Energy Storage Investment and Operation in Efficient Electric Power Systems.

• Author: Junge, Cristian

1. INTRODUCTION

   Driven importantly by concerns about climate change, variable renewable energy (VRE) resources, mainly wind and solar, are becoming increasingly important sources of electricity in many regions. Because the maximum output of VRE generators is variable and imperfectly predictable, however, increased penetration of VRE generation makes it more difficult for power system operators to match supply and demand at every instant. As the costs of storage, particularly lithium-ion (Li-ion) battery storage, have declined rapidly, storage has emerged as a potentially attractive, carbon-free solution to problems posed by increased VRE penetration (Patel 2018). Policy-makers in the U.S. and the E.U. have accordingly encouraged the deployment of storage. The California Public Utilities Commission has been requiring load-serving entities to procure storage since the promulgation of statutory requirements in 2010 (Petlin et al 2018, California Public Utilities Commission n.d.). As of mid-2021, seven states have established storage targets, and they are under consideration in other states (DSIRE database n.d.). The U.S. Federal Regulatory Commission (2018) Order 841 is intended to open wholesale energy markets (and other wholesale markets) to merchant storage providers. (1) Similarly, The European Union's Clean Energy Package calls for competitive supply of storage (Glowacki 2020).

   In this essay, we explore what economic theory implies about the general properties of cost-efficient electric power systems in which storage performs energy arbitrage to help balance supply and demand. (2) We start from an investment planning model ultimately based on the work of Boiteux (1960, 1964) and Turvey (1968). (3) In models of this sort, constant returns to scale are generally assumed in generation, i.e., costs are assumed to be linear in the capacities and outputs (up to capacity) of each of several types of dispatchable generators. There are no startup costs or ramping constraints, which limit thermal generators' ability to change output. There are thus no non-convexities or links between time periods on the supply side. Similarly, the demand function in each period is independent of prices charged in other periods. Thus the multiple time periods in these models are linked only by the generation capacities that are chosen at the outset.

   It is important to note that these assumptions are not descriptive of systems in which coal or nuclear generation are important supply sources. Both technologies have significant economies of scale, giving rise to nonconvexities. In addition, coal and nuclear plants take time and incur costs to start up and ramp down (4), which breaks the independence among time periods. Power systems with these characteristics resist general algebraic analysis, and sophisticated numerical optimization tools have been developed to permit explicit multi-period analysis of particular cases. (5)

   For modern gas generators and VRE facilities, however, neither lumpiness nor startup or ramp down costs are nearly as important. Boiteux-Turvey-style models are thus reasonable approximations for systems without significant coal or nuclear generation. (6) 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 we are aware, Gravelle (1976) and Nguyen (1976) 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 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 also amounts to assuming that the amount of energy that can be stored is unbounded, since low-price and high-price periods may be far apart in time. Helm and Mier (2018) consider a dynamic model with a constant demand curve and non-stochastic renewable output that follows a regular cyclic trajectory. Schmalensee (forthcoming) considers a model with stochastic demand and alternating daytime and nighttime periods in which VRE generation is only available in the daytime periods.

   Here we follow Crampes and Trochet (2019) and Brown and Reichenberg (2021) and consider an explicitly dynamic Boiteux-Turvey-style model with perfect foresight. We follow most of the literature and assume constant returns to scale in storage as well as in generation. The perfect foresight assumption is of course 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), and it is not clear that general results are available.

   Section 2 presents the (linear) capacity planning and operations model employed, and Section 3 provides brief derivations of a number of known results for the sake of completeness. We show explicitly that the problem of maximizing overall social welfare in this model can be decomposed into the problems faced by profit-maximizing, perfectly competitive suppliers of each available technology, even when considering limited energy capacity of energy storage and ramping constraints for dispatchable generation. We demonstrate that marginal-cost-based dispatch for thermal generators is not generally optimal when ramping constraints are binding.

   Section 4 provides generalizations of recent results regarding optimal investment in, profitability of, and operation of individual storage technologies. We employ a generalized characterization of storage technologies that uses seven distinct parameters, including independent charging and discharging power capital costs and efficiencies and show that all deployed storage technologies break even at equilibrium under constant returns to scale.

   Section 5 presents an analytical framework that yields insight into efficient configurations and operations of systems employing multiple storage technologies and points to the importance of the relative costs of power capacity and energy storage capacity. Finally, Section 6 provides simulation results that illustrate the complexity of operating patterns of storage in systems with multiple storage technologies and supports the insights developed in Section 5. It shows that general analytical results of the "merit-order" variety are not available for storage, and demonstrates the value of frequency domain analysis via Fourier Transforms to characterize the cost-efficient operating regimes of each storage technology. Section 7 provides some concluding observations.

2. OPTIMA AND EQUILIBRIA

   We consider a linear T-period model with one dispatchable technology (which we will often refer to as gas), one VRE technology, and a single storage technology. The restriction to a single technology of each type in this section is simply to reduce notational clutter. In later sections we consider systems with multiple technologies of each type when appropriate. Throughout we abstract from storage's ability to supply frequency regulation and other ancillary services and to defer investment in transmission or distribution systems.

   Because our focus is on the supply of electricity, we assume perfectly inelastic demand for simplicity. That is, we assume that demand in period t is equal to the exogenous quantity [Q.sub.t], for prices below [omega], the value of lost load. Then total welfare, to be maximized, is given by

   W = [omega] [SIGMA][[Q.sub.t] - [L.sub.t]]-[[C.sub.G]G + [C.sub.R]R + [C.sup.A.sub.P][P.sup.A] + [C.sup.D.sub.P][P.sup.D] + [C.sub.E]E + v[SIGMA][g.sub.t] + [o.sup.A][SIGMA][A.sub.t] + [o.sup.D][SIGMA][D.sub.t]], (1)

   where [L.sub.t] is the non-negative lost load in period t. (7) Throughout, sums are over t from 1 to T, unless otherwise specified. (8)

   We assume constant returns to scale, so that we can work with the aggregate capacities and outputs of all facilities using the same technology. (9) From left to right the Cs in equation (1) are the T-period per-MW capital costs of dispatchable capacity, G, of renewable capacity, R, of charge power capacity of storage, [P.sup.A], and of discharge power capacity of storage, [P.sup.D], respectively. [C.sub.E] is the T-period per-unit capital cost of energy storage capacity, E.

   For a pumped hydro storage facility, for instance, [P.sup.A] would be the maximum rate at which water can be pumped into the uphill reservoir, measured by the instantaneous power consumption of the pumping system in MW, [P.sup.D] would be the maximum rate at which the facility can generate electricity, again in MW, and E would be the capacity of the reservoir. For convenience we assume that [P.sup.A], [P.sup.D], and E can be chosen independently, though for some storage technologies this may not be possible. For pumped hydro for instance, the same turbine is often used to pump water into the reservoir and to generate electricity when the water is released. (10)

   Also in equation (1), v is the (constant) marginal cost of dispatchable generation, g, is dispatchable generation in period t, [o.sup.A] is the variable operation and maintenance (O&M) cost per MWh used to charge storage, [A.sub.t] is MWh used to charge storage in period t, [o.sup.D] is the variable O&M per MWh discharged, and [D.sub.t] is MWh discharged from storage in period t. In the context of pumped hydro storage, one can think of [o.sup.A] and [o.sup.D] reflecting the marginal wear and tear caused by pumping water into the uphill reservoir and using water from that reservoir to generate electricity, respectively. In the case of battery storage systems, these parameters are best thought of as providing an approximation to the degradation caused by charging and discharging. (11)

   We consider maximization of W subject to a set of linear constraints...