We focused on two response variables: the number of outages which are equivalent to the number of customers without power in a city and the fraction of customers without power. GLM and GAM use Poisson and negative binomial distributions to assess the count of outages as they model discrete and non-negative variables. Random forests can model the fraction of households without power in a city, which is important to compare impact levels across cities.

The hurricane parameters considered for this study are 3 s gust wind speed and duration of strong winds over 20 m s-1, as in previous research . The overhead distribution system is the most vulnerable component of a power grid to high hurricane winds . The distribution lines and poles are often close to trees and do not have considerable setbacks. The uprooting of trees due to strong winds often propagates damage to distribution lines. The poles are designed to sustain wind speeds of around 20 m s-1 . Thus, winds above 20 m s-1 can cause substantial damage to the electric poles. The 3 s gust wind speed and duration of strong winds for the three hurricanes in the dataset were calculated based on a complete wind profile model for tropical cyclones by . We determined the wind speed for each city at its centroid. Figure  illustrates the variation of 3 s wind gusts in New Jersey during Hurricane Isaias. However, wind speed is an important factor causing outages, and any approximation in wind speed estimates could lead to errors in outage predictions. We provide detailed information in Sect. S1 in the Supplement to demonstrate that wind speeds at the city centroid can be a reasonable estimate in determining the city-wide outages.

Power grid patterns vary for different land use classes, resulting in different outage mechanisms. For example, rural areas can suffer larger power outages since they have radial grid patterns where component failures can propagate more than in cities with gridded patterns . We obtained National Land Cover Data (NLCD) available from the Multi-Resolution Land Characteristics Consortium, which is maintained by the United States Geographical Survey (USGS) (https://www.mrlc.gov/viewer/, last access: 21 September 2022). National Land Cover Data have inaccuracies in the thematic pixel classification that could introduce uncertainties in the land cover type. However, evaluating the effect of the inaccuracies in the thematic pixel classification on power outages is not within the scope of this paper. NLCD are available in raster format with a resolution of 30 m × 30 m. USGS has classified the original land cover data into 20 different classes. We have reclassified the NLCD into nine classes to match previous power outage models. The nine different major classes of land cover data are developed area, water area, barren land, forest area, scrub area, grasslands, pasture land, cultivated cropland, and wetlands. We utilized the spatial analyst in ArcGIS (a tool for geographic information systems) to clip the 30 m × 30 m land cover raster for each city. We used zonal analysis within ArcGIS to determine the percentage of area covered by the nine major land cover classes.

Precipitation and soil moisture have been extensively used in power outage models, e.g., , , and . These parameters have non-linear effects on power outages, as deviations below and above standard values can result in more outages. The poles and overhead distribution lines in the vicinity of trees are susceptible to falling trees due to strong hurricane winds. The wet soil conditions from high precipitation and soil moisture increase the likelihood of trees and electric poles uprooting from strong hurricane winds . Also, persistent drought conditions, e.g., low precipitation in the months before a hurricane, can weaken the roots of trees because of gaps in the soil layer, making trees more susceptible to strong winds .

Precipitation and soil moisture data are available from the variable infiltration capacity (VIC) model from the National Land Data Assimilation System Phase 2 (NLDAS2) . However, the limited temporal resolution of parameters required for computing soil moisture and precipitation could introduce errors in the final estimates of these variables . The limitations of the variable infiltration capacity (VIC) model from the National Land Data Assimilation System Phase 2 (NLDAS2) to get soil moisture and precipitation are beyond the scope of this study. Precipitation and soil moisture have been recorded each hour since 1979 with a resolution of 0.125∘ × 0.125∘. We used nearest-neighbor interpolation to obtain soil moisture and precipitation at the city centroid by taking the value available at the nearest point.

