Opportunities for Price Manipulation by Aggregators in Electricity Markets

Transcription

1 1 Opportuntes for Prce Manpulaton by Aggregators n Electrcty Markets Navd Azzan Ruh, Krshnamurthy Dvjotham, Nangjun Chen, and Adam Werman arxv: v1 [cs.gt] 21 Jun 216 Abstract Aggregators are playng an ncreasngly crucal role n the ntegraton of renewable generaton n power systems. However, the ntermttent nature of renewable generaton makes market nteractons of aggregators dffcult to montor and regulate, rasng concerns about potental market manpulaton by aggregators. In ths paper, we study ths ssue by quantfyng the proft an aggregator can obtan through strategc curtalment of generaton n an electrcty market. We show that, whle the problem of maxmzng the beneft from curtalment s hard n general, effcent algorthms exst when the topology of the network s radal (acyclc). Further, we hghlght that sgnfcant ncreases n proft are possble va strategc curtalment n practcal settngs. Index Terms Aggregators, renewables, optmal curtalment, market power, locatonal margnal prce (LMP). I. INTRODUCTION Increasng the penetraton of dstrbuted, renewable energy resources nto the electrcty grd s a crucal part of buldng a sustanable energy landscape. To date, the enttes that have been most successful at promotng and facltatng the adopton of renewable resources have been aggregators, e.g. as SolarCty, Tesla, Enphase, Sunnova, SunPower, ChargePont [1] [3]. These aggregators nstall and manage rooftop solar nstallatons as well as household energy storage devces and electrc vehcle chargng systems. Some have fleets wth upwards of 8 MW dstrbuted energy resources [4], [5], and the market s expected to trple n sze by 22 [6], [7]. Aggregators play a varety of mportant roles n the constructon of a sustanable grd. Frst, and foremost, they are on the front lnes of the battle to promote nstallaton of rooftop solar and household energy storage, pushng for wde-spread adopton of dstrbuted energy resources by households and busnesses. Second, and just as mportantly, they provde a sngle nterface pont where utltes and Independent System Operators (ISOs) can nteract wth a fleet of dstrbuted energy resources across the network n order to obtan a varety of servces, from renewable generaton capacty to demand response. Ths servce s crucal for enablng system operators to manage the challenges that result from unpredctable, ntermttent renewable generaton, e.g., wnd and solar. However, n addton to the benefts they provde, aggregators also create new challenges both from the perspectve of the aggregator and the perspectve of the system operator. On the sde of the aggregator, the management of a geographcally dverse fleet of dstrbuted energy resources s a dffcult algorthmc challenge. On the sde of the operator, the partcpaton of aggregators n electrcty markets presents unque The authors are wth the Department of Computng and Mathematcal Scences, Calforna Insttute of Technology, Pasadena, CA, USA e- mal: {azzan,dvj,ncchen,adamw}@caltech.edu Manuscrpt receved Month DD, YYYY; revsed Month DD, YYYY. challenges n terms of montorng and mtgatng the potental of exercsng market power. In partcular, unlke tradtonal generaton resources, the ISO cannot verfy the avalablty of the generaton resources of aggregators. Whle the repar schedule of a conventonal generator can be made publc, the downtme of a solar generaton plant and the tmes when solar generaton s not avalable cannot be scheduled or verfed after the fact. Thus, aggregators have the ablty to strategcally curtal generaton resources wthout the knowledge of the ISO, and ths potentally creates sgnfcant opportuntes for them to manpulate prces. These ssues are partcularly salent gven current proposals for dstrbuton systems. Dstrbuton systems (whch are typcally radal networks) are heavly mpacted by the ntroducton of dstrbuted energy resources. As a result, there are a varety of current proposals to start dstrbuton-level power markets (see, for example [8] [9]), operated by Dstrbuton System Operators (DSOs). A future grd may even nvolve a herarchy of system operators dealng wth progressvely larger areas, net load and net generaton. In such a scenaro, aggregators could end up havng a sgnfcant proporton of the market share, and such markets may be partcularly vulnerable to strategc bddng practces of the aggregators. Thus, understandng the potental for these aggregators to exercse market power s of great mportance, so that regulatory authortes can take approprate steps to mtgate t as needed. A. Summary of Contrbutons Ths paper addresses both the algorthmc challenge of managng an aggregator and the economc challenge of measurng the potental for an aggregator to manpulate prces. Specfcally, ths work provdes a new algorthmc framework for managng the partcpaton of an aggregator n electrcty markets, and uses ths framework to evaluate the potental for aggregators to exercse market power. To those ends, the paper makes three man contrbutons. Frst, we ntroduce a new model for studyng the management of an aggregator. Concretely, we ntroduce a model of an aggregator n a two-stage market, where the frst stage s the ex-ante (or day-ahead) market that decdes the generaton schedule based on the soluton to a securty-constraned economc dspatch problem and the second stage s the ex-post (or real-tme) market to perform fne adjustment va Locatonal Margnal Prces (LMPs) based on updated nformaton. Second, we provde a novel algorthm for managng the partcpaton of an aggregator n the two stage market. The problem s NP-hard n general and s a blevel quadratc program, whch are notorously dffcult n practce. However, we develop an effcent algorthm that can be used by the

