Modeling Solar Generation during Hurricanes †

Because solar generation adoption shows unprecedented growth, our power systems may extensively rely on solar generation infrastructure as the primary source of modern and clean energy in just a few decades. Despite this growth, very few studies have captured solar generation infrastructure’s behavior during natural disasters to understand their real beneﬁt for resilience. Here, we present an integrative methodology to quantify solar generation during hurricanes. The methodology combines a hurricane hazard model, solar irradiance quantiﬁcation, solar panel vulnerability, and a model for irradiance decay during hurricane conditions. We develop the irradiance decay model through a mixed-eﬀect regression on a dataset that merges historical Global Horizontal Irradiance and the revised data of Atlantic hurricane activity. The methodology is applied to 21 states in the Eastern U.S. for diﬀerent extreme events. Our results show that for events with return periods of up to 33 years, the loss in generation stems from cloud conditions during hurricanes. However, less frequent events can cause solar panel failure, especially in southern regions of the U.S., triggering complete loss of solar generation. Given that solar generation is expected to grow signiﬁcantly, these results advocate for higher standards in the structural design of solar panels.

with a GHI database from the National Renewable Energy Laboratory (NREL). 17 The analysis identifies hurricane features that best predict the intensity and extent of GHI decay. We fit different functional forms for GHI decay during hurricane conditions and highlight the best predictive model. Next, the paper proposes an integrative framework to quantify solar generation during hurricanes. We use synthetic storm data generated with statistical and physics-based tropical cyclone simulations for current atmospheric and oceanic environments. 19, 20 We utilize a physics-based model to estimate the storms' wind fields 21 and couple the fields with recently developed wind-based fragility functions for solar panels 22,23 to estimate wind damage to panels. Then, the integrative framework uses probabilistic models to transform estimates of GHI during normal conditions 17 to hurricane conditions. The transformation uses our proposed model for GHI decay during hurricanes in multiple regions of the U.S. at different times of the day and throughout the entire hurricane season. The integrative framework uses an algorithm based on Monte Carlo simulation to quantify the time-series of solar generation.
Finally, we apply the framework to quantify power generation from solar panels in the Eastern U.S.. We discuss the results as well as regional variations of the contributions of solar panels to power generation resilience through different counties. With these novel models and geographically-extensive case study, this paper lays the groundwork to quantify the resilience of power systems with solar infrastructure to hurricanes.

Material and Methods
Collecting data for GHI during previous hurricanes Hurricane conditions reduce solar irradiance intensity at the ground level over large geographical extents, limiting the ability of PV panels to harvest energy in communities. Figure 1 shows intense GHI decays during Hurricane Katrina in most regions within the radius (R34) at a wind speed of 17 ms −1 (34 knots), which reached 262 km. In some regions, intense decays extended to distances similar to the radii of the outermost closed isobar (ROCI), which reached 556 km. While Figure 1 shows only a snapshot for one hurricane demonstrating irradiance decays, we consistently observe the same trend in other hurricanes. In contrast to cloudless conditions of clear skies, which are associated with maximum solar generation, hurricanes cover extensive regions with different cloud structures from the eyewall to the rainbands. 24 These clouds absorb and scatter light, reducing direct incident radiation and generally leading to lower GHI and reduced solar panel generation. 25,26 Clouds that have high moisture density and vertical depth, i.e. optically thick clouds, can drastically reduce direct incident radiation. 27 Accordingly, hurricanes can significantly and rapidly lessen generation through optically thick cloud structures such as large cumulonimbus. However, hurricanes can also reduce generation significantly even with less optically thick cloud structures like stratiform clouds because they can cover large geographical extents.
To systematically investigate the effect of hurricanes on irradiance, we coupled a large dataset of GHI with historical hurricane data. We used the Physical Solar Model (PSM) version 3 from the National Solar Radiation Database (NSRDB) published by the National Renewable Energy Laboratory (NREL) to extract GHI with high spatial and temporal resolution. 17 The PSM combines satellite-derived atmospheric and land surface properties with radiative transfer models to solve solar radiation through the Earth's atmosphere. The PSM provides solar irradiance at a 4-km horizontal resolution for 30-minute intervals from 1998 to 2017. The PSM enable us to observe the GHI behavior at different timesnaps for different hurricanes since 1998 for multiple sites and under various hurricane conditions.

