Precautionary Energy Storage

Tunç Durmaz⇤

May 02, 2015

Abstract

Energy storage technologies were primarily used to take advantage of dispatchablesources of energy and demand variability, and the underlying economic analysis focusedmainly on pumped hydro storage. Today, increasing shares of renewable energy (RE)have drawn attention to the role of storage technologies in dealing with the vulnerabili-ties caused by renewables in meeting energy demand. With the presence of intermittentRE, however, not much consideration has been given to energy storage due to precaution-ary motives. In our study, we look at to what extent a convex marginal utility (prudence)and a convex marginal cost (frugality) can spur precautionary energy storage. We usea simple theoretical model of energy consumption and production with intermittent re-newable sources, thermal systems, and energy storage. First, we characterize the optimalsolution and demonstrate how prudence and frugality will lead to higher levels of energysaving due to intermittent RE. Furthermore, we demonstrate how the optimal allocationcan be decentralized through competitive markets. Our analysis indicates that prudenceand frugality exert an upward pressure on spot market energy prices through higher de-mand for energy storage, higher thermal energy generation, and lower consumption.Keywords: Precautionary energy storage; Renewable energy; Prudence; Frugality;Rational Expectations Equilibrium

1 Introduction

Renewable energy (RE) is one of the key solutions in combating the climate change problemand is high on the policy agenda, especially in the developed countries. Since the launch ofthe European Climate Change Policy in 2001, the European Union (EU) has been a leader inthe global action to mitigate climate change through its energy policies. Following the pro-posal for an energy and climate change package in January 2007, the member states agreedon a binding target to raise the share of RE in the EU’s total primary energy consumption to20% by 2020 (IEA, 2008). This agreement was then followed by a directive on the promotion

⇤NHH, Norwegian School of Economics, Tunc.Durmaz@nhh.no

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of the use of RE, which endorsed mandatory national targets consistent with the 20% share ofthe RE target.1 In 2014, the RE generation from wind, solar, geothermal and other renewablesources accounted for 12% of the electricity generation in Europe (IEA, 2014a), which washigher by more than 100% compared to the generation level in 2010.2 The correspondingfigures for the OECD and the world were 6% and 4%, respectively, in 2010. The U.S. En-ergy Information Administration projects that by 2020 these shares will reach 11% and 8%,respectively.3

Nevertheless, RE is inherently variable and uncertain, and the shares of RE generationthat get larger in time can cause serious vulnerabilities in meeting energy demand. A num-ber of strategies exist to deal with the challenges created by uncertain RE generation.4 Theuse of thermal dispatchable generation is a common way to smooth the operation of existingpower grids and enable a balance between demand and supply. However, this comes at a costbecause these units are expensive to operate and add carbon to the air. A demand response isanother way to enhance the grid’s resilience and enable a greater use of RE. Although the ideaof getting consumers to become active in the markets can be seen as a novel solution by many,the culture is changing with smart grids that allow consumers to access real-time knowledgeabout prices, be more responsive and control their power usage and consumption. The exist-ing technologies allow a demand response as a result of direct feedback and prices throughvarious devices. These direct feedback technologies aim at engaging the consumers with di-rect feedback on electricity consumption and prices. These technologies allow consumers torespond to changing prices, which, for example, is similar to consumers responding to chang-ing gasoline prices. Alternatively, when active engagement is not practical, consumers canhave smart appliances that react to prices based on criteria set by the consumer (Hamiltonet al., 2012). With sustained investments, it is projected that the smart grid will provide acommunications network for the energy industry by 2020; i.e., a system of interconnectedenergy networks similar to the Internet in terms of its provisions for business and personalcommunications (RMI, 2014).

Another way to enhance the reliability of the grid is energy storage, especially in periodsof peak demand. Energy storage technologies absorb and store energy for a period of timebefore releasing it to supply energy or power services. In simple terms, these technologiestake excess generation produced on a windy or sunny day, store the power in multiple places(from large hydro stations to home batteries, and everything in between), and supply powerwhen RE is inadequate and thermal energy is expensive to produce. Key benefits includeproviding balancing services, such as load following, supplying power during brief distur-bances, and serving as substitutes for network transmission and distribution upgrades (Wanget al., 2012). Currently, the costs of electricity storage technologies are rather high. However,

1Directive 2009/28/EC of the European Parliament and Council of April, 23 2009 on the promotion of theuse of energy from renewable sources, which amended and subsequently repealed Directives 2001/77/EC and2003/30/EC, available at http://eur-lex.europa.eu/legal-content/EN/ALL/?uri=CELEX:32009L0028.

2In 2010, the share of RE in the total electricity production was 5.5%. In calculating the shares we restrictedour attention to the EU member countries except Bulgaria, Croatia, Latvia, Malta and Romania.

3The ratios for the OECD and the world were calculated using Tables 13 and 14 in EIA (2013).4Intermittency means that RE generation depends on weather conditions and is non-controllable (Steffen

and Weber, 2013).

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with the development of better economic storage technologies with larger storage capacities,they will potentially become game changing technologies. Nevertheless, the energy storageindustry is experiencing strong growth and it is expected that the industry will have a globalnet worth of $10.8 billion in 2018 (RMI, 2014).

When consumer awareness regarding price changes becomes more common, will theeconomics of energy storage need a more proper assessment? Note that although new in-vestments and improvements in power grid technology allow the consumers to become moreresponsive, they also make them more vulnerable to price changes and consumption fluctu-ations due to intermittent RE. This may even be a more serious issue during peak demandperiods when electricity supply is likely to be a necessity. When consumers are responsive,and energy generators –in particular, dispatchable thermal energy generators– are responsi-ble to match electricity supply with demand, two precautionary motives can lead to a higherdemand for energy storage. One is prudence with respect to electricity consumption, whichis formally equivalent to a positive third derivative of the utility function. The other is fru-gality, which is formally equivalent to a convex marginal cost of thermal energy productionfunction. We refer to the property of a convex marginal cost function as frugality, since, inthe presence of uncertainty, it endows a cost minimizing producer with the same motivationsas that of a prudent consumer. In Section 1.1, we will motivate the properties of prudenceand frugality and give a first intuition as to why they encourage energy storage.

In this study, our aim is to show how prudence and frugality drive the need for precaution-ary energy storage. In doing so, we first look at a benevolent planner problem and examinehow storage decisions are influenced in the presence of a convex marginal utility (prudence)and a convex marginal cost (frugality). As the share of RE in an energy system influences theoptimal operation of thermal systems, which, in turn, affects precautionary energy storage,we consider different shares of energy generated from renewable sources of energy. Further,we demonstrate how the optimal allocation can be decentralized through competitive mar-kets, and discuss how current energy prices and the use of energy systems are influenced byprudence, frugality, the degree of intermittency, price elasticities, and RE capacity. With asmall share of RE sources, prudence and frugality can cause precautionary energy storageto varying degrees. Similarly, the degree of intermittency boosts energy storage to varyingdegrees when both prudence and frugality are present. Even in the absence of prudence, wedemonstrate that frugality can still allow for precautionary storage and vice versa. Higherdemand and supply elasticities diminish the effect of prudence and frugality, respectively, onenergy storage. For a highly elastic demand, consumption adjustment becomes a good sub-stitute for energy storage and lower the need for precautionary energy storage. When energysupply is more price elastic, the flexibility offered by thermal systems in generating energybecomes a better substitute for storing energy.

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1.1 Motivations for prudence and frugality

Prudence

Let us explain what it means to be prudent in our framework. Consider a consumer witha utility function, U(Q), which is increasing, U 0

> 0, and concave, U 00< 0, in electricity

consumption, Q.5 Suppose that the consumer is exposed to a zero-mean consumption risk,x . The difference between certain and expected utility is given by

k(Q) ⌘ U(Q)� E[U(Q+ x)].

Due to the Jensen’s inequality, k(Q), is positive if U(Q) is concave. In other words, uncer-tainty is costly for the consumer when the he/she is risk averse.

A consumer is said to be prudent with respect to electricity consumption if the cost of un-certainty, k(Q), decreases as consumption, Q, increases. In differential terms, this is equiva-lent to k

0(Q), given by

k

0(Q) = U

0(Q)� E[U 0(Q+ x)],

being negative, which is ensured by the convexity of the marginal utility (i.e., U 000> 0).

Again, this results from the Jensen’s inequality. As consuming stored energy is one way toincrease Q, and thus, to decrease the cost of uncertainty, U 000

> 0 –i.e., prudence– gives aprima facie argument for energy storage.

Focusing on income lotteries, the evidence for prudence can be found in the experimen-tal research literature. In line with the prediction of precautionary saving theory, Noussairet al. (2014) indicate that the majority of individual decisions is consistent with prudence.6Crainich et al. (2013) provide theoretical arguments to show that prudence is more prevalentthan risk aversion, as risk lovers can also demonstrate it. This prediction is shown to hold inEbert and Wiesen (2014) and Deck and Schlesinger (2014). Accordingly, prudence may be amore universal trait, which suggests that narrowing down risk preferences to the second-ordermay obscure valuable information. There are also empirical studies such as Chavas and Holt(1996) and Guiso et al. (1996) that support prudence. Carroll and Samwick (1998) indicatethat wealth holdings are positively and significantly related to income uncertainty.7

5We assume a quasi-linear utility function. Thus, U(Q) is the monetary value of utility (or surplus) that isderived from consuming Q kilowatt-hour (kWh) of electricity. In economic theory, using such preferences is astandard assumption when discussing issues related to a single market in a general equilibrium framework.

6Noussair et al. (2014) also argue that the degree of prudence has implications in a wide range of economicapplications such as bargaining, bidding in auctions, rent seeking, discounting, sustainable development andclimate change, and tax compliance.

7Carroll and Kimball (2008) argue that, although there is evidence for prudence, it is measured differentlywith different data; i.e., the degree of the same motive indicates changes among different data sets.

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Frugality

In this subsection, we shall expound frugality. Consider a producer with a smooth increasingcost function C(Q), where Q is the level of production. Suppose that the firm faces a zero-mean production risk, x. Here, x represents the variation in the residual demand that the firmhas to match with its supply. The difference between the expected and the certain cost ofproduction is as follows:

⇢(Q) ⌘ E[C(Q+ x)]� C(Q).

Due to the Jensen’s inequality, the firm is exposed to a penalty of uncertainty when C

00>

0 (i.e., the cost function is convex). In other words, increasing marginal cost implies thatuncertainty is costly for the firm: ⇢(Q) > 0.

A producer is said to be frugal with respect to energy generation if the cost of uncertainty,⇢(Q), increases as production, Q, increases. This is equivalent to ⇢0(Q), given by

⇢

0(Q) ⌘ E[C 0(Q+ x)]� C

0(Q),

being positive, for which the convexity of the marginal cost (i.e., C 000> 0) is sufficient. Once

again, this results from the Jensen’s inequality. As storing enegy is one way to decrease Q

and, thus, to decrease the cost of uncertainty, C 000> 0 –i.e., frugality– provides a second

prima facie reason for energy storage.

Cecchetti et al. (1997) analyze production and inventory data and conclude that firmsface a convex marginal cost curve. Indeed, a convex marginal (production) cost curve has atransparent economic interpretation, which indicates that it becomes increasingly expensiveto make large and positive changes to meet the residual demand. Moreover, Cecchetti et al.(1997) adds that a firm is capacity constrained when faced with a convex marginal cost curve.

