Security-Constrained MIP formulation of Topology Control Using Loss-Adjusted Shift Factors

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1 1 Security-Constrained MIP formuation of Topoogy Contro Using Loss-Adjusted Shift Factors Evgeniy A. Godis, Graduate Student Member, IEEE, Michae C. Caramanis, Member, IEEE, C. Russ Phibric, Senior Member, IEEE, Aesandr M. Rudevich, Member, IEEE, Pabo A. Ruiz, Member, IEEE. Abstract Foowing up on our previous wor, we formuate a inear, oss-adjusted, shift factor mixed integer program MIP to co-optimize generation and networ topoogy. Whie both the Bθ and shift factor topoogy contro TC formuations ead to production cost saving, we showed that the shift factor formuation performs better for sma to medium switchabe sets. In this paper we extend the origina TC shift factor formuation to incude margina osses. We derive oss-adjusted shift factors and show that both osses and fows can be updated ineary with a change in topoogy by taing advantage of fow-canceing transactions FCTs. The margina oss formuation we present in this paper cosey resembes that of most maret engines. In doing so, we aim to better approximate the AC power fow and to generate topoogies eading to production cost savings whie maintaining feasibiity subject to AC security constrained power fow SCOPF constraints. I. NOMENCLATURE Vectors are indicated by ower case bod, matrices by upper case bod, and scaars by ower case itaic characters indexed appropriatey. Upper imits are indicated by an over-bar, and ower imits by an under-bar. Diagona matrices are denoted with a tide. Sensitivities are indicated with Gree characters. Indices m, n Nodes., Lines. m Line from node. n Line to node. τ Contingent topoogies. Contingent Topoogy-Dependent Parameters and Variabes For contingent topoogy τ, f τ Vector of rea power fows on transmission eements. g τ Bias from inearization of transmission fows. f τ, f τ Vectors of transmission imits. F τ, F τ Diagona matrices of transmission imits. v τ Vector of fow-canceing transactions. Ψ τ Shift factor matrix. ˆΨ τ Loss-adjusted shift factor matrix. Ψ M τ Shift factor matrix associated with monitored ines. Ψ S τ Shift factor matrix associated with switchabe ines. Φ SS τ PTDF matrix of switchabe ines for transfer between switchabe ine terminas. Φ MS τ PTDF matrix of monitored ines for transfer between switchabe ine terminas. ˆΦ τ Loss-adjusted PTDF matrix. ψτ m Eement of Ψ for ine, node m. φ τ PTDF of ine for a transfer across ine. o τ LODF of ine for the outage of ine. µ τ, µ τ Monitored faciities shadow prices. α τ, α τ Switchabe faciities shadow prices. Contingent Topoogy-Independent Parameters and Variabes 1 Vector of ones. 0 Vector of zeros. I Identity matrix. z Vector with the state of transmission ines. c Vector of noda generation variabe cost. p Vector of noda generation. Vector of noda oads. x Line oss factors. I Line current. R Line resistance RMS vaue. V Nomina Line votage RMS vaue. d Normaized vector that aocates tota transmission osses to busses. Fu topoogy without contingencies. f 0 Reference fows from topoogy of a oad fow soution. x 0 Line oss factors for a reference fow vector f 0. b 0 Bias factor from inearization of transmission osses in topoogy. g 0 Bias factor from inearization of transmission fows in topoogy. Vector of noda oads. λ Power baance shadow price. η Loss equation shadow price. Ψ S Matrix of a Ψ S τ. Ψ M Matrix of a Ψ M τ. O MS LODF matrix of monitored branches for the outage of switchabe branches. M Sufficienty arge number. INTRODUCTION In recent years there has been a significant interest in cooptimization of transmission and generation in power system operation. Appications have varied from corrective contro [1] [3] to security enhancements [4], [5], oss minimization [6], [7] and more recenty to production cost savings under economic dispatch [8] [10] and unit commitment UC [11], [12]. Previous wor has shown that even under a fu security-constrained OPF, significant production cost savings

