A Stochastic Electricity Market Clearing Formulation with Consistent Pricing Properties
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Operations Research manuscript
A Stochastic Electricity Market Clearing Formulationwith Consistent Pricing Properties*
Victor M. Zavala
Department of Chemical and Biological Engineering, University of Wisconsin-Madison1415 Engineering Dr, Madison, WI 53706 { [email protected] } Kibaek Kim, Mihai Anitescu
Mathematics and Computer Science Division, Argonne National Laboratory9700 South Cass Avenue, Argonne, IL 60439 { [email protected],[email protected] } John Birge
The University of Chicago Booth School of Business5807 South Woodlawn Avenue, Chicago, IL 60637 [email protected]
We argue that deterministic market clearing formulations introduce arbitrary distortions between day-aheadand expected real-time prices that bias economic incentives and block diversification. We extend and analyzethe stochastic clearing formulation proposed by Pritchard et al. (2010) in which the social surplus functioninduces penalties between day-ahead and real-time quantities. We prove that the formulation yields pricedistortions that are bounded by the bid prices, and we show that adding a similar penalty term to trans-mission flows and phase angles ensures boundedness throughout the network. We prove that when the pricedistortions are zero, day-ahead quantities converge to the quantile of real-time counterparts. The undesiredeffects of price distortions suggest that stochastic settings provide significant benefits over deterministic onesthat go beyond social surplus improvements. We propose additional metrics to evaluate these benefits.
Key words : stochastic, electricity, network, market clearing, pricing
History : This paper was first submitted on May, 2014; last revised on October 25, 2015.
1. Introduction
Day-ahead markets enable commitment and pricing of resources to hedge against uncertainty indemand, generation, and network capacities that are observed in real time. The day-ahead mar-ket is cleared by independent system operators (ISOs) using deterministic unit commitment (UC)formulations that rely on expected capacity forecasts, while uncertainty is handled by allocatingreserves that are used to balance the system if real-time capacities deviate from the forecasts. Alarge number of deterministic clearing formulations have been proposed in the literature. Repre-sentative examples include those of Carri´on and Arroyo (2006), Gribik et al. (2011), and Hobbs * Preprint ANL/MCS-P5110-0314 a r X i v : . [ q -f i n . E C ] O c t avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing Article submitted to
Operations Research ; manuscript no. (2001). Pricing issues arising in deterministic clearing formulations have been explored by Wanget al. (2012), Galiana et al. (2003), and O’Neill et al. (2005).In addition to guaranteeing reliability and maximizing social surplus, several metrics are mon-itored by ISOs to ensure that the market operates efficiently. For instance, as is discussed in Ott(2003), the ISO must ensure that market players receive economic incentives that promote par-ticipation (give participants the incentive to follow commitment and dispatch signals). It is alsodesired that day-ahead and real-time prices are sufficiently close or converge. One of the reasons isthat price convergence is an indication that capacity forecasts are effective reflections of real-timecapacities. Recent evidence provided by Bowden et al. (2009), however, has shown that persistentand predictable deviations between day-ahead and real-time prices (premia) exist in certain mar-kets. This can bias the incentives to a subset of players and block the entry of new players andtechnologies. The introduction of purely financial players was intended to eliminate premia, butrecent evidence provided by Birge et al. (2013) shows that this has not been fully effective. Onehypothesis is that virtual players can exploit predictable price differences in the day-ahead marketto create artificial congestion and benefit from financial transmission rights (Joskow and Tirole2000).Prices are also monitored by ISOs to ensure that they do not run into financial deficit (a situa-tion called revenue inadequacy) when balancing payments to suppliers and from consumers. Thisis discussed in detail in Philpott and Pritchard (2004). In addition, ISOs might need to use upliftpayments and adjust prices to protect suppliers from operating at an economic loss. This is neces-sary to prevent players from leaving the market. As discussed by O’Neill et al. (2005), Morales et al.(2012), and Wang et al. (2012); uplift payments can result from using incomplete characterizationsof the system in the clearing model. Such characterizations can arise, for instance, in the presenceof nonconvexities and stochasticity.Achieving efficient market operations under intermittent renewable generation is a challenge forthe ISOs because uncertainty follows complex spatiotemporal patterns not faced before (Constan-tinescu et al. (2011)). In addition, the power grid is relying more strongly on natural gas andtransportation infrastructures, and it is thus necessary to quantify and mitigate uncertainty inmore systematic ways (Liu et al. (2009), Zavala (2014)).
A wide range of stochastic formulations of day-ahead market clearing and operational UC proce-dures has been previously proposed. In operational UC models, on/off decisions are made in advance(here-and-now) to ensure that enough running capacity is available at future times to balancethe system. The objective of these formulations is to ensure reliability and maximization of social avala, Kim, Anitescu, and Birge:
Stochastic Market Clearing with Consistent Pricing
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Operations Research ; manuscript no. surplus (cost in case of inelastic demands) in intra-day operations. Examples include the works ofTakriti et al. (1996), Wang et al. (2008), Constantinescu et al. (2011), Jin et al. (2014), Ruiz et al.(2009), Bouffard et al. (2005), Papavasiliou and Oren (2013). These studies have demonstratedsignificant improvements in reliability over deterministic formulations. However, these works donot explore pricing issues.Stochastic day-ahead clearing formulations have been proposed by Kaye et al. (1990) and Wongand Fuller (2007). Kaye et al. (1990) analyse day-ahead and real-time markets under uncertaintyand argue that day-ahead prices should be set to expected values of the real-time prices. This price consistency ensures that the day-ahead market does not bias real-time market incentives inthe long run. Such consistency also avoids arbitrage as is argued by Khazaei et al. (2013).Wongand Fuller (2007) propose pricing schemes to achieve cost recovery for all suppliers (i.e., paymentscover the suppliers production costs). The pricing schemes, however, rely only partially on dualvariables generated by the stochastic clearing model which are adjusted to achieve cost recovery.Consequently, these procedures do not guarantee dual and model consistency.Morales et al. (2012) propose a stochastic clearing model to price electricity in pools with stochas-tic producers. Their model co-optimizes energy and reserves and they prove that it leads to revenueadequacy in expectation. In addition, they prove that prices allow for cost recovery in expectationfor all players (i.e., no uplifts are needed in expectation) but pricing consistency is not explored.Pritchard et al. (2010) propose a stochastic formulation that captures day-ahead and real-timebidding of both suppliers and consumers. The formulation maximizes the day-ahead social surplusand the expected value of the real-time corrections by considering the possibility of players’ biddingin the real-time market. The authors prove that the formulation leads to revenue adequacy inexpectation and provide conditions under which adequacy will hold for each scenario. The authorsdo not explore pricing consistency and economic incentives.Khazaei et al. (2013) propose a stochastic equilibrium formulation in which players bid param-eters of a quadratic supply function to maximize the expected value of their profit function whilethe ISO uses these parameters to solve the clearing model and generate day-ahead and real-timequantities and prices. It is shown that the equilibrium model generates day-ahead prices thatconverge to expected value prices and thus achieve consistency. It is also shown that day-aheadquantities converge to expected value quantities and a small case study is presented to demonstratethat the formulation yields higher social surplus and producer profits compared to deterministicclearing. The proposed formulation uses a quadratic supply function and quadratic penalties fordeviations between day-ahead and real-time quantities. No network and no capacity constraintsare considered. avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing Article submitted to
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Morales et al. (2014) propose a bilevel stochastic optimization formulation that uses forecastcapacities of stochastic suppliers as degrees of freedom. Using computational studies, they demon-strate that their framework provides cost recovery for all suppliers and for each scenario. Theauthors, however, do not discuss the effects of the modified pricing strategy on consumer payments(the demands are treated as inelastic) and no theoretical guarantees are provided. In particular, itis not guaranteed that a set of day-ahead capacities and prices exist that can achieve cost recoveryfor both suppliers and consumers in each scenario. While plausible, we believe that this requiresfurther evidence and theoretical justification.
In this work, we argue that deterministic formulations generate day-ahead prices that are distortedrepresentations of expected real-time prices. This pricing inconsistency arises because solving a day-head clearing model using summarizing statistics of uncertain capacities (e.g., expected forecasts)does not lead to day-ahead prices that are expected values of the real-time prices. We argue thatthese price distortions lead to diverse issues such as the need of uplift payments as well as arbitraryand biased incentives that block diversification. We extend and analyze the stochastic clearingformulation of Pritchard et al. (2010) in which linear supply functions for day-ahead and real-timemarkets are used. The structure of this surplus function has the key property that yields boundedprice distortions. We also prove that when the price distortion is zero, the formulation yields day-ahead quantities that converge to the quantile of their real-time counterparts. In addition, we provethat the formulation yields revenue adequacy in expectation and yields zero uplifts in expectation.We provide several case studies to demonstrate the properties of the stochastic formulation.The paper is structured as follows. In Section 2 we describe the market setting. In Section 3 wepresent deterministic and stochastic formulations of the day-ahead ISO clearing problem. In Section4 we present a set of performance metrics to assess the benefits of the stochastic formulation overits deterministic counterpart. In Section 5 we present the pricing properties of the formulation.In Section 6 we present case studies to demonstrate the developments. Concluding remarks anddirections of future work are provided in Section 7.
2. Market Setting
We consider a market setting based on the work of Pritchard et al. (2010) and Ott (2003). A set ofsuppliers (generators) G and consumers (demands) D bid into the day-ahead market by providingprice bids α gi ≥ i ∈ G and α dj ≥ j ∈ D , respectively. If a given demand is inelastic, we set thebid price to α dj = V OLL where
V OLL denotes the value of lost load, typically 1,000 $/MWh.Suppliers and consumers also provide estimates of the available capacities ¯ g i and ¯ d j , respectively. avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing
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Operations Research ; manuscript no. We assume that these capacities satisfy 0 ≤ ¯ g i ≤ Cap gi and 0 ≤ ¯ d j ≤ Cap dj where 0 ≤ Cap gi < + ∞ is the total installed capacity of the supplier (its maximum possible supply) and 0 ≤ Cap di < + ∞ is the total installed capacity of the consumer (its maximum possible demand). The cleared day-ahead quantities for suppliers and consumers are given by g i and d j , respectively. These satisfy0 ≤ g i ≤ ¯ g i and 0 ≤ d j ≤ ¯ d j .Suppliers and consumers are connected through a network comprising of a set of lines L and aset of nodes N . For each line (cid:96) ∈ L we define its sending node as snd ( (cid:96) ) ∈ N and its receiving nodeas rec ( (cid:96) ) ∈ N (we highlight that this definition of sending node is arbitrary because the flow cango in both directions). For each node n ∈ N , we define its set of receiving lines as L recn ⊆ L and itsset of sending lines as L sndn ⊆ L . These sets are given by L recn = { (cid:96) ∈ L | n = rec ( (cid:96) ) } , n ∈ N (1a) L sndn = { (cid:96) ∈ L | n = snd ( (cid:96) ) } , n ∈ N . (1b)Day-ahead capacities ¯ f (cid:96) are also typically estimated for the transmission lines. We assume thatthese satisfy 0 ≤ ¯ f (cid:96) ≤ Cap f(cid:96) . Here, 0 ≤ Cap f(cid:96) < + ∞ is the installed capacity of line (its maximumpossible value). The cleared day-ahead flows are given by f (cid:96) such that − ¯ f (cid:96) ≤ f (cid:96) ≤ ¯ f (cid:96) . The flows f (cid:96) are determined by the line susceptance B (cid:96) and the phase angle difference between two nodes of theline. Day-ahead capacities θ n , ¯ θ n are estimated for each node n ∈ N . The cleared day-ahead phaseangles are given by θ n such that θ n ≤ θ n ≤ ¯ θ n for n ∈ N . We define the set of all suppliers connectedto node n ∈ N as G n ⊆ G and the set of demands connected to node n as D n ⊆ D . Subindex n ( i )indicates the node at which supplier i ∈ G is connected, and n ( j ) indicates the node at which thedemand j ∈ D is connected. We use subindex i exclusively for suppliers and subindex j exclusivelyfor consumers.At the moment the day-ahead market is cleared, the real-time market conditions are uncertain.In particular, we assume that a subset of generation, demand, and transmission line capacities areuncertain. We further assume that discrete distributions comprising a finite set of scenarios Ω and p ( ω ) denote the probability of scenario ω ∈ Ω. We also require that (cid:80) ω ∈ Ω p ( ω ) = 1. The expectedvalue of variable Y ( · ) is given by E [ Y ( ω )] = (cid:80) ω ∈ Ω p ( ω ) Y ( ω ). If Y ( ω ) is scalar-valued, the quantilefunction Q is defined as Q Y ( ω ) ( p ) := inf { y ∈ R : P ( Y ( ω ) ≤ y ) ≥ p } . (2)Moreover, the median is denoted as M [ Y ( ω )] = Q Y ( ω ) (0 .
