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Bayesian Updating of Solar Panel Fragility
Curves and Implications of Higher Panel
Strength for Solar Generation Resilience
Luis Ceferino1,2,3, Ning Lin3,4, Dazhi Xi4
1Department of Civil and Urban Engineering, New York University
2Center for Urban Science and Progress, New York University
3The Andlinger Center for Energy and the Environment, Princeton University
4Department of Civil and Environmental Engineering, Princeton University
This paper has been submitted to the journal of Reliability Engineering & System Safety for publication.
Corresponding author: L. Ceferino. Address: 370 Jay St, Room 1309, Brooklyn, NY 11201. Email:
ceferino@nyu.edu
Bayesian Updating of Solar Panel Fragility Curves and Implications of Higher Panel Strength for 1
Solar Generation Resilience 2
Luis Ceferino1,2,3, Ning Lin3,4, Dazhi Xi4
3
1Department of Civil and Urban Engineering, New York University 4
2Center for Urban Science and Progress, New York University 5
3The Andlinger Center for Energy and the Environment, Princeton University 6
4Department of Civil and Environmental Engineering, Princeton University 7
8
ABSTRACT 9
Solar generation can become a major and global source of clean energy by 2050. Nevertheless, few 10
studies have assessed its resilience to extreme events, and none have used empirical data to 11
characterize the fragility of solar panels. This paper develops fragility functions for rooftop and 12
ground-mounted solar panels calibrated with solar panel structural performance data in the Caribbean 13
for Hurricanes Irma and Maria in 2017 and Hurricane Dorian in 2019. After estimating hurricane 14
wind fields, we follow a Bayesian approach to estimate fragility functions for rooftop and ground-15
mounted panels based on observations supplemented with existing numerical studies on solar panel 16
vulnerability. Next, we apply the developed fragility functions to assess failure rates due to hurricane 17
hazards in Miami-Dade, Florida, highlighting that panels perform below the code requirements, 18
especially rooftop panels. We also illustrate that strength increases can improve the panels' structural 19
performance effectively. However, strength increases by a factor of two still cannot meet the 20
reliability stated in the code. Our results advocate reducing existing panel vulnerabilities to enhance 21
resilience but also acknowledge that other strategies, e.g., using storage or deploying other generation 22
sources, will likely be needed for energy security during storms. 23
Keywords: solar panels, fragility functions, hurricane hazards, Bayesian update, structural 24
reliability 25
1. INTRODUCTION 26
As the world transitions towards cleaner energy sources, the power system infrastructure is rapidly 27
changing. In 2019, installations of solar generators accounted for 40% of the electric generating capacity 28
installed in the United States (Perea et al., 2019). Market and government projections state that solar 29
generation will be 20–30% of the global electricity by 2050 (Shah & Booream-Phelps, 2015; Sivaram & 30
Kann, 2016; Solaun & Cerdá, 2019; The International Renewable Energy Agency, 2018). As a result, the 31
resilience of the power system infrastructure is also changing. First, the design standards or the level of 32
exposure of solar energy generating infrastructure can differ from current generation infrastructure. For 33
example, engineers design nuclear plants or dams with risk category IV for safety in nuclear and 34
hydroelectric generation, source of 20% and 7% electricity generation in the United States (U.S. Energy 35
Information Administration, 2021). This category provides the highest structural reliability levels in the 36
ASCE7-16 design code since failure “could pose a substantial hazard to the community” (American Society 37
of Civil Engineers, 2017). In contrast, engineers can design solar panels following conventional reliability 38
levels for rooftops, i.e., risk category II. Engineers can design them with even lower levels, i.e., risk category 39
I, if the solar installation structural failure “represents low risk to human life in the event of failure” as for 40
large ground-mounted installations in remote locations (Cain et al., 2015). Moreover, by design, the solar 41
generators themselves must be placed outdoors and are directly exposed to extreme loads such as high 42
winds. This exposure level is markedly different from existing generating units typically within protective 43
infrastructure. For example, natural gas, source of 40% of the electricity in the United States (U.S. Energy 44
Information Administration, 2021), is transported in pipes underground and is processed in power plants 45
with several key equipment within buildings, protecting them from winds. As solar generation becomes a 46
key source of our energy production, we need a better understanding of its resilience to natural hazards and 47
its ability to provide sufficient and reliable power during extreme load conditions. 48
Fragility functions describe the likelihood of damage (or failure) due to an extreme load, e.g., earthquake 49
shaking, hurricane wind. The development of fragility functions for energy generation components is 50
essential to understand the risk profile of power systems (Bennett et al., 2021; Winkler et al., 2010; Zhai et 51
al., 2021). However, lack of data has prevented the assessment of panel vulnerability to extreme loads, 52
hindering our ability to understand the resilience of future power grids. Due to the lack of solar panel failure 53
data or appropriate experimental tests, Goodman (2015) used simplified numerical structural assessment to 54
propose the first solar panel fragility functions. The analysis focused on yielding onset of rooftop panel 55
racks due to high wind loads. Due to the lack of better models, its fragility function has also been applied 56
to ground-mounted solar panels (Bennett et al., 2021; Watson, 2018). To the best of the authors’ knowledge, 57
data-driven assessments of solar panel vulnerability have not been conducted. 58
In this paper, we fill this research gap by analyzing a novel dataset of solar panel structural performance in 59
60 sites in the Caribbean after the 2017 and 2019 hurricane seasons. This dataset captures these storms’ 60
severe impact on renewable infrastructure, especially in Puerto Rico (Kwasinski, 2018). We use this dataset 61
to propose the first data-driven fragility curves for both rooftop and ground-mounted solar panels. Through 62
a Bayesian approach, we supplement this empirical dataset with numerically driven fragility functions by 63
