A Bound Strengthening Method for Optimal Transmission Switching in Power Systems

Transcription

1 1 A Bound Strengthening Method for Optima Transmission Switching in Power Systems Saar Fattahi, Javad Lavaei,+, and Aper Atamtürk Industria Engineering and Operations Research, University of Caifornia, Berkeey + Tsinghua-Berkeey Shenzhen Institute, University of Caifornia, Berkeey. Abstract This paper studies the optima transmission switching (OTS) probem for power systems, where certain ines are fixed (uncontroabe) and the remaining ones are controabe via on/off switches. The goa is to identify a topoogy of the power grid that minimizes the cost of the system operation whie satisfying the physica and operationa constraints. Most of the existing methods for the probem are based on first converting the OTS into a mixed-integer inear program (MILP) or mixed-integer quadratic program (MIQP), and then iterativey soving a series of its convex reaxations. The performance of these methods depends heaviy on the strength of the MILP or MIQP formuations. In this paper, it is shown that finding the strongest variabe upper and ower bounds to be used in an MILP or MIQP formuation of the OTS based on the big-m or McCormick inequaities is NP-hard. Furthermore, it is proven that uness P = NP, there is no constant-factor approximation agorithm for constructing these variabe bounds. Despite the inherent difficuty of obtaining the strongest bounds in genera, a simpe bound strengthening method is presented to strengthen the convex reaxation of the probem when there exists a connected spanning subnetwork of the system with fixed ines. The proposed method can be treated as a preprocessing step that is independent of the sover to be ater used for numerica cacuations and can be carried out offine before initiating the sover. A remarkabe speedup in the runtime of the mixed-integer sovers is obtained using the proposed bound strengthening method for mediumand arge-scae rea word systems. I. INTRODUCTION In power systems, transmission ines have traditionay been considered uncontroabe infrastructure devices, except in the case of an outage or maintenance. However, due to the pressing needs to boost the sustainabiity, reiabiity and efficiency, power system directors ca on everaging the fexibiity in the topoogy of the grid and co-optimizing the production and topoogy to improve the dispatch. In the ast few years, Federa Energy Reguatory Commission (FERC) has hed an annua conference on Increasing Market and Panning Efficiency through Improved Software [1] to encourage research on the deveopment of efficient software for enhancing the efficiency of the power systems via optimizing the fexibe assets (e.g., transmission switches) in the system. Furthermore, The Energy Emai: {fattahi, avaei, atamturk}@berkeey.edu This work was supported by the ONR YIP Award, DARPA YFA Award, AFOSR YIP Award, NSF CAREER Award, and an ARL Grant. A. Atamtürk was supported, in part, by grant FA from the Office of the Assistant Secretary of Defense for Research and Engineering. Parts of this paper have appeared in the conference paper [22]. Poicy Act of 2005 expicity addresses the difficuties of siting major new transmission faciities and cas for the utiization of better transmission technoogies [2]. Unike in the cassica network fows, removing a ine from a power network may improve the efficiency of the network due to physica aws. This phenomenon has been observed and harnessed to improve the power system performance by many authors. The notion of optimay switching the ines of a transmission network was introduced by O Nei et a. [3]. Later on, it has been shown in a series of papers that the incorporation of controabe transmission switches in a grid coud reieve network congestions [4], serve as a corrective action for votage vioation [5] [7], reduce system oss [8], [9] and operationa costs [10], improve the reiabiity of the system [11], [12] and enhance the economic efficiency of power markets [13]. We refer the reader to Hedman et a. [14] for a survey on the benefits of transmission switching in power systems. However, the identification of an optima topoogy, namey optima transmission switching (OTS) probem, is a non-convex combinatoria optimization probem that is proved to be NP-hard [15]. Therefore, brute-force search agorithms for finding an optima topoogy are often inefficient. Most of the existing methods are based on heuristics and iterative reaxations of the probem. These methods incude, but are not restricted to, Benders