2 2 can be achieved. In [13] we introduced a nove ossess, inear MIP formuation to efficienty mode topoogy contro TC using fow-canceing transactions FCTs. This shift factor TC formuation is compact and scaabe, especiay when the set of candidate switchabe ines is sma reative to the number of monitored transmission eements. Reative to previous Bθ impementations of TC, the size of the shift factor formuation is a function of the number of contingencies times the number of monitored and switchabe transmission eements as opposed to number of contingencies times a transmission eements, which can be significant for arge system. Additionay, the use of FCTs eeps the inear structure of the MIP and avoids re-cacuation of the shift factor matrix with changes in transmission topoogy. In [13] we compare the computationa performance of the FCT-based formuation to the previousy used B-θ formuation and find that the FCT-based formuations resut in significanty smaer soution times. Athough the MIP formuation is inear, we observe exponentia growth in computationa time. For the IEEE 118- bus test case, a fu N 1 security constrained SC OPF with more than 24 switchabe ines becomes impractica to sove. To address this issue, we tested various heuristics such as price difference, ine profit and tota cost derivative 1 to identify candidate ines for switching and showed that through a combination of these heuristics we can achieve simiar cost savings to the MIP whie maintaining tractabe soution times. Further, consistent with previous pubications, most of the savings are reaized with ony a few ine openings and these initia savings are amost fuy captured by appropriate use of heuristics. We again note that the FCT-based approach avoids the need to recacuate the shift factor matrix with each topoogy change and is more efficient than the B-θ formuation in simuating these heuristics. Whie the ossess DC formuation shows promising resuts, most modern power marets incude a inearized mode of osses in their maret cearing agorithm. Therefore, for Independent System OperatorsISOs to adopt topoogy contro, it is important to incorporate margina osses into the TC MIP formuation. Additionay, whie TC provides benefits under the DC SCOPF context it must utimatey satisfy AC OPF constraints. If the DC optimized topoogy is not feasibe with respect to AC constraints, it is often time consuming to restore feasibiity whie maintaining production cost savings. Incuding osses in the TC MIP formuation shoud ead to a coser approximation of the AC OPF and thus reduce the occurrence of AC-infeasibe soutions. The rest of this paper has six sections. Section II summarizes fow-canceing transactions and the ossess TC MIP formuation. Section III introduces a formuation with osses and Section IV formuates the TC MIP with osses. Section V discusses LMP decompositions with osses and Section VI compares the performance between formuations with and without osses and to the fu AC OPF. Finay, Section VII concudes. 1 See [14], [15] for detais. II. FCTS AND THE LOSSLESS SHIFT FACTOR TC MIP To motivate the importance of FCTs we first show the difficuty arising from a naïve shift factor impementation of TC by deriving the fow constraints. Consider first a transaction pair of injection/withdrawa of v MW from m to n. The change in fow on ine due to this transaction can be expressed using the power transfer distribution factor PTDF [16] as f = φ v = ψ m v The change in fow on ine per unit of fow on ine after ine is disconnected is caed the ine outage distribution factor LODF [16] and is defined as o = 1 φ o = 1 φ,, φ 1 Hence, after ine is disconnected the change in fow on ine is f = ψ m 1 ψ m f 1 where the superscript denotes the disconnection of ine. Using the definition of fow as f = Ψp and 1 we have f = f + f ψ p + = ψ m 1 ψ m ψ p 2 where ψ denotes the row of Ψ corresponding to ine. Using our notation of z e = 1 to denote that ine is cosed and z = 0 to denote that it is disconnected, we can express the change in ψ conditiona on the opening of ine using 2 as ψ z = ψ m 1 ψ m ψ 1 z We can see that using this shift factor in an OPF formuation woud mae the probem non-inear since [ ψ z ]p is not inear in the decision variabes, z, p. The ey behind the formuation deveoped in [13] is that it aows us to use FCTs to sove a inear MIP under the shift factor formuation. FCTs are a common too for deriving ine outage distribution factors [16] and are defined as pairs of injections and withdrawas at the end of ines to be opened that have the same impact on a fows in the rest of the system as actuay opening these ines. If we open a ine the impact this opening wi have on the fow on any other ine is f = o f 3 Aternativey, if we want to repicate this impact by using a FCT, the change in fow on ine due to an injection/withdrawa v across ine is f = φ v 4