5) and satisfies M [ Y ( ω )] = argmin m E [ | Y ( ω ) − m | ] , (3) avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing Article submitted to
Operations Research ; manuscript no. where | · | is the absolute value function.In the real-time market, the suppliers can offer to sell additional generation over the agreed day-ahead quantities at a bid price α g, + i ≥
0. The additional generation is given by ( G i ( ω ) − g i ) + where G i ( ω ) is the cleared quantity in the real-time market and 0 ≤ ¯ G i ( ω ) ≤ Cap gi is the realized capacityunder scenario ω ∈ Ω. Real-time generation quantities are bounded as 0 ≤ G i ( ω ) ≤ ¯ G i ( ω ). Here,( X − x ) + := max { X − x, } . The suppliers also have the option of buying electricity at an offeringprice α g, − i ≥ G i ( ω ) − g i ) − over the agreed day-aheadquantities. Here, ( X − x ) − = max {− ( X − x ) , } .Consumers provide bid prices α d, − j ≥ D j ( ω ) − d j ) + in the real-timemarket, where D j ( ω ) is the cleared quantity and 0 ≤ ¯ D j ( ω ) ≤ Cap dj is the available demand capacityrealized under scenario ω ∈ Ω. We thus have 0 ≤ D j ( ω ) ≤ ¯ D j ( ω ). Consumers also have the optionof selling the demand deficit ( D j ( ω ) − d j ) − at price α d, + j ≥ F (cid:96) ( ω ) and satisfy − ¯ F (cid:96) ( ω ) ≤ F (cid:96) ( ω ) ≤ ¯ F (cid:96) ( ω ).Here, ¯ F (cid:96) ( ω ) is the transmission line capacity realized under scenario ω ∈ Ω and satisfies − Cap f(cid:96) ≤ ¯ F (cid:96) ( ω ) ≤ Cap f(cid:96) . Uncertain line capacities can be used to model N − x contingencies or uncertaintiesin capacity due to ambient conditions (e.g., ambient temperature affects line capacity). The clearedphase angles in the real-time market are given by Θ n ( ω ) such that θ n ≤ Θ n ( ω ) ≤ ¯ θ n for n ∈ N .We also define day-ahead clearing prices (i.e., locational marginal prices) for each node n ∈ N as π n . The real-time prices are defined as Π n ( ω ) , ω ∈ Ω.
3. Clearing Formulations
In this section, we present energy-only day-ahead deterministic and stochastic clearing formulations.The term “energy-only” indicates that no unit commitment decisions are made. We consider thesesimplified formulations in order to focus on important concepts related to pricing and paymentsto suppliers and consumers. Model extensions are left as a topic of future research.
In a deterministic setting, the day-ahead market is cleared by solving the following optimizationproblem. min d j ,g i ,f (cid:96) ,θ n (cid:88) i ∈G α gi g i − (cid:88) j ∈D α dj d j (4a)s.t. (cid:88) (cid:96) ∈L recn f (cid:96) − (cid:88) (cid:96) ∈L sndn f (cid:96) + (cid:88) i ∈G n g i − (cid:88) i ∈D n d i = 0 , ( π n ) n ∈ N (4b) f (cid:96) = B (cid:96) ( θ rec ( (cid:96) ) − θ snd ( (cid:96) ) ) , (cid:96) ∈ L (4c) − ¯ f (cid:96) ≤ f (cid:96) ≤ ¯ f (cid:96) , (cid:96) ∈ L (4d) avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing
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Operations Research ; manuscript no. ≤ g i ≤ ¯ g i , i ∈ G (4e)0 ≤ d j ≤ ¯ d j , j ∈ D (4f) θ n ≤ θ n ≤ θ n , n ∈ N . (4g)The objective function of this problem is the day-ahead negative social surplus. The solution ofthis problem gives the day-ahead quantities g i , d j , flows f (cid:96) , phase angles θ n , and prices π n . Thedeterministic formulation assumes a given value for the capacities ¯ g i , ¯ d j , ¯ f (cid:96) , θ n , and θ n . Because theconditions of the real-time market are uncertain at the time the day-ahead problem (4) is solved,these capacities are typically assumed to be the most probable ones (e.g., the expected value or forecast for supply and demand capacities) or are set based on the current state of the system (e.g.,for line capacities and phase angle ranges). In particular, it is usually assumed that ¯ g i = E [ ¯ G i ( ω )],¯ d j = E [ ¯ D j ( ω )], and ¯ f (cid:96) is the most probable state. One can also assume that ¯ g i = Cap gi and ¯ d j = Cap dj ,and ¯ f (cid:96) = Cap f(cid:96) . Such an assumption, however, can yield high economic penalties if the day-aheaddispatched quantities are far from those realized in the real-time market. Similarly, one can alsoassume conservative capacities (e.g., worst-case). In this sense, the day-ahead capacities ¯ g i , ¯ d j , ¯ f (cid:96) can be used as mechanisms to hedge against risk, as experienced ISO operators do to allow for asafety margin. Doing so, however, gives only limited control because the players need to summarizethe entire possible range of real-time capacities in one statistic. In Section 4 we argue that thislimitation can induce a distortion between day-ahead and real-time prices and biases revenues.When the capacities become known, the ISO uses fixed day-ahead committed quantities g i , d j , f (cid:96) , θ n , to solve the following real-time clearing problem.min D j ( · ) ,G i ( · ) ,F (cid:96) ( · ) , Θ n ( · ) (cid:88) i ∈G (cid:0) α g, + i ( G i ( ω ) − g i ) + − α g, − i ( G i ( ω ) − g i ) − (cid:1) (5a)+ (cid:88) j ∈D (cid:0) α d, + j ( D j ( ω ) − d j ) − − α d, − j ( D j ( ω ) − d j ) + (cid:1) (5b)s.t. (cid:88) (cid:96) ∈L recn F (cid:96) ( ω ) − (cid:88) (cid:96) ∈L sndn F (cid:96) ( ω ) + (cid:88) i ∈G n G i ( ω ) − (cid:88) j ∈D n D j ( ω ) = 0 , (Π n ( ω )) , n ∈ N (5c) F (cid:96) ( ω ) = B (cid:96) (Θ rec ( (cid:96) ) ( ω ) − Θ snd ( (cid:96) ) ( ω )) , n ∈ N (5d) − ¯ F (cid:96) ( ω ) ≤ F (cid:96) ( ω ) ≤ ¯ F (cid:96) ( ω ) , (cid:96) ∈ L (5e)0 ≤ G i ( ω ) ≤ ¯ G i ( ω ) , i ∈ G (5f)0 ≤ D j ( ω ) ≤ ¯ D j ( ω ) , j ∈ D (5g) θ n ≤ Θ n ( ω ) ≤ θ n , n ∈ N . (5h)The objective function of this problem is the real-time negative social surplus. The solution of thisproblem yields different real-time quantities G i ( ω ) , D j ( ω ), flows F (cid:96) ( ω ), phase angles Θ n ( ω ), andprices Π n ( ω ) depending on the scenario ω ∈ Ω realized. avala, Kim, Anitescu, and Birge:
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Motivated by the structure of the day-ahead and real-time market problems, we consider thestochastic market clearing formulation:min d j ,D j ( · ) ,g i ,G i ( · ) ,f (cid:96) ,F (cid:96) ( · ) ,θ n , Θ n ( · ) ϕ sto := (cid:88) i ∈G E (cid:2) α gi g i + α g, + i ( G i ( ω ) − g i ) + − α g, − i ( G i ( ω ) − g i ) − (cid:3) + (cid:88) j ∈D E (cid:2) − α dj d j + α d, + j ( D j ( ω ) − d j ) − − α d, − j ( D j ( ω ) − d j ) + (cid:3) + (cid:88) (cid:96) ∈L E (cid:2) ∆ α f, + (cid:96) ( F (cid:96) ( ω ) − f (cid:96) ) + + ∆ α f, − (cid:96) ( F (cid:96) ( ω ) − f (cid:96) ) − (cid:3) + (cid:88) n ∈N E (cid:2) ∆ α θ, + n (Θ n ( ω ) − θ n ) + + ∆ α θ, − n (Θ n ( ω ) − θ n ) − (cid:3) (6a)s.t. (cid:88) (cid:96) ∈L recn f (cid:96) − (cid:88) (cid:96) ∈L sndn f (cid:96) + (cid:88) i ∈G n g i − (cid:88) i ∈D n d i = 0 , ( π n ) n ∈ N (6b) f (cid:96) = B (cid:96) ( θ rec ( (cid:96) ) − θ snd ( (cid:96) ) ) , (cid:96) ∈ L (6c) (cid:88) (cid:96) ∈L recn ( F (cid:96) ( ω ) − f (cid:96) ) − (cid:88) (cid:96) ∈L sndn ( F (cid:96) ( ω ) − f (cid:96) ) + (cid:88) i ∈G n ( G i ( ω ) − g i ) − (cid:88) j ∈D n ( D j ( ω ) − d j ) = 0 , ( p ( ω )Π n ( ω )) ω ∈ Ω , n ∈ N (6d) F (cid:96) ( ω ) = B (cid:96) (Θ rec ( (cid:96) ) ( ω ) − Θ snd ( (cid:96) ) ( ω )) , ω ∈ Ω , (cid:96) ∈ L (6e) − ¯ F (cid:96) ( ω ) ≤ F (cid:96) ( ω ) ≤ ¯ F (cid:96) ( ω ) , ω ∈ Ω , (cid:96) ∈ L (6f)0 ≤ G i ( ω ) ≤ ¯ G i ( ω ) , ω ∈ Ω , i ∈ G (6g)0 ≤ D j ( ω ) ≤ ¯ D j ( ω ) , ω ∈ Ω , j ∈ D (6h) θ n ≤ Θ n ( ω ) ≤ θ n , ω ∈ Ω , n ∈ N . (6i)The stochastic setting provides a natural mechanism to anticipate the effects of day-ahead deci-sions on real-time market corrections. This property gives rise to several important pricing andpayment properties, as we will see in the following section.The above formulation is partially based on the one proposed by Pritchard et al. (2010). Wehighlight the following features of the model: • The real-time prices (duals of the network balance (6d)) have been weighted by their cor-responding probabilities. This feature will enable us to construct the Lagrange function of theproblem in terms of expectations. • The network balance in the real-time market is written in terms of the residual quantities( G i ( ω ) − g i ), ( D j ( ω ) − d j ), and flows ( F (cid:96) ( ω ) − f (cid:96) ). This feature will be key in obtaining consistentprices and it emphasizes the fact that the real-time market is a market of corrections. • We assume that the real-time quantity bounds ¯ G i ( ω ) , ¯ D j ( ω ) , ¯ F (cid:96) ( ω ) are independent of theday-ahead quantities. avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing
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Operations Research ; manuscript no. The differences between the proposed formulation and the one presented by Pritchard et al.(2010) are the following. • The formulation does not impose bounds on the day-ahead quantities, flows and phase angles.In Section 5 we will prove that the penalization terms render bounds for the day-ahead quantities,flows and phase angles redundant (see Theorem 7). • The parameters ∆ α f, + (cid:96) , ∆ α f, − (cid:96) , ∆ α θ, + , ∆ α θ, − > • We allow for randomness in the transmission line capacities. In Section 5 we will see that doingso has no effect on the underlying properties of the model. • We assume that the stochastic problem has relative complete recourse. That is, there exist afeasible real-time recourse decision for any day-ahead decision.We refer to the solution of the stochastic formulation (6) as the here-and-now solution to reflectthe fact that a single implementable decision must be made now in anticipation of the uncertainfuture and that day-ahead quantities and flows are scenario-independent. We also consider the(ideal, non-implementable) wait-and-see (WS) solution. For details, refer to Birge and Louveaux(1997). In the WS setting, we assume that the capacities for each scenario are actually known atthe moment of decision. In other words, we assume availability of perfect information. In order toobtain the WS solution, the clearing problem (6) is solved by allowing first-stage decisions g i , d j , f (cid:96) to be scenario-dependent. It is not difficult to prove that in this case, each scenario generatesday-ahead prices and quantities that are equal to real-time counterparts because no correctionsare necessary. We denote the expected social surplus obtained under perfect information as ϕ stoW S .