decomposition [10], [12], branchand-bound and cutting-pane methods [16], [17], genetic agorithms [7], and ine ranking [18], [19]. Recenty, another ine of work has been devoted to strong convexification techniques in soving mixed-integer probems for power systems [20] [22]. In this work, the power fow equations are modeed using the we-known DC approximation, which is the backbone of the operation of power systems. Despite its shortcomings for the OTS in some cases [23], the DC approximation is often considered very usefu for increasing the reiabiity, performance, and market efficiency of power systems [14]. The OTS consists of disjunctive constraints that are biinear and nonconvex in the origina formuation. However, a of these constraints can be written in a inear form using the so-caed big-m or McCormick inequaities [24], [25]. This formuation of OTS is referred to as the inearized OTS in the seque. A natura question arising in constructing the OTS formuation is: how can one find optima vaues for the parameters of the big-m or McCormick inequaities? An optima choice for these parameters is important for two

2 2 reasons: 1) they woud resut in stronger convex reaxations of the probem, and hence, fewer iterations in branch-andbound or cutting-pane methods, and 2) a conservative choice of these parameters woud cause numerica and convergence issues [26]. Hedman et a. [11] point out that finding the optima vaues for the parameters of the inearized OTS may be cumbersome, and, therefore, they impose restrictive constraints on the absoute anges of votages at different buses at the expense of shrinking the feasibe region. In this work, it is proven that finding the optima vaues for the parameters of the MILP or MIQP formuations of the OTS using either big-m or McCormick inequaities is NP-hard. Moreover, it is shown that there does not exist any poynomiatime agorithm to approximate these parameters within any constant factor, uness P = NP. This new resut adds a new dimension to the difficuty of the OTS; not ony is soving the OTS as a mixed-integer noninear program difficut, but finding a good inearized reformuation of this probem is NP-hard as we. In practice, system operators consider fexibiity ony for a set of key transmission ines. In order to maintain the reiabiity and security of the system, often a set of transmission ines are considered as fixed and the fexibiity in the network topoogy is imited to the remaining ines. An impicit requirement is that the network shoud aways remain connected in order to prevent isanding. One way to circumvent the isanding issue in the optima transmission switching probem is to incude additiona security constraints in order to keep the underying network connected at every feasibe soution [27], [28]. However, this new set of constraints woud ead to the overcompication of an aready difficut probem. Therefore, in practice, many energy corporations, such as PJM and Exeon, consider ony a seected subset of transmission ines as fexibe assets in their network [29], [30]. In this paper, it is proven that the OTS with a connected spanning fixed subnetwork is sti NP-hard but one can find non-conservative vaues for the parameters of the big-m or McCormick inequaities in the inearized OTS. In particuar, a simpe bound strengthening method is presented to strengthen the inearized formuation of the OTS. The proposed method is based on soving a simpe shortest path probem on a weighted graph. This method can, therefore, be integrated as a preprocessing step into any numerica sover for the OTS. Despite its simpicity, it is shown through extensive case studies on the IEEE 118-bus system and different Poish networks that the incorporation of the proposed bound strengthening method eads to substantia speedup in the runtime of the sover. II. PROBLEM FORMULATION Consider a power network with n b buses, n g generators, and n ines. This network can be represented by a graph, denoted by G(B, L), where B is the set of buses indexed from 1 to n b and L is the set of ines indexed as (i, j) to represent a connection between buses i and j. In order to streamine the presentation, we assume an arbitrary direction for each ine of the power system. Define N + (i) as the set of endpoints of the outgoing ines at bus i. In particuar, N + (i) is defined as {j B (i, j) L}. Simiary, define N (i) as the set of endpoints of the incoming ines at bus i. In particuar, we have N (i) {j B (j, i) L}. Denote G = {1, 2,..., n g } as the set of generators in the system. Furthermore, et N g (i) be