On the economics of electrical storage for variable renewable energy sources

The use of renewable energy sources is a major strategy to mitigate climate change. Yet Sinn (2017) argues that excessive electrical storage requirements limit the further expansion of variable wind and solar energy. We question, and alter, strong implicit assumptions of Sinn's approach and find that storage needs are considerably lower, up to two orders of magnitude. First, we move away from corner solutions by allowing for combinations of storage and renewable curtailment. Second, we specify a parsimonious optimization model that explicitly considers an economic efficiency perspective. We conclude that electrical storage is unlikely to limit the transition to renewable energy.


Introduction
In the 2015 Paris Agreement, the world agreed on ambitious targets for reducing greenhouse gas emissions to combat climate change (United Nations, 2015). The use of renewable energy sources is a major strategy for decarbonizing the global economy. As the potentials of hydro, biomass or geothermal energy are limited in many countries, wind power and solar photovoltaics (PV) play an increasing role.
For example in Germany, often considered as a frontrunner in the use of variable renewable energy sources, the government plans to expand the share of renewable energy in gross electricity consumption to at least 80% by 2050, compared to 36% in 2017 and only around 3% in the early 1990s (BMWi, 2018). Closing this gap requires a massive further expansion of wind and solar power.
Opposed to dispatchable technologies like coal-or natural gas-fired power plants that can produce whenever economically attractive, electricity generation from wind and solar PV plants is variable: it depends on exogenous weather conditions, the time of day, season, and location (Edenhofer et al., 2013;Joskow, 2011). At the same time, maintaining power system stability requires to continuously ensure that supply meets demand. The potential temporal mismatch of supply and demand raises two fundamental questions: how to deal with variable renewable energy at times when there is too much supply, and how to serve demand at times when supply is scarce (cf. also . Evidently, electrical storage can provide a solution, for instance in the form of batteries or pumped-hydro storage plants, allowing to shift energy over time. In a recent analysis,  argues that electrical storage requirements may become excessive and could thus impede the further expansion of variable wind and solar power in Germany. Based on historic time series of electricity demand and variable renewable energy supply, he illustrates that without storage a fully renewable electricity supply would imply not using 61% of the possible power generation from wind and solar generators. In contrast, to avoid any "waste" of renewable energy, storage requirements to take up renewable surplus energy 1 quickly rise to vast numbers. Under such a strategy, current German storage installations would not allow a share of wind and solar PV in electricity demand greater than 30%. 2 And for a fully renewable electricity supply, storage requirements would be more than 400 times as high as the currently installed German pumped-hydro storage capacity, and also much higher than the entire European potential to build such plants (eSTORAGE, 2015).
These considerations deserve merit as they illustrate important properties of variable renewable energy sources. As Sinn is considered to be one of the most influential economists in Germany, 3 , his conclusions can also be expected to be widely received both in policy and academic circles. This is indicated by the fact that the article was listed among the top downloads 4 from European Economic Review for several months. Downloads are strongly positively correlated with citations (Hamermesh, 2018) and thus serve as an early indicator and proxy for academic impact. As regards public impact, Sinn's analysis was covered by several influential German newspapers and magazines 5 , and it achieved by the time of writing an Altmetric attention score of 40, which means the article is in the top 5% of all research outputs scored by Altmetric. 6 Yet the approach is based on strong implicit assumptions, two of which are particularly questionable. First, it only considers two extreme cases in which either all surplus energy is stored or none. In turn, either storage needs are excessive or an excessive share of the available renewable energy is not used. An economically efficient solution is likely to be located in between, i.e., combines some amount of storage and some renewable curtailment. Second, it does not explicitly consider an economic efficiency perspective. Sinn's approach minimizes the storage energy capacity under the constraint that renewables must satisfy a specified proportion of annual electricity demand. Yet an economically efficient solution would seek to minimize the cost to reach that specified proportion of renewables. Such a solution trades off the costs of investments into storage plants, renewables that may get curtailed at times, and other assets in the power market.
We address, and alter, these implicit assumptions and show that their effects are significant. Both results and conclusions change substantially. When moving away from corner solutions, storage needs are up to two orders of magnitude lower in a framework otherwise identical to . Using a parsimonious optimization model with a more suitable economic objective function which leads to first-best solutions, we also find moderate storage requirements. They are even lower if we consider a future broadening of the electricity sector, that is, an additional and flexible use of renewable electricity in other sectors.
Throughout the paper, we provide the economic intuition of what drives storage requirements and use. Variable renewable energy sources are not only variable in supply, they are also nearly free of variable cost. A wind or solar PV plant generates electricity whenever the wind blows or the sun shines without requiring any fuel. Curtailment of renewable energy denotes the operation of a wind or PV plant below its actual temporary generation potential, that is, neither consuming the actually available renewable energy in the moment of generation nor storing it for later use. Analogously, also conventional power plants do not generate electricity at full capacity at all times.
The rationale is the following: if electricity demand is satisfied, electrical storage can be used to take up renewable surplus energy. Yet integrating increasing amounts of such surpluses requires disproportionately growing storage capacities which are not valuable at most times (Denholm and Hand, 2011;. Thus, a corner solution avoiding any curtailment likely leads to inefficiently high storage requirements.
Instead, an efficient solution seeks to balance investments into storage, renewables that get curtailed at times, and other capacities to minimize the total cost of providing electricity.
The remainder of this paper proceeds as follows: In Section 2, we show that Sinn's results are outliers compared to the established literature. We then replicate his findings using open data and an open software tool in Section 3. In Section 4, we extend the basic model to target solutions between the two extreme cases. In Section 5, we devise a parsimonious model to endogenously determine optimal storage and renewable capacities as well as renewable curtailment levels. In Section 6, we discuss further relevant factors and flexibility options that influence storage needs.
Section 7 concludes that electrical storage requirements are not likely to limit the transition to renewable energy.

