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. Shaner et al. (2018) examined combined energy generation and storage systems in the United States and in specific subregions, with both wind and solar generation. For solar-only generation, a system with g ≈ 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 ≈ 1.3 and a month of storage. 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.

Tong et al. (2020) developed optimized energy systems for the continental United States over a range of storage costs, using the same underlying model as in Shaner et al. (2018). For inexpensive storage, they found g ≃ 2.2, while more expensive storage required an increase in the capacity to g ≃ 2.7. Concomitantly, the amount of storage dropped from about 5 days of load to 1 day. Budischak et al. (2013) determined optimal solutions for a power network in the eastern United States 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: 1) clarify the relation between input physical parameters and the performance of a combined generation/storage system, and 2) 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 that were based on simplified solar irradiation statistics. Gordon and Zoglin (1986) assumed a deterministic day-night profile, while ignoring daily and seasonal fluctuations. Bucciarelli (1984); Bucciarelli (1986) and 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 Gordon (1987) gave explicit formulas for the storage capacity required to achieve a given reliability. Markvart (1996) included the effects of seasonality in generation and/or load but did not treat random weather fluctuations. 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. Here we develop such a theory.

We begin by first outlining basic features of the solar-flux data for the St. Louis region, which typifies those of the entire United States. 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.

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: 1) the distribution of energy for each day of the year, and 2) 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 (Figure 3A); 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.

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 (1986). To quantify these correlations, we start with the 40-year sequence of normalized daily energies {r _j ≡ E _j /⟨E _j ⟩}, where E _j is the energy per unit area on the jth day of the year and ⟨E _j ⟩ is its average, with j ranging ³ from 1 to 14,600 (40 × 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 ≈ 0.6157 over the range of 1–16 days (Figure 3B). 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) ∝ q ⁿ for all n.

It is now convenient to shift the time origin so that the year begins on July 1. The solar energy E ₁ on July 1 (now day 1) is given by E 1 = ⟨ E 1 ⟩ + 3 σ 1 rand − 1,1 (1) where rand(−1, 1) is a uniformly distributed random number between −1 and 1. The factor 3 ensures that the standard deviation that arises from the uniform distribution matches that of the actual data, and σ ₁ the standard deviation on July 1 (see Figure 1).

To determine the solar energy on successive days j, we define the indicator function I _j for j ≥ 1 as follows: For j = 1 I ₁ = 1 − 2Θ[rand(0, 1) − 0.5], where Θ is the Heaviside function. Thus I ₁ equals +1 or −1, each with probability 1/2. For j > 1, I j = I j − 1 { 1 − 2 Θ rand 0,1 − q } . (2) Thus I _j , which also takes the values ±1 only, has the same sign as I _j−1 with probability q. For each successive day j > 1, the solar energy E _j is given by E j = ⟨ E j ⟩ + I j 3 σ j rand 0,1 , (3) where σ _j the standard deviation on the jth day of the year. This algorithm results in the deviation of the solar energy from the average on the jth day, E _j − ⟨E _j ⟩, having the same sign as E _j−1 − ⟨E _j−1⟩ 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 Figure 3B. 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 Figure 2.

This approach serves our purposes better than the “Moving-Average” models, “Auto-Regressive” models, or combinations thereof that have often been used to model insolation data Inman et al. (2013). These approaches begin with an uncorrelated random process of a given distribution, and use it a starting point for building correlated sequences of daily insolation values. However, there is no guarantee that the daily insolation values have a physically reasonable distribution. For example, if the input distribution is Gaussian, then the insolation values on some days may be negative because of the tails in the distribution. The present method guarantees a physically reasonable distribution of insolation values. Furthermore, it incorporates the daily variations of the width of the insolation distribution. This is important because it is the width of the distribution around the winter solstice that is crucial for the reliability of the system.

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 jth day of the year is a random variable that changes daily according to s j + 1 = s j + E j − L , (4) where L is the daily load, subject to the constraint that s _j can never exceed S (Figure 4). Since the model treats only the total energy in a day, it does not include the diurnal variation mentioned in Section 2.2 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 ɛ is defined as the fraction of simulated years for which failure occurs.

