The Role of Strategic Load Participants in Two-Stage Settlement Electricity Markets
TThe Role of Strategic Load Participants in Two-Stage SettlementElectricity Markets
Pengcheng You, Dennice F. Gayme, and Enrique Mallada
Abstract
Two-stage electricity market clearing is designed to maintain market efficiency under ideal conditions, e.g., perfect forecastand nonstrategic generation. This work demonstrates that the individual strategic behavior of inelastic load participants in atwo-stage settlement electricity market can deteriorate efficiency. Our analysis further implies that virtual bidding can play arole in alleviating this loss of efficiency by mitigating the market power of strategic load participants. We use real-world marketdata from New York ISO to validate our theory.
I. I
NTRODUCTION
Electricity markets are designed to complement physical power systems by utilizing prices or other monetary incentives tomotivate efficient system operation. Wholesale electricity markets generally consist of two-stage settlement. The first stageis a day-ahead market where participants buy or sell electricity through bids or offers on an hourly basis. An independentsystem operator (ISO) determines the hourly generation and load schedules along with the corresponding day-ahead clearingprices for the next day. The second stage is a real-time market where participants trade in the same way at the real-timeclearing prices on a smaller timescale, usually every five minutes, to offset any discrepancy between day-ahead commitmentsand actual generation/load.The day-ahead and real-time markets are tightly coupled via time-varying supply, demand and prices [1]. The two-stagesettlement is designed to maintain equal day-ahead and real-time prices such that no speculator is able to perform arbitrage,i.e., to enforce the so-called no-arbitrage condition. However, the two stages are settled separately in practice and identicalprices in the day-ahead and real-time markets are therefore not directly enforced [2]. The difference between a day-aheadprice and its real-time counterpart is technically termed a price spread . Any nonzero spread is generally considered a loss ofefficiency [3]. Situations that result in systematic nonzero spreads include external factors, such as load forecast errors [4],non-scheduled generator shutdowns or line maintenance, as well as internal market power generally exercised by strategicgenerators [5].Transactions that are not intended for physical fulfillment in real time but holding financial positions for arbitrage arereferred to as virtual bids . Virtual bids primarily consist of decrement bids that buy electricity in the day-ahead market withthe obligation to sell back the same amount in the real-time market, as well as increment offers that work exactly in theopposite way [6]. See [7]–[10] for various examples of virtual bidding strategies. Virtual bidding is a valuable componentof the two-stage settlement design that contributes to increasing market liquidity and mitigating market power by allowingextra asset-free participants to compete in electricity markets. This practice has proven, through both real observation [11]and theoretical analysis [3], [12]–[14], to improve market efficiency by driving day-ahead and real-time prices to converge.Despite the aforementioned studies, little attention has yet been paid to the strategic behavior of load participants inelectricity markets, which may also play a role in degrading market efficiency. The load side is usually less regulated due toits inelasticity, which leaves load participants more freedom to make strategic decisions. Conceptually, even with inelasticdemand, a load participant enjoys the flexibility of two-stage settlement, which potentially enables it to exercise marketpower.In this paper, we look at the role of strategic inelastic load participants that take advantage of the two-stage settlementmechanism. We first establish a simple two-stage settlement market model that assumes (fully regulated) nonstrategicgeneration to characterize the inherent connection between the day-ahead and real-time markets. The strategic behaviorof load participants is then analyzed through a Cournot game. We further extend the framework to accommodate decrementbids in virtual bidding as a special case of strategic inelastic load participation in electricity markets. Real-world marketdata from New York ISO (NYISO) are employed for validation.Our analysis has multiple implications.
First , the proposed market model unveils the underlying mechanism that relates theno-arbitrage condition with market efficiency while maintaining realistic market settlement conditions such as the day-aheadcleared load being approximately equal to the total load for efficiency.
Second , we identify adverse impacts of strategicbehavior by inelastic load participants that induces negative spreads and deteriorates efficiency in electricity markets, despiteperfect forecast and nonstrategic generation.
Third , we show that virtual bidding is an effective solution to alleviating theloss of market efficiency caused by strategic load participants.