4. ISO Performance Metrics for Market Clearing
In this section, we discuss some objectives of the ISOs from a market operations standpoint anduse these to motivate a new set of metrics to quantify performance of deterministic and stochasticformulations. We place special emphasis on the structure of the social surplus function and on theissue of price consistency . We provide arguments as to why price consistency is a key property inachieving incentives. We argue that deterministic formulations do not actually yield price consis-tency and hence result in a range of undesired effects such as biased payments, revenue inadequacy,and the need for uplifts.We highlight that we define different metrics based on market behavior in expectation. A practicalway of interpreting these expected metrics is the following: assume that the market conditionsof a given day are repeated over a sequence of days and we collect the results over such period avala, Kim, Anitescu, and Birge:
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Operations Research ; manuscript no. by using each day as a scenario. We then compute a certain metric (like the social welfare) toperform the comparisons between the stochastic and deterministic clearing mechanisms to evaluateperformance. In this sense, market behavior in expectation can also interpreted as long run marketbehavior.
Consider the combination of the day-ahead and real-time costs for suppliers and consumers, C gi ( ω ) = + α gi g i + α g, + i ( G i ( ω ) − g i ) + − α g, − i ( G i ( ω ) − g i ) − (7a) C dj ( ω ) = − α dj d j + α d, + j ( D j ( ω ) − d j ) − − α d, − j ( D j ( ω ) − d j ) + . (7b)We define the incremental bid prices as ∆ α g, + i := α g, + i − α gi , ∆ α g, − i := α gi − α g, − i , ∆ α d, + j := α d, + j − α dj and ∆ α d, − j := α dj − α d, − j . To avoid degeneracy, we require that the incremental bid prices are positive:∆ α g, + i , ∆ α g, − i , ∆ α d, + j , ∆ α d, − j > Theorem 1.
Assume that the incremental bid prices are positive. The cost functions for suppliersand consumers can be expressed as C gi ( ω ) = α gi G i ( ω ) + ∆ α g, + i ( G i ( ω ) − g i ) + + ∆ α g, − i ( G i ( ω ) − g i ) − , i ∈ G , ω ∈ Ω (8a) C dj ( ω ) = − α dj D j ( ω ) + ∆ α d, + j ( D j ( ω ) − d j ) − + ∆ α d, − j ( D j ( ω ) − d j ) + , j ∈ D , ω ∈ Ω . (8b) Proof
Consider the cost function for suppliers C gi ( ω ) = α gi g i + α g, + i ( G i ( ω ) − g i ) + − α g, − i ( G i ( ω ) − g i ) − = α gi g i + ( α gi + ∆ α g, + i )( G i ( ω ) − g i ) + − ( α gi − ∆ α g, − i )( G i ( ω ) − g i ) − = α gi g i + α gi ( G i ( ω ) − g i ) + − α gi ( G i ( ω ) − g i ) − + ∆ α g, + i ( G i ( ω ) − g i ) + + ∆ α g, − i ( G i ( ω ) − g i ) − = α gi g i + α gi ( G i ( ω ) − g i ) + ∆ α g, + i ( G i ( ω ) − g i ) + + ∆ α g, − i ( G i ( ω ) − g i ) − = α gi G i ( ω ) + ∆ α g, + i ( G i ( ω ) − g i ) + + ∆ α g, − i ( G i ( ω ) − g i ) − . The last two equalities follow from the fact that X − x = ( X − x ) + − ( X − x ) − . The same propertyapplies to C dj ( ω ) (using the appropriate cost terms). (cid:3) We say that the incremental bid prices are symmetric if ∆ α g, + i = ∆ α g, − i and ∆ α d, + j = ∆ α d, − j .Denote the symmetric prices by ∆ α gi := ∆ α g, + i = ∆ α g, − i and ∆ α dj := ∆ α d, + j = ∆ α d, − j . Corollary 1.
If the incremental bid prices are symmetric, then the cost functions for suppliersand consumers can be expressed as C gi ( ω ) = α gi G i ( ω ) + ∆ α gi | G i ( ω ) − g i | , i ∈ G , ω ∈ Ω (9a) C dj ( ω ) = − α dj D j ( ω ) + ∆ α dj | D j ( ω ) − d j | , j ∈ D , ω ∈ Ω . (9b) avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing
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Consider the cost function for suppliers C gi ( ω ) = α gi G i ( ω ) + ∆ α g,i ( G i ( ω ) − g i ) + + ∆ α g,i ( G i ( ω ) − g i ) − = α gi G i ( ω ) + ∆ α gi | G i ( ω ) − g i | , because of the fact that | X − x | = ( X − x ) + + ( X − x ) − . The same property applies to C dj ( ω ) (usingthe appropriate cost terms). (cid:3) Definition 1 (Social Surplus).
We define the expected negative social surplus (or social surplus for short) as ϕ := E (cid:34)(cid:88) i ∈G C gi ( ω ) + (cid:88) j ∈D C dj ( ω ) (cid:35) = ϕ g + ϕ d , (10)where ϕ g , ϕ d are the expected supply and consumer costs , ϕ g := E (cid:34)(cid:88) i ∈G C gi ( ω ) (cid:35) = (cid:88) i ∈G (cid:0) α gi E [ G i ( ω )] + ∆ α g, + i E [( G i ( ω ) − g i ) + ] + ∆ α g, − i E [( G i ( ω ) − g i ) − ] (cid:1) (11a) ϕ d := E (cid:34)(cid:88) j ∈D C dj ( ω ) (cid:35) = (cid:88) j ∈D (cid:0) − α dj E [ D j ( ω )] + ∆ α d, + j E [( D j ( ω ) − d j ) − ] + ∆ α d, − j E [( D j ( ω ) − d j ) + ] (cid:1) . (11b)This particular structure of the expected social surplus function was noticed by Pritchard et al.(2010) and provides interesting insights. From Equation (11), we note that the expected quan-tities E [ G i ( ω )], E [ D j ( ω )] act as forecasts of the day-ahead quantities and are priced by usingthe day-ahead bids α gi , α jd (first term). This immediately suggests that it is the expected clearedquantities G i ( ω ) , D j ( ω ) and not the capacities ¯ g i , ¯ d j that are to be used as forecasts, as is donein the day-ahead deterministic formulation (4). The second and third terms penalize deviationsof the real-time quantities from the day-ahead commitments using the incremental bid prices.More interestingly, Corollary 1 suggests that when the incremental bid prices are symmetric (i.e.,∆ α g, + i = ∆ α g, − i and ∆ α d, + j = ∆ α d, − j ), day-ahead quantities will tend to converge to the medianof the real-time quantities if the expected social surplus function is minimized. A deterministicsetting, however, cannot guarantee optimality in this sense because it minimizes the day-ahead andreal-time components of the surplus function separately . In particular, the expected social surplusfor the deterministic formulation is obtained by solving the day-ahead problem (4) followed by avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing Article submitted to
Operations Research ; manuscript no. the solution of the real-time problem (5) for all scenarios ω ∈ Ω. The day-ahead surplus and theexpected value of the real-time surplus are then combined to obtain the expected surplus ϕ .A deterministic setting can yield surplus inefficiencies because it cannot properly anticipate theeffect of day-ahead decision on real-time market decisions. For instance, certain suppliers can beinflexible in the sense that they cannot modify their day-ahead supply easily in the real-time mar-ket (e.g., coal plants). This results in constraints of the form g i = G i ( ω ) or d j = D j ( ω ) , ω ∈ Ω. Thisinflexibility can trigger inefficiencies because the operator is forced to use expensive units in thereal-time market (e.g., combined-cycle) or because load shedding is needed to prevent infeasibilities.Most studies on stochastic market clearing and unit commitment have focused on showing improve-ments in social surplus over deterministic formulations. In Section 6 we demonstrate that evenwhen social surplus differences are negligible, the resulting prices and payments can be drasticallydifferent. This situation motivates us to consider alternative metrics for monitoring performance.We note that the objective function of the stochastic clearing formulation (6) can be written as ϕ sto = ϕ + (cid:88) (cid:96) ∈L E (cid:2) ∆ α f, + (cid:96) ( F (cid:96) ( ω ) − f (cid:96) ) + + ∆ α f, − (cid:96) ( F (cid:96) ( ω ) − f (cid:96) ) − (cid:3) + (cid:88) n ∈N E (cid:2) ∆ α θ, + n (Θ n ( ω ) − θ n ) + + ∆ α θ, − n (Θ n ( ω ) − θ n ) − (cid:3) , (12)where ϕ is the expected negative surplus function defined in (10). Consequently, if∆ α f, + (cid:96) , ∆ α f, − (cid:96) , ∆ α θ, + , ∆ α θ, − are sufficiently small, we have that ϕ sto ≈ ϕ . We seek that the day-ahead prices be consistent representations of the expected real-time prices.In other words, we seek that the expected price distortions (also known as expected price premia ) π n − E [Π n ( ω )] , n ∈ N be zero or at least in a bounded neighborhood. This is desired for variousreasons that we will explain. Definition 2 (Price Distortions).
We define the expected price distortion or expected pricepremia as M πn := π n − E [Π n ( ω )] , n ∈ N . (13)We say that the price is consistent at node n ∈ N if M πn = 0. In addition, we define the nodeaverage and maximum absolute distortions , M πavg := 1 |N | (cid:88) n ∈N |M πn | (14a) M πmax := max n ∈N |M πn | . (14b) avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing
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Operations Research ; manuscript no. Pricing consistency is related to the desire that day-ahead and real-time prices converge, as isdiscussed by Ott (2003). Note, however, that it is unrealistic to expect that day-ahead and real-time prices converge in each scenario. This is possible only in the absence of uncertainty (capacityforecasts are perfect such as in the perfect information setting). Any real-time deviation in capacityfrom a day-ahead forecast will lead to a deviation between day-ahead and real-time prices. Itis possible, however, to ensure that day-ahead and real-time prices converge in expectation . Thissituation also implies that any deviation of the real-time price from the day-ahead price is entirelythe result of unpredictable random factors . This is also equivalent to saying that day-ahead pricesconverge to the expected value of the real-time prices.Pricing consistency cannot be guaranteed with deterministic formulations because the day-aheadclearing model forecasts real-time capacities, not real-time quantities. Consequently, players areforced to “summarize” their possible real-time capacities in single statistics ¯ d j , ¯ g i , ¯ f (cid:96) . Expected val-ues are typically used. This summarization, however, is inconsistent because it does not effectivelyaverage real-time market performance as the structure of the surplus function (11) suggests. Infact, as we show in Section 5, expected values need not be the right statistic to use in the day-aheadmarket. This is consistent with the observations made by Morales et al. (2014). In addition, wenote that certain random variables might be difficult to summarize (e.g., if they follow multimodaland heavy-tailed distributions). For instance, consider that there is uncertainty about the stateof a transmission line in the real-time market (i.e., there is a probability that it will fail). In adeterministic setting it is difficult to come up with a ”forecast” value for the day-ahead capacity¯ f (cid:96) in such a case. As argued by Kaye et al. (1990), we can justify the desire of seeking price consistency by analyzingthe payments to the market players. The payment includes the day-ahead settlement plus thecorrection payment given at real-time prices, as is the standard practice in market operations. Formore details, see Ott (2003) and Pritchard et al. (2010).
Definition 3 (Payments).