3 3 Equating 3 and 4 gives φ o f = φ v 1 φ f = φ v f v = 1 φ exacty as shown in [13]. The resuting SCOPF formuation with TC is C = min p,v,z c p 6 s.t. 1 p = 0, 7 5 p p p, 8 f M τ Ψ M τ p + Φ MS τ v τ f M τ, τ 9 F S τ z ΨS τ p + Φ SS τ I v τ F S τ z, τ 10 M 1 z v τ M 1 z, τ 11 z {0, 1}, 12 We refer to probem 6-12 as the ossess shift factor TC formuation. The set of switchabe ines S denotes the set of ines that are enforced and are candidates for switching whie the set of monitored ines M denotes the set of ines that are enforced and may not be switched. For each opened ine S, z = 0 and the corresponding FCT is unrestricted. For a such ines, constraints 10 become a set of equaity constraints forcing the fow between opened ines and the rest of the system to zero, thus defining a simutaneous system of inear equations for the corresponding FCTs, which fa out directy from the inear formuation. Soving this system is the same as appying the principe of superposition [17] to 2 5: vτ S = fτ S I Φ SS 1 τ 13 For a singe ine, 13 is identica to 5 and substituting this vaue of v into equation 9 impacts fows on monitored ines exacty as in 2. Hence, FCTs give us a way to ineary repicate an update to the shift factor matrix whie co-optimizing transmission and generation. III. LINEARIZED LOSS MODEL FORMULATION Resistive osses are a quadratic function of current fowing on each transmission ine: Loss = I 2 R = f 2R cos 2 ϕ V 2 f 2R V 2, 14 where ϕ is the ange difference between votage and current and the approximation in the ast equaity depends on the assumptions that reactive power fows can be ignored votage and current are in phase, ϕ = 0. To incorporate a inear approximation of osses into the DC OPF we perform the standard Tayor series expansion around a base fow f 0 : Loss b 0 + Loss f f 0 2 Under the assumption that the set S is non-isanding. f 15 where for a ine, Loss f Using 16 we can aso express Loss as Loss = 1 2 = 2R V 2 f = x 16 Loss f 17 f Equating 17 and 15 for f = f 0 we can derive the bias term b 0 as Loss 0 = 1 2 x0 f 0 = b 0 + x 0 f 0 b 0 = 1 2 x0 f 0 Therefore, for any fow vector f, we write osses as Loss = x 0 f 1 2 f 0 18 The term x 0 is referred to as the vector of ine oss factors. The oss formuation we present here is simiar to one used in rea marets e.g. [18] where osses are incuded in the energy baance constraint and in the fow constraints via a noda aocation of Losses represented beow by the normaized vector d. Litvinov et a. [19] showed that the advantage of this formuation compared to other approaches is that ine osses and fows are reference bus independent. The formuation beow is sighty different from the one described in [19] and therefore, we wi repeat the proof of reference bus independence in the Appendix. Without oss of generaity, contingency constraints are excuded. 3 min p c p 19 s.t. 1 p = Loss 20 Loss = x 0 g 0 + Ψp d Loss 1 2 f 0 21 f g 0 + Ψp d Loss f 22 p p p 23 We wi refer to constraints as Formuation L1. In the fow constraints 22 the d vector aocates Loss to busses. Without this term, osses woud be baanced at the reference bus and thus the formuation woud be reference bus dependent. There are many ways to seect d and we wi not deve into this probem here. An intuitive approach, and the one we assume in this paper, is to set d n = n m m n, which aocates osses to oad busses in proportion to their contribution to tota oad. 3 Note that ony osses in the base topoogy are incuded in the energy baance equation, osses in contingent topoogies ony impact contingent fows.

