Coordinated Transaction Scheduling in Multi-Area Electricity Markets: Equilibrium and Learning

06/05/2020 ∙ by Mariola Ndrio, et al. ∙ 0

Tie-line scheduling in multi-area power systems in the US largely proceeds through a market-based mechanism called Coordinated Transaction Scheduling (CTS). We analyze this market mechanism through a game-theoretic lens. Our analysis characterizes the effects of market liquidity, market participants' forecasts about inter-area price spreads, transaction fees and interaction of CTS markets with financial transmission rights. Using real data, we empirically verify that CTS bidders can employ simple learning algorithms to discover Nash equilibria that support the conclusions drawn from the equilibrium analysis.

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1 Introduction

Different parts of an interconnected power grid are controlled and managed by different system operators (SOs). We call the geographical footprint within each SO’s jurisdiction an area, and transmission lines that interconnect two different areas as tie-lines. Efficient scheduling of power flows over tie-lines is paramount to improve market efficiency and exploit geographically diverse renewable resources. Tie-lines are capable of supplying a significant portion of each area’s electricity demand. For example, the New York ISO (NYISO) and ISO New England (ISO-NE) share nine tie-lines with approximately 1800 MW capacity, capable of supplying 12% of New England’s and 10% of New York’s demand as of 2009 according to White and Pike (2011). Even though tie-lines are important assets, they have been historically under-utilized as evidenced by the persistent price differences between regional markets (see White and Pike (2011)

). The economic loss from inefficient tie-line schedules has been estimated at $784 million between NYISO and ISO-NE in 2006-10 (see

White and Pike (2011)), the burden of which has been ultimately borne by end-use customers. What causes such inefficiencies? There are a number of factors that include the inherent uncertainty about power requirements when tie-lines are scheduled prior to delivery time points, the lack of coordination and appropriate information exchange among the SOs, ad hoc use of proxy buses in deciding the schedule and transaction fees.

Conceptually, power flows over tie-lines should be determined through a joint economic dispatch problem geared towards maximizing the efficiency of the interconnected power grid as a whole. However, historical and legal reasons render such an aggregation of market information from different areas at a central location untenable. Naturally, a considerable effort has been made to solve the joint dispatch problem in a distributed fashion, focusing on primal (see Bakirtzis and Biskas (2003), Zhao et al. (2014)) and dual decomposition methods (see Conejo and Aguado (1998), Jie Chen et al. (2004)). In such methods, SOs exchange information among themselves to compute the optimal tie-line schedule. This theoretical coordination mechanism, referred to as Tie Optimization (TO) in White and Pike (2011), proved challenging to implement in practice. It was perceived as requiring the SOs to trade directly with each other, violating their financial neutrality, in lieu of the earlier market-based, albeit inefficient, process for scheduling tie-line flows. Instead, many SOs adopted variants of Coordinated Transaction Scheduling (CTS), e.g., see FERC (2012, 2016), that sought to blend the earlier market-based tie-line scheduling with the theoretically optimal TO, after receiving approval from FERC. CTS is a market mechanism in which external market participants submit bids and offers to import or export from one area to the other. The schedule is computed using both participants’ offers as well as the SOs’ forecasts about price differences. CTS market design is predicated on the simple premise that arbitrage opportunity will attract more participants, whose profit motivation will ultimately shrink that opportunity, pushing the schedule closer to the theoretically optimum. CTS has certainly improved tie-line scheduling as per Monitor (2019), Economics (2019), but significant inefficiencies remain. Motivated by these inefficiencies, we analyze the impacts of strategic interactions among CTS market participants on the performance of these markets through a game theoretic study. We provide palpable insights on the consequences of an illiquid market, errors in SOs’ price forecasts and transaction fees on market efficiency. We remark that the use of proxy buses as trading locations results in the so-called ‘loop flow’ problem (see Cvijic and Ilic (2014)) that negatively impacts CTS market performance. We refer the reader to Guo et al. (2018) for mechanisms to tackle this problem, and instead focus on the repercussions of strategic interaction among market participants in this paper.

