An Exact Algorithm for Power Grid Interdiction Problem with Line Switching

Transcription

1 1 An Exact Agorithm for Power Grid Interdiction Probem with Line Switching Long Zhao, Student Member, IEEE, and Bo Zeng, Member, IEEE, Abstract Power grid vunerabiity anaysis is often performed through soving a bi-eve optimization probem, which, if soved to optimaity, yieds the most destructive interdiction pan with the worst oss. As one of the most effective operations to mitigate deiberate outages or attacks, transmission ine switching recenty has been incuded and modeed by a binary variabe in the ower eve decision mode. Because this bi-eve probem is a chaenging nonconvex discrete optimization probem, no exact agorithm has been deveoped, and ony a few recent heuristic procedures are avaiabe. In this paper, we present a nove trieve reformuation of this bi-eve probem, as we as its singeeve equivaent form, and describe a finitey-convergent cutting pane agorithm to derive an exact soution. Numerica resuts are provided and benchmarked against existing ones, which confirms the quaity of soutions and the computationa efficiency of our agorithm. Index Terms power grid interdiction, power fow, transmission ine switching, mixed integer bi-eve programming, cutting pane agorithm Indices and Sets NOMENCLATURE i Generator Demand Transmission ine n Node m Aias of n A Set of attack decisions C Set of switching decisions H C 1 where h {0,..., H} U Subset of {0,..., H} Parameters x M F Pi min Pi max D n K θ Decision variabes Reactance at ine A big number for ine Fow imit at ine Minima generation of generator i Maxima generation of generator i Demand at node n Cardinaity of attack Maximum difference of connected phase anges L. Zhao and B. Zeng are with the Department of Industria and Management Systems Engineering, University of South Forida, Tampa, FL E-mai: ongzhao@mai.usf.edu, bzeng@usf.edu d n θ n f mn z p n i w Satisfied demand at node n Phase ange at node n Power fow on ine from node m to n Line switching, 0 if is switched off Generation eve of generator at node n Attack, 0 if ine is removed by attacker I. INTRODUCTION The power grid interdiction probem often is modeed as an attack-defend mode (or sometime as a defend-attackdefend mode, if some defending decisions can be made before attacks) or a eader-foower game [8, 25, 27, 9]. The attacker seeks to minimize the satisfied oad demand by removing up to K transmission ines; after the attack, the system operators try to mitigate the oss by adusting noda generation, oad shedding, and other dispatching parameters. Identifying such a group of ines whose remova wi ead to the severe oss of demand is critica for system daiy operations and ongterm security. For exampe, imited protective or hardening resources can be aocated to those ines to enhance the reiabiity of the system or reduce the probabiity of successfuness of the attacks. Aso, those critica ines are of interest in the transmission network capacity expansion probem to better baance risk and economic advantages. In fact, when K = 1 and 2, the probem is cosey reated to N 1 and N 2 security criteria adopted in the power industry. Because of its significance to the power industry, emergency panning centers, and homeand security, as we as its compexity from the underining network/fow characteristics, an extensive amount of research effort has been dedicated to the power grid interdicting probem since As a resut, many agorithms have been deveoped to address variants with different considerations and scaes. Sameron et a. [25] formuate the probem of optima interdiction of an eectric power network as a bi-eve programming probem and deveop a heuristic agorithm to sove the probem. Later, they extend the mode to mutipe periods with the considerations of repair times of the power grid after attack and the demand variation over time [27]. They propose and impement a goba Benders decomposition method that can sove arge-scae probems with high-quaity soutions in a reasonabe time. In parae, Motto et a. [22] convert the bi-eve power interdicting probem into an equivaent singe-eve mixed integer programming (MIP) mode through duaizing the ower-eve inear programming (LP) probem, which reduces the formuation from bieve to singe-eve, and inearizing resuting binary-binary or binary-continuous terms, which finay yieds an MIP mode. They aso empoy Karush-Kuhn-Tucker (KKT) optimaity

