1

Coordinated Transaction Scheduling in Multi-AreaPower Systems with Strategic Participants

Mariola Ndrio Subhonmesh Bose Ye Guo Lang Tong

Abstract—In this paper we focus on the tie-line schedulingproblem in multi-area power systems. We discuss two schedulingschemes: the theoretical method of tie optimization (TO) andthe state of the art—coordinated transaction scheduling (CTS).Through game-theoretic analysis we compare the outcome ofCTS with that of TO and show that the TO result can beachieved provided sufficient liquidity in the market. When CTSmarket participants learn to bid/offer over time, simulationresults confirm that increased market liquidity leads to moreefficient tie-line schedules.

Index Terms—coordinated transaction scheduling, seams is-sues, game theory

I. INTRODUCTION

Different parts of the electric power grid are controlled bydifferent system operators (SOs) that aim to ensure nondis-criminatory access to power suppliers in the grid. The SOassumes operational control of the transmission facilities overa specified geographical area and administers the wholesalemarkets for electricity within its footprint. Although each areais operated independently, the power grids are interconnected.Every moment, power flows from one region to the othervia transmission lines, which we call tie-lines. These tie-linesare valuable assets as they are capable of transferring largeamounts of power across areas and cover significant portionof their demand. To put this into perspective, the tie-linesat the interface between NYISO and ISO-NE have a totalcapacity of 1800 MW, which is approximately 12% of NewEngland’s electricity consumption and 10% of New York’s[1]. As such, tie-line scheduling is important to enable theseamless and efficient operation of the grid as well as toharness geographically dispersed renewable energy resources.

Tie-lines have been historically underutilized as evidencedby persistent price differences between regional markets [1].Many factors contribute to inefficient schedules, includinglack of appropriate coordination between SOs and ineffectivemarket rules and procedures. Multiple pairs of SOs have takensteps to directly address these so-called seams issues throughthe design of tie-line scheduling mechanisms [2], [3]. Thefirst option is tie optimization (TO) that requires the SOs toexchange their effective supply stacks and jointly determinethe optimal tie-line schedule. The implementation of this

M. Ndrio and S. Bose are with the Department of Electrical and ComputerEngineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801.Ye Guo is with Tsinghua-Berkeley Shenzhen Institute, Shenzhen China,518071. L. Tong is with the School of Electrical and Computer Engineering,Cornell University, Ithaca, NY 14853. Emails: {ndrio2,boses}@illinois.edu,guo-ye@sz.tsinghua.edu.cn, lt35@cornell.edu. The work was partially sup-ported by Power System Engineering Research Center (PSERC).

scheduling scheme runs counter to the requirement that eachSO be independent of any financial/commercial interests in itsadministrative region. Furthermore, it amounts to upending theearlier market-based process for tie-line scheduling.

Coordinated Transaction Scheduling (CTS) is the state-of-the-art market-based tie-line scheduling procedure espousedby multiple SO pairs, e.g. ISO-NE and NYISO, NYISO andPJM as well as PJM and MISO [2], [3]. CTS aims to strikea healthy balance between the earlier market-based schemeand a purely SO-driven TO scheme. CTS market participantsbid to buy power at the proxy bus in one area and offer tosell exactly the same amount at the proxy bus in the otherarea. While these participants are virtual, in the sense thatthey neither produce nor consume any power, their bids/offersinfluence the scheduling of power from one area to the other.

In this paper, we aim to answer the question: how does thestrategic interaction among the CTS market participants affectthe efficiency of tie-line scheduling. To this end, we model theCTS market as a game, and analyze its equilibrium propertiesto answer the above question. We begin by discussing TOand CTS in Section II. Our competition model, shown inSection III, draws on parameterized supply function gamesstudied in [4]–[6] that have proven effective to study organizedwholesale electricity markets. We characterize the outcomeof the CTS market as Nash equilibria of the CTS game inSection IV. Our analysis shows that (i) a liquid CTS marketwill achieve the efficient tie-line schedule computed by TOand, (ii) the market participant with the highest budget canassume a pivotal role. On the other hand, an insufficientlyliquid CTS market results in efficiency loss. In Section V, weprovide numerical simulations to illustrate the key insights. Weconclude the paper with directions for future work in SectionVI.

