Surrogate Affine Approximation based Co-optimization of Transactive Flexibility, Uncertainty, and Energy

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1 TO APPEAR ON IEEE TRANS. ON POWER SYSTEMS. CITATION INFORMATION: DOI /TPWRS , IEEE TRANSACTIONS ON POWER SYSTEMS 1 Surrogate Affie Approximatio based Co-optimizatio of Trasactive Flexibility, Ucertaity, ad Eergy Hogxig Ye, Seior Member, IEEE Abstract This study presets a approach to co-optimizatio of trasactive flexibility, eergy, ad optimal ijectio-rage of Variable Eergy Resource (VER). Flexibility receives immese attetio, as it is the essetial resource to accommodate VERs i moder power systems. With a ovel cocept of trasactive flexibility, the proposed approach proactively positios the flexible resources ad optimizes the demad of flexibility. A Surrogate Affie Approximatio (SAA) method is proposed to solve the problem with variable ifiite-costrait rage i polyomial time. It is show that SAA is more optimistic tha the traditioal affie policy i power literature. The SAA method is also applicable to the search for secure ijectiorage of VER, which is ofte heuristically determied i idustry give the latest system iformatio. I practice, VER geeratio beyod the secure ijectio-rage has to be curtailed, eve if its cost is lower tha the margial price. The proposed techique helps accommodate more VERs securely ad ecoomically by icreasig secure ijectio-rage. The model ad the solutio approach are illustrated i the six-bus system ad IEEE 118-bus system. Idex Terms Flexibility ad Ucertaity, Secure Ijectio- Rage, Dispatchable Reewables, Surrogate Optimizatio, Affie Policy, Electricity Market NOMENCLATURE Idices i idex of fully-cotrollable uit l idex of trasmissio lie idex of bus/aggregated VER geerator t idex of time Sets ad otatios C( ) cost fuctio. C c ( ) for geeratio cost of fully cotrollable geerator; C v ( ) for cost of VER geeratio; C d ( ) for the beefit of load; C f ( ) for the beefit of upward/dowward flexibility diag( ) diagoal matrix whose diagoal etries are elemets of vector G() set of uits located at bus J set of rows i (4d) max(, ) elemet-wise maximum operator set of buses N b This work is supported i part by the U.S. Natioal Sciece Foudatio Project Number ECCS ad Clevelad State Uiversity Faculty Research & Developmet Program. H. Ye is with the Clevelad State Uiversity, Clevelad, OH 44115, USA. ( h.ye@csuohio.edu). Digital Object Idetifier /TPWRS N g N l R x R x y T U(u) Ũ set of geerators set of lies set of real x-vectors set of real x y matrices set of time itervals ucertaity set, a fuctio of flexibility u surrogate ucertaity set (costat) Costats A, b abstract matrix ad vector for costrait (2a)-(2d) B, C, E, d abstract matrices ad vector for costrait (3a)-(3d) c, f abstract coefficiet vectors for x ad u F, H, h abstract matrices ad vector for costrait (2e)-(2f) F l brach flow limit N b umber of buses N g umber of uits umber of trasmissio lies N l N r Pi mi, Pi max R up i, Rdow i T δ Γ l, Variables D G G c 2018 IEEE. umber of rows i (4d) miimum ad maximum geeratio outputs uit rampig up/dow limits (MW/miute) umber of time itervals timespa of oe iterval shift factor for lie l ad bus load demad at bus geeratio adjustmet matrix, G R Ng N b surrogate geeratio adjustmet matrix, G R Ng N b J SAA, J AP optimal values of problem (SAA-P) ad (TAP- P) P i output of fully cotrollable geerator i ˆP i (v) re-dispatch of fully cotrollable geerator i, a fuctio of v P i,t output of fully cotrollable geerator i at time t i the exteded multi-period model s(u LB, u UB ) surrogate fuctio s(u LB, u UB ) : R 2N b R 2N b u LB, u UB dowward ad upward flexibility (allowed dowward ad upward deviatios from VER s perspective), u LB R N b, u UB R N b u u = [(u LB ), (u UB )] U LB, U UB matrix diag(u LB ) ad diag(u UB )

2 2 TO APPEAR ON IEEE TRANS. ON POWER SYSTEMS. CITATION INFORMATION: DOI /TPWRS , IEEE TRANSACTIONS ON POWER SYSTEMS U LB, U UB upward ad dowward flexibility (allowed dowward ad upward deviatio of VER output) v realized VER output vector, v R N b V realized VER power output V s scheduled VER power output V, V lower ad upper boud of allowed VER power ijectio at the secod stage, [ V, V ] is the optimal ijectio-rage of VER V f forecast expectatio of VER output V f forecast lower boud of VER output x abstract vector deotig geeratio dispatch, VER output, ad load demad y(ɛ) corrective actio, a fuctio of ɛ ŷ(ɛ) surrogate corrective actio, a fuctio of ɛ ɛ, ɛ ucertaity at bus ad ucertaity vector, ɛ R N b π, π LB, π UB o-egative matrix of auxiliary multipliers I. INTRODUCTION VARIABLE eergy resources (VERs), such as solar ad wid power, have experieced rapid growth i the last decades. I the U.S., wid ad solar capacity have icreased by 100% ad 900%, respectively, betwee 2009 ad 2015 [1]. At the ed of 2016, the solar ad utility-scale wid geeratio capacity reach 42.4GW ad 81.3GW, respectively. Compared with traditioal fossil fuel-fired uits, VER geerators are ot fully cotrollable. They brig more variability ad ucertaity i the power system. I the U.S., a importat task of the Idepedet System Operator (ISO) or Regioal Trasmissio Orgaizatio (RTO) is to make the short-term geeratio schedule, which is to supply the load respectig physical limits ad security costraits. I the power commuity, the Uit Commitmet (UC) problem is defied as fidig the optimal uit ON/OFF status, ad the Ecoomic Dispatch (ED) is to fid the most cost-efficiet geeratio output schedule [2], [3]. Whe the peetratio of VERs reaches the certai level, ISOs/RTOs have to mitigate the adverse impacts of the variability ad ucertaity from VER geerators. The flexible resources, such as atural gasfired uit with large rampig rate, demad respose, ad eergy storage, are the ideal assets to achieve this goal by providig flexibility. VER geerators are ot fully cotrollable, ad their geeratios are ofte treated as ucertai parameters. Thus, with the rapid growth of VERs, the schedulig problems cosiderig ucertaity become active research topics i moder power systems. I the literature, two of the cadidate approaches to hadlig ucertaities are stochastic programmig ad robust optimizatio [4] [8]. Sceario-based stochastic programmig approaches ofte model a umber of scearios to get the cost expectatio ad reserve flexible resources by utilizig Probability Desity Fuctio (PDF). However, due to the computatioal itractability, may approaches oly cosider a small portio of scearios usig sample-reductio techiques. Recetly, the chace-costraied stochastic approach is also employed to solve optimal power flow problems [9]. I robust optimizatio-based approaches, probability iformatio is ot required ad its solutio is supposed to be immue to ay ucertaity i the predefied ucertaity set. It ofte requires efforts i solvig the NP-hard max-mi problems to obtai a robust ad