# Optimal Storage for Solar Energy Self-Sufficiency

Anders E. Carlsson  Department of Physics, Washington University, St. Louis,
Missouri  S. Redner  Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New
Mexico, 87501

###### Abstract

We determine the energy storage needed to achieve self sufficiency to a given
reliability as a function of excess capacity in a combined solar-energy
generation and storage system. Based on 40 years of solar-energy data for the
St. Louis region, we formulate a statistical model that we use to generate
synthetic insolation data over millions of years. We use these data to monitor
the energy depletion in the storage system near the winter solstice. From this
information, we develop explicit formulas for the required storage and the
nature of cost-optimized system configurations as a function of reliability
and the excess of generation capacity. Minimizing the cost of the combined
generation and storage system gives the optimal mix of these two constituents.
For an annual failure rate of less than 3%, it is sufficient to have a solar
generation capacity that slightly exceeds the daily electrical load at the
winter solstice, together with a few days of storage.

Moving away from fossil fuels to renewable energy is a crucial step to
minimize the extent of global warming. Because renewable energy sources, such
as wind and solar, are intermittent, achieving a 100% renewable scenario
requires either a large excess generation capacity, a substantial amount of
storage, or a judicious mixture of the two. Understanding the nature of is
tradeoff between excess capacity and storage is crucial for the design and
optimization of effective renewable energy systems. Understanding the factors
that determine the tradeoff will improve our grasp of the right balance
between the uncertain costs of generation and storage in the future.

This tradeoff is characterized by two fundamental parameters: the _generation
factor g_ , the ratio of the average annual generation capacity to the annual
load, and the storage capacity $S$, the number of days of electrical load that
reside in a storage system. Various simulation studies have given scattered
values for the optimal mix of $g$ and $S$ values, with little understanding of
how they depend on physical parameters. Ref. Shaner et al. (2018) examined
combined energy generation and storage systems in the US and in specific
subregions, with both wind and solar generation. For solar-only generation, a
system with $g\approx 2.1$ and a storage equivalent $S$ of 4 days of load was
virtually 100% reliable, defined as the fraction of total energy demand that
was met by renewables plus storage. However, equally high reliability was
obtained with $g\approx 1.3$ and a month of storage. Ref. Heide et al. (2010)
focused on the case $g=1$, with a mix of solar and wind energy and found that
a storage $S$ of 1.1–2.5 months was required.

Ref. Tong et al. (2020) developed optimized energy systems for the continental
US over a range of storage costs, using the same underlying model as in Ref.
Shaner et al. (2018). For inexpensive storage, they found $g\simeq 2.2$, while
more expensive storage required an increase in the capacity to $g\simeq 2.7$.
Concomitantly, the amount of storage dropped from about 5 days of load to 1
day. Ref. Budischak et al. (2013) determined optimal solutions for a power
network in the eastern US with disparate storage modalities. The requisite $g$
values were in the range of 2.5–2.9 and $S$ between 0.3–3 days, depending on
the type of storage. Related studies Heide et al. (2011); Rasmussen et al.
(2012); Jacobson et al. (2015); Cebulla et al. (2017) added easily
dispatchable renewable sources, such as hydroelectric power, which reduced the
required storage.

Given the range of the these predictions about optimal configurations, a need
exists for an analytical theory that would: (a) clarify the relation between
input physical parameters and the performance of a combined generation/storage
system, and (b) help constrain the parameters of this system to guide the
realm of feasibility. In this work, we construct such a theory that is based
on an idealized, but general model that faithfully incorporates the actual
solar irradiation statistics, including seasonality and day-to-day
correlations. This theory allows us to specify the nature of an optimal
generation/storage system and make explicit predictions about its cost and
reliability. Although optimal systems will in general include both wind and
solar energy, we treat only solar energy in order to obtain a theoretically
tractable model. We believe that the general features of our results will hold
for mixed systems as well.

Our model extends previous analytical theories based on simplified solar
irradiation statistics. Ref. Gordon and Zoglin (1986) assumed a deterministic
day-night profile, while ignoring daily and seasonal fluctuations. Refs.
Bucciarelli Jr (1984, 1986); Gordon (1987) included daily, but not seasonal
weather variations, and day-to-day correlations in some cases. They found that
the failure probability decays exponentially with increasing storage capacity,
and Ref. Gordon (1987) gave explicit formulas for the storage capacity
required to achieve a given reliability. Ref. Markvart (1996) included the
effects of seasonality in generation and/or load but did not treat random
weather fluctuations. Ref. Egido and Lorenzo (1992) used an empirical fit to
reliability simulations based on historical weather data (including
seasonality), and found an exponential relationship between generation and
storage. However, a principled theory that quantitatively treats the
combination of stochastic daily weather fluctuations, day-to-day correlations,
and seasonality does not yet seem to exist.

We begin by first outlining basic features of the solar-flux data for the St.
Louis region, which typifies those of the entire US. We then introduce our
data-driven model and use it to develop analytic formulas for the failure rate
and storage capacity needed to achieve a given reliability. We use these to
calculate the generation and storage capacities of a combined system that
minimizes the cost and yet is extremely reliable. We verify our predictions
based on simulations of millions of years of synthetic data.

##  I Empirical Background

###  I.1 Solar Flux Data

To illustrate the issues and as a preliminary to develop our model, we first
present and analyze data for the solar flux on a 270 km $\times$ 270 km region
centered on St. Louis over the 40-year period 1980–2019. This region is large
enough that its energy needs can be met by covering a small fraction of the
total land area with solar panels, but small enough that power transmission
across the region is nearly lossless and instantaneous. The solar data, from
the MERRA-2 dataset Molod et al. (2015a), is in the form of the energy flux
for each hour of the day from 1980–2019 (see Appendix B). We determine the
daily incident energy per unit area by multiplying each hourly energy flux by
the number of seconds in an hour, and then adding these values over a single
day. This gives an average daily solar energy per unit area that ranges
between roughly 8–25 MJ/m2 from the winter minimum to the summer maximum, with
daily extrema of 1.53 MJ/m2 and 32.1 MJ/m2 over the 40 years of data (Figs. 1
and 2). For simplicity in our analysis, the data for February 29 in leap years
are dropped, so that our results are based on the 40-year period 1980–2019 in
which all years consist of 365 days.

Figure 1: Average daily energy $\langle E\rangle$ per unit area from 1980–2019
on the St. Louis region (black), and the daily standard deviation
$\langle\sigma\rangle$ in this quantity over the same period (red). These data
are smoothed by averaging over 45 consecutive days.

The average daily solar energy is roughly sinusoidal, with the maximum at day
200 (July 18, nearly one month after the summer solstice) and the minimum at
day 357 (December 23, a few days after the winter solstice). The standard
deviation in the daily solar energy also has a systematic time dependence that
ranges between 1.5–5.5 MJ/m2, with maximal fluctuations occurring in the early
spring. Near the winter solstice, the magnitude of the fluctuations is about
35% of the mean value. On the minimum-insolation day, $E\approx 7.95$ MJ/m2
and $\sigma\approx 2.79$ MJ/m2. These numbers, which will play a central role
in our ensuing analysis, are based on averaging the daily energy data over a
45-day window.

Figure 2: Insolation fluctuations. The black curve is smoothed 45-day average
daily energy per unit area on the St. Louis region, 1980–2019 and the one
standard deviation range is dashed curve. Also shown is the daily data for
1980 (blue) and a typical realization of our synthetic data (red). The upper
dashed blue line corresponds to the load with $f\\!=\\!1$ (see text for the
definition of $f$) and the lower corresponds to $f\\!=\\!1.5$. The difference
between the daily data and the dashed lines gives the daily energy
surplus/deficit for these two $f$ values.

To illlustrate the influence of fluctuations on the daily insolation data,
Fig. 2 shows the daily solar flux on the St. Louis region for the single year
1980. For later convenience in our theoretical and numerical modeling, the
time origin has been shifted so that the year begins on July 1. The basic
feature of these data is that daily solar fluctuations significantly perturb
the average annual cycle. Around the winter solstice, which is the most
crucial time of the year for the reliability of a combined solar generation
and storage system, the minimum solar insolation is roughly a factor 5 less
than the _average_ maximum solar insolation. If the storage system is nearly
depleted near the winter solstice, multiple overcast days can quickly lead to
a system failure. Thus day-to-day fluctuations in solar flux play an important
role in determining the optimal tradeoff between generation and storage.