Literature review
Researchers from various fields have addressed the nexus of variable renewable energy and storage. Several review papers highlight different perspectives: the economic and regulatory challenges of integrating variable renewable energy sources (Perez-Arriaga and Batlle, 2012), features of techno-economic models required to generate policy-relevant insights (Pfenninger et al., 2014), 7 and the role of long-term storage (Blanco and Faaij, 2018). A synthesis of model-based analyses suggests that electrical storage requirements for renewable energy integration are generally moderate. They may only increase substantially in scenarios approaching a fully renewable energy system .
For Germany, Sinn (2017) derives electrical storage capacity needs of 2, 100 gigawatt hours (GWh) (5, 800 GWh, 16, 300 GWh), corresponding to 0.42% (1.15%, 3.23%) of yearly electricity demand, to achieve combined shares of wind and solar power of 50% (68%, 89%). To put these numbers into perspective, we compare Sinn's results with other studies on future electricity systems with high shares of variable 7 As techno-economic models, we classify numerical bottom-up electricity market simulation models that explicitly incorporate relevant technical constraints.

renewables.
Also for Germany,  determine optimal storage requirements in long-run scenarios. For 68% (78%, 88%) variable renewables, 8 they arrive at 55 GWh (159 GWh, 436 GWh) storage, corresponding to 0.01% (0.03%, 0.09%) of annual demand. Further results on storage needs for the German energy transition are available among policy studies (Fraunhofer, 2014). A particularly influential study concludes that hardly any additional storage investments are necessary in Germany and Europe in the short and medium term (Pape et al., 2014): In 2050 scenarios with European shares of variable renewables around 40% (45%, 55%), additional storage capacity between around 14 and 650 GWh is needed (corresponding to 0.00% to 0.02% of annual demand), largely located outside Germany. A long-term climate policy study commissioned by the German environmental ministry (BMUB) also finds that around 170 GWh of pumped-hydro storage (0.02% to 0.03% of annual demand) suffice to achieve variable renewable shares between 83% and 91% (Repenning et al., 2015).
For Europe,  derive cost-minimal storage capacities corresponding to about 0.08% (0.28%) of yearly demand to achieve 74% (85%) variable renewables in a setting with equal contributions of wind and solar power. Using the same numerical model,  derive larger storage needs of around 1% of yearly load at a variable renewables share of 80% in a transmission-constrained European scenario. This number decreases to 0.5% in case of increasing transmission capacity. In a recent long-term scenario commissioned by the German energy ministry (BMWi), storage capacities in the size of 0.01% of yearly demand are enough to achieve a pan-European renewable share of 65% (BMWi, 2017).
For the U.S.,  find that integrating up to 55% variable renewables in 2030 does not require any electrical storage. Instead, pan-U.S. geographical balancing, facilitated by transmission investments, mitigates the variability of wind and solar power. For the Pennsylvania-New Jersey-Maryland mar-8 Corresponding to overall renewables shares of 80%, 90%, and 100%. ket,  conclude that a large stock of electric vehicle batteries, corresponding to 0.3% of yearly overall demand, would enable pushing the renewables share to 99.9% in 99.9% of all hours. Using stationary batteries would-at higher overall cost-require even less storage capacity.  show that a fully renewable (wind and solar power contributing 90%) U.S. energy system covering all end use sectors would be possible with an electrical storage capacity smaller than 0.1% of yearly electricity demand. 9 For Texas, papers with different approaches also conclude on moderate storage requirements: capacities corresponding to around 0.02% (0.06%, 0.14%) of annual demand would suffice to integrate a combined share of wind and solar PV of 55% (70%, 80%) with relatively low renewable curtailment Denholm and Hand, 2011;.  derive optimal storage deployment of 0.10% of annual demand for a 66% wind power share.
Based on our review, Figure 1 plots the shares of variable renewable energy against storage energy requirements, normalized by yearly energy demand, and contrasts them with Sinn's findings. 10 It also includes the results of this paper from Section 5. Two findings stand out: first, storage needs disproportionately grow with higher renewables shares. Therefore, the vertical axis is provided in a logarithmic scale. This is driven by the distribution of surplus energy, which has high peaks in only a few hours of the year and is very small or zero in most other hours. Second, storage requirements found in the literature are considerably lower than those calculated by -often by at least an order of magnitude.
What drives these large differences? One important factor is renewable curtailment; the data labels in Figure 1 provide the numbers in percent of the annually available renewable energy. Sinn focuses on corner solutions without renewable curtailment as illustrated by the outer line: the storage has to take up every potential 9 This number includes 13 GWh of thermal storage coupled to concentrating solar thermal power generation. In addition, the optimal solution includes substantial heat storage capacities.
10 See Appendix A for more detailed information. Complementary to storage energy capacity in Figure 1, Figure A.1 in Appendix A also provides additional information on storage power ratings.   kilowatt hour of surplus energy that could be generated by wind power and solar PV generators, which leads to strongly increasing storage requirements for growing shares of variable renewable energy sources. By contrast, the literature agrees that a complete integration of variable renewable energy is not desirable (see, in particular, Ueckerdt et al., 2017).
A combination of renewable capacity oversizing and some temporary curtailment substitutes storage expansion when imposing economic efficiency criteria, such as finding a technology portfolio for least-cost renewable energy supply. In equilibrium, the marginal effects of adding another unit of storage and adding another unit of renewable generation that gets curtailed at times are equal. A social planner would thus trade off storage against renewable curtailment and other options that can provide flexibility. Accordingly, renewable curtailment is not necessarily inefficient.
Beyond curtailment, further options on both the supply and demand sides can provide flexibility for variable renewable energy sources and thus substitute for electrical storage . These comprise geographical balancing (Fürsch et al., 2013;Haller et al., 2012;, demand-side management (Pape et al., 2014;, and the flexible use of renewables in other sectors such as heat or mobility .
To be concise, we largely abstract from such options in our analysis, as in Sinn's original framework. We further discuss this avenue in Section 6.
Our literature review highlights two main insights: first, Sinn's findings are outliers compared to the consensus of established studies. Second, his extreme findings are driven by not considering relevant economic trade-offs concerning the provision of flexibility, in particular by neglecting potential renewable curtailment.

