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When the wind of change blows, build batteries? Optimum renewable generation and energy storage investments. Christian Kaps Simone Marinesi Serguei Netessine March 8, 2020 Abstract Renewables have become the cheapest energy sources in most of the world but their generation remains variable and difficult to predict. However, recent technology advances have rendered large- scale electricity storage economically viable, thus mitigating the renewable intermittency issue and enabling combinations of e.g. wind and batteries to replace fossil fuel power plants. Yet, it is not well understood what the optimal capacity combination of these new technologies is. We study this by developing a two-product newsvendor model of a utility’s strategic capacity investment in renewable generation and storage to match demand while using fossil-fuel backup in case of shortages. Although today’s storage investments are mostly lithium-ion batteries, we find that for longer duration storage, capacity decisions are driven more by per-unit-cost than efficiency thus favoring less expensive, less efficient technologies such as thermal. We determine the existence of an adoption threshold beyond which investment into storage increases sharply. Furthermore, we prove renewables and storage to be complements and establish optimal capacity guidelines based on market and technology parameters. Yet, based on real-life data from Europe and the US, we show that in settings with two separate investors in renewable generation and storage, the latter will typically be under-invested in as the market under-values the replacement of conventional generation through marginally free renewables. Lastly, if pollution externalities were priced-in, multi-GWh storage would become optimal in all studied markets. 1 Introduction As societies, utilities, and regulators increasingly move towards renewable energies they face the challenge of supply intermittency. The two main renewable sources with large-scale capacity ad- ditions globally are wind and solar. Yet, generation capacities are not created equal and while conventional power plants run over 90% of scheduled time, wind turbines are producing electricity at varying rates and the sun does not always shine. Simultaneously, demand for energy is largely price-inelastic, cannot be backlogged, and is itself cyclical across days, weeks and seasons. Hence, when shifting energy grids towards more renewable generation, one needs to find ways to match demand with increasingly intermittent supply. Budischak et al. (2013) identify four options to manage such variable generation: 1) geographical expansion, 2) diversifying resources 1 3) storage, and 4) fossil backup. Points 1 and 2 reduce the effect magnitude through diversification, but are often infeasible for geographical reasons, while 3 and 4 eliminate it through supplementary investments. In this paper, we are interested in the latter two cases given their applicability in all possible market scenarios and the fact that there have been four concurrent developments in the last years that make a grid-level-storage approach sought-after, technically feasible and potentially profitable. The first is political in nature, with many national and regional governments enacting regula- tion that requires minimum renewable energy generation ratios in future decades. These include California, aiming at 100% renewables until 2045 (Bill 2015), Germany (Bundestag 2016) set to achieve 50% by 2030, and China which has committed to 35% by the same year (Shen 2018). 1 Combining different renewable technologies with loosely or negatively correlated generation to reduce the intermit- tency and volatility of the joint output 1

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Page 1: When the wind of change blows, build batteries? Optimum ...€¦ · multi-GWh storage would become optimal in all studied markets. 1 Introduction As societies, utilities, and regulators

When the wind of change blows, build batteries? Optimum

renewable generation and energy storage investments.

Christian KapsSimone MarinesiSerguei Netessine

March 8, 2020

Abstract

Renewables have become the cheapest energy sources in most of the world but their generationremains variable and difficult to predict. However, recent technology advances have rendered large-scale electricity storage economically viable, thus mitigating the renewable intermittency issue andenabling combinations of e.g. wind and batteries to replace fossil fuel power plants. Yet, it is notwell understood what the optimal capacity combination of these new technologies is. We studythis by developing a two-product newsvendor model of a utility’s strategic capacity investmentin renewable generation and storage to match demand while using fossil-fuel backup in case ofshortages. Although today’s storage investments are mostly lithium-ion batteries, we find that forlonger duration storage, capacity decisions are driven more by per-unit-cost than efficiency thusfavoring less expensive, less efficient technologies such as thermal. We determine the existence ofan adoption threshold beyond which investment into storage increases sharply. Furthermore, weprove renewables and storage to be complements and establish optimal capacity guidelines basedon market and technology parameters. Yet, based on real-life data from Europe and the US, weshow that in settings with two separate investors in renewable generation and storage, the latterwill typically be under-invested in as the market under-values the replacement of conventionalgeneration through marginally free renewables. Lastly, if pollution externalities were priced-in,multi-GWh storage would become optimal in all studied markets.

1 Introduction

As societies, utilities, and regulators increasingly move towards renewable energies they face thechallenge of supply intermittency. The two main renewable sources with large-scale capacity ad-ditions globally are wind and solar. Yet, generation capacities are not created equal and whileconventional power plants run over 90% of scheduled time, wind turbines are producing electricityat varying rates and the sun does not always shine. Simultaneously, demand for energy is largelyprice-inelastic, cannot be backlogged, and is itself cyclical across days, weeks and seasons. Hence,when shifting energy grids towards more renewable generation, one needs to find ways to matchdemand with increasingly intermittent supply.

Budischak et al. (2013) identify four options to manage such variable generation: 1) geographicalexpansion, 2) diversifying resources1 3) storage, and 4) fossil backup. Points 1 and 2 reduce theeffect magnitude through diversification, but are often infeasible for geographical reasons, while3 and 4 eliminate it through supplementary investments. In this paper, we are interested in thelatter two cases given their applicability in all possible market scenarios and the fact that therehave been four concurrent developments in the last years that make a grid-level-storage approachsought-after, technically feasible and potentially profitable.

The first is political in nature, with many national and regional governments enacting regula-tion that requires minimum renewable energy generation ratios in future decades. These includeCalifornia, aiming at 100% renewables until 2045 (Bill 2015), Germany (Bundestag 2016) set toachieve 50% by 2030, and China which has committed to 35% by the same year (Shen 2018).

1Combining different renewable technologies with loosely or negatively correlated generation to reduce the intermit-tency and volatility of the joint output

1

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The second trend is the ever-decreasing cost for renewable energies, with wind generation costsdown 40% compared to 2009 levels and photovoltaic prices down by 70-80% respectively (IRENA2017). This has made renewable energy per unit of production (typically expressed in cost perMegawatt-hours (MWh)) increasingly competitive.

Third, carbon emissions are under scrutiny in international treaties, such as the Paris and Ka-towice climate accords (EU Commission 2015) and related strategies are investigated by economicscholars and institutions (Nordhaus 1994) (World Bank 2017). Regardless of eventual emissionenforcement mechanisms, renewable energies can substitute carbon-intensive, conventional tech-nologies.

Lastly, the cost of energy storage has been decreasing simultaneously, making industry-scalestorage economically viable. Tesla showcased in 2017 that multi-MWh batteries can be built andoperated profitably (Koziol 2017) and the industry’s exploitation of value stacking (using batteriesin multiple energy-related functions) improves their cost-benefit ratio. Currently, several multi-100MWh projects are under construction, partially designed to replace former power plants such asthe Moss Landing Power Plant in California2. Consequently, the International Energy Agency pre-dicts the global energy storage market to grow by 16% annually until 2030 (Cozzi and Gould 2018).

In combination, these trends make renewable energy grids not only politically desirable, buteconomically attainable while potentially offering lower long-term costs and higher sustainabilityat the same time. Hence, managing the operational aspect of supplying customers with electricity24/7, especially with sporadic generation is of utmost relevance.

In this paper, we propose a stylized, analytical model with two stages to study the capacity in-vestment decision in storage and renewable generation. In a first stage, a monopoly utility providerdecides on a combination of renewable generation capacity and storage to satisfy demand, whilewe assume back-up plants to already exist as the current generating technology. The subsequentsecond stage occurs after the decision is made and represents the realization of generation andstorage utilization over the lifetime of the investment.

With this model, we can answer multiple broad questions: What is the strategic trade-off be-tween the two choices (renewable plus storage vs. fossil fuel backup)? How does emission-pricingchange the optimal investment decision? What kind of storage technology is required to enable arenewables-first grid?

Interestingly and contrary to researchers (Kittner et al. 2017, Diouf and Pode 2015) and policymakers (Tsiropoulos I. 2018) suggesting that lithium batteries, with their high efficiency and marketpenetration, may be the future technology of choice, we find that cost-per-unit matters morewhen storing electricity across multiple hours or days thus favoring cheaper, less efficient storagetechnologies, such as thermal. Empirically, we find that such high-renewable-generation, large-storage electricity markets are attainable, however for renewable plus storage to be cheaper thantoday’s market prices, thermal storage and wind generation would both need to become 30-40%cheaper relative to 2019 levels.

Furthermore, we are able to confirm our prediction that emission tax levels positively correlatewith both storage and renewable generation investment as fossil fuel generation gets penalized.Yet, while our model supports this hypothesis, when calibrating it with real-life data to obtainparameter estimates, it additionally shows that the threshold tax rate beyond which large-scalestorage becomes the most profitable outcome differs widely by market - from none required inGermany over $50 for ERCOT3 to $100 at PJM’s4 optimum. The lower a market’s demandfluctuations and back-up costs (fossil fuel prices), the higher the carbon tax would need to be todrive a transition.

Lastly, we tested the same model under a competitive investment scenario, where separateinvestors individually decide on generation and storage investments to see how that outcome differsfrom the monopoly case described above. We find that even if one allows a transfer price to bepaid from one party to the other, the competitive dynamic results in under-investment by bothparticipants driven by a lack of alignment between generator profit and social value of replacingfossil generation.

2https://www.prnewswire.com/news-releases/vistra-energy-to-develop-300-megawatt-battery-storage-project-in-california-300674786.html

3Texas’ independent electricity system operator (ISO)4North Eastern US’s regional transmission organization (RTO)

2

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To the best of our knowledge, this paper is the first to explicitly model the storage capacityinvestment decision to complement the technology choice between renewable generation and fossilback-up in an operations context. Additionally, achieving the strategic goal of fulfilling the overallenergy demand without assuming the existence of a market in which the agent is a price-taker isnovel in the operations literature, thereby allowing this model to be applied in off-grid or small-scalecases as well. Methodologically, our contribution is to show the behavior of capacity investmentin renewable and storage under changing technology parameters and electricity demand structurefor both, monopoly and competitive cases. Furthermore, we provide numerical estimates of howfuture technology research or policy changes may impact the optimal investment decisions acrossseveral markets.

2 Literature Review

Given the broad relevance of renewable energy and storage, our paper is at the intersection ofmultiple research streams and operational topics. At its core, the investment case deals with theintricacies of capacity management under uncertainty, an area for which Van Mieghem (2003)provide an excellent review. This includes the classic decision of long-term investment facingmarket variability (Arrow 2017), but also how decisions change when different options of fulfillingdemand are available (Shumsky and Zhang 2009) as well as how flexibility impacts such capacitychoice (Boyabatlı and Toktay 2011). Wang et al. (2013) point out that such investment decisionsare increasingly common as many industries are changing production and distribution practices tobecome more sustainable.

Thematically, this paper relies on energy research that includes work by Chao (2011) on effi-cient pricing in electricity markets; emission cost’ impact on profitability and technology choice(Drake et al. 2016); the effect of net-metered energy on utility’s profitability (Sunar and Swami-nathan 2018) and incentive-compatible remuneration contracts (Kildegaard 2008) which becomesrelevant in the competitive case to avoid market-based under-investment. Additionally, there isa broad field of research on the technical feasibility of renewable grids, from comparing differenttypes of storage (Dunn et al. 2011) over cost-minimal combinations of technologies to achieve highrenewable penetration (Budischak et al. 2013) to the long-term impact of large-scale wind energydeployment (Miller and Keith 2018).

Generally, there is a divide in the operations community on how to best approach the inherentcomplexity of storage, given that it always requires multiple periods, at least one source of stochas-ticity and keeping track of the ”inventory” of the storage unit, i.e. the charge. On one end of thespectrum, Jiang et al. (2014) employ large-scale models and efficient algorithms to optimize overlarge parameter spaces to establish lower bounds on algorithmic solution quality. Similarly, (Kimand Powell 2011) use parametric models to derive optimal energy commitment conditions in theelectricity market. However, it is difficult to extract high-level implications from such computer-guided analyses beyond the individual optimization case. Even if solutions are obtainable in closedform in these papers, they typically do not easily lend themselves to human interpretation andmake it thus difficult to base decisions on them.

Hence, the second avenue of Aflaki and Netessine (2017) is arguably closer in spirit to thispaper. Aflaki and Netessine (2017) employ a higher level of abstraction and aim to derive general-izable, strategic investment insights for renewables using a Newsvendor approach. They concludethat in the presence of renewable intermittency, an increasing renewable generation share mighteven increase carbon emissions due to carbon intensive back-up plants. Analogously, Kok et al.(2017) use a Newsvendor-style model to solve an investment capacity problem between conven-tional and renewable energy sources. They find that flexible, conventional sources and renewablesare complements.

Yet, to the best of our knowledge, there currently exists no paper that considers the strategicrole of storage investments. While there are some operational papers on storage in the context ofrenewable energy, they have a different scope. Qi et al. (2015) look at the combination of grid-interconnection and storage to improve dispatchability of an individual wind farm. They are ableto show the existence of lower and upper bounds for storage sizes but focus more on the grid anddeployment aspect than the storage aspect and do not investigate storage’s overall market impact.Luo et al. (2015) calculate the optimal battery capacity in a similar wind park setting, but thepaper is simulation-based and focuses on a single market participant without any bargaining power.

3

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Another paper that investigates the role of storage is by Schill and Kemfert (2011) who focus on theeffect of pumped hydro in the German oligopoly market. This paper finds that pumped hydro doesnot affect a participants market power and its storage capacity is generally underutilized. Strategicinvestment analysis was not a part of the paper. Lastly, Song et al. (2012) discuss storage on theindividual project level, with an emphasis on the state-of-charge of a battery, but don’t consideran entire energy market, back-up costs or the existence of alternative generation technologies.

One paper that looks at optimal energy storage capacity investment is Avci et al. (2014). Herehowever, storage capacity is analyzed in the context of an electric vehicle charging station wherenot the combined or total charge was considered but the optimal number of replacement batteriesfor a recharge station was investigated. The authors employ a repair process model to capture therecharging process, as typically used in the spare-part literature (Muckstadt 2004).

This paper therefore expands the existing operations literate on energy storage by present-ing a way to jointly model energy storage and intermittent renewable generation capacity whileconsidering conventional sources as back-up capacity, charging/discharging efficiency and emissionprices.

3 Model

With our model we aim to capture the strategic trade-off between intermittent renewables combinedwith storage and fossil fuel back-up. Operationally, the decision is between two technologies, acheaper and less predictable (renewable) technology and a more expensive, yet always availablealternative. Storage can then be conceived of as a costly means of reducing the variability of theformer option.