To obtain cost-optimized system configurations within the simulations for a given value of ɛ, we set up a double mesh of the normalized generation capacity values f _i and the storage capacity values S _j . We evaluate the system cost for each mesh point (f _i , S _j ) as C i , j = C s S i / L + C g f j , as well as the failure probability ɛ _i,j . Then we find the pair (i, j) with the lowest C i , j value subject to the constraint that ɛ _i,j ≤ ɛ; f _i and S _j define the optimized system configuration.

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 (Figure 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 jth 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.

We now determine the optimal configuration of the combined system by minimizing the cost function: C = C s S f L + C g f . (13) 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 1 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 C / d f = 0 to give d S f d f f * , S * = − L C g C s . (14) Thus the optimal system configuration depends only on the ratio of storage to generation cost, C _s /C _g , once ɛ is specified. (The relation between the ratio C _s /C _g and conventional measures of storage and generation costs is given in Supplementary Section S7). The details of this minimization are given in Supplementary Section S8, from which the optimal solar farm size is determined from f * = 1 + 2 λ 0 Γ C s / C g r 0 − 1 2 C s / C g r 0 − 1 , (15) where the dimensionless parameter r ₀ is given by r 0 = 2 λ 0 2 L Γ ⁡ ln ε 0 / ε . (16) The optimal storage value S* is then obtained by substituting f* in Eq. 12.

Here, and in what follows, we use ɛ = 0.03 (failure about once every 33 years) because this value roughly corresponds to the accepted standard of a load loss of 1 day per 10 years FERC (2011). For this value of ɛ, 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. 15 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.

Our theoretical predictions for (f*, S*) agree well with simulation results shown in Figure 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.

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 Bucciarelli (1984); Bucciarelli (1986) and 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.

• We have shown for the first time that in the presence of seasonal variations, the failure probability decays nearly exponentially with increasing storage and generation capacity (Eq. 11). Previous work that found an exponential decay had ignored seasonal variations Bucciarelli (1984); Bucciarelli (1986); Gordon (1987).

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 1 day of energy load. With current technology, this cost ratio, C _s /C _g , is roughly 0.3. From Figure 6, the optimal configuration is then given by (f*, S*) ≈ (1.4, 1.3), which implies an overall system cost of 1.4 C _g + 1.3C _s + 0.6C _s ≈ $147 Billion (where the last term incorporates the diurnal storage need), consistent with Tong et al. (2020). As the storage cost decreases, the optimal generation capacity also decreases until the limiting case of (f*, S*) ≈ (1.0, 5), with overall system cost ≈ $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 ≲ 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 [EIA (2017), Proest (2021)]. Based on prices in the recent past, the fuel cost per year of operation is about $2.5 Billion [Constellation (2020), EIA (2021)]. However, gas prices have increased by a factor of three recently [Trading Economics (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 Tong et al. (2020), while 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). 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 ≲ 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 Supplementary Section S9). D = 2 π 365 L 1 − f 3 / 2 ≈ 164 L 1 − f 3 / 2 . (24) 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 C = 1 20 f C g + $ 7 M × 164 L 1 − f 3 / 2 + 10 L → $ α f + β 1 − f 3 / 2 Billion , (25) with α = 3.75 and β = 1.15. In the last line, we ignore the contribution that is independent of f. The crucial ingredient is β/α ≈ 0.307, the relative cost of natural gas to solar. With increasing β (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.

We developed our mathematical methods for the specific case of the climate in the St Louis region. The same approach can be applied to any geographic region. We expect the general aspects of our findings, such as the nearly-exponential dependence of the failure rate on the storage and excess generation capacities Eq. (11), to hold generally. Different geographic regions will then differ primarily via the parameter values, in two ways: 1) Variations in C _s /C _g . Regions at higher latitudes have lower insolation at the winter solstice, which increases C _g since more solar panel area is needed to satisfy the load. The increased generation cost will shift the optimal system toward more storage. 2) Variations in λ ₀, Γ, and τ. The variations in Γ are the most straightforward. It will be smaller in regions with large relative insolation fluctuations; since these fluctuations are mainly due to cloud cover, Γ will be smaller in cloudy regions. This will lead to increased storage and excess-generation capacity requirements. Broadly based systematic studies based on the general formalism developed here will further clarify the geographical variation of system configuration and cost.