This work was supported by ARO through contract W911NF-17-1-0092, US DoE EERE award de-ee0008006, NSF through grants CNS 1544771, EPCN1711188, AMPS 1736448 and CAREER 1752362, and Johns Hopkins University Discovery Award.P. You, D. F. Gayme and E. Mallada are with the Whiting School of Engineering, Johns Hopkins University, Baltimore, MD 21218, USA { pcyou,mallada,dennice } @jhu.edu a r X i v : . [ m a t h . O C ] S e p he rest of the paper is organized as follows. Section II introduces our electricity market model. The role of strategicbehavior by inelastic load participants is then analyzed in Section III. Empirical validation using real-world data follows inSection IV. Section V concludes the paper.II. E LECTRICITY M ARKETS M ODEL
In this section we describe the proposed electricity market model for the two-stage settlement mechanism. Consider anelectricity market that consists of a day-ahead market and a real-time market. Assume that the generation side is highlyregulated and all generators are non-strategic, i.e., they reveal their true cost functions . The generators are categorized intotwo sets based on whether they are sufficiently fast to participate in the real-time market. Let F and S respectively denotethe sets of fast-responsive and slow-responsive generators. Slow-responsive generators can only participate in the day-aheadmarket while fast-responsive generators are able to participate in both markets. For a fast-responsive generator i ∈ F thatoutputs an amount of power x fi ≥ , we assume a quadratic cost function of the form C fi ( x fi ) := α fi x f i + β fi x fi , (1)where α fi > and β fi are constant cost coefficients. Similarly, we denote the cost function of a slow-responsive generator j ∈ S by C sj ( x sj ) := α sj x s j + β sj x sj , (2)where x sj ≥ , α sj > and β sj are defined accordingly. We then define the associated vectors as x f := ( x fi , i ∈ F ) and x s := ( x sj , j ∈ S ) . A. Two-Stage Settlement
The two-stage settlement mechanism meets a total inelastic load of d > by clearing it efficiently but separately in theday-ahead and real-time markets. Denote the day-ahead cleared portion as d DA and the real-time cleared portion as d RT such that d = d DA + d RT . In the slow-timescale day-ahead market, all of the generators in F and S are involved to clearthe load d DA based on the following: Day-ahead market clearing problem min x f ,x s ≥ (cid:88) i ∈F C fi ( x fi ) + (cid:88) j ∈S C sj ( x sj ) (3a) s . t . (cid:88) i ∈F x fi + (cid:88) j ∈S x sj = d DA : λ DA , (3b)where λ DA is the dual Lagrange multiplier for the equality constraint (3b). Due to strong convexity, (3) has a uniqueminimizer which we denote as ( x f ∗ , x s ∗ ) . Since (3b) is affine, the KKT conditions suggest that all of the participatinggenerators should have an identical marginal cost that equals the optimal dual Lagrange multiplier : λ DA = α fi x f ∗ i + β fi = α sj x s ∗ j + β sj , ∀ i ∈ F , j ∈ S , (4)where we abuse λ DA to denote its optimum. λ DA , technically termed the shadow price in economics [15], is the minimumprice to incentivize generators to output the desired amount of power, which captures marginal generation cost.Combining (3b) and (4) results in λ DA = α DA d DA + β DA , (5a)where α DA := (cid:88) i ∈F α fi + (cid:88) j ∈S α sj − , (5b)and β DA := (cid:88) i ∈F α fi + (cid:88) j ∈S α sj − (cid:88) i ∈F β fi α fi + (cid:88) j ∈S β sj α sj . (5c) In real electricity markets, piecewise linear generation offers are made as a proxy for true generation cost functions, which are assumed to be knownby the ISO here for ease of analysis. For illustration purposes, throughout this paper we restrict our considerations to the case where the constraints x fi ≥ , i ∈ F and x sj ≥ , j ∈ S arenot binding. ere α DA and β DA serve as the aggregate pricing coefficients. The expressions in (5) implicitly reflect the elasticity ofsupply , defined as the responsiveness of the quantity of power supplied to a change in its price, in the day-ahead market.Basically, given the market price λ DA , the generators are willing to output a total amount of power d DA . In other words,to clear the load d DA in the day-ahead market, the clearing price needs to be set at λ DA .The fast-timescale real-time market clears in the same way as the day-ahead market except that only fast-responsivegenerators in F are involved. Note that these generators have also participated in the day-ahead market and have alreadybeen scheduled to output x f ∗ . Therefore, in order to clear the load d RT , the real-time market solves the following optimizationproblem: Real-time market clearing problem min δx f (cid:88) i ∈F C fi ( x f ∗ i + δx fi ) (6a) s . t . (cid:88) i ∈F δx fi = d RT : λ RT . (6b)Here δx fi denotes the scheduled output adjustment from x f ∗ for generator i ∈ F and δx f := ( δx fi , i ∈ F ) . λ RT is the(optimal) dual Lagrange multiplier for the equality constraint (6b). Note that the cost of fast-responsive generators in theday-ahead market, i.e., (cid:80) i ∈F ( α fi x f ∗ i + β fi x f ∗ i ) , should be subtracted from the objective function to represent the exacttotal cost for clearing the real-time load d RT . We ignore this constant term for brevity.We denote the unique minimizer of (6) as δx f ∗ and deduce the following from the KKT conditions: λ RT = α fi ( x f ∗ i + δx f ∗ i ) + β fi = α fi δx f ∗ i + λ DA , ∀ i ∈ F , (7)where the second equality follows directly from (4). Substituting (7) into (6b) yields λ RT = α RT d RT + β RT , (8a)where α RT := (cid:32)(cid:88) i ∈F α fi (cid:33) − (8b)and β RT := λ DA . (8c)Here α RT and β RT are the aggregate pricing coefficients that embody the elasticity of supply in the real-time market.Meanwhile, (8) also unveils the inherent correlation between the day-ahead and real-time prices: the latter deviates from theformer to account for the real-time cleared load. See Fig. 1. Formally, the price spread between the day-ahead and real-timeprices is λ DA − λ RT = − (cid:32)(cid:88) i ∈F α fi (cid:33) − d RT . (9) Day-ahead loadReal-time loadDay-ahead/real-time price Negative spread0 ! " $ " $ %& ' %& ' " ! %& ( %& Fig. 1: Correlation between day-ahead and real-time prices.
Remark Notably, according to (5), (8), we always have α RT > α DA > , as Fig. 1 illustrates, due to a smaller subsetof generators involved in the real-time market. This is consistent with the observation that real-time prices are more volatilehan day-ahead prices, since the real-time market typically has a smaller price elasticity of supply than the day-ahead market,i.e., the quantity of power supply in the real-time market is less sensitive to a change in its price than that in the day-aheadmarket. B. Market Efficiency
We next formalize our definition of market efficiency. Given all the available generators in F and S , we define marketefficiency as the minimum of social cost to meet the total inelastic load d , which is specifically realized by solving thefollowing: Social cost minimization problem min x f ,x s ≥ (cid:88) i ∈F C fi ( x fi ) + (cid:88) j ∈S C sj ( x sj ) (10a) s . t . (cid:88) i ∈F x fi + (cid:88) j ∈S x sj = d : λ, (10b)i.e., jointly optimizing the dispatch of all the generators across the two markets. We define the (optimal) dual Lagrangemultiplier λ for the equality constraint (10b) and denote the unique minimizer of (10) by ( x f , x s ) . The KKT conditionsrequire λ = α fi x f i + β fi = α sj x s j + β sj , ∀ i ∈ F , j ∈ S , (11)i.e., equal marginal cost for all of the participating generators, to achieve efficiency.Recall that the day-ahead price equals the marginal cost of slow-responsive generators in (4) while the real-time priceequals the marginal cost of fast-responsive generators in (7). By comparing (4) and (7) with the indicator of market efficiency(11), we arrive at the following theorem: Theorem In the two-stage settlement electricity market, efficiency can only be realized when λ DA = λ RT = λ (12)i.e., the day-ahead and real-time prices equalize, which further implies d DA = d, d RT = 0 . (13)Theorem 1 matches exactly the intuition of the two-stage settlement design: all (forecast) load should be cleared in the day-ahead market while the real-time market accounts for any load deviation from the forecast. It also suggests that efficiency isconsistent with the no-arbitrage condition between the two-stage markets, guaranteed by the zero spread from (12), whichis necessary for the market model to be realistic.Similar models for the two-stage settlement mechanism have been used in [4], [16], [17]. However, our simple modelfurther addresses several issues that are missing in these previous works, e.g., the fact that the day-ahead cleared loadshould equal the total load is not accounted for in [16], [17]; the correlation between the no-arbitrage condition and marketefficiency is not demonstrated in [4]. III. S TRATEGIC L OAD P ARTICIPANT
Given the two-stage settlement mechanism, an electricity market should clear all of the load in the day-ahead market andzero load in the real-time market in order to achieve efficiency. However, we observe that in the NYISO market there is anobvious positive bias for real-time loads throughout the year of 2018, as shown in Fig 2, which cannot be accounted forby uncertainties. We attribute this loss of efficiency to strategic behavior by inelastic load participants and next investigatetheir market power by taking advantage of the two-stage settlement mechanism. Ideal assumptions of perfect forecast andnonstrategic generation are made to focus our attention on the impact of strategic load. As we will see below, our analysisextends naturally to accommodate the role of virtual bidding.