2 2 conditions of the ower-eve LP probem to achieve a simiar transformation [5, 3]. Using the duaity of LP and inearization techniques, Janarassuk and Linderoth [19] aso convert a stochastic network interdiction probem into an equivaent MIP. Due to the stochastic structure, they are abe to appy an L-shaped decomposition technique with a samping-based approach to sove their probem. To reduce the computationa compexity, Bier et a. [7] deveop a greedy-based agorithm to derive interdiction strategies, and report a set of vunerabe transmission ines that are different from those in [25]. Different from the above cassica mathematica programming based approach, graph theory-based methods, in conunction with KKT conditions, are deveoped in [12, 11, 23] to sove power grid interdicting probem. As a resut, instances of argescae power grids can be soved approximatey within a short time [24]. It is noted the ower-eve probems (or defending probems) are assumed to be LP probems in a the aforementioned studies. This assumption is essentia for them in that duaity theory and KKT conditions of LP pay the centra roe in formuation transformations or agorithm deveopment. Recenty, transmission ine switching has been anayticay studied in order to reduce dispatch cost in power system scheduing. It is observed that up to 25 percent dispatch savings can be achieved [13, 16, 17, 20]. Given the fact that the topoogy of the power grid wi be changed if one or more transmission ines are attacked or disconnected, transmission ine switchings can aso be incorporated into system operator s post-disruption decision for a better mitigation effect. For exampe, in PJM, Specia Protection Schemes have incuded transmission ine switching as one operation during contingencies [18]. Nevertheess, ineswitching decisions are made with pre-defined rues, which are not anaytica and coud be ess effective. Recenty, this idea, i.e., incuding ine switching into the ower eve decision probem, is quantitativey studied by Arroyo and Fernandez [4] and Degadio et a. [10]. Because it is necessary to represent ine-switching decisions by binary variabes, the ower-eve probem becomes non-convex, and strong duaity theory and KKT conditions are not vaid anymore. As a resut, the previousy-deveoped techniques to convert two-eve to singe-eve are not appicabe. Given the difficuties to sove this probem exacty, two heuristic procedures, a genetica agorithm and a muti-start Benders decomposition method, are deveoped to identify high-quaity soutions [4, 10]. A numerica study shows that the atter coud identify optima soutions for some instances [10]. Nevertheess, the power grid interdiction probem with ine switching remains an open probem for researchers in the power systems as we as operations research communities. On the other hand, some recent research on robust optimization coud be used to sove this open probem. To provide a soution method to this chaenging probem that is of critica interest to the power industry, governmenta organizations, and security agencies, we deveoped an exact agorithm based on strategies used to sove 2-stage robust optimization probems [29, 30]. Specificay, unike existing work that either tries to derive a singe-eve equivaent formuation or directy deas with its two-eve structure, we reformuated the origina probem into a tri-eve structure, which separates binary ine switching variabes from other continuous decision variabes. Then, a cutting pane method within a master-subprobem framework can be used to dynamicay convert the tri-eve formuation into a (growing) singe-eve formuation, which can be readiy soved by off-the-shef MIP sovers. Because the whoe procedure invoves (provides) an upper bound and a ower bound, the quaity of the best feasibe soution found so far can be estimated, and a user-defined toerance can be suppied to achieve a computationa tradeoff, which provides a fexibe mechanism for system operators in practice. Our maor contributions are isted as foows: (i) We provide an equivaent tri-eve formuation of the bieve power grid interdicting probem by separating transmission ine switching from the defending decision set. This tri-eve formuation provides a framework to eiminate the difficuties brought by binary ine switching decisions. (ii) We deveop a finitey-convergent cutting pane agorithm to exacty sove the tri-eve formuation. The whoe procedure does not depend on externay-suppied parameters and is easy to impement. (iii) A set of preiminary computationa resuts is presented. Through benchmarked against existing ones in [10], our resuts show that the deveoped agorithm can derive optima soutions with significanty-reduced computation time. So, it greaty improves our soution capabiity on this chaenging probem. The paper is organized as foows. In Section II, we present the bieve formuation of power grid interdicting probem with transmission ine switching. In Section III we give the equivaent tri-eve formuation and the cutting pane agorithm with a master-subprobem framework. In Section IV we report the resuts of the computationa study. Section V concudes the paper and discusses future research directions. II. BILEVEL ATTACK-DEFEND MODEL In the foowing, we present a bi-eve min-max formuation for the power grid interdicting probem with transmission ine switching. The higher-eve decision is made by the attacker, which seeks to minimize the served oad by removing transmission ines. Then, after some ines are disrupted by the attacker, system operator soves the ower decision probem to compute the optima operations, incuding ine switching and other dispatching operations, to maximize the oad that can be served. Note that an essentiay simiar mode is proposed in [10], where the bi-eve formuation takes max-min format, i.e., both attacker and system operator consider the unmet oad demand. In remainder of this paper, a vector of variabes is in the bodface of the corresponding variabe, and ˆ denotes a fixed decision variabe. min d n w A (1) st. θ m θ n x f mn + (1 z w )M 0, (2) θ m θ n x f mn (1 z w )M 0, (3)

3 3 F z w f mn z w F, (4) p n i Pi max, i (5) d n D n, (6) f n + p n i = f n + d n, n (7) p n i 0, d n 0, f mn, θ n free, z {0, 1} (8) where A = {w {0, 1}, (1 w ) K}. As refected in obective function (1), the attacker can remove no more than K transmission ines trying to minimize the served demand. After the attack, the system operator tries to maximize the served demand (or equivaenty minimize the oss) by adusting network topoogy through ine switching, phase anges and generation eves. Constraints (2-3), simiar to those proposed in [1, 26], are direct current (DC) power fow approximations that foow Kirchhoff s Laws with additiona attack and switching decisions. The parameter M is a sufficienty arge number to be specified ater. If the ine is removed by the attacker, i.e., w = 0, or disconnected by the operator, i.e., z = 0, the two inequaities wi be satisfied regardess of the difference between two phase anges θ m and θ n at bus towers connected by ine. Otherwise, i.e. w = z = 1, the two constraints wi form the traditiona power fow equation θ m θ n x f mn = 0 for ine. In [10], one equation z w (θ m θ n x f mn ) = 0 is formuated to capture the power aws as we as attack/switching decisions. Ceary, constraints (2-3) can be treated as a resut of inearizing the aforementioned constraint. Constraint (4) forces the power fow on a transmission ine to be zero when the ine is attacked or disconnected; otherwise, the fow wi be restricted within [ F, F ] [17, 10]. Based on the oint restriction