Hurricane dataset
We compiled hurricane data from the revised Atlantic hurricane database (HURDAT2). 18 The data contain multiple hurricane features and span several decades; however, key spatial information including hurricanes' radii is only available since 1998. The hurricane data include ROCI, the radius of maximum wind (RMW), radius at wind speeds of 17 ms −1 (R34, 34 knots) and 33 ms −1 (R64, 64 knots), hurricane category, and maximum wind speeds. The hurricane data have a 3-hour temporal resolution, which is coarser than the PSM temporal resolution; thus, we reduced the granularity of the GHI dataset from 30 minutes to 3 hours and matched the hurricane recording times. After performing a preliminary assessment to estimate the geographical extent impacted by the hurricane, we collected GHI records from the 4×4-km spatial grid within two times ROCI from the hurricane center, which reached several hundreds of kilometers for massive storms.
We analyzed 22 landfalling hurricanes whose geneses were in the North American basin and whose lifetime maximum intensity reached a category of at least 3 to filter out the disproportionately large number of storms that do not reach high intensities. The 22 hurricanes cover an extensive geographical region of our assessment ( Figure S1). These hurricanes have a wide variety of conditions, with maximum wind speeds up to 80 ms −1 (category 5), ROCI from 200 km to above 800 km, RMW up to 250 km, and radii at circulating wind speeds of 0 (R0) from 200 km to above 2000 km ( Figure S2). HURDAT2 omitted R0, the shortest distance where hurricane circulating wind effects dissipate entirely. 1 Thus, we estimated R0 with a wind profile model that captures the radial structure of tropical cyclones. 21 Key features for predicting GHI during hurricanes To characterize GHI decay under different hurricane conditions, we define I h as GHI during a hurricane. Previous research shows that GHI has strong temporal and spatial variability during normal conditions, i.e., no hurricane. 15, 28 We account for such variability and characterize GHI deviations from normal conditions in the logarithm space as whereĪ represents the median of the GHI under normal conditions at the same location and at the same time of the year as I h . We used 20 years of GHI data to estimateĪ for all the geographical extent covered by the hurricanes using a 3-hour temporal resolution. We assume that at each time of the day, GHI has approximately the same distribution for a given month. As a result, we used approximately 600 instead of 20 data points to estimate the GHI medians. For example, to estimate GHI at 10 a.m. in June, we lumped the data of its days from 1998 to 2017. We observe that for sites farther from the center of the hurricane, the median of δ h approaches zero, implying that the site is outside the area where hurricanes reduce GHI, i.e.,Ī = I h .
We analyzed GHI during the 22 hurricanes to estimate the samples δ h and understand GHI behavior during different hurricane conditions. Because our focus was only on times of the day when communities can generate energy, we only included in our analysis daytime data where and whenĪ > 10 W-h/m 2 , which finally resulted in ∼28M data points. Figure   2 shows δ h as a function of distance from the site to the hurricane's center and category.  Additionally, we find that high hurricane intensity exacerbates GHI decay. To focus on sites with the largest hurricane decay and cover areas within hurricane eyewalls, we analyzed sites located at 100km or less from the hurricane center. Figure 2b shows a decaying trend between hurricane category C and averaged δ h values, indicating that more intense hurricanes induce larger reductions in solar irradiance. A similar trend is observed between δ h and maximum winds V ( Figure S3a) because V has high colinearity with C as the latter variable is an increasing step function of V . Thus, we see that the linear fit performs very similarly with R 2 of nearly 0.11 (ρ = -0.34) in both cases. Lower irradiance levels for higher hurricane categories are also consistent with recent evidence on satellite-derived cloud microphysical features during hurricanes. 31 There are larger regions with higher cloud optical thicknesses associated with large and thick cloud structures such as cumulonimbus during hurricane maturity and intensification rather than during hurricane development or dissipation.
To capture hurricane size effect, we evaluated the relationship between different relevant hurricane radii and both the intensity and geographical extent of GHI decay. To study whether GHI decays are larger for bigger hurricanes, we analyzed the relationship between δ h and ROCI, RMW, and R0, respectively. We observe that hurricane size does not intensify GHI decay as linear fits have low R 2 values of 0, 0.05, and 0.02, respectively ( Figure S3).
To study how hurricane size correlates with the geographical extent of GHI decay, we analyzed the relationship between GHI and distance to the storm's center normalized by the hurricane size. We normalized d by four hurricane size metrics, ROCI, RMW, R0, and R34, where R34 is the radius at which the maximum wind speed is 34 knots, the minimum speed for the event to be categorized as a tropical storm. We split the data by hurricane category because C showed predictive power for hurricane decay intensification ( Figure 2).
When the distance is normalized by ROCI and R34, we generally observe better fitting performance than for the absolute distance, with improved performance for higher hurricane categories ( Figure S4 and S7). We estimated that a linear fit between R = d/ROCI and δ h has an R 2 of 0.38 for category 5, almost twice the value found for absolute distance ( Figure   2a). For R = d/R34, R 2 values show comparably good fitting performance to using ROCI as normalizing distance (Table S1). The slopes of linear fits are steeper for higher categories, further demonstrating that the intensity of the hurricane intensifies GHI decay ( Figure S4 and S7). As discussed earlier, this feature of GHI decay is driven by optically thicker cloud structures occurring during hurricane maturity and intensification. Distances normalized by RMW and R0 give lower performance, which, however, still illustrate how the effect of the hurricane on irradiance dissipates for large enough values of d ( Figure S5 and S6).
The analysis also shows that the regions with GHI decay easily extend beyond RMW and R34 as they only define hurricanes' inner-core circulation (Table S1). In contrast, it also shows that the regions with significant GHI decay do not reach R0 but are close to being bounded by ROCI. Thus, these observation suggests that the outer structure and radial extent of circulation bounded by ROCI is coupled with the cloud structures absorbing and reflecting light during hurricanes.