Replication and intuition
We first replicate the central results of Sinn's analysis, using a spreadsheet tool and open-source input data. Following recent discussions on good practice in the field of energy research (Pfenninger, 2017;Pfenninger et al., 2018), we provide our tools and all input parameters under a permissive open-source license in a public repository 11

Focus of our analysis
In our replication, we focus on the central Section 6 in   12 Sinn's Sections 7 and 8 illustrate storage-reducing effects of geographical balancing. These are related to smoother aggregate demand and renewable supply patterns when considering multiple countries at a time and access to flexible hydro capacities in Norway and the Alps. Yet Sinn makes a range of strong assumptions, for example, on an unchanged geographical distribution of renewables. Sinn's Section 9 provides largely qualitative reflections on sector coupling aspects. We quantitatively analyze this in our Section 5.4. storage needs. However, there is no economic or technical reason for this kind of smoothing. It seems to be inspired by the notion that renewable generators should mimic the characteristics of conventional power plants. In this case, additional backup capacities (referred to as "double structures") would be obsolete. However, it is not clear why using existing backup power plants should be less desirable than installing additional electrical storage; the approach is silent about any efficiency or optimality criteria. The lack of practical relevance is illustrated by the fact that resulting storage requirements cannot be empirically observed in countries with high variable renewables shares like Denmark, Ireland or Spain. 13

Replication using open data and an open software tool
We derive input data from the Open Power System Data platform, which collects and provides European electricity market data from official sources (OPSD, 2017).
Input parameters comprise hourly time series of realized German electricity demand and availability of onshore wind power and solar PV, defined as capacity factors between zero and one. Electricity demand enters the analysis as inelastic, that is, we do not fit any demand curves. This assumption follows  and is also standard in much of the literature. 14 Capacity factors are calculated by relating historic hourly feed-in to installed renewable generation capacity in respective hours. (2017), all input data is taken from the base year 2014.

