Competitive Energy Storage and the Duck Curve.

AuthorSchmalensee, Richard

    In 2008, modelers at the National Renewable Energy Laboratory predicted that increased penetration of behind-the-meter solar photovoltaic (PV) generation in the area served by the California Independent System Operator (CAISO) would depress net demand in the middle of the day and increase ramping requirements in the late afternoon (Denholm et al., 2008). A few years later, the predicted hourly pattern of grid generation net of wind and (especially) solar was christened "the duck curve." As PV penetration has increased, the duck's back in mid-day has deepened, and a greater increase in output from other sources has been required in late afternoons, when solar generation drops off and residential load increases (in what can be described as the duck's neck). (1) The traditional solution to this problem would be to build and use more gas turbines or combined cycle plants that can increase output rapidly.

    However, building more fossil-fueled generators is inconsistent with the goal in California and elsewhere of reducing carbon dioxide emissions. As the costs of storage, particularly lithium-ion battery storage, have declined, storage has emerged as a potentially attractive, carbon-free alternative way of offsetting diurnal declines in solar generation (Patel, 2018). And, since the promulgation of statutory requirements in 2010, in part to facilitate the integration of solar and other variable renewable generation, (2) the California Public Utilities Commission has been requiring load-serving entities to procure storage (Petlin et al. 2018, California Public Utilities Commission n.d.). Storage targets have also recently been established in Massachusetts, Nevada, New Jersey, New York, and Oregon, and they are under consideration in other states.

    This state-level reliance on mandates contrasts with an apparent preference at the federal level to rely on competition to drive investment in storage facilities. As solar penetration has increased, intra-day price differences have also increased. (3) This suggests that with sufficient solar penetration, competitive storage providers could find it profitable to buy at mid-day when prices are low and sell a few hours later as solar generation begins to drop off and prices become high, thus mitigating or perhaps solving the duck curve problem. The U.S. Federal Regulatory Commission (2018) has recently issued Order 841, which is intended to open wholesale energy markets (and other wholesale markets) to merchant storage providers. (4) Similarly, The European Union's Clean Energy Package, most recently modified in 2019, calls for competitive supply of storage (Glowacki, 2020) and, in contrast to California's mandate, restricts ownership by distribution system operators. In both the U.S. and the EU, efforts are ongoing to reach agreement on exactly how to define markets and establish tariffs to ensure that storage providers have access to wholesale markets on appropriate terms. (5)

    The FERC and EU policies rest on the presumption that energy markets can provide at least approximately optimal incentives for competitive investment in storage as well as generation. (6) This essay explores the validity of this presumption in the context of the duck curve by investigating the properties of a Boiteux (1960, 1964)-Turvey (1968)-style model of an electric power system augmented with the addition of storage. (7) Models in this tradition, and the model developed here, assume constant returns to scale, stochastic and (generally) inelastic demand, and multiple dispatchable generation technologies without significant startup costs or minimum generation levels. If shortages occur, the system is assumed not to collapse, and price is assumed to rise to the value of lost load. (8)

    There are a number of ways that storage has been added to models of this sort. In the earliest formal treatments of storage in this context of which I am aware, Gravelle (1968) and Nguyen (1968) consider two-period--peak and off-peak--models and simply assume that an unlimited amount of the quantity being sold can be transferred between adjacent periods at a constant per-unit cost. Several authors, including Steffen and Weber (2013) and Korpas and Botterud (2020) have added storage to timeless Boiteux-Turvey-style models by assuming that power can be purchased whenever the price of energy is low and resold whenever the price is high. This amounts to assuming that energy storage capacity is effectively infinite, since low-price and high-price periods may be far apart in time. Other authors, including Crampes and Trochet (2019), Brown and Reichenberg (2020) and Junge et al. (2021) have introduced time explicitly but assumed perfect foresight. While these models yield a number of general results regarding investment in and operation of storage facilities under competition, the perfect foresight assumption is strong and eliminates the precautionary demand for storage. Relaxing that assumption, however, requires explicitly modeling the relevant stochastic processes, as demonstrated by Geske and Green (2019). This limits the generality of results that can be obtained. In perhaps the model closest to the one presented here, Helm and Mier (2018) consider a dynamic model with a constant demand curve and non-stochastic renewable output that follows a regular cyclic trajectory.