II. MECHANISMS FOR TIE-LINE SCHEDULING

We begin by describing TO and CTS—two different mech-anisms for tie-line scheduling. For ease of presentation, weconsider a stylized two-area power system, shown in Figure1, that are connected via a single tie-line with the inter-area power flow denoted by Q. Furthermore, to simplify theexposition we ignore the effects of total transfer capability onthe tie-lines.

A. Tie optimization

The core concept of TO is that SOs determine the tie-lineschedule in the same way they optimize power flows in the in-ternal transmission lines. Specifically, each SO computes their

2

Area a Area b

~ ~Q

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Fig. 1: The stylized two-area power system.

supply stacks by solving an area-wise parametric economicdispatch by varying the amount of power flowing on the tie-line. An example of supply stacks is shown in Figure 2. Thesupply stack of area a represents the incremental dispatch costof delivering power at its side of the interface. Similarly, thesupply stack of area b represents the decremental dispatch costof reduced supply, shown in descending order. In this example,the optimal direction of the tie-line schedule is from area ato b since for zero scheduled flow, area b operates at higherdispatch costs than area a. The TO schedule, denoted by QTO,is at the level where dispatch costs at the border become equal,or where the supply stacks intersect. This tie-line scheduleminimizes the aggregate dispatch costs across the two areas.In effect, TO requires SOs to trade directly with each otheron behalf of the market participants in their respective areas,which makes the SOs active participants of trade rather thanfinancially neutral market operators. The current practice ofCTS, described next, relies on virtual traders whose offers/bidsare utilized together with the supply stacks to arrive at the tie-line schedule.

supply stack of area a – supply stack of area b

TTC

$/energy

flow to SOa flow to SOb

supply stack of area b

supply stack of area a

supply stack of interface bids

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Fig. 2: Illustration of the TO and CTS scheduling mechanisms.

B. Coordinated Transaction Scheduling

CTS markets have interface bids that consist of threeelements: the minimum price (spread) the bidder is willingto accept, the maximum quantity to be transacted and thedirection of the trade, i.e., the source and sink. A CTS marketparticipant can offer to transport power across areas withoutphysically consuming or producing it. They have no obligationfor the physical power delivery. The transaction for a CTSbidder is purely financial.

Under CTS, a coordinator (often one of the SOs) poolsthe virtual bids at the proxy buses and the supply stacksfrom both operators to assemble the aggregate interface supply

stack, shown in Figure 2. All the bids indicating the optimaldirection are stacked from lowest to highest price to createtheir own “supply curve”. The price spread curve is derivedby subtracting the supply stack of area a from that of area b.The CTS schedule, denoted by QCTS, is set at the intersectionof the interface supply stack and the price spread. An interfacebid is accepted if its offer price is less than the price spreadat the tie-line schedule. Therefore, all interface bids to the leftof the CTS schedule are accepted; all bids to the right are not.In the next section, we model the CTS mechanism as a gameand compare the equilibrium outcome with that in TO.

Remark 1. Multi-area power systems often share more thanone tie-line. The CTS offers are cleared against demandsfor inter-area transport and price differences at proxy buseswithin each area. Proxy buses define trading locations, andtransactions cleared at proxy buses may differ from physicalpower flows owing to Kirchhoff’s laws, leading to the so-calledloop flow problem. Consideration of such effects is out of thescope of this paper. We rather focus on the effect of strategicinteraction in CTS markets and how its outcome compareswith that under the efficient TO scheme. Related work can befound in [7], [8].

III. GAME-THEORETIC MODEL OF CTS MARKET

We model the interaction among virtual bidders in the CTSmarket as a game, and characterize its Nash equilibrium.The equilibrium analysis provides insights into the marketoutcome, given the strategic incentives of its participants. Theensuing analysis only considers a CTS market between theadjoining footprints of two SOs that share a single tie-line,allowing us to sidestep complications that arise due to loopflows and proxy-bus approximations. Similar derivations witha more detailed model is relegated to future work.

For the model system shown in Figure 1 we assume thedispatch costs to be quadratic. Therefore, the supply stacksare linear functions of the tie-line schedule. Subtracting thesupply stacks of area a from that of area b, we obtain a demandfunction for transport of power from area a to b. Consider thedemand function, given by

Q(p) :=1

β(α− p). (1)

That is, area b will import Q MW from area a at a pricedifference of p $/MW at the border buses.