of constraints (2-4), the parameter M can be specified [1, 26], and the maxima difference of two phase anges at buses m, n connected by a ine, θ = θm max θn min, can be impicity incorporated. To be specific, assume θ is expicity given and a ine is avaiabe (that is, θ m θ n x f mn = 0), then (i) if F θ/x, θ n θ m θ wi aways be reductant; (ii) if F > θ/x, we repace F by a new vaue of θ/x and then the difference of anguar separation, i.e., θ m θ n, coud be automaticay restricted. Meanwhie, we can have a vaid vaue θ + x F for parameter M. Therefore, unike the mode in [10], we do not expicity incude phase ange imits. Constraint (5) describes the aggregated generation imit at buses with generators. Simiary, constraint (6) guarantees that the satisfied demand does not exceed the nomina demand vaue at oad buses. The nonnegativity constraints on generation and demand variabes ensure that generation/demand wi aways be generation/demand. Finay, constraint (7) is the traditiona node baance equation such that the infow and outfow of any node are equa [10, 26, 17, 1, 10]. III. AN EXACT SOLUTION APPROACH THROUGH A TRI-LEVEL REFORMULATION In this section, we first present a tri-eve formuation of the power grid interdiction probem and its singe-eve equivaent form. Then, we describe an exact agorithm using cutting pane strategy that dynamicay buids and soves the singe-eve equivaent form. A. A Tri-eve Formuation and Its Singe-Leve Equivaent Form Note that the obective function in the origina bi-eve formuation in (1-7) can be equivaenty expressed as min d n = min max d n w A w A z C max z,p,f,θ,d max p,f,θ,d where C is the binary set incuding a possibe ine switching decisions. Hence, for any given attack ŵ A and switching decision ẑ, the remaining probem, which is actuay a dispatching probem, is a pure LP probem. Furthermore, this LP probem is aways feasibe in that the soution with {p, f, θ, d} = 0 is feasibe in a cases. Therefore, the strong duaity hods, and the maximization dispatching probem can be equivaenty repaced by its minimization dua probem. Next, we present the corresponding dua probem in (9-14), where λ 1 and λ 2, λ 3 and λ 4, λ 5 and λ 6, λ 7 are dua variabes for constraints (2-7), respectivey. Note that n i@n and denote the bus n with generator i or demand, respectivey. Aso, = m and = m denote the transmission ine with bus m as the start and end bus, respectivey. min + + i λ 1 (1 ẑ ŵ )M + λ 3 F ẑ ŵ + λ 5 i P max i + λ 2 (1 ẑ ŵ )M λ 4 F ẑ ŵ λ 6 D n (9) st. λ 5 i + λ 7 n i@n 0, i (10) λ 6 λ 7 1, (11) x λ 1 x λ 2 λ 3 + λ 4 λ 7 m + λ 7 n = 0, (m n) (12) (λ 2 λ 1 ) + (λ 1 λ 2 ) = 0, m (13) =m = m λ 7 free, λ 1,..., λ 6 0 (14) Then, we can reformuate the bieve mode into an equivaent tri-eve mode; see Proposition 1 in the foowing. Proposition 1. The bieve probem defined in (1-8) is equivaent to the foowing tri-eve programming probem: min max min λ 1 w A z C λ 1,...,λ 7 (1 z w )M + λ 2 (1 z w )M + st. (10 14) λ 3 F z w + λ 4 F z w + i λ 5 i P max i + λ 6 D n (15) We can interpret this tri-eve mode as foows: the attacker makes attack decisions first, then the system operator responds with ine switching decisions to determine system configuration, and the attacker again makes a set of recourse decisions

4 4 represented by (λ 1,..., λ 7 ) after ine switching decisions are discosed. It is worth pointing out that for the grid interdiction probem (in either bi-eve or tri-eve formuations), the virtua soution we need is an optima attack pan w. Once it is determined, other decisions are determined through soving an MIP probem. Athough such a transformation from a bi-eve mode into a tri-eve mode is anti-intuitive, it provides a mechanism to isoate the ine switching set from other decisions. In particuar, based on a soution strategy presented to sove tri-eve robust optimization probem [29, 30], a singe-eve equivaent form actuay can be deveoped by enumerating a possibe the system operator s responses, which constitute a finite set, given the