We introduce CTS in Section 2. Then, we model CTS as a game among arbitrage bidders who compete through scalar-parameterized transport offers in Section 3. Our game formulation is inspired by supply function competition models considered in Johari and Tsitsiklis (2011), Xu et al. (2016). We establish the existence of Nash equilibria for this game under mild assumptions and study the impact of various factors on the nature of said equilibria in Sections 4-6 to offer insights into the CTS market. First, we show that when transaction costs (levied on a per-megawatt hour basis on bidders) are absent, then a highly liquid CTS market is efficient. Market efficiency degrades with liquidity shortfall, exhibiting bounded efficiency loss for intermediate liquidity and unbounded losses in low liquidity regimes. Second, with transaction costs, CTS fails to eradicate the price spread between adjacent markets even with a liquid market, implying that such costs undercut the vision behind the market design. Third, we show that SOs’ estimate of the price spread plays a central role in the efficiency of CTS markets in that bidders have limited ability to correct the effects of SOs’ forecast errors. Fourth, portfolios of financial transmission rights (FTR) held by CTS bidders can impact CTS market outcomes, revealing the dependency of the efficiency of these inter-area markets on other energy derivatives. Our equilibrium analysis reveals how the strategic incentives in CTS markets are oriented but does not illustrate if bidders can learn equilibrium behavior through repeated participation in these markets. We simulate repeated play using historical data from the NYISO–ISO-NE market and demonstrate that our conclusions from equilibrium analysis continue to hold in a statistical sense in our numerical experiments. All proofs are relegated to the Appendix.

2 The CTS Mechanism

CTS is a market-based mechanism for tie-line scheduling that replaced an earlier market-based structure in an effort to streamline the bidding and scheduling process. Among the important changes, CTS unified the bid submission and clearing process among the neighboring SOs, reduced the tie-line schedule duration from one hour to 15-minute intervals, and decreased time delays among bidding, scheduling, and power delivery. To illustrate the economic rationale of the CTS mechanism we consider throughout a stylized two-area power system, shown in Figure 1, connected via a single tie-line with the inter-area power flow denoted by . Each SO computes their supply stacks by solving an area-wise parametric economic dispatch by varying the amount of power flowing on the tie-line. An example of supply stack is shown in Figure 1. The stack of area represents the incremental dispatch cost of delivering power at its side of the interface. Similarly, the stack of area represents the decremental dispatch cost of reduced supply, shown in descending order. Since scheduling happens prior to power delivery, these stacks are based on SOs’ forecasts. In this example, the optimal direction for the power flow is from area to since for zero scheduled flow, area operates at higher dispatch costs than area . At the level where dispatch costs at the border become equal or where the supply stacks intersect, is the TO schedule, denoted by . This tie-line schedule minimizes the aggregate dispatch costs across the two areas, and it serves as our theoretical benchmark to compare CTS performance. However, TO requires SOs to trade directly with each other on behalf of the market participants in their respective areas, which may be viewed as the SOs becoming active participants of trade rather than financially neutral market operators. Instead, the current practice of CTS relies on virtual traders whose offers/bids are utilized together with the supply stacks to arrive at the tie-line schedule.

Figure 1: Illustration of the TO and CTS scheduling mechanisms.

CTS market participants in practice submit ‘interface’ bids that consist of three elements: the minimum price difference between the proxy buses in the two areas the bidder is willing to accept, the maximum quantity to be transacted and the direction of the trade, i.e., the source and the sink. A CTS market participant is a virtual bidder in that she can offer to transport power across areas without physically consuming or producing it. They only participate in the scheduling process, bearing no obligation for physical power delivery; the transaction is purely financial.

Under CTS, one of the SOs pools the virtual bids at the proxy buses and the supply stacks from both operators to assemble the aggregate interface supply stack, shown in Figure 1. All the bids indicating the optimal direction are stacked from lowest to highest price to create their own “supply curve”. The price spread curve is derived by subtracting the supply stack of area from that of area . The CTS schedule, denoted by , is set at the intersection of the interface supply stack and the price spread. An interface bid is accepted if its offer price is less than the price spread at the tie-line schedule. Therefore, all interface bids to the left of the CTS schedule are accepted; all bids to the right are not. All cleared interface bids are settled at the real-time proxy bus LMPs and there are not uplift credits or debits associated with tie-line schedules. In the next section, we extract a theoretical model for CTS and characterize its outcome under strategic interactions of interface bidders against the outcome of TO.