The payments to suppliers and from consumers in scenario ω ∈ Ω aredefined as follows: P gi ( ω ) := g i π n ( i ) + ( G i ( ω ) − g i )Π n ( i ) ( ω )= g i ( π n ( i ) − Π n ( i ) ( ω )) + G i ( ω )Π n ( i ) ( ω ) , i ∈ G , ω ∈ Ω (15a) P dj ( ω ) := − d j π n ( i ) − ( D j ( ω ) − d j )Π n ( j ) ( ω )= d j (Π n ( i ) ( ω ) − π n ( i ) ) − D j ( ω )Π n ( j ) ( ω ) , j ∈ D , ω ∈ Ω . (15b) avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing Article submitted to
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We say that the expected payments are consistent if they satisfy E [ P gi ( ω )] = + E (cid:2) G i ( ω )Π n ( i ) ( ω ) (cid:3) , i ∈ G (16a) E (cid:2) P dj ( ω ) (cid:3) = − E (cid:2) D j ( ω )Π n ( j ) ( ω ) (cid:3) , j ∈ D , (16b)where E [ P gi ( ω )] = + g i M πn ( i ) + E (cid:2) G i ( ω )Π n ( i ) ( ω ) (cid:3) , i ∈ G (17a) E (cid:2) P dj ( ω ) (cid:3) = − d j M πn ( j ) − E (cid:2) D j ( ω )Π n ( j ) ( ω ) (cid:3) , j ∈ D . (17b)If the prices are consistent at each node n ∈ N , the expected payments are consistent. Thisdefinition of consistency is motivated by the following observations. The price distortion is factoredin the expected payments. From (17) we see that price distortions (premia) can bias benefitstoward a subset of players. In particular, if the premium at a given node is negative ( M πn < M πn >
0, the oppostive holds true. Thissituation can prevent consumers from providing price-responsive demands. We can thus concludethat price consistency ensures payment consistency with respect to suppliers and consumers. Inother words, M πn = 0 implies (16).Kaye et al. (1990) argue that setting the day-ahead prices to the expected real-time prices (priceconsistency) is desirable because it effectively eliminates the day-ahead component of the market.Consequently, the market operates (in expectation) as a pure real-time market. This situationis desirable because it implies that the day-ahead market does not interfere with the incentivesprovided by real-time markets. This is particularly important for players that benefit from real-time market variability (such as peaking units and price-response demands). This also implies thatthe ISO does not give any preference to either risk-taking or risk-averse players. We also highlightthat price consistency does not imply that premia do not exist; they can exist in each scenario butnot in expectation.Deterministic formulations can yield persistent price premia that benefit a subset of players orthat can be used for market manipulation. For instance, consider the case in which a wind farmforecast has the same mean but very different variance (uncertainty) for several consecutive days.If the expected forecast is used, the day-ahead prices will be consistently the same for all days,thus making them more predictable and biased toward a subset of players. While the use of risk-adaptive reserves can help ameliorate this effect, this approach is not guaranteed to achieve priceconsistency. avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing
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Operations Research ; manuscript no. From (17) we see that if the premium at a given node is negative ( M πn < Definition 4 (Wholeness).
We say that suppliers and consumers are whole in expectation if E [ P gi ( ω )] ≥ E [ C gi ( ω )] , i ∈ G (18a) − E (cid:2) P dj ( ω ) (cid:3) ≤ − E (cid:2) C dj ( ω ) (cid:3) , j ∈ D . (18b)If the players are not made whole, they can leave the market and this can hinder diversification.Uplift payments are routinely used by the ISOs to avoid this situation (Galiana et al. 2003, Baldicket al. 2005). Uplifts can result from inadequate representations of system behavior such as non-convexities (O’Neill et al. 2005) or, as we will see in Section 6, can result from using inadequatestatistical representations of real-time market performance in deterministic settings. Consequently,uplift payments are a useful metric to determine the effectiveness of a given clearing formulation. Definition 5 (Uplift Payments).
We define the expected uplift payments to suppliers and con-sumers as M Ui := − min { E [ P gi ( ω )] − E [ C gi ( ω )] , } , i ∈ G (19a) M Uj := − min (cid:8) E [ P dj ( ω )] − E [ C dj ( ω )] , (cid:9) , j ∈ D . (19b)We also define the total uplift as M U := (cid:88) i ∈G M Ui + (cid:88) j ∈D M Uj .We highlight that our setting is convex and we thus only consider uplifts arising from inadequatestatistical representations. An efficient clearing procedure must ensure that the ISO does not run into financial deficit. In otherwords, the ISO must have a positive cash flow (payments collected from consumers are greaterthan the payments given to suppliers). We consider the following expected revenue definition, usedby Pritchard et al. (2010), to assess performance with respect to this case. avala, Kim, Anitescu, and Birge:
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Definition 6 (Revenue Adequacy).
The expected net payment to the ISO is defined as M ISO := E (cid:34)(cid:88) i ∈G P gi ( ω ) + (cid:88) j ∈D P dj ( ω ) (cid:35) = (cid:88) i ∈G E [ P gi ( ω )] + (cid:88) j ∈D E [ P dj ( ω )] . (20)We say that the ISO is revenue adequate in expectation if M ISO ≤
5. Properties of Stochastic Clearing
In this section, we prove that the stochastic clearing formulation yields bounded price distortionsand that these distortions can be made arbitrarily small. In addition, we prove that day-aheadquantities are bounded by real-time quantities and that they converge to a quantile of the real-timequantities when the distortions are zero. Further, we prove that the formulation yields revenueadequacy and zero uplifts in expectation.
We begin our discussion with a single-node formulation (no network constraints) and then gen-eralize the results to the case of network constraints. The single-node formulation has the form:min d j ,g i ,G i ( · ) ,D ( · ) (cid:88) i ∈G E (cid:2) α gi G i ( ω ) + ∆ α g, + i ( G i ( ω ) − g i ) + + ∆ α g, − i ( G i ( ω ) − g i ) − (cid:3) + (cid:88) j ∈D E (cid:2) − α dj D j ( ω ) + ∆ α d, + j ( D j ( ω ) − d i ) − + ∆ α d, − j ( D j ( ω ) − d j ) + (cid:3) (21a)s.t. (cid:88) i ∈G g i = (cid:88) j ∈D d j ( π ) (21b) (cid:88) i ∈G ( G i ( ω ) − g i ) = (cid:88) j ∈D ( D j ( ω ) − d j ) ω ∈ Ω ( p ( ω )Π( ω )) (21c)0 ≤ G i ( ω ) ≤ ¯ G i ( ω ) , i ∈ G , ω ∈ Ω (21d)0 ≤ D j ( ω ) ≤ ¯ D ( ω ) , j ∈ D , ω ∈ Ω . (21e)This formulation assumes infinite transmission capacity. In this case, the entire network collapsesinto a single node; consequently, a single day-ahead price π and real-time price Π( ω ) are used.We state that the partial Lagrange function of (21) is given by L ( d j , D j ( · ) , g i , G i ( · ) , π, Π( · ))= (cid:88) i ∈G E (cid:2) α gi G i ( ω ) + ∆ α g, + i ( G i ( ω ) − g i ) + + ∆ α g, − i ( G i ( ω ) − g i ) − (cid:3) avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing
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Operations Research ; manuscript no. − (cid:88) j ∈D E (cid:2) α dj D j ( ω ) − ∆ α d, + j ( D j ( ω ) − d i ) − − ∆ α d, − j ( D j ( ω ) − d j ) + (cid:3) − π (cid:32)(cid:88) i ∈G g i − (cid:88) j ∈D d j (cid:33) − E (cid:34) Π( ω ) (cid:32)(cid:88) i ∈G ( G i ( ω ) − g i ) − (cid:88) j ∈D ( D j ( ω ) − d j ) (cid:33)(cid:35) . The contribution of the balance constraints can be written in expected value form if we weight theLagrange multipliers of the balance equations (prices) by the probabilities p ( ω ). Theorem 2.
Consider the single-node stochastic clearing problem (21) , and assume that the incre-mental bid prices are positive. The price distortion M π = π − E [Π( ω )] is bounded as − ∆ α + ≤ M π ≤ ∆ α − , (22) where ∆ α + = min (cid:26) min i ∈G ∆ α g, + i , min j ∈D ∆ α d, + j (cid:27) (23a) and ∆ α − = min (cid:26) min i ∈G ∆ α g, − i , min j ∈D ∆ α d, − j (cid:27) . (23b) Proof
Since ( X − x ) − = ( X − x ) + − ( X − x ), we have the partial Lagrange function L ( d j , D j ( · ) , g i , G i ( · ) , π, Π( · ))= (cid:88) i ∈G E (cid:2) α gi G i ( ω ) + ∆ α g, + i ( G i ( ω ) − g i ) + + ∆ α g, − i ( G i ( ω ) − g i ) + − ∆ α g, − i ( G i ( ω ) − g i ) (cid:3) − (cid:88) j ∈D E (cid:2) α dj D j ( ω ) − ∆ α d, + j ( D j ( ω ) − d i ) + + ∆ α d, + j ( D j ( ω ) − d i ) − ∆ α d, − j ( D j ( ω ) − d j ) + (cid:3) − π (cid:32)(cid:88) i ∈G g i − (cid:88) j ∈D d j (cid:33) − E (cid:34) Π( ω ) (cid:32)(cid:88) i ∈G ( G i ( ω ) − g i ) − (cid:88) j ∈D ( D j ( ω ) − d j ) (cid:33)(cid:35) . The stationarity conditions of the partial Lagrange function with respect to the day-ahead quan-tities d j , g i are given by0 ∈ ∂ d j L = (∆ α d, + j + ∆ α d, − j ) ∂ d j E [( D j ( ω ) − d j ) + ] + ∆ α d, + j + π − E [Π( ω )] j ∈ D (24a)0 ∈ ∂ g i L = (∆ α g, + i + ∆ α g, − i ) ∂ g i E [( G i ( ω ) − g i ) + ] + ∆ α g, − i − π + E [Π( ω )] i ∈ G . (24b)Rearranging (24a), we obtain − ∆ α d, + j − π + E [Π( ω )]∆ α d, + j + ∆ α d, − j ∈ ∂ d j E [( D j ( ω ) − d j ) + ] j ∈ D (25a) − ∆ α g, − i + π − E [Π( ω )]∆ α g, + i + ∆ α g, − i ∈ ∂ g i E [( G i ( ω ) − g i ) + ] i ∈ G . (25b) avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing Article submitted to
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From the property ∂ x ( X − x ) + = − X > x
X < x [ − ,
0] if X = x , (26)we have ∂ x E [( X ( ω ) − x ) + ] = (cid:8) E (cid:2) − X ( ω ) >x + a X ( ω )= x (cid:3) : a ∈ [ − , (cid:9) = {− P ( X ( ω ) > x ) + a P ( X ( ω ) = x ) : a ∈ [ − , } , = { η : − P ( X ( ω ) ≥ x ) ≤ η ≤ − P ( X ( ω ) > x ) } . (27)Since − ≤ − P ( X ( ω ) ≥ x ) ≤ − P ( X ( ω ) > x ) ≤
0, we have ∂ x E [( X ( ω ) − x ) + ] ⊆ [ − , . (28)From (25) and (28), we have − ≤ − ∆ α d, + j − π + E [Π( ω )]∆ α d, + j + ∆ α d, − j ≤ − ≤ − ∆ α g, − i + π − E [Π( ω )]∆ α g, + i + ∆ α g, − i ≤ . (29b)The above relationships are equivalent to − ∆ α d, + j ≤ π − E [Π( ω )] ≤ ∆ α d, − j (30a) − ∆ α g, + i ≤ π − E [Π( ω )] ≤ ∆ α g, − i . (30b)or, equivalently, − ∆ α + ≤ M π ≤ ∆ α − . (cid:3) The price distortion is bounded above by the smallest of all ∆ α g, − i and ∆ α d, − j and bounded belowby the largest of all − ∆ α g, + i and − ∆ α d, + j . The bounds are denoted by ∆ α − and − ∆ α + , respectively.This implies that if we let ∆ α + and ∆ α − be sufficiently small, then we can make the price distortion M π arbitrarily small . Note that the bound is independent of the cleared quantities, which reflectsrobust behavior. Moreover, the upper bound depends on the incremental bid prices ∆ α g, − i and∆ α d, − j only, while the lower bound depends on ∆ α g, + i and ∆ α d, + j only. Boundedness of the pricedistortion also eliminates the day-ahead component of the suppliers and consumer payments andthus achieves payment consistency. We highlight that Theorem 2 assumes the positive incrementalbid prices. Otherwise, the solution can be degenerate.We now prove that the day-ahead quantities d j , g i obtained from the stochastic clearing modelare implicitly bounded by the minimum and maximum real-time quantities. avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing
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Operations Research ; manuscript no. Theorem 3.
Consider the single-node stochastic clearing problem (21) , and assume that the incre-mental bid prices are positive. The day-ahead quantities are bounded by the real-time quantitiesas min ω ∈ Ω D j ( ω ) ≤ d j ≤ max ω ∈ Ω D j ( ω ) , j ∈ D min ω ∈ Ω G i ( ω ) ≤ g i ≤ max ω ∈ Ω G i ( ω ) , i ∈ G . Proof
Consider the following two cases: • Case 1: The price distortion hits the lower bound for demand j ; we thus have π − E [Π( ω )] = − ∆ α d, + j . This implies 0 ∈ ∂ d j E [( D j ( ω ) − d j ) + ] from (25a), and hence P ( D j ( ω ) > d j ) = 0 and P ( D j ( ω ) ≤ d j ) = 1 from (26). This implies that d j ≥ D j ( ω ) , ∀ ω ∈ Ω and d j ≥ min ω ∈ Ω D j ( ω ). • Case 2: The price distortion hits the upper bound for demand j ; we thus have π − E [Π( ω )] =∆ α d, − j . This implies − ∈ ∂ d j E [( D j ( ω ) − d j ) + ] from (25a), and hence P ( D j ( ω ) ≥ d j ) = 1 from (26).This implies that d j ≤ D j ( ω ) , ∀ ω ∈ Ω and d j ≤ max ω ∈ Ω D j ( ω ).We thus conclude that d j is bounded from below by min ω ∈ Ω D j ( ω ) and from above bymax ω ∈ Ω D j ( ω ). The same procedure can be followed to prove that g i is bounded from below bymin ω ∈ Ω G i ( ω ) and from above by max ω ∈ Ω G i ( ω ). (cid:3) The implicit bound on the day-ahead quantities d j , g i is a key property of the stochastic modelproposed because it implies that we do not have to choose day-ahead capacities ¯ g i , ¯ d j (e.g., sum-marization statistics). These are automatically set by the model through the scenario information.This is important because, as we have mentioned, obtaining proper summarizing statistics forcomplex probability distributions might not be trivial.We now prove that if the price distortion is zero, the day-ahead quantities converge to quantiles of the real-time quantities. Theorem 4.