2 2 aggregators n radal networks to approxmate the optmal curtalment strategy and maxmze ther proft (Secton IV). Note that the algorthm s not just relevant for aggregators; t can also be used by the operator to asses the potental for strategc curtalment. The key nsght n the algorthm s that the optmzaton problem can be decomposed nto local peces and be solved approxmately usng a dynamc programmng over the graph. Thrd, we quantfy opportuntes for prce manpulaton (va strategc curtalment) by aggregators n the two stage market (Secton III). Our results hghlght that, n practcal scenaros, strategc curtalment can have a sgnfcant mpact on prces, and yeld much hgher profts for the aggregators. In partcular, the prces can be mpacted up to a few tens of $/MWh n some cases, and there s often more than 25% hgher proft, even wth curtalments lmted to 1%. Further, our results expose a connecton between the proft achevable va curtalment and a new market power measure ntroduced n [1]. B. Related Work Ths paper connects to, and bulds on, work n four related areas: () Quantfyng and mtgatng market power, () Cyberattacks n the grd, () Algorthms for managng dstrbuted energy resources, and (v) Algorthms for blevel programs. Quantfyng market power n electrcty markets: There s a large volume of lterature that focuses on dentfyng and measurng market power for generators n an electrcty market, see [11] for a recent survey. Early works on market power analyss emerged from mcroeconomc theory suggest measures that gnore transmsson constrants. For example, [12] ntroduced the pvotal suppler ndex (PSI), whch s a bnary value ndcatng whether the capacty of a generator s larger than the supply surplus, [13] later refned PSI by proposng resdental supply ndex (RSI). RSI s used by the Calforna ISO to assure prce compettveness [14]. The electrcty relablty councl of Texas uses the element compettveness ndex (ECI) [15], whch s based on the Herfndahl-Hrschmann ndex (HHI) [16]. Market power measures consderng transmsson constrants have emerged more recently. Some examples nclude, e.g., [17] [21], and [22]. Interested readers can refer to [23], whch proposes a functonal measure that unfes the structural ndces measurng market power on a transmsson constraned network n the prevous work. In contrast to the large lterature dscussed above, the lterature focused on market power of renewable generaton producers s lmted. Exstng works such as [1] and [24] study market power of wnd power producers gnorng transmsson constrants. The key dfferentator of the work n ths paper s that the use of the Locatonal Margnal Prce (LMP) framework, whch s standard practce n the electrcty market [25], [26], allows ths work to offer nsght about market power of aggregators when transmsson capacty s lmted. Cyber-attacks n the grd: The model and analyss n ths paper s also strongly connected to the cyber securty research communty, whch has studed how and when a malcous party can manpulate the spot prce n electrcty markets by compromsng the state measurement of the power grd va false data njecton, e.g., see [27] [31]. In partcular, [29], [3] shows that f a malcous party can corrupt sensor data, then t can create an arbtrage opportunty. Further, [27] shows that such attacks can mpact both the real tme spot prce and future prces by causng lne congestons. In ths paper, we do not allow aggregators to corrupt the state measurements of the power system, rather we consder a perfectly legal approach for prce manpulaton: strategc curtalment. However, strategc curtalment n the ex-post market can gan extra proft to the detrment of the power system, whch s a smlar mechansm to those hghlghted n cyber attack lterature. Techncally, the work n ths paper makes sgnfcant algorthmc contrbutons to the cyber-attack lterature. In partcular, the papers mentoned above focus on algorthmc heurstcs and do not provde formal guarantees. In contrast, our work presents a polynomal tme algorthm that provably maxmzes the proft of the aggregator. Algorthms for managng dstrbuted energy resources: There has been much work studyng optmal strateges for managng demand response and dstrbuted generaton resources to offer regulaton servces to the power grd. Ths work covers a varety of contexts. For example, researchers have studed frequency regulaton [32] [33] and voltage regulaton (or volt-var control) [34] [35]. A separate lne of work has been work on desgnng ncentves to encourage dstrbuted resources to provde servces to the power grd [36] [37]. However, the current paper s dstnct from all the work above n that we study strategc behavor by an aggregator of dstrbuted resources. Pror work does not model the strategc manpulaton of prces by the aggregator. Algorthms for blevel programs: The optmzaton problem that the strategc aggregator solves s a blevel program, snce the objectve (aggregator s proft) depends on the locatonal prces (LMPs). The LMPs are constraned to be equal to optmal dual varables arsng from economc dspatch-based market clearng procedure. These types of problems have been extensvely studed n the lterature, and fall under the class of Mathematcal Programs wth Equlbrum Constrants (MPECs) [38]. Even f the optmzaton problems at the two levels