4 4 IV. TC MIP FORMULATION WITH LOSSES As we saw in section II, FCTs ineary impact fows in the same way as updating the shift factor matrix. With the introduction of osses, however, these FCTs woud no onger be baanced since the injection at one end of the ine is not equa to the withdrawa at the other end. In the case of osses we must redefine FCTs as the oss-adjusted canceing fows that need to be introduced so that the effect from these fows is the same as actuay opening the ines 4. Fortunatey, we can sti retain the same framewor of the ossess shift factor TC probem. To do this, we first derive the oss-adjusted shift factor matrix, ˆΨ and oss-adjusted PTDF matrix, ˆΦ, by expicity expressing fows in terms of osses. Re-arranging 21 we have Loss = x0 g 0 + Ψp f 0 2 The fow equation can thus be expressed as f = g 0 + Ψ p d x0 g 0 + Ψp f 0 2 = g 0 Ψd x0 g 0 f Ψp Ψd x0 Ψp = ĝ 0 + Ψ I dx0 Ψ p f = ĝ 0 + ˆΨp 24 from which we see that the oss-adjusted shift factor matrix and fow bias are ˆΨ = Ψ I dx0 Ψ ĝ 0 = g 0 Ψd x0 g 0 f 0 2 For competness the fow bias g 0 can be cacuated as: g 0 = f 0 Ψp 0 0 d Loss 0 = f 0 Ψp 0 0 dx 0 f 0 2 = I + Ψd x0 2 f 0 Ψp 0 0 The oss-adjusted PTDF can now be expressed as: ˆφ = ˆΨ m ˆΨ n = ψ m ψ m x 0 ψ d ψ n ψn x 0 ψ d = φ φ x 0 ψ d where ψ m denotes the coumn of Ψ corresponding to bus m and φ denotes the coumn of Φ corresponding to ine. 4 These FCTs wi be adjusted to account for osses across a transmission ines, measured at the receiving ends. Athough this derivation is not necessary for the MIP formuation it provides some intuition for how the PTDF is adjusted for osses. We shoud note that athough the oss-adjusted shift factors and PTDF matrices depend on an initia dispatch and set of fows, they can neverthess be pre-cacuated from the origina shift factor matrix using ony matrix mutipication, which woud be fast even for arge systems. With these oss-adjusted shift factors and PTDFs we can now express oss-adjusted FCTs simiary to 13. In addition, since the Loss equation is a function of fows, we appy FCTs to update Loss for ine openings. Partitioning transmission osses into osses from monitored and switchabe ines respectivey, we have: Loss M = x 0 g 0 M 1 2 f 0M + Ψ M p d Loss MS + ˆΦ v τ0 Loss S = x 0 g 0 S 1 2 f 0S + Ψ S p d Loss SS + I ˆΦ v τ0 Finay, the fu topoogy contro DC SCOPF MIP with osses is: C = min p,v,z c p 25 s.t. 1 p Loss = 0 26 Loss = Loss M + Loss S 27 Loss M = x 0M ĝ 0M 1 2 f 0M + ˆΨ M p + ˆΦ MS v τ0 Loss S = x 0S ĝ 0S 1 2 f 0S + ˆΨ S p + SS ˆΦ Iv τ0 f M τ ĝ 0 τ + ˆΨ M τ p ˆΦ MS τ v τ f M τ, τ 30 F S τ z ĝ0 τ + ˆΨ S τ p + ˆΦSS τ I v τ F S τ z, τ 31 M 1 z v τ M 1 z, τ 32 p p p 33 z {0, 1}, S 34 We refer to probem as the oss-adjusted shift factor TC formuation. Note that the d Loss term in the above formuation is repaced via the equivaence reationship shown in 24. Generay, ISOs may ony monitor a subset of a transmission ines for osses. This means that Loss wi be aggregated over a set that is smaer than M S. In the extreme case when no ines in S are monitored for osses, constraint 29 woud be empty. If additionay the end nodes of ines in S are not oad busses d m, d n = 0, constraints 31 woud reduce to 10 and FCTs woud be cacuated independent of osses. V. LOCATIONAL MARGINAL PRICES AND LOSSES In this section we derive LMPs under the oss-adjusted shift factor TC formuation. By definition the LMP of probem equas the derivative of the Lagrangian with respect to a

5 5 change in noda oad. Let the Lagrangian mutipiers or shadow prices associated with constraints 26, combined constraints 27-29, 30 and 31 be denoted by λ, η, µ and µ, and α and α respectivey, where the atter four shadow prices are the coection of the respective contingent topoogy shadow prices. Using these shadow prices, LMPs are given by M LMP = λ1 λ ˆΨ x 0M S + ˆΨ x 0S ˆΨ M µ µ ˆΨ S α α = = λ1 λ ˆΨ x 0 ˆΨ M µ µ ˆΨ S α α 35 where ˆΨ M and ˆΨ S are matrices that consist of the coection of ˆΨ M τ and ˆΨ S τ, for a contingent topoogies τ. To better understand the meaning of LMPs under this MIP formuation we consider two aternate non-mip formuations [20] where we fix z = z optima vaues from the MIP