Goodman (2015). Combining these different information sources results in more robust estimates of 64
fragility function parameters than those based on either observation or numerical simulation. We present 65
an algorithm based on Metropolis-Hastings (MH) Monte Carlo Markov Chain (MCMC) to solve the 66
Bayesian update with computational efficiency. Additionally, the Bayesian approach explicitly 67
characterizes the uncertainty in the fragility functions’ parameters, which is critical to account for the 68
uncertainty of key risk metrics, e.g., panels’ annual rate of failure. 69
Next, this paper shows an application of the developed fragility functions by assessing the structural 70
reliability of solar panels in Miami-Dade, Florida, to hurricanes. Our assessment combines our new fragility 71
functions and hurricane hazard modeling for mainland United States (Marsooli et al., 2019). Finally, this 72
paper explores the value of increasing solar panel strength to, for example, assess its effectiveness in 73
reducing annual failure rates and meet different ASCE7-16 standards for structural reliability. This paper 74
contributes to the body of literature on the risk of modern power systems to extreme events by providing 75
the first data-informed fragility functions for solar panels and a holistic assessment of their reliability to 76
hurricanes. 77
2. SOLAR PANEL STRUCTURAL PERFORMANCE DATA 78
2.1. Panel damage data 79
Our dataset is an extended version of the “Solar Under Storm” reports’ panel failure dataset (Burgess et al., 80
2020; Burgess & Goodman, 2018). The initial dataset consists of 26 sites primarily located in residential 81
buildings in Puerto Rico for rooftop panels. “Solar Under Storm” focuses on reporting main failure 82
mechanisms in rooftop installations with qualitative descriptions of failure modes in the Caribbean after the 83
large hurricanes Irma and Maria in 2017 and Dorian in 2019. The study reports frequent failures in racks 84
and the clips that attach the panel to the racks (Burgess et al., 2020). Unlike Goodman (2015), which covers 85
the early serviceability damage state, i.e., yielding onset in racks, the identified damage conditions in the 86
dataset introduce a damage state of structural collapse (Figure 1). 87
(a) Residential rooftop panels
(b) Large ground-mounted panels
Figure 1. Example of solar panel damage in dataset. (a) Rooftop panels: Clip failures in the bolt
connection between panels and racks (red arrows) lead to panel uplift (see bolts in circle with zoom-in
view). Clamp failures (see rectangle with zoom-in view) lead to blown racks (see red line where a rack
used to be placed) (Burgess et al., 2020). (b) Large ground-mounted panels: Satellite imagery shows the
scale of the wind damage in comparison to the pre-hurricane view in the rectangle (National
Oceanographic and Atmospheric Administration, 2021). In large-scale failures, multiple failure modes
were found, including debris impact from damaged panel arrays.
Because the “Solar under Storm” dataset focuses on failed rooftop panels, we extended the dataset to cover 88
panels that survived the hurricanes. The data extension is critical to properly fit fragility functions with data 89
representing various panels’ structural performance. By surveying local engineers in Puerto Rico, we 90
extended the dataset to 46 sites. Supplementary Table 1 shows the list of the rooftop solar panels, their 91
geographical coordinates, and their failure mode, e.g., Figure 1a. Out of the 46 sites, 46% experienced clip 92
(clamp) failures, 17 % racking failures, 4% roof attachment failures, and 50% either rack or connection or 93
roof attachment failure. Most panels underwent damage due to debris impact (65% in the initial dataset). It 94
is important to note that debris failure was primarily part of a cascading mechanism with projectiles 95
originating from the damaged panels themselves. Figure 2a shows a map with all the panel installation sites, 96
indicating clip, racking, or attachment failures. The plot also shows that Hurricane Irma, Maria, and 97
Dorian’s tracks were near the sites. 98
For ground-mounted solar panels, we surveyed reports and newspapers to determine panels’ failures in 99
large sites. Utility-scale solar installations are primarily ground-mounted, each one composed of hundreds 100
or thousands of panels. Thus, their failures often have media coverage. We visually verified the installation 101
damage with high-resolution satellite imagery from the National Oceanic and Atmospheric Administration 102
(NOAA) (National Oceanographic and Atmospheric Administration, 2021) and Google Earth Satellite 103
Imagery. We obtained information for 14 large panel installations with 13 MW of capacity on average in 104
the Caribbean for Hurricanes Irma and Maria in 2017. The “Solar Under Storm” study also surveyed a few 105
of these installations, but it did not report the installations’ geographical coordinates to preserve the 106
confidentiality of the sites (Burgess & Goodman, 2018). Supplementary Table 2 shows the list of these 107
ground-mounted solar panels, their geographical coordinates, capacity, and the percentage of failed panels 108
(see site failure in Figure 1b). 36% of sites reported significant failures in more than 50% of their panels, 109
including the Humacao Solar Farm with 40 MW of capacity, the second largest solar farm in Puerto Rico 110
(Institute for Energy Research, 2018). Figure 2b shows installations indicating the sites with significant 111
failures, i.e., more than 50% of failed panels. The reported failures included clip (clamp) failures, racking 112
fractures and buckling, bolt shear failure, and bolt loosening (Burgess & Goodman, 2018). We observed 113
evidence of cascading structural failures triggered by debris from damaged panels in large sites, suggesting 114
that damage in a few panels can progress quickly to massive failures. This observation is consistent with 115
the cascading failures of clip (T-clamps) fractures found in the more detailed post-hurricane structural 116
inspections (Burgess & Goodman, 2018). 117
(a) Residential rooftop panels
(b) Large ground-mounted panels
Figure 2. Solar panel sites in collected dataset after Hurricanes Irma and Maria in 2017 and
Hurricane Dorian in 2019. The lines indicate the hurricane tracks, and the panels with failures (clip,
racking, rooftop attachment) and without failures are highlighted in the map. Failure in the panel
array is defined as either clip, racking, or roof attachment (in case of rooftop panels) failures in more
than 50% of the panels.
