outages exacerbated by interdependency is to improve the operational flexibility of the power grid. A flexible power grid should embrace diversified sources with distributed locations and versatile operation schemes. For example, distributed energy resources (DER) can provide energy to local loads and thus reduce the risk of outages caused by interruption of distant electricity transmission. Energy storage can also quickly supply energy to local or nearby loads. Moreover, microgrids characterized as local power grid with diverse energy sources and flexible operation modes can easily connect to or isolate from the main power grid, which is resilient to external threats and sustainable.



Real data used in this work consist of two data sets: customer power outage data and HRRR weather data, which are both fine resolution over space and time, and at a large scale. The data cover three major service territories across four states in the East Coast of the United States, including Massachusetts, Georgia, North Carolina, and South Carolina, spanning more than 155,000 square miles, with an estimated population of more than 33 million people in 2019. These service territories are divided by 351, 665, 115 disjointed geographical units, respectively, defined by township, zipcode, or county.

Customer power outage data in this work are collected by web crawlers from Massachusetts government [2], Georgia Power [32], and Duke Energy [2], respectively, since December 31th, 2017 until August 31th, 2020. The data record the number of customers without power supplies in each geographical unit in every 15 minutes. There are 351 units and 2,755,111 customers in Massachusetts, 665 units and 2,571,603 customers in Georgia, and 115 units and 4,305,995 customers in North Carolina and South Carolina. The HRRR model [6] is a National Oceanic and Atmospheric Administration (NOAA) real-time 3-km resolution, hourly updated, cloud-resolving, convection-allowing atmospheric model, initialized by 3-km grids with 3-km radar assimilation. Radar data is assimilated in the HRRR model every 15 minutes over a 1-hour period adding further detail to that provided by the hourly data assimilation from the 13-km radar-enhanced rapid refresh. In this work, we select 34 variables in the HRRR model (Supplementary Table 1) that characterize the near-surface atmospheric activities, which are directly linked to the weather impacts on power grids. To match the spatial resolution with the customer power outage data, we aggregate the regional weather data in the same unit by taking the average. To reduce noise in both the outage data and the weather data, we also aggregate the number of customer power outages and the value of weather variables in each unit by three hours (length of time slots).

We studied three typical extreme weather events between 2018 and 2020, during which customers in these three service territories have been significantly affected, and number of outages as well as weather information in each geographical units have been recorded in our data set. These three events are described as follows. March 2018 nor’easters: The March 2018 nor’easters include three powerful winter storms occurred in March 2018 that caused major impacts in the Northeastern, Mid-Atlantic and Southeastern United States. During its peak time, more than 14% customers (385,744) in Massachusetts remained without power supply and the outage ratios for nearly 30% of the units in the region (104) are higher than 50%. Hurricane Michael: Hurricane Michael was a very powerful and destructive tropical cyclone occurred in October 2018 that became the first Category 5 hurricane to strike the contiguous United States since Andrew in 1992. About 7.5% customers (193,018) in Georgia have been severely affected by the storm and lost their power, and the outage rates for about 25% of the units in Georgia (166) are higher than 50%. Hurricane Isaias. Hurricane Isaias was a destructive Category 1 hurricane occurred in August 2020 that caused extensive damage across the Caribbean and the East Coast of the United States, including a large tornado outbreak. During its peak time, about 13% customers in North Carolina and South Carolina lost their power, and the outage ratios for nearly 9% units in these two states are higher than 50%. For the three service territories, we determine the studied periods based on the power outage records and the associated weather data, including derived radar reflectivity, composite radar reflectivity, and wind speed. As the result, March 1st, 2018 to Marcch 16th, 2018 is considered as the period affected by the March 2018 nor’easters for service territories in Massachusetts; October 5th, 2018 to October 20th, 2018 is considered as the period affected by the Hurricane Michael for service territories in Georgia; August 1st, 2020 to August 10th, 2020 is considered as the period affected by the Hurricane Isaias for service territories in North Carolina and South Carolina. To fairly compare the resilience’s differences under the same condition, for each event, we also consider the rest of the days without the impact of extreme weather in the same month as daily operations (Supplementary Table 2).