Consider the stochastic clearing problem (21) , and assume that the incremental bidprices are positive. If the price distortion is zero at the solution, then d j = Q D j ( ω ) (cid:32) ∆ α d, − j ∆ α d, + j + ∆ α d, − j (cid:33) , j ∈ D (31a) g i = Q G i ( ω ) (cid:18) ∆ α g, + i ∆ α g, + i + ∆ α g, − i (cid:19) , i ∈ G . (31b) Proof
From (27) and (29a) we have that if π − E [Π( ω )] = 0, then − P ( D j ( ω ) ≥ d j ) ≤ − ∆ α d, + j ∆ α d, + j +∆ α d, − j ≤ − P ( D j ( ω ) > d j ), and thus P ( D j ( ω ) < d j ) ≤ ∆ α d, − j ∆ α d, + j +∆ α d, − j ≤ P ( D j ( ω ) ≤ d j ). Thisimplies (31a) from (2). The same argument holds for (31b). (cid:3) Corollary 2.
If the incremental bid prices are symmetric then d j = M ( D j ( ω )) , j ∈ D and g i = M ( D j ( ω )) , i ∈ G . avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing Article submitted to
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Proof
The proof follows from Corollary 1 and Theorem 4. (cid:3)
This result implies that the day-ahead quantities d j , g i cannot in general be guaranteed to con-verge to the expected values of the real-time quantities E [ D j ( ω )] , E [ G i ( ω )]. Such convergence canonly be guaranteed when the quantile and mean coincide. These observations thus imply that the expected value is not necessarily the only statistic that can be used for the capacities in theday-ahead market. We now prove that the stochastic formulation yields zero uplifts in expectation. Revenue ade-quacy is not considered because this is a single-node problem. We use the strategy followed byMorales et al. (2012). For this discussion, we denote a minimizer of the partial Lagrange function(subject to the constraints (21d) and (21e)) as d ∗ j , D j ( · ) ∗ , g ∗ i , G i ( · ) ∗ , π ∗ , Π ∗ ( · ). Because the problemis convex, we know that the optimal prices π ∗ , Π ∗ ( · ) satisfy( d ∗ j , D j ( · ) ∗ , g ∗ i , G ∗ i ( · )) = argmin d j ,D j ( · ) ,g i ,G i ( · ) L ( d j , D j ( · ) , g i , G i ( · ) , π ∗ , Π ∗ ( · )) s.t. (21d) − (21e) . (32)Moreover, at fixed π ∗ , Π ∗ ( · ), the partial Lagrange function can be separated as L ( d j , D j ( · ) , g i , G i ( · ) , π ∗ , Π ∗ ( · )) = (cid:88) i ∈G L gi ( g i , G i ( · ) , π ∗ , Π ∗ ( · )) + (cid:88) j ∈D L dj ( d j , D j ( · ) , π ∗ , Π ∗ ( · )) , (33)where L gi ( g i , G i ( · ) , π ∗ , Π ∗ ( · )) := E [ C gi ( ω )] − E [ P gi ( ω )] , i ∈ G (34a) L dj ( d j , D j ( · ) , π ∗ , Π ∗ ( · )) := E [ C dj ( ω )] − E [ P dj ( ω )] , j ∈ D . (34b)Consequently, one can minimize the partial Lagrange function by minimizing (34) independently. Theorem 5.
Consider the single-node clearing problem (21) , and let the assumptions of Theo-rem 2 hold. Any minimizer d ∗ j , D j ( · ) ∗ , g ∗ i , G i ( · ) ∗ , π ∗ , Π ∗ ( · ) of (21) yields zero uplift payments inexpectation: M Ui = 0 , i ∈ G (35a) M Uj = 0 , j ∈ D . (35b) Proof
From Definition 5, it suffices to show that E [ P gi ( ω )] − E [ C gi ( ω )] ≥ i ∈G and E [ P dj ( ω )] − E [ C dj ( ω )] ≥ j ∈ D . For fixed π ∗ , Π ∗ ( ω ), the candidate solu-tion d j = D j ( · ) = g i = G i ( · ) = 0 is feasible for (32) with values L gi ( g i , G i ( · ) , π ∗ , Π ∗ ( · )) =0 , i ∈ G and L d ( d j , D j ( · ) , π ∗ , Π ∗ ( · )) = 0 , j ∈ D . Because the candidate is suboptimal we have L gi ( g ∗ i , G ∗ i ( · ) , π ∗ , Π ∗ ( · )) ≤ L gi ( g i , G i ( · ) , π ∗ , Π ∗ ( · )) = 0 and L dj ( d ∗ j , D ∗ j ( · ) , π ∗ , Π ∗ ( · )) ≤
0. The result fol-lows from equations (34) and the definition of M Ui and M Uj in (19). (cid:3) avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing
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Operations Research ; manuscript no. Having established some insights into the properties of the stochastic model, we now turn ourattention to the full stochastic problem with network constraints (6) and generalize our results.It is well known that stochastic formulations yield a better expected social surplus. This followsfrom the well-known inequality (see Birge and Louveaux (1997)): ϕ stoW S ≤ ϕ sto ≤ ϕ det . (36)This follows from the fact that the stochastic formulation will lead to a lower recourse cost (real-timepenalty costs) than will the deterministic solution because the deterministic day-ahead problemdoes not anticipate recourse actions. The wait-and-see setting can perfectly anticipate real-timemarket conditions and therefore its real-time penalties are zero. This makes it the optimal, butnonimplementable policy.We now establish boundedness of the price distortions throughout the network. To establish ourresult, we need the following definitions. We rewrite equations (6c) and (6e) as f (cid:96) = (cid:88) n ∈N B (cid:96)n θ n ∀ (cid:96) ∈ L , (37a) F (cid:96) ( ω ) = (cid:88) n ∈N B (cid:96)n Θ n ( ω ) ∀ ω ∈ Ω , (cid:96) ∈ L , (37b)where B (cid:96)n = B (cid:96) if n = rec ( (cid:96) ) , − B (cid:96) if n = snd ( (cid:96) ) , . We note that the above definitions imply that B (cid:96),rec ( (cid:96) ) = B (cid:96) and B (cid:96),snd ( (cid:96) ) = − B (cid:96) . Moreover we havethat, (cid:88) (cid:96) ∈L recn f (cid:96) = (cid:88) (cid:96) ∈L recn (cid:0) B (cid:96) θ rec ( (cid:96) ) − B (cid:96) θ snd ( (cid:96) ) (cid:1) = (cid:88) (cid:96) ∈L recn (cid:0) B (cid:96),rec ( (cid:96) ) θ rec ( (cid:96) ) + B (cid:96),snd ( (cid:96) ) θ snd ( (cid:96) ) (cid:1) = (cid:88) (cid:96) ∈L recn (cid:0) B (cid:96)n θ n + B (cid:96),snd ( (cid:96) ) θ snd ( (cid:96) ) (cid:1) . Using similar observations we have that, (cid:88) (cid:96) ∈L sndn f (cid:96) = (cid:88) (cid:96) ∈L sndn (cid:0) B (cid:96),rec ( (cid:96) ) θ rec ( (cid:96) ) + B (cid:96)n θ n (cid:1) . (38)Substituting the flows f (cid:96) , F (cid:96) ( ω ) by their corresponding phase angle expressions and using the aboveproperties we have that the stochastic clearing problem (37) can be written as avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing Article submitted to
Operations Research ; manuscript no. min d j ,D j ( · ) ,g i ,G i ( · ) ,θ n , Θ n ( · ) (cid:88) i ∈G E (cid:2) α gi G i ( ω ) + ∆ α g, + i ( G i ( ω ) − g i ) + + ∆ α g, − i ( G i ( ω ) − g i ) − (cid:3) + (cid:88) j ∈D E (cid:2) − α dj D j ( ω ) + ∆ α d, + j ( D j ( ω ) − d j ) − + ∆ α d, − j ( D j ( ω ) − d j ) + (cid:3) + (cid:88) (cid:96) ∈L E ∆ α f, + (cid:96) (cid:32) (cid:88) n ∈N B (cid:96)n (Θ n ( ω ) − θ n ) (cid:33) + + ∆ α f, − (cid:96) (cid:32) (cid:88) n ∈N B (cid:96)n (Θ n ( ω ) − θ n ) (cid:33) − + (cid:88) n ∈N E (cid:2) ∆ α θ, + n (Θ n ( ω ) − θ n ) + + ∆ α θ, − n (Θ n ( ω ) − θ n ) − (cid:3) (39a)s.t. (cid:88) (cid:96) ∈L recn (cid:0) B (cid:96)n θ n + B (cid:96),snd ( (cid:96) ) θ snd ( (cid:96) ) (cid:1) − (cid:88) (cid:96) ∈L sndn (cid:0) B (cid:96),rec ( (cid:96) ) θ rec ( (cid:96) ) + B (cid:96)n θ n (cid:1) + (cid:88) i ∈G n g i − (cid:88) i ∈D n d i = 0 , ( π n ) ∀ n ∈ N (39b) (cid:88) (cid:96) ∈L recn (cid:2) B (cid:96)n (Θ n ( ω ) − θ n ) + B (cid:96),snd ( (cid:96) ) (cid:0) Θ snd ( (cid:96) ) ( ω ) − θ snd ( (cid:96) ) (cid:1)(cid:3) − (cid:88) (cid:96) ∈L sndn (cid:2) B (cid:96),rec ( (cid:96) ) (cid:0) Θ rec ( (cid:96) ) ( ω ) − θ rec ( (cid:96) ) (cid:1) + B (cid:96)n (Θ n ( ω ) − θ n ) (cid:3) + (cid:88) i ∈G n ( G i ( ω ) − g i ) − (cid:88) j ∈D n ( D j ( ω ) − d j ) = 0 , ( p ( ω )Π n ( ω )) , ∀ ω ∈ Ω , n ∈ N (39c) − ¯ F (cid:96) ( ω ) ≤ (cid:88) n ∈N B (cid:96)n Θ n ( ω ) ≤ ¯ F (cid:96) ( ω ) , ∀ ω ∈ Ω , (cid:96) ∈ L (39d)0 ≤ G i ( ω ) ≤ ¯ G i ( ω ) , ∀ ω ∈ Ω , i ∈ G (39e)0 ≤ D j ( ω ) ≤ ¯ D j ( ω ) , ∀ ω ∈ Ω , j ∈ D (39f) θ n ≤ Θ n ( ω ) ≤ θ n , ∀ ω ∈ Ω , n ∈ N . (39g)We consider the partial Lagrange function of (39) as L ( d j , D j ( · ) , g i , G i ( · ) , θ n , Θ n ( · ) , π n , Π n ( · ))= (cid:88) i ∈G E (cid:2) α gi G i ( ω ) + (cid:0) ∆ α g, + i + ∆ α g, − i (cid:1) ( G i ( ω ) − g i ) + − ∆ α g, − i ( G i ( ω ) − g i ) (cid:3) − (cid:88) j ∈D E (cid:2) α dj D j ( ω ) − (cid:0) ∆ α d, + j + ∆ α d, − j (cid:1) ( D j ( ω ) − d j ) + + ∆ α d, + j ( D j ( ω ) − d j ) (cid:3) + (cid:88) (cid:96) ∈L E (cid:0) ∆ α f, + (cid:96) + ∆ α f, − (cid:96) (cid:1) (cid:32) (cid:88) n ∈N B (cid:96)n (Θ n ( ω ) − θ n ) (cid:33) + − ∆ α f, − (cid:96) (cid:88) n ∈N B (cid:96)n (Θ n ( ω ) − θ n ) + (cid:88) n ∈N E (cid:2)(cid:0) ∆ α θ, + n + ∆ α θ, − n (cid:1) (Θ n ( ω ) − θ n ) + − ∆ α θ, − n (Θ n ( ω ) − θ n ) (cid:3) − (cid:88) n ∈N π n (cid:88) (cid:96) ∈L recn (cid:0) B (cid:96)n θ n + B (cid:96),snd ( (cid:96) ) θ snd ( (cid:96) ) (cid:1) − (cid:88) (cid:96) ∈L sndn (cid:0) B (cid:96),rec ( (cid:96) ) θ rec ( (cid:96) ) + B (cid:96)n θ n (cid:1) + (cid:88) i ∈G n g i − (cid:88) j ∈D n d j avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing
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Operations Research ; manuscript no. − E (cid:88) n ∈N Π n ( ω ) (cid:88) (cid:96) ∈L recn (cid:2) B (cid:96)n (Θ n ( ω ) − θ n ) + B (cid:96),snd ( (cid:96) ) (cid:0) Θ snd ( (cid:96) ) ( ω ) − θ snd ( (cid:96) ) (cid:1)(cid:3) − (cid:88) (cid:96) ∈L sndn (cid:2) B (cid:96),rec ( (cid:96) ) (cid:0) Θ rec ( (cid:96) ) ( ω ) − θ rec ( (cid:96) ) (cid:1) + B (cid:96)n (Θ n ( ω ) − θ n ) (cid:3) + (cid:88) i ∈G n ( G i ( ω ) − g i ) − (cid:88) j ∈D n ( D j ( ω ) − d j ) (cid:33)(cid:35) . (40)We define the subset ¯ N ⊆ N containing all nodes at which at least one supplier or consumer isconnected. We also define the subset L n := L recn ∪ L sndn . Theorem 6.