As in Sinn
To achieve an exogenously specified share of renewable electricity, the time series of capacity factors is scaled up until renewables meet the targeted share δ of annual demand. 15 Renewables satisfy demand either contemporaneously, that is in the hour of generation, or in a following hour facilitated by storage. To this end, we impose 13 Especially the Irish electricity system has a high supply of wind power and at the same time only few options for intertemporal balancing. The share of wind energy in Ireland was above 20% in 2016, with wind capacities somewhat above 2, 800 megawatts (SEAI, 2017). The only pumpedhydro storage plant had a capacity of somewhat below 300 megawatts and the interconnector to Great Britain a capacity of 500 megawatts. Compare also OPSD, 2017.
14 Assuming an inelastic short-run electricity demand appears appropriate because hourly wholesale price signals are so far not passed through to the majority of final consumers. We briefly discuss more flexible demand in Section 6. Storage needs rise sharply if more renewable electricity must be integrated. While current German pumped-hydro storage installations of somewhat below 0.04 terawatt hours (TWh) would suffice to fully integrate almost 30% renewable electricity, even moderate further increases in renewables would substantially drive up storage requirements. For 50% variable renewables, they already amount to 2.1 TWh, that is, they are two orders of magnitude higher. As Sinn does not provide his data and calculations open-source, we cannot trace back the small numerical differences to our findings to a specific reason; presumably, they arise due to slight differences in the input data.
16 See Table 1 in .

(Non-)Robustness
We address the robustness of findings in a sensitivity analysis using different base years. Both the time series of demand and the availability of renewable energy may change substantially between years. To this end, we repeat the basic analysis   The expressions load and demand can be used interchangeably. 18 We also use the base year 2014 in the remainder of this analysis, as in Sinn (2017)

Power-oriented renewable curtailment
The RLDCs suggest that curtailment of renewable surpluses may reduce storage requirements. We first devise a strategy that allows curtailment of all renewable energy surpluses beyond a defined threshold. This threshold can be interpreted as the power capacity of a storage (in megawatt, MW); this is the energy the storage can take up per hour-as opposed to the energy capacity the storage can accommodate in total (in megawatt hours, MWh). This distinction is an essential characteristic of any electric storage technology and missing in . In case of pumped-hydro storage, the power capacity describes the power of the pumps or of the turbine to generate electricity; the energy capacity indicates the volume (in energy terms) of the storage basin.
We extend the basic model by a renewable curtailment threshold, equivalent to the power capacity of the storage. If hourly surplus generation is below the threshold, it is channeled into the storage; all renewable surplus energy beyond the threshold is curtailed. Otherwise, the model is identical to Section 3 and storage use remains myopic. To gain some intuition, Figure 5 shows RLDCs for 80% renewables in final demand and a curtailment threshold of 44.1 gigawatt (GW), which leads to curtailment of 5% of maximum yearly renewable generation. The threshold is indicated by the horizontal solid gray part of the RLDC after renewable curtailment on the right-hand side. Area D represents all renewable surplus that is curtailed. The storage shifts the remaining surplus energy from area A to area B. The remaining energy demand, area C, is supplied by other means. This renewable curtailment strategy avoids storing the most excessive surplus events.  We iterate through combinations of minimum renewable requirements and maximum allowed renewable curtailment. In doing so, the spreadsheet model endogenously determines the curtailment threshold-or storage power capacity-such that no more renewable energy is curtailed than maximally allowed. Figure 6 shows the results. It is evident that increasing levels of renewable curtailment lead to lower storage requirements. The decrease is close to linear though somewhat convex. For instance, while a complete integration of 50% variable renewable electricity triggers 2.1 TWh storage energy capacity, allowing curtailment of 5% of the annual renewable generation reduces storage needs to 0.3 TWh.
Specifically, we provide solutions that lie between the two extremes "no renewable curtailment" and "no storage", which Sinn (2017)    curtailment is allowed. The numbers are identical to those that we replicate from Sinn's approach (compare Figure 2 and the right panel of his Table 1). The horizontal axis shows renewable curtailment levels for the corner solution if no storage is allowed.
Here, we also replicate Sinn's findings on "efficiency losses" that are given in the left panel of his Table 1. For instance, for 50% renewables, our model returns a renewable curtailment between 6.0 and 6.5%, as indicated by the point where the solid gray line intersects the horizontal axis in Figure 6. For the same case, Sinn determines an "efficiency" of 93.8%, which corresponds to curtailment of 6.2%. 21 Thus, we provide a solution space combining curtailment and storage that lies between the two extreme cases. For other base years, results are qualitatively unchanged; however, they exhibit great variation concerning the level of required storage.
To achieve the same share of renewable energy in final demand, the required renewable capacities are necessarily higher when allowing for curtailment, that is, if some of the available energy is not used. However, this increase is moderate, as Figure 7 shows. For instance, achieving 50% renewable energy in final demand requires 214 GW renewables without curtailment. With 5% curtailment, the necessary renewable capacities are somewhat higher, at 226 GW.   Yet renewable curtailment does not increase the necessary backup capacities to supply the remaining residual electricity demand after storage. They are no larger than in the case without renewable curtailment. Inspecting the left-most part of the RLDCs in Figures 4 and 5, there is also no reason to assume so. 22