In Puerto Rico, where 50% and 59% of the inspected rooftop and ground-mounted panels were located, 118
wind design levels range from 63 to 72 m/s and from 57 to 76 m/s for structures with risk categories I and 119
II, respectively (American Society of Civil Engineers, 2017). As mentioned earlier, the ASCE7-16 requires 120
solar panels on residential buildings to be designed with a risk category of II. Ground-mounted solar panels 121
can be designed with a risk category I since they “represent low risk to human life in the event of failure”. 122
While the structural design levels for ground-mounted solar panels are lower, our described findings 123
reported fewer sites with large failures than rooftop panels (50% versus 60%). For further assessment, we 124
analyzed the wind conditions that the panels experienced. 125
2.2 Wind conditions 126
We obtained the hurricanes’ tracks, their radii of maximum wind, and maximum winds from the revised 127
HURDAT2 Atlantic hurricane database (Landsea & Franklin, 2013). We estimated axisymmetric winds 128
circulating counterclockwise based on a tropical cyclone wind profile model (Chavas et al., 2015). We then 129
combined these circulating winds with the estimated background winds (Lin et al., 2012) to calculate the 130
resulting asymmetric wind fields for each hurricane. For smoothness, we interpolated HURDAT2 3-hour 131
timesteps and thus the corresponding wind fields to obtain maximum wind at each panel site for every 10 132
minutes (Supplementary Figure 1). 133
For evaluation, we compared the resulting wind estimates to the hourly wind records from the NOAA 134
National Centers for Environmental Information (2001)’ Global Integrated Surface Dataset during 135
Hurricane Maria from the weather station at the San Juan International Airport in Puerto Rico (Figure 3). 136
No other stations reported wind data from Puerto Rico for the event. Unfortunately, wind data were not 137
gathered for the most intense period; nevertheless, data during and before August 20th, 2017, show that our 138
wind estimates and records closely follow each other. During August 20th, both datasets showed rapid wind 139
intensification, at least up to the ~30 m/s, when records stopped. Our estimates indicate that winds peaked 140
at 60 m/s on August 20th, 2017. 141
Figure 3. Comparison of wind estimates from Chavas et al. (2015), (C15), and the wind records from
NOAA National Centers for Environmental Information (2001), (NCEI), at the San Juan International
Airport.
Using a multiplicative factor from an empirical formula (Vickery & Skerlj, 2005), we converted the 1-m 142
sustained wind estimates at the panel sites to 3-second gusts to be compatible with the wind load metrics 143
for structural design (American Society of Civil Engineers (ASCE), 2017). Failures in rooftop panels were 144
caused by gusts starting at 73 m/s, with a mean in all sites of 81 m/s (Figure 4a). Failures in ground-mounted 145
panels were caused by gusts starting at 83 m/s, with a mean of 91 m/s (Figure 4b). The solar panel dataset 146
is suitable for assessing fragility functions as it contains ranges of gusts where failure occurrence has large 147
variability (Figure 4). For example, between 70 and 90 m/s, several sites with rooftop panels experienced 148
both failure and no failure. Getting data in this range is critical for fragility functions to appropriately 149
capture the uncertainties in panel failure when transitioning from low winds to high winds. 150
(a) Rooftop panel
(b) Ground-mounted panel
Figure 4. 3-s gust distributions for panels with (black) and without (gray) damage. The data are shown
as points and the empirical probability density functions are estimated using a Gaussian kernel
3. BAYESIAN FRAMEWORK FOR FRAGILITY FUNCTION UPDATES 151
3.1. Fragility function 152
Fragility functions with lognormal shape are typically used to model infrastructure’s damage due to wind 153
hazards and multiple other extreme loads (Ellingwood et al., 2004; Shinozuka et al., 2000; Straub & der 154
Kiureghian, 2008). Its shape is given by 155
(;,)=
ln ()ln ()
(1)
where is the probability of panel failure due to a wind gust , is the wind gust with a failure probability 156
of 50%, is a normalizing factor, and is the cumulative standard normal distribution function. defines 157
the width of the transition range between winds with low and high failure probability, and it is a measure 158
of aleatory uncertainty in the vulnerability analysis. For example, a value of 0 would be equivalent to a 159
deterministic assessment, where the panel would fail after a given wind threshold. 160
We follow a Bayesian approach to fit solar panels’ fragility functions due to two key factors. 161
• The Bayesian formulation can represent fragility functions’ significant epistemic uncertainties 162
through random fragility function parameters, and . Treating and as random variables rather 163
than deterministic parameters allows for the propagation of their uncertainty to solar panel damage 164
predictions in risk analysis. 165
• The Bayesian approach allows for the combination of multiple sources of information to improve 166
the fragility function characterization. The dataset presented in this paper provides the opportunity 167
for a data-driven, probabilistic description of panel failure. However, the number of samples is not 168
high, e.g., 46 and 14 for rooftop and ground-mounted panels, respectively. Thus, through the 169
Bayesian approach, we use Goodman (2015)’s numerical assessment as prior information and then 170
combine it with the dataset for the final fragility function estimates. 171
In the Bayesian formulation, the posterior distribution (,|) after combining both data sources is 172
(,|) =
(|,)(,)
(
|,
)
(,)dd
(2)
where ={,, … , } is the vector containing the failure information at each site, thus {0,1} 173
equals zero if the panel did not fail and one if it failed, and is the number of sites, i.e., equal to 46 and 14 174
for rooftop and ground-mounted panels, respectively. The limit state for rooftop panel failure is defined as 175
extensive damage, including clip, racking, or roof attachment failures. Hereafter, we refer to this damage 176
state as panel failure. The limit state for failure in the large ground-mounted panels is defined as extensive 177
damage, including clip and racking failures, in more than 50% of their panels. (|,) is the likelihood 178
function of observing the dataset for fixed values of and , and (,) is the prior distribution of and 179 . 180
3.2. Prior 181
As in the Bayesian approach, and from Equation (2) are random variables rather than deterministic 182
values. Additionally, and can only be positive numbers. Accordingly, we use lognormal distributions 183
to model the prior. The probability density functions (pdfs) of () and () are given by 184
()=
exp
(
)
(3)
()=
exp
(4)
where and are hyperparameters defining the logarithmic mean and standard deviation of . and 185 are hyperparameters defining the logarithmic mean and standard deviation of . For simplicity, we 186
assume and are independent. Thus 187
(,) = ()()
(5)
For Bayesian assessments, the data supporting the prior distribution need to be independent of the data used 188
for the parameter update. Thus, the selection of Goodman (2015)’s fragility function is appropriate for this 189
study. The numerical assessment is based on code-conforming rooftop panels designed for wind conditions 190
in Atlanta, Georgia. The uncertainty in the fragility function stems from the stochastic velocity pressure 191
