Optimal Storage for Solar Energy Self-Sufficiency

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.

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.

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.

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.

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

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

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

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

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

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 $\Gamma=(0.89)\times(11.35/L)=10.1/L$.

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

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:

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 $\varepsilon$:

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

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).

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

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:

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

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

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.

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

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

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

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.