Energy-oriented renewable curtailment
While the power-oriented storage strategy-curtailing renewable surplus whenever it exceeds a defined threshold-seems plausible, it may not be optimal with 22 Backup capacities could only be smaller under the no-curtailment-regime if there was an extended number of hours with high surpluses, directly followed by hours with the highest residual loads. This case is rather unlikely; also, we cannot observe it in our data. In Section 5.3, we show how storage can lower the need for backup capacities. respect to finding the smallest required storage energy capacity. Given historic input data, it turns out that extended periods of renewable surpluses in contiguous hours determine the maximum energy capacity of the storage, and not single periods with the highest surplus generation. For instance, storing a moderate surplus in ten consecutive hours may require more storage than storing an extreme surplus event in one hour.  Therefore, we alternatively implement an energy-oriented renewable curtailment strategy. The storage operational pattern remains myopic and is identical to the above cases; however, renewable curtailment occurs if and only if the storage is fully loaded. Thus, it targets a minimum energy capacity requirement. Again, we iterate through minimum renewable requirements and maximum renewable curtailment constraints to explore the solution space and endogenously determine minimum storage capacities.
To provide some intuition, Figure 8 shows the resulting residual load duration curves for the case of 80% renewables and a maximum curtailment of 5% of the annual renewable energy. Curtailed energy, area D, is identical to the one under the power-oriented renewable curtailment strategy, area D in Figure 6. However, renewable curtailment is concentrated in hours in which surpluses trigger the highest storage requirements; these are not necessarily the hours with the highest surplus generation. Storage shifts the remaining surplus generation, area A, to hours with positive residual load, area B. Note that renewable curtailment and the storage operational pattern are still myopic and deterministic, that is, they do not require perfect foresight: surplus energy is charged into the storage as long as there are free capacities, and is curtailed otherwise. The stored energy serves residual load as soon as it is positive again. Maximum allowed renewable curtailment in percent 40% variable renewables 50% variable renewables 60% variable renewables 70% variable renewables 80% variable renewables  For instance, while a complete integration of 50% variable renewable electricity requires 2.1 TWh storage energy capacity, allowing for 5% renewable curtailment reduces storage needs to 0.019 TWh, or 19 GWh. This is one order of magnitude lower than under the power-oriented curtailment strategy, two orders of magnitude lower than without renewable curtailment, and less than the pumped-hydro power capacity installed in Germany by 2018. Allowing 8% curtailment, 44 GWh of storage, slightly more than installed in Germany by 2018, would suffice to reach 70% variable renewable energy.

Cost-minimal storage requirements
The data-driven analysis in Section 4 alters an important implicit assumption of Sinn's approach: plausible solutions lie between the two corner solutions of no renewable curtailment or no storage. However, the approach is still unlikely to result in an efficient market outcome because of the objective function used. From an economic perspective, finding least-cost solutions is relevant, that is, cost-minimal combinations of conventional and renewable generation, renewable curtailment, and electrical storage. 24 As we also include a stylized representation of conventional generators, we explicitly address what Sinn refers to as "double structure buffering." In this context, both storage energy capacity (in MWh) and storage power capacity (in MW) matter with respect to costs. 25 To find optimal solutions, we employ a stylized and parsimonious numerical optimization model. 26 We provide the source code and all input data under a permissive open-source license in a public reposi-23 Also here, renewable curtailment does not increase backup needs. Likewise, allowing renewable curtailment goes along with slightly higher renewable capacities necessary to achieve the imposed renewables share in final demand. By construction, numbers are identical to the power-oriented curtailment strategy.
24 For the sake of conciseness and traceability, we still neglect other potential sources of flexibility, such as load shifting or dispatchable biomass. We illustrate the effects of such flexibility options in . 25 For simplicity, we assume identical charging and discharging capacity. 26 The model is derived from our established and more detailed open-source model DIETER; see  for an exposition. The model is implemented in the General Algebraic Modeling System (GAMS).

tory. 27
The economic optimization approach addresses both challenges of renewable energy integration. First, it delivers an efficient solution to the trade-off how much and when renewable surplus energy to curtail, and how much and when to store. This corresponds to the right-hand side of the residual load duration curve. Second, it determines efficient conventional, renewable, and storage capacities to serve demand at any point in time; this corresponds to the left-hand side of the residual load duration curve. The results of the cost minimization model can be interpreted as long-run equilibria under the assumption of perfect competition and complete information.
The model thus mimics a first-best social planner approach.