soution. In the first formuation, which we refer to as the Static MIP probem, we re-abe a ines with z = 1 as beonging to monitored set M 5 and resove probem The LMPs for the Static MIP probem are of the same form as 35. In the second formuation, which we refer to as the LP- Equivaent, we recacuate the shift factor matrix given z and sove a standard SCOPF probem where the base topoogy has the ines with z = 0 removed. The shift factor matrix given z is expressed as [16] ˆΨ M = ˆΨ M + ÔMS ˆΨS, 36 where ÔMS is the oss-adjusted ine outage distribution factor matrix indicating the impact of switched ines on monitored ines for each contingent topoogy. The LMPs in the LP- Equivaent probem are defined in the standard manner as LMP M = λ1 λ ˆΨ x 0M ˆΨ M µ µ. 37 With z = z, the soution to the Static MIP and LP- Equivaent probems are identica 6 and, therefore, the LMPs and shadow prices associated with fow imits on transmission eements given our reabeing of ines in the Static MIP probem must aso be identica. LMP = LMP 38 µ = µ. 39 Substituting 35 and 37 into 38 and canceing the energy component yieds, M λ ˆΨ x 0M S + ˆΨ x 0S + ˆΨ M µ µ + ˆΨ S α α = M λ ˆΨ x 0M + ˆΨ M µ µ. 40 Furthermore, substituting 36 and 39 into 40 and appropriatey canceing ie terms yieds, ˆΨ S S α α = λ ˆΨ Ô MS x 0M S λ ˆΨ x 0S + ˆΨ S Ô MS µ µ As mentioned previousy, for a ines with z = 1, v = 0 for a topoogies and constraints 31 reduce to constraints We do not recacuate ine oss factors in the LP-Equivaent probem as this woud yied a different soution. Finay, by substituting 41 into 35, canceing ie terms and grouping appropriatey, we see that the LMP derived from the oss-adjusted TC MIP formuation is LMP = λ1 λ ˆΨ M S + ˆΨ Ô MS x 0M ˆΨ M S + ˆΨ Ô µ MS µ 42 As an aside, we observe that equation 41 has a oss component, which is ony reevant in topoogy. By appropriatey grouping terms by topoogy we can partition 41 into two components: and α τ0 α τ0 = ÔMS µ τ0 µ τ0 + λ Ô MS x 0M x 0S 43 α τ α τ = ÔMS τ µ τ µ τ τ 44 Expressions are simiar to the reationship between shadow prices for switchabe and monitored ines reported in [20]. This reationship provides a generaization of the tota derivative concept for ine openings introduced in [14], where α τ α τ refects the margina vaue positive or negative of ine switching. As shown in 43, this vaue consists of both congestion and oss components. The congestion component is the scaar product of shadow prices for monitored constraints and LODFs of the open ine on these constraints. The oss component is the difference between: the impact of ine opening on osses in monitored faciities LODFs mutipied by the corresponding ine oss factors and, the oss factors of open ines. Whie in the ossess formuation topoogy change is never beneficia in the absence of transmission congestion congestion component equas zero, incorporating margina osses recognizes potentia benefits of topoogy contro due to a reduction in osses, which may be reaized in the absence of congestion. VI. SIMULATION RESULTS For a anayses we fixed a set of 16 switchabe ines to maintain tractabe run times as mentioned in the introduction and performed a Monte Caro simuation by randomy varying fue prices and wind capacity over 100 sampes 7. For each sampe we modeed the ossess DC SCOPF and the DC SCOPF with shift factors and bias terms adjusted for osses 8. Taing the optima topoogy from each of the two formuations we soved the AC SCOPF for each sampe to assess the feasibiity of the DC soutions and the p.u. congestion cost savings 9. Congestion cost savings for the DC and AC modes are cacuated reative to the DC and AC modes with no TC respectivey. Tabe I beow summarizes the resuts from 7 See [14] for detais on this approach. 8 In the ossess formuation the bias term is cacuated as g 0 = f 0 Ψp Per-unit p.u. congestion savings are cacuated as C MIP C base where C base C 0 C MIP is the system cost from the DC MIP or from the AC OPF based on this MIP, C base is the DC or AC system costs with no switching and C 0 is the DC or AC system cost with no enforced transmission constraints. C base C 0 represents the maximum savings possibe for any sampe.

