What Duality Theory Tells Us About Giving Market Operators the Authority to Dispatch Energy Storage.

• Author: Jiang, Yuzhou

1. INTRODUCTION

   An issue that is raised by integrating energy storage into electricity systems with competitive wholesale markets is the role of market operators (MOs) in determining energy-storage dispatch. Much of the extant literature, which Castillo and Gayme (2014); Sioshansi et al. (2022) survey, focuses on optimizing energy-storage operations from the perspective of the asset owner. However, there are benefits to having the operations of energy storage and other assets co-optimized. Pozo et al. (2014) assess the value of incorporating energy storage into a unit-commitment model, whereby a single entity co-optimizes energy-storage and generator operations. Weibelzahl and Martz (2018) examine the impacts on zonal pricing of incorporating energy-storage-operations decisions into MOs' market-clearing models. Despite these benefits of co-optimizing the operation of energy storage with other power-system assets, there is a concern that MOs' independence can be threatened if they make energy-storage-operations decisions. Sioshansi et al. (2012); Sioshansi (2017) note that a primary rationale behind this concern is that the operation of energy storage can affect the balance of the power system and wholesale-price formation.

   An example of this concern involves Lake Elsinore Advanced Pumping Station (LEAPS). LEAPS's developer, Nevada Hydro Corporation, proposed building the plant to relieve congestion in Southern California. Because of its transmission benefits, Nevada Hydro Corporation sought in its filing to Federal Energy Regulatory Commission (FERC) an arrangement whereby California ISO (CAISO) dispatches LEAPS to maximize its transmission-relief benefits. (1) In its decision, FERC concludes that having CAISO dispatch LEAPS is akin to CAISO owning and operating generation, which could threaten the independence that is required of MOs by impacting wholesale-price formation. Indeed, market independence is an explicit objection that CAISO raises in its filings in the LEAPS case. (2) This conclusion stems, in part, from CAISO making financially binding unit-commitment and dispatch decisions for generating units. Sioshansi et al. (2010) contrast the roles of MOs in making these decisions between different markets.

   This debate is reminiscent of questions raised during the 1990s around the proper role of MOs optimize use of transmission networks. Hogan (1992); Ruff (1994); Oren (1997) provide formative analyses of this aspect of electricity-market design. There are several benefits to having MOs optimize transmission-network use. First, Hogan (1992) demonstrates that having MOs determine transmission use to maximize social welfare is equivalent to minimizing economic rents on transmission networks. This equivalence means that market solutions yield short-run efficiency, dispatch support, and incentive compatibility in power-system operations. Second, Perez-Arriaga et al. (1995) analyze the congestion rents that are generated by market models that give control of transmission-network use. They show that when considering a fixed time span, these rents are equal to the cost of transmission investment if the dynamic capacity-expansion plan is optimal over the time span and there are neither economies of scale nor lumpiness in transmission investment. This finding means that MOs determining the use of transmission networks is consistent with social-welfare maximization and long-run efficiency in transmission investment.

   In this paper, we examine the incentive and efficiency implications of giving MOs operational control of energy storage. We study this question by adapting and extending the approaches that are taken by Hogan (1992); Perez-Arriaga et al. (1995) to analyze MOs optimizing transmission-network use. We take a two-prong approach to our analysis, which yields two policy-relevant market-design findings.

   First, we examine optimal-power-flow (OPF) models with and without energy-storage-operational decisions embedded within them. Comparing the dual problems of these two OPF models shows that incorporating energy storage into market-clearing models does not change fundamentally the price-formation process. So long as the market model maximizes social welfare or minimizes system cost, energy storage factors into market clearing and price formation analogously to an energy producer when it discharges and analogously to an energy consumer when it charges. Analysis of the dual problem of the OPF model with embedded energy storage shows that the market price is dispatch-supporting and incentive-compatible in the sense that energy storage is incentivized to comply with the market solution. This result stems from the convexity of the OPF model and means that MOs having operational control of energy storage provides the same short-run-efficiency properties that Hogan (1992) demonstrates for MOs determining the use of transmission networks.