induced by winds acting on the panel. It also models stochasticity in material strength and construction 192
quality. Goodman (2015)’s study is frequentist; thus, the parameters defining the fragility function in 193
Equation (1) are deterministic. The resulting fragility functions had a deterministic υ, gust for 50%-failure 194
probability, of 60 m/s and of 0.13 for a panel on a 30o roof. 195
To use these numerical evaluations as a prior distribution, we adjusted their wind design conditions to the 196
Caribbean. Taking San Juan, Puerto Rico, as a reference, we scaled up to represent a local solar panel 197
design using the ratio between the wind design values in San Juan and Atlanta. We consider a design with 198
risk category II (wind with a return period of 700 years) for rooftop panels and a risk category I (wind with 199
a return period of 300 years) for the ground-mounted panels (Cain et al., 2015). As a result, we used equal 200
to 85 m/s for rooftop panels and 81 m/s for ground-mounted panels. 201
We use the values of 85 m/s and 81 m/s to find the prior logarithmic means of () for the rooftop and 202
ground-mounted solar panels, respectively, since they are equal to the prior medians () in the lognormal 203
distributions. For the logarithmic means of (), we use Goodman (2015)’s value of 0.13 for both panel 204
types. The logarithmic standard deviations ( and ) are a measure of epistemic uncertainty as data can 205
reduce them. We consider the results from Goodman (2015)’s numerical assessment to be initial sound 206
data. Thus, we use it as an informative prior rather than using weakly informative or non-informative prior 207
(Gelman et al., 2013). Accordingly, we set and equal to 0.5. This value is similar to other Bayesian 208
assessments for vulnerability curves (Noh et al., 2017), and it accounts for the lack of information (e.g., 209
actual material strength or failure modes) in the numerical study in reproducing panel failure. 210
3.3. Likelihood of observing the data 211
Panel failure follows a Bernoulli distribution with probability that is a function of the wind. Considering 212
that the failures at different sites are independent, then the likelihood of observing failures or non-failures 213
in sites is given by 214
(|,)= (1)
(6)
where is one if the panel failed at the site or zero otherwise and is found from the fragility function in 215
Equation (1) with parameters and . 216
3.4. Posterior distribution 217
According to the Bayes rule for conditional probabilities, the posterior (,|) can be found in Equation 218
(2). The numerator is the product of the likelihood of observing the panel failures and the prior distribution. 219
The denominator is the integral of this product through the entire parameter space of and . Equations (5) 220
and (6) allow us to find the numerator in closed form, but the denominator requires a complex integration 221
that cannot be solved analytically. 222
3.5. Solving for the posterior distribution using MCMC 223
To overcome the challenge stemming from numerical integration, we followed an approach based on 224
MCMC (Liu, 2004). MCMC only requires evaluating a proportional function to the posterior distribution 225
rather than the posterior itself. Thus, we can find samples from the posterior and circumvent the evaluation 226
of the integral with MCMC since 227
(,|)(|,)(,)
(7)
We use the Metropolis-Hastings (MH) MCMC algorithm to define a Markov Chain (MC) that samples 228
from the posterior distributions of and . With the MH algorithm, we define the MC as a random walk 229
through the parameter space of and . To generate -th sample pair (,) of the posterior, we sample 230
a candidate (,) using the following uncorrelated bivariate normal distribution 231
(,)((),())
(8)
where () is the mean vector of the random walk, and it is equal to the last posterior sample 232
(,). () is the covariance matrix, equal to the diagonal matrix (), (). () 233
and () are calibrated values for an effective exploration of the high-probability regions, i.e., good 234
mixing. For this symmetrical random walk, the sample candidate (,) is accepted with probability 235
min(1, ), where 236
=
(|,)(,)
(|,)(,)
(9)
According to the MH properties, the MC has a stationary distribution, i.e., the resulting distribution when 237
the number of samples is sufficiently large, equal to the posterior distribution of and in Equation (2). 238
This algorithm was implemented to assess the posterior of the fragility function parameters for both rooftop 239
and ground-mounted panels. We ensured a good mixing by calibrating () and () such that the 240
average acceptance rate is around 25% as recommended in the literature (Chib & Greenberg, 1995; Robert, 241
2014). Using the MH MCMC analysis, we sampled 10,000 realizations of and from the posterior 242
distribution after a burn-in period containing 1,000 realizations. We selected the burn-in period after 243
verifying the MC stationarity (Supplementary Figure 2). 244
4. BAYESIAN UPDATES FOR FRAGILITY FUNCTIONS 245
4.1. Rooftop panels 246
We used the generated 10,000 samples to estimate the posterior distribution of the fragility function 247
parameters. For , the median varied from 85 m/s in the prior to 80 m/s in the posterior, its standard 248
deviation from 51 m/s to 5 m/s, and its logarithmic standard deviation from 0.5 to 0.07 (Figure 5a). The 249
similar prior and posterior medians show that the numerical analysis in Goodman (2015) is consistent with 250
the observations of wind in terms of the 50%-failure probability. The significant decrease (90%) in the 251
standard deviation reveals the importance of the solar panel dataset in decreasing the initial epistemic 252
uncertainties of . 253
For , the median varied from 0.13 in the prior to 0.32 in the posterior, its standard deviation from 0.08 to 254
0.11, and its logarithmic standard deviation from 0.5 to 0.30 (Figure 5b). The posterior median of is 255
almost three times the prior value. Such an increase reveals the inconsistency of the numerical analysis in 256
Goodman (2015) with the empirical data in terms of the aleatory uncertainty measured by . The numerical 257
analysis implies that the transition range between winds with low and high failure probabilities is narrow. 258
Conversely, previous empirical evidence (Roueche et al., 2017, 2018) suggests that the value of 0.13 is 259
too small to characterize the uncertainty in wind hazards, implying a wider transition range between winds 260
with low and high failure probabilities. This observation demonstrates the importance of empirical data to 261
calibrate numerical analysis. 262
We found a lack of correlation between and in the posterior as the Pearson’s coefficient between their 263
posterior samples was only 3x10. This result suggests independence between and , as assumed in the 264
prior. 265
The Bayesian update from the parameters’ prior distribution to the posterior distribution brings important 266
implications for the fragility function of rooftop solar panels. The mean fragility function, describing the 267
probability of panel failure, for the posterior distribution can be found as 268
[()] = (;,)
(,|)
(10)
Equation (10) uses the posterior (,|) as the distribution of and to find the posterior of [()]. 269
Replacing (,|) by the prior (,) will result in the prior [()]. 270
a) (rooftop panel)
b) (rooftop panel)
c) (ground-mounted panel)
d) (ground-mounted panel)
Figure 5. The prior and posterior distribution of and for rooftop solar panels. Samples from the
posterior distribution were used to depict the histogram, and Gaussian kernel was used to develop
each empirical pdf.