s lnear, the problem s known to be NP-hard [39]. Global optmzaton algorthms [4] can be used to solve ths problems to arbtrary accuracy (compute a lower bound on the objectve wthn a specfed tolerance of the global optmum). However, these algorthms use a spatal branch and bound strategy, and can take exponental tme n general. In contrast, solvers lke PATH [41], whle practcally effcent for many problems, are only guaranteed to fnd a local optmum. In ths paper, we show that for tree-structured networks (dstrbuton networks), an ɛ-approxmaton of the global optmum can be computed n tme lnear n the sze of the network and polynomal n 1 ɛ. II. SYSTEM MODEL In ths secton we defne the power system model that serves as the bass for the paper and descrbe how we model the way the Independent System Operator (ISO) computes the Locatonal Margnal Prces (LMPs). Locatonal margnal prcng s

3 3 adopted by the majorty of power markets n the Untes States [26], and our model s meant to mmc the operaton of twostage markets lke ISO New England, PJM Interconnecton, and Mdcontnent ISO, that use ex-post prcng strategy for correctng the ex-ante prces [25], [26]. We consder a power system wth n nodes (buses) and t transmsson lnes. The generaton and load at node are denoted by p and d respectvely, wth p = [ ] T p 1,..., p n and d = [ ] T d 1,..., d n. We use [n] to denote the set of buses {1,..., n}. The focus of ths paper s on the behavor of an aggregator, whch owns generaton capacty, possbly at multple nodes. We assume that the aggregator has the ablty to curtal generaton wthout penalty, e.g., by curtalng the amount of wnd/solar generaton or by not callng on demand response opportuntes. Let N a [n] be the nodes where the aggregator has generaton and denote ts share of generaton at node N a by p a (out of p ). The curtalment of generaton at ths node s denoted by α, where α p a. We defne our model for the decson makng process of the aggregator wth respect to curtalment n Secton III. Together, the net generaton delvered to the grd s represented by p α, where α j = j N a. The flow of lnes s denoted by f = [ ] T f 1,..., f t, where fl represents the flow of lne l: f = G(p α d), where G R t n s the matrx of generaton shft factors [42]. We also defne B R n t as the lnk-to-node ncdence matrx that transforms lne flows back to the net njectons as p α d = Bf. A. Real-Tme Market Prce At the end of each dspatch nterval, n real tme, the ISO obtans the current values of generaton, demands and flows from the state estmator. Based on ths nformaton, t solves a constraned optmzaton problem for market clearng. The objectve of the optmzaton s to mnmze the total cost of the network, based on the current state of the system. The ex-post LMPs are announced as a functon of the optmal Lagrange multplers of ths optmzaton. Mathematcally, the followng program has to be solved. mnmze f c T Bf (1a) λ, λ + : p Bf p + α + d p (1b) µ, µ + : f f f (1c) ν : f range(g) (1d) In the above, c s the offer prce for the generator. f l s the desred flow of lne l, and Bf = p + p α d, where p s the desred amount of change n the generaton of node. Constrant (1b) enforces the upper and lower lmts on the change of generatons, and constrant (1c) keeps the flows wthn the lne lmts. In practce, the generaton change lmts are often set to be constant values of p = 2 and p =.1, for all, [43]. The last constrant ensures that f l are vald flows,.e. f = G p for some generaton p. Varables λ, λ + R n +, µ, µ + R t + and ν R t rank(g) denote the Lagrange multplers (dual varables) correspondng to constrants (1b), (1c) and (1d). Note that the ISO does not physcally redspatch the generatons, and the optmal values of the above program are just the desred values. In fact, by announcng the (ex-post) LMPs, the ISO provdes ncentves for the generators to adjust ther generaton accordng to the goals of the ISO [26]. Defnton 1. The ex-post locatonal margnal prce (LMP) of node at curtalment level of α, denoted by λ (α), s λ (α) = c + λ + (α) λ (α). (2) We assume that there s a unque optmal prmal-dual par, and therefore the LMPs are unque. In general, there are several ways that ISOs ensure such condton, for nstance by addng a small quadratc term to the objectve. III. THE MARKET BEHAVIOR OF THE AGGREGATOR The key feature of our model s the behavor of the aggregator. As mentoned before, aggregators have generaton resources at multple locatons n the network and can often curtal generaton resources wthout the knowledge of the ISO. Of course, such curtalment may not be n the best nterest of the aggregator, snce t means offerng less generaton to the market. But, f through curtalment, prces can be mpacted, then the aggregator may be able to receve hgher prces for the generaton offered or make money through arbtrage of the prce dfferental. In ths secton, we develop a model for a strategc aggregator, whch we then use n order to understand n what stuatons curtalment may be benefcal. A. Prelmnares To quantfy the proft that the aggregator makes due to the curtalment, let us take a look at the total revenue n dfferent producton levels. Defnton 2. We defne the curtalment proft (CP) as the change n proft of the aggregator as a result of curtalment: γ(α) = N a (λ (α) (p a α ) λ () p a ) (3) Note that the curtalment proft can be postve or negatve n general. We say a curtalment level α > s proftable f γ(α) s strctly postve. The curtalment proft s mportant for understandng when t s benefcal for the aggregators to curtal. Note that we are not concerned about the cost of generaton here, as renewables have zero margnal cost. However, f there s a cost for generaton, then that results n an addtonal proft durng curtalment, whch makes strategc curtalment more lkely. B. A Proft-Maxmzng Aggregator A natural model for a strategc aggregator s one that maxmzes curtalment proft LMPs and curtalment constrants. Snce LMPs are the soluton to an optmzaton problem themselves, the aggregator s problem s a blevel