We formulate the problem as a 2-stage, 2-product newsvendor. In stage one the utility decideson the optimal renewable generation - storage capacity combination. In stage two, demand andgeneration are realized over n stochastically identical periods and the demand is met by deployingthe capacity from stage 1, while supply shortages are met through the fossil fuel back-up.

Inherent in modeling storage is the need to consider at least two periods to allow for chargingand discharging to occur. Hence, each of the n periods represents one day, consisting of a nightsub-period followed by a day sub-period. This captures the main source of demand variation inelectricity markets and simultaneously provides structure to the storage decisions. Focusing onthese scenarios allows us to study the technology trade-offs without the need to assume replace-ment dynamics of adding renewable capacity to an existing electricity market or the impact ofintermittent generation on clearing prices.

Real-life examples of such a setting can be found in virtually all off-grid electricity use-cases,from islands using diesel-generators to fulfill inhabitants’ electricity needs to remote mines burn-ing gas to power operations. REIDS, a Singaporean-based project focuses on exactly the energytransition case we describe by realizing it for islands around Asia and Oceania (Choo 2017). Theirfocus is on electrifying or repowering off-grid islands with renewable micro-grids that only relyon diesel-generators as a last resort. On a larger scale, a related case can be seen in Taiwan’spush towards renewable energy with its 2017 ’Energy Development’ legislation, auctioning of 5.5GW of wind projects as well as introducing a cap-and-trade system. While still partially utilizingconventional generation, Taiwan aims to transition a quarter of the country’s energy demand torenewables (of Economic Affairs 2017). Other examples are a hydro-plus-storage micro-grid aimedto increase energy stability to the remote town of Cordova in Alaska Holly (2019) or a cocoa pro-cessing plant in Ijebu Imushin, Nigeria coupling solar and storage to ”significantly reduce operatingcosts” compared to previous diesel generators (Solarcentury 2019).

The paper proceeds as follows: Sections 3.1 and 3.2 introduce the model structure, section 3.3the monopolist’s objective function and analytical results. Section 3.4 develops the competitivecase and shows its analytical properties with section 3.5 stating potential extensions. Subsequentlychapter 4 discusses the data origins, while chapter 5 contains the empirical findings.

3.1 Demand structure

Each of the n periods in the model represents a 24-hour cycle that is further subdivided into twosub-periods, day and night. Let a be the fraction of the day sub-period with a deterministic demand

4

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DH occurring across the entire sub-period. Each day is followed by a night that lasts (1 − a)24hours with cumulative demand being DL, while DH ≥ DL. A split with a = 14

24 offers the closestfit to empirical observations in the studied markets of Germany, Texas and the North-Eastern US,but one could naturally assume different sub-period lengths.

3.2 Generation and storage technology

Let Qc be the capacity of the renewable generation investment (expressed in e.g. MW) andQ = 24Qc the maximum possible generation during a 24-hour period (in MWh). We assume thatthe renewable source’s generation is distributed uniformly QH ∼ U [0, aQ] during the day andQL ∼ U [0, (1− a)Q] during the night, independently of each other and independently of demand.Marginal generation costs are 0. While certainly a simplification of the true production curves,this depiction captures the essence of supply intermittency and renewable generation variability.This approximation is a more accurate representation of reality when renewable generation isbased on a natural factor that occurs along a continuum such as off-shore wind speeds and windpower generation and less accurate for dichotomous phenomena such as diurnal sunshine for photo-voltaic generation. Ceteris paribus, generation pdfs with more bunching around specific values (e.g.normal or two-point) result in storage capacity decisions that vary more quickly based on whetherthe modal utilization case (or the few most likely capacity utilizations) are profitable in expectation.The unit costs cQ are linear and distributed equally across all n periods, the assumed lifetime of awind turbine.

To account for different generation technologies, we re-scale these costs by the capacity factor5

r, so cQ = cunscaledQ ∗ 0.5/r. Note that potential renewable subsidies can be seamlessly priced into

this model by calculating the expected subsidies over the lifetime and adjusting the unit costsaccordingly. We presume Q ∈ [max(DH/a,DL/(1 − a)), (DH + DL)/0.5], which is the real-liferange for which daily storage is sensible - sufficient renewable generation so that there is a non-negative probability of generating more electricity than demanded but in expectation generationdoes not outstrip demand. In sum, the kind of generation we describe is a stylized wind park withfluctuating generation for which the size of the wind-park is the decision variable.

Furthermore, let K be the size of the storage, measured in power over time (e.g. MWh) itcan store. Unit costs cK are linear in MWh and are distributed equally across all n periods. Thestorage exhibits cycle efficiency 0 ≤ e ≤ 1, where 1 − e units of energy are lost in each charg-ing/discharging cycle. This efficiency is a core metric for storage technologies as a perfect systemwould not lose any energy in the charging/discharging process and return 100% of the originallystored energy. But among other things, secondary reactions in a battery and gradual temperaturedecline in thermal storage systems lead to energy dissipating in real-world installations. Next tounit cost, this factor is of utmost importance when choosing a storage solution. The storage sys-tem can be charged during a sub-period if a) there is more renewable generation than demand,and b) there is enough storage capacity to absorb the charge. We assume discharging to alwaysbe possible, which allows for a more tractable model, while limiting the maximum battery sizeto the daily demand DH . In combination with the bounds on the generation, this ensures thatoccurrences of generation outstripping demand in multiple consecutive periods, leading to chargedstorage without it being discharged are unlikely. From a real-life perspective, these bounds arenot restrictive as larger capacities than that would only become interesting in the long-term futurewith storage designated for multi-day, weekly or even seasonal use, which would require a differentmodeling approach all-together.

Given the fast-paced nature of the energy storage industry, we built the model to enable therepresentation of virtually any type of technology. The three main areas of advancement in storagetechnology are unit cost, cycle-efficiency and number of discharge cycles. Cost and efficiency aredirectly captured through parameters in the model, while discharge lifecycles are incorporatedby splitting the investment cost over the respective number of days/periods that correspond tothe anticipated lifetime. For renewable generation technology the key features are unit cost andcrucially intermittency, for which each model can be parameterized, while lifetime of the investmentcan be incorporated analogously to the storage case.

The back-up capacity has no initial investment cost as it is assumed to already exist, butthe marginal production cost is g per unit of energy. Imagine such back-up as a gas-fired powerplant that can respond quickly to changes in demand but requires natural gas for generation

5Average energy output relative to maximum possible capacity

5

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thereby incurring non-zero marginal cost. We assume this technology to be able to generateenough electricity to satisfy demand and to be always available.

3.3 Monopoly Case

We are solving the two-stage problem backwards, by calculating the second stage profit and thenfinding the optimal, first-stage capacity investment decision. The monopoly/joint objective func-tion for average daily profit can be expressed as a combination of five components, two for thenighttime, two for the daytime and one cost-term. Each unit of renewable generation saves thecost g of the fossil back-up it replaces. Given a maximum generation capacity choice Q, andthe aforementioned uniform distribution during a day sub-period, where the density function hasheight 1

aQ, one obtains the following revenue:

g

aQ

{∫ DH

0

Q dQ+

∫ aQ

DH

DH dQ

}(1)

The same logic holds true for the nighttime sub-period with the main difference that numberof hours and demands must be adjusted:

g

(1− a)Q

{∫ DL

0

Q dQ+

∫ (1−a)Q

DL

DL dQ

}(2)

These two expressions however only describe the case without any storage. Storage now is acostly technology that allows the usage of excess generation at some time to be carried over intoperiods of under-supply, as long as there is sufficient storage capacity. Therefore, each unit ofbattery discharge (charge times factor e) saves the back-up cost g. For day and night respectivelythe storage revenue then is:

ge

aQ

{∫ DH+K

DH

(Q−DH) dQ+

∫ aQ

DH+K

K dQ

}(3)

Again, the nighttime sub-period works similarly after adjusting the number of hours and thedemand:

ge

(1− a)Q

{∫ DL+K

DL

(Q−DL) dQ+

∫ (1−a)Q

DL+K

K dQ

}(4)

Combining the four revenue parts, adding the cost term for the generation capacity and forstorage, one reaches a final objective function of:

Π =g

aQ

{−DH

2

2− K2e

2−KeDH

}+

g

(1− a)Q

{−DL

2

2− K2e

2−KeDL

}+

g(DH +DL + 2Ke)− cKK − cQQ(5)

This derivation assumes all integrals to be positive, which is the case within the bounded rangeof Q considered. For a step-by-step derivation of the objective function, please refer to AppendixA which also contains the first and second partial derivatives (equations 12 - 16).

3.3.1 Complementarity and No Storage Case

The model is clearly concave as the second partial derivatives can easily be signed (see appendixA), resulting in a Hessian that is negative semi-definite as long as K and Q are non-negative.Furthermore, the cross-derivatives are positive indicating that generation and storage investmentsare complements - an increase in one increases the other’s optimal solution. As Q is multiplied withother parameters to obtain the optimal storage capacity K, we obtain the first part of theorem 1.

Subsequently, we first solve for the base case without any storage by setting K to zero and takingthe first partial derivative with respect to Q to obtain the following optimal solution. Beyond thatone can investigate in which cases the monopolist would decide to set K = 0.

Theorem 1 Renewable generation and energy storage are complementary investments. Yet, whilerenewable investment can be optimal without storage, storage without intermittent generation isnever optimal.

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The optimal generation capacity without storage in the monopoly case is:

Q∗No storage =

√g

acQ

D2H

2+

g

(1− a)cQ

D2L

2

which occurs when (2− cKge )Qa(1− a) ≤ DH(1− a) +DLa.

The optimal generation solution is a weighted sum of the day and night-time sub-period andtheir respective demands, where the square-root and the squared demand terms result in a quasi-linear relationship. Storage is not invested in when the financial benefit from substituting back-upcost through costly back-up (2− cK

ge ) for a fraction of the renewable portfolio a(1−a)Q is less thanthe duration-weighted demands.

3.3.2 Storage Case

Now, allowing for positive values of K, one can obtain optimality conditions via the two first partialderivatives w.r.t to Q and K.

Theorem 2 The optimal generation and storage investment capacities in the monopoly case are:

Q∗ =

√g

acQ

{DH

2

2+K2e

2+KeDH

}+

g

(1− a)cQ

{DL

2

2+K2e

2+KeDL

}

K∗ = max((2− cKge

)(a(1− a)Q)−DH(1− a)−DLa, 0)

Again, the generation capacity decision is a weighted sum of the day and night-time sub-periodand their respective demands to which the storage parameter is now added. While a square-rootat first sight, demand parameters and the storage decision are present in squared terms, resultingin order-of-magnitude linear effects of increased storage and demand, as long as the increase occursapproximately simultaneously across both sub-periods. The other decisive factor for generationcapacity is the balance between the cost of the renewable compared to the back-up it replaces asevident from the g

cQterm. We will investigate how a change in g effects generation decisions in

real-life scenarios, e.g. by imposing a carbon tax on fossil fuel plants.

Conversely, the storage investment decision is largely driven by the ratio of back-up replace-ment benefit to unit cost although storage efficiency has to be factored in as well now (2 − cK

ge ).

Interestingly, storage is less beneficial the more unequal the day/night distribution becomes, asevident from the a(1 − a) factor, which is maximized by a = 0.5. This reflects the intuition thatstorage can be most profitably used when it is continuously shifting demand and thereby utiliza-tion is high, rather than sitting charged and idle to be used in rare occasions, thereby increasingcost-per-use.

Furthermore, the storage decision is contingent on the difference between renewable capacityand weighted demand to be large, which shows that storage is best used when maximum productioncapacity outstrips demand. Although we don’t allow for negative storage investments, zero storageinvestments occur when either storage costs are too high, renewable generation too low relative todemand or the replacement alternative too cheap.

Analytically, a more direct way of solving the objective function is to substitute in K or Q fromone objective function’s partial derivative into the other, e.g. for K as displayed below.

g

acQ

{DH

2

2+K2e

2+KeDH

}+

g

(1− a)cQ

{DL

2

2+K2e

2+KeDL

}=

(K +DH(1− a) +DLa)2

((2− cKge )a(1− a))2

(6)Beyond the optimal investment decisions by themselves it is also instructive to see how these

change with respect to the other model parameters to gauge how solutions would change acrossdifferent markets and how technological advancements might impact outcomes as well. The full setof comparative static derivatives can be found in appendix A.5 and are directionally as one wouldexpect.

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3.4 Competitive Case

Aside from the case in which all investment decisions are made by the same entity, we also considerthe model’s behavior in the case that one player invests into renewable generation and a secondplayer into energy storage. In this scenario it assumed that renewables are dispatched first andthat the generation investor is responsible to operate and pay for the backup when necessary.

In case of excess renewable generation, the generator will sell this energy at a per-unit price tto the storage investor. It is assumed that the generator can set this transfer price to optimize herown payoff, but one would reasonably expect t ∈ [0, ge]. The storage investors owner then storesthe energy and, when needed, sells it back to the generator minus the efficiency loss 1− e for g (org − ε) at which point there is marginal price indifference between using the fossil backup or thestored energy.

In this setup, the objective function for the generator looks similar to the aforementionedmonopoly objective function, solely with a different payoff per unit of storage which is now t.Similarly, the storage investor only generates the marginal ge − t whenever storage is dischargedand has to pay for its system. It shall be noted that the monopoly and the competitive cases areidentical in the absence of storage.

ΠGenerator =g

aQ

{−D2

H

2

}+

t

aQ

{−K2

2−DHK

}+

g

(1− a)Q

{−D2

L

2

}+

t

(1− a)Q

{−K2

2−DLK

}+

g(DH +DL) + 2tK − cQQ(7)

ΠStorage =ge− taQ

{−K2

2−DHK

}+

ge− t(1− a)Q

{−K2

2−DLK

}+ 2[ge− t]K − cKK (8)

For a step-by-step derivation of the objective functions, please refer to Appendix B which alsocontains the first and second partial derivatives (equations 48 - 54).

3.4.1 Generation, Storage and Equilibrium Existence

Before analyzing the competitive case analogously to the monopoly case above, we first have toensure the existence of an equilibrium outcome, which as a results of the two participants’ responsefunctions can be shown to be guaranteed. In order to achieve this, one needs to proof that bothparties’ second partial derivatives are negative across the entire, convex parameter space. If thisholds, the response functions will intersect at least once, hence at least one pure strategy Nashequilibrium exists (Debreu 1952). In our case, the second partial derivatives found in equations50 and 54 are both negative, hence the competitive case has at least one pure strategy Nashequilibrium.

To show uniqueness of the equilibrium, we follow the topology-based proof from Cachon andNetessine (2006), for which we have to show that the matrix of second and cross derivatives (foundin equations 50, 51, 54, 55) has a determinant greater than 0.