Soil moisture from NLDAS2 is available for three depths: 0–10, 10–40, and 40–100 cm. We calculated daily soil moisture for these depths by taking the average hourly readings. Soil moisture can vary at different geographical locations due to different soil types in different regions. We first normalized soil moisture to compute deviations from average values by computing percentiles. We fit Pearson type III distributions to the daily time series of soil moisture for all three layers to normalize the soil moisture across different geographies. We use maximum likelihood estimates (MLEs) to compute the parameters for Pearson type III distribution . Then, we evaluate the soil moisture percentile. We denote the soil moisture percentiles for three layers of soil at 0–10, 10–40, and 40–100 cm depth as CDF1, CDF2, and CDF3, respectively.

Precipitation data are represented in the form of the standard precipitation index (SPI) . SPI for the months before the storm's impact can also be used as a proxy for the dryness and wetness of the soil. For the current study, we calculated SPI for durations of 1, 3, 6, and 12 months by adding hourly time series data for precipitation. The following are three steps to compute SPI. First, we fit the Pearson type III distributions to the time series of precipitation using MLE. Second, we compute the percentile from the Pearson type III distribution. Third, we take the inverse of the calculated percentile using a standard normal distribution to get the SPI for each duration. Figure  illustrates the variation of SPI 1 month in New Jersey before the arrival of Hurricane Isaias.

We also included the expected precipitation after the hurricane makes landfall for the next 7 d, as heavy rain can lead to flooding resulting in clustered outages . The soil moisture percentiles and SPI values are obtained from the day before the hurricane impacts the power systems. This starting time allows for outage predictions to give an early warning to the utilities and community members and take precautionary steps before strong hurricane winds hit the area.

The effective root zone depth is defined as the depth of the soil from which plants and trees can effectively extract water and nutrients for growth (http://www.wood-database.com, last access: 21 September 2022). The more effective the root zone depth for trees, the less likely they will fail from strong hurricane winds . We add root zone depth as an input parameter for outage predictions because it could indicate the hazard from falling trees to the power lines. Root zone data are available from the United States Department of Agriculture (USDA) under Gridded Soil Survey Geographic at 30 m × 30 m resolution as raster data. The root zone depth at the city level is calculated as the average of the root zone in a city using the spatial analytic tool in ArcGIS. Given the resolution of available outage data at the city scale, we were not able to consider the variations in root zone depth, which limits the ability of the power outage model to consider the variation of tree root strength within a city.

USDA created National Insect and Disaster Risk Maps in 2012, with the area covered by trees at 240 m × 240 m as raster data. The raster tree data are used to calculate the percent of the area covered by trees at the city level using the spatial analytic tool in ArcGIS (). Figure  illustrates the distribution of percent treed area in New Jersey.

Previously, researchers have found that hurricane wind speeds (and thus damages) vary with surface topography . Additionally, varying elevations could also introduce variations in precipitations and wind speeds within a city. Thus, we use the median and mean elevation at the city centroid, using nearest-neighbor interpolation, as topographic variables to capture the changes in elevations across the cities. We obtained these parameters from the digital elevation model (DEM) at a 30 arcsec resolution scale developed by USGS as part of the DEM: Global Multi-resolution Terrain Elevation Data (GMTED2010) . Note that the resolution of available outage data at the city level limit our ability to account for the varying elevations within a city. Future studies with high-resolution outage data might account for the variations in elevation within a city.

Demographics data are available from the American Community Survey (ACS) (https://www.census.gov/programs-surveys/acs, last access: 21 September 2022). The ACS collects different demographic data for each US census tract. The ACS started data collection in 2010, and we have considered data from 2019. We obtained the population density as it indicates the number of distribution poles and system components exposed to winds .

Poisson regression models, a category of GLMs, are applicable for positive count data where observations are independent. Outages are modeled as a Poisson random variable: 1P(y;μ)=e-μμyy!, where y is the number of outages in a city. The Poisson distribution is described by the parameter μ, the mean number of outages in a city. A log link connects the parameter μ to the input variables, which assures that μ is greater than zero. 2ln(μ)=βX, where β is learned from the historical outages from extreme events, often through maximum likelihood estimation (MLE). MLE finds the value of β that maximizes the probability of observing the data. Readers can refer to for more information on MLE estimates for GLMs. We use glm package in R studio (https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/glm, last access: 21 September 2022) to fit the Poisson GLM to our power outage data.