Probabilistic model for GHI during hurricanes
To leverage well-established mixed-effects regression models, 33 we assume that ln(I h ) is Gaussian, i.e., I h is lognormaly distributed, during daytime, when generation is not negligible, i.e., I h > 0. Thus where I h is the GHI median, and h is a Gaussian random variable with zero mean that accounts for the variability of GHI during hurricanes in the logarithmic space. We also assume that hurricanes reduce median GHI from normal conditions to I h such that in the logarithmic space whereĪ is the median GHI during normal conditions, and f (R, C) is a reduction factor that is function of the normalized distance to the hurricane's center R and the hurricane category C. f uses both R and C because they demonstrated to have good predictive power for GHI decay in the previous section. Using the expression in Equation 1, then Through this explicit decomposition, we properly represent the high GHI temporal and temporal variability structrure as extensively discussed in previous research. 15,28 The mixedeffects regression has both fixed and random components. 33 As described previously, we estimated δ h t,j for around ∼28 M observations corresponding to multiple time steps and sites of GHI recordings during the 22 hurricanes in the NREL dataset. We preprocessed the data by removing sites at long distances from the hurricane center, where hurricanes did not have significant effect on GHI. We then balanced the observations across the hurricane categories and distances from the center to the sites (see Supporting Information). Following these constraints, we used ∼ 0.75 M data points for the analysis.
We estimated the model parameters using maximum likelihood estimation (MLE) for the non-linear mixed-effects regression with a Matlab package. The package uses an expectationmaximization algorithm to solve for the parameters of the fixed component in Equation 5 while accounting for the unobserved component of the regression in Equation 4. 36 We fitted the parameters for the four models considering the four previously analyzed normalization radii, ROCI, RMW, R0, R34.