optimal solutio. I the real-time market, ISOs/RTOs are supposed to obtai the optimal solutio withi several miutes, give the latest available iformatio, such as load ad VER forecast output. To address the computatioal challege, researchers have itroduced affie policy i UC ad ED problems [9] [11]. Affie policy ca be traced back to 1950s i the chace-costraied stochastic programmig [12]. The strict affie policy helps make the problem tractable. I idustry, a similar priciple is widely applied i Automatic Geeratio Cotrol (AGC) [13] that is desiged to balace the system frequecy ad the scheduled iterchage i secods. The mai differece is that the participatio factors are heuristically determied i AGC. Flexibility has received may attetios i recet years, such as [14] [21]. Flexibility is cosidered as part of the geeratio expasio problem [14], [16], [17]. A metric for flexibility is further itroduced, ad profits of flexibility providers are aalyzed i [17]. However, the iheret stochastic ature of reewable is ot cosidered. I [18], [19], the authors preset approaches to utilizig the reewables as flexible resources. [20] presets a framework for coordiatig available reserves usig tie-lie i multi-area. This author, Ge, Shahidehpour, ad Li s previous work presets a pricig scheme for flexibility i robust optimizatio [21], [22]. Realizig the icreasig value of flexibility i power systems with deep VER peetratio, several ISOs/RTOs have take actios to secure more flexible resources. For example, CAISO has icreased the reserve requiremets, ad both CAISO ad MISO have itroduced rampig products i the electricity market. At the same time, as flexibility i the system is fiite, reewable eergy spillage ofte occurs i the real-time market. Thus, ISO-NE proposes the DO- NOT-EXCEED (DNE) limit i the real-time market [23]. It gives clear dispatch sigal for each VER geerator, which is istructed to curtail the VER geeratio beyod the DNE limit. Recetly, researchers preset iterestig results o this topic by maximizig the orm of rage vector ad utilizig historical data [24], [25]. While the termiology flexibility could have may defiitios, i this work, flexibility is defied as the rage of power-ijectio-chage that the system ca accommodate usig available flexible resources withi the specified time. The flexible resources ca be either from the geeratio side or from Load Servig Etity (LSE) side. Furthermore, they ca be delivered to the desired destiatio respectig trasmissio costraits. I this paper, the upward (dowward) flexibility is defied as the maximum accommodable power-ijectiochage i upward (dowward) directio. Most literature of flexibility is o the supply side of flexible resource that is ofte secured by the system operator based o some heuristic requiremets. This work itroduces a ovel cocept, trasactive flexibility, to optimize both demad ad supply of flexibility. It eables the owers of flexible resource ad demaders of flexibility to maage flexibility actively.

3 Resultig Price YE: SURROGATE AFFINE APPROXIMATION BASED CO-OPTIMIZATION OF TRANSACTIVE FLEXIBILITY, UNCERTAINTY, AND ENERGY 3 The demader of flexibility is allowed to procure flexibility so that she is able to maage the ucertaity or variatio of power ijectio cosiderig the beefit. The proposed model optimally positios flexible resource ad maages flexibility demad via a two-sided market. The trasmissio reserves are implicitly held so that deliverability of flexible resources is guarateed. Eergy is ofte trasactive, i.e., it ca be bought ad sold i the electricity market. However, flexible resources, such as rampig product, ca oly be sold i the existig markets. I the existig literature, flexibility providers are etitled to credits for reservig rampig capability. This author, Ge, Shahidehpour, ad Li s previous research [22] further proposes to allocate the flexibility cost to the ucertaity source based o Ucertaity Margial Price (UMP) followig a cost causatio priciple. The ucertaity source is thus iclied to reduce the ucertaity level. Whe flexibility has become icreasigly valuable, three questios remai ope: 1) how to determie the flexibility amout while keepig ISO/RTO idepedet; 2) how to allocate flexibility to demader cost-effectively; 3) how to hadle the high flexibility demad whe the flexible resource is scarce. To address these problems, this paper proposes a cooptimizatio model where flexibility is treated as a commodity, ad a geeral polyomial solutio method. I the proposed model, flexibility ca be bought as well as sold. The VER s procuremet of flexibility is equivalet to sellig ucertaity at a egative price. To the author s best kowledge, this is the first time to itroduce the cocept of trasactive flexibility. Thus, this work maily focuses o the ew co-optimizatio model ad the ew solutio methodology. The reader is referred to [22] for the pricig schemes. The cotributios of this paper are summarized as follows. 1) This paper proposes a ew co-optimizatio model to maximize the total social welfare. A cocept, trasactive flexibility, is proposed. The model co-optimizes flexibility ad eergy keepig ISO/RTO idepedet. System security ad ecoomic efficiecy are guarateed. The model will motivate VER to be a autoomous ucertaity mitigator ad flexibility demader. The VERs submit the flexibility bid (either zero or positive) so that resources are proactively positioed for more VER itegratios. Ay VER geeratios withi a optimally determied ijectio-rage ca be ijected ito the grid. Eve with zero flexibility bid, it is still possible to fid a ED to accommodate more VERs, whe a multiplicity of ED occurs. 2) A geeral Surrogate Affie Approximatio (SAA) method is proposed to solve the problem that icludes decisio variables of ifiite-costrait rage. It is computatioally tractable. By solvig oe Liear Programmig (LP) or Quadratic Programmig (QP) problem, oe ca attai the optimal solutio i the proposed approach. It is proved that the optimality of the SAA is ever worse tha that of the traditioal affie policy i power literature. I may circumstaces, the SAA method fids a better solutio. The SAA method is also applicable to the DNE-limit search, which was oce regarded as a computatioally Price ($/MW) Upward Flexibility Demad Resultig Demad Upward Flexibility Supply Demad (MW) Fig. 1. A illustrative itersectio of demad ad supply curve for upward flexibility i a system without etwork cogestio. The ucertaity source submits demad curve. The shadow area is the social surplus. itractable oliear problem. Heuristic methods are ofte employed to solve it [23]. The SAA method ca eve fid better solutios by solvig oe LP problem. It helps itegrate more VERs. The rest of this paper is orgaized as follows. I Sectio II, the co-optimizatio model is developed with trasactive flexibility ad ucertaity. The SAA method is preseted i Sectio III. Sectio