Now let us explain how the production risk emerges for a fossil fuel power generator.Variations in energy demand are typically limited and more predictable compared with thevariations in supply (Nyamdash et al., 2010; Hansen, 2009; Hart et al., 2012; Ummels et al.,2007). However, due to the low operating cost of intermittent RE that leads to its earlier dis-patch (Denholm et al., 2010), the residual load is intermittent.8 Therefore, after accountingfor RE, a capacity constrained thermal dispatchable generator that has to supply the residualload can incur high operating costs especially during periods of peak demand and low renew-able energy generation. As a result, a frugal firm will intend to balance its limited supply andthe residual demand in such a way that it minimizes its expected cost.

As a real world example, consider Figure 1, which illustrates the electricity supply and8In promoting electricity produced from RE sources, Directive 2001/77/EC of the European Parliament and

the Council of September 27, 2001 prescribes that transmission system operators shall give priority to generatinginstallations using RE sources.

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demand curves for the NordPool Power Exchange.9 Due to their low marginal costs (zero fuelcosts) renewables appear near the bottom of the curve. They are followed by nuclear power,combined heat and power (CHP) plants. Condensing plants and gas turbines have the highestmarginal costs for generating energy.10 In the presence of intermittent RE generation, a lowwind power will cause a limited change in the energy price at night due to low generationcosts. During peak times, however, the increase in the electricity price will become muchmore significant, as each additional generating capacity is brought online and the cost of thiscapacity dramatically increases (Gravelle, 1976).

Figure 1: Supply and Demand Curves for the NordPool Power Exchange (EWEA, 2009, p.18).

The sequence of linear cost functions that appears in Figure 1 can be collectively approx-imated by a smooth curve, which would then represent a convex industry supply schedule. Inthis case, frugality becomes an industry trait and, in turn, can have economically significantimpact on production, consumption and storage decisions under uncertainty.

The remainder of the paper is organized as follows. Section 2 reviews the related litera-ture. In Section 3 we present the model and describe the benevolent planner solution given acertain and cheap supply of baseload power, and then given an intermittent RE supply withdifferent shares of RE generation in the energy system. In Section 4 we demonstrate how theoptimal allocation can be decentralized through competitive markets, and discuss the impli-cations of the precautionary motives. Section 5 concludes.

9The Nordic electricity exchange, Nord Pool Spot, is a power market that primarily serves the players in thewholesale market for electricity. It covers Denmark, Finland, Sweden, Norway, Estonia and Lithuania (NordPool Spot, 2011).

10Hydropower is not identified in the figure (EWEA, 2009).

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2 Related literature

The optimal dispatch of energy and energy storage in renewable and thermal energy systemswas addressed earlier in the operations research literature. In a model of hydroelectric andthermal systems, Little (1955) studies hydroelectric generation under uncertainty. Disregard-ing fluctuations in energy demand, the study determines optimal energy dispatch and waterstorage policies. Borrowing most of his assumptions from Little (1955), Koopmans (1957)calculates the optimal energy generation and storage policies in the presence of completecertainty.11 He shows how thermal energy generation and storage decisions are related to theenergy prices and storage rents.

However, with a few exceptions, the economics of pumped-hydropower (PSH), has notattracted many researchers so far. An early work on the economics of PSH is Jackson (1973),where the motivation to use PSH is due to the flexibility the technology offers as nuclearpower plants cannot be efficiently turned off and on. In his analysis, Jackson assumes thatstorage is always optimal and, hence, the technology always pumps water to an upper reser-voir. In contrast, Gravelle (1976) shows the conditions under which storage is efficient. As-suming that demand deterministically varies between off-peak and peak periods, he showsthat storage allows the substitution of less costly off-peak production for highly valued peakproduction. In return, more highly valued peak consumption is substituted for off-peak con-sumption. Horsley and Wrobel (2002) build on the framework given by Koopmans (1957)and, they study the optimal operation of existing PSHs and the valuation of energy and stor-age rents in the presence of uncertain inflows.

Crampes and Moreaux (2010) build their work on Jackson (1973) and Gravelle (1976).Unlike Horsley and Wrobel (2002), who assume an exogenously given demand and perfectlyefficient conversion, they investigate the optimal thermal energy dispatch and PSH when en-ergy demand varies deterministically between peak and off-peak periods and there are lossesin converting energy. Assuming a merit order in using thermal generators, the study firstcalculates a frontier that separates storage and no-storage solutions given technical condi-tions such as operation cost characteristics and energy losses. The authors then calculate thesocially optimal allocation given consumer preferences. When thermal generation is usedto pump water to an upper level reservoir, the welfare losses corresponding to this off-peakperiod is compensated by welfare gains in the peak period when stored water is used. In linewith Jackson (1973) and Gravelle (1976), the study then discusses the implementation of anoptimal energy dispatch in competitive markets where agents are price takers. The calcula-tions show that the peak and off-peak price differential is reduced when storage is feasible.

The literature on commodity storage has relevant implications for the economics of en-ergy storage. Wright and Williams (1982, 1984) examine the welfare effects of storage ina market with stochastic supply and indicate that the welfare effects of storage depend onthe specification of the inverse demand function (i.e., the curvature of the demand curve).Elasticity of supply is another factor that influences welfare. The authors introduce a param-

11Koopmans (1957) argues that the purpose is to develop concepts and tools that will be useful in a systematicanalysis of cases involving uncertainty.

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eter that is analogous to the coefficient of prudence (cf. Kimball, 1990) and argue that agentswill pay for a mean-preserving decrease in the variability of the commodity consumed whenthe relative prudence is bigger than one (Wright and Williams, 1984; Williams and Wright,1991). Given the storage and current production (i.e., the amount on hand), the authors de-rive a storage rule numerically. Accordingly, when the stored amount is less than a particularthreshold, all of the stored commodity will be consumed, and vice versa. Numerical simula-tions indicate that storage is more likely and the marginal propensity to store at the thresholdincreases when there is a higher degree of variability in supply (Wright and Williams, 1982).

Regarding the relationship between the degrees of variability in RE and energy storage,one finds similar results in the operations research and economics literature. Tuohy andO’Malley (2011) argue that intermittency increases the benefit driven from the flexibilityoffered by PSH and makes energy storage more attractive. Evans et al. (2013) demonstratethat water storage becomes more welfare-enhancing with higher uncertainty. Surely, it ispossible to find more discussions in the literature that connect the higher levels of storageto the higher degrees of variation in the RE supply. However, the role that precautionarymotives play is not elaborated upon adequately. Evans et al. (2013) assume a linear demandschedule (i.e., U 00

> 0 and U

000 = 0) and a convex supply schedule (i.e., C 00> 0 and C

000> 0)

for thermal energy generation. As we will show later in our study frugality will lead toprecautionary energy storage, unless capacity constraints are explicitly considered for eachthermal unit. Evans et al. (2013) do not address such a relationship. In Bobtcheff (2011),the cost of thermal energy is constant and not subject to any capacity constraints; i.e., hermodel disregards frugality. She numerically shows that a benevolent planner keeps morewater in a reservoir when faced with higher uncertainty and explains that this action is dueto prudence. However, she does not present a formal analysis. A first step in this regardcomes from Hansen (2009). He analyzes the effects of uncertainty on market performance inhydropower systems, and shows that competitive firms decrease hydroelectricity generationand save more water for the future when consumers are prudent (i.e., the inverse demand forelectricity is convex).

Our work is an extension of Crampes and Moreaux (2010) with intermittent RE. Insteadof assuming only a PSH system, we consider a generic storage technology that has the abilityto shift energy for long durations when necessary.12 Thus, we are interested in storage tech-nologies that are more suitable for energy management applications.13 As these applicationsinclude generating power over longer time periods, which require longer discharge times (i.e.,continuous discharge rates), we consider storage technologies that have the ability to shift thebulk of energy for a duration of several hours or more (Denholm et al., 2010).14 This way

12It is possible to assess different types of storage technologies by using different round-trip efficiencies.13When broadly categorized, one can classify energy storage applications as energy management and power

applications (Kim et al., 2012). Power applications correspond to a range of ramping and ancillary services thatdo not typically require a steady discharge for several hours. For systems driven by renewables, the dischargetime may extend up to an hour, which allows peaker plants to come online. Applications such as frequencyregulations, voltage stability and contingency spinning reserves enter this category (Kim et al., 2012; Lichtneret al., 2010). Flywheels, capacitors, supercapacitors, superconducting magnetic energy storage and severalbattery technologies are a few examples of storage technologies for power quality purposes.

14However, we can always make some inferences for applications that require short discharge times andhydropower systems.

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energy storage technologies can insulate the rest of the power grid from substantial changesin net supply.15 High energy batteries, pumped hydro (the most widely used form of elec-trical energy storage), and compressed air are the technologies for this type of applications(Denholm et al., 2010).16

Although we focus on uncertainty in RE only, we do not neglect variations in demand andemploy a deterministic demand that varies between off-peak and peak periods.17 However,even though we work with a deterministically varying demand function, it can be noted thatthe residual load is intermittent. This is due to the low operating cost of intermittent RE thatleads to its earlier dispatch. After accounting for RE, the net but intermittent load is then metby the peaking power plants or “peakers".

Our work discusses storage and no-storage solutions and indicates that intermittency canlead to a higher level of storage when agents are prudent and frugal. With a 100% share ofRE sources, where thermal systems are obsolete, our results are in line with Hansen (2009)where a higher level of energy generated from renewable sources is transfered into the nextperiod.18 However, for systems that have thermal systems that are continuously kept online,frugality can have a significant impact on precautionary energy storage. We further indicatethat higher degree of intermittency leads to higher energy storage, and the effects of prudenceand frugality can differ in magnitude due to endogenously assigned weights. Moreover, weexplain how the effects from prudence and frugality change in relation to the share of RE inthe energy system. By applying our findings to perfectly competitive markets, we can showthat precautionary motives lead to a higher willingness to pay for electricity today which inturn allows for a higher level of energy storage. While a higher price elasticity of demanddecreases the effect of prudence (i.e., consumption adjustment becomes a stronger substitutefor stored energy), a higher supply elasticity decreases the precautionary storage motive fromfrugality. This is because a higher elasticity of supply will make the intermittency problemless of an issue when the residual demand can be more easily met by the thermal systems.

15One example of energy management applications is the electric energy time shift, which means charginga storage device when electricity prices are low (e.g., storing excess wind power during periods of low energydemand) and then discharging the device when electricity prices are high (Lichtner et al., 2010; Kim et al.,2012).

16The interested reader may refer to Appendix A, which reviews some mature and promising storage tech-nologies for energy management applications.

17Compared with the variations in supply, the variations in demand tend to be limited and more predictable(Nyamdash et al., 2010; Hansen, 2009; Hart et al., 2012; Ummels et al., 2007). In Norway, the annual demandof electricity consumers has not varied by more than 6 terawatt-hours (TWh) between 1999 and 2009. Thecorresponding number for water inflows is between 85–140 TWh. This is the main reason that numericalmodels of the Nordic power market, such as the Balmorel model and the EMPS (EFI’s Multi-area Power-marketSimulator or “Samkjøringsmodellen” in Norwegian), consider inflow variations more than demand variations(Hansen, 2009).

18In Hansen (2009), hydroelectric power generation is reduced to transfer more water into the next period.