We solved Equation (10) by averaging all values for the suite of 10,000 fragility functions, obtained by 271
evaluating the 10,000 samples of and (Figure 6a). With this procedure, we incorporate and propagate 272
the uncertainty in and to the fragility function. The deterministic prior distribution in Goodman (2015) 273
was used to set up the prior medians’ hyperparameters. However, the resulting mean fragility function 274
([()]) from the Bayesian prior is different than its frequentist counterpart due to its parameters’ 275
uncertain nature. The difference is negligible for the wind with a 50%-failure probability (~85m/s for both). 276
Yet, it is significant for the wind with a 10% and 90%-failure probability (71 versus 43 and 100 versus 167 277
m/s). The wider wind range in the transition from a 10% to a 90%-failure probability in the Bayesian 278
assessment results from the uncertainty propagation from and (Figure 5a and 5b’s grey curves) to the 279
fragility function. 280
The posterior distribution changes the wind for 50%-failure probability only slightly (-5%), from 86 m/s in 281
the prior to 80 m/s in the posterior. The wind range that transitions from a 10% to a 90%-failure probability, 282
52 and 123 m/s, respectively, has a width that is 56% smaller than the prior. This reduction results from the 283
lower uncertainty on , whose standard deviation decreases from 51 m/s in the prior to 5 m/s in the posterior 284
(Figure 5a). Moreover, the posterior fragility function shows a significantly narrower confidence interval 285
than the prior fragility function. These results demonstrate the importance of the Bayesian approach to 286
capture and reduce large initial uncertainties through empirical data, not only in the fragility function 287
parameters ( and ), but also in the mean fragility function itself. 288
a) Rooftop panel
b) Ground-mounted panel
Figure 6. Fragility functions for random samples and according to their prior and posterior
distributions. The solid thicker lines indicate the expectation of the failure probability over the
parameters’ distribution, and the dashed lines indicate the mean plus and minus a standard deviation.
Goodman* is the deterministic fragility function adapted from Goodman (2015).
4.2. Ground-mounted panels 289
The distribution of shows that the median varies from 81 m/s in the prior to 90 m/s in the posterior, its 290
standard deviation from 50 m/s to 6 m/s, and its logarithmic standard deviation from 0.5 to 0.07 (Figure 291
5c). The posterior shows a significant reduction in the uncertainty of , with a standard deviation 87% lower 292
than that of the prior. Such a reduction is very close to the one found in rooftop solar panels, even though 293
the number of data points is one-third of the latter. 294
For , the median varies from 0.13 in the prior to 0.15 in the posterior, its standard deviation remains in 295
0.07, and its logarithmic standard deviation from 0.5 to 0.39 (Figure 5d). As a result, the posterior 296
distribution exhibits a slight shift to the right. The little variations in 's standard deviation and logarithmic 297
standard deviation suggest that the number of data points is insufficient to substantially reduce uncertainty. 298
Following the same procedure for the rooftop panels, we estimated the mean fragility function ([()]) 299
for ground-mounted solar panels (Figure 6b). Unlike the posterior fragility function for rooftop panels, the 300
posterior fragility function for ground-mounted panels has a higher wind value (+10%) for a 50%-failure 301
probability than its prior, 90 m/s versus 81m/s. This increase suggests that the panel installations for ground-302
mounted solar panels were structurally sounder than for rooftop panels, whose wind for 50%-failure 303
probability in the posterior was 5% less than in the prior. This better structural performance may result from 304
more code enforcement, better member and connection installation (e.g., avoiding loose bolts), or proper 305
inspections (Burgess et al., 2020; Burgess & Goodman, 2018). These panels are part of large installations 306
with massive investments from utility companies, which, unlike residential homes that install rooftop 307
panels, often have a budget for appropriate quality and control. 308
We found that the wind range that transitions from a 10% to 90% failure probability in the posterior, 73 309
and 116 m/s, is reduced in 64% from the prior, 41 m/s and 160 m/s. This narrower range is driven mainly 310
by the lower standard deviation in (Figure 5c). This reduction in the transition range is larger than that in 311
the case of the rooftop panels (Figure 6) because, unlike the rooftop panels, the ground-mounted panels’ 312
posterior of did not have a larger standard deviation than the prior. Furthermore, the posterior fragility 313
function shows a much narrower confidence interval than the prior fragility function. However, the 314
confidence interval is slightly wider than in rooftop panels because the ground-mounted panel dataset is 315
only a third of the rooftop panel dataset. 316
5. PANEL’S ANNUAL FAILURE RATE 317
To illustrate their application, we use our fragility functions to assess solar panel risk for hurricane winds 318
for Miami-Dade, Florida, as a case study. Miami-Dade is exposed to similar wind hazards in Puerto Rico. 319
For example, the risk category II design wind (700-year return period) in San Juan, Puerto Rico, is 71 m/s 320
(159 mph), whereas the design wind in Miami-Dade is 73 m/s (164 mph). Different standards for solar 321