3 Modeling the CTS Market as a game

We model the CTS market as a game among the virtual bidders who compete to transport power over a tie-line against an elastic inter-area price spread that varies with the power flow over the tie-line.111The study in White and Pike (2011) indicates that the primary interface between NYISO–ISO-NE was congested 0.3% of the hours eastbound and 1.2% of the hours westbound in 2009. Hence, to avoid unneccessary complication of the analysis and facilitate exposition, we ignore the total transfer capability of the tie-line in modeling the CTS game. Recall that LMPs at the proxy bus in each area comes from the solution to an area-wise economic dispatch problem, parameterized by the tie-line power flow . For areas and , denote these LMPs by and , respectively. Without loss of generality, let area export and area import power, and define

(1)

as the price spread between the areas. Assume without loss of generality that is strictly decreasing, concave and differentiable in with . Note that acts as the inverse demand function in a supply competition model with virtual bidders. Our framework adopts standard assumptions on the demand function that are employed in several supply function competition models in the study of electricity markets (see Green and Newbery (1992), Baldick et al. (2004), Rudkevich (2003)).

Consider virtual bidders in the CTS market. Let the -th bidder provide two parameters to the SOs with the understanding that she is willing to transport up to

(2)

amount of power from area to at a price spread of . Our transport offer is inspired by supply function competition models studied in Johari and Tsitsiklis (2011), Xu et al. (2016), Ndrio et al. (2020). Figure 2 reveals how the parameters affect the shape of the transport offer. Bidder is willing to transport a maximum quantity of , but at a minimum price spread of . The required price difference increases with the power transport and grows unbounded as the latter approaches . In effect, transporting power above requires an infinite price difference. The parameterized “hockey-stick” shaped transport offer in (2) is a smooth approximation to the one in practice where a player is willing to transport up to at a specified price difference. The realized price spread is uncertain and a higher exposes the player to a higher potential loss. Therefore, bidder expresses her total budget for potential losses or her liquidity in . Notice that since ’s express budget constraints for the bidders, we assume does not vary strategically in day-to-day transactions.

Figure 2: Parameterized interface bid of CTS market participant.

The family of transport offers in (2) allows market participants to have one-dimensional action spaces and has been shown to possess a number of attractive properties including bounded price of anarchy and price markup at the Nash equilibrium Johari and Tsitsiklis (2011), Xu et al. (2016). Moreover, they prohibit situations when market participants can bid above their means by explicitly incorporating maximum budget/capacity in the offer structure, which is not straightforward to do with e.g. linear supply functions Baldick et al. (2004). Other families of supply offer such as the (degenerate) pure price (Bertrand) or quantity (Cournot) competition are not suitable representations of the CTS interface bid. Although variations of the Bertrand model with capacity constraints may seem an attractive alternative, however, in such settings pure Nash equilibria may not exist Shubik (1959).

Given the liquidities , the choice of bids from the CTS bidders describe their willingness to transport power across the interface according to (2). Collect the liquidities and bids in and , respectively. The SOs allocate the aggregate tie-line schedule among virtual bidders, given as follows. They calculate as the allocations of the tie-line flow to the participants by solving

(3a)
subject to (3b)

where, denotes the set of real numbers and

is a vector of all ones. The above problem seeks a tie-line schedule where the offer stack for inter-area power transport from CTS market participants intersects the SOs’ estimated price spread function. The tie-line schedule is then given by

The transport offer in (2) enters the SOs’ problem through its implied cost of transport. This induced cost is calculated by equating the implied marginal cost curve to the transport offer. With this interpretation, the SOs’ flow allocation problem in (3) seeks to maximize the social welfare of an economy that is composed of the wholesale markets in areas and together with the CTS bidders (see Guo et al. (2018) for a similar interpretation of the CTS market objective). CTS identifies a single clearing price for its market as

(4)

The CTS flow allocation to bidder and the resulting CTS tie-line schedule are respectively given by

(5)
(6)

Note that when , we have when and for . To avoid difficulties due to a zero price, we adopt the convention from Johari and Tsitsiklis (2011)

Within our notational framework, TO determines the tie-line schedule as

(7)

that seeks to maximize , a measure of welfare for the wholesale markets in areas and . is given by the schedule determined by the no-arbitrage condition, i.e., where the price spread vanishes. Notice that equals the CTS market objective in (3) with . Thus, we expect CTS to emulate TO only when all bidders bid zero ’s. We now proceed to formally define the CTS game and characterize its Nash equilibrium to understand what bidding behavior we expect, given the bidders’ strategic incentives.