Recall from Figure 2 that tie-optimization (TO) yields atie-line schedule where the supply stacks in areas a and bintersect. In other words, the price difference is zero at theschedule. With the demand function described by (1), the tie-line schedule under TO is given by

QTO :=α

β. (2)

In this section, we model the CTS market as a game andcontrast its equilibrium outcome with that under TO. Our goalis to answer the question: can CTS markets yield the efficienttie-line schedule?

We model the CTS market with a collection of N marketparticipants, labeled 1, . . . , N ; henceforth, we refer to them

3

as players. Recall that each player offers to transport up toa certain capacity at a specified price difference between theareas in a given direction. Without loss of generality, assumethat all participants choose to transport in the same direction,i.e., from area a to area b. The offered pair of price differenceand transport quantity against the elastic inter-area demand fortransport in (1) fulfill a role similar to that of generators whooffer to supply electricity within their generation capacities ata specified price. This analogy motivates us to draw inspirationfrom prior work on game-theoretic analysis of organizedwholesale electricity markets to the study of CTS markets.

A. The transport offer of CTS market participants

Let each player provide two parameters θi, Ci to the SOswith the understanding that she is willing to transport up to

qi(p) := Ci −θip

(3)

amount of power from area a to b at price spread p. Figure 3reveals how the parameters θi, Ci affect the shape of the supplyoffer. Player i is willing to transport a maximum quantityof Ci, but at a minimum price of θi/Ci. The required pricedifference increases with the power transport, and approaches∞ as the latter approaches Ci. Transporting power aboveCi requires an infinite price difference. The parameterizedtransport offer in (3) is a smooth approximation to the onein practice, where a player is willing to transport up to Ci at aspecified price difference. A smooth approximation facilitatesthe analysis. Figure 3 illustrates how that approximation relatesto the offer format used in practice. Our transport offer isinspired by parametric supply function competition modelsconsidered in [4]. Notice that Ci multiplied by the realizedprice spread defines the payoff of player i, when θi is zero. Inpractice, the realized price spread is uncertain and a higher Ciexposes the player to a higher potential loss. Therefore, playeri expresses its total budget for potential losses or its liquidityin Ci.

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Interface bid of player i

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✓i

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CTS offer structure in practice Parameterized offer structure

Fig. 3: Parameterized interface bid of CTS market participant.

B. The CTS market clearing procedure

Given the players’ offers (θi, Ci) for i = 1, . . . , N , the SOssolve the following optimization problem to compute the tie-

line schedule:

maximizex∈RN

∫ 1ᵀx

0

(α− βq′) dq′ −N∑i=1

∫ xi

0

θiCi − qi

dqi. (4)

The above problem seeks a tie-line schedule where the offerstack for inter-area power transport from CTS market partic-ipants intersects the demand function in (1). The variable xidefines the quantity of power to be transported by player i,the collection of which we denote by x. The tie-line scheduleis then given by

QCTS := 1ᵀx∗,

where 1 is a vector of all ones and x∗ stands for x at optimalityof (4). The transport offer in (3) enters the SO’s problemthrough its implied cost of transport. Drawing parallels frommarket clearing procedures in organized wholesale electricitymarkets, the implied cost of transport is such that the resultingmarginal cost curve coincides with the transport offer. Thenature of the induced cost in (4) ensures that x∗i is alwaysnonnegative and less than Ci.

The transport procured from each player and the resultingtie-line schedule depend on the players’ offers. Denote themas x∗i (θθθ,C) and QCTS(θθθ,C), respectively. Here, θθθ and Crepresent the vector of θ’s and C’s, respectively.

IV. ANALYSIS OF THE CTS MARKET OUTCOME

The N players compete to provide transport of power fromarea a to b over the tie-line. Assume, henceforth, that playerstruly reflect their liquidities in C and model the CTS market asa game where players strategically choose θθθ. Player i’s payoffis then given by

πi(θi, θθθ−i) = p (QCTS(θθθ,C))x∗i (θθθ,C)

=[α− β1ᵀ

x∗(θθθ,C)]x∗i (θθθ,C), (5)

where θθθ−i is the collection of all θ’s, save player i’s. We haveused the notation

p(Q) = α− βQto denote the price difference with tie-line schedule Q asimplied by the demand function in (1).