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3 The model

Consider a closed economy in which there is a single-commodity (i.e., energy) that can besupplied using systems of thermal energy, RE and energy storage:

Q

st = Qdt + ztQct + St � ↵St+1.(1)

Q

st , the energy supply at time t, is composed of thermal energy, Qdt, RE, ztQct, and the

available level of stored energy St, minus the amount of energy transferred to the next period,↵St+1. Once there is an installed capacity, the unit cost of generating RE becomes so low thatwe consider it as zero (Ambec and Crampes, 2012; Evans et al., 2013). Thus, the renewablesystem operates at its capacity and Qc = Qc at all times.19 For ↵ > 1, 1/↵ is the round-trip efficiency parameter, which is the ratio of energy recovered to the initially stored energy.Hence, a certain percentage of stored energy is lost with time. zt 2 [0, 1], a parameterthat represents weather conditions such as wind, sun or rain, is an exogenous, independentand identically distributed (i.i.d.) random variable. It has a commonly known cumulativedistribution function, F (z), with a compact support [0, 1]. µ and �2 represent the mean andvariance of z, respectively. When there is no wind (zt = 0), there is no generation of RE(ztQc = 0), and vice versa. While zt is observed prior to making decisions in period t, zt+1

is uncertain and thus it is denoted by zt+1. We assume that energy supply, Qst , meets energy

demand, Qt, instantaneously.

Further in the analysis, we will work with

Z0 ⌘ z0Qc and Z1 ⌘ z1Qc

instead of z1Qc. This enables us to use a simpler notation. Accordingly,

µZ ⌘ µQc and �

2Z ⌘ �

2Q

2c

are the corresponding mean and variance, respectively, for the random variable Z1.

The cost function for thermal energy is assumed to be continuous; i.e., there is a con-tinuum of thermal generators. The marginal cost (unit cost) of fossil fuel energy generationis increasing, C 0(Qdt) > 0, C 00(Qdt) > 0 with C

000(Qdt) � 0, where C

0, C 00 and C

000 arethe first-, second- and third-order derivatives of the cost function, respectively. When themarginal cost is increasing, where C 00(Qdt) > 0 with C

000(Qdt) � 0, one can think of a uniquemerit order of using individual generators: initially the power plants with the lower marginalcosts of energy generation will be brought online (such as a coal-fired power plant), followedby more costlier ones (such as a natural gas power plant with carbon capture and storage).We assume that given the market price for energy, there is no constraint on the availabilityof Qdt, i.e., there is a large existing generating capacity portfolio that can meet the demandwhen RE is not adequate to supply the total load (Joskow, 2011; Bobtcheff, 2011). However,when C

000(Qdt) > 0 (i.e., the marginal cost is convex), one can think of an implicitly as-signed capacity constraint –an upper bound– on the fossil fuel energy generation that allows

19The only cost for the renewable system is the opportunity cost of not generating more than Qc.

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for the effect from convexity to dominate when high levels of thermal energy generation arerequired.

The unit cost of storing energy, 1/↵, which is a constant, is due to energy losses. Givenvarious storage technologies with differing round-trip efficiencies, we could have considereda unique merit order of using storage facilities as well. Although, such an assumption woulddiminish the level of energy storage and take our model one step closer to reality, it wouldnot affect our key results.

U(Qt) is the utility function over the kWh consumption of energy, where Qt denotesenergy consumption. It is assumed that U 0

> 0, U 00 0 and U

000 � 0, where U

0, U 00 and U

000

are the first-, second- and third- order derivatives of the utility function, respectively. Thus,under perfect competition, the inverse demand schedule is downward sloping and convex.

In our analysis we assume that the power grids are smart in a way that the transmissionand distribution systems of electricity are added with digital sensors and remote controls(Ambec and Crampes, 2012; van de Ven et al., 2013; Evans et al., 2013). This assumptioninstantaneously lets the prices adjust, such that the energy supply meets the demand at alltimes. Thus, there is no overloading of the power grids. In the following subsection, we solvethe model from a benevolent planner perspective. We then turn to a decentralized setting andask whether competition leads to an optimal solution.

3.1 Two-period model

We consider a two-period model. In the first period, t = 0, the demand for energy is low.Let’s call this the off-peak period. In the final period, t = 1, we call it the peak period, thedemand is high. For h representing the peak period and l representing the off-peak period, thiscan be algebraically shown as U 0

h(Q) > U

0l (Q). Hence, for the same level of consumption,

Q, the benefit from consuming an additional amount of energy is strictly higher in the peakperiod. We characterize the peak period with ✏ > 0, which is subtracted from Q in the utilityfunction. Such a formulation shows a resemblance to the subsistence level of consumption:in the day time, the use of energy becomes so much more valuable that the agents cannotafford to consume less than ✏. Similar to U

0h(Q) > U

0l (Q), U 0(Q� ✏) > U

0(Q).20

The planner’s problem for the two-period model is then as follows:

max{Qj ,Qj ,Qdj

,S1}U(Q0)� C(Qd0) + E

hU(Q1 � ✏)� C(Qd1)

i(2a)

subject to Q0 = Qd0 + Z0 + S0 � ↵S1,(2b)

Q1 = Qd1 + Z1 + S1,(2c)Q0 � 0, Q1 � ✏ � 0, Qdj � 0, j = 0, 1(2d)S � S1, S1 � 0 and S0 � 0 given.(2e)

20This is always the case when U 00 < 0.

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As the weather for next period is uncertain, we use E[·] to denote the expected welfare inperiod 1. Expressions given by (2b) and (2c) are due to the fact that energy supply meetsthe demand instantaneously. Energy consumption (net of ✏) is positive and thermal energycan equal zero (i.e., become idle), when the RE generation is sufficiently high (c.f., Eq.(2d)).S is the capacity constraint for energy storage. When the storage capacity is reached, theremaining energy will be consumed. Stored energy cannot be negative; i.e., we cannot bor-row energy from the future to consume today. Throughout the study, we assume S0 = 0.This assumption does not change the main results of the study, which identify prudence andfrugality as the main drivers of precautionary storage. However, we shall comment on thepossible effects of S0 > 0 later in the study. For simplicity, we neglect discounting betweenthe first and final periods.

We solve the problem recursively. Given RE generation in the last period, Z1, and theavailable amount of stored energy, S1, the problem in period 1 is the following:

max{Q1,Qd1

}U(Q1 � ✏)� C(Qd1)(3a)

subject to Q1 = Qd1 + Z1 + S1,(3b)Q1 � 0, Qd1 � 0.(3c)

The first-order necessary condition for a maximum yields:21

U

0(Qd1 + Z1 + S1 � ✏) C

0(Qd1), with equality if Qd1 > 0.(4)

If the level of energy supplied by the renewable systems and energy storage is sufficientlyhigh such that the marginal utility will become less than the marginal cost of fossil fuelenergy, then no thermal energy will be produced: for U 0(Z1 + S1 � ✏) < C

0(0), Qd1 = 0.Otherwise, U 0(Qd1 + Z1 + S1 � ✏) = C

0(Qd1) and the thermal systems will be active. As aresult, one can calculate a threshold level, Z, such that when Z1 > Z, the thermal systemsbecome idle, and vice versa:22

Q

⇤d1 � 0 if Z1 Z,(5a)

Q

⇤d1 = 0 otherwise (i.e., Z1 > Z),(5b)

where we denote the optimal thermal energy decision by

Q

⇤d1 ⌘ Qd(Z1 + S1 � ✏).

Eq. (5a) demonstrates that when the weather conditions are bad and the level of RE is low,then thermal systems will be used to meet the residual demand.23 In contrast, (5b) expresses

21The second order condition for a maximum is satisfied by U 00(Q1 � ✏)� C 00(Qd1) < 0.22Using U 0(Z1 + S1 � ✏) < C 0(0), one can calculate Z as follows:

Z1 > Z ⌘ U 0�1(C 0(0))� S1 + ✏.

Note that Z depends on the decision variable S1. Given ✏, we assume that this condition is never violated.23We assume that there is no correlation between weather conditions and energy demand.

12

that when RE generation is high, then the thermal systems will be shutdown.

When RE generation cannot get beyond Z, Z1 Z –in other words, when such gener-ation is not plentiful– thermal systems will always be employed to supply the net load, andvice versa. Hence, when this is the case (i.e., there is an interior solution for thermal energy),the comparative statics provide the following:

@Q

⇤d1

@Z1=

U

001

C

001 � U

001

< 0,@Q

⇤d1

@S1=

U

001

C

001 � U

001

< 0,@Q

⇤d1

@✏

= � U

001

C

001 � U

001

> 0(6)

where U 001 = U

00(Q⇤1�✏) and C

001 = C

00(Q⇤d1). Comparative statics analysis indicates that when

RE generation is higher, thermal systems produce less energy. The same reasoning appliesto a higher level of stored energy. When ✏ is higher (hence, energy is valued even more inthe peak period), the thermal energy production level is high. This result is an implicationof the interior solution, where thermal energy is the marginal resource and supplies the extraamount of energy demand. In contrast, when Z1 > Z, the thermal systems are kept idle.Thus,

@Q

⇤d1/@Z1 = @Q

⇤d1/@S1 = @Q

⇤d1/@✏ = 0.

Given Q

⇤d1 , the maximum value function for period 1 is:

W1(Z1, S1, ✏) = U(Q⇤d1 + Z1 + S1 � ✏)� C(Q⇤

d1).(7)

The problem in period 0 is then the following:

max{Q0,Qd0

,S1}U(Q0)� C(Qd0) + E

hW1(Z1, S1, ✏)

i(8a)

subject to Q0 = Qd0 + Z0 � ↵S1,(8b)Q0 � 0, Qd0 � 0,(8c)S � S1, S1 � 0.(8d)

The first-order necessary condition for thermal energy at a maximum is:24

(9) U

0(Q⇤d0 + Z0 � ↵S1) C

0d(Q

⇤d0), with an equality if Q⇤

d0 > 0.

Using the maximum value function in (7) and the Envelope Theorem, the first-order con-24Similar to the period 1 problem, the second order condition for a maximum is satisfied: U 00(Q0) �

C 00d (Qd0) < 0.

13

dition with respect to S1 provides the following:25

U

0(Q⇤d0 + Z0) � �E

hU

0(Q⇤d1 + Z1 � ✏)

iif S

⇤1 = 0,(10a)

U

0(Q⇤d0 + Z0 � ↵S

⇤1) = �E

hU

0(Q⇤d1 + Z1 + S

⇤1 � ✏)

iif S > S

⇤1 > 0,(10b)

U

0(Q⇤d0 + Z0 � ↵S) �E

hU

0(Q⇤d1 + Z1 + S � ✏)

iotherwise

⇣i.e., S

⇤1 = S

⌘,(10c)

where � ⌘ 1/↵ is the round-trip efficiency parameter and Q

⇤d0 ⌘ Qd(Z0 � ↵S

⇤1) as there

is no stored energy initially. From the benevolent planner’s perspective, the willingness tostore energy is determined by the expected marginal utility of energy consumption in thenext period. For

Q

⇤0 ⌘ Q

⇤d0 + Z0 � ↵S

⇤1 and Q

⇤1 ⌘ Q

⇤d1 + Z1 + S

⇤1 � ✏,

when there is an expected loss from storing energy, i.e., if U

0(Q⇤0) > �E

hU

0(Q⇤1)i

forS1 = 0, then energy storage facilities will be kept idle. Otherwise, energy is going to be storeduntil its current and expected social values are equalized. When U

0(Q⇤0) < �E

hU

0(Q⇤1)i

forS1 = S, then energy will be stored at its capacity. Notice that when S0 > 0, the marginalcost of energy storage becomes lower. This makes it more likely that energy will be storedand transferred to the next period.