4 4 optmzaton problem. In order to be able to express ths optmzaton n an explct form, let us frst wrte the KKT condtons of the program (1). Prmal feasblty: Dual feasblty: Complementary slackness: p Bf p + α + d p f f f Hf = λ, λ +, µ, µ + (4a) (4b) (4c) (4d) λ + ((Bf) p + α + d p ) =, = 1,..., n (4e) λ ( p (Bf) + p α d ) =, = 1,..., n (4f) Statonarty: µ + l (f l f l ) =, l = 1,..., t (4g) µ l (f l f l) =, l = 1,..., t (4h) B T (c + λ + λ ) + µ + µ + H T ν = (4) Here H R (t rank(g)) t, and the range of G s the nullspace of H. Usng the KKT condtons derved above, the aggregator s problem can be formulated as follows. γ = maxmze α,f,λ,λ +,µ,µ +,ν γ(α) (5a) α p a, N a (5b) α j =, j N a (5c) (4) (5d) The objectve (5a) s the curtalment proft defned n (3). Constrants (5b) and (5c) ndcate that the aggregator can only curtals generaton at ts own nodes, and the amount of curtalment cannot exceed the amount of avalable generaton to t. Constrant (5d), whch s the KKT condtons, enforces the locatonal margnal prcng adopted by the ISO. Note that f one assumes that curtalments of a hgher amount than a lmt τ can be detected by the ISO, then we can smply replace p (a) n (5b) by τ. An mportant note about ths problem s that we have assumed the aggregator has complete knowledge of the network topology (G), and state estmates (p and d). Ths s, perhaps, optmstc; however one would hope that the market desgn s such that aggregators do not have proftable manpulatons even wth such knowledge. The results n ths paper ndcate that ths s not the case. C. Connectons between Curtalment Proft and Market Power Whle our setup so far seems dvorced from the noton of market power, t turns out that there s a fundamental relatonshp between market power and the noton of curtalment proft ntroduced above. As mentoned earler, there has been sgnfcant work on market power n electrcty markets, but work s only begnnng to emerge on the market power of renewable generaton producers. One mportant work from ths lterature s [1], and the followng s the proposed noton of market power from that work. Defnton 3. For α aggregator s defned as ( λ (α ) λ () η = λ (), the market power (ablty) of the ) ( ) α / p a In ths defnton the value of η captures the ablty of the generator/aggregator to exercse market power. Intutvely, n a market wth hgh value of η, the aggregator can sgnfcantly ncrease the prce by curtalng a small amount of generaton. Interestngly, the optmal curtalment proft s closely related to ths noton of market power. We summarze the relatonshp n the followng proposton, whch s proven n Appendx C. Proposton 1. If the curtalment proft γ s postve then the market power η > 1. Furthermore, the larger the curtalment proft s, the hgher s the market power. Ths proposton hghlghts that the noton of market power n [1] s consstent wth an aggregator seekng to maxmze ther curtalment proft, and hgher curtalment proft corresponds to more market power. IV. OPTIMIZING CURTAILMENT PROFIT The aggregator s proft maxmzaton problem s challengng to analyze, as one would expect gven ts blevel form. In fact, blevel lnear programmng s NP-hard to approxmate up to any constant multplcatve factor n general [44]. Furthermore, the objectve of the program (5) s quadratc (blnear) n the varables, rather than lnear. Ths combnaton of dffcultes means that we cannot hope to provde a complete analytc characterzaton of the behavor of a proft maxmzng aggregator. In ths secton, we begn wth the case of a sngle-bus aggregator and buld to the case of general mult-bus aggregators n acyclc networks. For the sngle-bus aggregator, the optmal curtalment can be found exactly, n polynomal tme. For the general case, we cannot provde an exact algorthm, but we do provde a practcal approxmaton algorthm for general mult-bus aggregators n acyclc networks (e.g. dstrbuton networks). A. An Exact Algorthm for a Sngle-Node Aggregator Even n the smplest case, when the aggregator has only a sngle node,.e. ts entre generaton s located n a sngle bus, t s not trval how to solve the aggregator s proft maxmzaton problem. (6)

5 5 B. An Approxmaton Algorthm for Mult-Bus Aggregators n Radal Networks Fg. 1. The LMP at bus as a functon of curtaled generaton at that bus. Shaded areas ndcate the aggregator s revenue at the normal condton and at the aggreaton. The frst step toward solvng the problem s already dffcult. In partcular, n order to understand the effect of curtalment on the proft, we frst need to understand how does curtalment mpact the prces an mpact whch s not monotonc n general. However, although LMPs are not monotonc n general, t turns out that n sngle-bus curtalment, the LMP s ndeed monotonc wth respect to the curtalment. The proof of the followng lemma s n Appendx A. Lemma 2. The LMP of any bus s monotoncally ncreasng wth respect to the curtalment at that bus. That s λ (α ) λ (α) f α > α, and α j = α j for all j = [n]\{}. A consequence of the above lemma s that the prce λ s a monotoncally