A. Single Load
We start with the simplest case where there is only one single inelastic load d to be cleared. It has the option to participatein either one of or both of the day-ahead and real-time markets to meet its demand. The participation of the load in thetwo markets affects the market clearing prices, which in turn determine its cost. For analysis purposes, we assume the loadhas full knowledge of the supply elasticity of both markets, i.e., it knows the exact values of α DA , β DA , α RT , β DA , e.g.,through estimates based on long-term historical data . Since the set of participating generators in an electricity market and their cost functions are usually stable subject to subtle changes in the long run, itis reasonable to argue those coefficients that characterize the dependence of market prices on the amount of cleared load are approximately constant andeasy to estimate. eb Apr Jun Aug Oct Dec
Time of year 2018 -50510152025
Load ( G W ) Day-ahead loadReal-time load
Fig. 2: Day-ahead and real-time cleared loads, NYISO, 2018.A strategic load will anticipate the impact of its decision on the two markets and minimize the expenditure of purchasingelectricity to meet its demand accordingly. Formally, it solves the following:
Single load cost minimization problem min d DA ≥ ,d RT λ DA ( d DA ) · d DA + λ RT ( d DA , d RT ) · d RT (14a) s . t . d DA + d RT = d. (14b) Theorem The optimal load participation for a single load in the two-stage settlement electricity market is uniquelydetermined by d DA ∗ = (cid:18) − α DA α RT (cid:19) d, d RT ∗ = α DA α RT d, (15)i.e., d > d DA ∗ > d > and d > d RT ∗ > . Therefore, λ RT > λ DA and a strictly negative spread, as defined in (9),follow. Proof:
First of all, we relax the constraint d DA ≥ and show that it is not binding at the optimum. Given the explicitexpressions of λ DA ( d DA ) and λ RT ( d DA , d RT ) in (5) and (8), we can substitute (14b) into (14a) to reorganize the objectivefunction in terms of d DA only: λ DA ( d DA ) · d DA + λ RT ( d DA , d RT ) · d RT = λ DA ( d DA ) · d DA + ( α RT d RT + λ DA ( d DA )) · d RT = ( α DA d DA + β DA ) d + α RT ( d − d DA ) = α RT d DA + ( α DA d − α RT d ) d DA + β DA d + α RT d . The unique minimizer of the above unconstrained optimization is straightforwardly obtained by the first-order optimalitycondition, i.e., (15). Recall α RT > α DA > , which implies d > d DA ∗ > d > and d > d RT ∗ > . The relaxedconstraint is satisfied and (15) is also the unique optimum of (14). Remark The negative spread indicates the loss of market efficiency caused by the strategic behavior of a singleinelastic load participant in the two-stage settlement electricity market. Meanwhile, the strictly positive load participation inthe real-time market coincides with the observation of positive bias for real-time loads in Fig. 2.The single-load case serves as a toy example. Next we proceed to characterize the general case with market competitionand analyze its impact on efficiency.