binary property of switching decision. The key step is that, for any possibe operator s decision ẑ (h) we create a new set of recourse decision variabes (λ 1(h),..., λ 7(h) ). Next, we present this singe-eve form. Proposition 2. The tri-eve mode defined above is equivaent to the singe-eve form defined in (16-22). min η (16) st. η + + i λ 5(h) i λ 6(h) λ 1(h) λ 3(h) (1 ẑ (h) w )M + F ẑ (h) w + λ 5(h) i Pi max + λ 4(h) λ 2(h) F ẑ (h) w (1 ẑ (h) w )M λ 6(h) D n, ẑ (h) C (17) + λ 7(h) n i@n 0, i, h (18) λ 7(h) 1,, h (19) x λ 1(h) x λ 2(h) λ 3(h) + λ 4(h) λ 7(h) m + λ 7(h) n = 0, (m n), h (20) (λ 2(h) =m λ 1(h) ) + (λ 1(h) = m λ 2(h) ) = 0, m, h (21) w A, η, λ 7(h) free, λ 1(h),..., λ 6(h) 0 (22) where {ẑ (h) } H h=0 = C. In the above singe-eve equivaent form, decision variabes are w and (λ 1(h),..., λ 7(h) ) for a h. We note that constraint (17) invoves terms that are products of one binary variabe, w, and one continuous variabe λ k(h) with k = 1,..., 4. So, it woud be natura to inearize those terms using standard inearization techniques so that professiona MIP sovers can be caed to sove this probem. Assume that M is a sufficienty arge number that provides an upper bound to optima vaues of λ k(h) in (9-14). It can be observed that a unit change of the right-hand side of any constraint in (2-4) wi incur some unserved oad or bring additiona served oad, both are bounded by Dn, i.e., the tota noda demand. Hence, M = Dn is a vaid upper bound for those dua variabes (recourse decision variabes). With M, we can have the foowing equivaent inear constraints. Proposition 3. Constraints defined in (17) can be inearized as (23-36). η + + α (h) α (h) (λ 1(h) ẑ (h) F ẑ (h) + ˆα (h) )M + µ (h) F ẑ (h) + i (λ 2(h) ẑ (h) λ 5(h) i Pi max ˆβ (h) )M λ 6(h) D n, ẑ (h) C (23) λ 1(h), h, (24) λ 1(h) (1 w )M, h, (25) α (h) w M, h, (26) β (h) β (h) λ 2(h), h, (27) λ 2(h) (1 w )M, h, (28) β (h) w M, h, (29) λ 3(h), h, (30) λ 3(h) (1 w )M, h, (31) w M, h, (32) µ (h) µ (h) λ 4(h), h, (33) λ 4(h) (1 w )M, h, (34) µ (h) w M, h, (35), µ (h), α (h), β (h) 0 (36) As a resut, the tri-eve mode, as we as the bi-eve one, can be reformuated as a inear mode. Coroary 1. The equivaent singe-eve inear MIP formuation of the tri/bi-eve power grid interdiction probem is the one obtained by repacing (17) with (23-36) in the formuation of (16-22). Ideay, the tri/bi-eve power grid interdiction probem coud be easiy soved using this singe-eve equivaent inear form. We note that, however, this singe-eve equivaent form is of theoretica interest ony. As a set of (λ 1(h),..., λ 7(h) ) and their corresponding constraints must be introduced for any possibe ẑ h in set C, which is actuay exponentia with respect to the number of ines in the grid; it is not feasibe to expicity ist this singe-eve form to sove the tri-eve probem for any rea instances. Instead of using the compete enumeration to obtain the equivaent form, we next describe a soution agorithm that makes use of a partia enumeration strategy and cutting pane method. Our computationa study shows that this agorithm is very effective in soving power grid interdiction probem. B. Agorithm Description Based on observations made in [29, 30] on soving the trieve 2-stage robust optimization probems, it is anticipated that ony a very sma part of C is critica in determining an optima soution. So, one idea is to start with a singeeve formuation with a sma subset of C (i.e., their variabes and constraints) and then graduay expand the formuation by incuding more significant components (i.e., their variabes