IV illustrates the model ad solutio approach usig a simple six-bus system ad a modified 118- bus system. Sectio V cocludes this paper. II. CO-OPTIMIZATION OF TRANSACTIVE FLEXIBILITY, UNCERTAINTY, AND ENERGY The flexibility demad has bee icreasig sigificatly with the growig VER peetratio i power systems. I may circumstaces, flexibility is the scarce resource, that may be eve more expesive tha eergy [27]. To optimally ad costeffectively positio flexible resource, this paper presets a model with trasactive flexibility, which ca be sold by the supplier, ad bought by the demader. Flexibility ca be either upward or dowward. Fig. 1 illustratively depicts a itersectio 1 of the demad ad supply curve for the upward flexibility. The itersectio is the optimal poit, which yields the resultig value of price ad amout of flexibility. The shadow area is the social surplus. This paper focuses o a ew model ad solutio methodologies. I the proposed model, the flexibility cosumer, such as VER, bids for flexibility, so that she ca use it for ucertaity accommodatio or rampig followig. I the meatime, the flexible resource ower gets paid for providig flexibility. The proposed market clearig model follows a cost causatio priciple. It provides a optio to address the challege of cost allocatio ad resource deficiecy for flexibility. O the other had, the system operator remais idepedet i this model. Similar to stochastic/robust literature, VER geerators at the same bus are aggregated as a sigle oe. For simplicity, a sigle-period ED is cosidered here i the real-time market. A extesio to multi-period ED will be briefly discussed later. Let deote the bus idex; Let V s deote the scheduled VER output; let U LB ad U UB deote the allowable dowward 1 Fig. 1 is used to illustrate the basic idea, ad the itersectio may be more complicated whe the lie cogestio exists i the system

4 4 TO APPEAR ON IEEE TRANS. ON POWER SYSTEMS. CITATION INFORMATION: DOI /TPWRS , IEEE TRANSACTIONS ON POWER SYSTEMS ad upward deviatio, respectively; Let V deote the realized VER output. Ay realized VER output V [V s U LB, V s + U UB ], N b ca be ijected ito the grid. This work cosiders a ew twostage model to maximize the social welfare, where flexible resources are to be proactively positioed at the first stage, ad they are used to accommodate ucertaities at the secod stage. It is so called wait-ad-see process. Whe the scheduled output V s, ad allowable deviatio for VER are determied, U LB ad U UB are viewed as the largest dowward ad upward ucertaity at bus by the system operator. If these ucertaities ca always be accommodated by deliverable flexible resource, it is defied i this work that the system has the dowward flexibility of U LB, ad the upward flexibility of U UB at bus. I the proposed model, VER geerator is allowed to procure flexibility at the first stage, so that it could iject more eergy (i.e., larger ijectio-rage) ito the grid at the secod stage whe ucertaity is realized. The scheduled VER output V s,t respects V s V f, V s V f U LB, N b, where V f ad V f are the forecast expectatio, ad the forecast lower boud of VER output, respectively. The above equatios guaratee that VER output ca always be accommodated. I the proposed co-optimizatio model, the objective is to miimize the total cost 2 (i.e., the egative of social welfare), which icludes cost of eergy, beefit of load, ad beefit of the ijectio-rage for VER geerators. Similar to rampig products [30], [31], the proposed model optimizes the basecase cost. Let C c ( ) ad C v ( ) deote cost fuctios of the covetioal geeratio ad VER geeratio respectively. Let C f ( ) ad C d ( ) deote beefit fuctios of flexibility ad load respectively. The, the objective fuctio of co-optimizatio ED model is formulated as C c (P i ) + C v (V s ) mi i C d (D ) It is subject to P i + V s = i F l ( Γ l, P mi i F l, l N l i G() D C f (U LB, U UB P i + V s D ) ) (1) (2a) (2b) P i P max i, i N g (2c) V s V f, N b, (2d) V s V f U LB, N b, (2e) V = V s U LB, V = V s + U UB, N b (2f) 2 It follows the curret practice i idustry. There are rich discussios o other objectives i literature, such as [28]. Amog them is removig cogestio reveue from social welfare [29]. O the cotrary, some researchers believe cogestio reveue should be couted as a part of the social surplus, as it is distributed to FTR holders [28]. Iterested readers are referred to [28], [29] for detailed discussios. ˆP i (v) + V = D, V [ V, V ] i F l ( ) Γ l, ˆP i (v) + V D F l, i G() (3a) (3b) V [ V, V ], l N l Ri dow δ ˆP i (v) P i R up i δ, V [ V, V ], i N g (3c) U LB, U UB, 0, N b (3d) V where P i, D, V s, U LB, U UB,, ad V V are decisio variables at the first stage. P i, D,, ad V V deote the output of Fully Cotrollable Geerator (FCG) i, load at bus, allowable lower ad upper boud of VER ijectio at bus, respectively. It is oted that the cotrollable load ca also be modeled as FCG. As it is optimally determied ad secure, [ V, V ] is called Optimal Ijectio-Rage (OIR). ˆPi (v) is the redispatch at the secod stage whe VER output v is revealed. Costat F l, Pi mi, Pi max, Ri dow, R up i, ad δ are, respectively, the trasmissio limit of lie l, lower ad upper bouds of geeratio of FCG i, dowward ad upward rampig rates, ad timespa. Costat Γ l, deotes the shift factor for lie l ad bus. Like C d beig submitted by LSE, C f ca be submitted by the demader of flexibility, i.e., VER. Equatio (2a) deotes the power balace costrait; (2b) deotes the trasmissio lie limit; (2c) represets the geeratio capacity. Equatio (2e)-(2f) are the costraits for VER geerators. They are well discussed at the begiig of this sectio. Equatio (3a)-(3c) deote the costraits for the ucertaity accommodatio at the secod stage whe the iformatio o VER output is revealed. The rampig costrait (3c) is eforced for the re-dispatch of FCG. III. SOLUTION APPROACH The co-optimizatio problem (1)-(3d) models ifiite costraits for the re-dispatch process, which is sometimes called recourse i robust optimizatio literature [32]. The upward ad dowward flexibility, procured by VERs, are decisio variables at the first stage. I this sectio, a geeral surrogate method is proposed to solve the problem of its kid. For brevity, the model is first rewritte i a compact form (P) mi x,u c x f u (4a) s.t. Ax b (4b) F u + Hx h Bx + Cy(ɛ) + Eɛ d, ɛ U(u) (4c) (4d) where variable x icludes the geeratio dispatch, VER s scheduled output, ad the load. The variable u deotes the flexibility procured by VER. It icludes the dowward deviatio boud u LB R N b ad the upward deviatio boud u UB R N b, where N b is the umber of buses. Thus, the system must maitai eough flexibility at the first stage such that it ca accommodate these deviatios of VER outputs at the secod stage. Equatio (4a) deotes the objective fuctio (1). It could be a liear or semi-positive quadratic fuctio. Equatio (4b) deotes (2a)-(2d). Equatio (2e)-(2f) are represeted by (4c). The re-dispatch costraits (3a)-(3c) are rewritte i (4d). The fuctio y(ɛ) : R N b R Ng