###  I.2 Generation and storage costs

Our determination of the optimal system configuration is based on two key
costs: the cost $C_{g}$ of the generation capacity to supply the daily
electrical energy load $L$ at the winter solstice (based on the average
insolation on that day), and the cost $C_{s}$ of energy storage to cover one
day of electrical load. This daily load of the St. Louis region is111This is
obtained from the continental US yearly electricity consumption of 4$\times
10^{12}$ kWh (https://www.statista.com/statistics/201794/us-electricity-
consumption-since-1975/), dividing by 365 to get consumption per day and then
by 100 (taking the St. Louis region to contain about 1% of the US population)
roughly $4\times 10^{14}$ J, or $1.1\times 10^{8}$ KWh. This corresponds to an
average power usage of 4.6 $\times 10^{6}$ kW.

It is conventional to express the cost of solar panels in dollar per watt.
Using the current solar panel cost of $1.50/W Energysage (2020), the cost of
generation is thus $C_{g}\approx$ $75 billion. This cost grows roughly
linearly with the area of the solar farm222There is little economy of scale
for a large solar farm Renewable Energy World (2015). While the cost per solar
panel decreases as the number of installed panels increases, there are
additional costs associated with transmitting solar power from the farm to end
users. These transmission costs largely negate the installation economy of
scale; such costs do not exist for rooftop solar panels for household use..
For 20% efficient solar cells (close to the best that are currently available
Solar Reviews (2020)), the required solar farm area is $4\times
10^{14}\text{J}/(0.20\times 7.95\times 10^{6}\text{J}/\text{m}^{2})\approx
2.5\times 10^{8}$ m${}^{2}\equiv A_{0}$. This roughly corresponds to a 16 km
$\\!\times\\!$ 16 km square. The cost of a solar farm of area $fA_{0}$ will
therefore be $f\,C_{g}$. The excess generation capacity, $(f-1)L$, is a
fundamental metric of the generation system. Using the current price of
$1.25/m2 AG Web (2019) for rural land in the region, the land cost of the
solar farm is approximately $300 million; this is negligible compared to the
solar panel costs and will be ignored.

The cost to store one day of electrical energy load for the St. Louis region
at the current price of $200/KWh is $C_{s}=1.1\times
10^{8}\text{kWh}\times\$200/\text{kWh}\approx$ $22 billion Ziegler et al.
(2019). It is convenient to measure the capacity $S$ of the storage system in
units of the daily electrical energy load in the St. Louis region. We define a
storage system of capacity $S$ as one that can supply $S$ days of electrical
load to this region. The cost of this storage system therefore is $C_{s}S/L$.

Since roughly 60% of a 24-hour period is dark at the winter solstice in the
St. Louis region and total electrical energy use is roughly time independent
in the winter US Energy Information Administration (2020a, b), there is a
baseline storage need of 60% of the daily load to cover the energy use when it
is dark. If there were no day-to-day fluctuations in the solar flux, this
baseline storage, together with the solar energy gathered during the day by a
solar farm of area $A_{0}$ could fully supply the regional electrical energy
needs during a 24-hour period at the solstice, and thus throughout the year.

The existence of insolation fluctuations has several essential consequences.
First, the optimal area of the solar farm must be larger than $A_{0}$ and the
storage capacity must be larger than the 60% of daily energy use that is
needed to deal with the regular diurnal fluctuations. Second, we will see that
it is impractical to achieve 100% reliability with this combined solar
generation and electrical storage system. Thus it is necessary to balance the
tradeoff between reliability and cost. Establishing how generation capacity
and storage combine to achieve a given reliability, and understanding the
tradeoff between reliability and cost, are primary goals of this work. We will
find that the optimal cost system configuration is determined by the ratio of
storage to generation costs, $C_{s}/C_{g}$. The above numbers give roughly 0.3
for this ratio. Since storage costs are rapidly decreasing Ziegler et al.
(2019), we will explore the consequences of potential future storage cost
reductions by up to a factor of 7.

##  II Synthetic Data and Simulations

Because of the substantial day-to-day fluctuations in the solar flux, the 40
years of available data are too sparse to determine the reliability of a
combined solar farm/storage system with statistical significance. To formulate
a generally applicable theory, we first construct synthetic daily insolation
data that faithfully incorporates the annual trends, the daily fluctuations,
and the day-to-day correlations that are present in the solar flux data for
the St. Louis region. The simple and direct algorithm that we use to construct
these data allows us to readily generate time series for millions of years.
From these, we obtain statistically meaningful results about the reliability
and cost of a combined solar power generation and storage system.

To construct the synthetic data, we require two additional features beyond the
average daily incident energy and its standard deviation: (a) the distribution
of energy for each day of the year, and (b) the day-to-day energy
correlations. The energy distributions away from the winter solstice are
irrelevant when $f>1$ because there will be ample solar energy plus stored
energy to meet the daily load on any given day that is not near the solstice.
It is only near the winter solstice that the daily energy distributions become
relevant. However, 40 years of data are too sparse to accurately represent
these distributions. To obtain daily energy distribution data of reasonable
quality, we aggregate these distributions over symmetric time ranges of 15,
31, and 45 days around the minimum solar-energy day (day 357). These
distributions are nearly the same for the three time ranges (Fig. 3(a)); this
justifies using a universal shape for the daily energy distribution near the
winter solstice. For simplicity, we replace the actual and somewhat
triangular-shaped distribution by a uniform distribution whose width is chosen
to be the same as that of the data.

Figure 3: (a) The probability distribution $P(E)$ for the daily energy per
unit area over symmetric time intervals around the minimum-energy day (day
357). (b) Probability $C(n)$ that the ratios $r_{j}\equiv E_{j}/\langle
E_{j}\rangle$ are all greater than or all less than 1 over $n$ consecutive
days. Also shown is the exponential best fit to the data in the range $n\leq
13$, $C(n)\propto q^{n}$, with $q=0.6157$.

There are also day-to-day correlations in the energy flux that reflect the
well-known feature that the weather on consecutive days is more likely to be
similar than different Bucciarelli Jr (1986). To quantify these correlations,
we start with the 40-year sequence of normalized daily energies
$\\{r_{j}\equiv E_{j}/\langle E_{j}\rangle\\}$, where $E_{j}$ is the energy
per unit area on the $j^{\text{t}h}$ day of the year and $\langle
E_{j}\rangle$ is its average, with $j$ ranging333There is a small error in the
correlation function because we drop the data for the 10 leap days. from 1 to
14600 ($40\times 365$). We first determine the length of strings of
consecutive days for which the ratios $r_{j}$ are either all greater than 1 or
all less than 1. We then obtain the probability distribution $C(n)$ for the
number of consecutive days $n$ where all the $r_{j}$ are greater than 1 or
less than 1.

In the absence of correlations in the daily solar flux, the string length
distribution would decay in $n$ as $C(n)=(1/2)^{n}$. However, the actual
correlations decay as $q^{n}$, with $q\approx 0.6157$ over the range of 1–16
days (Fig. 3(b)). Beyond 16 days, the correlations decay more slowly still.
However, the frequency of such long strings of 16 days or longer is roughly
once every 6 years. In generating our synthetic data, we ignore these
extremely rare events and use the simple exponential decay $C(n)\propto q^{n}$
for all $n$.

It is convenient to shift the time origin so that the year begins on July 1.
We first determine the solar energy on July 1 (now day 1). This energy is
given by a random number that is uniformly distributed in the range
$[E_{1}\\!-\\!\sqrt{3}\,\sigma_{1},E_{1}\\!+\\!\sqrt{3}\,\sigma_{1}]$, where
$E_{1}$ is the average energy and $\sigma_{1}$ is the standard deviation on
July 1 (see Fig. 1). The factor $\sqrt{3}$ ensures that the standard deviation
that arises from the uniform distribution matches that of the actual data. For
each successive day $j$, the solar energy is again chosen uniformly from the
range $[E_{j}\\!-\\!\sqrt{3}\,\sigma_{j},E_{j}\\!+\\!\sqrt{3}\,\sigma_{j}]$,
but augmented by the following construction to incorporate the energy
correlations. The sense of the deviation from the average energy on the
current day $j$ (either $E_{j}/\langle E_{j}\rangle>1$ or $E_{j}/\langle
E_{j}\rangle<1$) is the same as the sense of the deviation on the previous day
with probability $q$. This persistent random-walk construction Weiss and Rubin
(1983); Weiss (1994) ensures that the string length distribution
asymptotically decays as $q^{n}$, as in Fig. 3(b). The synthetic data
accurately mimic both the annual variation, as well as the day-to-day
fluctuations of the incident energy, as illustrated by a typical realization
of synthetic daily energies in Fig. 2.