The variance in a Poisson distribution is equal to the mean, i.e., Var(y)=μ. Thus, the variance grows as μ increases. However, previous research has found that outage variance from historical data is significantly bigger than the mean , a phenomenon that is known as overdispersion in Poisson regressions . Overdispersion may arise from the interdependence of output variables, especially when they happen in clusters . Poisson distributions represent counts of events, e.g., customers without power, that are independent . In contrast, multiple outages in the city can happen due to the failure of the same power grid components . Thus, outage counts are not independent.

Negative binomial GLM is a hierarchical model which can account for overdispersion effects in power outage count predictions . Negative binomial GLMs are based on a Poisson–gamma mixture distribution; i.e., the outage count y is distributed as a Poisson random variable 3P(y;μ,τ)=e-μτ(μτ)yy!, where μ is a factor that when multiplied by τ equals the mean of the Poisson distribution. τ is an additional random variable to account for extra variance, with a mean equal to 1 and distributed as gamma 4P(τ;k)=(1/k)1/kΓ(1/k)τ1/k-1e-τ/k, where k is the overdispersion parameter and Γ() is the gamma function; thus, the variance of τ equals k. After marginalizing the random variable τ, 5P(y;μ,k)=Γ(y+1/k)Γ(y+1)Γ(1/k)μμ+1/ky1-μμ+1/k1/k, which is equivalent to a negative binomial distribution with a variance of μ+kμ2. This variance is higher than the one in the Poisson GLM with one term that is proportional to μ2. Thus, negative binomial GLMs account for significantly higher variances. μ is parameterized as in Eq. (). Then, β and k are estimated through MLE using the glm package in R studio (https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/glm, last access: 21 September 2022).

Researchers have also developed zero-inflation outage prediction models to improve statistical performance for unbalanced data, e.g., when there are a lot of data points with no outages . The zero-inflation model has two levels of predictions . The first level can be a logistic regression or a decision tree model to check if there is at least one power outage . The first level model predicts “0” in case of no outages and “1” in case of at least one outage. The second level is the regression model predicting the number of outages for cases where the prediction was “1” at the first level. In this paper, we do not fit zero-inflated models, as our data are balanced; i.e., we observe at least one outage in each city.

We trained Poisson and negative binomial GLMs to predict the outage counts. The predictions are based on the seven input features mentioned in the feature selection section. All the input features are significant at a p value of 2×10-6. We compared the statistical performance of the Poisson and negative binomial GLMs (Table ). We have a total of 1528 training data points with seven input variables and one additional slope constant. Thus, the residual degree of freedom for each model is 1520.

The high value of residual deviance, relative to the degree of freedom, in the Poisson GLM shows large overdispersion . Thus, using this new outage dataset, we confirm that the variance in historical outages largely exceeds the mean value. A negative binomial GLM has a low residual deviance value compared to the Poisson model and is more similar to the degrees of freedom, indicating that a negative binomial GLM can handle overdispersion in power outage predictions more satisfactorily.

RDEV2 in Table  is a measure of the deviance explained by the fitted model compared to the null model. The null model predicts the average of observed outages (y‾) for all the cities irrespective of the input parameters. Rψ2 quantifies the amount of overdispersion explained by the additional variability introduced as a parameter in Eqs. (4) and (5) for the negative binomial fitted model. The RDEV2 is higher for the negative binomial GLM, also suggesting the negative binomial's better statistical performance. The Rψ2 for the negative binomial GLM is 0.69, which means the model can capture 69 % of variability by considering the additional level of uncertainty in the form of Poisson–gamma mixture given by Eq. () for outage counts. The reported value of Rψ2 for the negative binomial GLM in this paper is comparable to the values presented by , i.e., ∼0.8. However, we observe a lower value of RDEV2 compared to previous literature, i.e., ∼ 0.6. The lower value of RDEV2 may be due to the use of fewer parameters, e.g., 7 in this study versus 20 in , as RDEV2 always increases with more predictors. For example, we get a value of 0.48 for RDEV2 when all the input variables in Table  are included, but we considered fewer parameters to avoid correlated features and enhance the generalization of these models. We may also get a lower value of RDEV2 because we have a generalized dataset covering different regions in the US, and previous models were applied to data from smaller regions.