5 YE: SURROGATE AFFINE APPROXIMATION BASED CO-OPTIMIZATION OF TRANSACTIVE FLEXIBILITY, UNCERTAINTY, AND ENERGY 5 is a image of ucertaity ɛ R N b, where N g ad N b are umbers of FCGs ad VER geerators, respectively. It represets the corrective actios (or re-dispatch) of FCGs whe the ucertaity is revealed. The ucertaity is defied as ɛ = V V s, N b. I other words, ɛ is a deviatio of realized VER geeratio V from the scheduled oe V s. The ucertaity set is defied as U(u) {ɛ R N b : u LB ɛ u UB }, where u LB 0 ad u UB 0. A. Surrogate Affie Approximatio Problem (P) is computatioally itractable due to the ifiite costraits (4d). O the other had, U(u) is a fuctio of u ad its extreme poits are ukow. Hece, problem (P) caot be solved directly by extreme poit-based approaches, which are ofte used to hadle ifiite costraits i robust optimizatio literature [7], [32], [33]. The problem of DNElimit search has a similar structure [23]. It was viewed as a itractable oliear problem. Hece, ISO-NE uses a heuristic method to set re-dispatch strategy, istead of fidig the optimal oe. Next, it is show why the traditioal affie policy is itractable to solve problem (P). Cosider the traditioal affie policy y(ɛ) = Gɛ, (5) where G R Ng N b is the matrix of affie policy. It maps the ucertaity to re-dispatch. With the restricted recourse Gɛ, the jth row of costrait (4d) becomes { } max (CG + E) 0 ɛ j ɛ + B j x d j, j J (6) s.t. u LB ɛ u UB where ( ) j deotes the jth row of matrix/vector. It is oted that u LB ad u UB are o-egative vectors. J is the set of rows i (4d). Equatio (6) shows that value of (CG + E) j ɛ + B j x d j is ever greater tha zero for ay ɛ respectig u LB ɛ u UB. I other words, whe ɛ is revealed, the corrective actio Gɛ respects the system-wide costraits. Followig strog duality [34], equatio (6) is exactly recast as CG + E + π LB π UB = 0 (7a) Bx d + π LB u LB + π UB u UB 0 (7b) π LB, π UB 0 (7c) where x, G, π LB, π UB, u LB ad u UB are variables. Equatio (6) is equivalet to (7a)-(7c), a approximatio to the origial costrait (4d). As a side ote, affie policy is ofte employed i stochastic programmig ad robust optimizatio literature due to its computatioal tractability [9], [11], [26], [35]. However, equatio (7b) is computatioally itractable due to biliear term π LB u LB ad π UB u UB resulted from the traditioal affie policy. Next, a ew SAA method is proposed to solve the problem i polyomial time. Fudametally, the SAA method itroduces ew variables to replace biliear term π LB u LB ad π UB u UB. The ew costraits without oliear terms are surrogates to the origial oes. The proposed techique does ot relax ay costraits, eve better, it icreases the freedom degree of affie policy. While techiques are differet, the termiology surrogate ca date back to 1990s i Lagragia Relaxatio literature [36]. Authors i [36] propose to update Lagragia multipliers by solvig oly some rather tha all subproblems. To elimiate the biliear terms i (7b), two surrogates are desiged i the SAA method. The origial ucertaity set U(u) is the uderlyig cause of oliearity. The proposed two surrogates are thus used to replace U(u). Oe is a surrogate ucertaity set, ad the other is a surrogate fuctio. Defie the ew surrogate ucertaity set Ũ { (δ LB, δ UB ) R 2N b : 0 δ LB 1, 0 δ UB 1 }. (8) The surrogate ucertaity set Ũ is costat. Defie the surrogate fuctio s(u LB, u UB ) : R 2N b R 2N b s(u LB, u UB ) diag(u UB )δ UB diag(u LB )δ LB, where diag( ) is a diagoal matrix whose diagoal etries are elemets of vector. The the image of Ũ uder the surrogate fuctio s(u LB, u UB ) is { } Û diag(u UB )δ UB diag(u LB )δ LB : (δ LB, δ UB ) Ũ. A lemma is established as follows regardig the primitive set ad the image of the surrogate set. Lemma 1. The image of the ucertaity set Ũ uder the surrogate fuctio s(u LB, u UB ) is equivalet to U, i.e., U(u) = Û. The proof is trivial. Lemma 1 reveals that ay origial ucertaity poit i U ca be replaced with its image i the surrogate set Û. Based o Lemma 1, propositios are established as follows cocerig the computatioal tractability. Propositio 1. Let U LB = diag(u LB ) ad U UB = diag(u UB ). The, costrait (4d) is rewritte as [ ] [ ] δ Bx + Cŷ(δ LB, δ UB ) + E U LB U LB UB d, δ UB (9) (δ LB, δ UB ) Ũ, where ŷ(δ LB, δ UB ) : R 2N b R Ng is the surrogate re-dispatch fuctio of ucertaity (δ LB, δ UB ). Accordig to the defiitio of Û, [ [ ] U LB U UB] δ LB Û, (δ LB, δ UB ) Ũ. Followig Lemma 1, (x, ulb, u UB ) respectig (9) must be feasible for (6). Propositio 2. Cosider a surrogate affie fuctio [ ] δ LB ŷ(δ LB, δ UB ) = Ĝ. (10) δ UB δ UB

6 6 TO APPEAR ON IEEE TRANS. ON POWER SYSTEMS. CITATION INFORMATION: DOI /TPWRS , IEEE TRANSACTIONS ON POWER SYSTEMS Followig the strog duality, the surrogate affie approximatio of (4d) is [ ] CĜ + E U LB U UB π 0 (11a) Bx d + π 1 0 (11b) π 0 (11c) where x, Ĝ, π, u LB ad u UB are variables. Compared to (7a)-(7b), equatio (11a)-(11c) do ot have ay oliear terms. Biliear term π LB u LB ad π UB u UB i (7b) are replaced with liear term π 1 i (11b). Equatio (11a)-(11c) are liear ad computatioally tractable. Followig Propositio 2, the SAA model (SAA-P) J APP = mi x,u,ĝ s.t. c x f u (4b) (4c), (11a) (11c) is formulated, ad it ca be solved usig moder LP solvers. I problem (SAA-P), the decisio variables are geeratio dispatch, VER s scheduled output, load demad, upward/dowward flexibility, ad surrogate affie policy. Surrogate affie policy Ĝ i problem (SAA-P) is used for re-dispatch oce the ucertaity is revealed. After optimal Ĝ is attaied, the re-dispatch ca be writte as [ ] [ ( δ LB max (U LB Ĝ = δ Ĝ ) 1 ɛ, 0 ) ] UB max ( (U UB ) 1 ɛ, 0 ) (12) = Ĝ [ (U LB ) (U UB ) 1 ] [max ( ɛ, 0) max (ɛ, 0) ].