We aim to analyze the properties of the market outcomeresulting from the strategic interaction among the CTS marketparticipants. To that end, we characterize the Nash equilibriaof the game among these players, seeking to maximize theirpayoffs through their offers θθθ, given their liquidities C. Anoffer profile θθθNE constitutes a Nash equilibrium if

πi(θNEi , θθθNE−i

)≥ πi

(θi, θθθ

NE−i)

for all θi ≥ 0. That is, no player has an incentive for aunilateral deviation from the equilibrium offer. In our firstresult, we characterize the equilibrium offer profile for CTSand compare the resulting tie-line schedule with that in tie-optimization.

Proposition 1. The CTS market always admits a Nash equi-librium. Moreover, suppose that C1, . . . , CN are distinct withCm as the maximum. If 1ᵀC−m ≥ α/β, then

θθθNE = 0, QCTS

(θθθNE,C

)=α

β.

4

Otherwise,

θNEi =

{β4C

2m − 1

4β p2(1ᵀC), if i = m,

0, otherwise,

QCTS

(θθθNE,C

)=

1

2

(α

β+ 1

ᵀC−m

).

The above result characterizes the result of strategic inter-action in CTS markets in terms of the Nash equilibria of itsgame formulation; it provides a comparison of that outcomewith that under TO 1. Recall that TO defines the tie-lineschedule QTO = α/β, at where the supply stacks of the twoareas intersect. TO defines the efficient outcome that the twoSOs can achieve together without an inter-area market. It isalso the outcome that a joint clearing of the two organizedmarkets in the two SOs’ footprints would yield. According toProposition 1, the strategic incentives of the CTS market drivesthe market outcome to the efficient tie-line schedule dictatedby TO, as long as the CTS market participants have enoughliquidity. More precisely, whenever the liquidity of all but theplayer with maximum budget is enough to cover the transportof power required by TO, the CTS market will emulate TO.That is, TO can be achieved in theory through the CTS marketmechanism if no single player is pivotal for efficient transport.For a CTS market with a large enough pool of players, onewould expect that the sum of C’s for the N−m players will beable to cover the demand for transport under TO, and hence,will result in an efficient schedule.

If the CTS market is not liquid enough, then the marketoutcome will yield a schedule different from that in TO. Thedifference between the schedules is given by

QTO −QCTS =1

2

(α

β− 1ᵀ

C−m

),

which is precisely half the shortfall of the aggregate liquidityof all but the player with the maximum budget from theefficient schedule QTO = α/β. In other words, the efficiencyloss grows with the lack of liquidity in the CTS market.

A. Proof of Proposition 1

The Hessian of the objective function of (4) can be shownto be negative definite in θθθ. Therefore, the objective functionis strictly concave and admits a unique maximizer x∗(θθθ,C).Hereon, we suppress the dependency on C for convenience.The first-order optimality conditions yield

x∗i (θθθ) = Ci −θi

α− β1ᵀx∗(θθθ). (6)

Summing over i = 1, . . . , N , we obtain

QCTS(θθθ) = 1ᵀx∗(θθθ)

=1

2β

(α+ β1

ᵀC−

√(α− β1ᵀC)

2+ 4β1ᵀθθθ

)=

1

2β

(α+ β1

ᵀC−

√p2 (1ᵀC) + 4β1ᵀθθθ

). (7)

1The main result holds more generally, but we only present it with distinctentries of C for ease of exposition.

Using the above relation in (6), we get

x∗i (θi, θθθ−i) = Ci −2θi

p (1ᵀC) +√p2 (1ᵀC) + 4β1ᵀθθθ

.

The payoff of player i then becomes

πi(θi, θθθ−i) =Ci2

(p(1ᵀC)+√p2 (1ᵀC) + 4β1ᵀθθθ

)− θi.

The above payoff is continuous in θθθ and strictly concave inθi. For θi > βC2

i , the payoff becomes negative, regardlessof θθθ−i. Therefore, it suffices to analyze the game where thestrategy space of the players is restricted to θi ∈ [0, βC2

i ]. Theexistence of a Nash equilibrium then follows from Rosen’sresult in [9, Theorem 1].