3.2 Energy storage in the absence of RE generation

As a special case, suppose that energy can only be produced using dispatchable thermalsystems and there is energy storage. From Eq. (10b) one can get the following as an interiorsolution for thermal energy in both periods; i.e., U 0(Q⇤

0) = C

0(Q⇤d0) and U

0(Q⇤1 � ✏) =

C

0(Q⇤d1):

C

0�Q

⇤d0

�

C

0�Q

⇤d1

� = �.(11)

Eq. (11) satisfies intertemporal efficiency. There is an equality between the marginal rateof transformation of off-peak energy into peak energy and the cost of energy transformation.A similar result can be seen in Crampes and Moreaux (2010), where ↵ (⌘ 1/�) is the levelof energy required to add one unit to the stock of energy in a water reservoir for use inperiod 1 when the demand is high. If the absolute value of the slope of the isocost curve,C

0(Q⇤d0)/C

0(Q⇤d1), is greater than �, no energy is stored in period 0. This is because the cost

of storage on the margin is bigger than its value in the peak demand period. In contrast, ifC

0(Q⇤d0)/C

0(Q⇤d1) < �, the available storage capacity is completely utilized.

Suppose that the energy system is now composed of baseload power plants as well asdispatchable thermal and energy storage systems. Baseload power plants, such as nuclear

25The second order condition for a maximum gives @2V (S1)/@S21 = ↵2U 00(Q0) + U 00(Q1 � ✏) < 0.

14

and coal-fired plants that produce at low marginal costs and are devoted to the production ofbaseload supply, have slow ramp rates and are not flexible to switch on and off. As it is moreeconomical to operate them at constant production levels, these power plants, in general,do not change their production to match changing energy demand. Thus, these plants arerather inflexible in practicing “load following,” and electric power companies try to operatethem at full output as much as possible (Denholm et al., 2010).26 Provided that Q⇤

d1 > 0, inthe presence of an inflexible baseload supply, where µZ denotes this capacity, we have thefollowing:

C

0�Q

⇤d0

�= �C

0�Q

⇤d1),(12)

where

Q

⇤d0 = Qd(µZ � ↵S

⇤1) and Q

⇤d1 = Qd(µZ + S

⇤1 � ✏).

Similarly, energy will not be stored when the storage unit is empty and the unit cost of storageis greater than that of the peak demand period. In contrast, the storage capacity is completelyutilized if the present cost of storing a unit of energy on the margin is less than its valueadjusted for the round-trip efficiency (�) in the next period (i.e., when C

0(Q⇤d0) < �C

0(Q⇤d1)).

Provided the marginal rate of transformation of off-peak energy into peak energy is equal tothe cost of energy transformation, ↵, some energy will be stored and available in the nextperiod.

3.3 Energy storage in the presence of RE generation

In this subsection, we augment our analysis of energy storage by considering intermittentrenewable energy. In this regard, let there be variable generation from RE systems. In lightof Eq. (5) we can write the expected marginal utility as follows:(13)E⇥U

0(Q⇤1 � ✏)

⇤= F (Z)E[U 0(Q⇤

1 � ✏)|Z1 Z] +�1� F (Z)

�E[U 0(Q⇤

1 � ✏)|Z1 > Z].

While E[U 0(Q⇤1 � ✏)|Z1 Z] represents the conditional expected marginal utility from con-

suming energy supplied by both the thermal and renewable systems, E[U 0(Q⇤1 � ✏)|Z1 > Z]

is the conditional expected marginal utility when consuming energy from only the renewablesystems. Thus, the latter corresponds to cases in which thermal systems are kept idle. Further,F (Z) is the probability of Z < Z and vice versa.

For power systems with renewable sources of energy, the availability of energy storagecan be crucial in dealing with the uncertain variability of RE (Doherty and O’Malley, 2005;Steffen and Weber, 2013). As we will show in the following subsections, the motivation tostore energy can be different according to the share of intermittent RE in the energy system.

26Load following describes situations where power plant generation can accommodate changes in energydemand.

15

3.3.1 Small share of RE

Generally, the existing energy systems worldwide can be characterized by small shares ofRE (Lund et al., 2012). Thus, even with very favorable weather conditions, the RE systemscannot produce enough energy to meet the total energy demand, and thermal dispatchablegeneration always supplies the residual load. In our model, this translates into F (Z) = 1;thus, Z1 Z (i.e., the energy generated from renewable sources can never be high enoughso that thermal systems are taken offline).

In studying the effect of energy storage on welfare, we start from a situation of certainty,as presented earlier when we considered baseload power (see Eq. (12)). We then introducesome noise x around µZ in period 1 that satisfies Z1 = µZ + x, E[x] = 0 and E[x2] = �

2Z .

Hence, Z1 represents the intermittent RE with mean µZ and variance �2Z . Our purpose here

is to determine whether the optimal level of energy storage under uncertainty (intermittency)is greater than the corresponding level without uncertainty. Let S+

1 be the optimal level ofenergy storage when Z1 = µZ in the future with certainty:

S

+1 = argmax U(Qd(Z0 � ↵S1) + Z0 � ↵S1)� C(Qd(Z0 � ↵S1)) +W1(µZ + S1 � ✏).

Thus, without any uncertainty, the only factor that leads to energy storage is the higher valu-ation of energy in the peak period due to ✏ > 0. Furthermore, suppose that S⇤

1 is the optimallevel of energy storage when there is uncertainty in RE generation:

S

⇤1 = argmax U(Qd(Z0 � ↵S1) + Z0 � ↵S1)� C(Qd(Z0 � ↵S1)) + E[W1(Z + S1 � ✏)].

Following these definitions, we present our first major result by Theorem 1:

Theorem 1. If F (Z) = 1, then for every µZ and x with E[x] = 0, S⇤1 � S

+1 if and only if:

UU000 + CC

000 � 0,(14)

where U ⌘ (C 003)/(C 00 � U

00)3 and C ⌘ (�U

003)/(C 00 � U

00)3.

Proof. The proof is provided in Appendix.

Therefore, the level of energy storage is larger if RE is intermittent than otherwise. Noticethat it is the weighted sum in Eq. (14) that matters for precautionary energy storage, where U and C are weights attached to U

000 and C

000, respectively. Thus, there can be precautionarystorage even if U 000

< 0 and C

000> 0 or C 000

< 0 and U

000> 0. However, our main focus is on

prudence and frugality. In this regard, Theorem 1 has a stronger corollary:

Corollary 1. U 000 � 0 and C

000 � 0 are sufficient for S⇤1 � S

+1 .

Therefore, if U 000 � 0 and C

000 � 0, then Eq. (14) holds and thus more storage is optimalunder uncertainty. When there is no prudence (U 000 = 0), frugality alone leads to precaution-ary energy storage. The same is true when C

000 = 0 and U

000 � 0.

16

As thermal energy is the marginal resource, thermal systems will supply the extra amountof energy for storage. Given Qd0 ⌘ Qd(Z0 � ↵S1), @Qd0/@S1 > 0, in the presence ofintermittency, the economy increases energy storage through further load smoothing; i.e.,generating more energy than the demand at the off-peak period, storing the additional amountenergy and using it in the peak period. A similar analysis for the off-peak energy consumptiongives @Q0/@S1 < 0. For Q+

d0and Q

+0 , which represent the fossil fuel energy generation and

energy consumption, respectively, in period 0, and there is no uncertainty in period 1, thisleads to the following corollary:

Corollary 2. S⇤1 � S

+1 implies Q⇤

d0 � Q

+d0

and Q

⇤0 Q

+0 .

Higher thermal energy and energy storage result in lower off-peak energy consumptionalong with a lower welfare in the initial period. However, by transferring the social surplusfrom the off-peak to peak period, a higher welfare in the future is expected to more thancompensate for this loss.27

The fact that a convex marginal utility allows for a higher level of energy storage withinthis framework is referred to as prudence.28 Although it was Kimball (1990) who coined theterm prudence, the analysis of precautionary demand for savings was done earlier by Leland(1968) and Sandmo (1970). Within an expected utility framework, they indicate that a riskyfuture income increases savings only if the third-order derivative of the utility function ispositive (i.e., the agent is prudent).

Frugality, C 000> 0, conversely, is not fully investigated in the literature. The fact that

frugality alone can lead to precautionary energy storage (i.e., U 000 = 0 and C

000 � 0) isa consequence of thermal stations following the load when RE and energy storage are notadequate to cover the optimal level of energy demand. However, if RE is low and there islittle or no stored energy, then this situation may become a rather costly process. In our setup,such costly decisions to produce thermal energy lead to precautionary storage.

Let us take the second-order Taylor approximation of the intertemporal efficiency condi-tion given by Eq. (10b) around the mean-level RE generation, µZ . This will allow us to makefurther use of our main result given by Theorem 1, and see how the degree of intermittencycan affect the level of energy storage. To be specific, the second-order approximation of Eq.(10b) is as follows:29

C

0�Q

⇤d0

�' �

hC

0�Q

⇤d1

�+

1

2�

2Z

⇣ UU

000(Q⇤1 � ✏) + CC

000�Q

⇤d1

�⌘i.(15)

27From Eq. (9), comparative statics analysis for the off-peak thermal energy generation and energy consump-tion provides:

@Q⇤d0

@S1=

�↵U 000

C 000 � U 00

0

> 0 and@Q⇤

0

@S1=

�↵C 000

C 000 � U 00

0

< 0.

28In other words, within the expected utility framework, prudence is formally equivalent to a convex marginalutility function.

29Refer to Appendix C for the calculations. Note that F (Z) = 1.

17

In Eq. (15) we use the fact that U 0 = C

0 for an interior solution of Qd1 . Further, �2Z is the

conditional variance when Z1 Z, and

Q

⇤1 = Q

⇤d1 + µZ + S

⇤1 and Q

⇤d1 = Qd(µZ + S

⇤1 � ✏)

are peak-period electricity consumption and thermal energy generation, respectively, that areevaluated at the mean-value of RE generation, µZ . Eq.(15) yields the intertemporal efficiencycondition, which is indeed Eq. (12) augmented with uncertainty and the precautionary mo-tives. In comparison to Eq. (12), we see that the expected marginal value of energy storagein the next period is larger. As a result, when Eq. (14) holds, a higher level of energy will bestored through dispatching more thermal energy in the initial period. For a higher degree ofintermittency, �2

Z , the level of stored energy will be more for an interior solution of energystorage.

On the other hand, it is optimal to store at the capacity S if the cost of storing an extraamount of energy in the off-peak period is lower than the expected marginal cost of producingthermal energy in the on-peak period. Therefore, precautionary motives cannot lead to anyfurther energy storage in such a case. In contrast, if there is no energy storage when there isno uncertainty (i.e., S+

1 = 0), but Eq. (14) holds, energy can be stored when some uncertaintyis introduced. Lastly, notice that for UU

000 + CC000 = 0, or U 000 = 0 and C

000 = 0, Eq. (15)boils down to Eq. (12). Thus, the level of energy will be identical to the one in the presenceof no intermittency, but a baseload power generation.