While virtual bidders do not incur any costs to physically transport power, many pairs of SOs levy transaction fees on a per-MWh basis, e.g., in CTS between NYISO and PJM, NYISO charges physical exports to PJM at a rate ranging from $4-$8 per MWh, while PJM charges physical imports and exports rates that average less than $3 per MWh. See Economics (2019) for details. For a willingness to transport MW of power from area to , assume that transaction cost equals , where is measured in $/MWh. Then, each bidder’s payoff equals the total revenue garnered less the transaction costs, formally given in

(8)

Formally, define as the CTS game among virtual bidders—henceforth referred to as players—who bid , given , and receive a payoff described by (8). Bidders selfishly seek to maximize their own payoffs, given their liquidities. A bid profile constitutes a Nash equilibrium of , if

for all . That is, no player has an incentive for a unilateral deviation from the equilibrium offer. We establish the existence of such an equilibrium profile in our first result.

Theorem 1 (Existence of Nash Equilibrium)

The CTS game admits a Nash equilibrium if satisfies

(9)

for .

Our proof relies on Rosen’s result in Rosen (1965) after we establish that is a concave game. Uniqueness of the equilibrium remains challenging to prove. However, in the next sections, we will establish uniqueness of the Nash equilibrium under a number of settings.

To explicitly characterize the Nash equilibrium we restrict our attention to affine price spreads

(10)

with to compute the equilibria and study its properties. It is straightforward to verify that as defined above satisfies (9) and hence, an equilibrium always exists for , according to Theorem 1. Indeed, the price spread can be shown to be affine in , when each area is represented as a copperplate power system, having a generator with quadratic generation cost and a fixed demand. This follows from properties of multiparametric quadratic programs in (Borelli et al. 2014, Theorem 7.6)

. To further justify our modeling choice, we perform a linear regression of New England’s LMP at the proxy bus (Roseton) as

, and obtain with an adjusted coefficient of 0.91, revealing an affine dependency of in . Encouraged by this data analysis, we now proceed to analyze the CTS market with strategic participants for the affine price spread model.

4 Impact of Liquidity in CTS Markets

Our first goal is to investigate the impacts of liquidity on the CTS scheduling efficiency. To isolate the effects of liquidity, neglect transaction fees and set . We define the efficiency of CTS as the ratio

where recall that measures the aggregate welfare of the wholesale markets in the two areas attained at a particular tie-line schedule. TO seeks to maximize this welfare with , while the outcome of CTS arises from the strategic interaction of the market participants. Our next result characterizes the equilibrium and provides key insights into the behavior of in different liquidity regimes.

Proposition 1

Consider the CTS game , where is the unique maximal budget in . Then, admits a unique Nash equilibrium given by

(11)

and for . Furthermore, we have

(12)

where .

Existence of the equilibrium follows from Theorem 1. The rest follows from analysis of the first-order equilibrium conditions. The result highlights that allocation and the efficiency vary widely with liquidity and the player with the maximal liquidity plays a rather central role in determining the outcome of the CTS market. To offer more insights, distinguish three different liquidity regimes. Identify the liquidity as high when , where the aggregate liquidity of all players but is sufficient to cover the efficient schedule . The intermediate liquidity occurs where the aggregate liquidity is different from by at most the liquidity of player , i.e., . Finally, the low liquidity regime is where . The outcome and the efficiency differ substantially across these regimes. Using the equilibrium profile, it is easy to see that the flow allocation is given by

where denotes the vector of liquidities of all players, except . Thus, all but player offer their maximum liquidity at equilibrium. These players benefit from being inframarginal, exploiting the bid of the marginal player . This behavior is reminiscent of the so-called ‘free-rider problem’ (see Fudenberg and Tirole (1991)). When the liquidity is too high or too low, player does not have enough market power and does not benefit from bidding nonzero , implying that she does not withhold from her maximal budget in her transport offer. In the intermediate liquidity case, player enjoys market power and her flow allocation can be shown to be the Cournot best response to this residual price spread . See Cai et al. (2019), Hobbs (2001) for details on Cournot competition.