5 5 and constraints) of C. Note from (16-22) that a singe-eve formuation with a subset of C, which is named the partia singe-eve formuation, yieds a ower bound to the actua optima vaue. Aso, any feasibe soution to the bi-eve formuation, which can be obtained for any attack pan, yieds an upper bound, given that the utimate obective function is minimization. Therefore, the expansion process can be terminated with an optima soution whenever upper and ower bounds match. This idea is impemented within a master-subprobem framework using a cutting pane strategy as foows. The same strategy is used to sove 2-stage robust optimization probems, which have been proven very efficient comparing to other soution methods [29, 30]. To simpify our exposition, we use the compact forms of a probems in this subsection. Specificay, et y denote the set of decision variabes, incuding ine switching and other dispatching variabes, made by system operator, then the ower eve decision mode given any attack ŵ A wi be max cy (37) st. By g Aŵ (38) variabe restrictions on y, (39) where c, g, B, A are appropriate vectors or matrices defined in (1-7). Simiary, for some known {z (h) } h U C, the partia singe-eve formuation is min η (40) st. Qη + Eλ (h) + Gw hẑ (h) + a, h U (41) w A (42) variabe restrictions on w, η, λ (h), (43) where h, a, Q, E, G are appropriate vectors or matrices defined in (18-21, 23-36). Next we describe the agorithm in steps. Since this soution approach is based on a tri-eve reformuation of the bi-eve probem, we name it Tri-eve reformuation (TLR) method for convenience. Steps of TLR Agorithm 1) Set LB =, UB = +, h = 0, U = and an optimaity toerance ɛ. 2) Sove the partia singe-eve formuation defined in (40-43) (as the master probem). Derive an optima soution (w h, η h, λ h ) and update LB = η h. 3) Sove the ower eve probem defined in (37-39) (as the subprobem) with ŵ = w h and update UB = min{ub, cy h }. 4) If UB LB ɛ, return w h as an optima attack pan and terminate. Otherwise, update U = U {h} and h = h + 1, create new recourse decision variabes λ (h) and add the corresponding constraints defined in (41) as cutting panes to the partia singe-eve probem. Go to step 2. Based on the resuts presented in [29] and the fact that C is a finite binary set, it foows directy that TLR is finitey convergent to an optima soution. To the best of our knowedge, it is the first exact agorithm deveoped to sove the power grid interdiction probem with ine switching. In fact, its TABLE I CONFIGURATIONS OF COMPUTING FACILITIES Patform TLR MSBD CPU 1 processor at 3GHz 4 processors at 2.6GHz RAM 3.25G 32G CPLEX Goba Opt. Toerance ɛ NA ɛ for each probem computationa performance is aso very promising, compared to the recent study on a muti-start Benders decomposition method [10]. IV. COMPUTATIONAL STUDY This section presents a case study based on the IEEE 24- bus reiabiity test system [15]. The system has 24 buses, 11 of which are equipped with generators and 16 which are oad buses. There are, in tota, 38 transmission ines, and the cardinaity of attacks is from 1 to 12. To benchmark against existing research in [10], we adopted their parameters, at our best efforts, to define the power grid. Specificay, parameters of power fow capacity F and noda demand D were assumed to be those used in [10], which were converted into per unit (pu) quantities with a 100 MVA base. Aso, the two ines connected with the same towers were treated as independent ines as in [10]. The reactance x of each transmission ine and the maxima generation eve p max at each node are not isted in [10], so we adopted those parameters directy from [15]. A parameters are isted in the appendix. Given that the restrictions of anguar differences are not presented in [10], we aimed to seect those parameters cose to practice. Observe that in previous research, different vaues were adopted for θ, such as 1.2 radian [17], 1 radian [1], etc. However, in practice it is extremey rare to ever see such anguar separation exceed 30 degrees [21]. Aso, the sufficient accuracy of DC simpification (2-3) of aternating current (AC) power fow reies on a sma-vaued θ as sin(θ) = θ pays a centra roe in the approximation process [2, 14]. Hence, θ is taken impicity to be 0.5 in this paper. The soution was impemented in C++ on a PC desktop, and the commercia sover IBM ILOG CPLEX was used as the MIP sover for a probems. In addition, we set bus 1 to be the