Figure 4: Schematic and not to scale dependence of the daily energy minus the
load near the winter solstice (blue curve), with three periods of below
average insolation ($a,c,e$) and two above-average periods ($b,d$). The extent
of the energy deficits and surpluses are shown by the blue and red shaded
areas. The green curve indicates the instantaneous storage $s(t)$ and the
green dotted line indicates full storage.

With this computational approach, we generate millions of years of synthetic
insolation data over a two-dimensional mesh of thousands of $(f,S)$ values. We
start with a full storage system, that is, $s=S$ on July 1. For a given pair
$(f,S)$, the stored energy $s_{j}$ on the $j^{\text{th}}$ day of the year is a
random variable that changes daily according to $s_{j+1}=s_{j}+E_{j}-L$, where
$L$ is the daily load, subject to the constraint that $s_{j}$ can never exceed
$S$ (Fig. 4). Since the model treats only the total energy in a day, it does
not include the diurnal variation mentioned in Sec. I.2 that will require an
additional constant storage requirement of $0.6L$.

The time evolution in the model defines a biased random-walk-like process on
the interval $[0,S]$, in which the bias corresponds to the difference between
the daily insolation and the daily load, and the day-to-day insolation
fluctuations correspond to random noise. System failure occurs whenever
$s_{j}$ reaches zero. The failure probability $\varepsilon$ is defined as the
fraction of years for which failure occurs. The cost-optimized system
configuration is defined as the pair of $(f,S)$ values that has the lowest
cost and a failure probability less than a specified value.

##  III Generation/Storage Tradeoff

Due to fluctuations in daily insolation, even a system with $f>1$ will be
insufficient to supply the electrical load unless storage is included. We will
construct a theory to determine the range of possible mixes of generation and
storage that achieve a given reliability. We treat only the constraints that
arise from storage-capacity limitations and not from power-delivery
limitations. We also assume 100% efficient storage, perfect power transmission
across the region, and a constant daily load.

The stylized time history of the insolation and stored energy near the winter
solstice (Fig. 4) also illustrates the tradeoffs involved in optimizing the
combined system. In this figure, the energy deficit during period $a$ is
larger than the surplus in the following period $b$. Thus full storage on the
$j^{\rm th}$ day is only partially replenished in period $b$. Conversely,
while the storage is fully replenished in period $d$ with a large solar
surplus, some of this surplus is wasted because of the limited storage
capacity (indicated by the cutoff in the red area). The optimal storage system
should maximize the energy returned to storage during surplus days near the
winter solstice, while minimizing cost.

###  III.1 Stored energy distribution

For a solar farm of area $A_{0}$, insolation tends to replenish the storage
during most of the year; this time range corresponds to what we term the
_strong-bias_ regime. Conversely, for an average day near the winter solstice,
the insolation roughly matches the load, so that the state of the storage
system change only slightly from day to day. We term this time range as the
_weak-bias_ regime. In an optimal design, the storage system should be nearly
depleted through the winter solstice. Otherwise, excess unused storage
capacity exists that increases the system cost without meaningfully increasing
its reliability.

For each day of the year, there is a day-specific average distribution of
energy in the storage system. We will determine these daily distributions over
a period around the winter solstice. From these distributions, we will
determine how the annual failure probability $\varepsilon$ depends on $f$ and
$S$. Sec. IV uses these relations to calculate the necessary storage and
generation capacity in a cost-optimized system configuration.

To compute the stored energy distribution on a single day, we first treat the
idealized situation of a strong and time-independent bias. Based on the biased
random-walk picture described above, the distribution of stored energy on the
$j^{\text{th}}$ day of the year, $P_{j}(s)$, attains the steady-state form

$\displaystyle P_{j}(s)=\mathcal{N}\big{(}e^{\lambda_{j}s}-1\big{)}\,,$ (1)

with normalization constant
$\mathcal{N}=\lambda_{j}/(e^{\lambda_{j}S}-1-\lambda_{j}S)\approx\lambda_{j}\,e^{-\lambda_{j}S}$
determined by requiring that $P(s)$ integrates to 1. A salient feature of Eq.
(1) is that the decay constant in the exponential is different for each day of
the year. (Related approaches for this distribution were given in Refs.
Bucciarelli Jr (1984, 1986); Gordon (1987).)

To begin, we determine the decay constant $\lambda$ for the case where the
bias is fixed. Then we incorporate the effect of a seasonal variation in this
bias, as well as the role of day-to-day correlations in the insolation, to
find the decay constants in the storage distribution for a range of days about
the winter solstice. From these results, we will compute the annual failure
probability.

When the bias is constant, the stored energy after each day changes by the
average solar energy surplus (or deficit), $(f-1)L$, plus or minus a uniform
random variable in the range
$[-\sqrt{3}f\,\widetilde{\sigma}\,L,\sqrt{3}f\,\widetilde{\sigma}\,L]$ to
account for day-to-day fluctuations. Here $\widetilde{\sigma}=\sigma/E$ is the
ratio of the standard deviation in the energy density to the energy density on
any given day (see Fig. 1). Thus the governing equation for $P(s)$ is

$\displaystyle
P(s)=\frac{1}{2\sqrt{3}f\,\widetilde{\sigma}\,L}\int_{s_{-}}^{s_{+}}P(s^{\prime})\,ds^{\prime}\,,$
(2)

where $s_{\pm}=s-(f-1)L\pm\sqrt{3}f\widetilde{\sigma}L$. This equation is
satisfied by the exponential form of Eq. (1), where the decay constant
satisfies

$\displaystyle\frac{\sinh{(\lambda_{\rm
cb}{\sqrt{3}f\widetilde{\sigma}L})}}{\lambda_{\rm
cb}{\sqrt{3}f\widetilde{\sigma}L}}=\exp{[\lambda_{\rm cb}(f-1)L]}.$ (3)

Here we write the decay constant as $\lambda_{\rm cb}=\lambda_{\rm cb}(f)$,
with subscript cb to emphasize that we specialize to the constant-bias case.
Equation (3) gives the dimensionaless quantity $\lambda_{\rm cb}L\to
2(f-1)/(f^{2}\widetilde{\sigma}^{2})$ for $f\to 1$, while $\lambda_{\rm cb}$
deviates slightly from linearity for larger $f$ (Fig. A1 in Appendix C). Over
the practical range of $1<f<1.5$, this dependence is accurately described by a
linear interpolation between $\lambda_{\rm cb}=0$ at $f=1$ and $\lambda_{\rm
cb}L\approx 5.675$ that arises by numerically solving Eq. (3) at $f=1.5$. Thus
we infer $\lambda_{\rm cb}L=\Gamma\,(f-1)$, with $\Gamma\approx 11.35$. As we
shall see, this linear interpolation allows us to construct an analytical
theory for the failure probability that incorporates both seasonality and
fluctuation effects.