We also trained the Poisson and negative binomial GAMs. Like for GLMs, GAMs are trained with the seven input features mentioned before. All the input features are significant at a p value of 2×10-6 for both models. Like for GLMs, the residual degrees of freedom for each model is 1520.

The residual deviance for Poisson GAM (Table ) is ∼ 38 % less than the one in Poisson GLM (Table ), which is a minimal improvement to overcome the large overdispersion, as the deviance is still significantly higher than the degrees of freedom. However, RDEV2 shows a more significant performance improvement as its value increases to 0.53 for Poisson GAM (Table ) over a value of 0.20 for Poisson GLM (Table ).

The negative binomial GAM has a low value of residual deviance, which indicates that the negative binomial GAM can handle overdispersion. Additionally, non-linear shapes from spline functions (Eq. ) for GAMs improve the outage predictions. The RDEV2 for the negative binomial GAM improves significantly to a value of 0.62 (Table ) over the 0.29 in the negative binomial GLM (Table ). The observed value of 0.99 for Rψ2 is similar to the one reported by in their GAM model development for power outage predictions. The RDEV2 for GAMs is not available in the previous literature, so further comparisons with previous work could not be made.

We calibrated the random forest model to predict the fraction of customers without power. We performed the hyperparameter tuning using the GridSearch tool in Python with cross-validation to select the best input parameters for the random forest. The hyperparameter tuning resulted in a mean cross-validation R2 of 0.52. However, we obtained a training R2 of 0.84 when fitting the random forest with the tuned hyperparameters on the training data. The high training R2 compared to cross-validation R2 represents potential overfitting in the random forest model. We further tuned the model parameters by reducing the maximum depth. We obtained a cross-validation R2 of 0.48 and training R2 of 0.63. Finally, we obtained an R2 of 0.48 on the hold-out test. The number of randomly grown trees in the selected random forest model is 500. Figure  shows the predicted fractional outages against the observed outages using a tuned random forest (RF). RF does not generalize well to the outages due to Hurricane Isaias, which had low wind speeds (∼36 m s-1) in New Jersey but still caused outages to 60 % of consumers in 213 cities (out of 565) in New Jersey. This effect is visible in Fig. , where RF underestimated the fractional outages for severely affected cities in New Jersey.

The R2 for the random forest model cannot be compared to different R2 statistics calculated for GLM and GAM models, as the output for random forest (fraction of outages) differs from GLM and GAM models (outage counts). Also, we cannot calculate RDEV2 and Rψ2 statistics for random forest as there is no underlying probability distribution assumed for random forest predictions.

We present the variable importance in the random forest model in Fig. . We calculated the variable importance by training a base model with all the input features and a permuted model resulting from training different random forests and removing one feature at a time from the base model. We ranked importance by finding the variable that leads to the largest difference in the mean squared error between the base (full) model and the permuted (reduced) model. We present the normalized importance factors in decreasing order of importance (Fig. ).

We found that maximum wind speed is the most important parameter in the random forest model (Fig. ), which coincides with our findings from a simple linear regression in Fig. . Precipitation is the second most important variable, with a relative importance of 0.33, as trees can more easily be torn out from wetter soil. Population density is the third most important variable in outage predictions since it is a proxy for cities’ density of transformers exposed to winds.