ncreasng starcase functon of α, for any bus, as depcted n Fg. 1. Further, the bndng constrants of (1) do not change n the ntervals, and thus the dual varables reman the same,.e., the LMPs reman constant. When a constrant becomes bndng/non-bndng, the LMP jumps to the next level. In Fg. 1, the blue and red shaded areas are proft at the normal condton and at the curtalment, respectvely. The dfference between the two areas the curtalment proft. In partcular, f the red area s larger than the blue one, the aggregator s able to earn a postve curtalment proft on bus. The optmal curtalment α also happens where the red area s maxmzed. It should be clear that the optmal curtalment always happens at the verge of a prce change, not n the mddle of a constant nterval (otherwse t can be ncreased by curtalng less). Gven the knowledge of the network and state estmates, t s possble to fnd the jump ponts (.e. where the bndng constrants change) and evaluate them for proftablty. Therefore, f there are not too many jumps, an exhaustve search over the jump ponts can yeld the optmal curtalment. Based on ths observaton, we have the followng theorem, whch s proven n Appendx B. Theorem 3. The exact optmal curtalment for an aggregator wth a sngle bus, n a network wth t lnes, can be found by an algorthm wth runnng tme O(t ). Clearly, ths approach does not extend to large mult-bus aggregators. The followng sectons use a dfferent, more sophstcated algorthmc approach for that settng. In ths secton, we show that the aggregator proft maxmzaton problem, whle hard n general, can be solved n an approxmate sense to determne an approxmately-feasble approxmately-optmal curtalment strategy n polynomal tme usng an approach based on dynamc programmng. In partcular, we show that an ɛ-approxmaton of the optmal curtalment proft can be obtaned usng an algorthm wth runnng tme that s lnear n the sze of the network and polynomal n 1 ɛ. Before we state the man result of ths secton, we ntroduce the noton of an approxmate soluton to (5) n the followng defnton. Defnton 4. A soluton (α, f, λ, λ +, µ, µ +, ν) to (5) s an ɛ-accurate soluton f the constrants are volated by at most ɛ and γ (α) γ ɛ. Note that, f one s smply nterested n approxmatng γ (as a market regulator would be), the ɛ-constrant volaton s of no consequence, and an ɛ-accurate soluton of (5) suffces to compute an ɛ-approxmaton to γ. Gven the above noton of approxmaton, our man theorem s the followng (proof s n Appendx D): Theorem 4. An ɛ-accurate soluton to the optmal aggregator curtalment problem (5) for an n-bus radal network can be found by an algorthm wth runnng tme cn ( ) 1 9 ɛ where c s a constant that depends on the parameters p a, B, d, p, f, f. On a lnear (feeder lne) network, the runnng tme reduces to cn ( ) 1 6. ɛ We now gve an nformal descrpton of the approxmaton algorthm. Consder a radal dstrbuton network wth nodes labeled [n], (where 1 denotes the substaton bus, where the radal network connects to the transmsson grd). Radal dstrbuton networks have a tree topology (they do not have cycles). We denote bus 1 as the root of the tree, and buses wth only one neghbor as leaves. Every node (except the root) has a unque parent, defned as the frst node on the unque path connectng t to the root node. The set of nodes k that have a gve node as ts parent are sad to be ts chldren. It can be shown that the strategc curtalment problem on any radal dstrbuton network can be expressed as an equvalent problem on a network where each node has maxmum degree 3 (known as a bnary tree, see Appendx D). Thus, we can lmt our attenton to networks of ths type, where every node has a unque parent and at most 2 chldren. For a node, let c 1 (), c 2 () denote ts chldren (where c 1 =, c 2 = s allowed snce a node can have fewer than two chldren). We use the shorthand p net () = f c1() + f c2() f (p α d ) (4a) reduces to p p net () p where f 1 = and f =. The matrx H n (4c) s an empty matrx (the nullspace of the matrx B s of dmenson ), so ths constrant can be

6 6 Fg. 2. The representaton of a bnary tree. For any node, and ts chldren denoted c 1 (), c 2 (). dropped. Usng ths addtonal structure, the problem (5) can be rewrtten (after some algebra) as: maxmze λ,f,α n λ (p a α ) =1 (7a) α p a, [n] (7b) p p net () p, [n] (7c) f f f, [n] \ {1} (7d) c, f p net () = p λ = c, f p < p net () < p, [n] (7e) c, f p net () = p, f f = f λ cj() λ =, f f < f < f, [n], j = 1, 2 (7f), f f = f where λ s the LMP at bus. Note that we assumed that there s some aggregator generaton and potental curtalment at every bus (however ths s not restrctve, snce we can smply set p a = at buses where the aggregator owns no assets). Defne x = (λ, f, α ). (7) s of the form: n max g (x ) for some functons g (.) x:h (x,x c1 (),x c2 ()), [n] =1 and h (.). Ths form s amenable to dynamc programmng, snce f we fx the value of x, the optmzaton problem for the subtree under s decoupled from the rest of the network. Set κ n (x) =, defne κ for < n recursvely as: 2 j=1 g c j() ( xcj() ) + κ (x) = max x c1 (),x c2 () h ( (x,x c1 (),x c2 ()) κ cj() xcj()) Then, the optmal value can be computed as = max x κ 1 (x) + g 1 (x). However, the above recurson requres an nfnte-dmensonal computaton at every step, snce the value of κ needs to be calculated for every value of x. To get around ths, we note that the varables λ, f, α are bounded, and hence x can be dscretzed to le n a certan set X such that every feasble x s at most δ(ɛ ) away (n