(13) Above two equality equatios aturally itroduce two ways to calculate re-dispatch whe ucertaity ɛ is realized. Equatio (12) shows oe way that requires the calculatio of the surrogate ucertaity. I cotrast, (13) shows the other way without such calculatio. Oe ca calculate Ĝ [ (U LB ) (U UB ) 1 without ucertaity ɛ iformatio i advace. It is preprocessig of the surrogate affie policy. B. Optimality of SAA I Sectio III-A, the SAA method is itroduced to solve the co-optimizatio model. I this part, its optimality is aalyzed by beig compared with that of the covetioal affie policy. Propositio 3. Deote the feasible regio of (x, u) i (SAA- P) as F saa. Cosider the traditioal affie policy model (TAP-P) J AP = mi x,u,g s.t. ] c x f u (4b) (4c), (7a) (7c), ad deote the feasible regio of (x, u) i (TAP-P) as F tap, the F tap F saa always holds. Proof. Assume global optimal solutio (x, u, G ) to itractable problem (TAP-P) is attaied from a oracle. The surrogate affie policy is costructed as Ĝ = [ G U LB G U UB ]. Feasible Regio of (x,u) i Problem (SAA-P) Feasible Regio of (x,u) i Problem (TAP-P) Fig. 2. Compariso of feasible regios of the covetioal affie policy ad the surrogate affie policy-based approaches. Problem (P) Affie Policy Surrogate Affie Policy Noliear Itractable Liear Tractable Optimal Value J1 J AP Optimal Value J2 J SAA J SAA J Fig. 3. Compariso of covetioal affie policy ad surrogate affie policy. Give ay ucertaity ɛ [ u LB, u UB ], a surrogate ucertaity ca be costructed as δ LB = max ( (U LB ) 1 ɛ, 0 ), δ UB = max ( (U UB ) 1 ɛ, 0 ), where max(, ) returs a compoet-wise maximum vector. (x, u, Ĝ ) is a feasible poit to problem (SAA-P). Therefore, F tap F saa. Propositio 3 idicates that the feasible regio of (x, u) i SAA-based model is ever smaller tha that i the covetioal affie policy-based model. Due to the higher freedom degree of the surrogate affie policy, it is possible that the feasible regio of the covetioal affie policy-based model is a strict subset of that of the SAA-based model. This case is illustrated i Fig. 2. The followig lemma shows the relatio betwee the optimal values obtaied usig these two methods. Lemma 2. If problem (SAA-P) ad (TAP-P) are feasible, the J SAA J AP holds, where J SAA ad J AP deote the optimal values of problem (SAA-P) ad (TAP-P), respectively. The proof is trivial give Propositio 3. If F tap F saa holds, the J SAA < J AP holds sometimes. It is show i [37] that separatig the ucertaity ito upward ad dowward parts ca improve the solutio. Fig. 3 illustrates a compariso betwee two methods. I the SAA method, oe oly eeds to hadle tractable liear costraits. I cotrast, oe has to deal with a itractable oliear problem i the covetioal affie policybased method. C. Extesio to Multi-period ED The co-optimizatio model ad solutio approach is readily exteded to the multi-period ED. Cosider period idex t = {1,, T }. The objective fuctio (1) ad costrait (2a)-(3c) are repeated over all periods. The geeratio output is subject to rampig up/dow costraits R dow i δ P i,t P i,t+1 R up δ, i I, t = 1,, T 1. i AP

7 YE: SURROGATE AFFINE APPROXIMATION BASED CO-OPTIMIZATION OF TRANSACTIVE FLEXIBILITY, UNCERTAINTY, AND ENERGY 7 G1 1 G2 2 L1 3 VER1 V VER L2 L3 G3 V TABLE I FULLY CONTROLLABLE GENERATORS IN SIX-BUS SYSTEM P mi* P max* R dow /R up*** S1-IC ** S2-IC ** S3-IC ** G G G * MW; ** icremetal cost, $/MWh; *** MW/Iterval TABLE II COMPARISON OF FINDING OPTIMAL FLEXIBILITY IN THREE CASES IN THE SIX-BUS SYSTEM Fig. 4. The oe-lie diagram for the six-bus system Costraits betwee ˆP i,t ad ˆP i,t+1 ca also be eforced. Equatio (4d) is geeral eough to iclude these rampig costraits, ad the proposed solutio approach is still applicable. It is oted ISO/RTO rus ED tool o a rollig basis, ad oly issues dispatch sigal for the first iterval i practice. To simplify the policy, G ca be restricted to deped oly o the most recetly revealed iformatio v t. A multi-stage model is thus attaied sice the re-dispatch decisio at t ca be made based o all available iformatio at time t. I the proposed SAA method, the resultig problem (SAA-P) is covex ad has a similar structure with the covetioal affie policy-based multi-stage model [38]. D. Applicatio i DNE-limit Search The proposed SAA method ca also be directly applied to the problem of DNE-limit search i ISO-NE. The DNE-limit search is the sequetial optimizatio, where the regular ED problem is first solved, ad the a problem maximizig the DNE-limit is formulated give the ED solutio. It has several advatages to apply the SAA method i the DNE-limit search. Firstly, the SAA method gets optimal participatio factors i polyomial time. Secodly, SAA provides a adjustmet matrix with higher freedom degree for the cotrollable geerators. More specifically, the optimal participatio factors for upward ucertaities ca be differet from those for dowward ucertaities. Thirdly, due to tractability of SAA, oe ca solve two problems sequetially to get a larger ijectio-rage whe a multiplicity of ED occurs. The first oe is a regular ED problem, whose optimal value will be eforced as a costrait i the secod problem, i.e., DNE-limit search. Alteratively, oe ca also get the ED ad DNE-limit i oe shot by solvig problem (SAA-P) with a special f that has small elemets. These advatages i retur help itegrate more VERs. IV. CASE STUDIES The simulatios were performed i a six-bus system ad a modified IEEE 118-bus system to illustrate the proposed model ad SAA method. All cases were solved usig CPLEX 12.7 o a PC with 2.6 GHz Itel Core i7. A. Six-bus system Fig. 4 shows the oe-lie diagram of the six-bus system. There are three FCGs, two VER uits, ad three loads. Decisio Variable Parameter Case 1 (sequetial opt.) up./dow. flexibility G, ED Case 2 (sequetial opt.) up./dow. flexibility, G ED Case 3 (Co-opt.) up./dow. flexibility, G, ED Table I shows parameters of three FCGs. All geerators are committed. For simplicity, oly a sigle-period schedulig problem was cosidered. Three scearios, S1, S2, ad S3, were cosidered with differet Icremetal Costs (ICs) for FCG. Forecast outputs of VER1 ad VER2 are 16 MW ad 10 MW, respectively. The upward deviatios are 16 MW ad 14 MW; the dowward deviatios are 15 MW ad 8 MW, respectively. The followig case studies were performed i the six-bus system: Case 1: Sequetial optimizatio with fixed AP. Case 2: Sequetial optimizatio with variable AP. Case 3: Co-optimizatio with variable AP. I the six-bus system, flexibility is held for the deviatio of VER output. Hece, it is gauged by allowable ijectio-rage for VERs. Table II shows the mai differeces i three cases. I Case 1, flexibility/ijectio-rage for VER was calculated give G ad ED solutio. I Case 2, the ijectio-rage ad G are decisio variables give ED solutio. As the search of maximum ijectio-rage for VERs fiishes i two steps, it is the so-called sequetial optimizatio i Case 1 ad 2. I Case 3, ijectio-rage, ED solutio, ad G were optimized simultaeously. The trasmissio costraits were relaxed i Case 1 ad 2, so that it is easy to illustrate the basic ideas. For compariso, the trasmissio costraits were eforced i Case 3. 1) Case 1: I this case, there are two steps to determie the ijectio-rage for VER or flexibility. The first step is to solve a classic ED model that determies the FCG power outputs. The secod step is to fid the maximal secure ijectio-rage for VER, give the ED solutio to the first problem. I Case 1, the costrait (2e) was dropped. This case is similar to the DNE limit search i ISO-NE [23]. Table III shows the ED solutios i three scearios with differet ICs. It is observed that the total cost rises with icreasig ICs. The lowest cost is $2684 i the sceario with ICs, $10/MWh, for G1, G2, ad G3. The scheduled VER outputs are 16 MW ad 10 MW, respectively. They reach the limits (i.e., forecast values) accordig to costrait (2d), as VER geerators IC, $0/MWh, is much cheaper tha other geerators IC, $10/MWh. I this case, the affie adjustmet matrix is