Consider the stochastic clearing model (39) and assume that the incremental bidprices are positive and that ∆ α f, + (cid:96) , ∆ α f, − (cid:96) , ∆ α θ, + n , ∆ α θ, − n > , (cid:96) ∈ L , n ∈ N . The price distortions M πn , n ∈ N are bounded as − ∆ ¯ α + n ≤ M πn ≤ ∆ ¯ α − n , n ∈ ¯ N , − ∆ α + n ≤ M πn ≤ ∆ α − n , n ∈ N \ ¯ N , where ∆ ¯ α + n = min (cid:26) min i ∈G n ∆ α g, + i , min j ∈D n ∆ α d, + j , ∆ α + n (cid:27) , n ∈ ¯ N , ∆ ¯ α − n = min (cid:26) min i ∈G n ∆ α g, − i , min j ∈D n ∆ α d, − j , ∆ α − n (cid:27) , n ∈ ¯ N and ∆ α + n = (cid:80) (cid:96) ∈L n (cid:0) ∆ α f, + (cid:96) + (1 − B (cid:96)n )∆ α f, − (cid:96) (cid:1) + ∆ α θ, + n (cid:80) (cid:96) ∈L n B (cid:96) , n ∈ N , ∆ α − n = (cid:80) (cid:96) ∈L n B (cid:96)n ∆ α f, − (cid:96) + ∆ α θ, − n (cid:80) (cid:96) ∈L n B (cid:96) , n ∈ N . Proof
The stationarity conditions of the partial Lagrange function with respect to the day-ahead quantities g i , d j and phase angles θ n are given by0 ∈ ∂ d j L = (∆ α d, + j + ∆ α d, − j ) ∂ d j E [( D j ( ω ) − d j ) + ] + ∆ α d, + j + π n ( j ) − E (cid:2) Π n ( j ) ( ω ) (cid:3) j ∈ D (41a)0 ∈ ∂ g i L = (∆ α g, + i + ∆ α g, − i ) ∂ g i E [( G i ( ω ) − g i ) + ] + ∆ α g, − i − π n ( i ) + E (cid:2) Π n ( i ) ( ω ) (cid:3) i ∈ G (41b)0 ∈ ∂ θ n L = (cid:88) (cid:96) ∈L n (cid:0) ∆ α f, + (cid:96) + ∆ α f, − (cid:96) (cid:1) ∂ θ n E (cid:32) (cid:88) m ∈N B (cid:96)m (Θ m ( ω ) − θ m ) (cid:33) + + (cid:88) (cid:96) ∈L n B (cid:96)n ∆ α f, − (cid:96) + (cid:0) ∆ α θ, + n + ∆ α θ, − n (cid:1) ∂ θ n E (cid:2) (Θ n ( ω ) − θ n ) + (cid:3) + ∆ α θ, − n − (cid:88) (cid:96) ∈L recn B (cid:96)n − (cid:88) (cid:96) ∈L sndn B (cid:96)n ( π n − E [Π n ( ω )]) n ∈ N , (41c) avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing Article submitted to
Operations Research ; manuscript no. where we recall that n ( i ) is the node at which supplier i is connected and n ( j ) is the node at whichdemand j is connected. Following the same bounding procedure used in the proof of Theorem 2,we obtain − ∆ α d, + j ≤ M πn ( j ) ≤ ∆ α d, − j , j ∈ D− ∆ α g, + i ≤ M πn ( i ) ≤ ∆ α g, − i , i ∈ G . Rearranging (41c), we obtain − (cid:88) (cid:96) ∈L n B (cid:96)n ∆ α f, − (cid:96) − ∆ α θ, − n + (cid:88) (cid:96) ∈L recn B (cid:96)n − (cid:88) (cid:96) ∈L sndn B (cid:96)n M πn ∈ S n , (42)where S n := (cid:88) (cid:96) ∈L n (cid:0) ∆ α f, + (cid:96) + ∆ α f, − (cid:96) (cid:1) ∂ θ n E (cid:32) (cid:88) m ∈N B (cid:96)m (Θ m ( ω ) − θ m ) (cid:33) + + (cid:0) ∆ α θ, + n + ∆ α θ, − n (cid:1) ∂ θ n E (cid:2) (Θ n ( ω ) − θ n ) + (cid:3) (43)Because ∂ θ n E (cid:104)(cid:0)(cid:80) m ∈N B (cid:96)m (Θ m ( ω ) − θ m ) (cid:1) + (cid:105) ⊆ [ − ,
0] and ∂ θ n E (cid:2) (Θ n ( ω ) − θ n ) + (cid:3) ⊆ [ − , S n ⊆ (cid:34) − (cid:88) (cid:96) ∈L n (cid:0) ∆ α f, + (cid:96) + ∆ α f, − (cid:96) (cid:1) − ∆ α θ, + n − ∆ α θ, − n , (cid:35) , (44)and therefore, from (42) and (44), − (cid:88) (cid:96) ∈L n (cid:0) ∆ α f, + (cid:96) + ∆ α f, − (cid:96) (cid:1) − ∆ α θ, + n − ∆ α θ, − n ≤ − (cid:88) (cid:96) ∈L n B (cid:96)n ∆ α f, − (cid:96) − ∆ α θ, − n + (cid:88) (cid:96) ∈L recn B (cid:96)n − (cid:88) (cid:96) ∈L sndn B (cid:96)n M πn = − (cid:88) (cid:96) ∈L n B (cid:96)n ∆ α f, − (cid:96) − ∆ α θ, − n + (cid:88) (cid:96) ∈L n B (cid:96) M πn ≤ . (45)Hence, we have ∆ α + n ≤ M πn ≤ ∆ α − n . Because ∆ ¯ α + n and ∆ ¯ α − n are the smallest incremental bid pricesat node n ∈ ¯ N , we obtain the bound − ∆ ¯ α + n ≤ M πn ≤ ∆ ¯ α − n , n ∈ ¯ N . (cid:3) The price distortion is bounded for every node ∈ N . Moreover, if the penalty parameters∆ α f, + (cid:96) , ∆ α f, − (cid:96) , ∆ α θ, + n , ∆ α θ, − n are made arbitrarily small, then the price distortion at every nodebecomes arbitrarily small. We now state results that are natural extensions of Theorems 3 and 4. Theorem 7.
Consider the stochastic clearing model (39) , and let the assumptions of Theorem 6hold. The day-ahead quantities and phase angles are bounded by the real-time quantities and phaseangles as min ω ∈ Ω D j ( ω ) ≤ d j ≤ max ω ∈ Ω D j ( ω ) , j ∈ D avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing
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Operations Research ; manuscript no. min ω ∈ Ω G i ( ω ) ≤ g i ≤ max ω ∈ Ω G i ( ω ) , i ∈ G min ω ∈ Ω F (cid:96) ( ω ) ≤ f (cid:96) ≤ max ω ∈ Ω F (cid:96) ( ω ) , (cid:96) ∈ L min ω ∈ Ω Θ n ( ω ) ≤ θ n ≤ max ω ∈ Ω Θ n ( ω ) , n ∈ N . Proof
For the suppliers and demands, we can use the same procedure used in the proof ofTheorem 3. The bounds on the day-ahead flows and phase angles follow the same argument aswell. We use the definition (43) for simplicity. Consider the following two cases: • Case 1: The price distortion hits the lower bound for node n ; we thus have M πn = − ∆ α + n . Thisimplies that − (cid:80) (cid:96) ∈L n (cid:0) ∆ α f, + (cid:96) + ∆ α f, − (cid:96) (cid:1) − ∆ α θ, + n − ∆ α θ, − n ∈ S n from (42), and hence we have − ∈ ∂ θ n E (cid:2) (Θ n ( ω ) − θ n ) + (cid:3) (46a) − ∈ ∂ θ n E (cid:32) (cid:88) n ∈N B (cid:96)n (Θ n ( ω ) − θ n ) (cid:33) + , (cid:96) ∈ L n . (46b)From (26), equation (46a) implies that P (Θ n ( ω ) ≥ θ n ) = 1, and equation (46b) implies that P ( (cid:80) n ∈N B (cid:96)n Θ n ( ω ) ≥ (cid:80) n ∈N B (cid:96)n θ n ) = P ( F (cid:96) ( ω ) ≥ f (cid:96) ) = 1 for (cid:96) ∈ L n . Therefore, we have θ n ≤ Θ n ( ω ) , ω ∈ Ω and θ n ≤ max ω ∈ Ω Θ n ( ω ). Similarly, f (cid:96) ≤ F (cid:96) ( ω ) , ∀ ω ∈ Ω and f (cid:96) ≤ max ω ∈ Ω F (cid:96) ( ω ) for (cid:96) ∈ L n . • Case 2: The price distortion hits the upper bound for node n ; we thus have M πn = ∆ α − n . Thisimplies 0 ∈ S n from (42), and hence we have0 ∈ ∂ θ n E (cid:2) (Θ n ( ω ) − θ n ) + (cid:3) (47a)0 ∈ ∂ θ n E (cid:32) (cid:88) n ∈N B (cid:96)n (Θ n ( ω ) − θ n ) (cid:33) + , (cid:96) ∈ L n . (47b)From (26), equation (47a) implies that P (Θ n ( ω ) ≤ θ n ) = 1, and equation (46b) implies that P ( (cid:80) n ∈N B (cid:96)n Θ n ( ω ) ≤ (cid:80) n ∈N B (cid:96)n θ n ) = P ( F (cid:96) ( ω ) ≤ f (cid:96) ) = 1 for (cid:96) ∈ L n . Therefore, we have θ n ≥ Θ n ( ω ) , ω ∈ Ω and θ n ≥ min ω ∈ Ω Θ n ( ω ). Similarly, f (cid:96) ≥ F (cid:96) ( ω ) , ∀ ω ∈ Ω and f (cid:96) ≥ min ω ∈ Ω F (cid:96) ( ω ) for (cid:96) ∈ L n . (cid:3) Theorem 8.
Consider the stochastic clearing problem (39) , and let the assumptions of Theorem6 hold. If the price distortions M πn , n ∈ N are zero at the solution, then d j = Q D j ( ω ) (cid:32) ∆ α d, − j ∆ α d, + j + ∆ α d, − j (cid:33) , j ∈ D (48a) g i = Q G i ( ω ) (cid:18) ∆ α g, + i ∆ α g, + i + ∆ α g, − i (cid:19) , i ∈ G . (48b) avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing Article submitted to
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Proof
For (48a) and (48b), we can use the same procedure used in the proof of Theorem 4. (cid:3)
Corollary 3.
