Another Step Towards Equilibrium Offers in Unit Commitment Auctions with Nonconvex Costs: Multi-Firm Oligopolies.

• Author: Duggan, Joseph E.

1. INTRODUCTION

   The organization of wholesale electricity markets is a question of immense importance. However, market designs of this nature remain an open area of inquiry. Two commonly utilized market designs are centrally and self-committed markets. In both market designs, generators offer their supply into a uniform-price auction that is operated by an independent third party, which we refer to hereafter as the market operator (MO). What distinguishes these market designs is the format of the offers and the way in which generator-operating decisions are made.

   In a centrally committed market, each generator submits a complex offer, which contains the generator's complete cost and operating-constraint information, to the MO. The MO uses the complex offers that it receives to make financially binding decisions regarding the commitment (i.e., on/off status) of each generator and its dispatch (i.e., production levels). (1) The MO makes these decisions by solving a unit commitment model, which is formulated normally as a mixed-integer optimization problem. Sheble and Fahd (1994); Baldick (1995); Hobbs et al. (2001); Padhy (2004) provide discussions and surveys of unit commitment and dispatch models.

   Conversely, in a self-committed market generators make their own commitment decisions while dispatch decisions are left to the MO. A generator that decides to commit itself (i.e., switch on) submits a simple offer, which specifies the minimum price at which it is willing to produce and sell energy. The MO does not decide which units to commit. However, the MO does determine the dispatch of each generating unit on the basis of the simple offers that are submitted. This is done by stacking the simple offers in merit order into a supply function, which is intersected with electricity demand to determine each generator's production level and the market-clearing price.

   Centrally committed markets are commonplace in North America, whereas much of the rest of the world, including Australia and Western Europe, employ self-committed markets. Because of this regional divergence in designs, the debate regarding their relative merits is ongoing. Cramton (2017) notes that 'the exchange model [self commitment] operating in much of Europe is improving, but still falls far short in pricing transmission congestion both within and across countries.' Furthermore, while recognizing the benefits of the self-committed design in environments without non-convex costs, Cramton (2017) notes that a centrally committed market design 'better handles non-convexities and is simpler for participants.' Moreover, many European markets include now limited non-convex constraints, such as block bids, in their dispatch markets, which complicate price formation. These types of constraints are included to provide generators a mechanism with which to enforce some of their complicated operating constraints in dispatch solutions. Thus, some European markets are considering whether tighter co-ordination, for instance through a centrally committed design, may be beneficial. Imran and Kockar (2014) provide a comparison of market designs in North America and Europe.

   The primary stated benefit of a centrally committed market is that the MO is in the best position to make commitment decisions. This is because the MO co-optimizes the commitment and dispatch of all units using all of the cost and constraint information that pertain to their operation. A self-committed market may be inefficient because generators making commitment decisions independently may not achieve the same level of co-ordination. On the basis of these observations, Ruff (1994); Hogan (1994) advocate centrally committed markets over self-committed designs.

   However, in a centrally committed market the MO must rely on cost and constraint data that are provided by generators in their complex offers. Generators have weak incentives (at best) to provide true information in their offers. For example, Oren and Ross (2005) show that generators can misstate their ramping limits profitably. These misstated ramping limits reduce system efficiency because the MO commits and dispatches more expensive units mistakenly (i.e., on the basis of incorrect constraint information). Further to this point, Mufloz et al. (2018) examine the longterm efficiency of investments that are made within a market environment. They demonstrate that markets that rely on audited cost information for dispatch and pricing can reduce social welfare relative to bid-based market designs. This is because a cost-based market design can distort investment incentives. To illustrate this phenomenon, they discuss the example of opportunity costs diverging from direct fuel costs due to energy or start-up limits.

   Another shortcoming of a centrally committed market design is price formation and generator cost recovery. Price formation in a self-committed market is relatively straightforward as the market-clearing price is given by the intersection of aggregate demand and supply. Generators must structure their offers to ensure that they recover all of their operating costs. In a centrally committed market, the wholesale energy price is set normally based on the value of the dual variables that are associated with the load-balance constraints in the unit commitment model. (2)

   The issue with pricing energy on the basis of these dual variables in a centrally committed market is that generators that are committed and dispatched may not recover all of their costs. This is a well known shortcoming of linear pricing in a setting with non-convexities and Scarf (1990, 1994) provides analyses and illustrative examples of this phenomenon. This economic confiscation can be overcome by some form of discriminatory pricing. O'Neill et al. (2005) propose addressing the confiscation by pricing the non-convexities explicitly. This is done by adding constraints to a linear relaxation of the unit commitment model that fix the commitment decisions to their optimal values. The dual variables that are associated with these added constraints are used to supplement energy payments that are made using the dual variables that are associated with the load-balance constraints. Sioshansi (2014) shows, however, that the dual variables that are associated with the commitment decisions may be negative. As such, inframarginal generators may 'lose' their inframarginal rents through the supplemental payments and be constrained to earn zero profits. Convex hull pricing is discussed as another approach to addressing the complications that non-convexities raise. Instead of starting with a convex dispatch problem and adding back non-convexities for pricing, convex hull pricing uses a 'convexified' unit commitment problem from the outset for operational planning and pricing. Schiro et al. (2016) provide an in-depth discussion of convex hull pricing.

   In practice, revenue adequacy is ensured in centrally committed markets by giving generators linear payments for energy (based on the dual variables that are associated with the load-balance constraints) and make-whole payments. Make-whole payments provide any generator that would operate at a net loss on the basis of the costs that are specified in its complex offer (which may differ from its true costs) a supplemental payment that is equal exactly to its revenue shortfall. The rationale behind a make-whole payment is that absent the mechanism, generators have strong incentives to overstate their costs in their offers to ensure that they are not operated at a loss. Moreover, an MO economically confiscating a generating unit may run afoul of the owner's property rights. Makewhole payments are undesirable, however, because they are discriminatory and their costs must be recovered by the MO, which is done typically by uplifting to consumers.

   Given these shortcomings of the centrally committed market design, there are advocates of self-committed markets. Elmaghraby and Oren (1999) argue that a self-committed market is more efficient than a centrally committed design, when all of the incentive problems and uplift payments are taken into account. Despite the tradeoffs between centrally and self-committed markets being an area of interest and discussion, there is very little formal and systematic examination of the two designs in the extant literature. Sioshansi and Nicholson (2011) compare offering incentives under the two market designs for a symmetric duopoly. They derive equilibrium offering strategies under the two market designs and show that if certain conditions are met, the two designs are expected-cost-equivalent (when accounting for strategic profit-maximizing offer behavior by the firms). Alvaro Riascos et al. (2016) conduct an econometric analysis of the Colombian electricity market, which transitioned from a self-committed to a centrally committed design in 2009. They find that while productive efficiency was increased after the transition, marginal cost markups and prices were higher after 2009 than they would have been absent the market-design change. They attribute this result to the possibility that firms are able to exercise market power more freely after the transition to a centrally committed design.

   This paper adds to this literature by building on the work of Sioshansi and Nicholson (2011) and examining a general symmetric oligopoly. We characterize equilibrium offering strategies under the two market designs. Importantly, we find that the expected-cost equivalence between the two designs fails to hold in the oligopoly with more than two firms. The loss of expected-cost equivalence occurs because with more than two firms there are low-demand states in which generating firms earn strictly positive profits under a self-committed market design. This cannot happen in low-demand states with a centrally committed market. Conversely, in a duopoly, firms are restricted to earning zero profits under both market...