The model
The numerical model minimizes the total costs of satisfying electricity demand in every hour h of a year. The objective function (1) sums the products of specific investment costs κ i and capacity entry N of storage, differentiated by energy N se and power N sp , renewables N r , and conventional capacity N c . Throughout the model, upper-case Roman letters indicate variables.
For conciseness, we consider one stylized technology for variable renewables which aggregates the generation patterns of onshore wind power and solar PV, 28 and two stylized conventional technologies c ∈ {base, peak}, parameterized to lignite and natural gas plants. Base generators incur high capacity costs and low variable costs, and vice versa for peak plants. We parameterize storage according to pumped-hydro storage, which features relatively high costs for power capacity κ i,sp and relatively low costs for energy capacity κ i,se . By focusing on pumped-hydro storage, we follow the narrative in . Moreover, it is a mature technology and its cost struc- The concrete numbers are provided in the Zenodo repository.
The market clearing condition (2) makes sure that price-inelastic electricity demand d h in every hour is satisfied either by renewable generation G r h , conventional 30 According to current German legislation, owners of wind and PV plants generally receive a subsidy payment for each megawatt hour of energy generated, also when being curtailed, to recover their investments. Analogously, the model's objective function accounts for investment costs of wind and PV plants irrespective whether generation is curtailed at times or not. In 2016, curtailment amounted to 2.3% of the renewable energy generation under the German subsidy scheme, however, entirely mandated by the electricity network operators to ease congestion (BNetzA and BKartA, 2017). 31 We do not aim to derive detailed projections for a future electricity market in Germany here. Our stylized analysis only focuses on relevant drivers influencing an cost-optimal storage capacity. Therefore, we abstract from further economic and technological details. generation or generation from storage.
The storage level S h in any hour equals the storage level in the previous hour h − 1, plus the energy charged to storage − → S h minus the energy discharged ← − S h , both corrected by efficiency losses (4a), which are identical to the spreadsheet approach.
Capacity constraints impose that the hourly energy charged or discharged does not exceed the installed pump or turbine capacity (4b-4c) and that the storage level never exceeds the installed energy storage capacity (4d). Further, we require an identical storage level in the first and last period of the analysis.
To explore rising shares of renewable electricity, we exogenously impose a min-imum share of yearly final demand to be satisfied by renewables, δ ∈ [0, 1], for reasons of convenience imposed as maximum share of conventional energy (5). We explore minimum renewables shares between 25% and 90% in five percentage points increments.
The model is a linear program and solved numerically to global optimality. The result is a cost-minimal combination of renewable, conventional, and storage capacity investments as well as their optimal hourly dispatch. Specifically, two strategies can increase the required minimum share of renewables: the use of storage to integrate surpluses, or larger renewable capacities plus curtailment. The model solves this trade-off endogenously. Note that, opposed to the myopic models in Section 4, this approach requires the assumption of perfect foresight to optimally schedule the release of energy from the storage. 32

Intuition
To again provide some intuition before discussing numerical results, Figure 10 plots the resulting RLDCs for the 80% renewables case. If capacity entry is costly with respect to both storage power and storage energy, the optimal solution combines both channels analyzed in Section 4. Areas D 1 and D 2 represent the curtailed renewable energy. The kink to the right is driven by power-oriented curtailment, compare When increasing the targeted renewables share, the optimal storage energy capacity grows much faster than the optimal storage power capacity. For 50% renewables, 35 GWh energy are accompanied by about 6 GW storage power. Dividing energy by power yields an energy-to-power (E/P) ratio of about 6 hours. The E/P ratio is an important metric to characterize a storage technology and reflects its temporal layout: a 6 hours storage is a typical short-to-medium-term storage to compensate diurnal fluctuations, such as of solar PV generation. If it is completely charged, it can generate electricity for 6 hours at maximum power rating. 36 For higher renewables shares, the E/P ratio increases to reach about 19 hours for 90% renewables. This highlights the importance of considering both rather inexpensive storage energy and 34 Residential battery storage coupled to prosumage-oriented solar PV installations constitutes a notable exception . 35 Figure B.1 in Appendix B provides results for alternative base years. 36 The German pumped-hydro storage fleet has an E/P ratio of about 7 hours. rather expensive storage power separately. Moreover, no need for a true long-term storage arises, that is, storing energy for weeks or months.
Optimal endogenous renewable curtailment also grows with higher minimum renewables shares. As such, the economics of renewable electricity provide no reason why curtailment should be avoided. It can be more efficient not to use available renewable energy at times despite costly investment into wind and PV plants. The optimal solution combines conventional plants, storage, and renewables, part of which being curtailed at times.