panel installation and code enforcement might be in place in Miami-Dade, especially for rooftop panels, 322
which performed worse than ground-mounted panels. However, more data collection efforts will be needed 323
to confirm whether panels in mainland United States have fundamentally different structural behavior than 324
those in the Caribbean. Due to the lack of these datasets, here we use our fragility functions from the 325
Caribbean to study solar panels’ reliability and resilience performance in Miami-Dade; analysis for other 326
regions can be similarly performed. 327
A study site in the mainland United States is chosen to leverage a synthetic hurricane database with 5018 328
landfalling storms in the United States generated from a statistical-deterministic tropical cyclone (TC) 329
model (Marsooli et al., 2019). These synthetic hurricanes account for current climate conditions (from 1980 330
to 2005) according to the National Center for Environmental Prediction (NCEP) reanalysis. The 5018 331
synthetic storms correspond to ~1485 years of storm simulation. The model that generates these storms 332
consists of three stages: a genesis model; a beta-advection TC motion model; and a dynamical TC model 333
that captures how environmental factors influence TC development (Emanuel et al., 2008). The model 334
solves the synthetic storms’ tracks, maximum sustained winds, and radii of maximum winds, and we use 335
its results at 2-hour intervals. We estimated the wind fields by combining the storm’s axisymmetric winds 336
circulating counterclockwise (Chavas et al., 2015) and background winds (Lin et al., 2012). The synthetic 337
hurricanes were evaluated with observations by Marsooli et al. (2019). 338
We determine the annual rate of panel failure by combining the wind simulations with the Bayesian 339
fragility functions. The rate defines the average number of events leading to panel failures in a given year 340
assuming a Poisson process. In a frequentist analysis, the fragility function parameters and are fixed. 341
Thus, (,) can be estimated as 342
(,) = (;,)
(11)
where is the annual exceedance probability of wind speed. It is the average number of events that result 343
in winds exceeding a threshold in a given year under a Poisson process of storm arrivals, and it can be 344
estimated from the synthetic storms. In our Bayesian framework, and are random variables. Thus, is 345
also a random variable. Accordingly, its probability density function p() can be found as 346
p() = (,|) (;,)
(12)
where () is the Dirac delta function on (;,)
. Equation (12) uses the posterior (,|) 347
as the distribution of and to find the posterior of p(). Replacing (,|) by the prior (,) will 348
result in the prior p(). The expected value of , E[], can be found as: 349
E[] = (;,)
(,|)
(13)
Explicitly evaluating E[] and particularly p() is computationally challenging by traditional numerical 350
integration. Thus, we used Monte Carlo analysis due to its simplicity to find such estimates. Using the 351
10,000 Monte Carlo samples of prior and posterior fragility functions, we estimated the prior and posterior 352
of (Figure 7). 353
a) Rooftop panel
b) Ground-mounted panel
Figure 7. Probability density function p
() of the annual probability of failure rate of solar panels.
Samples from the Monte Carlo simulations were used to fit empirical pdfs with a Gaussian kernel.
Our results indicate a marked decrease in uncertainty for in the posterior. The posterior standard 354
deviation and logarithmic standard deviation for rooftop panels are 1.2 × 10/yr and 6.3 × 10 , 355
whereas the priors’ ones are 5.1 × 10/yr and 1.82. The posterior standard deviation and logarithmic 356
standard deviation for ground-mounted panels are 1.7 × 10/yr and 5.5 × 10, whereas the priors’ ones 357
are 5.7 × 10/yr and 1.87. This uncertainty decrease in the annual failure rate is consistent with the 358
observed posterior fragility function uncertainty reductions for rooftop and ground-mounted panels (Figure 359
6). 360
For rooftop panels, the posterior E[] is 1.3 × 10/, i.e., return period of 75 years. Under the 361
assumption of a Poisson process, this rate results in a 48% probability of failure in 50 years. This rate is 362
equivalent to a 33% failure probability in 30 years, often considered the usable panel service time. The 363
reliability index, defined as the inverse of the cumulative standard normal distribution function on the 364
survival probability, i.e., one minus the failure probability, in 50 years, is 0.04. This reliability is 365
significantly lower than the current standards in the ASCE7-16. Using results from a recent study 366
(McAllister et al., 2018), we estimated that a structure designed for winds with a 700-year return period 367
(risk category II) should have a reliability index of 2.3 in 50 years, i.e., failure rate of 2.3 × 10/. Thus, 368
our findings show that the structural reliability of rooftop solar panels in our dataset was significantly below 369
current code standards if similar panels are adopted in Miami-Dade. These results are consistent with the 370
observed structural deficiencies in the installation and design of panels with failures in the dataset, e.g., 371
insufficient connection strength, lack of vibration-resistant connections (Burgess et al., 2020). Thus, 372
significant gains in reliability could be achieved by increasing quality and control during design and 373
installation. 374
For ground-mounted panels, the posterior E[] is 2.0 × 10/, i.e., return period of 504 years. This rate 375
is equivalent to a 9% and a 6% probability of failure in 50 and 30 years, respectively. The reliability index 376