Spatio-temporal non-homogeneous Poisson process.

The occurrences of customer power outages in a service territory are modeled as a multivariate non-homogeneous Poisson process. Assume there are units in the service territory and consider time slots in the time horizon that spans the entire event. Let be the number of customer power outages in unit at time . We model the occurrence of customer power outages in each unit and at arbitrary time as a Poisson process, i.e., , where the outage occurrence rate in unit at time can be characterized by .

Assume that the occurrence of customer power outages can be possibly induced by either direct impact of accumulation of weather effects in the past or indirect impact from previous outages occurred in the connected units. First, let be the value of weather variable in unit at time . We assume the accumulation of weather variable in unit at time is defined as the weighted sum of weather variable for the past time slots, i.e., , where is the decay rate for weather variable . We denote the accumulation of all weather variables in unit at time

as a vector

, where is the total number of weather variables. Then we define the power network in the service territory as a directed graph , where is a set of vertices representing the units, and

is a set of directed edges (ordered pairs of vertices), which represents underlying connections between units. Following the spatio-temporal model suggested by

[16], we define the outage occurrence rate as:

where is the planning vulnerability of unit . Function returns the weather-induced outage occurrence rate, approximated by a deep neural network parametrized by given the accumulation of weather effects in the past as input. The architecture of the neural network is described in Supplementary Fig 9. Function is the triggering effects on unit at time from previous outages occurred in unit at time . There are many possible forms of functions . Here we adopt one of the most commonly used form, which assumes the triggering effects decay exponentially over time [33]:

where captures the decay rate of the influence; note that the kernel function integrates to one over . We assume each edge is associated with a non-negative weight indicating the correlation between unit and . The larger the weight is, the more likely the unit will be influenced by unit . We also assume ; thus, can be regarded as the recovery rate of unit . In particular, the term captures the dynamics of the restoration process of unit and the term captures the influence of unit to unit . We also disallow the presence of loops between two arbitrary units, i.e., if .

The model can be estimated by maximizing the log-likelihood of the parameters, which can be solved by stochastic gradient descent

[21]. Denote the set of all the parameters , , , , and as . Denote the corresponding parameter space as , where is the parameter space of the neural network defined in . Denote all the observed customer power outages and weather data in the studied service territory and time horizon as and . Our objective is to find the optimal by solving the following optimization problem:

subject to

where denotes the log-likelihood function.