8 8 TO APPEAR ON IEEE TRANS. ON POWER SYSTEMS. CITATION INFORMATION: DOI /TPWRS , IEEE TRANSACTIONS ON POWER SYSTEMS TABLE III GIVEN ECONOMIC DISPATCHES FOR SEQUENTIAL OPTIMIZATION WITH DIFFERENT INCREMENTAL COSTS IN CASE 1 AND 2 G1(MW) G2(MW) G3(MW) VER1(MW) VER2(MW) Cost ($) S S S TABLE IV FLEXIBILITY FOR VERS (MW) Up.VER1 Up.VER2 D.VER1 D.VER2 S S S , (14) which was determied proportioally accordig to the rampig rate of FCG. Although it is o-optimal, a similar strategy is used i idustry, as fidig the optimal adjustmet matrix was regarded as computatioally itractable i the traditioal affie policy-based method. As a side ote, the SAA method ca fid the optimal adjustmet matrix i polyomial time. Based o the ED solutio ad the adjustmet matrix (14), it is trivial to calculate corrected dispatches ad ijectio-rages for VER. For istace, if the realized VER outputs are 17 MW ad 12 MW, respectively, the adjusted output vector i S1 is = [ ] The allowed deviatio rages for the VER are show i Table IV. It is observed that the system-wide upward deviatio rage is 11.5 MW (i.e., 11.5 = ) i S1. Colum Up. VER1 ( D.Ver1 ) is the upward (dowward) deviatio rage for VER1. However, while G2 reaches its lower boud (i.e., 10 MW) i S2, ad G3 reaches its lower boud (i.e., 0 MW) i S3, the deviatio rages are 0 MW i both S2 ad S3, give the ED solutio ad the adjustmet matrix (14). 2) Case 2: Similar to Case 1, the ED problem ad ijectio-rage search problem were solved sequetially, but with variable affie adjustmet i this case. Give the ED solutio, the proposed SAA method was used to fid the secure ijectio-rage. Table VI shows the secure ijectio-rages for VER i Case 2. It is observed that the system-wide ijectiorage (or flexibility) is larger tha that i Case 2. Next, detailed discussios ad aalysis of this advatage will be preseted. Some limitatios of sequetial optimizatio, such as limitatio of ijectio-rage ad ifeasibility, will be discussed. First, the larger ijectio-rage partially comes from the optimality of the affie adjustmet. Case 2 has the same ED solutio with Case 1, as both employ the same ED model. I cotrast, the adjustmet matrix is a decisio variable i Case 2. I the problem of optimizig ijectio-rage/deviatio rage/flexibility, the same coefficiet for VER1 ad VER2 were used. I the SAA method, oe ca attai the optimal solutio by solvig a LP problem. Table V presets the optimal surrogate affie adjustmet matrices i sceario S1, S2, ad TABLE V OPTIMAL SURROGATE AFFINE ADJUSTMENT MATRICES IN CASE 2 (MW) S1 * S2 * S3 * δ1 UB δ2 UB δ1 LB δ2 LB δ1 UB δ2 UB δ1 LB δ2 LB δ1 UB δ2 UB δ1 LB δ2 LB G G G * 3 4 matrix. TABLE VI OPTIMAL ALLOWED DEVIATION RANGE FOR VER IN CASE 2 (MW) Up.VER1 Up.VER2 D.VER1 D.VER2 S S S S3, ad Table VI presets the largest allowed deviatios for VER i differet scearios. Table V shows three 3 4 matrices. The re-dispatch ca be calculated based o (12)-(13) accordig to Table V ad VI. For example, cosider realized VER1 ad VER2 output beig 10 MW ad 13 MW, respectively. The, the output vector after re-dispatch i S1 is = } 14 }, 17 } 14 } max{0, max{0, max{0, max{0, where (16 10)/17 ad (13 10)/14 are the surrogate ucertaities from VER1 ad VER2, respectively. By comparig the data i Table IV ad VI, oe ca fid that the deviatio rage is larger i Case 2, eve Case 1 ad 2 have the same ED. For example, i sceario S1, the total upward deviatio rage icreases to 22 MW (i.e., 22 = ) from 11.5 MW. At the same time, the dowward deviatio rage also icreases to 17 MW from 11.5 MW, although up to 16 MW is useful. Secod, the larger ijectio-rage for VER is partly because of the higher dimesio of affie policy i SAA method. As each ucertaity was separated ito virtual positive ad egative compoets, various affie coefficiets ca be utilized for re-dispatch whe ucertaities fall ito differet regios. I Case 1, the affie adjustmet matrix is i the space of R 3 2. I cotrast, the affie adjustmet matrix is i the space of R 3 4 i Case 2. Take sceario S2 as a example. As G2 s output is at its lower boud 10 MW, it caot further lower its output. Therefore, G2 is ot able to provide the dowward flexible resource. I Case 1, although G2 has the ability to provide the upward reserves, the adjustmet coefficiets are all zeros limited by its zero dowward reserves. I cotrast, G2 s coefficiet of surrogate ucertaity (egative compoet) for VER1 is 6, accordig to colum δ1 LB for sceario S2 ad row G2 i Table V. At the same time, the allowed dowward deviatio for VER1 is 6 MW, ad the correspodig surrogate ucertaity is 1 = 6/6. It meas that all the upward reserve of G2 will be utilized for the ucertaity maagemet accordig to equatio (10) (i.e., 6 = ). Cosequetly, the system has more flexibility to accommodate VER output i Case 2. However, although the SAA method helps get the optimal secure ijectio-rage, the largest possible ijectio-rage for

9 YE: SURROGATE AFFINE APPROXIMATION BASED CO-OPTIMIZATION OF TRANSACTIVE FLEXIBILITY, UNCERTAINTY, AND ENERGY 9 TABLE VII EDS FROM THE CO-OPTIMIZATION WITHOUT TRANSMISSION CONSTRAINTS IN CASE 3 G1(MW) G2(MW) G3(MW) VER1(MW) VER2(MW) Cost ($) S S S TABLE VIII OPTIMAL ALLOWED DEVIATIONS OF VER GENERATION WITHOUT TRANSMISSION CONSTRAINTS IN CASE 3 Up1 Up2 D1 D2 Up a D b S S S a Chage of allowed upward deviatios of VER geeratio from Case 2. b Chage of allowed dowward deviatios of VER geeratio from Case 2. VER is still costraied by the available flexible resources. These flexible resources are determied as byproducts of solvig the ED problem. Accordig to Table VI, differet EDs lead to various ijectio-rages. For example, the total dowward deviatio i sceario S1 is 17 MW, which is larger tha that of 11 MW i sceario S3. Oe ca observe similar treds for the total upward deviatios i differet scearios. The simulatio results also show that the secure ijectiorage obtaied from the sequetial optimizatios may be ifeasible i reality. The ifeasibility is due to the fact that the VER geerator ca oly spill power, but ot produce electricity larger tha its maximum available power. Whe the available VER power is smaller tha the lower boud of the secure ijectio-rage, it is impossible to eforce the lower boud limit for VER geerator. For example, the allowed dowward deviatio rage is 11 MW i sceario S3 accordig to Table VI. It idicates power produced by VERs should be at least 15 MW (i.e., 15 = ). However, there is a possibility that the available VER power is 3 MW (i.e., 3 = ). I this case, the secure dowward rage is ifeasible. The similar defect exists i the DNE limit proposed by ISO-NE [23]. 