Seasonality causes the steady-state distribution of stored energy to be
slightly different for each successive day of the year; thus we now write this
distribution as $P_{j}(s)$, with $j$ indexing the individual day. We first
determine $P_{\rm min}(s)$ on the minimum-insolation day, with $\lambda_{\rm
min}$ the decay rate on this day. This decay rate would equal 0 when $f\to 1$
within the above constant-bias description. However, our simulations show that
the storage distribution still has a nearly exponential form even when $f=1$
(see Appendix D). Thus we need to postulate a functional form for the decay
constant on the minimum-insolation day that interpolates smoothly between the
limiting cases of a value $\lambda_{0}$, which we will determine, when $f=1$
and $\Gamma(f-1)$ when $f-1$ is not small. A simple form that satisfies these
criteria is

$\displaystyle\lambda_{\rm
min}(f)=\tfrac{1}{2}\big{[}\Gamma(f-1)+\sqrt{(4\lambda_{0}^{2}+\Gamma^{2}(f-1)^{2}}\big{]}\,.$
(4)

We also need the steady-state storage distributions $P_{j}(s)$ and their
associated decay rates $\lambda_{j}(f)$ on a range of days around the minimum-
insolation day. To obtain these distributions, we use the fact that the
average daily generated solar energy $E_{j}$ on days near the winter solstice
is well described by the quadratic $E_{j}/L=f+f(j-j_{\rm min})^{2}/\tau^{2}$,
where $j_{\rm min}$ is the day of minimum insolation and $\tau=72$ (in units
of days) is determined by fitting to the 40-year average insolation data. For
each day, the effect of the additional bias as one moves away from the
minimum-insolation day is equivalent to increasing $f$ by $f(j-j_{\rm
min}^{2})/\tau^{2}$. Thus the day-specific decay constant is

$\lambda_{j}(f)=\lambda_{\rm min}(f)+f\,\Gamma\;(j-j_{\rm
min})^{2}/\tau^{2}\,.$ (5)

Finally, we need to account for correlations in the daily insolation. To
include these effects, we perform stochastic simulations of a system with
constant bias $f=1.5$ (taken to be typical of the high-$f$ regime) and
constant $\widetilde{\sigma}=0.351L$. Logarithmic plots of $P(s)$ obtained
both with and without correlations confirm the exponential behavior of $P(s)$,
and show that including correlations reduces the decay parameter by $11$%.
Thus we take $\Gamma=(0.89)\times(11.35/L)=10.1/L$.

###  III.2 The failure probability

From the distribution of storage for each day of the year, we now determine
the annual failure probability $\varepsilon$ of the storage system. We first
estimate the failure probability $\varepsilon_{j}$ on each day $j$, and then
add these daily failure probabilities over a time range that includes the
winter solstice, to obtain the annual failure probability.

The day-specific failure probability for the $j^{\text{t}h}$ day of the year
in the strong-bias limit is

$\displaystyle\varepsilon_{j}=\frac{1}{2\sqrt{3}f\widetilde{\sigma}_{j}L}\int_{0}^{\delta
s}P_{j}(s)(\delta s-s)ds=A_{j}\,e^{-\lambda_{j}(f)S}\,.$ (6)

That is, we integrate the storage distribution $P_{j}(s)$ over the energy
range $\delta s=\sqrt{3}f\widetilde{\sigma}_{j}L-(f-1)L$, for which for the
storage system can be completely depleted within a single low-insolation day,
multiplied by the probability $(\delta s-s)/2\sqrt{3}f\widetilde{\sigma}_{j}L$
that starting with stored energy $s$, a negative energy step actually depletes
the storage system. The expression for $A_{j}$ is written in the SM and
$\lambda_{j}(f)$ is the day-specific decay constant in Eq. (5).

To calculate the annual failure probability $\varepsilon$, we the sum the
daily failure probabilities in Eq. (6) over the range of days where the
quadratic dependence of the decay rate in Eq. (5) applies, under the
assumption that these daily failure probabilities are all independent. Because
of the quadratic time dependence of $\lambda_{j}(f)$ in (5), we convert the
sum over a finite range of days $j$ around the insolation minimum to the
following Gaussian integral over an infinite time range (see Appendix E for
details), in which days far from the minimum give negligible contributions:

$\displaystyle\varepsilon=\sum_{j}\varepsilon_{j}$
$\displaystyle\simeq\int_{-\infty}^{\infty}A(t)\,e^{-\lambda(t,f)S}\,dt$
$\displaystyle=\frac{B}{\sqrt{\lambda_{\rm min}(f)S}}\;e^{-\lambda_{\rm
min}(f)\,S},$ (7)

where we replace the index $j$ by the continuous time $t$, and $B$ is defined
in Appendix E.

We now invert this expression to solve for the required storage as a function
of the reliability $\varepsilon$. In Appendix F, we show that the following
approximate expression accurately describes the dependence of $S$ on $f$ and
$\varepsilon$:

$\displaystyle S(f,\varepsilon)$
$\displaystyle=\frac{\ln\left(\varepsilon_{0}/\varepsilon\right)}{\lambda_{\rm
min}(f)}$
$\displaystyle=\frac{2\ln\left(\varepsilon_{0}/\varepsilon\right)}{[\Gamma(f-1)+\sqrt{(4\lambda_{0}^{2}+\Gamma^{2}(f-1)^{2}}]}\,,$
(8)

where $\varepsilon_{0}$ is defined in Appendix F.

Figure 5: Dependence of the storage $S$ needed for a failure probability
$\varepsilon$, on generation capacity factor $f$. Circles indicate simulation
data and the curves give the theoretical result of Eq. (III.2). $L$ is the
daily load.

Equation (III.2) illustrates the two key features of the tradeoff between
generation and storage capacities: (a) The storage $S$ depends logarithmically
on $\varepsilon$; thus a small increase in storage capacity substantially
increases the combined system reliability. (b) A small increase in $f$ beyond
1 substantially decreases the required storage capacity (Fig. 5).

##  IV Cost Optimization

We now determine the optimal configuration of the combined system by
minimizing the cost function

$\displaystyle{\cal C}=C_{s}\,\frac{S(f)}{L}+C_{g}\,f\,.$ (9)

Here again $C_{g}=\$75$ billion is the cost of a solar farm whose area $A_{0}$
is just sufficient to supply the daily electrical load $L$ of the St. Louis
region during an average insolation day at the winter solstice, while
$C_{s}=\$22$ billion is the cost of a storage system that can supply one day
of electrical load for the region. As mentioned previously, the cost of the
generation system is assumed to be linear in its area, so the cost of a solar
farm of area $fA_{0}$ will be $C_{g}f$. Similarly, a storage system that
supplies an energy $S$ will have a cost $C_{s}S/L$, under the assumption that
the cost of storage is also linear in its capacity.

To find the optimal parameters $(f^{*},S^{*})$ in the minimum cost
configuration, we set $d\mathcal{C}/df=0$ to give

$\displaystyle\frac{dS(f)}{df}\Big{|}_{f^{*},S^{*}}=-L\,\frac{C_{g}}{C_{s}}.$
(10)

Thus the optimal system configuration depends only on the ratio of storage to
generation cost, $C_{s}/C_{g}$, once $\varepsilon$ is specified. (The relation
between the ratio $C_{s}/C_{g}$ and conventional measures of storage and
generation costs is given in Appendix G.) The details of this minimization are
given in Appendix H, from which the optimal solar farm size is determined from

$\displaystyle
f^{*}=1+\frac{2\lambda_{0}}{\Gamma}\frac{C_{s}/(C_{g}r_{0})-1}{\sqrt{2C_{s}/(C_{g}r_{0})-1}}\,,$
(11)

where the dimensionless parameter $r_{0}$ is given by

$\displaystyle
r_{0}=\frac{2\lambda_{0}^{2}L}{\Gamma\ln{(\varepsilon_{0}/\varepsilon})}~{}.$
(12)

The optimal storage value $S^{*}$ is then obtained by substituting $f^{*}$ in
Eq. (III.2).

Here, and in what follows, we use $\varepsilon=0.03$ (failure about once every
33 years) because it roughly corresponds to the accepted standard of a load
loss of one day per ten years sta (2011). For this value of $\varepsilon$,
$r_{0}=0.038$. If the cost ratio $C_{s}/C_{g}$, which currently is roughly
0.3, were to become less than 0.038, then Eq. (11) gives $f^{*}<1$ and indeed
$f^{*}$ would not be defined if $C_{s}/C_{g}$ became less than 0.019. In this
regime, our theory no longer applies, but is also unlikely to be reached by
reductions in storage cost in the foreseeable future.

Figure 6: (top) Optimal storage value $S^{*}$ and generation capacity $f^{*}$,
as functions of cost ratio $C_{s}/C_{g}$. (bottom) Ratio $R$ of storage cost
to excess generation cost, and the system cost $\mathcal{C}$, as functions of
$C_{s}/C_{g}$. In both panels, circles are simulation points, while solid
lines are the theoretical predictions of Eqs. (III.2) and (11).