If the incremental bid prices are symmetric from Corollary 1, then d j = M ( D j ( ω )) , j ∈ D , and g i = M ( G i ( ω )) , i ∈ G . We treat the penalty terms purely as a means to constrain the day-ahead flows and phase anglesand induce the desired pricing properties. Our results indicate that this can be done with no harmby allowing ∆ α f, + (cid:96) , ∆ α f, − (cid:96) , ∆ α θ, + n , ∆ α θ, − n to be sufficiently small. Moreover, making these arbitrarilysmall guarantees that the expected social surplus of the stochastic problem (12) satisfies ϕ sto ≈ ϕ .The alternative is to simply impose day-ahead bounds of the forms (4d) and (4g) and to eliminatethe penalty terms on the flows and phase angles. In this case, however, we cannot guarantee thatthe price distortions are bounded, as we illustrate in the next section. In addition, similar to thecase of day-ahead quantities, imposing day-ahead bounds on flows would require us to choose aproper statistic for the bounds of flows and phase angles, which might not be trivial to do.We now prove revenue adequacy and zero uplift payments in expectation for the network-constrained formulation. We denote a minimizer of the partial Lagrange function (40) (subject tothe constraints (6f)-(6h)) as d ∗ j , D ∗ j ( · ) , g ∗ i , G ∗ i ( · ) , θ ∗ n , Θ ∗ n ( · ) , π ∗ n , Π ∗ n ( · ). Because the problem is convex,we know that the prices π ∗ n , Π ∗ n ( · ) satisfy( d ∗ j , D j ( · ) ∗ , g ∗ i , G ∗ i ( · ) , θ ∗ n , Θ ∗ n ( · )) = argmin d j ,D j ( · ) ,g i ,G i ( · ) ,θ n , Θ n ( · ) L ( d j , D j ( · ) , g i , G i ( · ) , θ n , Θ n ( · ) , π ∗ n , Π ∗ n ( · ))s.t. (6f) − (6h) . Moreover, at π ∗ n , Π ∗ n ( · ), the partial Lagrange function can be separated as L ( d j , D j ( · ) , g i , G i ( · ) , θ n , Θ n ( · ) , π ∗ n , Π ∗ n ( · )) = (cid:88) i ∈G L gi ( g i , G i ( · ) , π ∗ n , Π ∗ n ( · )) + (cid:88) j ∈D L dj ( d j , D j ( · ) , π ∗ n , Π ∗ n ( · )) + L θ ( θ n , Θ n ( · ) , π ∗ n , Π ∗ n ( · )) . (49)where the first two terms are defined in (34) and L θ ( θ (cid:96) , Θ (cid:96) ( · ) , π ∗ n , Π ∗ n ( · )) = (cid:88) (cid:96) ∈L E ∆ α f, + (cid:96) (cid:32) (cid:88) n ∈N B (cid:96)n (Θ n ( ω ) − θ n ) (cid:33) + + ∆ α f, − (cid:96) (cid:32) (cid:88) n ∈N B (cid:96)n (Θ n ( ω ) − θ n ) (cid:33) − + (cid:88) n ∈N E (cid:2) ∆ α θ, + n (Θ n ( ω ) − θ n ) + + ∆ α θ, − n (Θ n ( ω ) − θ n ) − (cid:3) − (cid:88) n ∈N π n (cid:88) (cid:96) ∈L recn (cid:0) B (cid:96)n θ n + B (cid:96),snd ( (cid:96) ) θ snd ( (cid:96) ) (cid:1) − (cid:88) (cid:96) ∈L sndn (cid:0) B (cid:96),rec ( (cid:96) ) θ rec ( (cid:96) ) + B (cid:96)n θ n (cid:1) − E (cid:88) n ∈N Π n ( ω ) (cid:88) (cid:96) ∈L recn (cid:2) B (cid:96)n (Θ n ( ω ) − θ n ) + B (cid:96),snd ( (cid:96) ) (cid:0) Θ snd ( (cid:96) ) ( ω ) − θ snd ( (cid:96) ) (cid:1)(cid:3) avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing
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Operations Research ; manuscript no. − (cid:88) (cid:96) ∈L sndn (cid:2) B (cid:96),rec ( (cid:96) ) (cid:0) Θ rec ( (cid:96) ) ( ω ) − θ rec ( (cid:96) ) (cid:1) + B (cid:96)n (Θ n ( ω ) − θ n ) (cid:3) (50)Consequently, one can minimize the partial Lagrange function by minimizing (34) and (50) inde-pendently. Theorem 9.
Consider the stochastic clearing problem (39) , and let the assumptions of Theorem6 hold. Any minimizer d ∗ j , D j ( · ) ∗ , g ∗ i , G i ( · ) ∗ , θ ∗ n , Θ ∗ n , π ∗ n , Π ∗ n ( · ) of (39) yields zero uplift payments forall players and revenue adequacy in expectation: M Ui = 0 , i ∈ G , (51a) M Uj = 0 , j ∈ D , (51b) M ISO ≤ . (51c) Proof
For fixed π ∗ n , Π ∗ n ( · ), by the separation of the partial Lagrange function, the zero upliftpayments directly result from Theorem 5. At fixed π ∗ n , Π ∗ n ( · ) we also note that θ n = Θ n ( · ) = 0is a feasible candidate solution for the maximization of L θ ( θ n , Θ n ( · ) , π ∗ n , Π ∗ n ( · )) and that, at thissuboptimal point, this term is also zero.If the flow balances (6b) and (6d) hold, we have0 = − (cid:88) n ∈N π n (cid:88) (cid:96) ∈L recn (cid:0) B (cid:96)n θ n + B (cid:96),snd ( (cid:96) ) θ snd ( (cid:96) ) (cid:1) − (cid:88) (cid:96) ∈L sndn (cid:0) B (cid:96),rec ( (cid:96) ) θ rec ( (cid:96) ) + B (cid:96)n θ n (cid:1) + (cid:88) i ∈G n g i − (cid:88) j ∈D n d j − E (cid:88) n ∈N Π n ( ω ) (cid:88) (cid:96) ∈L recn (cid:2) B (cid:96)n (Θ n ( ω ) − θ n ) + B (cid:96),snd ( (cid:96) ) (cid:0) Θ snd ( (cid:96) ) ( ω ) − θ snd ( (cid:96) ) (cid:1)(cid:3) − (cid:88) (cid:96) ∈L sndn (cid:2) B (cid:96),rec ( (cid:96) ) (cid:0) Θ rec ( (cid:96) ) ( ω ) − θ rec ( (cid:96) ) (cid:1) + B (cid:96)n (Θ n ( ω ) − θ n ) (cid:3) + (cid:88) i ∈G n ( G i ( ω ) − g i ) − (cid:88) j ∈D n ( D j ( ω ) − d j ) (cid:33)(cid:35) . (52)Consequently, for any arbitrary set of prices π n , Π n ( · ), we have − (cid:88) n ∈N π n (cid:88) (cid:96) ∈L recn (cid:0) B (cid:96)n θ n + B (cid:96),snd ( (cid:96) ) θ snd ( (cid:96) ) (cid:1) − (cid:88) (cid:96) ∈L sndn (cid:0) B (cid:96),rec ( (cid:96) ) θ rec ( (cid:96) ) + B (cid:96)n θ n (cid:1) − E (cid:88) n ∈N Π n ( ω ) (cid:88) (cid:96) ∈L recn (cid:2) B (cid:96)n (Θ n ( ω ) − θ n ) + B (cid:96),snd ( (cid:96) ) (cid:0) Θ snd ( (cid:96) ) ( ω ) − θ snd ( (cid:96) ) (cid:1)(cid:3) − (cid:88) (cid:96) ∈L sndn (cid:2) B (cid:96),rec ( (cid:96) ) (cid:0) Θ rec ( (cid:96) ) ( ω ) − θ rec ( (cid:96) ) (cid:1) + B (cid:96)n (Θ n ( ω ) − θ n ) (cid:3) avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing Article submitted to
Operations Research ; manuscript no. = (cid:88) n ∈N π n (cid:32) (cid:88) i ∈G n g i − (cid:88) j ∈D n d j (cid:33) + E (cid:34) (cid:88) n ∈N Π n ( ω ) (cid:32) (cid:88) i ∈G n ( G i ( ω ) − g i ) − (cid:88) j ∈D n ( D j ( ω ) − d j ) (cid:33)(cid:35) = (cid:88) i ∈G n π n ( i ) g i + E (cid:34) (cid:88) i ∈G n Π n ( i ) ( ω ) ( G i ( ω ) − g i ) (cid:35) − (cid:88) j ∈D n π n ( j ) d j − E (cid:34) (cid:88) j ∈D n Π n ( j ) ( ω ) ( D j ( ω ) − d j ) (cid:35) = M ISO . Therefore, we have0 ≥L θ ( θ ∗ n , Θ ∗ n ( · ) , π ∗ n , Π ∗ n ( · )) ≥ − (cid:88) n ∈N π n (cid:88) (cid:96) ∈L recn (cid:0) B (cid:96)n θ n + B (cid:96),snd ( (cid:96) ) θ snd ( (cid:96) ) (cid:1) − (cid:88) (cid:96) ∈L sndn (cid:0) B (cid:96),rec ( (cid:96) ) θ rec ( (cid:96) ) + B (cid:96)n θ n (cid:1) − E (cid:88) n ∈N Π n ( ω ) (cid:88) (cid:96) ∈L recn (cid:2) B (cid:96)n (Θ n ( ω ) − θ n ) + B (cid:96),snd ( (cid:96) ) (cid:0) Θ snd ( (cid:96) ) ( ω ) − θ snd ( (cid:96) ) (cid:1)(cid:3) − (cid:88) (cid:96) ∈L sndn (cid:2) B (cid:96),rec ( (cid:96) ) (cid:0) Θ rec ( (cid:96) ) ( ω ) − θ rec ( (cid:96) ) (cid:1) + B (cid:96)n (Θ n ( ω ) − θ n ) (cid:3) = M ISO , where the second inequality holds because (cid:88) (cid:96) ∈L E ∆ α f, + (cid:96) (cid:32) (cid:88) n ∈N B (cid:96)n (Θ n ( ω ) − θ n ) (cid:33) + + ∆ α f, − (cid:96) (cid:32) (cid:88) n ∈N B (cid:96)n (Θ n ( ω ) − θ n ) (cid:33) − + (cid:88) n ∈N E (cid:2) ∆ α θ, + n (Θ n ( ω ) − θ n ) + + ∆ α θ, − n (Θ n ( ω ) − θ n ) − (cid:3) ≥ . (cid:3) We highlight that the introduction of the penalty terms for flows does not affect revenue adequacyand cost recovery because the partial Lagrange function remains separable for fixed prices.
6. Computational Studies
In this section, we illustrate the different properties of the stochastic model. We also demonstratethat the stochastic model outperforms the deterministic one in all the metrics proposed. We alsoseek to highlight stochastic formulations provide benefits that go beyond improvements in socialsurplus. The optimization problems considered in this section were solved using
CPLEX-12.6.1 .All models can be accessed at http://zavalab.engr.wisc.edu/data . We first consider System I sketched in Figure 1. The system has two deterministic suppliers onnodes 1 and 3 and a stochastic supplier on node 2. The stochastic supplier has three possiblecapacity scenarios G ( ω ) = { , , } MWh of equal probabilities p ( ω ) = { / , / , / } . For avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing
Article submitted to
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Scheme of System I. ¯ G = 50 ↵ = 20 ¯ G = 50 ↵ = 10 ¯ D = 100 ↵ = 10001 2 32 ¯ G ( ! ) = { , , } ↵ = 1 ¯ F = 25 ¯ F = 50 deterministic clearing, the day-ahead capacity limit ¯ g for the wind supplier will be set to 50 MWh,the expected value forecast. The demand in node 2 is deterministic at a level of 100 MWh. Weuse α d = V OLL =1000$/MWh as the bid price and an incremental bid price of ∆ α d = 0 . α gi for the suppliers are { } $/MWh, and the incremental bid prices ∆ α gi are { } $/MWh. The transmission capacities of lines 1 → → F → = { , , } MWh and ¯ F → = { , , } , respectively, with the penalty of ∆ α f = 0 . α θn = 0 . ϕ g as well as prices and quantities. Because the demand is deterministic, thesurplus for the consumers ϕ load is a constant. Consequently, we show only ϕ g . For the deterministicsetting, the expected supply surplus ϕ g is $835, and the day-ahead prices π n are { } $/MWh.The price difference between the first two nodes results from the binding day-ahead flow for line1 → n ( ω ) are { } , { } , and { } $/MWh with expected value E [Π n ( ω )] = { } . There is a strongdistortion in the prices, indicated by the metrics M πavg = 109 and M πmax = 325.We now analyze the clearing of the stochastic formulation. The day-ahead prices are { } and the real-time prices are { } , { } , { } with expected value { } .The price distortion metrics M πavg , M πmax are both zero. The stochastic WS (wait-and-see) solutionis consistent in that it leads to no corrections of quantities in the real-time market and it yields thesame day-ahead and real-time prices. Thus, we can guarantee convergence of day-ahead and real-time prices for each scenario only in the presence of perfect information . We note that the expected avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing Article submitted to
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Table 1
System I. Comparison of quantities, prices, and social surplus. g i π n G i ( ω ) Π n ( ω ) E [Π n ( ω )] ϕ g M πmax { , , } { , , } Deterministic { , , } { , , } { , , } { , , } { , , }
835 325 { , , } { , , }{ , , } { , , } Stochastic { , , } { , , } { , , } { , , } { , , }
835 0 { , , } { , , }{ , , } { , , } { , , } { , , } Stochastic-WS { , , } { , , } { , , } { , , } { , , }
800 NA { , , } { , , } { , , } { , , } Table 2
System I. Comparison of suppliers and ISO revenues. E [ P gi ( ω )] E [ C gi ( ω )] M ISO
Deterministic { -7219 ,569 } { } -8400Stochastic { } { } -12719Stochastic-WS { } { } -9546 surplus as well as the day-ahead and real-time quantities for the stochastic and deterministicformulations are the same. The reason is that the the deterministic and stochastic formulations havethe same primal solution. This situation might lead the practitioner to believe that no benefits areobtained from the stochastic formulation. The prices obtained, however, are completely different.Hence one can see that arguments based on social surplus do not fully capture the benefits ofstochastic formulations.The different prices obtained with both formulations lead to drastically different payment dis-tributions among the market participants. As seen in Table 2, for the deterministic setting thesuppliers obtain expected payments E [ P gi ( ω )] of $ { } . The wind supplier receives neg-ative payments, and requires an uplift to enable cost recovery. In this case, the expected cost E [ C gi ( ω )] for the wind supplier is $52 and thus requires an expected uplift M Ui of $7,271. Forthe stochastic formulation, the expected payments are $ { } . The wind supplier haspositive payments and no uplift is required. This situation illustrates that the stochastic settingallocates resources efficiently. Note that all formulations are revenue adequate in expectation. Remark 1.