   Second, we examine a stylized energy-storage-investment model and compare the cost of energy-storage investment to energy-storage rents that are engendered by the solution of an OPF model that has energy-storage-operational decisions embedded within it. We show that if a power system has a socially optimal amount of energy-storage capacity, marginal energy-storage rents equal marginal energy-storage-investment costs. This result means that MOs having operational control of energy storage yields the same long-run investment-efficiency properties that Perez-Ar-riaga et al. (1995) demonstrate for MOs determining the use of transmission.

   Taken together, our work shows that giving MOs operational control of energy storage provides the same short-run properties (e.g., efficiency, dispatch support, and incentive compatibility) and long-run efficiency that MOs determining the use of transmission provides. On the basis of our findings, we argue that the price-formation and market-independence concerns that are raised in the case of LEAPS and similar proposed energy-storage projects are unwarranted. Indeed, we find that giving MOs operational control of energy storage raises no new market-design issues as compared to MOs determining the use of transmission or making operational decisions for generating units.

   The remainder of this paper is organized as follows. Section 2 provides the formulation of the stylized OPF model that we analyze and its dual problem. Section 3 provides our theoretical results. Sections 4 and 5 demonstrate the properties of the stylized OPF model using a simple example and real-world case study, respectively. Section 6 concludes and provides a discussion of the market-design implications of our work.

2. MODEL FORMULATION

   This section presents the formulation of a multi-period OPF model, which is assumed to have hourly time-steps, and its dual problem. The model is multi-period because energy storage couples decisions between hours. This model and its dual underlie our analysis of MOs having operational control of energy storage. The model is an idealized example of a perfect market, which is known to be efficient. Mas-Colell et al. (1995) provide a detailed treatment of these efficiency results, which we paraphrase. According to the first fundamental theorem of welfare economics, if preferences are locally non-satiated, then a competitive equilibrium is Pareto optimal. Furthermore, the second fundamental theorem of welfare economics states that if each consumer has convex preferences and each firm has a convex production set, then there is a price vector that gives a competitive equilibrium to support any Pareto-optimal allocation. These theorems have two technical requirements, which underlie our model. First, there must not be any information asymmetry. Second, economic agents must be price-taking. In the context of our work, the OPF model provides a competitive equilibrium in which supply equals demand. The dual problem allows us to demonstrate the dispatch-support and incentive-compatibility properties of the prices that are given by a competitive equilibrium.

   We begin our model formulation by defining the following notation.

   2.1 Indices, Sets, and Parameters

   B set of transmission buses [B.sub.g] transmission bus at which generator g is located [B.sub.i] transmission bus at which energy storage i is located [b.sub.n] willingness-to-pay for energy of transmission-bus-n customers ($/MWh) [c.sub.g] operating cost of generator g ($/MWh) [D.sub.n,t.sup.max] [D.sub.n,t.sup.max] maximum hour-t demand at transmission bus n (MW) [F.sub.l.sup.max] [F.sub.l.sup.max] capacity of transmission line l (MW) g g generator index G G set of generators [G.sub.n] [G.sub.n] set of generators that are connected to transmission bus n [H.sub.i] [H.sub.i] energy-carrying capacity of energy storage i (h) i i energy storage index I I set of energy-storage devices [K.sub.g.sup.max] [K.sub.g.sup.max] production capacity of generator g (MW) I I transmission-line index L L set of transmission lines m,n m,n transmission-bus indices [P.sub.i.sup.max] [P.sub.i.sup.max] power capacity of energy storage i (MW) [S.sub.n] [S.sub.n] set of energy-storage devices that are connected to transmission bus n t t time index T T set of hours within model horizon [[eta].sub.i] [[eta].sub.i] round-trip efficiency of energy storage i (p.u.) [[pi].sub.n,l] [[pi].sub.n,l] transmission-bus-n /transmission-line-l shift factor (p.u.) 2.2 Decision Variables

   [e.sub.n, t] [e.sub.n, t] hour-t net injection of power from transmission bus n into the transmission network (MW) [h.sub.i,t] [h.sub.i,t] hour-t discharging rate of energy storage i (MW) [L.sub.n t] [L.sub.n t] hour-t load at transmission bus n that is served (MW) [r.sub.i, t] [r.sub.i, t] hour-t charging rate of energy storage i (MW) [S.sub.i, t] [S.sub.i, t] ending hour-t state of energy (SOE) of energy storage i (MWh) [x.sub.g, t] [x.sub.g, t] hour-t power output of generator g (MW) 2.3 OPF Model

   The OPF model is formulated as:

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