Our theoretical predictions for $(f^{*},S^{*})$ agree well with simulation
results shown in Fig. 6. Over the range of cost ratios shown, $S^{*}$ varies
from 1 to 7 days, while $f^{*}$ varies from near 1 to 1.4. To gain insight
into the dependences of $(f^{*},S^{*})$ on system parameters, it is helpful to
focus on the limits of expensive and inexpensive storage:

###  IV.1 Expensive storage limit

We define this limit as $C_{s}/(C_{g}r_{0})\gg 1$, where (11) reduces to

$\displaystyle f^{*}\simeq
1+\frac{\lambda_{0}}{\Gamma}\sqrt{\frac{2C_{s}}{C_{g}r_{0}}}=1+\sqrt{\frac{\ln{(\varepsilon_{0}/\varepsilon)}}{\Gamma
L}\frac{C_{s}}{C_{g}}}\,.$ (13a) When storage is expensive, the combined
system favors generation over storage. Consequently, $f$ becomes large, so
that we can drop the $\lambda_{0}$ term in Eq. (III.2) to give $\displaystyle
S^{*}\simeq\frac{\ln\left(\varepsilon_{0}/\varepsilon\right)}{\Gamma(f^{*}-1)}=\sqrt{\frac{\ln{(\varepsilon_{0}/\varepsilon)}L}{\Gamma}\frac{C_{g}}{C_{s}}}\,.$
(13b)

Thus $f^{*}$ and $S^{*}$ have inverse dependences on the cost ratio
$C_{g}/C_{s}$. Combining Eqs. (13a) and (13b), the ratio $R$ of the _total_
storage cost to the _excess_ generation capacity cost is particularly simple:

$\displaystyle R\equiv\frac{C_{s}S^{*}/L}{C_{g}(f^{*}-1)}=1\,.$ (14)

As shown in Fig. 6, this ratio is already close to 1 when $C_{s}/C_{g}$
exceeds 0.2.

From Eqs. (9) and (14), we may write the total system cost in the equivalent
forms

$\displaystyle\mathcal{C}$
$\displaystyle=C_{g}+2C_{s}\frac{S^{*}}{L}=C_{g}+2C_{g}(f^{*}\\!-\\!1)$
$\displaystyle=C_{g}+2\sqrt{\frac{\ln(\varepsilon_{0}/\varepsilon)C_{g}C_{s}}{\Gamma
L}}\,.$ (15)

The first term in each of these forms is the “bare” cost of the generation
system that would be adequate in the absence of insolation fluctuations. The
second term represents the additional system cost that is needed to mitigate
the effect of fluctuations. For $\varepsilon=0.03$ and $C_{s}/C_{g}$ in the
range $[0.1,0.3]$, this additional cost is roughly 50–80% of $C_{g}$ or
$40–$60 Billion. Equation (15) also shows that the additional system cost due
to fluctuations increases only as $\sqrt{\ln(1/\varepsilon)}$. Thus, for
example, to reduce $\varepsilon$ from $0.03$ to $0.003$, the excess cost needs
to be increased by less than 15%.

Equation (15) also provides an explicit way to decide whether it is more cost
effective to invest in reducing the generation cost or the storage cost. The
quantity $C_{s}(\partial{\mathcal{C}}/\partial C_{s})$ gives the sensitivity
of the system cost to a given fractional reduction in $C_{s}$, while
$C_{g}(\partial{\mathcal{C}}/\partial C_{g})$ plays a similar role for
$C_{g}$. From (15), we find

$\displaystyle\frac{C_{s}(\partial{\mathcal{C}}/\partial
C_{s})}{C_{g}(\partial{\mathcal{C}}/\partial
C_{g})}=\frac{\sqrt{\ln{(\varepsilon_{0}/\varepsilon)}C_{s}C_{g}/\Gamma
L}}{C_{g}+\sqrt{\ln{(\varepsilon_{0}/\varepsilon)}C_{s}C_{g}/\Gamma L}}.$ (16)

This cost sensitivity ratio is about $0.3$ when $C_{s}/C_{g}=0.3$. Thus a 30%
reduction in storage cost has about the same impact as a 10% reduction in
generation cost.

###  IV.2 Inexpensive storage limit

We define this limit by $C_{s}/(C_{g}r_{0})\\!-\\!1\ll 1$. Expanding Eq. (11)
and the denominator of Eq. (III.2) to first order in this quantity, we obtain

$\displaystyle f^{*}$ $\displaystyle\simeq
1+\frac{2\lambda_{0}}{\Gamma}\left(\frac{C_{s}}{C_{g}r_{0}}\\!-\\!1\right)=1\\!-\\!\frac{2\lambda_{0}}{\Gamma}\\!+\\!\frac{\ln{(\varepsilon_{0}/\varepsilon)}}{\lambda_{0}L}\frac{C_{s}}{C_{g}}$
(17a) $\displaystyle S^{*}$
$\displaystyle=\frac{2\lambda_{0}L}{\Gamma}\frac{C_{g}}{C_{s}}$ (17b)

As $C_{s}/C_{g}$ approaches $r_{0}$, $f^{*}\to 1$ while $S^{*}$ approaches a
constant value, so the excess system cost becomes dominated by the storage
cost, as shown in Fig. 6.

Combining (17a) and (17b), the total system cost is now

$\displaystyle\mathcal{C}$ $\displaystyle\simeq
C_{g}+\frac{\ln(\varepsilon_{0}/\varepsilon)C_{s}}{\lambda_{0}L}\,.$ (18)

For $C_{s}/C_{g}=0.04$, which is the smallest cost ratio value that we
simulated, the additional cost due to weather fluctuations, (${\cal
C}-C_{g}$), is roughly $\frac{1}{4}C_{g}$. Here, the additional system cost
due to fluctuations increases as the logarithm of the inverse failure
probability. Thus to reduce $\varepsilon$ from $0.03$ to $0.003$ requires a
significant increase in the additional cost of about 30%.

The relative influence of cost reductions in storage versus generation is
similar to that in the expensive-storage limit. The cost sensitivity ratio now
becomes

$\displaystyle\frac{C_{s}(\partial{\mathcal{C}}/\partial
C_{s})}{C_{g}(\partial{\mathcal{C}}/\partial
C_{g})}=\frac{\ln{(\varepsilon_{0}/\varepsilon)}}{\lambda_{0}L}\frac{C_{s}}{C_{g}},$
(19)

which is about $0.2$ at $C_{s}/C_{g}=0.04$. Thus a 50% reduction in storage
cost now has about the same impact as a 10% reduction in generation cost. In
both the limits of expensive and inexpensive storage, reducing the generation
cost has more impact on the overall cost than reducing storage cost.

##  V Discussion

We developed an analytic theory to determine the optimal mix of solar
generation and storage that minimizes the overall system cost and achieves a
given reliability. This system is specified by $f^{*}$, the ratio of the solar
farm area to the area of a farm that fully supplies the electrical load $L$
for the St. Louis region on the _average_ minimum insolation day, and $S^{*}$
the capacity of the storage system, measured in units of daily load. Our
modeling extends the work of Refs. Bucciarelli Jr (1984, 1986); Gordon (1987)
by including seasonal insolation variations, a more realistic distribution of
daily energies, and day-to-day correlations in the insolation. Based on a
quasi-steady-state approximation for the fill level of the storage system, we
obtained the following key results:

  * •

The dependence of the failure probability on storage capacity $S$ (Eq. (7)).
By ignoring seasonal variations, Refs. Bucciarelli Jr (1984, 1986); Gordon
(1987) found that the failure probability $\varepsilon$ decays exponentially
in $S$. We find that that including seasonal variations gives a weak
correction to this result. The near-exponential dependence of $\varepsilon$ on
$S$ suggests that the seasonal variation is slow enough that the steady-state
form that we used for the storage distribution $P(s)$ is accurate.

  * •

The dependence of the storage required to achieve a given reliability, on the
insolation fluctuations and and the generation capacity. We find that the
storage need is an increasing function of daily insolation fluctuations, since
they reduce $\Gamma$ in Eq. (III.2). Without excess generation capacity
($f\\!=\\!1$), storage of almost a week of load is required to achieve a
failure probability $\varepsilon$ less than $0.03$ (Fig. 5). The required
storage decreases rapidly when $f$ increases from $1$. The functional form
that we obtain for the decrease (Eq. (III.2)) differs from both the
logarithmic dependence found in Ref. Gordon (1987) and the exponential one
found in Ref. Egido and Lorenzo (1992).