One can clearly sign all derivatives as long as ge−t > 0, which is a trivial condition as otherwisethe storage owner would loose money even if the technology cost was zero. One is then left withproving the relative sizes of the derivatives result in a positive determinant.∣∣∣∣∣ ∂

2ΠQ

∂Q2

∂2ΠQ

∂Q∂K∂2ΠK

∂K∂Q∂2ΠK

∂K2

∣∣∣∣∣ > 0 (9)

We thus have to show that∂2ΠQ

∂Q2∂2ΠK

∂K2 >∂2ΠQ

∂Q∂K∂2ΠK

∂K∂Q , for which we analyze the magnitude of

the derivatives. The difference of the derivative products even when t = g (see equation 57) ispositive, and increasing in the difference between g − t, so the determinant is positive and theequilibrium thus unique.

Theorem 3 For a given transfer price in the competitive case, there exists one, unique equilib-rium between the generation and storage investors’ behavior. The optimal generation and storageinvestment capacities in the competitive case with a given transfer price are:

Q∗ =

√g

acQ

{D2H

2

}+

t

acQ

{K2

2+DHK

}+

g

(1− a)cQ

{D2L

2

}+

t

(1− a)cQ

{K2

2+DLK

}

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K∗ = max

(a(1− a)Q(2− cK

ge− t)− (1− a)DH − aDL, 0

)We provide the optimal solution for both decision variables as a function of parameters only in

appendix B.2, but they become unwieldy and are less interpretable as the above-presented solutionswhere the decision parameters are dependent on each other. Although we are not able to solvefor the optimal transfer price t in an interpretable closed-form, we can shine some light on thedynamics that shape the equilibrium, for which the generator would choose the t that maximizesits profit. The transfer price is bounded by [0, ge− cK

2−DHaQ − DL

(1−a)Q

] as we show in equation 59. At

both ends of these bounds, the generator invests in as much capacity as in a non-storage case.If t = 0 this occurs, because there is no profit in selling the electricity to the storage owner andthe generator is indifferent between storage discharge and back-up generation. Yet, assuming thatstorage is at least profitable for t = 0 (otherwise the competitive market has the same outcomefor every transfer price), there will be non-zero storage. The generator’s profit first increases in tthrough the revenue paid by the storage owner revenue. But as t becomes too high, the marginalincrease in price does not compensate for the marginal decrease in storage capacity, thus ultimatelyreturning the generator capacity and profit to the no-storage case at its upper bound. The storageowner’s profits decrease in t, although the capacity may slightly increase at first and then startdropping as t becomes too high. Concrete examples of how such solutions look like are providedin the empirical section.

The storage investor is now one step removed from the back-up her technology replaces andrather than the absolute value of g, the margin of ge − t is now the driving force in deciding theinvestment quantity.

Taken together, the transfer price t can be seen as a mechanism to divide the social benefit ofstorage (replacement of g) between the generator and the storage investor.

Theorem 4 In a competitive situation in which the generator sets the transfer price, the storageinvestment capacity will be lower than in the monopoly case for the same renewable capacity as2− cK

ge−t < 2− cKge , thus decreasing the storage margin relative to the monopoly case. Hence, if the

storage investor were to set the price, it would be set to 0. As t < ge, the generation capacity willalso be lower in the competitive case than in the monopoly.

The remainder of the solution is structurally similar to the monopoly case. The now price-and-time-weighted discrepancy between renewable capacity and demand relative to the transfer pricesand storage costs drive the optimal investment. The aforementioned factors of day/night durationsimilarity, technology efficiency and unit-cost operate in the same manner. Additionally, if thestorage margin including the transfer price compensates for the money lost through inefficiencyand unit cost 2(ge− t) ≥ cK , more renewables will result in more storage.

In order to better understand the model dynamics in the competitive model, we follow Bernsteinand Federgruen (2003) to provide the comparative statics in appendix B.3 . We find interestingthreshold behavior for certain parameter changes (e.g. e and DL) where opposite effects competein determining the direction of the equilibrium quantity changes. In the case of the efficiencyparameter, more efficient storage requires less capacity for the same discharge quantity, but atthe same time storage can be operated more profitably. For the night-time demand, a decrease involatility and an increase in total energy use pull the equilibrium into different directions. In theappendix, we provide thresholds beyond which the different behaviors are observable.

3.5 Extensions

After having developed the main model, we will now provide multiple extensions that move towardsmore realism at the expense of slightly reduced tractability. We first introduce the extensions thatrequire smaller changes while subsequently presenting a more involved model that uses a differentintermittency assumption, before juxtaposing models and computer simulation results in chapters4 and 5.

3.5.1 Accounting for Emissions

One of the major drivers of change in the energy sector in the 21st century has been the increasedscrutiny of pollution mainly in the form of carbon emissions. Carbon emissions are a classic ex-ample of market failure through the inability to appropriately price-in negative externalities. Aseconomic theory predicts, if negative externalities are not accounted for, the good in question will

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be over-utilized and the market outcome will be inefficient - in the energy case this means toomuch carbon intensive generation and pollution than socially optimal. At the same time, manylocal and national efforts have been undertaken to put a price on carbon, whether through taxa-tion or cap&trade systems, designed to move towards more sustainability. So far however, theseframeworks have only regulated individual industries or small parts of businesses and the resultingmarket carbon prices are so low that they would require very high discounting of future costs andrisks to be considered properly priced (Glanemann et al. 2020). Even more importantly, withoutinternalizing emission costs, energy storage can actually increase carbon costs by owners utilizingit for energy arbitrage and nonstrategic deployment (SGIP 2017).

Rather than debating what the right process is to charge for emissions, we will allow for anycarbon price to be reflected in the model and later analyze what would happen to the solutions ifcarbon was priced closer to what many (e.g. the United Nations or the World Bank) believe to bethe true social cost of emissions. Let cC denote the additional cost of one ton of C02 equivalentemissions. If this is set to a value cC >> 0 it has multiple consequences in our model.

Most obviously, it raises the costs of the back-up technology as this technology uses fossil fuels.Given that wind’s emissions are on the order of 40-90 times lower than gas or coal, we do notconsider them explicitly. If one assumes the pollution intensity of the backup technology to be vG,the back-up cost has to increase at least from gNew = gOld + cCvG.

Second, the different storage technologies’ manufacturing emissions are priced in as well. LetvK be the manufacturing carbon intensity per MWh of storage, which are then divided over thetechnologies expected lifetime and impact storage cost in this manner. For contemporary batterytechnologies this adds 150 tons/MWh across all days, or 0.030 tons/MWh/day. For the thermalbattery, the choice and procurement of the storage substrate (e.g. salt) is the biggest driver foremissions, but CO2 emissions below 60 tons/MWh were achieved, which divided by the lifetimeresults in 0.005 tons/MWh/day.

Third, increasing energy prices will change the fossil fuel mix and electricity prices (whichwe introduce next). Yet, because especially private consumers are price-inelastic in their energydemand, we use the heuristic that electricity prices increase by the average plant emissions timestax, which takes into account a region’s existing power generation portfolio in the short run. Whatbecomes obvious here is that the gas back-up plant technology at high emission prices would becheaper across the board than existing coal-based plant portfolios. So even disregarding willingnessto pay or prices altogether - high emission taxes will drive coal out of the market in favor of naturalgas, as is already happening in the US and other countries today.

Thus, by accounting for emission costs, the back-up capacity becomes relatively more expen-sive than the storage options, thereby increasing the optimal capacity investments in storage andrenewable energy. However, the effect magnitude is dependent on the other parameters and theoptimal solution in the no-tax case. With this extension though policy makers can gauge whatlevel of emission pricing would result in which investment change.

3.5.2 Electricity Prices

In the basic form of our model, the only energy related price is the back-up cost that renewablesreplace (e.g. a gas plant). Because of the high required degree of flexibility, this type of fossil fuelgeneration is more expensive though than current levels of energy prices. As an intermediate step,before modeling a full electricity market, the introduction of e.g. average prices during day andnight pH , pL with pH ≥ pL allows for a more granular pricing distinction and also can highlight thehigh-value proposition of storage. The prices here are understood as a willingness to pay for a goodat a certain time not as a market clearing mechanism. One could then envision that renewablegeneration is remunerated at the respective market prices per sub-period, while storage yields theoriginal rate g, because of its added value of flexibility.

In such a setting, pairing storage and renewable generation becomes more valuable and in thecompetitive case one obtains a partial flavor of energy arbitrage where the storage investor buyscheaper at night and sells during the day. While interesting conceptually and certainly insightfulto investigate academically, the added assumptions of effectively needing three fix prices (low, high,flexible) is typically only seen in markets with time-of-use contracts.

Including different electricity prices changes the renewable generation investment, for which theoptimal solution then is not only the duration-adjusted demand, but duration and price weighted.For the no-storage case, if the average price remains the same, generation capacity would not beaffected. Yet, in the storage case, the outcome would change as it is conceivable that charging a

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battery at high prices only to discharge at low ones would not be profitable, thus reducing the usecase of the battery and reducing its optimal investment which indirectly impacts the renewablegeneration capacity choice.

3.5.3 Transfer Contracts

Another conceivable alteration of the model is to augment the competitive market contract. Whilegoverned through only one fix parameter t in the original case, a natural extension is to also havetwo separate transfer prices during day and night periods tL and tH , which give the generator finercontrol over the payoff structure.

Common real-life practices, such as capacity premiums that the storage investor receives re-gardless of utilization can already be easily accommodated by the current model, by deducting thepremium from the storage cost term, effectively making the variable generation cheaper.

If one favors external validity over closed-form expressions, more intricate pricing schemes areconceivable, with e.g. flat-rate charges for the first half of a stored charge and higher rates fordeeper discharging or a transfer price that varies with the demand realization per period.

Suppose transfer price contracts were developed to account for the electricity price structure,one could design them to avoid socially detrimental outcomes. In the aforementioned case ofcharging at high prices and discharging at low ones, adjusting transfer prices to electricity pricescould increase storage utilization, thus reducing contract inefficiencies and moving the competitiveoutcome closer to the monopoly scenario.

3.5.4 Discharge Probability

So far, we restricted renewable generation to the most relevant cases from a large-scale storageperspective, where renewable energy production can outstrip production occasionally, but notin expectation, thus making it likely for storage discharge to be possible in most cases. In themedium/long-term future energy systems are imaginable in which renewable generation capacityis much larger than mean demand, therefore creating scenarios in which a stored charge may notbe dischargeable for several periods.

For structural results, one could simply multiply the storage terms in the objective function bya discharge probability that is decreasing in Q. Additionally, one could impose a numerical form(e.g. linear) based on discharge probabilities derived through simulations. Good approximationsof said probabilities would allow for the removal of the upper bounds on Q and K as in caseswith large batteries and high renewable generation the constraint discharge occurrences could becaptured adequately, thereby widening the investigate-able parameter range.

3.5.5 Binary Intermittency Model

Until now, the model captures the renewable’s intermittency through a uniform distribution overall possible generation outcomes. This represents an understanding of intermittency as uncertaintyover which ’intensity’ state will be realized. With certainty there will be some wind generation,but it is uncertain if it will be at 26% 87% or any other % of capacity. Yet, papers such as (Aflakiand Netessine 2017) conceive of intermittency as the binary chance of a generation source to beavailable or not. This represents the most extreme form of intermittency from a generation/storageperspective but simultaneously allows for easier aggregation of states as each sub-period only hastwo possible outcomes. As mentioned in section 3.2, the choice of intermittency interpretation islargely influenced by the renewable situation considered.

Combining this different intermittency choice and some of the above extensions, we introducea second model of storage that allows different insights, but only offers concise analytic results intwo out of three possible cases which we will discuss in the following.

In this binary intermittency interpretation, the renewable has a chance of r ∈ [0, 1] to produceat full capacity and otherwise produces nothing at every point in time. This, again, is a simpli-fication of true production curves, but focuses more on the potential of no energy production atany given point in time than on aggregate production over time as in the main model. If onethen shrinks the sub-period during which one evaluates the binary on/off decision, in the limit thedistribution during a day or night subperiod approaches a normal distribution. To that end, letbL ∼ N(r(24−a), (24−a)r(1−r)), bH ∼ N(ra, ar(1−r)) be zero-truncated, normal pdfs capturingthe amount of time the renewable source is producing during an a hour day sub-period and a (24-a)

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hour night sub-period.

Because this model has a stronger focus on hours of production, we also express generation,demands and storage in hours. Let Q be the nameplate capacity of the renewable generationinvestment (expressed in e.g. MW) with linear unit costs cQ that are distributed across thetechnology’s lifetime. Let DL and DH be the deterministic hourly demands during the night andday sub-period respectively, again with DL ≤ DH . The energy prices for renewable generation arepL ≤ pH during night and day, respectively.

Have K denote the size of storage measured in hours of production it can store when chargedduring the night (i.e. Q−DL). Unit costs cK are distributed equally across all n periods and cycleefficiency e is analogous to the main model. Because this model is built around the availabilityof generation and storage in hours, not just in the aggregate, we also model discharge probability.For simplicity, discharging is only possible in the sub-period that follows the period in which thestorage was charged. Furthermore, discharging is virtually always possible for small values of Qand r, but becomes less probable as the renewable generation capacity and up-time increase, whichwe capture through a linearly decreasing probability for all Q ≥ DH . For notational convenience,let V = Q−DL

Q−DH. All energy that cannot be met by generation or storage is fulfilled by the backup

source at a cost of g per unit of energy. This leads to the following objective function, which’sderivation is explained in detail in appendix C.2.

ΠQ,K =(24− a)

{DL(pL − (1− r)g)− rg(DL −Q)+

}+

g (Q−DH)+ e[KV −

∫ KV

0

BH(x)dx](1−

√r(

(Q−DH)+

DL+DH

r −DH

))+ +

a

{DH(pH − (1− r)g)− rg(DH −Q)+

}+

g (Q−DL)+ e[K −

∫ K

0

BL(x)dx](1−

√r(

(Q−DH)+

DL+DH

r −DH

))+ −

cQQ− (Q−DL)+cKK

(10)

As the battery is only able to charge when enough generation capacity exists, this modeldecomposes into three cases, depending on Q.

Case 1: If Q ≤ DL, there is less renewable generation capacity than demand at night, soeven if the intermittent source is producing, fossil backup is needed, and no charge can ever beaccumulated. Hence this is a zero-battery case that is fully computable with expected values.

Case 2: If Q is chosen so that DL ≤ Q ≤ DH the renewable generation capacity is large enoughto satisfy all demand at night, while its generating electricity, any excess capacity may be storedto be discharged during the day. The installed renewables are not sufficient however to cover theentire demand during the day, so available storage needs to be discharged or back-up capacity usedeven during daytime generation.

Case 3: If generation capacity is set to DH ≤ Q, the battery can always be charged whenthe renewable source is generating electricity. This case is closest in spirit to the off-grid real-lifescenarios, given that such systems are designed to over-produce during their up-times and utilizepreviously charged electricity if the wind does not blow or the sun does not shine.