3) Case 3: I this case, the SAA method was employed to co-optimize the trasactive flexibility, ucertaity, ad eergy. By solvig oe LP problem, oe ca get OIRs for VER, as well as the optimal power output of FCG (i.e., ED). The followig cotext will illustrate how the proposed cooptimizatio model addresses the issues revealed i sequetial optimizatio models. The results will verify its beefits i the social welfare ad the VER ijectio-rage. For compariso purposes, the co-optimizatio model was formulated without trasmissio lie costraits first, but with the feasibility costrait (2d). Accordig to Table VII, the cost i sceario S1 remais $2684, ad VERs are scheduled to geerate power at the forecast values i Case 3. However, accordig to Table VII ad Table VI, the total upward deviatio for VERs icreases by 1 MW (i.e., 1 = ) i sceario S1 from Case 2 to Case 3. Similarly, the dowward deviatio for VERs rises by 6 MW (i.e., 6 = ) i TABLE IX ALLOWED UPWARD DEVIATION OF VERS WITH DIFFERENT FLEXIBILITY BIDS IN SCENARIO S3 IN CASE 3 WITHOUT TRANSMISSION CONSTRAINTS VER1 VER2 Up-1 Up-2 G1 G2 G3 Cost ($) sceario S1. It suggests that the co-optimizatio model helps accommodate more VER power eve though the total cost is the same. That is because ED problem, as a LP problem, ofte has multiple optimal solutios. Co-optimizatio also addresses the ifeasibility issue i the sequetial optimizatios. Feasibility of VER ijectio-rage comes at the expese of slightly higher total cost whe biddig of flexibility is zero. For example, the total cost i sceario S3 is icreased by $36 = $2762 $2726 from Case 2 to Case 3, accordig to Table III ad VII. I Case 3, the VER geerators ca iject power withi the rage of [3 MW, 44 MW] (i.e., 3 = , 44 = ) as show i Table VII ad VIII. I cotrast, the rage is [15 MW, 42 MW] (i.e., 15 = , 40 = ) accordig to Table III ad VI. More importatly, [3 MW, 44 MW] is a feasible rage. Therefore, the operator ca guaratee that ay VER power withi the OIR will be securely accommodated. A importat questio the arises: is it possible to further icrease the secure ijectio-rage of VERs (flexibility) or lower the total cost? Next, this questio will be aswered, ad several iterestig observatios will be highlighted for the model with flexibility bids. Firstly, the social welfare icreases whe VER geerators are allowed to bid for flexibility. I Table IX, allowed upward deviatios, colum Up-1 for VER1 ad Up-2 for VER2, are differet with various flexibility bids i sceario S3. The bids are show i colum VER1 ad VER2 i Table IX. The colum Cost i Table IX shows the total cost. For example, accordig to the 2d row i Table IX, if the flexibility bids of VER1 ad VER2, respectively, are $4/MW ad $5.1/MW, the the allowed upward deviatios for VER1 ad VER2 are 4 MW ad 14 MW, respectively. Compared to data i the 3rd row i Table VII, the total social welfare is icreased by $72 = $2762 $2690 i Case 3, where the flexibility bids are itroduced. Secodly, the secure ijectio-rage for VERs ca be expaded whe the bid reaches a certai level. For example, i the last two rows of Table IX where biddigs are greater tha $5/MW, the system-wide allowed upward deviatio icreases to 23 MW (i.e., 23 = = ) from 18 MW. I fact, 23 MW is the highest possible upward deviatio (i.e., 23 = ). By comparig colum G3 i Table VII ad Table III, oe ca observe that the icrease of flexibility is due to the higher output of G3 i Case 3. If the upward deviatio occurs, the system operator ca lower G3 s output to 0 MW so that G3 provides the additioal 5 MW dowward reserve. However, oly whe the flexibility bid is larger tha

10 10 TO APPEAR ON IEEE TRANS. ON POWER SYSTEMS. CITATION INFORMATION: DOI /TPWRS , IEEE TRANSACTIONS ON POWER SYSTEMS TABLE X EDS AND ALLOWED UPWARD DEVIATIONS FOR VERS WITH FLEXIBILITY BIDS IN SCENARIO 3 WITH TRANSMISSION CONSTRAINTS VER1 VER2 Up1 Up2 G1 G2 G3 Cost Price ($/MW) (40.71, 5.86) (15.02, 1.99) Upward Flexibility (MW) a = 0.10 a = 0.15 a = 0.20 a = 0.25 a = 0.30 a = 0.35 a = 0.40 TABLE XI EDS AND ALLOWED UPWARD DEVIATIONS FOR VERS WITH FLEXIBILITY BIDS IN SCENARIO 3 WITH TRANSMISSION CONSTRAINTS VER1 VER2 Up1 Up2 G1 G2 G3 Cost $5/MW (i.e., 5 = 18 13), G3 will icrease its output, ad G2 will reduce its output. It verifies that itroducig bid ca help the system hold more flexibility. Thirdly, VER1 ad VER2 compete for the upward flexibility. By comparig the bids ad the allowed upward deviatios i Table IX, oe ca observe a iterestig poit that the VER geerator with larger bid always has the larger allowed deviatio. For example, whe the bids from VER1 ad VER2 are $4/MW ad $6/MW, respectively, the procured upward flexibility by VER2 is 14 MW, which is 10 MW (i.e., 10 = 14 4) higher tha that of VER1. 14 MW is also the highest possible deviatio of VER2. It suggests the cooptimizatio ecourages VERs to bid for flexibility based o the beefit. From the system s perspective, flexibility will be positioed cost-effectively. Now, cosider the impacts of trasmissio lie costraits. Table X presets the simulatio results with eforced trasmissio lie costraits. By comparig EDs i Table X ad IX, oe ca observe that G1 produces less electricity. I cotrast, G2 geerates more electricity. That is due to the cogestio of lie 1-4. Moreover, the total cost icreases whe the trasmissio lie costrait is eforced. For example, whe bids are $4/MW from both VER geerators, G1 s scheduled output decreases by MW (i.e., = ), ad G2 s output icreases by MW (i.e = ). The total cost is $ , which is icreased by $ (i.e., = ). The allowed upward deviatio does ot chage whe the trasmissio lie costraits are eforced i this case. To aalyze the impact o flexibility/oir for VERs, VER1 was moved from bus 5 to bus 1. The results are show i Table XI. By comparig the colum G1, G2, ad G3 i Table XI ad X, oe ca observe that G1 s output icreases ad G2 s output decreases if VER is moved to bus 1. It is observed that whe flexibility bids from VER1 ad VER2 Fig. 5. Upward flexibility U UB procured by VER27 whe the load is 3060 MW. Solid lies deote biddig strategies (2aU UB + b), which chage with the icreasig a. Red dots deote the optimal solutios with the icreasig a. are $6/MW ad $5.1/MW, respectively, the allowed deviatio of VER1 is decreased to 12 MW from 16 MW accordig to Table XI ad X, although system-wide flexibility remais 23 MW. If VER1 icreases its bid to $11/MW, the the allowed deviatio of VER1 decreases back to 16 MW. At the same time, the output of G1 decreases to MW from MW. It suggests that the cogestio of lie 1-3 prevets the larger power ijectio at bus 1. From the system s poit of view, the beefit of providig upward flexibility at bus 1 is larger tha that of usig the cheap eergy from G1, (i.e., = 5.9 > 3 = 13 10). Therefore, the upward flexibility is re-distributed betwee VER1 ad VER2, ad G2 supplies more loads i this case. B. Modified 118-bus System The modified IEEE 118-bus system cosists of 54 geerators, 186 lies, ad 91 loads. The VER geerators are located at 18 buses. The detailed data for the model ca be foud at bus