B. Load-Side Cournot Competition
We extend the above analysis to the case with multiple individual strategic loads, e.g., different local utility companies ina market. Let L := { , , . . . , L } be the set of these loads. Each load l ∈ L can independently determine its participation, (cid:126)d l := ( d DAl ≥ , d RTl ) , in the day-ahead and real-time markets in order to satisfy its inelastic demand d l with d DAl + d RTl = d l , l ∈ L . (16) Note that on the load side we constrain nonnegative load participation in the day-ahead market to maintain the identity as a load. et (cid:126)d := ( (cid:126)d l , l ∈ L ) be the aggregate decisions for all of the loads. Further denote the aggregate decisions for all of theloads except load l as (cid:126)d − l . Suppose that all of the loads are aware of the mechanism that determines market prices, i.e., λ DA = α DA d DA + β DA , λ RT = α RT d RT + β RT , (17)where d DA := (cid:80) l ∈L d DAl and d RT := (cid:80) l ∈L d RTl . Define d := (cid:80) l ∈L d l as the total load to be cleared. Each load l willaim to minimize its expenditure of purchasing electricity from the two markets to meet demand given other loads’ decisions,i.e., min (cid:126)d l c l ( (cid:126)d l ; (cid:126)d − l ) := λ DA ( (cid:126)d ) · d DAl + λ RT ( (cid:126)d ) · d RTl (18a) s . t . (16) . (18b)These loads compete in quantities of participation in the two markets that affect market clearing prices and seek tominimize individual cost, which can be formalized as a Cournot game: Load-side Cournot game
Players : each load l ∈ L ; Strategies : load participation (cid:126)d l in the day-ahead and real-time markets to satisfy (16); Costs : expenditure of purchasing electricity c l ( (cid:126)d l ; (cid:126)d − l ) . Definition (cid:126)d ∗ is a Nash equilibrium of the load-side Cournot game if it satisfies c l ( (cid:126)d ∗ ) ≤ c l ( (cid:126)d l ; (cid:126)d ∗− l ) for any (cid:126)d l , ∀ l ∈ L .At a Nash equilibrium, no load has the incentive to deviate from its current decision unilaterally, given other loads’ decisions.In order to characterize the Nash equilibrium of the load-side Cournot game, we first propose the following lemma: Lemma There do not exist equilibria of the load-side Cournot game where d DA ∗ l = 0 for some l ∈ L .Refer to the appendix for the proof. Given Lemma 1, the possibility of Nash equilibria with any of the constraints d DAl ≥ , l ∈ L binding is excluded and we next prove the existence and uniqueness of the Nash equilibrium of the load-sideCournot game by ignoring these constraints: Theorem In the two-stage settlement electricity market, there exists a unique Nash equilibrium of the load-side Cournotgame, characterized by d DA ∗ l = (cid:18) − Lα DA ( L + 1) α RT (cid:19) d l + α DA ( L + 1) α RT (cid:88) k ∈L\{ l } d k ,d RT ∗ l = Lα DA ( L + 1) α RT d l − α DA ( L + 1) α RT (cid:88) k ∈L\{ l } d k , (19) for ∀ l ∈ L . Proof:
Given (16) and (17), the expenditure function c l ( (cid:126)d l ; (cid:126)d − l ) of each load l in (18) can be rewritten explicitly interms of d DAl only as follows. λ DA d DAl + λ RT d RTl = ( α DA (cid:88) k ∈L d DAk + β DA ) d DAl + ( α RT ( d − (cid:88) k ∈L d DAk ) + β RT )( d l − d DAl )= ( α DA (cid:88) k ∈L d DAk + β DA ) d l + α RT ( d − (cid:88) k ∈L d DAk )( d l − d DAl )= α DA d l d DAl + α RT (cid:88) k ∈L\{ l } ( d k − d DAk )( d l − d DAl ) + α RT ( d l − d DAl ) + α DA (cid:88) k ∈L\{ l } d DAk d l + β DA d l , where the second equality follows from β RT = λ DA . Given Lemma 1 and the strict convexity of the expenditure function c l ( (cid:126)d l ; (cid:126)d − l ) in d DAl , the Nash equilibrium of the load-side Cournot game can be characterized by imposing the first-orderoptimality condition on all the loads, i.e., for ∀ l ∈ L , α DA d l − α RT (cid:88) k ∈L\{ l } ( d k − d DA ∗ k ) − α RT ( d l − d DA ∗ l ) = 0 , (20)or equivalently, d DA ∗ l = (cid:18) − α DA α RT (cid:19) d l + 12 (cid:88) k ∈L\{ l } ( d k − d DA ∗ k ) . (21)Note that the first term on the right-hand side of (21) is exactly the individual optimum without any competitors in (15),while the second term represents the influence of competition. Intuitively, if other loads participate more in the real-timemarket, load l will increase its participation in the day-ahead market to hedge against rising real-time prices.Combining (21) for all l ∈ L naturally yields the unique solution (19). We can readily observe d DA ∗ l > , which isconsistent with Lemma 1. The theorem follows.y summing (19) over L and reorganizing the expression, we are able to derive the following: Corollary At the Nash equilibrium of the load-side Cournot game, the total day-ahead load and real-time load arerespectively (cid:88) l ∈L d DA ∗ l = (cid:18) − α DA ( L + 1) α RT (cid:19) (cid:88) l ∈L d l , (cid:88) l ∈L d RT ∗ l = α DA ( L + 1) α RT (cid:88) l ∈L d l , (22)which implies d > d DA ∗ > LL +1 d and L +1 d > d RT ∗ > . Therefore, λ RT > λ DA and a strictly negative spread follow. Remark Notably, the optimal load participation (15) in the single-load case is a special case of (22) where L = 1 .Corollary 1 generalizes the conclusion to multi-load cases, and specifically it states that the strategic behavior of loadparticipants even with inelastic demand can reduce market efficiency by taking advantage of the two-stage settlementmechanism. However, as the number of load participants L increases, the total day-ahead load approaches the total loadand the spread diminishes towards zero, meaning the restoration of market efficiency. This is consistent with the intuitionthat when there are infinite participants, the individual impact on market prices becomes negligible and therefore the marketpower of each strategic load vanishes, which drives the market to be competitive. C. The Role of Virtual Bidding
Virtual bidding is an essential part of competitive electricity markets as it mitigates market power. Virtual bidders profitfrom arbitrage on nonzero spreads. As analyzed above, we have demonstrated that systematic negative spreads can resultfrom the strategic behavior of load participants. However, through an extended analysis of the prior load-side Cournotcompetition, we now show decrement bids in virtual bidding that act like load participation play an important role in drivingthese spreads to zeros.In particular, consider a set V := { , , . . . , V } of virtual bidders. They individually determine their participation ( d DAv , d
RTv ) , v ∈ V to compete in the day-ahead and real-time markets in pursuit of arbitrage. However, they differ from realload participants l ∈ L in that no real demand needs to be satisfied, i.e., d v = 0 , v ∈ V . The following theorem characterizesthe involvement of these virtual bidders in the load-side Cournot game: Theorem In the two-stage settlement electricity market, there exists a unique Nash equilibrium of the load-side Cournotgame with virtual bidders, where the virtual bids are given by d DA ∗ v = α DA ( L + V + 1) α RT (cid:88) l ∈L d l ,d RT ∗ v = − α DA ( L + V + 1) α RT (cid:88) l ∈L d l , (23)for ∀ v ∈ V .Here d DA ∗ v > represents a decrement bid. Corollary At the Nash equilibrium of the load-side Cournot game with virtual bidders, the total day-ahead load andreal-time load are respectively (cid:88) l ∈L d DA ∗ l + (cid:88) v ∈V d DA ∗ v = (cid:18) − α DA ( L + V + 1) α RT (cid:19) (cid:88) l ∈L d l , (cid:88) l ∈L d RT ∗ l + (cid:88) v ∈V d RT ∗ v = α DA ( L + V + 1) α RT (cid:88) l ∈L d l , (24) which implies d > d DA ∗ > L + VL + V +1 d and L + V +1 d > d RT ∗ > . As the number of virtual bidders V goes to infinity, thetotal day-ahead load approaches the total load and the spread converges to zero. Remark Virtual bidders have the incentive to arbitrage over the negative spread resulting from the strategic behaviorof load participants, which in turn contributes to alleviating the loss of market efficiency by driving the two-stage marketprices to equalize.