The tie-line schedule at the equilibrium of is given by

When liquidity is high, coincides with , implying that CTS yields the SOs’ intended outcome. In other words, perfect competition arises as a result of strategic incentives. In the intermediate liquidity regime, CTS suffers welfare loss due to strategic interaction. The loss, however, is bounded; strategic behavior cannot cripple the welfare under perfect competition by more than 25%. When the liquidity is low, the lower bound on can be small. However, in this case lack of efficiency is not due to strategic interactions but rather due to the very low market liquidity.

4.1 Learning equilibria through repeated play

Nash equilibria characterize how the incentives of market participants are oriented. However, the power of said equilibria to predict market outcomes may appear limited in that players are endowed with intelligence over their opponents’ payoff and the system conditions to compute such an equilibrium. In practice, players interact repeatedly exploring the market environment while facing a noisy reward. Motivated to investigate if players can learn equilibria through repeated play, we study the game dynamics where bidders adopt action-value methods (see Sutton and Bart (2018)

) to update their bids. More precisely, we implement an upper confidence bound (UCB) algorithm for each bidder. In such a setting, each player is agnostic to the presence of other players and the SOs’ clearing process, i.e., they endogenize these as part of the environment that yields a random reward. UCB is a popular reinforcement learning algorithm that achieves logarithmic regret per

Auer and Ortner (2010), L. Lai and Robbins (1985) in static environments and balances between exploration and exploitation. In each round (an instance of a CTS market), each player selects the action that has the maximum observed payoff thus far plus some exploration bonus.

The game proceeds as follows. At each round, each bidder chooses from a finite set of actions . Each bidder maintains a vector of average rewards from each action and the number of times each action is chosen, where denotes the set of naturals. Here, the reward equals the revenue less the transaction cost from the CTS market. Bidders initialize by selecting every action (possible bid from ) at least once. Upon bidding at a certain round, say she receives the reward from the CTS market. Then, the bidder updates and as

(13)

Then, the bidder bids the action , where

(14)

The parameter controls the degree of exploration. The larger the , the player is eager to explore actions that have not been tried often enough. The smaller the , the player tends to choose an action largely based on the average reward seen thus far.

(a) Subfigure 1 list of figures text
(b) Subfigure 2 list of figures text
(a) Subfigure 1 list of figures text
(b) Subfigure 2 list of figures text
Figure 3: Plot of cumulative percentage of times the Nash action is chosen across 3000 games for bidders 1 () and 5 (). Bidder 5 is marginal for (a) and inframarginal for (b). After 3000 games, bidders 1-5 respectively select in (99.9, 92.1, 99.9, 99.6, 99.2) games in (a) and (90.1, 99.9, 86.4, 92.4, 88.2) games in (b).
Figure 4: Comparison of tie-line schedules and price spreads for a highly () and intermediately liquid () CTS market.
Figure 3: Plot of cumulative percentage of times the Nash action is chosen across 3000 games for bidders 1 () and 5 (). Bidder 5 is marginal for (a) and inframarginal for (b). After 3000 games, bidders 1-5 respectively select in (99.9, 92.1, 99.9, 99.6, 99.2) games in (a) and (90.1, 99.9, 86.4, 92.4, 88.2) games in (b).

We utilize historical CTS data from the NYISO and ISONE markets to compute the affine price spread that yields MW. We consider repeated play of the CTS game with five participants, first with and then with . The first example corresponds to an intermediate liquidity regime with . The second example belongs to the high liquidity category for which . In our simulations, we use following (Sutton and Bart 2018, Chapter 2). Each CTS bidder chooses from ten ’s in that includes the optimal actions. Figure 4 shows percentages of optimal actions selected by bidders in a total of 3000 games for the high and intermediate liquidity regimes.