The optimization problems for System I are highly degenerate, and thus multiple dualsolutions (i.e., prices in our context) are available. We highlight that the solutions reported in thissection were obtained from the barrier method (without crossover) implemented in
CPLEX-12.6.1 ,which would provide central point solutions. However, we also report the solutions obtained fromdifferent linear programming algorithms in Appendix. avala, Kim, Anitescu, and Birge:
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Operations Research ; manuscript no. We now demonstrate that adding bounds onthe day-ahead flows and phase angles, as opposed to adding penalty terms, can affect the pricingproperties of the stochastic model. The price distortions π − E [Π( ω )] obtained using day-aheadbounds (without the penalty term) are {− . , , } while those obtained with the penalty termusing a penalty of ∆ α f = ∆ α θ = 0 .
001 (without the bounds) are { , , } . The penalty term achievesthe desired pricing property. The day-ahead flows obtained with the penalty terms are { , } ;these are the medians of the real-time flows which are { , } for scenario 1, { , } for sce-nario 2, and { , } for scenario 3. This also implies that the day-ahead flows are bounded andtherefore day-ahead bounds are redundant. Similarly, the day-ahead phase angles obtained withthe penalty terms are {− . , − . , − . } ; these are the medians of the real-time phase angles {− . , − . , − . } , {− . , − . , − . } , and {− . , − . , − . } for scenarios 1, 2 and 3,respectively. This suggests that the day-ahead bounds bias the statistics. We experiment the effect that the incrementalbid prices have on the price distortion. Consider the case in which the demand in the central nodeis also stochastic and with scenarios D ( ω ) = { , , } . We set the incremental bid price forthe stochastic supplier ∆ α g to 1 .
0. For the demand incremental bid prices ∆ α d = 0 . , . , . M πmax are 0 . , . , . We now consider the more complex system presented in Figure 2. This is an adapted version ofthe system presented in Pritchard et al. (2010). The system has two stochastic suppliers in nodes2 and 4, three deterministic suppliers in nodes 1, 3, and 5 and one stochastic demand in node 6.The demand is treated as inelastic. The demand follows a normal distribution with mean 250 andstandard deviation 50. We use sample average approximation for solving this model and generate25 scenarios of the demand with equal probabilities. The stochastic suppliers can have 5 possiblecapacities { , , , , } MW. Each scenario represents one of the 25 different permutationsfrom all possible capacities. The bid prices α gi for the suppliers are { , , , , } $/MWh,and the incremental bid prices ∆ α gi are { , . , , . , } . We set α d = V OLL = 1000 $/MWhand ∆ α d = 0 . α f(cid:96) = ∆ α θn = 0 . avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing Article submitted to
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Figure 2
Scheme of System II. ¯ G = 100 ¯ G = 100 ¯ G = 50 ↵ = 100 ↵ = 100 ↵ = 200 ↵ = 1000 ¯ D ( ! ) ⇠ N (250 , ↵ = 1 ↵ = 1 ¯ G ( ! ) = { , , , , } ¯ G ( ! ) = { , , , , } ¯ F = 50¯ F = 50¯ F = 50 ¯ F = 25 ¯ F = 25¯ F = 25 ¯ F = 100¯ F = 100¯ F = 100 Table 3
System II. Comparison of day-ahead prices and surplus with deterministic and stochasticformulations. ϕ M πn π n Deterministic -139569 { , − , − , − , − , } { , − , , , , } Stochastic -139569 {− . , , , , , } { , − , , , , } Stochastic-WS -139737 NA NA
The results are presented in Tables 3 and4. We first note that the price distortion for the deterministic setting is large, reaching values aslarge as 158 $/MWh. Also note that the distortion (premia) is small and positive in nodes 1 and 6and large and negative in the other nodes. This is inefficient because it biases incentives towards asubset of players. The system is overly optimistic about performance in the real-time market wheremultiple scenarios exhibit transmission congestion, but the deterministic setting cannot foreseethis. The stochastic formulation has almost the same expected social surplus as the deterministicformulation, but the price distortion is eliminated.In Table 4 we see that payments for both formulations are similar except for the 4th supplier,which is a stochastic supplier. This supplier receives a negative payment and requires uplift underdeterministic clearing. The uplift is eliminated by using the stochastic formulation. The expectedpayments collected with the stochastic here-and-now solution are close to those of the perfectinformation solution. avala, Kim, Anitescu, and Birge:
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System II. Comparison of suppliers and ISO revenues with deterministic andstochastic formulations. E [ P gi ( ω )] E [ C gi ( ω )] M ISO
Deterministic { , , , − , } { , , , , } -129111Stochastic { , , , , } { , , , , } -116885Stochastic-WS { , , , , } { , , , , } -117518 Table 5
System II. Day-ahead, quartiles of real-time quantities, and mean ofreal-time quantities for suppliers. ∆ α g, + i ∆ α g, − i Quantity Gen 1 Gen 2 Gen 3 Gen 4 Gen 50 . α gi . α gi g i
90 0 0 20 50 Q G i ( ω ) (0 .
25) 90 0 0 20 50 E [ G i ( ω )] 84.6 0.4 5.7 19.7 481 . α gi . α gi g i Q G i ( ω ) (0 .
5) 91.7 0 0 20.8 50 E [ G i ( ω )] 84.6 0.4 5.7 19.7 481 . α gi . α gi g i Q G i ( ω ) (0 .
75) 91.7 0 2.5 20.8 50 E [ G i ( ω )] 84.6 0.4 5.7 19.7 48 In Table 5 wepresent the day-ahead quantities g i convergent to a quantile of the real-time quantities. The day-ahead quantities g i with the quartiles (i.e., quantiles at p = 0 . , . , .
75) of the real-time quanti-ties are compared with the means E [ G i ( ω )] of the real-time quantities. Recall that Q G i ( ω ) (0 .
5) = M [ G i ( ω )]. We use asymmetric incremental bid prices ∆ α g, + i , ∆ α g, − i as set in Table 5. As can beseen, convergence is achieved for all suppliers. We now consider the case in which there are random linefailures. We consider 25 scenarios and assume that each one of the lines 1 →
2, 2 →
6, 3 → →
6, and 2 → avala, Kim, Anitescu, and Birge: Stochastic Market Clearing with Consistent Pricing Article submitted to
Operations Research ; manuscript no.
Table 6
System II. Comparison of suppliers and ISO revenues with deterministic andstochastic formulations under transmission line failtures. E [ P gi ( ω )] E [ C gi ( ω )] M ISO
Deterministic { , , , , } { , , , , } Stochastic { , , , , } { , , , , } -39691Stochastic-WS { , , , , } { , , , , } -39965 Table 7
IEEE-118 System. Comparison of suppliers andISO revenues with deterministic and stochastic formulations. M U ϕ g M πavg M πmax M ISO
Deterministic -151 36531 0.270 2.331 -2306Stochastic 0 36527 0.001 0.003 -2009Stochastic-WS 0 36410 0 0 -1982
We now demonstrate the properties of the stochastic setting in a more complex network. TheIEEE-118 system comprises 118 nodes, 186 lines, 91 demand nodes, and 54 suppliers. We assumethat three stochastic suppliers are located at buses 10, 65 and 112, and that each supplier hasan installed capacity of 300 MWh. This represents 14% of the total generation capacity. We alsoassume that a generation level for a given stochastic supplier follows a normal distribution withmean 300 MWh and standard deviation 150 MWh. The total generation capacity is 7,280 MW,and the total load capacity is 3,733 MW. We use sample average approximation and generate 25scenarios for the stochastic suppliers. We use 10% of the generation cost for the incremental bidprices ∆ α gi . The demands are assumed to be deterministic, and we set ∆ α dj = 0 . α f(cid:96) = ∆ α θn = 0 . M πmax = 0 .
7. Conclusions and Future Work
We have demonstrated that deterministic market clearing formulations introduce strong and arbi-trary distortions between day-ahead and expected real-time prices that bias incentives and blockdiversification. We present a stochastic formulation capable of eliminating these issues. The formu-lation is based on a social surplus function that accounts for expected costs and penalizes deviationsbetween day-ahead and real-time quantities. We show that the formulation yields day-ahead prices avala, Kim, Anitescu, and Birge:
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System I solutions from primal simplex method π n Π n ( ω ) E [Π n ( ω )] E [ P gi ( ω )] M ISO M πmax { , , } Deterministic { , , } { , , } { , , } { , − , } -8400 327 { , , }{ , , } Stochastic { , , } { , , } { , , } { , , } -250 0 { , , }{ , , } { , , } Stochastic-WS { , , } { , , } { , , } { , , } -8417 NA { , , } { , , } that are close to expected real-time prices. In addition, we show that day-ahead quantities convergeto the quantile of real-time counterparts.Future work requires extending the model in multiple directions. First, it is necessary to capturethe progressive resolution of uncertainty by using multi-stage models and to incorporate rampingconstraints and unit commitment decisions. Second, it is necessary to construct formulations thatdesign day-ahead decisions that approach ideal wait-and-see behavior. Morales et al. (2014) demon-strate that this might be possible to do by using bi-level formulations, but a more detailed analysisis needed. Third, the proposed stochastic model is computationally more challenging than existingmodels available in the literature because it incorporates the detailed network in the first-stage.This leads to problems that much larger first-stage dimensions which are difficult to decomposeand parallelize. Consequently, scalable strategies are needed. Finally, it is necessary to exploreimplementation issues of stochastic markets such as effects of distributional errors. Appendix. System I Solutions from Different LP Algorithms
Since System I problem is degenerate, multiple dual solutions are available. We report the solu-tions of System I obtained from different LP algorithms. Tables 8 and 9 present the solutions fromprimal simplex method and dual simplex method, respectively, from using
CPLEX-12.6.1 . Notethat we present the dual solutions (i.e., prices) only, since all the primal solutions (i.e., day-aheadand real-time quantities) are identical. We also note that in all cases we have price converges. Thedeterministic settings result in the positive price distortions.
Acknowledgments
This work was supported by the U.S. Department of Energy, Office of Science, under Contract DE-AC02-06CH11357. We thank Mohammad Shahidehpour for providing the data for the IEEE-118 system. The firstauthor thanks Paul Gribik, Tito Homem-de-Mello, and Bernardo Pagnoncelli for technical conversations andAntonio Conejo for providing references to existing work. avala, Kim, Anitescu, and Birge:
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Table 9
System I solutions from dual simplex method π n Π n ( ω ) E [Π n ( ω )] E [ P gi ( ω )] M ISO M πmax { , , } Deterministic { , , } { , , } { , , } { , − , } -8400 326 { , , }{ , , } Stochastic { , , } { , , } { , , } { , , } -183 0 { , , }{ , , } { , , } Stochastic-WS { , , } { , , } { , , } { , , } -8333 NA { , , } { , , } References
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