Modeling solar generation during hurricanes
We propose an integrative algorithm that couples our proposed GHI decay model with synthetic hurricane simulations and a fragility function for rooftop solar panels. We used a synthetic dataset wtih 5018 physically possible landfalling storms in the U.S. generated from a statistical-deterministic tropical cyclone (TC) model. 19 The model accounts for current Additionally, at each time step, we estimate R0 based on both the radius of maximum wind and maximum wind using a TC wind field profile model that connects the inner storm structure to the outer structure. 21 We estimate ROCI using the expression ROCI = 0.18 × R0 + 226 (km), which was obtained conducting a regression on the historic hurricane data.
Wind fields are estimated by combining axisymmetric winds circulating counterclockwise from the TC wind profile model 21 and the background wind field. 37 Previously, this synthetic storm model has been extended to quantify TC surge 20 and TC rainfall, 38 demonstrating its versatility for multiple hurricane hazard assessments. Here, we extend the applicability of the TC model to quantify solar generation during hurricanes.

Fragility function of solar panels
We used a fragility function developed according to current structural design standards for solar panels. 22 The wind measure in the fragility function was transformed from 3-second gust to 1-minute sustained wind speeds to make it compatible with the synthetic hurricane data using an empirical formula. 39 Hurricanes of category 3, starting at maximum wind speeds of 50 ms −1 (3-second gusts of 60 ms −1 ), can induce failure with a likelihood higher than 50% ( Figure S10).
Algorithm to estimate cumulative solar generation during hurri- where Φ(.) is the standard normal cumulative distribution function, max t (w t,j ) is the maximum wind that the solar panel at site j experiences during the hurricane, andw and β equal 58 ms −1 (3-second maximum wind) and 0.3, respectively ( Figure S8). Then, for a site j, a realizations j is sampled from Equation 9. Next, we explicitly model the panel failure time because this key variable will account for the energy that the panel will be able to generate before becoming nonfunctional. If there is panel failure, i.e.,s j = 1, we model failure time τ with the probability density function g τ (t). To account for higher likelihoods of failure when winds are more intense, we consider that g τ (t) is proportional to the time-varying failure likelihood due to different wind conditions at the site throughout the hurricane. Thus, g τ (t) ∝ p t,j , where p t,j can be estimated from Equation 8b. Accordingly, at site j, a realizationτ j is sampled from g τ (t) ifs j = 1, or it is assigned ∞ ifs j = 0, i.e., when the hurricane does not cause panel failure. At each time step t, the panel's time-varying functionality statusx t,j is estimated as After assessing solar panel functionality, the algorithm samples GHI realizations. Following Equations 3 and 6, In the logarithmic space, h accounts for spatiotemporal variability in GHI during hurricanes. Under our initial assumption that hurricanes only modify the GHI logarithmic mean, h remains the same as normal-conditions . Thus Following the lognormality assumption for GHI under normal conditions, I h can be estimated by transforming GHI during normal conditions to GHI during hurricane conditions Based on the assumption that hurricanes only modify the GHI logarithm mean, Equation 12 enables us to leverage well-defined GHI normal condition statistics throughout the entire U.S. 17 with a clean and simple formula to find decayed GHI during hurricanes. For each site j and time t, a realization of GHI during normal conditions (Ĩ t,j ) is sampled and adjusted to hurricane conditions using f (R t,j , C t ) as Next, the powerq t,j generated at time t and site j is estimated per area of installed solar panel A with efficiency E, the ratio between the amount of electricity the panel produces and the amount of solar energy it absorbs from the sun. Thus Finally, the cumulative energyQ t,j generated is updated by adding the product betweeñ q t,j and dt, the interval between time steps.