nfnty-norm sense) from some pont n X (Lemma 5). The dscretzaton error can be quantfed, and ths error bound can be used to relax the constrant to h (x, x +1 ) ɛ guaranteeng that any soluton to (5) s feasble for the relaxed constrant. Ths allows us to defne a dynamc program (algorthm 1). γ Algorthm 1 Dynamc programmng on bnary tree S { : c 1 () =, c 2 () = } κ (x) x X, S whle S n do S { S : c 1 (), c 2 () S} S, x X : κ (x) max x 1 X c 1 (),x 2 X c 2 () h (x,x 1,x 2) ɛ S S S end whle γ max x X1 κ 1 (x) + g 1 (x) j=1,2 g cj() ( x j ) + κcj() (x ) The algorthm essentally starts at the leaves of the tree and proceeds towards the root, at each stage updatng κ for nodes whose chldren have already been updated (stoppng at root). Along wth the dscretzaton error analyss n Appendx D, ths essentally concludes Theorem 4. It s worth notng that prevous work on dstrbuton level markets have used AC power flow models (at least n some approxmate form) due to the mportance of voltage constrants and reactve power n a dstrbuton system [45]. Our approach extends n a straghtforward way to ths settng as well, as the dynamc programmng structure remans preserved (the KKT condtons wll smply be replaced by the correspondng condtons for the AC based market clearng mechansm). V. THE IMPACT OF STRATEGIC CURTAILMENT Usng the algorthms developed n the prevous sectons, we can now move to characterzng the potental mpact of strategc curtalment. We do ths by frst provdng an llustratve example of how curtalment leads to a larger proft for a smple sngle-bus aggregator n a small, 6-bus network. Then, we consder curtalment n larger networks and show the global effect of strategc behavor. We use IEEE 14-, 3-, and 57-bus test cases, and ther enhanced versons from NICTA Energy System Test case Archve [46]. Bus 4 35 $/MWh 2-3 MW LMP4=35 $/MWh Bus 1 2 $/MWh 2-4 MW LMP1=2 $/MWh 85 MW 25 MW Bus 5 23 $/MWh 2-25 MW LMP5=28.7 $/MWh 375 MW 1/1 MW -65.2/9 MW 3 MW Generator 15 MW 25.2/1 MW 25 MW 1/1 MW -9/9 MW 73 MW 3 MW -31/9 MW Bus 2 25 $/MWh 2-2 MW LMP2=25 $/MWh 397 MW Load -96/1 MW -76.4/9 MW Bus 6 Bus 3 22 $/MWh 2-3 MW LMP3=25 $/MWh 3 MW 28 MW 24 $/MWh 2-4 MW LMP6=24 $/MWh s 2$/MWh, generaton output range s 2 4 MW. In the gven operatng condton, the actual load at bus 1 s 15 MW and the actual generaton output s 375 MW. As a result, 1 MW power flows from bus 1 to 2, reachng the lne capacty 1 MW. A negatve power flow value means the power flows n the opposte drecton of the defned lne orentaton, e.g. 96 MW power flows from bus 3 to 2 n lne l 23, whose capacty s 1 MW. code for the add-then-remove algorthm s Algorthm 1, where the lnes 2 to 5 correspond to addng loosenng constrants whle lnes 6 to 14 for removng tghtenng constrants. C. A case study and nsghts To brng out ntutons, we mplement the proposed algorthm n an example 6 bus system n Fg. 1, where G 1 s B Globecom Communcaton and Informaton System Securty Symp Fg. 3. The 6-bus example network from [27], used to llustrate the effect of curtalment. A. An Illustratve Example We use a 6-bus example network from [27], n whch the amounts of generaton are 375.2, 73., 299.6, 84.8, C hypothetcally 2 reducng t n the ex-post LMP problem LMP at the tagged bus s re IV. ATTACK I In ths secton, we realze usng LR attack, where df pendng on the attackers s c estngly, we also fnd that has mpact to future electrc A. LR attack to real-tme L Suppose that a false data data a =(a 1,a 2,..,a m ) n Then, the receved measurem z = H From (3), the ML estmate and the estmate of lne flo wth the actual state estmat x ˆx = Pa. Meanwhle, th r = z H x =(I Load redstrbuton (LR) The attack vector of a LR a a p = and e T a d =al generaton and demand see attacks do not compromse because generaton measure protected and can be verfe between ISOs and utlty c and load measurements are w

7 7 LMP ($/MWh) Normal Curtalment Bus No. Fg. 4. The locatonal margnal prces for the 6-bus example before and after the curtalment. 25., MW (See Fg. 3). The loads and the orgnal offer prces for the generators can be found n the fgure. In ths case, the lnes l 12, l 14 and l 56 are carryng ther maxmum flow, and the real-tme LMPs are 2., 25., 25., 35., 28.7, 24. $/M W h, respectvely. Assume that the aggregator owns node 1 and ams to ncrease ts proft by curtalng the generaton at ths node. It can be seen that by decreasng the generaton at node 1 by.15, from MW to MW, the bndng and non-bndng constrants n problem (1) change, and as a result the ISO determnes the new LMPs as 25.8, 25., 25., 35., 3.6, 24. $/M W h. Fg. 4 shows the LMPs, before and after the curtalment. The curtalment proft s γ = = 2172 $ / h,whch means that the aggregator has been able to ncrease ts proft by 2172 $/h. For dspatch ntervals of length 15 mnutes, ths amount s around $543 per dspatch nterval. B. Case Studes %age error n optmal value Tme (s) Fg. 5. The dfference from the optmal soluton as a functon of the runnng tme of the algorthm, on a 9-bus network wth 1% curtalment allowance. We smulate the behavor of aggregators wth dfferent szes,.e. dfferent number of buses, n a number of dfferent networks. We use the IEEE 14-, 3-, and 57-bus test cases. Snce studyng market manpulaton makes sense only when there s congeston n the network, we scale the demand (or equvalently the lne flows) untl there s some congeston n the network. In order to examne the proft and market power of aggregator as a functon of ts sze, we assume that the way aggregator grows s by sequentally addng random