Extension: flexible sector coupling (power-to-x)
In future low-carbon energy systems, renewable electricity supply considered as surplus energy in the above framework is likely to be highly valuable for new uses.
Merging electricity, heating, and transport sectors can not only provide flexibility for integrating variable renewables into the power market, but can also contribute to decarbonizing these other sectors (Mathiesen et al., 2015). This concept, often referred to as sector coupling, comprises using renewable electricity, for example, for residential heating (see Bloess et al., 2018, for an overview of the recent literature) or for electric mobility (Richardson, 2013). Moreover, renewable electricity can also be used to produce other energy carriers, such as hydrogen or synthetic gaseous or liquid fuels, by means of electrolysis (Schiebahn et al., 2015). Such sector coupling options are often referred to as power-to-x (or P2X). They can lower electrical storage requirements if the new loads are sufficiently flexible and the additional demand can be shifted to periods in which renewable availability is high.
To illustrate this avenue in our model, we add a stylized additional electricity demand of a generic power-to-x technology with a capacity of n x = 50 GW and 2, 000 full-load hours to our model, which corresponds to an additional annual energy demand of d x = 100 TWh. 37 For instance, this could be a fleet of electric vehicles: assuming a yearly electricity demand of 2, 000 kWh per vehicle, this would correspond to 50 million vehicles. Alternatively, the additional demand may come from flexible electric heaters, or from electrolyzers converting renewable surplus electricity into hydrogen. For simplicity, we do not further restrict the timing of this additional consumption; it is only limited by the installed power capacity, i.e. the power-to-x demand is assumed to be highly flexible. In line with the literature, we generally assume that it is less costly to store x, for instance heat or synthetic fuels, than electrical energy, rendering the timing of the power-to-x generation more flexible. Moreover, we require the additional demand to be satisfied entirely by additional renewables. To this end, we augment Equation (3b) to where the variable − → X h is the hourly power-to-x demand that is optimized endogenously. Two further equations restrict the hourly demand by the installed capacity n x (6a) and require that annual power-to-x demand d x is satisfied over the year (6b). Otherwise, all model assumptions and equations are identical to Sec-tion 5.1.  The rationale is the following: in the 50% renewables case, wind and solar capacities rise from 223 GW to 298 GW to supply part of the additional power-to-x demand. Another part of the additional demand is satisfied by renewable electric-ity previously curtailed or stored. Accordingly, both storage needs and renewable curtailment rates are substantially lower in most cases. 38 As we assume power-to-x to be perfectly flexible, the diminishing effects on storage requirements and renewable curtailment constitute an upper bound for less flexible real-world applications. However, they illustrate that flexible sector coupling could substantially drive down electrical storage needs. If additional flexible electricity demand emerges that contributes to decarbonizing other energy sectors, then renewable surplus generation becomes a valuable resource. 39

Discussion
Departing from Sinn (2017), we implement small but relevant changes to move his the setup away from corner solutions. The impact is substantial and reduces storage needs up to two orders of magnitude. At the same time, our analysis remains stylized and tractable. In the following, we highlight further important points that researchers should consider when analyzing storage needs to integrate variable renewable electricity, both against the background of Sinn's analysis and the large body of academic literature .
First, the definition of efficiency should be clarified when it comes to variable renewable energy sources.  seems to refer to inefficiency as both curtailment of renewables and storage use as such; the first motivated by avoiding waste, the second by avoiding backup capacities (referred to as "double structure buffering"). 40 However, a welfare economic approach should rather target the least-cost 38 To be transparent, results strongly depend on the assumptions on capacities and full-load hours of the generic power-to-x technology. See Figure D.1 in Appendix D. The effect of power-to-x on storage needs is zero or negative in this setting up to 4, 000 full load hours, which is a considerably high value. With the parameterization used here, the largest effect on electrical storage emerges between 1, 500 and 2, 000 full-load hours. 39 Likewise, more variable generation and, accordingly, wholesale market prices, may incentivize also the current electricity demand to become more temporally flexible in the long-run. This would have an analogous mitigative effect on storage needs. 40 See, for instance, Table 1 or Figure 8 in . Moreover, the notion of "wasting energy" from renewables is questionable. On that note, one may also consider electricity not produced by conventional plants, that is, full-load hours smaller than 8, 760, as "waste". In both cases, marginal costs are zero. provision of electricity for given minimum renewable energy constraints. The result is a combination of conventional and renewable plants, storage, and curtailment of a certain amount of renewable energy. Why the one or the other should be "inefficient" is unclear. In the optimum, the marginal cost of further expanding storage, renewables that are curtailed at times, and conventional capacities is equal. Which shares of renewable energy are optimal when also considering external costs, for instance, arising from climate change, local emissions or land use change, whether the market achieves this solution, or which regulatory measures are required is another issue left for analysis and discussion elsewhere.
Second, electrical storage has values beyond arbitrage. It can provide firm capacity and thus reduce the need for backup plants (see Section 5.2), provide balancing reserves and other ancillary services to maintain power system stability, and may also help mitigating grid constraints.
Third, other types of energy storage are relevant beyond pumped-hydro. These comprise batteries, which could become much cheaper in the future (Schmidt et al., 2017), 41 or power-to-gas storage. Specifically, different storage technologies have different costs for power and energy capacities. While batteries are relatively cheap in power, they are expensive in energy, and vice versa for power-to-gas. Thus, electrical storage technologies have different optimal E/P ratios: batteries are generally suited for short-term storage of a few hours, pumped hydro for around six to ten hours, and power-to-gas for longer periods. An optimal deployment of different storage types can address different types of renewable fluctuations, for example, intra-hourly, diurnal or even seasonal . Such a differentiated storage fleet also tends to be smaller and cheaper.
Fourth, scaling up historical feed-in time series of a fixed proportion of wind and solar power tends to over-estimate flexibility requirements. Both market forces and the renewables support scheme in Germany tend to incentivize renewable generation when prices are higher and supply is, accordingly, scarce. Such system-friendly renewables comprise wind turbines that dis-proportionally produce electricity when wind speeds are low, both due to their location and technical layout (May, 2017). A similar argument holds for solar PV panels, which may be directed such that they generate more electricity in morning or afternoon hours. Even more relevant, offshore wind turbines have much smoother generation patterns and higher full-load hours than onshore wind parks. 42 The sensitivity toward the base year in Section 3 further illustrates the relevance of appropriate input data choices. Ideally, analyses should be based on bottom-up weather data covering as many years as possible (Staffell and Pfenninger, 2016).
Fifth, further electricity market integration across national borders generally yields smoother residual load. When balanced over greater geographical areas, the variability of wind, solar, and demand tends to be evened out Fürsch et al., 2013;Haller et al., 2012;. This results in smoother residual load patterns and lower storage requirements.
Finally, a temporally more flexible demand can also substitute electrical storage (Denholm and Hand, 2011;Pape et al., 2014;. If a more variable electricity supply triggers more volatile prices, and these prices are passed through to consumers, then the demand side should have increasing incentives to consume more flexibly and profit from arbitrage gains.