3.3.2 100% RE

We now consider an energy generation portfolio that contains a 100% RE system. Iceland’selectricity sector with minor contributions from thermal system is an example of this extremecase. With hydropower accounting for 74% and geothermal for 26%, electricity generationwas produced solely from renewables in 2010 (IEA, 2013). One can also consider Norway’shydropower system, which generates almost the entire country’s electricity and power needs.In 2010, approximately 95% of Norway’s electricity was generated from hydropower (IEA,2013). Although our focus here is on variable and intermittent RE, in the absence of thermalsystems our model can easily be converted to a model with a reservoir hydroelectric system.For example, suppose Z0 is the current flow of water to a reservoir and Z1 is the uncertainflow in the next period. Let S1 be the water that is stored in the reservoir for use in thepeak period. Disregarding losses due to surface evaporation and leakages (i.e., ↵ = � = 1),Q0 = Z0�S1 is the consumption of energy that is generated by letting the water flow throughwater turbines.

For our model, a 100% RE system is equivalent to F (Z) = 0. Thus, Z1 > Z alwaysand there is no thermal energy dispatch. In other words, there is zero dispatchable energycapacity, which implies an infinite marginal cost beyond that state (cf. Eq. (5)).

Theorem 2. If F (Z) = 0, then for every µZ and x with E[x] = 0, S⇤1 � S

+1 if and only if

U

000 � 0.

18

Proof. The proof is similar to that of Theorem 1, except that for t = 0, 1, Q

⇤dt = 0.

Thus, for an economy with a 100% RE system, prudence is necessary and must be suffi-cient for precautionary energy storage. The economy increases energy storage by consumingless electricity generated by renewable sources in the initial period. Theorem 2 leads to thefollowing corollary:

Corollary 3. S⇤1 � S

+1 implies Q⇤

0 Q

+0 .

Consequently, some social surplus will be transferred to the peak period using energystorage technology. In a hydroelectric system with reservoirs, this will result in lower energygeneration in the off-peak period and therefore a higher level of water transfer to the peakperiod. Hence, a higher volume of water that is kept in the reservoir due to precautionarymotives is equivalent to generating a lower level of electricity from hydropower.

By taking a second-order Taylor approximation of Eq. (10b) the intertemporal efficiencycondition for the mean-level RE generation provides the following:30

(16)U

0�Q

⇤0

�' �

hU

0�Q

⇤1 � ✏

�+

1

2�

2ZU

000�Q

⇤1 � ✏

�i.

For an interior solution of the energy storage, the marginal (consumption) cost of storingenergy (U 0

/�) equals the expected marginal value of energy. If, however, the marginal costof storing an extra unit of energy is greater than the expected marginal value of energy inthe peak-period, the no energy is stored. It is also possible that when U

000 � 0 and all thestorage capacity is completely utilized, the net expected value of energy in the next periodis bigger than the unit cost of storing energy in the initial period. Lastly, a higher degree ofintermittency increases energy storage whenever U 000 � 0.

3.3.3 Large share of RE

The European Commission presented a roadmap that can deliver greenhouse gas emissionsreductions by 80%–95% by 2050 compared with 1990.31 According to the roadmap, theelectricity sector will play a crucial role in the transition toward a low carbon economy, andit is expected that the RE target of 75%–85% and approximately 100% will be achieved by2030 and 2050, respectively. Such large shares of RE sources will most likely meet the energyload under favorable weather conditions and allow the thermal systems to be taken offline forsome periods. Translated to our model, this means that, depending on the weather conditions,Z1 > Z and the thermal systems will become idle or Z1 < Z and the thermal systems willmeet the residual load.

30Refer to Appendix C for the calculations. Note that F (Z) = 0.31EC European Commission (2007) “Communication from the Commission to the European Parliament,

the Council, the European Economic and Social Committee and the Committee of the Regions, A roadmapfor moving to a competitive low carbon economy in 2050”, COM(2011) 112 final, available at: http://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX:52011DC0112.

19

Theorem 3. For every µZ and x with E[x] = 0, S⇤1 � S

+1 if and only if:

F (Z)� UU

000(Q⇤1 � ✏) + CC

000(Q⇤d1)

�+�1� F (Z)

�U

000(Q⇤1 � ✏) � 0.(17)

Proof. The proof is given in Appendix.

Note that for a variable (or a function) k, k and k denote that k is evaluated at the condi-tional means E[Z1|Z1 Z] and E[Z1|Z1 > Z], respectively. Taking a second-order Taylorapproximation on the RHS in Eq. (10b) gives the following intertemporal efficiency condi-tion:

(18)U

0(Q⇤0) ' �

hF (Z)

⇣U

0(Q⇤1 � ✏) +

1

2�

2Z

� U U

000 + C C000�

⌘

+�1� F (Z)

��U

0(Q⇤1 � ✏) +

1

2�

2ZU

000�i.

In opposition to an economy with a small share of RE, we see that frugality can have arelatively lower impact on precautionary energy storage. This is especially the case when thethermal plants are less frequently brought online; i.e., when F (Z) is small.

4 Competitive market equilibrium with energy storage

In this section, we want to show that the planner solution can be decentralized through com-petitive markets. This will enable us to see the role of prices in coordinating the energy mar-ket that we have been discussing thus far. However, this task stipulates a well-defined marketequilibrium concept in which agents have rational expectations. As there are no externalitiesin the model, the planner solution will coincide with the competitive rational expectationsequilibrium (REE).

On the demand side, we assume that all consumers have identical preferences. This al-lows us to model their behavior by a representative consumer. Specifically, the first-ordernecessary conditions for the consumer problem yield the following:

(19)U

0(Q⇤0) = P

⇤0 ,

U

0(Q⇤1 � ✏) = P

⇤1 ,

where Q

⇤t ⌘ Q(P ⇤

t ) is the aggregate demand function for energy where Pt > 0. Eq. (19)corresponds to the optimization problem of a representative consumer with quasilinear prefer-ences. In economic theory, using such preferences is a standard assumption when discussingissues related to a single market in a general equilibrium framework.32 Price taking compet-itive behavior in Eq. (19) implies an instantaneous adjustment of consumption in responseto changes in the market price. In energy markets, this condition typically requires that the

32This approach can be justified in the absence of income effects (see Mas-Colell et al., 1995, chap. 10),which we do not consider in our study.

20

consumers are equipped with high-tech equipment that informs them about the spot price orthey have access to software that can switch electrical appliances on and off, given the pricein the market (Ambec and Crampes, 2012).

Regarding the production side of the economy, fossil-fuel and RE generators and en-ergy storage firms are price-taking competitors. There is a continuum of RE generators withmeasure normalized to one. Given that the unit cost of generating energy is so low –whichwe consider as zero– the RE generation is price inelastic. Thus, given Pt > 0, the profitmaximization problem of each RE generator in both periods is given by

maxqcit

⇡cit = P

⇤t ztqcit subject to qci � qcit � 0,

where qci is the installed capacity of RE generator i. Obviously, for t = 0, 1, the profitmaximizing production is qci. The total RE generation then satisfies

Q

⇤ct ⌘ Qc(P

⇤t ) = Qc =

Z 1

0

qcidi.(20)

In the thermal energy industry, there is a unique merit order of thermal power plants withmeasure normalized to one. Given Pt, thermal energy generation extends up to the thermalunit for which the marginal cost of generating thermal energy equals the market price.33 Theprofit maximization problem of each thermal energy generator is as follows:

maxqdjt

⇡djt = P

⇤t qdjt � cjqdjt , subject to qdjt � qdjt � 0,(21)

where cj > 0 is a constant and qdjt is the energy generation from thermal unit j. The firstorder necessary condition of profit maximization for a generator is then

(22)P

⇤t cj if q

⇤djt = 0,

P

⇤t = cj if qdj > q

⇤djt > 0,

P

⇤t � cj if q

⇤djt = qdj.

Given that q⇤djt ⌘ qdjt(P⇤t ) is the profit maximizing level of energy that thermal generator j is

willing to supply at price P ⇤t , the thermal industry (aggregate) supply function in both periods

is

Q

⇤dt ⌘ Qd(P

⇤t ) =

Z 1

0

q

⇤djtdj.

We characterize the energy storage sector by a continuum of energy storage firms withidentical technologies. In the economy, these firms are the ones that bear the uncertainty inthe future price of energy. When a storage firm decides to store more energy, it rationally an-ticipates the future energy price based on the available information (i.e., the supply schedule

33For the units that are brought online earlier, the individual marginal costs equal the market price minus theshadow prices of the individual capacity constraints.

21

of the thermal energy industry, the aggregate demand schedule, the processes that affect theweather, and –in turn– the RE generation). With the ultimate motivation to maximize profits,energy storage firms apply the principle of rational behavior to the acquisition and process-ing of information and the formation of anticipations. In this sense, they are rational profitmaximizers. When storage firms are fully aware of the economic implications of intermit-tency, they will, for example, change their energy storage levels in anticipation of the effectsfrom intermittent RE rather than wait for these effects to occur in the electricity market. Byanticipating the effects of uncertainty in RE, and, thus, the period 1 price, the net anticipatedprofit for energy storage firm ` from storing s`1 is as follows:

⇡

a`1 = �P

a1 s`1 � P

⇤0 s`1 ,(23)

where P

a1 and P

⇤0 are the anticipated and current equilibrium spot prices, respectively. Each

storage firm maximizes its anticipated profits subject to a non-negativity constraint, s`1 � 0,and a capacity constraint, s` � s`1 . As each storage firm shares the same rational expecta-tions with every other firms, the anticipated price is not indexed by a particular storage firm.The first-order condition for the maximization problem yields the following:

(24)@⇡

a`1

@s`1

= �P

a1 � P

⇤0 0, with equality if s

⇤`1 > 0,

= �P

a1 � P

⇤0 � 0, otherwise s

⇤`1 = s.

The fact that these firms share the same rational expectations and, therefore, anticipate thesame market clearing future price indicates that, in equilibrium, the anticipated profit from amarginal unit of energy storage cannot be positive.34 Otherwise, profit-seeking entrepreneurswould eliminate any type of disequlibria by adjusting the individual levels of energy storage.This allows us to describe Eq. (24) as the condition for market equilibrium rather than thefirst-order condition for an energy storage firm’s optimization problem. The relationshipbetween the industry level of energy storage and the associated expected profit can then besummarized by

(25)P

⇤0 � �P

a1 , S

⇤1 = 0,

P

⇤0 = �P

a1 , S > S

⇤1 > 0,

P

⇤0 �P

a1 , S

⇤1 = S,

where

S

⇤1 =

Z 1

0

s

⇤`d` and S

⇤1 ⌘ S1(P0, P

a1 ).

In the sense that the energy storage firms’ expectations are rational and make informedpredictions of future prices, the subsequent market prices that arise from the decisions basedon these expectations will confirm their anticipations. Hence, the expected price will be

34This holds for an interior solution. If the industry storage capacity binds, the storage firms will makepositive profits.

22

consistent with the level of energy storage that is governed by the anticipated price. Then, ina REE

P

a1 = E[P ⇤

1 ],(26)

i.e., the anticipated price is confirmed in equilibrium.