data.xlsx. The sesitivity aalysis was performed with various upward ad dowward flexibility bids. The simulatio for the multi-period model was also performed. The quadratic curves were employed to simulate the flexibility beefit. For istace, the beefit of upward flexibility was assumed as a (U UB ) 2 + b (U UB ) + c, where a, b ad c are coefficiets. The icremetal beefit is 2a U UB + b. By chagig a, oe ca easily modify beefit curves. Two scearios were cosidered. Uit ON/OFF stats were assumed determied i advace. I the first sceario, the load is 3030 MW. As the load is low, oly a small umber of uits are committed. Figure 5 illustrates a set of the biddig curves with differet a. For simplicity, it was assumed that all VER uits use the same a, ad the bids for the dowward flexibility were set to 0 for all VER uits. Geerator VER27 is located at bus 27. It is observed that VER ca procure more flexibility if its bid price is high. For example, if a = 0.10, the the bid of VER27 is 0.2U UB +14. I this case, the red dot (40.71, 5.86) is the optimal poit, which represets that VER27 ca purchase MW flexibility at the price of $5.86/MW. Alteratively, settig a to 0.40 idicates that VER27 willigess-to-pay is lower for upward flexibility. I this case, the optimal poit is (15.02, 1.99), which meas that VER27 purchases MW

11 YE: SURROGATE AFFINE APPROXIMATION BASED CO-OPTIMIZATION OF TRANSACTIVE FLEXIBILITY, UNCERTAINTY, AND ENERGY 11 Price ($/MW) TABLE XII PROCURED FLEXIBILITY WITH VARIOUS BIDS (LOAD: 3060 MW) a *** Cost ** Scheduled * Up. Flex. * Dow. Flex. * OIR * *** $/MW 2 ** $ * MW (28,0) (34.42, 0.23) (47.29, 4.54) (41, 2) Upward Flexibility (MW) a = 0.10 a = 0.15 a = 0.20 a = 0.25 Fig. 6. Upward flexibility U UB procured by VER27 whe the load is 6060 MW. Solid lies deote biddig strategies (2aU UB +b) that chage with the icreasig a. Red dots deote the optimal solutios with the icreasig a. flexibility at the price of $1.99/MW. The eergy price at this locatio is aroud $12/MWh. The simulatio results idicate that VERs ca iject more clea eergy ito the grid by purchasig flexibility at a low price. Table XII shows the system-wide iformatio with icreasig a. Colum Up. Flex. deotes system-wide upward flexibility. For example, a = 0.1 idicates that VER geerators prefer to purchase more upward flexibility. The total cost is $ , ad upward flexibility is MW, ad system-wide OIR is MW. Data i Table XII idicates that system-wide upward flexibility ad the total cost (social welfare) are mootoically decreasig (icreasig) with a. If a icreases to 0.4 from 0.1, upward flexibility decreases by MW (i.e., = ) to MW. As a result, OIR is reduced to MW. I the secod sceario, the load was icreased to 6060 MW, ad more uits were committed olie (UC was determied i advace). Figure 6 shows the procured flexibility by VER27 with icreasig a. It depicts a similar tred as Figure 5. If VER27 is willig to pay more, the it ca procure more upward flexibility. A iterestig observatio is that VER27 ca get 28 MW flexibility free of charge. It idicates that flexible resources are free products i some circumstaces. At the same time, VER27 ca also procure MW flexibility at $1.8/MW, which is $4.06/MW (i.e., 4.06 = ) cheaper tha that i the first sceario. It reveals that more flexibility will be available whe more uits are o-lie. A importat tred is revealed that procured flexibility is a mootoically decreasig fuctio of a i geeral. It idicates that the more VERs are willig to pay, the higher flexibility they ca secure. Cosequetly, VERs ca iject more eergy ito the grid. O the other had, the total social welfare icreases. The model ca be exteded to the multi-period ED problem. ISOs/RTOs ofte perform multi-period ED problems o a rollig basis. To reduce the computatioal burde, the adjustmet matrix was set to a diagoal block matrix. I retur, the corrective actios of the FCGs are determied by the VER output deviatio at the curret iterval. For a 24-period problem, LP solver was able to get the solutio withi 281 secods. If iactive lie costraits were removed based o [39], the solutio time was reduced to 18 secods, which is 6.4% of the origial time. V. CONCLUSION With the growig VERs peetratio, it becomes importat to maage the demad of flexibility i the moder electric grid. This work provides a optio to address these questios: 1) how to determie the flexibility amout while keepig ISO/RTO idepedet; 2) how to allocate flexibility to demader cost-effectively; 3) how to hadle the high flexibility demad whe the flexible resource is deficiet. A model is proposed to optimize trasactive flexibility, ucertaity, as well as eergy. I this work, flexibility is defied as the chage rage of power ijectio that the system ca accommodate usig available flexible resources withi a specified time. A ovel SAA method is proposed to solve the problem i polyomial time. It is proved that its solutio is eve better tha the origial affie policy-based method used i the power literature. The SAA method ca also be applied to the DNE-limit search i the idustry. The simulatio results show the co-optimizatio approach icreases the social welfare ad proactively positios flexibility for VER accommodatio. As costraits i the SAA method are liear ad covex, the geeral acceleratio techiques for a LP/QP problem ca be used directly i the proposed approach. By itroducig decisio variables ad costraits associated with UC [3], [22], oe ca exted the co-optimizatio model to the UC problems. It will be a iterestig future work, as UC may positio more flexible resources. The proposed model is covex, ad Slater s coditio holds. Thus, the strog duality follows. Therefore, it is possible to derive margial prices for eergy ad upward/dowward flexibility based o Lagragia multipliers. It is worth metioig that the deliverability of flexible resources will result i cogestio compoets i margial prices. Whe the eergy loss is igored, the prices cosist of the Lagragia multipliers for the eergy balace costraits ad trasmissio lie costraits. This author, Ge, Shahidehpour, ad Li have doe some work o pricig scheme with o-dispatchable reewables [22], [26]. The priciples i [22] are applicable to the model i this work where the curtailmet of VER geeratio is cosidered. For example, the defiitio of Ucertaity Margial Price ca be exteded to the margial price for flexibility. Iterested readers are referred to [22], [26] for details. With the fair treatmet for ucertaity ad flexibility, VERs iclie to purchase flexibility cost-effectively. That requires VERs to self-maage ucertaities or self-optimize their resources. I retur, the

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He was hoored Outstadig Reviewer for IEEE Trasactios o Power Systems ad IEEE Trasactios o Sustaiable Eergy. He received Sigma Xi Research Excellece Award at Illiois Istitute of Techology i 2016.