for 50 years is 1.3. According to the ASCE7-16, the reliability index for a structure designed for winds with 377
a 300-year return period (risk category I) is 1.9, i.e., failure rate of 6.1 × 10/ (McAllister et al., 2018). 378
Thus, our results indicate that ground-mounted panels also have lower reliability than required by the 379
current code standards. These results are also consistent with previously reported structural deficiencies in 380
ground-mounted panels in the Caribbean, e.g., undersized racks, and undersized or under-torqued bolts 381
(Burgess & Goodman, 2018). Nevertheless, the contrast between rooftop and ground-mounted panel 382
performance indicates that the latter had a significantly higher structural reliability than the former. 383
6. STRONGER SOLAR PANELS FOR GENERATION RESILIENCE 384
385
6.1. Assessing structural reliability and generation in stronger panels 386
We assessed panels’ strength increases by factors of 1.25, 1.50, 1.75, and 2.0. This wide range of strength 387
increases accounts for addressing various panel installations and design deficiencies reported in the 388
Caribbean. Existing studies already point to cost-effective solutions to correct these deficiencies, e.g., 389
torque checks on bolts, well-designed clips (Burgess et al., 2020; Burgess & Goodman, 2018). 390
This range also covers increases in strength for critical infrastructure. Hospitals and fire stations require 391
that their buildings’ structural and non-structural components have higher strength for continuous 392
operations in a disaster emergency response. Accordingly, solar panels serving these facilities must be 393
designed with a risk category IV, higher than for panels on residential (risk category II) or utility-scale (risk 394
category I) installations. For example, the wind design in Miami-Dade is 69 m/s (154 mph) for a risk 395
category I and 81 m/s (182 mph) for a risk category IV. The difference represents a strength factor of 1.40 396
as the design force is proportional to the square of the design wind. 397
For our assessment, we multiplied the posterior samples of by the square root of the strength increase 398
factors, i.e., 1.12, 1.22, 1.32, 1.41. We let samples remain the same to limit the increase in uncertainty, 399
i.e., the transition from low-failure-probability to high-failure-probability winds. The resulting fragility 400
functions are shifted to the right of the posterior functions in Figure 6, reducing the likelihood of panel 401
failure (Figure 8). For example, the mean failure probability when rooftop panels undergo gusts of 60 402
m/s decreases from 0.19 to 0.12, 0.08, 0.05, and 0.04 for the strength factors of 1.25, 1.50, 1.75, and 2.0, 403
respectively. Similarly, the mean when ground-mounted panels undergo gusts of 80 m/s decreases from 404
0.23 to 0.09, 0.04, 0.02, and 0.01. 405
406
407
a) Rooftop panel
b) Ground-mounted panel
Figure 8. Mean fragility functions for panels with increases in strength. The factors that multiply each
sample are equal to the square root of the strength factors in the labels. The dashed curves indicate
the wind annual exceedance rates. The x-axis represents 3-s gusts.
Using Monte Carlo sampling, we estimated p() for the different increases in strength (Figure 9). 408
Expectedly, increases in strength shift p() to the left as they reduce the resulting annual rate of failure. 409
We also found E[] and assessed the corresponding panels’ structural reliability (Table 1). The increases 410
in strength are effective at decreasing E[]. The strength factor of two reduces E[] by a factor of 3.9 and 411
2.5 for rooftop and ground-mounted panels, respectively. A more modest strength factor of 1.25 also 412
effectively decreases panel failure risk, reducing E[] by ~50% and ~70% for rooftop and ground-mounted 413
panels, respectively. Nevertheless, our results indicate that the reliability indexes for these stronger panels 414
are still below the ASCE7-10 targets even for a risk category I, i.e., 1.9. 415
Table 1. Annual probability of panel failure and reliability indexes (for 50
years) for different increases in strength
Strength Factor
Rooftop panel
Ground-mounted panels
E[
] (1/yr)
Reliability
index
E[
] (1/)
Reliability
index
1.0
0.0132
0.04
0.0020
1.30
1.25
0.0089
0.36
0.0012
1.58
1.50
0.0061
0.63
0.0010
1.66
1.75
0.0043
0.87
0.0009
1.73
2.0
0.0034
1.01
0.0008
1.77
These results highlight large structural vulnerabilities in solar panels since they do not reach code-level 416
reliability even if their strength is increased twice. These results suggest that existing lack of structural 417
design and limited inspections in the panel installations were significant (Burgess et al., 2020; Burgess & 418
Goodman, 2018). High vulnerability to hurricane winds has been noted previously in buildings. For 419
example, a previous study in Southern Florida determined that roof-to-wall connections with 3-8d toenails 420
in wooden residential buildings have an annual failure rate between 0.005-0.024 (Li & Ellingwood, 2006). 421
These rates are comparable to the rooftop panels in our case study and below the performance of ground-422
mounted panels (Table 1). Furthermore, roof panels with 6d nails @ 6/12 in. on these buildings showed 423
even poorer performance, with higher annual failure rates of 0.077-0.137. 424
425
a) Rooftop panel
b) Ground-mounted panel
Figure 9. Probability density function of the annual failure rate of solar panels for different increases
in panel strength. The labels indicate the strength factor increase.