The advantage of this model is that is allows one to tease apart the important difference be-tween overall storage size and storage capacity in hours relative to storage output. Its disadvantageis that the most interesting, yet most complicated case 3 cannot be solved in closed-form anymoreand requires numerical solving. However, we show proof in appendix C.2 that it is a quasi-concavemaximization problem that is well behaved and derive the optimal solutions for cases 1 and 2. Thecompetitive model can be built from here analogously to the original model.

This model not only shows how to practically implement the aforementioned extensions, butoffers a way of thinking about generation not as maximum power across a period, but as installedcapacity and casts storage in the context of ”hours of charge” it can hold. This is closer inspirit to how utilities and electricity managers talk about investment capacities, but more rigid inforcing the battery to have a certain power output (Q−DL) - a decision we abstract from in the

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original model. Especially the case-based thinking can be helpful in understanding under whichcircumstances storage is being used, when conceiving of it as technology to shift load between nightand day. Empirically, the normal distribution and the inclusion of the discharge probability termnaturally lead to different results compared to the original model as was discussed in the extensionsection. Storage investment cluster more around 0 and the distribution means (3-4) which impactsthe renewable capacity decisions as well. Additionally, because hours of storage and power ofstorage (Q−DL) can be analyzed separately, one sees that batteries grow in size mainly throughan increase in the generation-demand-differential (i.e. being able to charge more excess energy atonce), not through being able to be charged for more hours, as even with high emission taxes, 5-6hours of capacity are not surpassed.

4 Data

After analyzing the theoretical properties of the model and suggesting possible extension, we willuse empirical data from various regions across the world to see what our model would suggest withreal life parameters and how changing technology and policy would impact storage and generationcapacity decisions. Most of the analyses will be done with the original monopoly decision modelas it allows to investigate the socially optimal investment without needing to rely on a specificenforcement mechanism or redistribution contract or selection of who gets to set a transfer price.

4.1 Market data

The energy demand and price data were obtained from the organizations coordinating energytransmission in their regions. ERCOT in Texas6, PJM in the US’s Northeast7, and the Bundesnet-zagentur in Germany8. ERCOT was selected, because it is the most independent, market-friendlyand least-integrated energy market in the US given Texas’ desire to not have a federally regulatedenergy market. We chose Germany because of the country’s leadership position in pursuing theEnergiewende (energy transition) towards renewables with more than 40% of the country’s elec-tricity coming from renewable sources. Lastly, we selected PJM as a potentially difficult regionfor storage - with low renewable penetration, high levels of grid interconnection integration andhistorically low prices combined with a lot of carbon-free, nuclear generation.

The data has hourly granularity, is available for at least the past five years and is publiclyavailable from either the website or upon registration to the respective company’s data service. Theemission data was obtained through the same channel for PJM, through the US Energy InformationAdministration for ERCOT 9 and from the EU climate initiative ”Covenants of Mayors for Climateand Energy” for Germany 10. The utilized historic market demand and price data is summarizedin table 1. For the purposes of this paper the first 4 rows are of most importance, while the pricedata will only be used to attain a lower bound in a subsequent analysis.

Table 1: Historic energy demand and price data from three marketsUnit ERCOT Germany PJM

Demand Day (DH) MWh 910,000 1,008,000 1,386,000Demand Night (DL) MWh 350,000 480,000 840,000Back-Up Cost (g) $ 50 75 38Back-Up t CO2/MWh (vG) tons/MWh 0.41 0.41 0.41Price Day (pH) $ 37 58 25Price Night (pL) $ 23 46 19Average CO2 emissions tons/MWh 0.73 0.62 0.51

6http://www.ercot.com/mktinfo7https://www.pjm.com/markets-and-operations/etools/data-miner-2.aspx8https://www.smard.de/home9https://www.eia.gov/electricity/state/

10https://www.eumayors.eu/IMG/pdf/technical_annex_en.pdf

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4.2 Storage data

Parameter estimates for the storage technologies were provided by the proprietary research of Kraft-block, a German energy storage start-up and were validated against publicly available sources, suchas Fu et al. (2018) and reports of contemporary storage installations as well as Larcher and Taras-con (2015) for storage manufacturing emissions. The wind generation assumptions are in line withthe global 2018 renewable analysis in Anuta et al. (2019). Table 2 contains the utilized storageand renewable generation values.

Often, the first thing that comes to mind when considering storage are batteries. With highdegrees of efficiency and wide use-cases from cellphones to cars to industrial storage they are oneof technologies with the largest capacity installations in recent years. The major downside of thisclass of storage is the high unit-cost and the resource-intensive production process. Lithium-ionbatteries will be the high cost - high efficiency technology we analyze. A competitor to this isthermal energy storage - systems in which energy is stored as heat in various conductive materialsranging from sand over concrete or salt to oils. Typically, these storage solutions have lowerlevels of efficiency than batteries but are also less expensive to build and allow for high-densitystorage (stored energy/volume). To make the technologies comparable, we adjusted the cost forthe expected lifetime of the technology, given that batteries have typically a few thousand chargingcycles while thermal solutions may last a multiple of that.

Table 2: Storage technology and renewable generation dataBattery Thermal

$/MWh 330,000 100,000Lifetime in days 5,475 10,950$/MWh / day 60 9Efficiency 90% 45%t CO2/MWh 150 80t CO2/MWh / day (vK) 0.030 0.005

Wind

$/MW 2,300,000Lifetime in days 10,950$/MWh/day11 12.5Gen. Prob. (r) 35%

5 Results

5.1 Model Simulation Comparison

In the following, we will always compare the model results with a simulation in which the generationpatterns are drawn from the uniform distributions across 30.000 day/night periods. We expect tosee the largest discrepancies for cases in which the model predicts generation storage close to itsupper bounds, given that these are the cases where the discharge is always possible assumption isviolated most often. The possible range of parameters we consider are Q ∈ [max(DH/a,DL/(1−a)), (DH + DL)/0.5] and K ∈ [0, DH]. This covers the entire range of having enough generationto at least have a non-zero probability to outstrip demand to generation enough renewable energyto meeting all demand in expectation as well as allowing for storage to be large enough to storeall the energy of the following sub-period.

5.1.1 Battery Versus Thermal

We start in the simplest case, with current technologies and no emission tax across all threemarkets:

Two things become apparent when analyzing these initial results. First, battery technologyat current levels does not seem to be profitable for large-scale energy storage, with virtually allmarkets’ results for model and simulation returning no storage and lower generation than in thethermal case. For the latter technology significant storage investment can be observed in Germanywith the highest back-up costs, while PJM with the lowest costs remains without storage. Secondly,while returning almost identical numbers in the no storage case, the model’s predictions deviatesup by several hundred MWh for generation and storage alike in the high-storage cases.

11adjusted for intermittency r

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Table 3: Model and simulation solutions for all markets and technologies (in ’000 MWh)Market Model Battery Simulation Battery Model Thermal Simulation Thermal

ERCOT Q∗ 1,851 1,835 2,220 1,885ERCOT K∗ 0 0 280 100GER Q∗ 2,755 2,628 2,976 2,703GER K∗ 44 0 554 350PJM Q∗ 2,753 2,751 2,753 2,751PJM K∗ 0 0 0 0

A Word on translating Q and K into capacity investments in MWh As Q isexpressed in maximum, intermittency-adjusted generation over a day and wind-park projects com-monly compared in MW of installed capacity, it shall be noted that one can be translated into theother through Q/24/r ∗ 0.5 = Qc, where Qc is the aforementioned MW capacity that needs to beinstalled. Please refer to the model section 3.2 for more details. K directly refers to the storagein MWh with the implicit assumption that it can discharge with at least DH/(24a) per hour - therate at which electricity is demanded during the day.

Germany’s 2,700 GWh would then equal an installed renewable capacity of 160 GW, whichcompares to 120 GW currently installed (BMWi 2020). For the other two markets, the modelgeneration and the actual generation is further apart with 1,900 GWh, 113 GW in Texas basedon the model and 40 GW installed capacity as well as 2,750 GWh, 164 GW in PJM according tothe model and 30 GW installed capacity. This however does not represent a flaw in the modelbut merely shows the rapid cost-improvements of renewables over the past years. While Germanysubsidized wind and solar heavily in the past to achieve such high renewable capacities, the USmarkets only now ramp up their exploration and construction of these technologies.

The storage capacities with respect to batteries are very close to what is currently deployedin the market - very little capacity if anything, at least not on the GWh scale. Only Germany ismodeled to have some storage, but in reality, the country is special in this regard. Its high levelof grid interconnection to European countries allows it to run high levels of renewables alreadytoday with less storage than predicted as it imports/exports energy across national borders - a”geographical” way of storage. Similarly to the generation findings, the storage results are basedon 2019 prices, yet building an electricity grid takes decades, so current capacity levels reflecthistoric prices. As can be seen from battery not being invested in - expensive storage, even at highefficiencies is not profitable. Comparing the thermal storage investments then to today’s levels isnot instructive as the other two markets with their currently low levels of renewable do not yethave the need for such storage nor historically have they had access to low-cost thermal. Thecapacities can be seen as guidance though to size daily storage for grids of the future.

5.1.2 Profit vs. capacity investment

In the following, we want to understand when the model and simulation results differences instorage investment occur and whether the difference is only for capacities or also has a profitimpact. Using ERCOT’s Texas market as an example, we fix the renewable generation at variouspoints across the studied range and see what model and simulation select as optimal. We thendefine delta capacity as the absolute difference between simulation and model result for storagecapacity divided by the daily demand (DH). Delta profit is the relative difference in profit if onewere to simulate the model’s parameter choices versus the optimal simulation results.

The key take-away from this analysis is that the model, despite resulting in higher optimalstorage decisions for high-emission tax and high-renewable generation cases does not differ bymore than 5% in daily profit compared to the simulation. Furthermore, the model and simulationagree on capacities within less than 1-2% for most larger parts of the parameter space, especiallyin the thermal case.

It seems to be the case that model and simulation both converge on the optimal area of theparameter space, but for a fix renewable generation the model chooses relatively more storagecapacity. This however is not merely a cost-factor, but also increases revenue simultaneouslythereby impacting profitability less than one might expect at first.

This investigation showcases how well the model performs within the previously set bounds,

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(a) Battery (b) Thermal

Figure 1: ERCOT Model vs. Simulation differences in storage capacity and profit for different renew-able generation and emission taxes.

which is the core empirical range for the renewable transition we try to study. In said range,sufficient renewable generation is available to warrant storage, but so little that a lack of storageleads to supply shortages that need to be covered through the back-up source.

With these two comparisons, we feel confident to rely on the model for the following sections.

5.2 Storage results monopoly case

We now discuss the different possible monopoly outcomes and investigate how said results changesunder different model parameters. For ease of exposition, we assume no carbon tax and thermalstorage in all cases. In a subsequent chapter, we then discuss the precise effect of carbon emissionstaxation and battery storage on optimal decisions, while we just depict the overall process for now.

In figure 2 we depict the optimal solution for each of the two partial derivatives with respectto various realizations of the other parameter.

(a) ERCOT (b) Germany (c) PJM

Figure 2: Monopoly optimization

The general derivative pattern across all three markets is similar, but the different magnitudeof g leads to very different optimal realizations. The left graph shows that in Texas, with g of $50,the two partial derivatives intersect within the parameter space, while for PJM (right graph) withthe lowest g of $38, the optimum solution is a corner case with K=0, while in Germany with thehighest g of $75 the other corner-case is observed where the upper limit of storage and generationis optimal.

5.3 Carbon tax and its impact on battery technologies

Judging from the previous section’s results, the back-up replacement cost level can have a significanteffect on market outcomes. In our original assumption of g, we looked at what the marginal cost offlexible generation for these markets had been in periods without generation shortages. However,average market prices are cheaper in today’s electricity markets and renewables are only seldomthe predominant fuel source.

We therefore want to understand how taxing carbon emissions costs at $0, $25, $50 $75 and$100 per metric ton would change the optimal outcomes in current grid system conditions. Theseprices reflect a realistic range, from the lowest cap-and-trade prices for carbon emissions up to the

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cost of the national CO2 levy in Switzerland (approximately $100 per ton12), irrespective of whatone believes to be the true social cost of emissions. We therefore calculate the g values based ontoday’s average prices while taking into account the market’s emission intensities as stated in table1. In the following paragraphs tons will always refer to CO2-equivalent emissions.

Furthermore, it could be that different prices and markets favor different storage technologychoices. Thus, we run the analysis with two technologies to illustrate the trade-off current storageinvestors have to consider - lower unit cost and lower efficiency (thermal) or higher unit cost andhigher efficiency (batteries).

(a) ERCOT (b) Germany (c) PJM

Figure 3: Monopoly optimization outcomes under different emission prices

Multiple dynamics become apparent:First, regardless of market, between $50 - $75 of emission tax leads to large scale renewable

and storage investments, indicating that even the low-emission, low-cost market of PJM wouldtransition from nuclear, coal and gas to renewable-first at these levels. Simultaneously, for marketswith already high costs, the emission prices mainly effect the magnitude of storage investment forthe battery case, but do not change renewable or thermal storage investment.

Second, the thermal storage is always larger than the battery storage, even beyond what thedifference in efficiency would lead one to expect. As the thermal technology loses more electricityin a charge/discharge cycle, one would expect it to be larger than the battery by about factor two(0.9/0.45) for the same desired output, as a function of both technologies’ efficiency. But becausethe thermal storage becomes cost-effective at much lower back-up prices, it gets invested in beforethe battery. ERCOT in sub-figure 3a) is a case in point for this behavior, where the cheaper thermalstorage is not only growing faster than battery, but also drives renewable investment earlier. Onlyat $100 can the battery catch up in capacity. Similarly insightful is the PJM market in sub-figure3c), where overall low prices render the battery solution cost prohibitive even for the high emissioncases.

It has to be noted that these multi-GWh storage scenarios are for predominantly renewablemarkets and do not fulfill the same purpose as the few MW for 2-4 hours that are currently deployedfor frequency regulation and ramping support.

Third, the largest discrepancy between the model without tax and real life is Germany (3b).While predicted installed renewable capacity almost matches Germany’s (BMWi 2017), the real-life level of storage is not nearly as high as found to be optimal. This, as explained above, is atleast partially due to Germany’s grid connections to other countries which it can use to balanceits grid in lieu of storage.

Yet, the general idea that at moderate emission prices storage of renewables is more profitablethan conventional back-up plants holds across the three studied regions.