Supplementary Material

Figure 9: Architecture of the neural network.
No. Variable Description
1 REFC Maximum / Composite radar reflectivity [dB]
2 RETOP Echo Top [m], 0[-] CTL=”Level of cloud tops”
3 VERIL Vertically-integrated liquid [kg/m], 0[-] RESERVED(10) (Reserved)
4 VIS Visibility [m], 0[-] SFC=”Ground or water surface”
5 REFD-1000m Derived radar reflectivity [dB], 1000[m] HTGL=”Specified height level above ground”
6 REFD-4000m Derived radar reflectivity [dB], 4000[m] HTGL=”Specified height level above ground”
7 REFD-263k Derived radar reflectivity [dB], 263[K] TMPL=”Isothermal level”
8 GUST Wind speed (gust) [m/s], 0[-] SFC=”Ground or water surface”
9 TMP-92500pa Temperature [C], 92500[Pa] ISBL=”Isobaric surface”
10 DPT-92500pa Dew point temperature [C], 92500[Pa] ISBL=”Isobaric surface”
11 UGRD-92500pa u-component of wind [m/s], 92500[Pa] ISBL=”Isobaric surface”
12 VGRD-92500pa v-component of wind [m/s], 92500[Pa] ISBL=”Isobaric surface”
13 TMP-100000pa Temperature [C], 100000[Pa] ISBL=”Isobaric surface”
14 DPT-100000pa Dew point temperature [C], 100000[Pa] ISBL=”Isobaric surface”
15 UGRD-100000pa u-component of wind [m/s], 100000[Pa] ISBL=”Isobaric surface”
16 VGRD-100000pa v-component of wind [m/s], 100000[Pa] ISBL=”Isobaric surface”
17 DZDT Verical velocity (geometric) [m/s], 0.5-0.8[’sigma’ value] SIGL=”Sigma level”
18 MSLMA MSLP (MAPS System Reduction) [Pa], 0[-] MSL=”Mean sea level”
19 HGT Geopotential height [gpm], 100000[Pa] ISBL=”Isobaric surface”
20 UGRD-80m u-component of wind [m/s], 80[m] HTGL=”Specified height level above ground”
21 VGRD-80m v-component of wind [m/s], 80[m] HTGL=”Specified height level above ground”
22 PRES Pressure [Pa], 0[-] SFC=”Ground or water surface”
23 TMP-0m Temperature [C], 0[-] SFC=”Ground or water surface”
24 MSTAV Moisture Availability [%], 0[m] DBLL=”Depth below land surface”
25 SNOWC Snow cover [%], 0[-] SFC=”Ground or water surface”
26 SNOD Snow depth [m], 0[-] SFC=”Ground or water surface”
27 TMP-2m Temperature [C], 2[m] HTGL=”Specified height level above ground”
28 SPFH Specific humidity [kg/kg], 2[m] HTGL=”Specified height level above ground”
29 UGRD-10m u-component of wind [m/s], 10[m] HTGL=”Specified height level above ground”
30 VGRD-10m v-component of wind [m/s], 10[m] HTGL=”Specified height level above ground”
31 WIND Wind speed [m/s], 10[m] HTGL=”Specified height level above ground”
32 LFTX Surface lifted index [C], 50000-100000[Pa] ISBL=”Isobaric surface”
33 USTM U-component storm motion [m/s], 0-6000[m] HTGL=”Specified height level above ground”
34 VSTM V-component storm motion [m/s], 0-6000[m] HTGL=”Specified height level above ground”
Table 1: Weather effects
Basic information Extreme weather event Daily operation
Units Unit type Customers Time period Average outages Max outages Time period Average outages Max outages
MA 351 township 2,755,111 2018-03-01 2018-03-15 2393 19964 2018-03-16 2018-03-31 49 1096
GA 665 zipcode 2,571,603 2018-10-05 2018-10-20 359 7182 2018-10-21 2018-11-05 38 1659
NC & SC 115 county 4,305,995 2020-07-31 2020-08-10 2198 92424 2020-08-11 2020-08-31 474 6178
Table 2: Data sets on the customer power outages used in this study.
Figure 10:

Empirical customer power outage ratio distribution in three major service territories. Black dash lines indicate the moment when the number of customer power outages in the system reaches the peak during the incident. Red and green dotted lines indicate the moment when the weather intensities (REFD and WIND, respectively) reach their peak during the extreme weather incidents.

Figure 11: Scatter plots of duration of restoration state during five extreme weather events and daily operations. Each bubble corresponds to the duration of restoration state of a geographical unit. The horizontal and vertical coordinates correspond to the start and end time of the restoration, respectively. The distance of the bubbles above the diagonal line indicates the length of restoration state. The diagonal dash line represents the length of restoration state is zero. Bubble sizes represent the maximum number of customer outages occurred in the corresponding unit. The colours represent different service regions.
Figure 12: Unit-level predictions units in (a) Massachusetts, (b) Georgia, and (c) North Carolina and South Carolina. Black lines represent the real customer power outages in the certain area and green shaded regions represent the three-hour ahead prediction for customer power outages in the same area using our model.