Having depicted the formation of expectations and the response of competitive energystorage firms to current and anticipated prices and the qualitative relationship between priceand profit maximization for each storage firm, we can make formulate the following defini-tion:

Definition 1. REE is a price vector, P = {P ⇤0 , P

⇤1 , P

a1 }, and an allocation vector,

Q = {Q⇤0, Q

⇤1, Q

⇤c0 , Q

⇤c1 , Q

⇤d0 , Q

⇤d1 , S

⇤1}, that solve Eqs. (19), (20), (22), (24), and (26), such

that markets clear: Q⇤0 = Qd(P

⇤0 )+Z0�↵S1(P

⇤0 , P

a1 ) and Q

⇤1 = Qd(P

⇤1 )+ Z1+S1(P

⇤0 , P

a1 ).

In equilibrium, the prices, P ⇤0 and P

⇤1 , are implicitly defined by

P

⇤0 ⌘ P

�Q

⇤d(P

⇤0 ) + Z0 � ↵S

⇤1(P

⇤0 )�,

P

⇤1 ⌘ P

�Q

⇤d(P

⇤1 ) + Z1 + S

⇤1

�,

respectively, where we recall that Z0 ⌘ z0Qc and Z1 ⌘ z1Qc. In the presence of uncer-tainty, the storage firms will increase the amount of stored energy until the net expectedprice, �E[P ⇤

1 ], equals the current spot price of energy.35 This additional demand for energystorage will either be met by increased thermal energy and reduced current consumption, orby only the reduction of the current consumption.36

As the REE quantities in our problem corresponds to the allocation dictated by the benev-olent social planner, we carry forward our results from the previous section. Consider thatthe share of RE in the energy system is small. From today’s perspective, this setup is morefrequently observed than systems with large shares of RE. To observe the implications of asmall share of RE for competitive pricing, we arrange Eq. (15) as in the following:

(27) P0 ' �

1 +

1

2�

2Z

✓ U

U

000

U

0 + C

C

000

C

0

◆�P1,

where from Eq. (19) P0 = U

0(Q⇤0), and P1 = U

0(Q⇤1� ✏) is the energy price in period 1 at the

mean level of RE generation, µZ ⌘ E[Z1]. RHS is the product of � and the expected marketprice of energy in period 1, which is indeed the market price that corresponds to µZ , adjustedfor the degree of intermittency, �2

Z , and the weighted sum of U

000/U

0 and C

000/C

0. In theabsence of intermittency (i.e., �Z = 0) Crampes and Moreaux (2010) indicate that P0 = �P1

35Note also that for P ⇤0 > �E[P ⇤

1 ], S⇤1 = 0, i.e., when the net expected price is below the current price, then

the energy storage is zero. If the capacity constraint in the energy storage industry is met, then the net expectedfuture price is above the current price at storage capacity: P ⇤

0 �E[P ⇤1 ], S

⇤1 = S.

36Corollary 2 and Corollary 3, respectively.

23

and that there is still peak load pricing.37 In the presence of intermittent RE, the current price,P0, becomes higher when Eq. (14) holds. Our intuition is that the precautionary motives spurenergy storage and increase the current price.

All else equal, if S0 > 0, the current price of electricity becomes lower. This, in turn,allows for a higher transfer of energy into the peak period and augments the spot market pricein period 0 until it equals the net expected price or until the capacity constraint is reached. Insuch a case, storage would be effective in eliminating an excessively high supply of energy.A similar result can be found in Wright and Williams (1982), where storage is primarily aninsurance against shortages and very high prices.

It is possible to differentiate the demand- and the supply-side effects in Eq. (27). Thedemand side refers to the ratio given by U

000/U

0. Conversely, the supply side of the market isrepresented by C

000/C

0. Let us first consider the case where there is no frugality (C 000 = 0).

If C 000 = 0, one can rearrange Eq. (27) to obtain the following:

(28)P0 ' �

1 +

1

2

⇣�Qc

Q1

⌘2

U

⇠

pr

⌘d

�P1,

where ⇠pr ⌘ �Q1U 000

U 00 is the relative prudence, and Q1 ⌘ Qd(µZ + S

⇤1 � ✏) + µZ + S

⇤1

and ⌘d ⌘���dQ/QdP/P

��� are energy consumption and the price elasticity of energy demand thatcorrespond to the mean RE generation, respectively. This leads to the following proposition:

Proposition 1. If U 000> 0, C 000 = 0 and F (Z) = 1, then in a REE with energy storage, P0 is

augmented by a lower ⌘d and a higher �, ⇠pr , �Z and U .

This proposition indicates that if the consumers are prudent but the thermal generatorsare not frugal (C 000 = 0), then –in a REE with energy storage and small share of RE– a higherround-trip efficiency parameter, � and a higher weight on the demand side, U , raise the ex-pected price of energy. This is because a more efficient storage technology and a strongereffect from the demand side create arbitrage opportunities that lead to a higher demand forenergy storage and, in turn, increase the current price of energy. Moreover, given that theagents are prudent (⇠pr > 0), a relatively higher sensitivity of energy demand to changes inenergy price in the peak period (i.e., a higher ⌘d) causes a lower energy price. This is dueto the fact that consumption adjustment becomes a stronger substitute than energy storagewhen the price elasticity of demand is higher. A similar result can be found in the commoditystorage literature. Wright and Williams (1982, 1984) show that higher demand elasticity de-creases the scope for commodity storage. Furthermore, one can see that the higher sensitivityto prices diminishes the impact of prudence on precautionary energy storage.

The data, however, indicates that the relative price response is rather low. The short-runresidential own-price elasticity of electricity demand in absolute value is estimated at 0.3

37The prices are closer together when energy can be stored, which is a consequence of the constraint thatstorage must not be negative.

24

(EPRI, 2008).38 The same number averaged for potential system peak hours for the summermonths is estimated to be 0.15 (Taylor et al., 2005). Surveying the evidence from the recentexperiments with dynamic pricing of electricity, Faruqui and Sergici (2010) report that theown price elasticities in peak usage range from 0.02 to 0.10.

Although electricity consumers exhibit low price elasticities, this may indeed be relatedto the difficulties in accessing complete information. In other words, the households may beprice inelastic because they simply lack complete information. When households are pro-vided with real-time information regarding energy usage, Jessoe and Rapson (2014) demon-strate that they become more price sensitive. There evidence suggests that when a set ofresidential electricity customers is exposed to real-time feedback on electricity consumptionvia smart meters (in this case, in-home displays), a much larger reduction in electricity con-sumption in response to higher prices is exhibited.39 In this case, the role of prudence inaugmenting the demand for energy storage may become weaker over time as consumers gainbetter access to real-time information.

�Qc/Q is an indicator that shows the degree of deviations in RE compared with Q. Ifthe standard deviation of RE generation is very small with respect to energy consumptioncorresponding to the mean-level RE, intermittency is less of a problem for the economy.Thus, the precautionary motives may not result in a high level of energy storage. However, ifthe relative deviations in RE generation are high, the energy consumption may exhibit strongdeviations as well. In this case, the economy benefits from a higher level of energy storage.A higher demand for energy storage will lead the storage firms to purchase and store moreenergy, while increasing the energy price.

Suppose now that the thermal industry is characterized by a convex marginal cost functionand, hence, it can be exposed to significant cost variations when responding to changes in thenet load. Furthermore, suppose that U 000 = 0; i.e., the demand function is linear. One canrewrite Eq. (27) to obtain the following:

(29)P0 ' �

1 +

1

2

⇣�Qc

Qd

⌘2

C

⇠

fr

⌘s

�P1,

where Qd ⌘ Qd(µZ+S

⇤1�✏) is thermal energy generation at the mean level of RE generation

and ⇠fr ⌘ QdC000

C00 and ⌘s ⌘ dQd/Qd

dP/Pare the coefficient of relative frugality and the elasticity

of thermal energy supply, respectively. Here C

0 is the price calculated by the inverse sup-ply curve (industry marginal cost function) for thermal energy. This leads to the followingproposition:

38“Price elasticity estimates based on consumer behavior for a single utility over a relatively short time series(1–5 years) are considered short-run under the assumption that the stock of electricity devices does not changeappreciably, so customers are limited to actions and behaviors associated with the available technology" (EPRI,2008, p. 20).

39Jessoe and Rapson (2014) explain that the experience with smart meters facilitates consumer learning andallows the households to improve their decision making in relation to changing electricity prices. Although ouradditive utility framework excludes all forms of habit formation and we focus mainly on the short-run decisions(i.e., we disregard capacity decisions), Jessoe and Rapson indicate that, in the long run, the experience withsmart meters leads to habit formation and lowers the use of electricity.

25

Proposition 2. If U 000 = 0, C 000> 0 and F (Z) = 1, then in a REE with energy storage, P0 is

augmented by a lower ⌘s, and a higher �, ⇠pf , �Z and C .

Proposition 2 indicates that, when the price schedule is linear, the firms are characterizedby frugality and the share of RE is low, a higher frugality induces a higher level of energystorage and, therefore, a higher current energy price. Moreover, a higher elasticity of supply(i.e., a more responsive thermal energy generation) causes a lower level of energy storage.Hence, both the supply- and demand-side elasticities have similar effects. If, however, thesupply elasticity is low, this situation will create extra incentives to store energy. Think of abaseload power plant, which has low supply elasticity due to its poor flexibility in adjustingits output. For such a firm, it is not very cost effective to vary its supply. Therefore, it storesmore energy in the presence of uncertainty.

If the standard deviation in RE generation scaled by Qd is high, then the energy supplycan significantly deviate from Qd and result in costly attempts by the thermal energy industryto supply the desired level of energy when the energy has a higher value. Therefore, there willbe a higher level of energy storage in equilibrium. In contrast, if �Qc is small, the deviationsin RE generation become less of an issue and diminish the level of stored energy due toprecautionary motives.

Finally, let us now consider that the consumers are both prudent and frugal. Further, Eq.(30) shows that there is a higher demand for storage services that will cause higher currentenergy prices than those in Eqs. (28) and (29):

(30)P0 ' �

1 +

1

2

⇣��Qc

Q1

�2 U

⇠

pr

⌘d+��Qc

Qd

�2 C

⇠

fr

⌘s

⌘�P1.

The interpretation follows from Propositions 1 and 2. An interesting feature of Eq. (30) is re-lated to the weights assigned to the demand- and supply-side effects, U and C , respectively.Miranda and Helmberger (1988) show that price variability is more sensitive to the demandelasticity than the supply elasticity, which, when translated to our case implies that the effectof demand elasticities on the current energy price has a greater weight; i.e., U is greater than C . For energy markets, this effect can be an interesting problem, and an empirical investi-gation may supply crucial information when pricing energy to attain the first-best dispatch.Furthermore, recall the possible relationship between a better access to real-time informa-tion and a weaker role for prudence (see our earlier discussion under Proposition 1). Whensuch a relationship becomes more significant, it will place a relatively greater importance onfrugality and its impact on the level of energy storage and energy prices.

When we consider an electricity market with 100% RE (e.g., an economy with only hy-droelectric power generation), then for U = 1, Proposition 1 follows. Hence, for U = 1,the pricing equation is given by Eq. (28). The interpretation regarding how an equilib-rium level of energy storage is affected by the degree of intermittency, the price elasticityof demand and the coefficient of relative prudence remains the same. However, note thatthe energy storage firms obtain the desired level of energy from RE generators instead ofpurchasing energy from the thermal energy industry.