In the intermediate regime, the pivotal and inframarginal players act in a rather ‘greedy’ fashion, exploiting their optimal action north of of the games. This implies that the observed reward from playing the optimal action is large enough, even as the exploration bonus of other actions increases. Bidder 5 loses her role as the marginal player when the liquidity is high. In this regime, players are slower to discover their optimal actions, although selection percentages are north of of the games. Our numerical experiments clearly demonstrate that even in a setting where players know little to nothing about the game setting, they are able to discover and play equilibrium actions (in majority of the games) through repeated play. This experiment lends credence to the conclusions from our equilibrium analysis. Indeed, in Figure 4 remains close to unity and price spreads are below MWh in most games for a highly liquid CTS market. A liquidity reduction of around has palpable effects on market performance, although in aggregate, the players have the capacity to meet . In particular, the price spread for intermediate liquidity is more than MWh higher than the highly liquid case and remains well below . This experiment highlights how rise of pivotal players exercising market power exploiting the lack of liquidity can impact market performance.

5 Interactions with Financial Transmission Rights (FTRs)

CTS performance may be influenced by potential uneconomic bidding that aims to benefit financial positions whose value is tied to CTS outcomes, such as FTRs. Price manipulation that involves uneconomic virtual transactions has emerged as a central policy concern for FERC, as shown by several high-profile enforcement cases that ended in multi-million dollar settlements (see Ledgerwood and Pfeifenberger (2013)). Here, we investigate the CTS performance when any subgroup of market participants hold FTR positions. An FTR is a unidirectional financial instrument, defined in megawatts, from a source node to a sink node. One unit of an FTR entitles its holder a payment equal to the difference between the LMPs at the sink and the source nodes (see Apostolopoulou et al. (2013), Bushnell (1999)). We focus on FTR positions that negatively impact CTS.

Denote by , the FTR megawatt position of CTS bidder from an internal node inside area to the CTS trading location. Let denote the LMP at node in area . Recall that we have assumed so far that has an affine dependence on , the amount that flows from bus to bus . In general, will also depend on . Assume a similar affine dependence

for an internal node . Albeit simplistic, this model is enough to reveal the impact of FTRs on CTS markets. The payoff of bidder from her FTR positions then becomes , where the sum is taken with ranging over buses within area . To illustrate the coupling between FTR positions and CTS market, consider the joint payoff from them for bidder in

(15)

where depends on CTS market clearing with bids and liquidities . Formally, call this game with payoffs in (15). Here, , , collect the respective variables across all internal buses. Our next result characterizes the market outcome with FTR positions.

Proposition 2

The game admits a unique Nash equilibrium if is elementwise nonnegative, for which the tie-line schedule at the equilibrium is

where for and is the only player with maximal .

Our proof again appeals to Rosen’s result and the rest relies on analyzing the first-order conditions for equilibrium. The result reveals that the bidder with maximum combined CTS and FTR position emerges as the pivotal player in this market. Moreover, dictates that less power is scheduled to flow in the tie-line when bidders have such FTR positions. This results from the incentives of the pivotal player who benefits from higher prices at the importing region ’s proxy bus as that yields a higher FTR payoff. In fact, the difference in the tie-line schedules with and without FTR grows with the difference between and that is directly proportional to the FTR positions. Opposite conclusions can be drawn if we consider players with FTR positions that source at area ’s proxy bus.

The following example illustrates the shift in market power and scheduling efficiency when participants hold FTRs. Consider the CTS market in Section 4.1 where the fifth bidder is pivotal in the intermediate liquidity regime. At the equilibrium, MW. Assume that the first bidder holds an FTR MW to an internal bus for which and , while the rest of players do not have FTR portfolios. Then, . Notice that bidder one emerges as the new marginal bidder and has incentive to bid in a way that leads to less power being scheduled to flow into area . Indeed, the new tie-line schedule is MW, MW less than CTS without FTRs, falling even shorter of MW.

6 Impact of Forecast Errors and Transaction Costs

Our analysis of the CTS game so far has assumed that players and the SOs have perfect forecasts into the price spread function. In practice, tie-line scheduling takes place with a lead time to power delivery, meaning that there is an inherent uncertainty in the price spread when these markets are convened. To model this uncertainty, assume that the SOs conjecture an affine price spread function

with . The SOs use this spread to clear the CTS market as in (3). Let the realized price difference be

with . Then, the TO schedule and the optimal tie-line schedule, respectively, are given by

Modeling the uncertainty explicitly at the time of scheduling reveals that may not equal , the ex-post optimal tie-line schedule. Our interest lies in analyzing if strategic behavior of bidders in the CTS market can correct the errors in SOs’ forecasts. Do bidders draw the outcome closer to than or do they drive it further away as a result of their strategic interaction? We answer this question through a game-theoretic study. We also derive insights into how non-zero transaction fees () affect these conclusions.