Conclusions
The use of renewable energy is a major strategy to mitigate greenhouse gas emissions, reduce fossil fuel imports, and create a sustainable energy system. However, integrating growing shares of variable wind and solar power in electricity markets poses increasing challenges. Electrical storage is an important-albeit not the only-option to address the mismatching time profiles of variable renewable supply and electric demand. In a recent analysis,  calculates storage needs in a German setting and finds vastly growing electrical storage requirements, already for renewable supply shares only moderately greater than currently the case in Germany. Based on these findings, he suggests that electrical storage may limit the further expansion of variable renewable energy sources. We also illustrate that electrical storage needs may decrease further if the electricity sector is broadened to also include flexible additional demand, for example related to heating, mobility or hydrogen production. While we demonstrate that such powerto-x options may substantially change the picture, further and more detailed research on this avenue would be desirable.
All things considered, we conclude that electrical storage requirements do not limit the further expansion of variable renewable energy sources.  (2017) does not explicitly mention any storage power capacities, we use the numbers from our replication of his calculations presented in Section 3. We also include our findings from the cost minimization approach. For comparability across studies, we normalize storage power capacities by dividing them by the system peak load. The system peak load arises in the hour with the highest demand. For Germany, it amounted to about 79 GW in 2014. As in the case of storage energy, the literature finds much lower storage power requirements than   Figures 1 and A.1 require information not explicitly provided in several of the underlying studies. We calculate or infer missing data. Further, we select the most relevant cases. In the following, we provide additional information: • BMWi, 2017: Aggregated values for Germany and rest of Europe from Basisszenario; peak load is not provided, but inferred from the ratio between peak load and yearly demand from .
Appendix C. Storage requirements in the optimization model: sensitivity with lithium-ion batteries Figure C.1 plots cost-optimal storage energy and power capacities as well as curtailment rates for different shares of renewables in final demand for a sensitivity where the storage technology is parameterized to lithium-ion batteries instead of pumped hydro. Here, costs for storage power capacity are lower and costs for storage energy capacity are higher compared to the other calculations presented in this paper.
In addition, the round-trip efficiency of lithium-ion batteries is higher than those of pumped-hydro storage. Three findings emerge: first, qualitative results are unchanged. Also here, storage energy requirements are substantially lower than in . Second, absolute storage energy deployment is lower compared to the case with pumped-hydro storage for most renewable shares, driven by higher specific costs. For instance, for 50% variable renewables, they amount to 11 GWh energy and 4 GW power, opposed to 35 GWh and 6 GW in the pumped hydro case. In contrast, the optimal renewable curtailment rate is higher: for instance, for 50% renewables, it amounts to 6%, Appendix D. Power-to-x: sensitivity with respect to different configurations Figure D.1 plots optimal storage energy capacities for different capacities and full-load hours of the generic power-to-x technology. Specifically, medium full-load hours between around 1, 000 and 3, 500 can trigger substantially lower storage needs.
For lower full-load hours, power-to-x demand can largely be satisfied from renewable surpluses otherwise curtailed, so there is little or no effect on optimal storage capacity. For very high full-load hours, storage needs increase again. This is because of the increasing mismatch of the time profiles of additional power-to-x demand and renewable availability, which triggers dis-proportional renewable capacity expansion, and in turn increases renewable curtailment and the optimal amount of electrical storage.