26

Lastly, for the general case when thermal systems are occasionally shut down, we havethe following:

(31) P0 ' F (Z)P0 + (1� F (Z))P0,

where P0 and P0 are given by Eqs. (30) and (29), respectively. Note that while P0 correspondsto the cases when thermal systems are active, P0 corresponds to the cases when they are keptidle due to high RE generation. Furthermore, note that the higher the F (Z) becomes, thelower the impact of frugality on the current spot price of electricity, and vice versa.

5 Conclusion

Increased energy generation from renewable sources of energy is one of the key solutionsin combating the climate change problem and is high on the policy agenda, especially inthe developed countries. As RE is inherently variable and uncertain, a larger future shareof RE generation can cause serious vulnerabilities in meeting energy demand and makingrenewables unreliable baseload contributors. In this regard, policymakers and utility plannershave demonstrated interest in energy storage technologies as a way to enhance the resilienceand reliability of the power grid.

Even though energy storage is addressed in many studies in the literature, the extent towhich precautionary motives can spur energy storage is not well known. In designing co-herent energy policies and making utility planning decisions, both governments and powerutilities can benefit from knowledge regarding the direct and indirect impacts precaution-ary motives can have on electricity prices, and electricity generation and storage decisions.The model we develop in this study provides a simple setup to assess these impacts due toprudence (convex marginal utility) and frugality (convex marginal cost). We first look at aplanning problem and find the conditions when energy storage increases social welfare. Fur-ther, we turn to a decentralized setting and examine how, in the presence of precautionarymotives, the use of the energy storage technology affects the current market price of electric-ity, thermal energy generation, and electricity consumption.

Our analysis indicates that for economies with a small share of RE, prudence and frugal-ity cause precautionary energy storage to varying degrees. Even in the absence of prudence,frugality can still contribute toward precautionary storage. With higher levels of RE gener-ation, the impact of frugality diminishes. However, it does not vanish until the share of REgeneration in the energy system reaches 100%.

Decentralization of the optimal allocation through competitive markets allows us see theways that prudence and frugality can affect energy prices, which, in turn, help to coordinatethe energy market. When prudence and frugality are present, there is precautionary energystorage and a higher current energy price. The standard deviation in RE boosts energy stor-age to varying degrees. Considering prudence, the effect will depend on the ratio of thestandard deviation in RE to energy consumption. The demand for energy storage and the

27

current energy price are positively related to this share. In other words, if this share is small,intermittency is less of a concern and results in a relatively lower level of energy storage andenergy price. Regarding frugality, what matters is the ratio of the standard deviation of RE tothermal energy generation. If this share is high, the degree of intermittency will increase thecurrent energy price through a higher demand for energy storage.

We find that when agents are prudent and there is intermittent RE, energy storage andprices are negatively related to price elasticity of demand. This is a consequence of the factthat consumption adjustment becomes a poor substitute for energy storage when consumersare less reactive to changes in the energy price. As the experimental evidence indicates, whenhouseholds are provided with real-time information regarding energy usage via smart meters,they become more responsive to price changes. A higher responsiveness of consumers, inturn, can diminish the role of prudence in augmenting the demand for energy storage. Whenenergy supply is less price elastic, dispatchable thermal energy generation becomes a poorersubstitute for energy storage. In this case, the intermittent residual load will lead to a higherdemand for energy storage due to frugality. Furthermore, both the demand- and supply-side effects; i.e., the effects coming from prudence and frugality, respectively, are weighteddifferently by the endogenously determined weights.

One way to extend our study is to incorporate energy generation and storage capacityinvestment decisions into the model. Similar to the approach adopted by Ambec and Cram-pes (2012), a central planner can first determine various capacities and then choose how todispatch them for each state of supply and demand. While the capacity decisions would beassociated with long-term commitments in the economy, the latter, which corresponds to thepresent study, would be related to the short-term decisions constrained by the installed capac-ities. It would then be interesting to analyze how precautionary motives can impact capacitydecisions. Furthermore, considering market imperfections and pollution externalities, inves-tigating the relationship between precautionary motives and energy generation and storagedecisions will be interesting. As we showed how energy prices can be influenced by energystorage in the presence of precautionary motives, testing the empirical relevance of our resultswill also be interesting.

AppendicesA Storage technologies for energy management applications

Pumped-storage hydropower (PSH)

Presently, PSH and compressed-air energy storage are considered the most mature methodsfor electricity storage. Of the 140 gigawatts (GW) of large-scale energy storage that arecurrently installed in the electricity grids worldwide, over 99% corresponds to PSH. Theremaining installed capacity for electricity storage constitutes 976 megawatts (MW) (IEA,2014c).

28

PSH, which is an example of mechanical energy storage, comprises thermal and hydraulictechnologies. The hydraulic aspect of the system requires two reservoirs with differing ele-vations. The uphill reservoir (i.e., a mountain lake) is used to generate electricity by allowingthe water to fall and turn the blades of turbines to meet the peak-load demand. The lower-level reservoir is used to collect water, which is then pumped to the uphill reservoir usingthermal systems during the off-peak periods (IEA, 2014c). PSH plants can achieve round-trip efficiencies that exceed 75% and capacities that exceed 20 hours of discharge capacity(Denholm et al., 2010).

Compressed-air energy storage (CAES)

CAES, which is another example of mechanical energy storage, use off-peak electricity tocompress air in airtight underground storage caverns or storage tanks. The compressed airis then released in peak periods to help power the natural gas-fired turbines, which generateelectricity using significantly less natural gas. Similar to PSH plants, CAES plants are alsoappropriate for load leveling because they can be constructed with capacities of a few hun-dred MWs and discharged over 8-20 hours. The current global installed CAES capacity isapproximately 440 MW (IEA, 2014c). Among the two CAES systems in commercial oper-ation, the facility in Huntorf, Germany, can operate with efficiencies up to 55% (Al-Khouryand Bundschuh, 2014).

For CAES technologies, new design proposals, such as advanced adiabatic compressed-air energy storage (AA-CAES), have the potential to reach efficiencies up to 70%, primarilythrough reducing natural gas usage. The notable difference between an AA-CAES plant andan existing CAES plant is that instead of dissipating the heat produced by the compressionprocess into the environment, it is transferred by heat exchangers and stored in heat storagesites. During discharge, the stored heat is released into the compressed air and prevents theturbines from freezing; thus, no gas combustion is needed to heat the compressed air. Thisdesign not only makes AA-CAES plants more efficient but also makes CO2 neutral. However,this option still needs to be deployed at scale (Oberhofer and Meisen, 2012).

Hydrogen storage

Another storage technology for long-term energy applications is hydrogen storage. In thissystem, the excess electricity from RE can be converted into hydrogen through the electrol-ysis of water and, with further processing, into methane.40 Methane and hydrogen can bestored in natural gas grids. Although it is recommended that the volume of stored hydrogenshould be limited to 5% to preserve the pipelines and gas turbines. These storage technologieshave significant potential due to their high energy density, quick response times, and poten-tial for use in large-scale energy storage applications (IEA, 2014c; IEC, 2011; Oberhofer and

40Methanation (CH4) of hydrogen is the next step once the limit for hydrogen is reached. It requires carbonfrom carbon dioxide (CO2). Methane is the main component of natural gas (95%). The bound CO2 will beemitted during re-electrification, making the whole process CO2 neutral.

29

Meisen, 2012). 41 Although there have not been any commercial hydrogen storage systemsused for RE systems, various R&D projects have already been successfully executed overthe last few decades. One example is the self sufficient island of Utsira, Norway. Anotherexample is a hybrid power plant in Enertrag, Germany (IEC, 2011).42

Although PSH, CAES and hydrogen storage require a suitable geography and face highupfront investment costs, the fact that they are capable of storing huge amounts of energy,offering sustainable energy storage possibilities, and having a rather fast response time makesthem an essential part of energy storage technology. However, this does come at a cost.PSH has a significant and negative impact on the environment and requires a huge watersource. Even though AA-CAES is capable of reaching efficiencies nearly as high as PSH, itrequires sealed storage caverns and, therefore, competes against other storage needs relatedto natural gas and hydrogen. Furthermore, the system is only economical up to one day ofstorage. Hydrogen storage, like PSH, is capable of storing energy for several days or months.However, the technology has a very low efficiency (30%–40%) that is not expected to exceed50% in the future.43 Furthermore, the technology requires a well-established natural gas grid.

Batteries

Although, the most obvious alternative for energy storage will be the use of “high-energy”batteries, however, only if they are more affordable. They are considered to be a strongcandidate to solve various problems because of their small spatial requirements, mechanicalsimplicity, and flexibility in sitting (Wang et al., 2014). When put on a desirable cost trajec-tory, such grid-scale storage systems will be the key in enabling renewable systems to operatewith fewer subsidies or government regulations.44

Sodium-sulfur (NaS) batteries, a high temperature battery, constitute 304 MW of the glob-ally installed electricity storage capacity. Excluding PSH, this represents 31% of the globallyinstalled energy storage capacity. The technology is already in operation in countries suchas Japan, Germany, France, the U.S., and UAE (IEC, 2011). As an example, the Rokkasho

41Europe has gigantic storage capacities. For example, even with the recommended 5% limit for hydrogen,the German power grid alone offers storage capabilities of 1.8TWh, which is 25 times more than the currentshare of pumped hydro plants. Similarly, methane can be injected into the natural gas grids, making it possibleto use the rest of the available storage capacity. However, methanation decreases the efficiency to 30%-40%overall. (Oberhofer and Meisen, 2012).

42Production of hydrogen through wind turbines could also be beneficial for electric cars that use hydrogenas fuel. An advanced hydrogen economy would push the electrification of the automobile fleet, as hydrogencars are about to go into mass production (Oberhofer and Meisen, 2012).

43Although PSH has a very low efficiency, the fact that wind turbines are often curtailed to prevent the gridfrom overloading can make it beneficial to employ this technology. The main current obstacle for its practicalimplementation is the cost. The electrolysis of water and the methanation process are still more expensive thanpurchasing conventional natural gas (Oberhofer and Meisen, 2012).

44For many batteries, there is a considerable overlap between energy and power quality management appli-cations. As batteries have short response times, those designed for energy management applications can alsoprovide other types of services, such as frequency regulation, that necessitate low discharge times (Denholmet al., 2010). However, it is worth mentioning that continuous cycling requirement can significantly reducebattery life.

30

Futamata Project in Japan, which is located near the 51MW Rokkasho wind farm, comprises34MW of NaS batteries and enables peak shaving (and load leveling). The batteries arecharged at night and then used to supply electricity to the grid along with the wind powerduring peak times (O’Malley, 2008).

With a reduction in cost over time, lithium-ion batteries, which have high energy densitiesand low energy loss rates, and easily accessible lithium and graphite sources have significantpotential for storing large amounts of energy. Battery efficiencies are 90%–95% and have avery slow loss of charge when not in use (a 5% self-discharge rate per month on average)(Oberhofer and Meisen, 2012). Although the current lithium-ion projects serve short-termpower needs (e.g., frequency regulation), projects under development fall into the long-runcategories and will have a more vital role in load leveling. Currently, the globally installedstorage capacity for lithium-ion batteries is 100MW (IEA, 2014c).