  * •

The cost of the optimal generation/storage system (Eqs. (15) and (18)).

  * •

A given percent reduction in the generation cost reduces the system cost by
three to five times more than the same percent reduction in the storage cost
(Eqs. (16) and (19).)

A fundamental ingredient in our cost calculations is the ratio of the cost
$C_{g}$ for a solar farm that can supply the daily load of the St. Louis
region on an _average_ insolation day at the winter solstice, to the cost
$C_{s}$ of storing one day of energy load. With current technology, this cost
ratio, $C_{s}/C_{g}$, is roughly 0.3. From Fig. 6, the optimal configuration
is then given by $(f^{*},S^{*})\approx(1.4,1.3)$, which implies an overall
system cost of $1.4\,C_{g}+1.3C_{s}+0.6C_{s}\approx\$147$ Billion (where the
last term incorporates the diurnal storage need), consistent with Ref. Tong et
al. (2020). As the storage cost decreases, the optimal generation capacity
also decreases until the limiting case of $(f^{*},S^{*})\approx(1.0,5)$, with
overall system cost $\approx$ $91 Billion, is reached after a 7-fold decrease
in storage. If storage costs are smaller still, the optimal value $f^{*}$
becomes less than one, a range where our theory is not valid. Below we outline
an approach to treat the range $f\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}1$.

A system cost of roughly $100 Billion seems staggering. However, we emphasize
that the long-term cost of a solar/storage system is likely cheaper than
natural gas power generation. The construction cost for the requisite 5 GW of
natural gas generation for the St. Louis region is roughly $4–5 Billion con
(2017, 2021). Based on prices in the recent past, the fuel cost per year of
operation is about $2.5 Billion gas (2020, 2021). However, gas prices have
increased by a factor of three recently gas (2022). Thus, assuming a 20-year
amortization, the cost of natural gas generation will lie between $55 Billion,
based on the average gas price in the previous decade, and $155 Billion, using
the current price. The renewable-energy systems modeled here will be cheaper
if the gas price is at the upper end of the range. This finding is consistent
with that of Ref. Tong et al. (2020), while Refs. Jacobson et al. (2015) and
Jacobson et al. (2022) found renewable systems to be even more cost-effective.
Our estimates neglect maintenance costs, but we anticipate that maintenance of
solar/storage will be cheaper than that of natural gas because the former has
almost no moving parts. The primary impediment to implementing a solar/storage
system is its huge upfront capital cost.

Within a 100% renewable system, costs can be reduced by deploying a mix of
solar and wind energy Shaner et al. (2018); Heide et al. (2010). Ref. Tong et
al. (2020) found that such a mix would reduce costs by about 50% relative to
solar-only generation. One advantage of solar/wind generation is that the wind
is typically stronger when it is overcast, so an energy deficit in one mode of
generation would be offset by a surplus in the other.

If one is willing to forgo 100% renewable energy generation, a solar/storage
system could be augmented by natural gas “peaker” plants that operate only
during solar energy deficit periods near the winter solstice. Because natural
gas generation plants are relatively cheap to build (as mentioned above), they
are well suited to being run for just a few days of the year. Thus consider a
composite system that consists of a solar farm of area $fA_{0}$, with
$f\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}1$, which is supplemented by a 5 GW natural gas peaker
plant.

In the absence of insolation fluctuations, the annual energy deficit for such
a solar farm is approximately (see Appendix I)

$\displaystyle\mathcal{D}=\frac{\sqrt{2}}{\pi}365L(1-f)^{3/2}\approx
164L\,(1-f)^{3/2}\,.$ (20)

Assuming that the fluctuation contribution to the energy deficit is constant,
and using a daily fuel cost of $7 Million (the price over the past decade),
the cost of a combined solar/peaker generation plant, amortized over the
assumed lifespan of 20 years, is

$\displaystyle C$
$\displaystyle=\tfrac{1}{20}\;fC_{g}+\$7\;\text{M}\times[164L\,(1-f)^{3/2}+10L]$
$\displaystyle\to\$[\alpha f+\beta\,(1-f)^{3/2}]\;\text{Billion}\,,$ (21)

with $\alpha=3.75$ and $\beta=1.15$. In the last line, we ignore the
contribution that is independent of $f$. The crucial ingredient is
$\beta/\alpha\approx 0.307$, the relative cost of natural gas to solar. With
increasing $\beta$ (increasing gas price), the cost-optimal value of $f$ will
increase. However, regardless of how expensive gas becomes, the cost-
minimizing system will always have $f<1$—in other words, some use of peakers
will be cost-effective. The optimal combination of generation, storage, and
peakers is a question to be determined by future analyses. Our analytic
results for the failure probability and required storage will aid such
efforts.

###### Acknowledgements.

SR thanks Dan Shrag for helpful advice and conversations, and acknowledges
partial financial support from National Science Foundation grants DMR-1910736
and EF-2133863.

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##  Appendix A Glossary

In spite of our minimalist theoretical modeling starting point, a large number
of parameters unavoidably arise. For the convenience of the reader, the most
important of these parameters and their numerical values are given in Table
LABEL:tab-g.

Table 1: List of the essential parameters used in this work.  Symbol | Definition | Value (if fixed)
---|---|---
$A_{0}$ | Area of solar farm that meets the daily electrical load at the solstice | $2.5\times 10^{8}\,\text{m}^{2}$
$f$ | Generation capacity relative to a solar farm of area $A_{0}$ |
$f^{*}$ | Cost-optimized value of $f$ |
$S$ | Capacity of the storage system |
$S^{*}$ | Cost-optimized value of $S$ |
$s(t)$ | Amount of energy in storage at time $t$ |
$L$ | Daily energy load of the St. Louis region |  $1.1\times 10^{8}$ kWh $\approx 4\times 10^{14}$ J
$E_{j}$ | Incident energy per m2 on $j^{\rm th}$ day of the year |
$\sigma_{j}$ | Standard deviation of $E_{j}$ |
$\widetilde{\sigma}_{j}$ | Scaled standard deviation of $E_{j}$ |
$E$ | Value of $E_{j}$ at minimum insolation, smoothed over 45 days | 7.95 MJ/m2
$\sigma$ | Value of $\sigma_{j}$ at minimum insolation, smoothed over 45 days | $2.79~{}\text{MJ/m}^{2}$
$\widetilde{\sigma}$ | Scaled value of $\sigma$ | 0.351
$q$ | Day-to-day correlation parameter of solar insolation | 0.6157
$P_{j}(s)$ | Probability that the storage system contains energy $s$ on the $j^{\rm th}$ day of the year |
$P_{\rm min}(s)$ | Value of $P_{j}(s)$ on the minimum-insolation day |
$\lambda_{j}$ | Decay constant in $P_{j}(s)\sim e^{\lambda_{j}s}$ |
$\lambda_{\rm cb}$ | Value of $\lambda_{j}$ for constant bias |
$\lambda_{\rm min}$ | Value of $\lambda_{j}$ on the minimum-insolation day |
$\lambda_{0}$ | Value of $\lambda_{min}$ for $f=0$ | $1.055/L$
$\Gamma$ | Coefficient in the linear interpolation $\lambda_{min}=\Gamma\times(f-1)$ | $10.1/L$
$C_{g}$ | Cost of generation system to supply the daily load at the insolation minimum | $75 Billion
$C_{s}$ | Cost of storage system that can store the daily load | $22 Billion
$\mathcal{C}$ | Total system cost |
$\varepsilon$ | Annual failure probability of the generation/storage system |
$\varepsilon_{j}$ | Failure probability on the $j^{\rm th}$ day of the year |
$\varepsilon_{0}$ | Eq. (F4), prefactor for the failure probability | 9.72
$r_{0}$ | Eq. (12), characteristic cost ratio scale | 0.038 (for $\varepsilon=0.03)$
$R$ | Eq. (14), ratio of storage cost to excess generation cost |

##  Appendix B Calculation of Weather Statistics

The weather data was taken from the MERRA-2 dataset Molod et al. (2015b) that
covers the period 1980 through 2019, using the “SWGDN” variable mer (2022)
(surface incoming shortwave flux, in W/m2). This dataset supplies hourly solar
resource data on a $0.5^{\circ}$ $\times$ $0.625^{\circ}$ mesh. We used a
subset of $7\times 5$ mesh points surrounding St. Louis to define the energy
incident on the St. Louis region. For each day, the hourly data was added over
these $7\times 5$ mesh points to obtain a total daily energy.