5.4 Renewable generation with hypothetical storage solutions

Aside from the question of optimum outcome and profitability, it is instructive to shed light onthe question of how storage solutions can drive renewable generation in the future. To this end,we tested hundreds of hypothetical battery technology combinations of % efficiency (40% - 100%in 5% increments) and unit cost ($5 to $70 in $5 increments) with the ERCOT parameters fromthe previous section for $0, $50 and $100 taxation. Here, we calculated for each hypotheticalsolution what the profit optimal generation and storage investment would be and then derivedwhat percentage of demand was met by renewable generation.

12https://www.bafu.admin.ch/bafu/de/home/themen/klima/fachinformationen/klimapolitik/co2-abgabe.html

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Because we are using the model, the renewable percentage is naturally limited below by theminimum generation which in the Texas case is at 56%. This however does not mean that 56%would be profit-optimal in all cases with these parameters but that the model chooses no storageand the least possible renewable generation so that unconstrained realization would likely leadto lower outcomes. Yet with the renewable energy transition in full swing and e.g. Germanygenerating over 45% of its power from renewable sources in H1 2019 (BDEW 2019), the morerelevant future generation scenarios are capture-able with the model.

(a) $0 emission tax (b) $50 emission tax (c) $100 emission tax

Figure 4: Renewable generation percentage in hypothetical storage technologies in the ERCOT market

All cases of figure 4 show that renewable generation adoption is not linear, but rather occursonce a minimum combined threshold of efficiency and cost is surpassed. Intuitively this happenswhen generation cost including the efficiency loss and storage cost combined are cheaper than thebackup they replace. Inspecting the behavior more carefully where the graph starts folding upgives an idea of the price/efficiency threshold necessary for storage to take off - a useful indicationfor storage companies considering how to improve their technologies. In the $50 dollar case of sub-figure 4b), this seems to be for batteries with efficiency of 0.9 at approximately half their currentcost (30$) and for less efficient technologies below $20.

Analytically, the threshold beyond which renewable energy adoption increases sharply is moredifficult to exactly pinpoint, however looking at the explicit solution of optimal storage in equation28 helps to identify the core driver. With the term (2 − ck

ge )2 being added in the numerator andsubtracted in the denominator, its value is of crucial importance for storage and hence generationinvestment. This sharp rise then occurs when 2 − ck

ge > 1, which quadratically increases thenumerator and decreases the denominator - in other words ck

ge < 1 is the threshold. Of course,this only translates into real-life investments if the other cost parameters are favorable - that isthe generation - backup cost ration g

cQneeds to be large enough, so the entire expression becomes

non-zero. The demands then are a scaling factor for the optimal decision based on the otherparameters.

Contrast the previous case’s finding with today’s no taxation case in sub-figure 4a), for which80-100% renewable-penetration is essentially out of reach. Only technologies that are far away fromtoday’s solutions result in more storage and generation investment than little above the minimumbounds.

It also becomes apparent when comparing the renewable percentage of the ’flat’ baselines with-out storage across the three tax levels that even if storage remains nonviable, increased generationwill occur regardless as it is cheaper to occasionally turn off (out of the wind) wind turbines thanmarginally burning fossil fuels as those are becoming more expensive. In sub-figure a) the baselineis at 56% while it shifts upwards to over 60% in sub-figure b) and to over 70% in sub-figure c) with100$ tax.

5.5 Renewable break-even at current prices without taxation

Another immediate question following the hypothetical technology section is to ask how muchcheaper wind and storage would need to become in combination to be profitable under today’smarket conditions. Especially understanding whether generation or storage costs are holding backprofitability and if there are any threshold values beyond which adoption rapidly changes. Toachieve this, we utilized the existing storage and generation technology and calculated optimalcapacity investment, given that they cost between 95%-50% of today’s cost.

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To calculate absolute levels of profit, we assume that today’s average prices are received anddemand can only be met with renewable generation, storage and the costly back-up that stillcosts (g). While moving away from the island case in which the model was originally situatedthis exercise can give an idea of how far away we are today from seeing unsubsidized, large-scalerenewable penetration even if pollution is not penalized.

(a) ERCOT (b) Germany (c) PJM

Figure 5: Net daily profits of optimal capacity investment given cheaper generation and storage costwith battery storage

(a) ERCOT (b) Germany (c) PJM

Figure 6: Net daily profits of optimal capacity investment given cheaper generation and storage costwith thermal storage

The graphs in figures 5 and 6 show the profit with respect to storage and generation cost, wherethe lowest daily profit occurs at current technology prices. For ERCOT (5a), a 30-35% decreasein storage and generation cost would make a wind plus battery set-up profitable, while for thethermal storage technology (6a) a decrease of approximately 25% would be required.

One can see that because of the already low per-unit costs for thermal, further cost decreaseshave less of an impact than on the battery technology - essentially today’s thermal technologywould be deployed at large scale, if renewable generation was sufficiently cheap - which cannotbe said for batteries as a storage medium. However, because battery is so much more expensive,it overall has lower profitability than thermal and would require larger price decreases to becomeprofitable.

PJM displays the dynamics already previously established, where battery storage is not investedin at 50% cost reductions and profitability is only driven through much cheaper generation ascan be seen in sub-figure (5c). Thermal storage and generation would both need to decrease incost by another 40-45% before becoming profitable (6c). Germany, displayed in sub-figures b),is the opposite where due to the high prices renewable- generation-only is already profitable andprofitability increases through cheaper prices but not by increased capacity investment.

Overall, before storage becomes deploy-able in most markets, generation also has to becomecheaper. If that were the case, thermal storage would not be limited by its cost but its efficiencyand battery storage would need to become less expensive as well to become profitable at largescale. Yet, with (Lazard 2018) reporting annual cost reductions of 12% for wind and 8% forbattery storage, the required cost decreases do not seem out of reach in the near-term future. Ofcourse, it has to be noted that transmission costs are not included in this analysis yet, but couldbe incorporated by adjusting cQ accordingly.

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5.6 Storage results competitive case

After analyzing the monopoly optimum, we now investigate a competitive market between twoinvestors. Competition between a generation investor and storage investor adds another layer ofcomplexity by changing how the profitability of storing energy is evaluated from an overall socialoptimum to two individual parties, which comes down to their operating margins.

With this change, the competitive model requires the introduction of a transfer price for energyprocured by the storage owner to the generator when charging, which is simultaneously an incen-tive mechanism for the generator to encourage larger generation capacity investment by not simplywasting over-generation electricity. As mentioned in the modeling section, the storage owner re-ceives g for every unit discharged. The transfer price t is naturally bounded between t ∈ [0, ge] tomake it desirable for both parties to accept the transfer price agreement.

First, we analyzed which transfer prices lead to solutions that are not equal to the non-storagecase for a given market and tax situation. Two things are important, understanding how muchthe capacities would change and which transfer price is most profitable for the generator who setst. We omit the case in which the storage owner sets t, as he would always set it to 0.

(a) ERCOT, $0 tax, Thermal (b) ERCOT, $50 tax, Thermal

Figure 7: Comparing ERCOT capacity decision and profit changes for different transfer prices underdifferent emission taxes

Figure 7a) shows the general behavior of capacities (black lines) and profits (yellow lines). Forthe storage investor, the profit decreases as t increases, but depending on how much more generationcapacity is added, storage capacity might also slightly increase (see sub-figure 7b), before droppingdown towards 0 when the increasing transfer price squeezes the storage investor’s ge− t margin.

Conversely, the results for the generator change with the transfer prices, but first capacitiesand profits increase in t, before first generation capacities peak, then generation profit and thenboth return back to the non-storage case as the transfer price makes storage noncompetitive.

It already becomes apparent from those two examples that the generator will not necessarilychoose prices that optimize joint (social) payoffs, but is likely going to set prices higher than that.Compare the combined profits in subfigure 7a), where both profit lines intersect at t between 4-5with a t of 6-7 at which the solid yellow line peaks. However, there are also other possible dynam-ics as we will see in the following two examples, using different market data, taxes and storagetechnologies.

The graph 8a) is the familiar finding for many of PJM’s and the battery scenarios, where nomatter which transfer price is set, the storage investor cannot profitably operate batteries and hencethere is no change in profits or capacities compared to the non-storage case for either investor.

The sub-figure 8b) shows how storage in the PJM market even at a hypothetical emission taxlevel of $100 is only profitable if the transfer price is very close to 0, i.e. the electricity with whichthe batteries would be charged have to be virtually free and the electricity the discharge replacesvery costly.

Overall this section shows that the exact dynamic of the competitive market and the resultingcapacity investments highly depends on the market demands and price of the back-up source. Con-

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(a) PJM, $0 tax, Battery (b) PJM, $100 tax, Battery

Figure 8: Comparing PJM capacity decision and profit changes for different transfer prices underdifferent emission taxes

cretely, it is to be expected that the transfer price is set higher than under joint profit optimization,which increased generator’s profits, decreases storage owners’ profit and decreases overall welfarerelative to the first best case.

6 Discussion

Our paper provides the first methodological approach in the Operations literature to study thelarge-scale capacity investment in storage to shift intermittent, renewable electricity across time,especially between daily peaks and lows. We show how a monopoly investor can use informationon demand, cost, technology and tax parameters to decide on the optimal level of fossil-free gen-eration and storage. Structurally, we find both investment technologies to be complements, butinterestingly evenly distributed demand across time to actually be beneficial for storage, as largelyrenewable grids favor high utilization of storage assets and adequate generation capacity matchingover energy arbitrage opportunities.

A related finding regarding the type of storage is that despite the current investment focus onbatteries, our models shows that for multi-GWh storage, lower-cost, lower-efficiency technologies,such as thermal, are more profitable, invested into sooner and co-located with more generation.This result is driven, by the decreasing added-value opportunities for storage (reserve capacity, fre-quency regulation) once total capacities moves into the realm of hours of demand. In these cases,the total discharge cost (renewable input plus efficiency loss plus storage cost) is the main decisionfactor. This effect is magnified by the unpredictable usage of a given storage unit on a givenday, where idling, expensive batteries lower the bottom line more than their thermal counterparts.From a social welfare perspective, we find that a monopoly is investing more into storage thancompeting investors, because the former accurately values the use of marginally free renewablescompared to costly, fossil-fuel backups, whereas the competitors in maximizing their own profit donot focus on the same dynamic and under-invest.

Because of this dynamic, carbon emission taxes boost storage and renewable adaptation byrendering the back-up alternative more costly. The extent to which a given tax level increasesfossil-free adaptation however depends on the cost of delivering this alternative generation option.In markets with low gas prices and low demand fluctuations such as the North-Eastern US, emissiontaxes of up to $75 dollars are required to drive a transition, while in countries with higher electricitycosts, such as Germany, much lower tax levels create the same effect.

Furthermore, we are able to show that storage profitability and hence adoption is not linear,but increases abruptly once a technology threshold is surpassed. Loosely speaking, this limit isreached when the unit cost of storage charged with the renewable source, including all idling costsand efficiency losses is cheaper in expectation than the alternatives ( cKge < 1). This can eitherbe achieved by low unit costs or high efficiency or a competitive combination of both factors.Empirically, the cost of thermal storage and generation would have to decrease by approximately

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30-40% (not accounting for transmission costs) relative to 2019 levels to make renewable plusstorage cheaper than today’s market prices.

It has to be noted though that these findings are based on a model that, in its core, tries tounravel the over-arching dynamics that drive macro-investment choices into storage and renew-able energies, not guide individual project decisions and hence simplifies the demand and pricingdynamics of a modern-day energy grid. Thus, additional layers of complexity could be added byconsidering stochastic demand and or costs, the interplay between energy market participants asthe grid is shifting over time as well as location choices and the grid itself with its physical con-straints. Likewise, the engineering and design challenges for storage installations are glanced overas we treat them as a modular investment with known capabilities.

These limitations simultaneously present ample opportunities for future research. Understand-ing how existing plant portfolios and market prices impact the adoption of storage capacity aswell as where to locate said investments within a grid and how to size the individual modularcomponents of the combined grid storage systems are relevant, challenging and open questions.

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Appendix A Monopoly Model

a = 1424 = hours day

hours night

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Π =g

aQ

{∫ DH

0

Q dQ+

∫ aQ

DH

DH dQ+

∫ DH+K

DH

(Q−DH) e dQ+

∫ aQ

DH+K

K e dQ

}

+g

(1− a)Q

{∫ DL

0

Q dQ+

∫ (1−a)Q

DL

DL dQ+

∫ DL+K

DL

(Q−DL) e dQ+

∫ (1−a)Q

DL+K

K e dQ

}−cKK − cQQ

=g

aQ

{DH

2

2+ (aQ−DH)DH +

K2e

2+ (aQ−DH −K)Ke

}+

g

(1− a)Q

{DL

2

2+ ((1− a)Q−DL)DL +

K2e

2+ ((1− a)Q−DL −K)Ke

}−cKK − cQQ

=g

aQ

{−DH

2

2− K2e

2−KeDH

}+

g

(1− a)Q

{−DL

2

2− K2e

2−KeDL

}+

g(DH +DL + 2Ke)− cKK − cQQ(11)

∂Π

∂Q=

g

aQ2

{DH

2

2+K2e

2+KeDH

}+

g

(1− a)Q2

{DL

2

2+K2e

2+KeDL

}− cQ

→ Q∗ =

√g

acQ

{DH

2

2+K2e

2+KeDH

}+

g

(1− a)cQ

{DL

2

2+K2e

2+KeDL

} (12)

∂2Π

∂2Q=− 2g

aQ3

{DH

2

2+K2e

2+KeDH

}− 2g

(1− a)Q3

{DL

2

2+K2e

2+KeDL

}(13)

∂2Π

∂Q∂K=

g

aQ2{Ke+ eDH}+

g

(1− a)Q2{Ke+ eDL} (14)

∂Π

∂K=

g

aQ{−Ke− eDH}+

g

(1− a)Q{−Ke− eDL}+ 2ge− cK

→ Keg

{1

aQ+

1

(1− a)Q

}= − ge

aQDH −

ge

(1− a)QDL + 2ge− cK

K1

a(1− a)Q= 2− DH

aQ− DL

(1− a)Q− cKge

→ K∗ = (2− cKge

)(a(1− a)Q)−DH(1− a)−DLa

(15)

∂2Π

∂2K=g

aQ(−e) +

g

(1− a)Q(−e) =

−ge(1− a)aQ

(16)

∂2Π

∂K∂Q=

g

aQ2{Ke+ eDH}+

g

(1− a)Q2{Ke+ eDL} (17)

H(Π) =

[(−) (+)(+) (−)

](18)

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H(Π) =[− 2g

aQ3

{DH

2

2+K2e

2+KeDH

}− 2g

(1− a)Q3

{DL

2

2+K2e

2+KeDL

}]

[−ge

(1− a)aQ]−

(g

aQ2{Ke+ eDH}+

g

(1− a)Q2{Ke+ eDL})2 < 0

→(1− a)