    To focus on a (necessarily) simplified version of the duck curve problem, the model considered here has alternating periods of two types, labeled daytimes and nighttimes, corresponding roughly to the duck's back and its neck. Renewable generation has positive, stochastic output only in daytime periods. Gas generation, which, for simplicity, stands in for the whole suite of dispatchable generation technologies, is assumed to be available in both daytime and nighttime periods. Short-term storage can be installed at a constant cost per unit of capacity, and storage involves a constant fractional round-trip loss of energy. (9) Demand in both days and nights is stochastic, constant within periods, and perfectly inelastic. (10) Section 2 presents these assumptions in more detail and introduces the notation used in what follows.

    Under constant returns, competitive generators' operating rules are simple: produce if and only if market price of energy is greater than or equal to marginal cost. In general, optimal charging or discharging of storage under competition depends on the current energy market price, the amount of energy in storage, and expectations regarding future energy prices. In general, it does not seem possible describe the behavior of competitive storage suppliers when storage is not fully discharged in each nighttime period without additional assumptions or (per Geske and Green (2019)) resorting to numerical methods. In the context of the duck curve, however, at least in the near term, imposing the restriction that storage is fully discharged in each nighttime seems reasonable. Doing so leads to three possible regimes relating the marginal cost of gas generation to expected nighttime prices, and Section 3 derives the three corresponding competitive operating rules that storage suppliers would follow. Section 3 also presents a sufficient consistency condition for each rule: if that condition holds and if competitive storage suppliers follow the corresponding operating rule, they will in fact find it optimal to sell all stored energy each nighttime.

    Section 4 considers minimization of expected total cost conditional on the algebraically simplest competitive operating rule derived in Section 3. Parallel analyses under the other two regimes are summarized in Sections A and B of the online Appendix. Section 5 discusses some implications of the results of this analysis.


    The daytime and nighttime periods in each day are assumed to be of equal length for convenience, and the probability distributions governing demand within daytime and nighttime periods and the output of renewable generation are assumed to be independent. Independence rules out weather-induced correlations, among other things, but it is not clear how to relax this strong assumption and maintain tractability. There are four technologies with constant returns to scale. Their capacities are initially assumed to be determined by a benevolent social planner interested in minimizing expected total cost. After the implications of that assumption have been explored, we consider whether a continuum of risk-neutral, perfectly competitive firms would provide the same capacities in long-run equilibrium:

    Gas stands in for all dispatchable fossil and nuclear technologies, has capacity G, per-day unit capacity cost [C.sub.G], and per-MWh operating cost c. Renewables have capacity R, per-day unit capacity cost [C.sub.R], and zero operating cost. Maximum renewable output is zero during the night and is equal to [theta]R MWh during the day, where 9 is a random variable with smooth distribution function H([theta]) on [[[theta].bar], 1], with [[theta].bar] > 0, and density function h([theta]). (11) It is assumed that renewable generation can be costlessly curtailed whenever daytime demand is less than available renewable supply. Scarcity operates when load exceeds capacity and there is lost load. Scarcity involves zero capital cost and has per-MWh variable cost v, the value of lost load. The probability of scarcity is assumed to be positive in both periods. (It must be positive in at least one period for gas to recover its capital cost under competition. (12)) Storage has capacity S, per-day unit capacity cost [C.sub.S], and round-trip efficiency [eta]

    Cost parameters are assumed to satisfy the following inequalities:


    The first two of these are familiar: gas has lower capital cost than renewables, and the value of lost load exceeds the incremental cost of gas generation. The third is necessary for gas to be economical. Gas capacity is...

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