Another way to store considerable amounts of energy can be achieved through liquid-metal batteries. These batteries require low cost of materials, are easy to assemble, and havethe potential for a long lifetime (Kim et al., 2012). When they become commercial, theiradvantages will include their grid-level storage capacities, small spacial requirements (andflexibility in locating them in the field), and long endurance.45 Some recent finding in theserespects can be found in Bradwell et al. (2012).46 A few liquid-metal battery prototypes willbe tested in 2015, followed by commercial versions in 2016 (Chesto, 2014).

Originally developed by NASA in the early 1970s, for long-term space flights, flow bat-teries (e.g., vanadium redox and zinc bromine) can store energy for hours or days with up toseveral megawatts of power. These batteries can discharge power for long durations, are ableto tolerate a high number of charge/discharge cycles and can easily be scaled up. Althoughthe current systems have a low energy density and are not commercially mature, the technol-ogy is expected to be a strong competitor in the grid-scale storage market (IEC, 2011).47

Lastly, lead-acid batteries, are still an important option for today’s storage systems, de-spite being the oldest known type of rechargeable battery and nearly having reached theend of their development (Oberhofer and Meisen, 2012). Globally, these batteries represent70MW of installed storage capacity (IEA, 2014c).48

45A levelized cost of 5¢ kWh�1cyc�1, where cyc stands for cycles, will allow liquid metal batteries to becompetitive with storage technologies such as PSH. As an example, a battery that costs $400 kWh�1 and has alife of 10,000 cycles and 80% energy efficiency meets the 5¢ kWh�1cyc�1 level (Kim et al., 2012).

46Bradwell et al. (2012) investigate the means to develop batteries that can be utilized in a wide array ofstationary storage applications. In doing so, the project aims at a low-cost, high-voltage system with suffi-ciently low levels of corrosion. The findings obtained by experimenting with an all-liquid battery comprisingmagnesium (Mg) and Antimony (Sb) liquid-metal electrodes indicate that the initial cell performance is promis-ing. Wang et al. (2014) propose an improved version of a liquid-metal battery system that is likely to put thetechnology on a more desirable cost trajectory.

47The globally installed storage capacity for redox-flow batteries is 10MW (IEA, 2014c).48Notrees Battery Storage Demonstration Project in Texas (North America’s largest battery storage project

at a wind farm) contains fast-response capabilities provided by advanced lead-acid batteries. The system isconfigured to provide 36 MW of peak output and has a total storage capacity of 24 MWh (IEA, 2014b).

31

B Proof of Theorem 1

To prove our main result, we will need the following lemma:

Lemma 1. If UU000 + CC

000 � 0, then increase in risk (or, the degree of intermittency) inthe sense of Rothschild and Stiglitz (1970) (RS) increases S1.

Proof. Let E and F represent the cumulative distribution functions of Y1 and Z1, respectively.Assume that F is a mean-preserving spread of E in the sense of RS. Thus, although bothsystems have the same average level of RE generation, the density function f(Z1) has moreweight in the tails, and is more risky. Then, for every non-decreasing concave function, �,we have the following:

Z b

a

�(y)dE(y) �Z b

a

�(x)dF (x).

Taking � ⌘ �U

0 yields the following:Z b

a

U

0(y)dE(y) Z b

a

U

0(x)dF (x).

Then, for b = Qc, a = 0, and U

0 = U

0(Qd1(J1 + S1 � ✏) + J + S1 � ") for J = Y1, Z1,Theorem 2(A) in RS states that an increase in risk leads to a higher S1 if U’ is convex in J ;i.e.,

@

2U

0

@J

2= U

000⇣@Qd1

@J

+ 1⌘2

+ U

00@2Qd1

@J

2� 0.(32)

Using Eq. (6) we get

(33)@

2Qd1

@J

2=

C

002U

000 � U

002C

000

(C 00 � U

00)3.

Substituting Eqs. (6) and (33) in Eq. (32) then gives

UU000 + CC

000 � 0.(34)

First, let us prove sufficiency using Lemma 1. Given S

+1 , let V (S+

, µZ) be the maxi-mum value function for the intertemporal optimization problem under certainty. Further, letE[V (S⇤

1 , Z1)] be the expected value of the maximum value function for the intertemporaloptimization problem when RE is uncertain. Given Z1 = µZ + x and E[x] = 0, the firstorder conditions with respect to S1 for V (S1, µZ) and E[V (S1, Z1)]; i.e., VS(S1, µZ) = 0 and

32

E[VS(S1, Z)] = 0, respectively, yield:

�U

0(Qd(Z0 � ↵S

+1 ) + Z0 � ↵S

+1 ) + �[U 0(Qd(µZ + S

+1 � ✏) + µZ + S

+1 � ✏)] = 0,

�U

0(Qd(Z0 � ↵S

⇤1) + Z0 � ↵S

⇤1) + �E[U 0(Qd(Z + S

⇤1 � ✏) + Z + S

⇤1 � ✏)] = 0.

If V (S+1 , Z1) is convex in Z1, then E[VS(S

+1 , Z1)] � VS(S

+1 , µZ) = 0, or equivalently,

E[U 0(Qd(Z1 + S

+1 � ✏) + Z1 + S

+1 � ✏] � U

0(Qd(µZ + S

+1 � ✏) + µZ + S

+1 � ✏). If

Lemma 1 holds and we take µZ and 0 as the mean and the variance of Y1, respectively, thenE[U 0(Qd(Z1 + S

+1 � ✏) + Z1 + S

+1 � ✏] � U

0(Qd(µZ + S

+1 � ✏) + µZ + S

+1 � ✏). Hence, the

marginal benefit of increasing energy storage is positive when S1 = S

+1 and, thus, S⇤

1 � S

+1 .

This ends the proof for sufficiency.

If S

⇤1 � S

+1 for every µZ and x with E[x] = 0, then this must also be true for small

zero-mean risks. The small risk allows us to focus on 2nd Taylor approximation around µZ :

VS(S+1 , µZ + x) ' VS(S

+1 , µZ) + xVSZ(S

+1 , µZ) +

1

2x

2VSZZ(S

+1 , µZ) + O

�x

3�

(35)

where VSZ = �

@U 0

@Z1, VSZZ = �

@2U 0

@Z21

(see Eqs. (32) and (33)) and O(x3) is the remainder.By assuming that the risk is small, we can ignore the remainder term. From the first ordercondition, VS(S

+1 , µZ) = 0. Taking the expectation of both sides yields:

E[VS(S+1 , µZ + x)] ' 1

2�

2ZVSZZ(S

+1 , µZ)(36)

For a small risk, if S⇤1 > S

+1 , then E[VS(S

+1 , µZ + x)] � 0. For E[VS(S

+1 , µZ + x)] � 0 to

be positive, VSZZ � 0 must be positive. One can calculate that VSZZ � 0 is equivalent to Eq.(34). This completes the proof for necessity.

C Applying a Second-Order Taylor Expansion

Let g(Z1)def= C

0(Q⇤d1) and h(Z1)

def= U

0(Q⇤1 � ✏), where and Q

⇤d1 = Qd(Z1 + S

⇤1 � ✏) and

Q

⇤1 = Q

⇤d1 + Z1 + S

⇤1 . Given S1 and ✏, a second-order Taylor series expansion around the

conditional means µZ ⌘ E⇥Z1|Z1 < Z

⇤for g(Z1) and µZ ⌘ E

⇥Z1|Z1 > Z

⇤for h(Z1) yield:

g(Z1) ' g(µZ) + (Z1 � µZ)g0(µZ) + (1/2)(Z1 � µZ)

2g

00(µZ),(37a)

h(Z1) ' h(µZ) + (Z1 � µZ)h0(µZ) + (1/2)(Z1 � µZ)

2h

00(µZ).(37b)

Here, g0(µZ) ⌘ C

00@Q

⇤d1/@µZ , h0(µZ) ⌘ U

00, g00(µZ) ⌘ C

000 �@Q

⇤d1/@µZ

�2+ C

00@

2Q

⇤d1/@

2µZ

and h

00(µZ) ⌘ U

000, where C

00 ⌘ C

00(Q⇤d1), U

00 ⌘ U

00(Q⇤1 � ✏), C 000 ⌘ C

000(Q⇤d1) and U

000 ⌘U

000(Q⇤1 � ✏).

For an interior solution for thermal energy, U 0 = C

0, one can calculate the second order

33

derivative for the optimal thermal energy decision using Eq. (6):

(38)@

2Q

⇤d1

@

2µZ

=C

002U

000 � U

002C

000

(C 00 � U

00)3.

Hence,

g

00(µZ) =C

003

(C 00 � U

00)3U

000 +�U

003

(C 00 � U

00)3C

000.(39)

Calculating the conditional expectations, i.e., E[g(Z1)|Z1 Z] and E[h(Z1)|Z1 > Z], givesthe following:

E[g(Z1)|Z1 Z] ' g(µZ) +1

2�

2zg

00(µZ),(40a)

E[h(Z1)|Z1 > Z] ' h(µZ) +1

2�

2zh

00(µZ),(40b)

where �2Z = E

h(Z1 � µZ)

2|Z1 Z

iand �

2Z = E

h(Z1 � µZ)

2|Z1 > Z

iare conditional

variances.

From Eqs. (13), (39), (40a) and (40b), and the fact that h00(µZ) ⌘ U

000, one can rewriteEq. (10b) using a second-order Taylor approximation as in the following:

(41)U

0(Q⇤0) = �

hF (Z)E[C 0(Q⇤

d1)|Z1 Z] +�1� F (Z)

�E[U 0(Q⇤

1 � ✏)|Z1 > Z]i

= �

F (Z)

✓C

0 +1

2�

2Z

� U U

000 + CC000�

◆+�1� F (Z)

�✓U

0 +1

2�

2ZU

000◆�

D Proof of Theorem 3

Following the proof of Theorem 1, if V (S+1 , Z1) is convex in Z1, then E[VS(S

+1 , Z1)] �

VS(S+1 , µZ) = 0, or equivalently, E[U 0(Qd(Z1 + S

+1 � ✏) + Z1 + S

+1 � ✏] � U

0(Qd(µZ +S

+1 � ✏) +µZ +S

+1 � ✏). Hence, the marginal benefit of increasing energy storage is positive

when S1 = S

+1 and there is uncertainty. Conducting a similar analysis will show that:

F (Z)� UU

000(Q⇤1 � ✏) + CC

000(Q⇤d1)

�+�1� F (Z)

�U

000(Q⇤1 � ✏) � 0.(42)

Therefore, if Eq. (42) holds, S⇤1 � S

+1 .

If S

⇤1 � S

+1 for every µZ and x with E[x] = 0, then this must also be true for small

zero-mean risks. Given that there is a small zero-mean risk allows us to focus only on thesecond-order Taylor approximation. This yields the following:

E[VS(S+1 , µZ + x)] ' F (Z)

1

2�

2ZVSZZ(S

+1 , µZ) + (1� F (Z))

1

2�

2ZVSZZ(S

+1 , µZ).(43)

34

For a small risk, if S⇤1 > S

+1 , then E[VS(S

+1 , µZ + x)] � 0. For E[VS(S

+1 , µZ + x)] � 0 to

be positive, F (Z)12 �2ZVSZZ(S

+1 , µZ) + (1� F (Z))12 �

2ZVSZZ(S

+1 , µZ) must be positive. One

can calculate that this is equivalent to U U000 + CC

000 � 0 and U

000 � 0. This completes theproof for necessity.

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