##  Appendix C Linear Interpolation Solution for $\lambda$ in Eq. (3)

Figure A1: Exact numerical solution of Eq. (3) (red) and the linear
interpolation used in our theory (dashed).

To investigate the nature of the solution to Eq. (3), we define
$z\equiv\sqrt{3}f\widetilde{\sigma}\lambda L$ and
$\alpha\equiv(f-1)/\sqrt{3}f\widetilde{\sigma}$. This transforms (3) to

$\displaystyle\text{sinh}(z)/z=e^{\alpha z}\,.$ (22)

For solving this equation numerically, we use the values $L=1.1\times 10^{8}$
kWh and $\widetilde{\sigma}=0.351L$. The numerical solution of this equation
is shown in Fig. A1. For $f-1\to 0$ or equivalently, for $\alpha\to 0$, the
solution for $z$ depends linearly on $f-1$. For $f$ in the range 1.1–1.3 the
dependence qualitatively deviates from strict linearity, but quantitatively
remains close to the linear interpolation in the range $1<f<1.5$ shown in Fig.
A1. This interpolation is obtained by matching the linear function with the
exact numerical solution of (22), $\lambda=5.675\ldots$, at $f=1.5$.

##  Appendix D Distribution of Storage at the Solstice

The theory outlined in Sec. III.1 ostensibly applies to the limit of strong
bias, namely $f-1$, not small. In contrast, our simulations indicate that the
storage distribution still has a nearly exponential form even when $f=1$, with
a non-zero decay constant $\lambda_{0}$, whose numerical value is $1.140/L$
(Fig. A2). This value of $\lambda_{0}$ is close to the value of $1.055/L$ that
we obtained by the fitting procedure outlined in Sec. III.1. This latter
numerical value of $\lambda_{0}$ is what we used in our analytic theory.

Figure A2: The distribution of stored energy on the minimum-insolation day,
averaged over $10^{7}$ years, on a semi-logarithmic scale.

##  Appendix E The Failure Probability

We compute the integral in Eq. (6) for the day-specific failure probability,
$\varepsilon_{j}$, in which we use the approximate normalization constant
$A_{j}\approx\lambda\,e^{-\lambda_{j}S}$ for $P_{j}(s)$. We thus obtain

$\displaystyle\varepsilon_{j}$
$\displaystyle=\frac{\mathcal{N}}{2\sqrt{3}f\widetilde{\sigma}_{j}}L\int_{0}^{\delta
s}(e^{\lambda_{j}s}-1)(\delta s-s)\,ds$
$\displaystyle\approx\frac{e^{-\lambda_{j}S}}{2\sqrt{3}f\lambda\widetilde{\sigma}_{j}L\lambda}\left(e^{\lambda_{j}\delta
s}-1-\lambda_{j}\delta s-\lambda_{j}^{2}\,\delta s^{2}/2\right)$
$\displaystyle\equiv A_{j}\,e^{-\lambda_{j}S}\,.$ (23)

Next we compute the annual failure rate $\varepsilon$ by adding the day-
specific failure probabilities over a range of days around the winter
solstice. The first step in this calculation is to convert the discrete sum to
an integral:

$\displaystyle\varepsilon=\sum_{j}\varepsilon_{j}$
$\displaystyle=\sum_{j}A_{j}\,e^{-\lambda_{j}S}\simeq\int_{-\infty}^{\infty}A(t)\,e^{-\lambda(t,f)S}\,dt\,.$

In the integral, we replace the discrete index $j$ by a dimensionless time
variable $t$ that is expressed in units of days, in both the constant $A$ and
in the decay rate $\lambda$. Thus the integral is dimensionless, as it must
be. To evaluate the integral, we exploit the fact that the amplitude $A_{j}$
is slowly varying with $j$, while the Gaussian factor (see Eq. 5) quickly
diminishes for times away from the solstice. Thus we make the approximation in
which we evaluate $A_{j}$ on the minimum insolation day and define this value
as $A_{\rm min}$. We then take this amplitude outside the integral, and use
Eq. (5) to write the dependence of the decay rate on each day to give

$\displaystyle\varepsilon$ $\displaystyle\simeq A_{\rm
min}\int_{-\infty}^{\infty}e^{-\lambda_{\rm min}(f)S-f\,\Gamma\,S(t-t_{\rm
min})^{2}/t_{0}^{2}}\;dt=\frac{B}{\sqrt{\lambda_{\rm
min}(f)S}}\;e^{-\lambda_{\rm min}(f)\,S},$

where $B=A_{\min}\;t_{0}\sqrt{\pi\lambda_{\rm min}(f)/f\Gamma}$.

##  Appendix F Solution to Eq. (7)

We want to invert (7), $\varepsilon=B\,e^{-\lambda_{\rm
min}S}/\sqrt{\lambda_{\rm min}S}$, and solve for $S$ as a function of
$\varepsilon$. By elementary steps, we rewrite this equation as

$\displaystyle 2\left(\frac{B}{\varepsilon}\right)^{2}=w\,e^{w}\,,$ (24)

where $w=2\lambda_{\rm min}S$. The solution for $w$ in this equation is the
Lambert function. In the relevant limiting case in which $B/\varepsilon\gg 1$,
$w$ has the asymptotic behavior

$\displaystyle w=2\lambda_{\rm
min}S\simeq\ln\left[2(B/\varepsilon)^{2}\right]-\ln\left\\{\ln\left[2(B/\varepsilon)^{2}\right]\right\\}\,.$
(25)

Solving for $S$ and neglecting higher-order terms then gives

$\displaystyle S\simeq\frac{\ln(B/\varepsilon)}{\lambda_{\rm
min}}-\frac{\ln\ln(B/\varepsilon)}{2\lambda_{\rm min}}\,.$ (26)

Because the double logarithm in Eq. (26) is very slowly varying in
$\varepsilon$, we replace the factor $\varepsilon$ in this double logarithm by
a constant that is equal to its value at $\varepsilon=0.003$, which is in the
middle of the range of reliability, $0.0003\leq\varepsilon\leq 0.03$ that we
investigated numerically. Using this last simplification, we rewrite (26) as

$\displaystyle S=\frac{\ln{(\varepsilon_{0}/\varepsilon)}}{\lambda_{\rm
min}},$ (27)

where $\varepsilon_{0}={B}/\sqrt{\ln{B/0.003}}$. This result, in combination
with Eq. (4), gives Eq. (III.2) in the main text.

##  Appendix G Calculation of $C_{s}/C_{g}$ from standard cost measures

The cost of energy generation and storage is typically quoted as dollars per
watt of generation capacity and dollars per kWh of capacity for storage.
Currently, the generation cost $c_{g}$ is roughly $1.50 per Watt and the
storage cost $c_{s}$ is roughly $200 per kWh. On the other hand, in this
manuscript we have quoted generation and storage costs, each of a specified
size, in units of dollars only. Here, we relate the cost ratio $C_{s}/C_{g}$
used in this manuscript to the more conventional cost ratio $c_{s}/c_{g}$.
This relation can be used to calculate the ratio $C_{s}/C_{g}$ in different
geographical regions, and does not depend on assumptions about solar panel
efficiency.