To isolate the impacts of uncertainty and transaction fees, we analyze the game under a simpler setting where the bidders are homogenous, each with liquidity and conjectured price spread with . Notice that bidders’ conjectured optimal schedule may be different from both and . We assume here that players share a common belief that the market operates at an intermediate liquidity where the aggregate liquidity is close to her conjectured optimal tie-line schedule , i.e.,

(16)

Under such an assumption, bidder conjectures the market price from bidding with liquidities to be

which yields the following perceived payoff for bidder .

(17)

Call the CTS game with conjectured price spreads , where satisfy (16) and the payoffs are given by (17). Assuming that all players offer based on an equilibrium profile for this game, the SOs then solve the CTS flow allocation problem in (3) with to ultimately compute the tie-line schedule. Our next result characterizes both a (symmetric) equilibrium profile and the resulting tie-line schedule.

Proposition 3

The CTS game admits a unique symmetric Nash equilibrium given by for , for which the tie-line schedule at equilibrium is

where .

Our proof leverages the result in (Chen et al. 2004, Theorem 3) and the analysis of first-order conditions for a symmetric equilibrium of the game. Notice that players bid solely based on their own conjectures. The tie-line schedule, however, depends on the conjectures of both the bidders and the SOs. This result will allow us to study the effect of price spread forecasts and transaction costs on the scheduling efficiency in the sequel.

The lack of knowledge of by the SOs and market participants prompts us to investigate whether CTS can yield a more efficient schedule than the pure SO-driven TO. Proposition 3 implies , meaning that CTS cannot yield a more efficient schedule than TO if . Hence, CTS can only outperform TO when the SOs’ forecast overestimates . In this regime, Figure 1 yields that is always closer to when . Outside of this setting, the outcome of CTS depends on the liquidity and conjectures of players. Specifically, if , defined in Figure 5, is closer to than , if

(18)

Such a premise appears to run counter to the intuition that TO is optimal. This situation can only arise under uncertainty where SOs make serious forecast errors in the expected price spread. Surprisingly, forecast errors are not that rare, according to Economics (2019), where the error in SOs’ point forecast for the price spread between NYISO and ISO-NE averaged MWh. Notice how, in this liquidity regime, the presence of transaction fees makes it harder to satisfy (18). This is intuitively correct since transaction fees drive the tie-line schedule toward smaller values, as established in Proposition 3.

Figure 5: Ability of market participants to correct SO’s forecast error depends on liquidity and transactions costs.

When , liquidity is sufficiently high and the presence of costs might improve scheduling efficiency since players bid higher prices to counter costs. Overall, players ability to correct SOs’ forecast is somewhat limited and relies on many qualifications, indicating that the SOs forecasts and systematic bias plays a vital role in scheduling efficiency. Moving bid submittal and clearing timelines closer to power delivery should improve the efficiency of CTS.

Figure 6: The trajectory of CTS schedules cleared against SO’s forecasted prices with error with and MWh.

Proposition 3 suggests that incentives of CTS bidders are aligned in a way that allows them to correct SOs’ forecast errors in some settings. Can players learn such equilibria through repeated play. We employ the learning framework in Section 4.1, where players have their bids cleared against that are perturbed from learned from historical data. That is, in every round, bidders receive reward from the ex-post price spread described by . The trajectory of tie-line schedules in Figure 6 with reveals that bidding behavior of players results in CTS schedules consistently closer to the ex-post optimal than TO. Despite the SO’s persistent forecast error, bidders ‘correct’ the tie-line schedule to an extent by seeking actions that maximize their observed reward.

The relation in (18) reveals that presence of nonzero transaction fees make it more difficult for CTS market to drive the outcome closer to the ex-post optimal as increases with . Bidders reacting to observed rewards with /MWh in Figure 6 yield a CTS schedule farther from , seeking actions that yield higher prices but smaller schedules. This result corroborates our theoretical finding that transaction fees impede bidders’ ability to correct SOs’ forecast errors.