buses to ts set (more or less lke the way e.g. a solar frm grows). Then at any fxed set of buses, t can choose dfferent curtalment strateges to maxmze ts proft. In other words, for each of ts nodes t should decde whether to curtal or not (assumng that the amount of curtalment has been fxed to a small porton). We assume that the total generaton of the aggregator n each bus s 1 MW and t s able to curtal 1% of t (.1 MW ). For each of the three networks, Fg. 6 shows the proft for a random sequence of nodes. Comparng the no-curtalment proft wth the strategc-curtalment proft, reveals an nterestng phenomenon. As the sze of the aggregator (number of ts buses) grows, not only does the proft ncrease (whch s expected), but also the dfference between the two curves ncrease, whch s the curtalment proft. More specfcally, the latter does not need to happen n theory. However n practce, t s observed most of the tme, and t ponts out that larger aggregators have hgher ncentve to behave strategcally, and they can ndeed gan more from curtalment. Fnally, to demonstrate the performance of our approxmaton algorthm on acyclc networks, we plot the suboptmalty gap versus runnng tme of the algorthm (Fg. 5) on a radal verson of the IEEE 9-bus network (taken from [47] ). It can be seen that wth the optmal soluton can be computed wth error n 1s. VI. CONCLUDING REMARKS Understandng the potental for market manpulaton by aggregators s crucal for electrcty market effcency n the new era of renewable energy. In ths paper, we characterzed the proft an aggregator can make by strategcally curtalng generaton n the ex-post market as the outcome of a blevel optmzaton problem. Ths model captures the realstc prce clearng mechansm n the electrcty market. Wth ths formulaton when the aggregator s located n a sngle bus, we have shown that the locatonal margnal prce s monotoncally ncreasng wth the curtalment; the proft drectly correlates wth market power defned n the lterature, and we have an exact polynomal-tme algorthm to solve the aggregators proft maxmzaton problem. The aggregator s strategc curtalment problem n a general settng s a dffcult b-level optmzaton problem, whch s ntractable. However, we show that for radal dstrbuton networks (where aggregators are lkely located) there s an effcent algorthm to approxmate the soluton up to arbtrary precson. Fnally, we show va smulatons on realstc test cases that, frst, there s potentally large proft for aggregators by manpulatng the LMPs n the electrcty market, and second, our algorthm can effcently fnd the (approxmately) optmal curtalment strategy. We vew ths paper as a frst step n understandng market power of aggregators, and more generally, towards market desgn for ntegratng renewable energy and demand response from geographcally dstrbuted sources. Wth the result of our paper, t s nterestng to ask what can the operator do to address ths problem. In partcular, how to desgn market rules for aggregators to maxmze the contrbuton of renewable energy yet mtgate the exercse of market power. Also, extendng the analyss to the case of multple aggregators n the market s another nterestng drecton for future research. REFERENCES [1] K. Bulls, Why SolarCty s succeedng n a dffcult solar ndustry, 212.

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9 9 A. Proof of Lemma 2 APPENDIX Let us take a look at the ISO s optmzaton problem (1), whch s a lnear program. It s not hard to see that the dual of ths problem s as follows. maxmze λ,λ +,µ,µ +,ν ( p + p α d) T λ + ( p + α + d p) T λ + + f T µ f T µ + B T (c + λ + λ ) µ + µ + + H T ν = λ, λ +, µ, µ + (8a) (8b) (8c) If one focuses on the terms nvolvng α for a certan, the objectve of the above optmzaton problem s n the form: ( p +p α d )λ +( p +α +d p )λ + plus a lnear functon of the rest of the varables (.e. the rest of λ, λ +, as well as µ, µ +, ν). There s no α n the constrants, and the frst two terms of ths objectve are the only parts where α appears (and wth opposte sgns). We need to show that f α s changed to α + δ for some δ >, then c + λ +new λ new c + λ + λ, where λ +new, λ new are the optmal solutons of the new problem. We prove ths n a general settng. Consder the followng two optmzaton problems. f = sup x 1,x 2 R x 3 R m f new = sup x 1,x 2 R x 3 R m T a 1 x 1 + a 2 x 2 + a3 x 3 (9a) s.t. (x 1, x 2, x 3 ) S (9b) T (a 1 δ)x 1 + (a 2 + δ)x 2 + a3 x 3 (1a) s.t. (x 1, x 2, x 3 ) S (1b) Assume that the optmal values of the problems are attaned at (x 1, x 2, x 3) and (x new 1, x new 2, x new 3 ), respectvely. We clam that x new 2 x new 1 x 2 x 1 (Ths precsely mples the LMP condton n our case,.e. λ +new λ new λ + λ ). Suppose by way of contradcton that x new 2 x new 1 < x 2 x 1. We know that a 1 x 1 + a 2 x 2 + a T 3 x 3 a 1 x 1 + a 2 x 2 + a T 3 x 3, (x 1, x 2, x 3 ) S. Therefore we have (a 1 δ)x new 1 + (a 2 + δ)x new 2 + a T 3 x new 3 = a 1 x new 1 + a 2 x new 2 + a T 3 x new 3 δx new 1 + δx new 2 a 1 x 1 + a 2 x 2 + a T 3 x 3 + δ(x new 2 x new 1 ) < a 1 x 1 + a 2 x 2 + a T 3 x 3 + δ(x 2 x 1) = (a 1 δ)x 1 + (a 2 + δ)x 2 + a T 3 x 3. The frst nequalty above follows from the fact that (x new 1, x new 2, x new 3 ) S. Now the above mples that (x new 1, x new 2, x new 3 ) s not the optmal soluton of (1), and t s a contradcton. As a result, x new 2 x new 1 x 2 x 1. B. Proof of Theorem 3 Snce we are n the sngle-bus curtalment regme, α has only one nonzero component. For the sake of convenence, we denote that element tself by a scalar α