Since the generation capacity of solar panels is given in watts, we have

$\displaystyle C_{g}=G\,c_{g}$ (28)

where $G$ is the generation capacity in watts required to satisfy the daily
load of the St. Louis region at the solstice and $c_{g}$ is the cost of the
panels per watt. The generation capacity $G$ is defined with respect to a
reference insolation intensity of $1000\,\text{W/m}^{2}$. Consequently, the
power generated at an insolation intensity $I$ is
$G\times(I/1000\,\text{W/m}^{2}$). Thus if the generation capacity $G$
generates an energy $L$ in a day at the solstice,
$L=G\times({\bar{I}}/1000\,\text{W/m}^{2})\times 86,400~{}\text{sec}$, where
$\bar{I}$ is the day-averaged insolation level at the solstice and 1 day =
86,400 sec. Then we have

$\displaystyle
G=\frac{L}{{86,400~{}\text{sec}}}\frac{1000\,\text{W/m}^{2}}{{\bar{I}}}\;.$
(29)

On the other hand $c_{s}$ is given in dollars per kWh, so that
$C_{s}=L(\text{in}~{}\text{kWh})\times c_{s}$. Combining this with Eqs. (28)
and (29) gives

$\displaystyle\frac{C_{s}}{C_{g}}$
$\displaystyle=\frac{1W}{1kWh}(86,400\text{sec})\frac{\bar{I}}{1000\,\text{W/m}^{2}}\frac{c_{s}}{c_{g}}=0.024\frac{\bar{I}}{1000\,\text{W/m}^{2}}\frac{c_{s}}{c_{g}},$
(30)

where we use $1~{}\text{kWh}=3.6\times 10^{6}\,\text{J}$. For the St. Louis
region, we use the data given in Sec. I.1, namely,
${\bar{I}}=(7.95\,\text{MJ/m}^{2})/(86,400~{}\text{sec})=91\,\text{W/m}^{2}$,
so that

$\displaystyle\frac{C_{s}}{C_{g}}={0.0022}\frac{c_{s}}{c_{g}}\,.$ (31)

For $c_{s}=$$200/kWh and $c_{g}$=$1.50/W, this gives $C_{s}/C_{g}\sim 0.3$.

##  Appendix H Derivation of Eq. (11)

We write Eq. (III.2) in the form

$\displaystyle
S=\frac{\left[\ln{(\varepsilon_{0}/\varepsilon)}/\lambda_{0}\right]}{x+\sqrt{1+x^{2}}}=\left[\ln{(\varepsilon_{0}/\varepsilon)}/\lambda_{0}\right](\sqrt{1+x^{2}}-x)\,,$
(32)

where $x=\Gamma(f-1)/2\lambda_{0}$. Using the chain rule, Eq. (10), and the
definition
$r_{0}=(2\lambda_{0}^{2}L)/\big{[}\Gamma\ln{(\varepsilon_{0}/\varepsilon})\big{]}$,
we find

$\displaystyle\frac{x}{\sqrt{1+x^{2}}}-1=-\frac{2\lambda_{0}L}{\Gamma}\frac{\lambda_{0}}{\ln{(\varepsilon_{0}/\varepsilon)}}\frac{C_{g}}{C_{s}}=-r_{0}C_{g}/C_{s}\,.$
(33)

Then

$\displaystyle\frac{x}{\sqrt{1+x^{2}}}=1-r_{0}C_{g}/C_{s}\,.$ (34)

Squaring both sides and solving for $x$ gives

$\displaystyle
x=\frac{1-r_{0}C_{g}/C_{s}}{\sqrt{2r_{0}C_{g}/C_{s}-r_{0}^{2}C_{g}^{2}/C_{s}^{2}}}=\frac{C_{s}/C_{g}r_{0}-1}{\sqrt{2C_{s}/C_{g}r_{0}-1}}\,.$
(35)

Using the definition of $x$ then gives Eq. (11).

##  Appendix I Solar Energy Deficit Near the Winter Solstice

For a solar farm of area $fA_{0}$ with $f\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}1$, with a
deterministic sinusoidal solar insolation time dependence, we estimate the
energy deficit over the course of a year. Under these assumptions the daily
energy produced by a solar farm of area $A$ is

$\displaystyle E(t)=(\overline{E}+\Delta E\,\cos\omega t)fA_{0}\,e\,,$ (36)

where $e$ is the efficiency of the solar panels. Reading off from Fig. 1, we
infer the values $\overline{E}\approx 16$MJ/m2 and $\Delta E\approx 8$MJ/m2.
For the purpose of a simple estimate, we take these numbers to be exact.

The energy produced each day by a solar farm of area $fA_{0}$ on the
$t^{\text{t}h}$ day of the year is

$\displaystyle E(t)=(\overline{E}+\Delta E\,\cos\omega t)fA_{0}\,e\,,$ (37)

and a shortage arises when $E(t)$ first becomes less than $L$. This event
happens at a time $t_{1}$ when

$\displaystyle(\overline{E}+\Delta E\,\cos\omega t)_{1}fA_{0}\,e=L\,,$ (38)

which occurs before the winter solstice (Fig. A3). Subsequently, the energy
deficit increases day by day until a time $t_{2}$ when $E(t)$ again exceeds
$L$.

Figure A3: Schematic time dependence of the daily solar energy minus the load
for sinusoidal insolation on a solar farm of area $A_{0}$ (blue) and of area
$fA_{0}$ (red), with $f\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}1$. The annual energy deficit $\mathcal{D}$. is shown
shaded

Notice that for $f=1$ the solar energy produced at the solstice matches the
electrical load; that is, $(\overline{E}-\Delta E\,)A_{0}\epsilon=L$. Using
this in (38) gives

$\displaystyle(\overline{E}+\Delta E\,\cos\omega
t_{1})fA_{0}\epsilon=(\overline{E}-\Delta E\,)A_{0}\epsilon\,.$

Solving for the time when the shortfall first occurs gives

$\displaystyle\cos\omega t_{1}=\frac{\overline{E}}{\Delta
E}\left(\frac{1}{f}-1\right)-\frac{1}{f}\,.$ (39)

If we again use $\overline{E}=16$MJ/m2 and $\Delta E=8$MJ/m2, then (39) gives
$\cos\omega t_{1}=\frac{1}{f}-2$. For $f\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}1$, we set
$f=1-\delta$, with $\delta\ll 1$, and obtain $\cos\omega
t_{1}\approx-1+\delta$.

We now set $t_{1}=\frac{1}{2}T-u$, where $T=365$ is the number of days in a
year, and we also shift the time origin so that $\frac{1}{2}T$ corresponds to
the winter solstice. Thus $u$ is the number of days before the solstice when a
solar energy deficit exists, which will be small when $\delta\ll 1$. With
these assumptions,

$\displaystyle\cos\omega
t_{1}=\cos\left[\frac{2\pi}{T}\left(\frac{T}{2}-u\right)\right]=\cos\pi\cos
z\approx-1+z^{2}/2\,,$

where $z\equiv 2\pi u/T$. Since $\cos\omega t_{1}$ also equals $-1+\delta$, we
have $u=T\sqrt{2\delta}/(2\pi)$, for the number of days before the solstice
that a solar farm of area $fA_{0}$ cannot supply the daily energy needs.
Between $t_{1}\equiv\frac{1}{2}T-u$ and $t_{2}\equiv\frac{1}{2}T+u$, or in the
time range $\Delta t\equiv
t_{2}-t_{1}\approx\frac{T}{\pi}\sqrt{2\delta}=\frac{T}{\pi}\sqrt{2(1-f)}$, the
daily solar energy is less than the energy load for each day. The annual
energy deficit in this time range is (see Fig. A3)

$\displaystyle\mathcal{D}=\int_{t_{1}}^{t_{2}}dt\left|E(t)-L\right|\,.$ (40)

Substituting $E(t)$ from (36), eliminating $A_{0}$ by the relation
$(\overline{E}-\Delta E\,)A_{0}\epsilon=L$, and again using
$\overline{E}=16$MJ/m2 and $\Delta E=8$MJ/m2, we obtain

$\displaystyle\mathcal{D}=L\int_{t_{1}}^{t_{2}}dt\left|\frac{(\overline{E}+\Delta
E\,\cos\omega t)f}{\overline{E}-\Delta E}-1\right|$ $\displaystyle\to
L\int_{t_{1}}^{t_{2}}dt\left|(2+\cos\omega t)f-1\right|$ $\displaystyle\approx
L\Delta t(1-f)=\frac{\sqrt{2}}{\pi}LT(1-f)^{3/2}\,.$ (41)

In the integrand, we evaluated the trigonometric function to first order for
$\omega t$ close to $\pi$, since both $t_{1}$ and $t_{2}$ are close to
$\frac{1}{2}T$ for $f\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}1$, to give the result (20).