6 6 TABLE I MEDIAN NUMBER OF LINE OPENINGS AND AVERAGE PER-UNIT CONGESTION COSTS SAVINGS WITH DIFFERENT LOSS MODELING ASSUMPTIONS OVER 100 SAMPLES Num. Lines Congestion Cost Mode Opened Savings with TC DC w/losses % DC Lossess % AC Based on DC w/losses N/A 20.66% AC Based on DC Lossess N/A 20.65% which we mae three ey observations. First, based on the AC SCOPF soutions, accounting for osses does not improve congestion cost savings. The AC OPF based on the ossadjusted DC MIP soution produces more savings in 53 of the sampes, compared to the AC OPF based on the ossess DC MIP, however, the magnitude in savings tends to be greater for the atter mode so that both AC soutions ead to about the same 20.65% p.u. congestion cost savings. Second, soving the AC SCOPF using the optima topoogies from the two DC formuations confirms that both DC formuations refect egitimate congestion cost savings. The ossess MIP formuation overstates the benefits from switching by an average of 1.44% in 76 sampes whie the oss-adjusted MIP understates them by 4.03% in 93 sampes. Both formuation, however, provide a good indication of potentia savings from topoogy contro. Finay, we observe that the oss-adjusted DC MIP tends to open 3 fewer ines than its ossess counterpart. Opening fewer ines is ceary preferabe; operating circuit breaers, athough not incuded expicity modeed, has a cost and, achieving the same savings with fewer openings is more efficient and reduces the number of discrete changes to the state of the system. Figure 1 beow shows the number of ine openings across a sampes. As shown in the figure, the oss-adjusted VII. CONCLUSION In this paper, we deveop a oss-adjusted MIP-based TC formuation and show that both osses and fows can be updated ineary with changes in topoogy. We prove that our formuation is reference bus independent and extend the notion of FCTs to account for osses by deriving oss-adjusted shift factors and PTDFs. We derive ocationa margina prices LMPs for the LP equivaent to the oss-adjusted TC MIP formuation and show how the margina vaue of transmission switching can be expressed in terms of a congestion and oss component. Through simuation, we anayze the impact of osses on the DC formuation and compare the optima topoogies from the ossess and oss-adjusted formuations by soving an AC SCOPF. We find that both DC formuations ead to amost identica savings when soving the AC SCOPF and whie the oss-adjusted formuation opens fewer ines both can be used reiaby to assess the benefits of topoogy contro. Future wor wi focus on adjusting oss factors not ony for topoogy changes but aso for re-dispatch. Athough this woud mae the probem non-inear, it may ead to a better soution in terms of the number of ine openings, as indicated by our resuts, or congestion cost savings. One approach for addressing the non-inearity is to iterate on the MIP soution, updating oss factors after each iteration, athough the practicaity of this approach aong with other aternatives needs to be studied. APPENDIX Proof that formuation L1 is reference bus independent To demonstrate that L1 is reference bus independent we modify the choice of reference bus by introducing a normaized vector w that assigns a weighting to each node in proportion to its contribution to the new distributed reference bus a singe reference bus n can be represented by setting w n = 1. As shown in [19], a weighting w modifies the shift factor matrix according to Substituting 45 into 21 gives Ψ w = Ψ Ψw1 45 Loss w = x 0 g 0 + Ψ Ψw1 p d Loss 1 2 f 0 = x 0 g 0 + Ψp d Loss 1 2 f 0 Fig. 1. Number of Lines Opened in DC MIP Formuations MIP opens between 1 and 9 fewer ines in 79 sampes, opens the same number of ines in 18 sampes, and opens ony 1 additiona ine in 3 sampes. x 0 Ψw1 p d Loss = Loss since 1 p d Loss = 0. Simiary, substituting 45 into 22 gives f w = g 0 + Ψ Ψw1 p d Loss = g 0 + Ψp d Loss Ψw1 p d Loss = f We have thus shown that the constraint set is reference bus independent. Further, since the objective function is reference bus independent, by definition, the shadow prices wi be reference bus independent as we.

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