throughout ths proof (no α s vector n ths proof). The proof conssts of the followng two peces: 1) From each jump pont, the pont where the next jump happens can be computed n polynomal tme, 2) There are at most polynomally (n ths case even lnearly) many jumps. Assumng that the soluton to the program (1) s unque, for any fxed value of α, exactly t of the constrants (1b,1c,1d) are bndng (actve). We can express these bndng constrants as Af = b(α), where A R t t s an nvertble matrx, and b(α) R t s a vector that depends on α. As long as the bndng constrants do not change, the matrx A s fxed and the optmal soluton s lnear n α (.e. f = A 1 b(α)). Then, for smplcty, we can express the soluton as f(α) = f + αa, for some t-vectors f and a. Now f we look at the non-bndng (nactve) constrants of (1), they can also be expressed as Ãf < b, for some matrx à and vector b of approprate dmensons. Insertng f nto ths set of nequaltes yelds Ãf + αãa < b, or equvalently α(ãa) < b (Ãf ), for all = 1, 2,..., (2n + 2t rank(g)). Now we need to fgure out that, wth ncreasng α, whch of the non-bndng constrants becomes bndng frst and wth exactly how much ncrease n α. If for some we have (Ãa), then t s clear that ncreasng α cannot make constrant bndng. If (Ãa) > then the constrant can be wrtten as α < b (Ãf ) (Ãa). Computng the rght-hand sde for all, and takng ther mnmum, tells us exactly whch constrant wll become bndng next and how much change n the current value of α results n that. The complexty of ths procedure s O(t ) for computng f and a, plus O(t(2n + 2t)) = O(nt + t 2 ) for computng the lowest bound among all the constrants. Hence the complexty s O(t ). The above procedure descrbes how the next jump pont can be computed effcently from the current pont. The exact same procedure can be repeated for reachng the subsequent jump ponts. All remans to show s the second pece of the proof, whch s that the number of jump ponts are bounded polynomally. To show the last part note that by ncreasng α, f a bndng constrant becomes non-bndng, t wll not become bndng agan. As a result, each constrant can change at most twce, and therefore the number of jumps s at most twce the number of constrants. Thus, the number of jumps

10 1 s O(n + t), and the overall complexty of the algorthm s O((n + t)t ) = O(t ). C. Proof of Proposton 1 From the defnton of γ(α ) = λ (α )(p a α ) λ ()p a t follows that γ(α ) λ ()(p a α ) = λ (α ) λ () pa p a α = 1 + λ (α ) λ () (1 α λ () p a ) λ (α ) λ () λ () = λ (α ) λ () λ () (1 + α p a ) α p a. (11) Therefore we have p a ( λ ()(p a γ(α λ (α ) ( ) ) λ () α ) = / α )α λ () p a 1 = η 1. Snce the left-hand sde parameters are all postve, f γ(α ) >, we can conclude that η > 1. Moreover, t s clear that the larger the value of γ(α ) s, the hgher the value of η s. Note that we used the approxmaton (1 α p ) 1 1+ α a p, snce the a curtalment s small wth respect to the generaton; however, the rght-hand sde expresson (11) s an upper bound on the left-hand sde anyway, and the result holds exactly. flow from ts parent s just splt n some way between ts chldren. Ths n fact enforces the flow conservaton constrant at b. Smlar constructon can be appled to any node of the tree wth more than two chldren, untl no such node exsts. It can be seen that the number of nodes n the new graph s stll lnear n n. So any tree can be transformed to a bnary one by the above procedure. For the rest of the analyss we focus on the ɛ- approxmaton of the dynamc program on the resultng bnary tree. The optmzaton problem (5) on a bnary tree, can be wrtten after some algebra as the followng. max λ,f,α n λ (p α ) =1 (12a) α p a, = 1,..., n (12b) p f c1() + f c2() f p + α + d p, = 1,..., n (12c) f f f, = 2,..., n (12d) { (λ c )(f c1() + f c2() f p + α + d p ) (λ c )(f c1() + f c2() f p + α + d p ) = 1,..., n (12e), D. Proof of Theorem 4 Lemma 5 (δ-dscretzaton). Gven a set C [L 1, L 1 ] [L k, L k ], there exsts a fnte set X such that z C z X, max 1 k z z δ and X contans at most V/δ k ponts, where V = k =1 (L L ) s a constant (the volume of the box). X s sad to be an δ-dscretzaton of C and wrtten as X (δ). Lemma 6 (Reducton to Bnary Tree). Any tree wth arbtrary degrees can be reduced to a bnary tree by ntroducng addtonal dummy nodes to the network. Proof. Take any node b n the tree wth some parent a and k > 2 chldren c 1,..., c k. There exsts m > such that 2 m < k 2 m+1 for some m. We wll show that ths subgraph can be made a bnary tree by ntroducng O(k) dummy nodes (n m levels) between b and ts chldren. The addtonal nodes and edges are defned as follows: up to m levels: b b, b b 1, b b, b b 1, b 1 b 1, b 1 b 11, b b, b b 1,..., b 11 b 111, b... c 1, b... c 2, b...1 c 3... Ths s transparently a bnary tree wth O(k) nodes. Each of the new nodes has zero njecton, and effectvely the ncomng { (λ λ cj())(f cj() f c j()) (λ λ cj())(f cj() f cj()), = 1,..., n, j = 1, 2 (12f) The constrants α p a and f f f, along wth a pror bound on lambda λ λ λ can be used to defne the box where x = (λ, f, α ) lves. Then an ɛ-accurate soluton s a soluton to the followng problem. n λ (p α ) (13a) max λ,f,α =1 p ɛ f c1() + f c2() f p + α + d p + ɛ, = 1,..., n (13b) { (λ c )(f c1() + f c2() f p + α + d p ) ɛ (λ c )(f c1() + f c2() f p + α + d p ) ɛ { (λ λ cj())(f cj() f c j()) ɛ (λ λ cj())(f cj() f cj()) ɛ = 1,..., n, (13c) = 1,..., n, j = 1, 2 (13d) Assumng a δ-dscretzaton of the constrant set, each of the constrants (as well as ɛ-accuracy of the objectve) mposes,