Random forest and negative binomial GAMs show superior performance in predicting the power outages caused by a hurricane. MSE (mean squared error) has been used to compare the performance difference statistic models, given as 8MSE=1n∑(y-y^)2, where y is the predicted value, y^ is the observed value, and n is the total number of observations. Thus, we report MSE on both negative binomial GAM and RF models to compare the performance of these models. We rescale the negative binomial GAM predictions by the total number of customers to compare the MSE values of negative binomial GAM and RF models at the same scale as fractional outages. MSE for the negative binomial GAM is 45.13, and MSE for RF is 0.06. Researchers should be careful in making the direct comparison for MSE values of the fraction-based RF model and the count-based negative binomial GAM model, as these models are optimized for a different set of response variables. The high MSE for the negative binomial arises from overestimating outages, which we discuss next.

GLM and GAM regressions can predict the mean number of outages in the city. The models have a lower bound of zero as both Poisson and negative binomial distributions predict the count of outages. However, there is no upper bound on the predicted number of outages. Hence, GLM or GAM models can predict more outages than the number of customers, resulting in an overestimation of power outages.

For illustration, we present the power outage predictions on 20 % hold-out test data for the negative binomial GAM (Fig. ). For 76 cities out of 382 (19.8 %) in test data, predicted outages are more than the number of customers, and the overestimation can be significant. For example, the model predicted outages as high as 16 times those of Rockleigh, New Jersey, customers. The average ratio of predicted outages over the number of customers in the cities that experienced overestimation was 5.2. The cities that experienced overestimation had smaller populations, with an average of 5962; e.g., Rockleigh had only 106 customers. In contrast, cities without overestimation had an average population of 30 058. Modelers could impose an upper bound on the predictions using the total number of customers as the maximum possible outage. However, this adjustment would violate the assumptions in the Poisson–gamma mixture model (Eq. ) and GAM link function (Eq. ).

Random forest predictions are an average value of the outages in the training data (Eq. ). Thus, unlike GLMs, random forest predictions are bounded by the minimum and maximum values of power outages in training data. Based on simple physics, one would expect more damage and more outages from higher wind speeds. In order to understand the influence of wind speeds on the power outages in the random forest regression, we plotted the partial dependence of the fraction of customers without power against wind speed. The partial dependence, g(xj) , of the input variable xj is given by 9g(xj)=1N∑i=1Ny^i(xj,xic), where N is the total number of observations, xic are the variables other than xj, and y^i is outage prediction (Eq. ) for the ith data point. To plot the partial dependence, we varied xj (wind for this case) and kept xic (input parameters other than wind speed) constant. We estimated outages by averaging all observations in training data plotted against the xj (wind speed).

For this assessment, we trained the random forest model on a reduced dataset, with only New York and New Jersey, and on a complete dataset, including Florida and Texas. We present the partial dependence of wind speed in Fig. , also including the distribution of wind speeds in the training data. Hurricanes of category 3 or higher bring wind speeds above 40 m s-1 that can significantly damage electric poles . However, they are significantly less observed in inland cities, especially in the northern United States, as storms often weaken in their transition to higher latitudes and after leaving the ocean. For example, only tropical storms with wind speeds of less than 33 m s-1 have impacted New York City for the past 20 years (https://coast.noaa.gov/hurricanes/#map, last access: 21 September 2022). For example, Superstorm Sandy (2012) transitioned to a tropical storm before impacting New York City (https://coast.noaa.gov/hurricanes/#map, last access: 21 September 2022). As per ASCE 7–10 wind hazard maps (https://hazards.atcouncil.org/, last access: 21 September 2022), a wind speed of 43 m s-1 has a return period of 100 years for New York City. Thus, it is very unlikely, and evident from Fig. , to get outage data in New York and New Jersey for winds above 43 m s-1.

These limited outage datasets have strong implications for the validity and extrapolability of outage models based on random forest regressions. Under the reduced dataset, i.e., with only New Jersey and New York, outage predictions increase as the wind speed increases from 20 to 40 m s-1. However, the fraction of outages reached a maximum of 0.58 at wind speeds of 40 m s-1 and does not increase with higher wind speeds. The random forest model cannot extrapolate for the higher winds, which limits the capability of random forest to make outage predictions for large hurricanes.