An equilibrium offer profile θθθNE satisfies

θNEi > 0 =⇒ ∂πi(θi, θθθ−i)

∂θi

∣∣∣∣θθθNE

= 0,

θNEi = 0 =⇒ ∂πi(θi, θθθ−i)

∂θi

∣∣∣∣θθθNE≤ 0,

(8)

where the above derivative is given by

∂πi(θi, θθθ−i)

∂θi

∣∣∣∣θθθNE

=βCi√

p2 (1ᵀC) + 4β1ᵀθθθNE− 1. (9)

Since the Ci’s are distinct, the derivative cannot vanish formore than one player. Therefore, θθθNE can have at most onenon-zero entry. If Cm = max{C1, . . . , CN}, then (9) implies

∂πm(θm, θθθ−m)

∂θm

∣∣∣∣θθθNE

>∂πi(θi, θθθ−i)

∂θi

∣∣∣∣θθθNE

(10)

for i 6= m. Equipped with the above relation, we infer from (8)that offer profiles at equilibrium satisfy θθθNE−m = 0. To computeθNEm , we consider two cases separately.• When 1ᵀC−m ≥ α/β, it is straightforward to verify thatθNEm = 0 together with θθθNE−m = 0 defines an equilibriumusing (8) and (9). Searching for an equilibrium with apositive θNEm , yields

θNEm =β2C2

m − p2(1TC)

4β< 0 (11)

contradicting the nonnegativity of θ’s. Therefore, θθθNE = 0defines the unique Nash equilibrium.

• When 1ᵀC−m < α/β, (8) can be utilized to rule out theall-zero equilibrium. The only possible equilibrium has anon-zero θNEm , given by (11), which in this case can beshown to be positive.

The rest follows from substituting the computed θθθNE in (7).

V. NUMERICAL EXPERIMENTS

The Nash equilibria in CTS markets from Proposition 1predict the market outcomes, given the strategic incentivesof the players. In practice, virtual traders repeatedly play insuch markets, and often employ simple offer strategies theyupdate over time, given their history of offers and rewards.We relegate the question of learning to offer in such marketsfor a future endeavor. In this section, we demonstrate througha numerical experiment that simple update rules for offers

5

(a)

pivotal player

(b) (c)

Fig. 4: Plots (a) and (b) show the trajectories of offered θθθ in the CTS markets with the update rule in (12) for the liquid and the illiquidmarket examples, respectively. Plot (c) shows the ratio of CTS vs. TO schedules for these two markets.

of CTS market participants results in market outcomes thatcorroborate the findings of Proposition 1; the tie-line scheduleis more efficient with a liquid market.

Consider N = 3 players, each of whom updates their θ’sas follows.

θk+1i = wi

(θki + riCi

)+ (1− wi)

(θki − r′iCi

), (12)

where wi, ri, r′i ∈ (0, 1). Based on (12), player i increases

(decreases) θi proportional to her liquidity depending on herpayoff outcome over a specified number of previous rounds.We construct the inter-area demand function using price andtie-line schedule data from the CTS market between ISO-NEand NYISO (obtained from NYISO’s website) over 11/1/2018- 11/3/2018, using linear regression as shown in Figure 5. Weclear the market using (4) against that demand function, per-turbed by a small additive noise to the price spread intercept,for which the average value of QTO is 1492.4 MW.

The CTS market participants are deemed agnostic to thedemand function, and follow the update rule as prescribed in(12). We study two cases:• Liquid market with C = (967, 892, 818), θθθNE = (0, 0, 0)• Illiquid market with C = (744, 446, 372), θθθNE =(3291, 0, 0).

Figures 4a and 4b provide the trajectories of θθθ offers of playersin these two cases. The plots reveal that θ’s remain closerto their equilibrium values for the liquid market than in theilliquid market. The player with the highest budget assumes apivotal role and garners the maximum profit, when the market

counter–economic schedules

under–utilization

Fig. 5: Price spread and tie-line schedules for the NE-NY interface.

is not liquid. Finally, QCTS/QTO remains close to unity inFigure 4c with a liquid market, implying that tie-line schedulefrom CTS is efficient in such markets. Lack of liquidity resultsin long excursions of the CTS schedule from the efficient one.

VI. CONCLUDING REMARKS

The game-theoretic analysis of the market-driven CTSscheme reveals that its incentives are aligned with the SO-driven TO. The preliminary simulations illustrate that highermarket liquidity improves scheduling efficiency. In futurework, we aim to investigate how the uncertainty in the inter-area demand affects the Nash equilibrium and the marketoutcome, and explore more sophisticated offer strategies ofCTS market participants.

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