(a) Subfigure 1 list of figures text
(b) Subfigure 2 list of figures text
Figure 7: Plot (a) depicts the time series of spread between NYISO and PJM proxy buses in 2018 (absolute mean = 8.92 $/MWh, std. deviation = 22.11 $/MWh). Plot (b) shows the same between NYISO and ISO-NE for the same year (mean = 0.44, absolute mean = 5.59 $/MWh, std. dev. = 18.14 $/MWh).

Notice that equilibrium bid grows with , per Proposition 3. With , bidders are reluctant to offer their entire liquidity. A similar result can be shown under more general settings of Theorem 1. This may prevent the price spread from converging to zero, even if the market is liquid. And, transaction fees make it less attractive for CTS bidders overall, hurting long-term liquidity of the CTS market. Figure (a)a indicates that the price spread in the CTS market between NYISO and PJM exhibits longer excursions from zero and higher volatility compared to that between NYISO and ISO-NE, depicted in Figure (b)b. The average absolute spread between NYISO and PJM is approximately $3.3/MWh higher than that between NYISO and ISO-NE. We surmise that transaction fees between NYISO and PJM and the lack thereof between NYISO and ISO-NE are largely responsible for this difference.

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Appendix A Proof of Theorem 1.

The proof aims to establish that is a concave N-person game and there exists a Nash equilibrium. First, we show that (9) is a sufficient condition for to be a convex function in . We utilize this result to prove that is continuous and convex in . For ease of exposition, throughout the proof we drop the dependence on and the subscript from . From (6) we have

(19)

We have and is continuous and strictly decreasing in . Hence, is invertible and its inverse is continuous and crosses at a unique value. Hence, is continuous in . To make the first term of (8) concave in for fixed, we require to be convex or

(20)

Then, is the inverse of a decreasing, convex function and is therefore convex itself. Moreover, given that and are continuous, then is continuous in . From (5) and (19), we can rewrite the CTS allocation as

with its first derivative satisfying

(21)

Therefore, the CTS allocation is strictly decreasing in . It remains to show that is convex in for fixed. Computing the second derivative we have

(22)

The last term in (22) is nonnegative by convexity of . Therefore, we require the sum of the remaining terms to be non-negative or

Differentiating both sides of (19) with respect to we have

(23)

Furthermore, from (19) we have

(24)

Note that since the price spread is decreasing in the tie-line schedule. Hence, is convex in for every player given . As such, the cost term is convex in . Moreover, is concave in since the price spread is concave and decreasing in and is convex in . It follows that the payoffs given by (8) is continuous in and concave in .

Notice that is decreasing in and approaches negative infinity as grows unbounded. Hence, there exists a unique threshold value above which the transport offer becomes negative. As such, players have no incentive to bid since that yields negative payoff. Consider the game with strategy spaces restricted to the compact interval . Then, any Nash equilibrium of is a Nash equilibrium of as well. We have established that is a concave N-person game where the strategy space of each player is a compact, convex, nonempty subset of . Applying Rosen’s existence theorem [Rosen, 1965, Theorem 1], we conclude that a Nash equilibrium exists for and therefore for as well.

Appendix B Proof of Proposition 1

Existence of the Nash equilibrium follows from Theorem 1. Solving (3) we find that is given by

(25)

The payoff for player is given by

(26)

The payoff is continuous in and strictly concave in . Note that becomes negative for . Hence, we restrict our attention to for a Nash equilibrium in the compact interval . A bid profile is a Nash equilibrium if and only if

(27a)
(27b)

where the above derivative is given by

(28)

From (28) we deduce that the payoff derivative cannot vanish for more than one player. Moreover, no player would bid since that yields negative payoff and each player profitably deviates by infinitesimally decreasing . From the previous discussion and the following observation

(29)

we conclude that . In search for positive we find that

  • If , then

    (30)
  • Otherwise, since (30) yields a negative value.

To prove the bounds on first note that the social welfare attains its maximum at with

(31)

Hence, in the high liquidity regime, i.e., , and . In the intermediate regime, the social welfare at is