Under the complete dataset, results improve by including outages in Florida and Texas. These states experience higher winds; e.g., their 100-year return-period winds are ∼70 m s-1 in contrast to the 43 m s-1 in New York (https://hazards.atcouncil.org/, last access: 21 September 2022). The reduced dataset (with only New York and New Jersey) did not have any data points with wind speeds above 40 m s-1. In contrast, the complete dataset (including Florida and Texas) had 88 cities (4.6% of data points) with winds greater than 40 m s-1 and 29 cities (1.5% of data points) with winds above 70 m s-1. With the complete dataset, the random forest predictions reach a maximum value of 0.76 for winds of 75 m s-1. While these results show improvement, they also show that the data are still insufficient to make the random forest model follow the physics of infrastructure failure and extrapolate predictions for high winds causing outages close to 100 %.

In catastrophic storms, we expect large outages with higher certainty, e.g., Hurricane Ida (2021) in Louisiana and Tropical Storm Fiona (2022) in Puerto Rico , close to 100 %. Structural models for power poles estimate failure probabilities close to 95 % for winds of 70 m s-1 (). Thus, the physics of infrastructure failure suggests that variance in outages should be smaller for catastrophic winds. To evaluate if existing models follow these principles from the physics of power infrastructure failure, we quantified the variance in predictions for Jersey City by varying the wind speed and keeping the other input variables unchanged (Fig. ).

As discussed previously, negative binomial GAMs capture the variability of outages better than Poisson models. Thus, we focus on the former models and estimate the variance according to Eq. (). Figure a shows the mean and 1 standard deviation interval for outage predictions with varying wind speeds. We normalized the GAM's predictions to show fractions and compare them to the random forest model. GAM's predictions go beyond 1, as discussed previously, but we truncated the y axis at 1 for comparison purposes. The linear relationship in the link function ensures that the variance (function of the square of mean outages from Eq. ) grows with wind speed. For example, for a wind of 40 m s-1, we have a standard deviation of 0.02, and for a wind of 70 m s-1, we have a standard deviation that is 15 times higher with a value of 0.3. Thus, the variance shows higher values as the predicted outage fraction approaches to 1. In fact, the variance is also unbounded in the negative binomial GAM and goes to infinity.

The random forest model can only predict the mean number of outages. Thus, it cannot evaluate variances. However, the quantile regression forest (QRF) can predict the outages at different confidence intervals, and we use it to quantify variance. The QRF uses the recorded observation at the leaf node to predict confidence intervals. These intervals are fully data-driven, as random forests do not assume any underlying probability distribution on predicted outages . We presented random forest prediction intervals in Fig. b. The random forests had a standard deviation of 0.45 for high winds (>70 m s-1), departing from the expected value of zero for catastrophic winds. Additional data could improve these random forest variance estimates. However, as mentioned earlier, infrastructure failure data are sparse.

Moreover, structural models predict no damage to power infrastructure at wind speeds lower than 10 m s-1 . Thus, we expect outage predictions closer to 0 with a higher degree of certainty. Negative binomial (and Poisson) GAMs handle this case well, as the variance is zero when the mean outage is zero (Eqs.  and ). In contrast, we found that random forests had a standard deviation of 0.66 for zero wind speeds, showing that they also have limitations to represent physics-informed variance at low wind speeds.

As standard deviation is high for outages with the variation of winds, and precipitation is the second most important variable in the RF model per Fig. , we explore the relationship between outage fraction and precipitation, based on Eq. (), to understand the non-linearity in outages. We show the partial dependence plot of precipitation in Fig. . The partial dependence plot of precipitation (Fig. ) shows non-linearity with an increase in the fraction of outages from 0.3 to 0.4 over the range of observed precipitation for the historical storms. However, we can observe from Fig. , similar to Fig. , that precipitation also explains limited variability in the outages in this case. Also, at zero wind speed and zero precipitation RF model predicts non-zero outages.