426
6.2 Will stronger panels increase generation resilience? 427
428
As demonstrated previously, increasing panel strength will increase its reliability. However, other critical 429
factors also play a significant role in solar generation resilience, i.e., the ability to generate sufficient 430
electricity during storms. First, solar generation can decrease even if panels remain structurally sound and 431
functional during a hurricane. Ceferino et al. (2021) demonstrated that hurricane clouds can reduce 432
irradiance and generation significantly through light absorption and reflection. For example, a category-4 433
hurricane can decrease the generation by more than 70%, even if the panels remain undamaged. Cloud-434
driven generation losses can last for days, although they will bounce back to normal conditions in an 435
undamaged panel as the hurricane leaves. 436
Failure of supporting infrastructure can also decrease generation resilience even if panels withstand extreme 437
wind loads. Increasing the strength of rooftop panels on vulnerable roofs will not increase the global 438
reliability of the residential energy system. Global reliability must consider that panels can fail in a 439
cascading failure triggered by roof uplift, damaging the panel or its connections. The weakest link will 440
control the reliability of this in-series system. As mentioned previously, roof-to-wall connections with 3-441
8d toe nails or roof panels with 6d nails @ 6/12” exhibited similar or poorer performance than vulnerable 442
rooftop panels (Li & Ellingwood, 2006). Strengthening panels on these roofs will substantially increase 443
their local reliability (Table 1), but it will increase global reliability only negligibly. Conversely, roofs with 444
H2.5 hurricane clips in roof-to-wall connections and 8d nails @ 6/12'' in roof panels will make roofs an 445
appropriate supporting system through higher reliability (Li & Ellingwood, 2006). Thus, our results 446
advocate for stronger panels but under a holistic assessment of global reliability. 447
Structurally sound rooftop panels have the intrinsic advantage of delivering power even if the primary grid 448
is down. When inverters are within buildings, occupants can use their locally generated energy during an 449
outage (Cook et al., 2020). Access to power can be vital for residential buildings, especially if heatwaves 450
following storms increase the demand for cooling (Feng et al., 2021). Access to energy is also pivotal to 451
sustaining emergency response operations for critical infrastructure such as hospitals or fire stations. 452
Communities can further utilize locally generated energy through energy sharing and microgrids to increase 453
households’ access to power after a disaster, even for those who did not install panels (Ceferino et al., 2020; 454
Patel et al., 2021). Nevertheless, solar panels will not replace the need for backup generation units for 455
resilience, especially for critical facilities, and fully charged behind-the-meter batteries must complement 456
them for power access during an emergency response. 457
Stronger panels will also increase power security at the utility level by avoiding massive structural failures 458
at the generation sites, as in Figure 1b. As noted previously, solar panels are directly exposed to wind. Poor 459
structural performance in utility companies’ solar installations could result in significant generation losses 460
and outages that can affect the disaster emergency response and recovery activities. Recently, Hurricane 461
Ida caused damage to the power system that resulted in ~1M outages in Louisiana, reducing electricity 462
access by more than 60% in more than ten parishes (counties), critically affecting the functionality of the 463
water system and delaying recovery (J. D. Goodman et al., 2021; Prevatt et al., 2021). While solar 464
generation losses could be potentially offset by other generating sources during an emergency response, 465
adopting vulnerable panels in our grid will be a missed opportunity to make our power systems resilient. 466
Solar is projected to be an important generation source in our future grids. Simultaneously, hurricanes are 467
projected to be more intense in the future climate (Knutson et al., 2020). Governments invest massively to 468
redesign the grid and transition to cleaner energy (The International Renewable Energy Agency, 2018). 469
Thus, our results advocate for governments to leverage this unique opportunity to change the grid’s risk 470
trajectory course by strengthening the infrastructure that will provide our future communities with energy 471
safety. 472
7. CONCLUSIONS 473
This paper presented the first data-driven fragility curves for solar panels under hurricane wind loads. The 474
article estimated the fragility curves using data on the structural performance of 46 rooftop panels in 475
residential buildings and 14 large ground-mounted solar panel arrays in utility generation sites. Solar panel 476
failure data was collected after Hurricanes Maria and Irma in 2017 and Hurricane Dorian in 2019 in the 477
Caribbean. Further, this paper assessed solar generation resilience and its improvements with stronger 478
panels. 479
We used a Bayesian approach to supplement the panel dataset with an existing numerical assessment of 480
panel failure. Using a Markov Chain Monte Carlo algorithm, we estimated the posterior distributions of 481
fragility parameters for the rooftop and ground-mounted panels separately. Our results show significant 482
reductions in epistemic uncertainty for (wind for a 50%-failure probability) in rooftop and ground-483
mounted panels with 90% and 87% decreases in the standard deviation. Using Monte Carlo, we then 484
propagated the uncertainty in the parameters to the fragility functions, showing significantly narrower 485
confidence intervals. This result highlighted the importance of characterizing fragility functions with 486
ground-truth data. 487
We combined our fragility functions with a hurricane hazard assessment in Miami-Dade, Florida, using 488
Monte Carlo simulations. Miami-Dade has similar hurricane hazards to Puerto Rico, where most damage 489
data was collected. Our estimates of the annual rate of panel structural failure indicated that the panels are 490
below the current structural reliability standards specified in ASCE7-16. These performance deficiencies 491
were particularly striking for rooftop panels (estimated failure rate of 1.3 × 10/ versus 2.3 × 10/ 492
in the code), whose documented installation issues and frequent lack of structural design made them 493
particularly vulnerable to high winds. 494
Finally, we analyzed the implications of building stronger solar panels by up to a factor of two due to 495
improvements in the panels’ installations, structural design, or higher structural requirements. We show 496
that increasing panel strength effectively reduces the annual failure rate. However, even the factor of two 497
is still insufficient to meet annual failure rates in the ASCE7-10 (reliability index of 1.9 for the lowest risk 498
category) for rooftop and ground-mounted panels (reliability indexes of 1.01 and 1.77). 499
As we transition towards cleaner energy sources and solar generation becomes an essential component of 500
our grid, ensuring its resilience is critical for our communities. Our paper argues that increasing panels’ 501
structural strength has critical implications for enhancing generation resilience during extreme storms. In 502
the context of growing hurricane hazards due to climate change, panels must at least meet existing code 503
structural performance standards. However, we also discuss that generation losses might arise even if 504
panels can sustain high wind speeds. Thus, we point out different plans, such as using backup power, 505
behind-the-meter storage, or sharing energy, to address such losses during hurricane emergencies in order 506
to sustain a proper response to hurricanes. 507
8. AKNOWLEDGMENTS 508
We thank Sanya Detweiler from the Clinton Foundation, Chris Needham and Solar Frank Oudheusden from 509
FCX Solar, and Christopher Burgess from the Rocky Mountain Institute for sharing their reports, photos, 510
and relevant documentation on solar panels’ structural performance after Hurricanes Maria and Irma in 511
2017 and Dorian in 2019. We also acknowledge the financial support by the Andlinger Center for Energy 512
and the Environment at Princeton University through the Distinguished Postdoctoral Fellowship. 513
Additionally, this research was also supported by the NSF Grant 1652448. The authors are grateful for their 514
generous support. 515
9. SUPPLEMENTARY INFORMATION 516
Supplementary Tables 1 and 2 and Supplementary Figure 1 to 4 can be found at: 517
https://tinyurl.com/mw224py5 518
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