Remark From (24), the real demand from load participants in the day-ahead market remains positive but actuallydecreases in the number of virtual bidders V , as captured below: (cid:88) l ∈L d DA ∗ l = (cid:18) − ( V + 1) α DA ( L + V + 1) α RT (cid:19) (cid:88) l ∈L d l . (25)ABLE I: Linear regression for λ DA = α DA d DA + β DA Estimate Standard error p -value RMSE R α DA < β DA < TABLE II: Linear regression for λ RT = α RT d RT + γλ DA + δ Estimate Standard error p -value RMSE R α RT < γ < δ -8.2569 0.4980 < IV. R
EAL -W ORLD D ATA V ALIDATION
We next employ real-world electricity market data from NYISO to verify the extent to which our model and analysisreflect real market conditions.
A. Electricity Market Model
Day-ahead and real-time loads and prices are collected for the whole year of 2018 . Uniform energy clearing prices areadopted instead of locational marginal prices since emphasis of our analysis is on the two-stage settlement mechanism ratherthan the physical constraints of power networks. Fig. 3 is a scatterplot of day-ahead prices with respect to day-ahead loads.As (5) suggests, a day-ahead price should be linear in the corresponding day-ahead load. The linear regression result inTable I shows that both of the pricing coefficients α DA and β DA are statistically significant. Fig. 4 is a scatterplot of negativespreads, i.e., λ RT − λ DA , with respect to real-time loads to justify the connection between the day-ahead and real-timeprices, identified in (9). A multiple linear regression of real-time prices on real-time loads and day-ahead prices is carriedout yielding the result in Table II, which confirms that the linearity approximately holds. As analyzed, the coefficient γ forday-ahead prices is almost 1 and α RT > α DA is observed. However, the proposed model cannot account for the negativeintercept δ . This could be caused by factors that our analysis neglects, such as strategic generation. Day-ahead load (GW) D a y - ahead p r i c e ( $ / M W h ) Fig. 3: Day-ahead price with respect to day-ahead load, NYISO, 2018.
B. Virtual Bidding
To validate our analysis of strategic load participants, we assess the special case of virtual bidding due to its significantand verifiable impact on market clearing. The mechanism of virtual bidding was officially introduced into the NYISO marketin November, 2001 [18]. We collected available data of real loads cleared in the day-ahead market and total actual loadsfor the several months around that time point to validate the deduction in Remark 5. It is reasonable to assume V = 0 prior Note that several periods of time, such as Jan. 1-9 and May 21-31, that exhibit extremely abnormal price elasticity of supply are removed.
Real-time load (GW) -200-150-100-50050100150200 N ega t i v e s p r ead ( $ / M W h ) Fig. 4: Negative spread with respect to real-time load, NYISO, 2018.to the introduction of virtual bidding while
V > thereafter. As a result, the proportion − ( V +1) α DA ( L + V +1) α RT is anticipated todiminish with virtual bidding introduced, which is precisely captured by the sudden drop in Fig. 5 . Jul Sep Nov Jan Mar
Time of years 2001/2002 P e r c en t ( % ) Fig. 5: Percentage of day-ahead real load in total load, NYISO, 2001/2002. The introduction of virtual bidding in November,2001 caused a sudden drop in this percentage. V. C
ONCLUDING R EMARKS
This paper develops a model for two-stage settlement electricity markets that explicitly characterizes the interconnectionbetween day-ahead and real-time markets. Given the model, we attribute systematic negative spreads in electricity marketsto the strategic behavior of inelastic load participants that takes advantage of the two-stage settlement mechanism. Wetherefore argue that strategic load participation in electricity markets should be taken into account in the characterizationof nonzero spreads, in addition to empirical factors like load forecast errors or market power of strategic generators. Ouranalysis generalizes to accommodate virtual bidding and demonstrates its role in improving market efficiency by mitigatingmarket power. Real-world market data from NYISO support our theory.Our model and analysis focus on strategic behavior by inelastic load participants only and are thus not able to account forother factors that can also result in loss of market efficiency. A more comprehensive framework is the subject of ongoingwork. R
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