{D2H

e+K2 + 2KDH

}+ a

{D2L

e+K2 + 2KDL

}<(

1

a{K +DH}+

1

(1− a){K +DL})2

→(1− a)

{D2H

e+K2 + 2KDH

}+ a

{D2L

e+K2 + 2KDL

}<(

1

a{K +DH}+

1

(1− a){K +DL})2

(19)

A.1 Monopoly Solution as a Function of Primitives Only

K∗ =(2− cKge

)(a(1− a)Q)−DH(1− a)−DLa

Q∗ =

√g

acQ

{DH

2

2+K2e

2+KeDH

}+

g

(1− a)cQ

{DL

2

2+K2e

2+KeDL

} (20)

K +DH(1− a) +DLa =(2− cKge

)(a(1− a)

√g

acQ

{DH

2

2+K2e

2+KeDH

}+

g

(1− a)cQ

{DL

2

2+K2e

2+KeDL

})

(21)

K2 + 2KDH(1− a) + 2KDLa+ 2DHDLa(1− a) +D2H(1− a)2 +D2

La2 =

(2− cKge

)2a2(1− a)2 g

cQ

[1

a

{DH

2

2+K2e

2+KeDH

}+

1

(1− a)

{DL

2

2+K2e

2+KeDL

}](22)

AssumecKge

< 2 and let (2− cKge

)2a2(1− a)2 g

cQ= f > 0

K2[1− f e2

(1

a+

1

1− a)] +K[2DH(1− a) + 2DLa− fe(

DH

a+

DL

1− a)]+

2DHDLa(1− a) +D2H(1− a)2 +D2

La2 − f(

D2H

2a+

D2L

2(1− a)) = 0

(23)

K2[1− f e2

(1

a+

1

1− a)] +K[2DH(1− a) + 2DLa− fe(

DH

a+

DL

1− a)]+

(DH(1− a) +DLa)2 − f(D2H

2a+

D2L

2(1− a)) = 0

(24)

K∗ =[−2DH(1− a)− 2DLa+ fe(DH

a + DL

1−a )]±√η2[1− f e2 ( 1

a + 11−a )]

η =[2DLa− fe(DH

a+

DL

1− a)]2 − 4[1− f e

2(1

a+

1

1− a)]

(DH(1− a) +DLa)2 + 2DH(1− a)− f(D2H

2a+

D2L

2(1− a))

(25)

Assuming a = 0.5

K2[1− 2fe] +K[(1− 2fe)(DH +DL)] + [(DH

2+DL

2)2 − f(D2

H +D2L)] = 0

→ K∗ =−(1− 2fe)(DH +DL)±

√[(1− 2fe)(DH +DL)]2 − 4([1− 2fe])[(DH

2 + DL

2 )2 − f(D2H +D2

L)]

2(1− 2fe)(26)

26

Page 27: When the wind of change blows, build batteries? Optimum ...€¦ · multi-GWh storage would become optimal in all studied markets. 1 Introduction As societies, utilities, and regulators

→ [(1− 2fe)(DH +DL)]2 − 4([1− 2fe])[(DH

2+DL

2)2 − f(D2

H +D2L)]

= (1− 2fe)(DH +DL)]2 − (1− 2fe)(DH +DL)2 + 4f(1− 2fe)(DH +DL)2

= (DH +DL)2[(1− 2fe)((1− 2fe) + 4f − 1)]

(27)

→ K∗ =DH +DL

2

−(1− 2fe)±√

(1− 2fe)((1− 2fe) + 4f − 1)

(1− 2fe)

K∗ =DH +DL

2

(−1±

√1 +

4f − 1

1− 2fe

)

=DH +DL

2

−1±

√√√√1 +

14 ((2− cK

ge )2 gcQ

)− 1

1− 18 (2− cK

ge )2 gcQe

=DH +DL

2

−1±

√√√√√1 +(g − cQ − cK

e +c2K

4ge2 ) 1cQ

(cQ + ge2 −

cK2 +

c2K8ge ) 1

cQ

=DH +DL

2

−1±

√√√√√1 +(g − cQ − cK

e +c2K

4ge2 )

(cQ + ge2 −

cK2 +

c2K8ge )

(28)

A.2 Monopoly Solution as a Function of Primitives Only - Trial 2

Assume a = 0.5

K∗ =(2− cKge

)(a(1− a)Q)−DH(1− a)−DLa

=(2− cKge

)Q

4− DH

2− DL

2

Q∗ =

√g

acQ

{DH

2

2+K2e

2+KeDH

}+

g

(1− a)cQ

{DL

2

2+K2e

2+KeDL

}Q∗ =

√g

cQ{D2

H +K2e+ 2KeDH +D2L +K2e+ 2KeDL}

(29)

(4K + 2(DH +DK)

2− cKge

)2

=g

cQ

{D2H +K2e+ 2KeDH +D2

L +K2e+ 2KeDL

}16K2 + 16K(DH +DK) + 4(DH +DL)2

(2− cKge )2

=g

cQ

{D2H +K2e+ 2KeDH +D2

L +K2e+ 2KeDL

}K2[

16

(2− cKge )2

− 2eg

cQ] +K[

16(DH +DL)

(2− cKge )2

− 2eg(DH +DL)

cQ] + [

4(DH +DL)2

(2− cKge )2

− g(D2L +D2

H)

cQ] = 0

K2 +K[DH +DL] +

4(DH+DL)2

(2− cKge )2

− g(D2L+D2

H)cQ

16(2− cK

ge )2− 2eg

cQ

= 0

K = −DH +DL

√(cK − 2eg)2(e(DH +DL)2 − 2(D2

H +D2L))

e(c2K − 4cKeg + 4eg(eg − 2cQ))

K = −DH +DL

2± (2eg − cK)

√(e(DH +DL)2 − 2(D2

H +D2L))

e((cK − 2eg)2 − 8cQeg)

(30)

A.3 Increase of investment in K

Get rid of constants and positive factors, take derivative

27

Page 28: When the wind of change blows, build batteries? Optimum ...€¦ · multi-GWh storage would become optimal in all studied markets. 1 Introduction As societies, utilities, and regulators

K∗ ∼

√√√√√1 +(− cKe +

c2K4ge2 )

12 (cQ + cK −

c2K4ge )

(31)

∂K∗

∂cK=∂K∗

∂e

→ 8cQg(cK − 2eg)

(c2K − 4egcK − 4cQeg)2

√2(

c2K

4e2g− cK

e )

−c2K

4eg +cK+cQ

+ 1

=cK((16cQ + 16cK)g2e2 + (−8cKcQ − 8c2K)ge+ c3K)√

2(c2K

4ge2− cK

e )

−c2K

4ge +cQ+cK

+ 1e2((4cQ + 4cK)ge− c2K)2

(32)

e28cQg(cK − 2eg) = cK((16cQ + 16cK)g2e2 + (−8cKcQ − 8c2K)ge+ c3K)

e2cQ(cK − 2eg) = cK((2cQ + 2cK)ge2 + (−cKcQ − c2K)e+c3K8g

)

e2cKcQ − 2e3cQg = cK(2e2cQg + 2e2cKg − ecKcQ − ec2K +c3K8g

)

e2cKcQ − 2e3cQg = cK(2e2cQg + 2e2cKg − ecKcQ − ec2K +c3K8g

)

(33)

28

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A.4 Decreasing Probability Term

Lets first assume that K =∞, because at where we take the derivative, it will not matter anyway.

Pr =

∫ aQQH=0

∫ (1−a)Q

QL=0min (min(QL −DL,K)+, (DH −QH)+) dQLdQH

1Q2a(1−a)∫ (1−a)Q

QL=0min((QL −DL)+,K)dQL

1(1−a)Q

=

∫ aQQH=0

∫ (1−a)Q

QL=0min((QL −DL)+, (DH −QH)+)dQLdQH

1Qa∫ (1−a)Q

QL=DLQL −DLdQL

(34)

∫ aQ

QH=0

∫ (1−a)Q

QL=0

min((QL −DL)+, (DH −QH)+)dQLdQH

=

∫ DH

QH=0

∫ (1−a)Q

QL=DL

min((QL −DL)+, (DH −QH)+)dQLdQH

=

∫ DH

QH=0

[∫ DH+DL−QH

QL=DL

(QL −DL)dQL +

∫ (1−a)Q

QL=DH+DL−QH

(DH −QH)dQL

]dQH

=

∫ DH

QH=0

[(QH −DH)2

2+ ((1− a)Q−DL −DH +QH)(DH −QH)

]dQH

=D3H

6− D2

H((3a− 3)Q+ 3DL + 2DH)

6

= −D2H((3a− 3)Q+ 3DL +DH)

6

(35)

∫ (1−a)Q

QL=DL

(QL −DL)dQL

=((1− a)Q−DL)2

2

(36)

Pr =D2H((3a− 3)Q+ 3DL +DH) 1

aQ

3((1− a)Q−DL)2

=D2H((3a− 3)Q+ 3DL +DH)

3((1− a)Q−DL)2aQ

∂Pr

∂Q=D2H(6(1− a)2Q2 − 3(1− a)(3DL +DH)Q+ 3DL +DH)

3aQ2((1− a)Q−DL)3

(37)

∂Pr(DH +DL)

∂Q=D2H(6(1− a)2(DH +DL)2 − 3(1− a)(3DL +DH)(DH +DL) + 3DL +DH)

3a(DH +DL)2((1− a)(DH +DL)−DL)3

=D2H(6(1− a)2(DH +DL)2 − 3(1− a)(3DL +DH)(DH +DL) + 3DL +DH)

3a(DH +DL)2((1− a)DH)3

(38)Now assume a = 0.5 and let DH = cDL, c > 1

=D2H(1.5(DH +DL)2 − 1.5(3DL +DH)(DH +DL) + 3DL +DH)

316 (DH +DL)2D3

H

=c2D2

L(1.5(1 + c)2D2L − 1.5((3 + c)DL)((1 + c)DL) + (3 + c)DL)

316 (1 + c)2D2

Lc3D3

L

=c2D2

L((1.5 + 3c+ 1.5c2)D2L − (4.5 + 6c+ 1.5c2)D2

L + (3 + c)DL)316 (1 + 2c+ c2 + c3)D5

L

=c2D3

L((−3− 3c)DL + 3 + c)316 (1 + 2c+ c2 + c3)D5

L

=16c2((−3− 3c)DL+ 3 + c)

3(1 + 2c+ c2 + c3)D2L

(39)

Ignore 3+c term in numerator because DL >> 3

= − 16(c2 + c3)

(1 + 2c+ c2 + c3)DL(40)

29

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A.5 Comparative statics

In this section, we report the sign of the derivatives with the assumption of a = 0.5. We knowfrom the cross-derivatives that both optimal capacities will move into the same direction, hence wecan analyze the simplified solution from equation 28 with respect to the parameters. The decisionparameter behave as one would expect.

Parameter Q∗ K∗

Back-up Cost g ↑ ↑Generation Cost cQ ↓ ↓Storage Cost cK ↓ ↓Storage Efficiency e ↑ ↑Demands DH , DL ↑ ↑Generation Capacity Factor r ↑ ↑

∂K∗

∂g=

2cQ(dL + dH)(2− e)(2eg − cK)(2eg + cK)

(4e2g2 + (−8cQ − 4cK)eg + c2K)2

√√√√ (2− cKeg

)2g

4cQ−1

1−e(2− cK

eg)2g

8cQ

+ 1(41)

∂K∗

∂cQ=− 2(dL + dH)(2− e)g(2eg − cK)2√√√√ (2− cK

eg)2g

4cQ−1

1−e(2− cK

eg)2g

8cQ

+ 1(8egcQ − 4e2g2 + 4cKeg − c2K)2(42)

∂K∗

∂cK=− 4cQ(dL + dH)(2− e)g(2eg − cK)

(c2K − 4egcK + 4e2g2 − 8cQeg)2

√√√√ g(2− cKeg

)2

4cQ−1

1−eg(2− cK

eg)2

8cQ

+ 1(43)

∂K∗

∂e=

(dL + dH)(2ge− cK)((8g3 − 8cQg2)e3 − (12cKg

2 + 4cKcQg)e2 + (16cKcQ + 6c2K)ge− c3K)

2e2(4g2e2 + (−8cQ − 4cK)ge+ c2K)2

√√√√ g(2− cKge

)2

4cQ−1

1−g(2− cK

ge)2e

8cQ

+ 1

(44)

cSimoneQ = cOriginalQ /24/2r =cOriginalQ

48r

∂K∗

∂r=

(dL + dH)(2ge− cK)((8g3 − 8cQg2)e3 + (−12cKg

2 − 4cKcQg)e2 + (16cKcQ + 6c2K)ge− c3K)

2e2(4g2e2 + (−8cQ − 4cK)ge+ c2K)2

√√√√ g(2− cKge

)2

4cQ−1

1−g(2− cK

ge)2e

8cQ

+ 1

(45)

∂K∗

∂DH=

√√√√ (2− cKeg

)2g

4cQ−1

1−e(2− cK

eg)2g

8cQ

+ 1− 1

2

(46)

∂K∗

∂DL=

√√√√ (2− cKeg

)2g

4cQ−1

1−e(2− cK

eg)2g

8cQ

+ 1− 1

2

(47)

Appendix B Competitive Model

Generation Investor Objective Function :

30

Page 31: When the wind of change blows, build batteries? Optimum ...€¦ · multi-GWh storage would become optimal in all studied markets. 1 Introduction As societies, utilities, and regulators

ΠQ =g

aQ

{∫ DH

0

QdQ+

∫ aQ

DH

DHdQ

}+

t

aQ

{∫ DH+K

DH

(Q−DH)dQ+

∫ aQ

DH+K

KdQ

}+

g

(1− a)Q

{∫ DL

0

QdQ+

∫ (1−a)Q

DL

DLdQ

}+

t

(1− a)Q

{∫ DL+K

DL

(Q−DL)dQ+

∫ (1−a)Q

DL+K

KdQ

}− cQQ

=g

aQ

{DH

2

2+ (aQ−DH)DH

}+

t

aQ

{K2

2+ (aQ−DH −K)K

}+

g

(1− a)Q

{D2L

2+ ((1− a)Q−DL)DL)

}+

t

(1− a)Q

{K2

2+ ((1− a)Q−DL −K)K

}− cQQ

=g

aQ

{−D2

H

2

}+

t

aQ

{−K2

2−DHK

}+

g

(1− a)Q

{−D2

L

2

}+

t

(1− a)Q

{−K2

2−DLK

}+

g(DH +DL) + 2tK − cQQ

(48)

∂ΠQ

∂Q=

g

aQ2

{D2H

2

}+

t

aQ2

{K2

2+DHK

}+

g

(1− a)Q2

{D2L

2

}+

t

(1− a)Q2

{K2

2+DLK

}− cQ

→ Q∗ =

√√√√√√√√g

acQ

{D2H

2

}+

t

acQ

{K2

2+DHK

}+

g

(1− a)cQ

{D2L

2

}+

t

(1− a)cQ

{K2

2+DLK

}(49)