reference bus, that is, θ 1 = 0. The differences between computing faciities used in this study and in [10] are isted in Tabe I. In the remainder of the section, to avoid confusion, we use MSBD to denote the muti-start Benders decomposition method and the associated research presented in [10]. Our computationa study incuded 12 instances, ranging from K = 1 to K = 12 for the attacker, basicay repicating those made in [10]. To provide a comparabe evauation, we adopted a strategy by [10] in our TRL impementation, which pursued a high-quaity initia attack by soving the bieve programming probem without transmission switching. As reported in [10], such a strategy can often acceerate the computationa speed. The comparison of the computationa performance with respect to MSBD is shown in Tabe II together with the number of iterations in TLR. From the soution quaity point of view, it is observed that TLR can

6 6 Bus 18 Bus 17 Bus 16 Bus 15 Bus 24 Bus 21 Bus 19 Bus 20 Bus 14 Synch. Cond. Bus 22 Bus 11 Bus 12 Bus 23 Bus 13 TABLE II ALGORITHM PERFORMANCE COMPARISON K Time (s) of MSBD [10] Time (s) of TLR Iterations of TLR * * * * * * * * * indicating quaity of the soution is unknown. TABLE III LOAD SHEDDING (MW) CAUSED BY ATTACKS Bus 3 Bus 9 Bus 10 Bus 4 Bus 5 Bus 1 Bus 2 Bus 7 Bus 6 Bus 8 K MSBD [10] TLR Fig. 1. Lines attacked Lines disconnected Best attack pan for K=6 and the optima switching derive optima soutions in a instances within a reasonabe time, whie MSBD cannot provide quaity guaranteed soutions in most instances. From the computationa speed point of view, TLR is comparabe with MSBD for easy instances, whie it performs an order of magnitude faster than MSBD for most chaenging ones. Actuay, for the most difficut instance where K = 6, as shown in Figure 1, the computationa time coud reduced to a few hundred ( 600) seconds if the acceerating strategy [10] is not empoyed. Note that initia attack pans obtained by the acceerating strategy are reported to be optima for instances K = 7, 11, 12 in [10]. In our study, we observe that initia attack pans are optima for instances K = 5, A possibe expanation is that some parameters, which are not reported in [10], are different from those used in the numerica study by Degadio et a. [10]. It is aso confirmed by resuts presented in Tabe III. Tabe III presents the maxima oad shedding in a instances. We mention that 168.5MW demand cannot be met even without any attacks, i.e., K=0, based on the parameters in our study. Optima attack pans are shown in Tabe IV, from which we can identify critica transmission ines for different Ks in N K vunerabiity consideration. For exampe, for K 2 ine is critica; for K 5, the important ines are 7-8, 12-23, 20-23A, and 20-23B, which actuay is the best attack pan when K is four; simiary, for cases with K 6, 12-23, 15-21A, 15-21B, 16-17, 20-23A, and 20-23B are, in genera, critica ines. V. CONCLUSION AND DISCUSSION In this paper, we proposed an equivaent tri-eve formuation to the power grid interdicting probem. Then, we derived its singe-eve equivaent form and deveoped an exact soution approach using a cutting pane strategy within a mastersubprobem framework. The formuations are nontrivia, and the agorithm is nove. In particuar, a set of preiminary computationa resuts show that it performs significanty better than existing work and is promising for deaing with rea-size power grids. TABLE IV BEST ATTACK STRATEGY k Best attack pan , , 20-23A, 20-23B 4 7-8, 12-23, 20-23A, 20-23B 5 7-8, 12-23, 15-24, 20-23A, 20-23B , 15-21A, 15-21B, 16-17, 20-23A, 20-23B 7 7-8, 12-23, 15-21A, 15-21B, 16-17, 20-23A, 20-23B 8 7-8, 12-23, 13-23, 15-21A, 15-21B, 16-17, 20-23A, 20-23B 9 1-3, 1-5, 2-4, 2-6, 3-24, 7-8, 12-23, 20-23A, 20-23B , 2-4, 2-6, 7-8, 12-23, 15-21A, 15-21B, 16-17, 20-23A, 20-23B , 1-5, 2-4, 2-6, 7-8, 12-23, 15-21A, 15-21B, 16-17, 20-23A, 20-23B , 1-5, 2-4, 2-6, 7-8, 12-23, 13-23, 15-21A, 15-21B, 16-17, 20-23A, 20-23B

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