Best-Fitted functions for GHI decay
We conducted mixed-effect regressions for all 16 combinations of functional forms and normalizing radii. We report all fitted parameters in the Supplementary Information. Additionally, we estimated the Akaike information criterion (AIC) 40 to evaluate the regressions' relative statistical performance. The model with f 4 and R = d/ROCI exhibits the best performance. We find that the selection of the functional form f did not modify the regression statistical performance to the degree of the selection of the normalizing radius. The performance of ROCI is followed by R34, and ROCI and R34 performed significantly better than RMW and R0 (see Supplementary Information). Figure 3 shows the best fit, that is, f 4 and R = d/ROCI, for different categories. The plot shows how GHI decays during hurricanes, with stronger effects closer to the hurricane center and for higher hurricane categories. These observations are consistent with the presence of optically thick cloud structures close to the hurricane center and during hurricane maturity and intensification as noted previously. The regression also shows that the decay consistently extends up to sites that are ∼1.3 times ROCI from the hurricane center, confirming the observation that the cloud structures and radial extent of hurricane circulation defined by ROCI are strongly coupled with the hurricane mechanism for high light absorption and reflection. Because this threshold (∼ 1.3) does not change significantly for different categories, hurricanes with low categories can cover more extensive regions with clouds that reduce GHI than hurricanes with high categories as long as they have larger ROCI. However, the level of the decay will be smaller for lower categories. We assessed cumulative solar energy generation for four days during a hurricane emergency using time steps of 2 hours. In Figure 4, we show a subset of cumulative generationŝ Q t,j for four coastal counties exposed to high hurricane hazard, Galveston (

Discussion
This paper has proposed the first framework to evaluate solar generation during hurricanes at a regional scale. The framework integrates four key pieces: hurricane hazard analysis, solar irradiance modeling, solar panel vulnerability, and a newly presented model to assess irradiance decay as a result of hurricane cloud conditions. This framework aims to be the foundation for assessing the vulnerability of modern power systems with solar generation infrastructure to hurricanes and a tool to test strategies and policies to increase their resilience.
The integrative framework has been presented through a proposed algorithm to model the time-series of solar generation during hurricanes. While the algorithm is a key contribution from this article, our scope also includes the development of a model to capture irradiance decay during hurricanes, a crucial piece of the framework, which to the authors' knowledge, has not been developed before. The irradiance decay model is based on an extensive assessment of GHI under 22 landfalling storms in the North American basin, which reached a category of at least 3 during their lifetime. The dataset conclusively shows that hurricanes reduce the GHI throughout their tracks. We confirmed that the distance from a site to the hurricane and its category are key predictors of the irradiance decay. We argue that the mechanism driving the decay is the formation of optically thick clouds in the eyewall, which often become thicker during hurricane intensification. These optically thick clouds, with high moisture density and vertical depth, reduce direct incident radiation by light absorption and reflection.
We fitted four functional forms that vary in complexity to represent irradiance decay using a mixed-effects regression. Multiple category-dependent features controlling the intensity and shape of decay were tested, and the best functional form was selected using AIC to demonstrate its suitable statistical performance. ROCI is shown to be an effective size metric for normalizing the distance in the functional forms of irradiance decay.
Finally, we described the algorithm to quantify solar generation during hurricanes. We apply the algorithm to 1217 counties belonging to 21 states in the Eastern region of the U.S..
We use cumulative generation at day four since landfall as a metric to measure resilience.
Our results show that generation during most storms with return periods shorter than three years will be distributed as normal conditions during summer. In contrast, events with return period of 10 years and 33 years will reduce generation significantly, by 37% and 48%, on average, respectively. Optically thick clouds that reflect and absorb light are the drivers of such reductions. Rarer events (with return periods of 333 and 1000 years) will reduce generation to a higher degree. While in the northern states, these extreme events will reduce generation due to optically thick clouds, in the southern states, they will likely trigger solar panel structural failure due to strong winds. As a result, southern regions face a higher risk of losing power generation, as northern regions can still generate at a reduced level if the panel has not failed. Solar generation is expected to become a pillar for our future power systems. Thus, our results show that for communities to rely on this pillar to deliver critical power during extreme events, higher standards for solar panel structural design are required.