∂2ΠQ

∂2Q=− 2g

aQ3

{D2H

2

}− 2t

aQ3

{K2

2+DHK

}−

2g

(1− a)Q3

{D2L

2

}− 2t

(1− a)Q3

{K2

2+DLK

} (50)

∂2ΠQ

∂Q∂K=

t

aQ2{K +DH}+

t

(1− a)Q2{K +DL} (51)

31

Page 32: When the wind of change blows, build batteries? Optimum ...€¦ · multi-GWh storage would become optimal in all studied markets. 1 Introduction As societies, utilities, and regulators

Storage Investor Objective Function :

ΠK =ge− taQ

{∫ DH+K

DH

(Q−DH)dQ+

∫ aQ

DH+K

KdQ

}+

ge− t(1− a)Q

{∫ DL+K

DL

(Q−DL)dQ+

∫ (1−a)Q

DL+K

KdQ

}− cKK

=ge− taQ

{−K2

2−DHK

}+

ge− t(1− a)Q

{−K2

2−DLK

}+ 2[ge− t]K − cKK

(52)

∂ΠK

∂K=ge− taQ

{−K −DH}+ge− t

(1− a)Q{−K −DL}+ 2[ge− t]− cK

=− ge− ta(1− a)Q

K − ge− taQ

DH −ge− t

(1− a)QDL + 2[ge− t]− cK

K∗ =− (1− a)DH − aDL + 2a(1− a)Q− cKa(1− a)Q

ge− t=− (1− a)DH − aDL + a(1− a)Q(2− cK

ge− t)

(53)

∂2ΠK

∂2K=− ge− t

aQ− ge− t

(1− a)Q

=− ge− t(1− a)aQ

(54)

∂2ΠK

∂K∂Q=ge− taQ2

{K +DH}+ge− t

(1− a)Q2{K +DL} (55)

The Hessian can then be easily signed:

H(Π) =

[∂2ΠQ

∂Q2

∂2ΠQ

∂Q∂K∂2ΠK

∂K∂Q∂2ΠK

∂K2

]=

[(−) (+)(+) (−)

](56)

32

Page 33: When the wind of change blows, build batteries? Optimum ...€¦ · multi-GWh storage would become optimal in all studied markets. 1 Introduction As societies, utilities, and regulators

We then want to show that |H| > 0

[− 2g

aQ3

{D2H

2

}− 2t

aQ3

{K2

2+DHK

}− 2g

(1− a)Q3

{D2L

2

}− 2t

(1− a)Q3

{K2

2+DLK

}]

[− ge− t(1− a)aQ

] >

[t

aQ2{K +DH}+

t

(1− a)Q2{K +DL}]

[ge− taQ2

{K +DH}+ge− t

(1− a)Q2{K +DL}]

→[2g

a

{D2H

2

}+

2t

a

{K2

2+DHK

}+

2g

(1− a)

{D2L

2

}+

2t

(1− a)

{K2

2+DLK

}] >

[t(1− a) {K +DH}+ ta {K +DL}][(1− a) {K +DH}+ a {K +DL}]

→[gD2

H

a+

2t

a

{K2

2+DHK

}+

gD2L

(1− a)+

2t

(1− a)

{K2

2+DLK

}] >

[tD2

H

a+

2t

a

{K2

2+DHK

}+

tD2L

(1− a)+

2t

(1− a)

{K2

2+DLK

}] >

t[((1− a)(K +DH) + a(K +DL))2]

→[1

a(DH +K)2 +

1

(1− a)(DL +K)2 >

[((1− a)(K +DH) + a(K +DL))2]

→[1

a(DH +K)2 +

1

(1− a)(DL +K)2 >

[((1− a)2(K +DH)2 + 2a(1− a)(K +DH)(K +DL) + a2(K +DL)2]

→[(DH +K)2(1

a− (1− a)2) + (DL +K)2(

1

(1− a)− a2) >

2a(1− a)(K +DH)2 > 2a(1− a)(K +DH)(K +DL)

→[(DH +K)2(1

a− (1− a)2 − 2a(1− a)) + (DL +K)2(

1

(1− a)− a2) > 0

(57)

B.1 Equilibrium calculations

Reformulating the storage investors optimal solution yields

Q =K + (1− a)DH − aDL

a(1− a)(2− cKge−t )

(58)

What does t have to be set to for K to be non-zero:

a(1− a)Q(2− cKge− t

)− (1− a)DH − aDL > 0

(2− cKge− t

) >DH

aQ+

DL

(1− a)Q

cKt− ge

>DH

aQ+

DL

(1− a)Q− 2

t < ge− cK

2− DH

aQ −DL

(1−a)Q

(59)

B.2 Competitive Solution as a Function of Primitives Only

Let l = a(1− a)(2− cKge−t )

K = lQ− (1− a)DH − aDL

K2 = l2Q2 − 2lQ(1− a)DH − 2lQaDL + (1− a)2D2H + a2D2

L + 2(1− a)aDHDL

33

Page 34: When the wind of change blows, build batteries? Optimum ...€¦ · multi-GWh storage would become optimal in all studied markets. 1 Introduction As societies, utilities, and regulators

K2 = l2Q2 − 2lQ[(1− a)DH + aDL] + (1− a)2D2H + a2D2

L + 2(1− a)aDHDL

Q∗ =

√g

acQ

{D2H

2

}+

t

acQ

{K2

2+DHK

}+

g

(1− a)cQ

{D2L

2

}+

t

(1− a)cQ

{K2

2DLK

}Q2 =

g

acQ

{D2H

2

}+

t

2a(1− a)cQK2 +

t

cQK

{DH

a+

DL

(1− a)

}+

g

(1− a)cQ

{D2L

2

}Q2 =

g

2cQ

{D2H

a+

D2L

(1− a)

}+

t

2a(1− a)cQK2 +

t

cQK

{DH

a+

DL

(1− a)

}Q2 =

g

2cQ

{D2H

a+

D2L

(1− a)

}+

t

2a(1− a)cQ

[l2Q2 − 2lQ[(1− a)DH + aDL] + (1− a)2D2

H + a2D2L + 2(1− a)aDHDL

]+

t

cQ[lQ− (1− a)DH − aDL]

{DH

a+

DL

(1− a)

}0 = Q2[

tl2

2a(1− a)cQ− 1]+

Q[tl

cQ

{DH

a+

DL

(1− a)

}− 2tl[(1− a)DH + aDL]

2a(1− a)cQ]+

g

2cQ

{D2H

a+

D2L

(1− a)

}+

t

2a(1− a)cQ[−(1− a)DH − aDL]

2+

t

cQ[−(1− a)DH − aDL]

{DH

a+

DL

(1− a)

}= Q2[

tl2

2a(1− a)cQ− 1]+

g

2cQ

{D2H

a+

D2L

(1− a)

}+

t

2a(1− a)cQ[−(1− a)DH − aDL]

2+

t

cQ[−(1− a)DH − aDL]

{DH

a+

DL

(1− a)

}(60)

Assume a = 0.5→ l = 0.25(2− cKge−t )

0 = Q2[2tl2

cQ− 1]+

g

cQ

{D2H +D2

L

}+

t

cQ[−DH −DL]

2+

t

cQ

[−D2

H −D2L − 2DLDH

]= Q2[

2tl2

cQ− 1] +

g

cQ[D2

H +D2L]− t

cQ[4DLDH ]

Q∗ =

√√√√ tcQ

[4DLDH ]− gcQ

[D2H +D2

L]

2tl2

cQ− 1

Q∗ =

√t [4DLDH ]− g[D2

H +D2L]

2tl2 − cQ=

√√√√ t [4DLDH ]− g[D2H +D2

L]

2t 14

2((2− cK

ge−t ))2 − cQ

=

√32t [DLDH ]− 8g[D2

H +D2L]

t(2− cKge−t )

2 − cQ(61)

K∗ = lQ− (1− a)DH − aDL = 0.25(2− cKge− t

)

√32t [DLDH ]− 8g[D2

H +D2L]

t(2− cKge−t )

2 − cQ− (1− a)DH − aDL

= (2− cKge− t

)

√2t [DLDH ]− 8g[D2

H +D2L]

t(2− cKge−t )

2 − cQ− (1− a)DH − aDL

(62)

B.3 Comparative statics

Following Bernstein and Federgruen (2003), we derive the signs of the comparative statics withrespect to some parameter α, through

34

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∂K∗

∂α= −

∂2ΠK

∂K∂α∂2ΠQ

∂2Q −∂2ΠK

∂K∂Q∂2ΠQ

∂Q∂α

|H|(63)

∂Q∗

∂α= −∂

2ΠK

∂2K

∂2ΠQ

∂Q∂α− ∂2ΠK

∂K∂α

∂2ΠQ

∂Q∂K(64)

with H being the Hessian of both profit functions for which the derivative was shown in equation9 to be positive, which is why we only present the numerator in this section. In the process,we will make extensive use of the partial derivatives already introduction in equations 49 - 55.It is interesting to see that for example increased efficiency (e) or may not always leading tomore investment, but for different reason. In the efficiency case, higher efficiency can mean lessinvestment, because the improved performance of the previous equilibrium capacity is more thansufficient in its upgraded condition (more efficiency, less energy lost, less storage and generationrequired). However, for the increase in night-time demand (DL), the capacity decisions can increaseor decrease depending on whether the effect of reduced demand volatility or potential saving ofmarginal back-up storage is larger. The condition columns introduces these constraints wherenecessary - again the comparative static investigate capacity investment, not profit, which especiallyfor t and g can have different directional effects.

Parameter Q∗ K∗ Condition

Back-up Cost g ↑ ↑ K(1−a)aQ + DL

(1−a)Q + DH

aQ < 2

Generation Cost cQ ↓ ↓Storage Cost cK ↓ ↓Storage Efficiency e ↑ ↑ K

(1−a)aQ + DL

(1−a)Q + DH

aQ < 2

Transfer Price t ↑ ↓ K(1−a)aQ + DL

(1−a)Q + DH

aQ > 2

Demands Day DH ↑ ↑Demands Day DL ↑ ↑ For Q∗: tDH > gDL, For K∗:

g(D2H−DLK−DHDL)+tDHK)

a − (g−t)(DLK)1−a < 0

Fraction day a ↑ lK

1−a +DH

a2 >Ka +DL

(1−a)2

∂K∗

∂g=− [

∂2ΠK

∂K∂g(−)− (+)(+)]

{> 0 if K

(1−a)aQ+ DL

(1−a)Q+ DH

aQ< 2

< 0 if K(1−a)aQ

+ DL(1−a)Q

+ DHaQ

> 2(65)

∂Q∗

∂g=− [(−)(+)−

∂2ΠK

∂K∂g(+)]

{> 0 if K

(1−a)aQ+ DL

(1−a)Q+ DH

aQ< 0

< 0 if K(1−a)aQ

+ DL(1−a)Q

+ DHaQ

> 0(66)

∂K∗

∂cQ= −[0− (

∂2ΠK

∂K∂cQ(−1))] = −

∂2ΠK

∂K∂cQ< 0 (67)

∂Q∗

∂cQ= −[

∂2ΠK

∂2K(−1))− 0] =

∂2ΠK

∂2K< 0 (68)

∂K∗

∂cK=− [(−1)(

∂2ΠK

∂K∂cK)− 0] =

∂2ΠK

∂K∂cK< 0 (69)

∂Q∗

∂cK=− [0− ((−1)

∂2ΠQ

∂Q∂K)] = −

∂2ΠQ

∂Q∂K< 0 (70)

∂K∗

∂e= −

∂2ΠK

∂K∂e

∂2ΠQ

∂2Q−

∂2ΠK

∂K∂Q

∂2ΠQ

∂Q∂e(71)

∂Q∗

∂e= −[0−

∂2ΠK

∂K∂e(+)] =

∂2ΠK

∂K∂e(+)

{> 0 if K

(1−a)aQ+ DL

(1−a)Q+ DH

aQ< 2

< 0 if K(1−a)aQ

+ DL(1−a)Q

+ DHaQ

> 2(72)

∂K∗

∂t= −[

∂2ΠK

∂K∂t(−)− (+)(+)]

{> 0 if K

(1−a)aQ+ DL

(1−a)Q+ DH

aQ> 2

< 0 if K(1−a)aQ

+ DL(1−a)Q

+ DHaQ

< 2(73)

∂Q∗

∂t= −[(−)(+)−

∂2ΠK

∂K∂t(+)]

{> 0 if K

(1−a)aQ+ DL

(1−a)Q+ DH

aQ> 2

< 0 if K(1−a)aQ

+ DL(1−a)Q

+ DHaQ

< 2(74)

35

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∂K∗

∂DH=− [

∂2ΠK

∂K∂DH

∂2ΠQ

∂2Q−

∂2ΠK

∂K∂Q

∂2ΠQ

∂Q∂DH]

=− (−ge− t

aQ)(−

2g

aQ3

{D2

H

2

}−

2t

aQ3

{K2

2+ DHK

}−

2g

(1− a)Q3

{D2

L

2

}−

2t

(1− a)Q3

{K2

2+ DLK

})+

(ge− t

aQ2{K + DH}+

ge− t

(1− a)Q2{K + DL})(

gDH + Kt

aQ2)

→(−DHKt

a−

D2Lg

(1− a)−

DLKt

(1− a)) + (

DHKg

a+

KgDH + DLDHg

(1− a)) > 0

(75)

∂Q∗

∂DH=[(

ge− t

(1− a)aQ)(gDH + Kt

aQ2)− (

ge− t

aQ)(

t

aQ2{K + DH}+

t

(1− a)Q2{K + DL})]

→(gDH

(1− a)a)− (

t

a{DH}+

t

(1− a){DL})] > 0

(76)

∂K∗

∂DL= −[−(

ge− t

(1− a)Q)(−

2g

aQ3

{D2

H

2

}−

2t

aQ3

{K2

2+ DHK

}−

2g

(1− a)Q3

{D2

L

2

}−

2t

(1− a)Q3

{K2

2+ DLK

})−

(ge− t

aQ2{K + DH}+

ge− t

(1− a)Q2{K + DL})(

gDL + Kt

(1− a)Q2)]

→ −[(D2

Hg

a+

t

a{DHK}+

t

(1− a){DLK})

DLKg + gDLDH

a+

DLKg

1− a]> 0if

g(D2H−DLK−DHDL)+tDHK)

a− (g−t)(DLK)

1−a< 0)

< 0ifg(D2

H−DLK−DHDL)+tDHK)

a− (g−t)(DLK)

1−a> 0)

(77)

∂Q∗

∂DL=− [(−

ge− t

(1− a)aQ)(gDL + Kt

(1− a)Q2)− (−

ge− t

(1− a)Q)(

t

aQ2{K + DH}+

t

(1− a)Q2{K + DL})]

→ −[(1− a)(gDL − tDH)]

{> 0 if tDH > gDL)< 0 if tDH < gDL)

(78)

∂K∗

∂a=− [(−

ge− t

(1− a)aQ)(

(K2

2+ DLK)t

Q2(1− a)2−

(K2

2+ DHK)t

Q2a2−

D2Hg

2Q2a2+

D2Lg

2Q2(1− a)2)−

((ge− t)K

Q(1− a)a2−

(ge− t)K

Q(1− a)2a+

(ge− t)DH

Qa2−

(ge− t)DL

Q(1− a)2)

(−2g

aQ3

{D2

H

2

}−

2t

aQ3

{K2

2+ DHK

}−

2g

(1− a)Q3

{D2

L

2

}−

2t

(1− a)Q3

{K2

2+ DLK

})]

(79)

∂Q∗

∂a=− [

∂2ΠK

∂2K

∂2ΠQ

∂Q∂a−

∂2ΠK

∂K∂a

∂2ΠQ

∂Q∂K] = [(−)(−)− (+)(−)]> 0 if

K1−a

+DH

a2 >Ka

+DL

(1−a)2

< 0 ifK

1−a+DH

a2 >Ka

+DL

(1−a)2

(80)

Appendix C Binary Intermittency Model

C.1 Model proofs

We will now formalize the optimal solutions and decision criteria where possible.

C.1.1 Case 1 - Low generation

Case 1 is for Q ≤ DL, with a battery-size of 0 as it could never be used. The objective functionand all first and second degree partial derivatives can be found in appendix C.2.2. With Q asthe only decision variable, the optimal solution can be found by inspecting the first derivative∂Π∂Q = 24rg− cQ which is constant, therefore leading to an optimal capacity investment of 0 or DL,depending on the sign of the derivative.

Theorem 5 The optimal generation capacity investment in the joint investment case 1 can befound by:

Q∗ =

{0, if 24rg − cQ ≤ 0

DL, if 24rg − cQ > 0

36

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Through the constant derivative, the solution is a border case dependent on the model param-eters that nevertheless is easily interpretable. If the average costs of the generation are lower thanthe expected savings from replacing the back-up plant, then invest into the renewable source upto the lower demand limit. Otherwise, do not invest.

C.1.2 Case 2 - Moderate generation

Case 2 in the parameter range of DL ≤ Q ≤ DLH features a battery, albeit one that is only chargedduring the night and discharged during the day. Given that Q ≤ DH , discharging is assumed toalways be possible. The case’s objective function and all derivatives can be found in appendixC.2.3.

From the first partial derivative with respect to (w.r.t.) Q, one can directly see that Q will beat either limit of the interval of Case 2 (see equation 89). However, this derivative now containsthe second choice parameter K. Turning to the storage decision, the second partial derivative w.r.t.K is always negative, hence setting the first partial derivative (equation 91) to 0 finds the uniqueoptimum K∗. Finding the optimal investment capacities therefore is a two-step process to whichstep one is finding the optimal K in hours:

Theorem 6 The optimal storage size in hours in the monopoly investment case 2 can be foundby:

K∗ =

{0, if ge < cK

F−1BL

(1− cKge ), if ge ≥ cK

Hence if at least the marginal unit of battery is less costly than the costly back-up it replaces(ge ≥ cK), one can find K∗ to satisfy K∗ = F−1

BL( cKge ). Otherwise, no battery is the profit-

maximizing choice.

Having found the optimal storage size in hours, the second step of finding optimal capacitiesinvolves substituting K∗ into the first partial derivative w.r.t. Q.

Theorem 7 The optimal generation capacity in the monopoly investment case 2 can be found by:

Q∗ =

{DL, if arg + g e(K∗ −

∫K∗

0BL(x)dx) < cQ + cKK

DH , if arg + g e(K∗ −∫K∗

0BL(x)dx) ≥ cQ + cKK

The intuition behind this solution follows directly from theorem 6. Given the optimal batterysize, do the savings from additional generation investment above the demand at night pay for themarginal battery and generation unit? If they do, invest all the way up to DH , if they don’t stayat DL and don’t use a battery. Note that investment savings are twofold: The generation saves theexpected back-up plant it replaces during the day, plus the combination of generation and storagesaves the backup-up cost they replace at night, which is the second term.

Despite being simpler than the full model, the second case already showcases some of the keytrade-offs. Generation has to be profitable in expectation, once the battery storage is included andbattery storage has to save at least its investment in back-up capacity costs. On top of that, if onewere to replace the assumption of deterministic demands by a demand distribution, the optimaldecisions would not necessarily be border cases anymore, but also lose their closed form.

C.1.3 Case 3 - High generation

Contrary to case 2, the battery can now also be charged during the day, and is then dischargedat night whenever the renewable source is not running. However, the probability of dischargeis decreasing in Q. This happens as with larger Q, the amount of energy that is charged duringgeneration times increases, ultimately making the accumulated charge in a sub-period so large thatit can only be partially discharged in the following sub-period. In the previous cases, this term wasassumed to be one, because of the lower bounds for Q. We simplify this discharge probability to belinear in Q - a simplification designed to keep the model tractable, as otherwise a term combiningthe two normal pdfs bL and bH would be required. Objective function and derivatives are detailedin appendix C.2.4.

In order for the model to then still be well-defined (to have non-negative discharge probability)one needs to impose an upper limit on Q of DL+DH

r , which in real-life terms just means that renew-able capacity investments above almost twice the total demand in expectation are not considered.

37

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By including discharge probabilities and a second way of utilizing the storage, the complexity isincreased significantly, making closed form solutions impossible to obtain as one now has to solvea system of equations. Yet by inspecting the derivatives’ signs and patterns, this case’s complexityis able to generate rich insights. The subsequent data and result chapters will exclusively discussthis case, but even at an initial glance one can see the core dynamics driving the model elements.

Most notably, the second partial derivatives for both decision parameters are negative acrossthe entire parameter space (equations 97, 98 and 100), guaranteeing the existence of at least oneoptimum.

Theorem 8 There exists at least one globally optimal solution to the monopoly capacity investmentdecision in case 3 that can be obtained by solving the systems of equations obtained through settingthe model’s two first partial derivatives equal to zero.

Next to the existence of an optimal decision, it is instructive to look at the strategic interactionbetween the two capacity decisions. Yet, when looking at equations 101 and 102 one notices thatthe objective function’s cross derivative contains both decision parameters so that cross derivativemay change sign across the parameter space, which indicates that renewables and storage can becomplements and substitutes depending on the specific realization.

C.2 Derivatives

C.2.1 Objective Function Simplification∫ K

0xbL(x)dx + K(1−BL(K)) = [xBL(x)]K0 −

∫ K

0BL(x)dx + K(1−BL(K))

= KBL(K)−∫ K

0BL(x)dx + K(1−BL(K)) = K −

∫ K

0BL(x)dx

(81)

C.2.2 Monopoly Objective Function Properties, Case 1

ΠQ =(24− a)

{DL(pL − (1− r)g)− rg(DL −Q)

}+ a

{DH(pH − (1− r)g)− rg(DH −Q)

}+cQQ (82)

∂Π

∂Q= 24rg − cQ (83)

∂Π2

∂Q2= 0 (84)

∂Π

∂K= 0 (85)

∂Π2

∂K2= 0 (86)

∂Π2

∂K∂Q= 0 (87)

C.2.3 Monopoly Objective Function Properties, Case 2

ΠQ,K =(24− a)

{DL(pL − (1− r)g)

}+ a

{DH(pH − (1− r)g)− rg(DH −Q)

}+

g (Q−DL) e[K −

∫ K

0BL(x)dx

]− cQQ− (Q−DL)cKK

(88)

∂Π

∂Q= arg + g e[K −

∫ K

0BL(x)dx]− cQ − cKK (89)

∂Π2

∂Q2= 0 (90)

∂Π

∂K=g e (Q−DL)(1−BL(K))− (Q−DL)cs → g e (1−BL(K)) = cs (91)

∂Π2

∂K2= −bL(K) g e (Q−DL) (92)

∂Π2

∂K∂Q= g e (1−BL(K))− cs (93)

38

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C.2.4 Monopoly Objective Function Properties, Case 3

ΠQ,K =(24− a)

{DL(pL − (1− r)g)

}+ g (Q−DH) e

[KV −

∫ KV

0BH(x)dx

](1−

√r(

Q−DH

DL+DHr

−DH

))+

a

{DH(pH − (1− r)g)

}+ g (Q−DL) e

[K −

∫ K

0BL(x)dx

](1−

√r(

Q−DH

DL+DHr

−DH

))− cQQ− (Q−DL)cKK

(94)

∂Π

∂Q= g e

{K −

∫ KV

0BH(x)dx−KBH(KV )

DL −DH

Q−DH−

√r

DL+DHr

−DH

[K(2Q−DH −DL)− 2(Q−DH)

∫ KV

0BH(x)dx−KBH(KV )(DL −DH)]

}+

g e[K −∫ K

0BL(x)dx][1−

√r(2Q−DL −DH)DL+DH

r−DH

]− cQ − cKK

(95)

∂Π2

∂Q2=g e

{−KBH(KV )

DL −DH

(Q−DH)2−

K

(Q−DH)2[BH(KV )(DH −DL) +

(DL −DH)2

Q−DHKbH(KV )]−

√r

DL+DHr

−DH

[2K − 2

∫ KV

0BH(x)dx− 2KBH(KV )(DL −DH)−K2 (DL −DH)2

(Q−DH)2bH(KV )]

}−

g e[K −∫ K

0BL(x)dx]

2√r

DL+DHr

−DH

=g e

{−

K2(DL −DH)2

(Q−DH)3bH(KV )−

√r

DL+DHr

−DH

[2K − 2

∫ KV

0BH(x)dx + 2KBH(KV )(DH −DL)−K2 (DL −DH)2

(Q−DH)2bH(KV )]

}−

g e[K −∫ K

0BL(x)dx]

2√r

DL+DHr

−DH

(96)

Want to show that ∂Π2

∂Q2 < 0

g e

{−

K2(DL −DH)2

(Q−DH)3bH(KV )− g e[K −

∫ K

0BL(x)dx]

2√r

DL+DHr

−DH

√r

DL+DHr

−DH

[2K − 2

∫ KV

0BH(x)dx + 2KBH(KV )(DH −DL)−K2 (DL −DH)2

(Q−DH)2bH(KV )]

}< 0

→g e

{g e[K −

∫ K

0BL(x)dx]

2√r

DL+DHr

−DH

>

−√r

DL+DHr

−DH

[2K − 2

∫ KV

0BH(x)dx + 2KBH(KV )(DH −DL)]

}(97)

This holds asK2(DL−DH )2

(Q−DH )3bH(KV ) > K2 (DL−DH )2

(Q−DH )2bH(KV )

√r

DL+DHr

−DH

, because 1Q−DH

>√r

DL+DHr

−DH

0 > −[2K − 2

∫ KV

0BH(x)dx + 2KBH(KV )(DH −DL)]

[2K −KV + 2KBH(KV )(DH −DL)] > 0

DH −DL >>2− V

2BH(KV )

(98)

Hence as long as the difference between low and high demand is within its usual order of magnitude, of severalthousand MWs, this model holds trivially. Only for very small K and V this may not hold, which are cases that falloutside the scope of this paper.

∂Π

∂K=g e (1−

√r(

Q−DH

DL+DHr

−DH

))(Q−DL)(1−BH(KV )) + g e (1−√r(

Q−DH

DL+DHr

−DH

))(Q−DL)(1−BL(K))− (Q−DL)cs

=g e (1−BH(KV ))− (

√rg e

DL+DHr

−DH

))(Q−DH)(1−BH(KV )) + g e (1−BL(K))−

(

√rg e

DL+DHr

−DH

))(Q−DH)(1−BL(K))− cS

(99)

∂Π2

∂K2=− g e (1−

√r(

Q−DH

DL+DHr

−DH

))(Q−DL)2

Q−DHbH(KV )−

g e (1−√r(

Q−DH

DL+DHr

−DH

)) (Q−DL)bL(K)

=− g e (1−√r(

Q−DH

DL+DHr

−DH

))((Q−DL)2

Q−DHbH(KV ) + (Q−DL)bL(K))

(100)

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∂Π2

∂Q∂K=g e

{1−BH(KV )V −

DL −DH

Q−DH[BH(KV ) + KbH(KV )V ]−

√r

DL+DHr

−DH

[(2Q−DH −DL)− 2(Q−DL)BH(KV )−BH(KV )(DL −DH)−KbH(KV )V (DL −DH)]

}+

g e[1−BL(K)][1−√r(2Q−DL −DH)DL+DH

r−DH

]− cS

(101)

∂Π2

∂K∂Q=g e [1−BH(KV )− (Q−DL)KbH(KV )

DL −DH

(Q−DH)2]−

(

√rg e

DL+DHr

−DH

)[(2Q−DH −DL)(1−BH(KV ))− (Q−DL)KbH(KV )DL −DH

(Q−DH)]+

g e (1−BL(K))− (

√rg e

DL+DHr

−DH

)(2Q−DH −DL)(1−BL(K))− cS

=g e [1−BH(KV ) + KV bH(KV )DH −DL

Q−DH]−

(

√rg e

DL+DHr

−DH

)[(2Q−DH −DL)(1−BH(KV )) + KV bH(KV )(DH −DL)]+

g e (1−BL(K))− (

√rg e

DL+DHr

−DH

)(2Q−DH −DL)(1−BL(K))− cS

(102)

C.2.5 Monopoly Objective Function Properties, Case 3, Q Fixed

ΠQ,K =(24− a)

{DL(pL − (1− r)g)

}+ g (Q−DH) e

[KV −

∫ KV

0BH(x)dx

](1−

√r(

Q−DH

DL+DHr

−DH

))+

a

{DH(pH − (1− r)g)

}+ g (Q−DL) e

[K −

∫ K

0BL(x)dx

](1−

√r(

Q−DH

DL+DHr

−DH

))− cQQ− (Q−DL)cKK

(103)

∂Π

∂K=g e (1−

√r(

Q−DH

DL+DHr

−DH

))(Q−DL)(1−BH(KV ))+

g e (1−√r(

Q−DH

DL+DHr

−DH

))(Q−DL)(1−BL(K))− (Q−DL)cs

=g e (1−√r(

Q−DH

DL+DHr

−DH

))(1−BH(KV ))+

g e (1−√r(

Q−DH

DL+DHr

−DH

))(1−BL(K))− cs

→ BH(KV ) + BL(K) = 2−cs

g e (1−√r(Q−DH )

DL+DHr

−DH

)

(104)

40