the indices of generators that are connected to bus i. Note that N g (i) may be empty for a bus i. The variabe p i corresponds to the active-power production of generator i G and the variabe θ i is the votage ange at bus i B. For every (i, j) L, the variabe f ij denotes the active fow from bus i to bus j. Consider the set of ines S L that are equipped with on/off switches and define the decision variabe x ij for every (i, j) S as the status of the ine (i, j). Let n s denote the cardinaity of this set. We refer to the ines beonging to S as fexibe ines and the remaining ines as fixed ines. Notice that the decision variabes p i, θ i, and f ij are continuous, whereas x ij is binary. For simpicity of notation, define the variabe vectors p [p 1, p 2,..., p ng ], (1) Θ [θ 1, θ 2,..., θ nb ], (2) f [f i1j 1, f i2j 2,..., f in j n ], (3) x [x i1j 1, x i2j 2,..., x ins j ns ], (4) where the ines in L are abeed as (i 1, j 1 ),..., (i n, j n ) such that the first n s ines denote the members of S. The objective function of the OTS is defined as i G g i(p i ), where g i (p i ) takes the quadratic form g i (p i ) = a i p 2 i + b i p i + c i. (5) with a i 0 or the inear form g i (p i ) = b i p i + c i. (6) for some numbers a i, b i, c i 0. In this paper, we consider both quadratic and inear objective functions, which may correspond to system oss and operationa cost of generators. Every in-operation power system must satisfy operationa constraints arising from physica and security imitations. The physica imitations incude the unit and ine capacities. Furthermore, the power system must satisfy the power baance equations. On the security side, there may be a cardinaity constraint on the maximum number of fexibe ines that can be switched off in order to avoid endangering the reiabe operation of the system. Let the vector d = [d 1, d 2,..., d nb ] coect the set of demands at a buses. Moreover, define p min i and p max i as the ower and upper bounds on the production eve of generator i, and fij max ine (i, j) L is associated with susceptance B ij. as the capacity of ine (i, j) L. Each Using the above notations, the OTS is formuated as the

3 3 foowing mixed-integer noninear probem: minimize g i (p i ) f,x,θ,p i G (7a) s.t. x ij {0, 1}, (i, j) S (7b) p min k p k p max k, k G (7c) fij max x ij f ij fij max x ij, (i, j) S (7d) fij max f ij fij max, (i, j) L\S (7e) B ij (θ i θ j )x ij = f ij, (i, j) S (7f) B ij (θ i θ j ) = f ij, (i, j) L\S (7g) p k d i = f ij f ji, i B (7h) k N g(i) j N + (i) j N (i) x ij r, (7i) (i,j) S where - (7b) states that the status of each fexibe ine must be binary; - (7c) imposes ower and upper bounds on the production eve of generating units; - (7d) and (7e) state that the fow over a fexibe or fixed ine must be within the ine capacities when its switch is on, and it shoud be zero otherwise; - (7f) and (7g) reate the fow over each ine to the votage anges of the two endpoints of the ine if it is in service, and it sets the fow to zero otherwise; - (7h) requires that the power baance equation be satisfied at every bus; - (7i) states that at east r fexibe ines must be switched on. Define F as the feasibe region of (7), i.e., the set of {f, x, Θ, p} satisfying (7b)- (7i). Due to space restrictions, we consider ony one time sot of the system operation. However, the techniques deveoped in this paper can aso be used for the OTS over mutipe time sots with couping constraints, such as ramping imits on the productions of the generators. As another generaization, one can consider a combined unit commitment and optima transmission switching probem, as formuated in Hedman et a. [10]. Henceforth, the term optima soution refers to a gobay optima soution rather than a ocay optima soution. III. LINEARIZATION OF OTS The aforementioned formuation of the OTS beongs to the cass of mixed-integer noninear programs. The noninearity of this optimization probem is, in part, caused by the mutipication of the binary variabe x ij and the continuous variabes θ i and θ j in (7f). However, since this noninear constraint has a disjunctive nature, one can use the big-m or McCormick reformuation technique to formuate it in a inear way. First, we consider the big-m method, and then show that the same resut hods for the McCormick reformuation scheme in the OTS. One can re-write (7f) for each fexibe ine (i, j) in the form B ij (θ i θ j ) M ij (1 x ij ) f ij B ij (θ i θ j )+M ij (1 x ij ) (8) for a arge enough scaar M ij, which resuts in the inearized OTS formuation. The above inequaity impies that if x ij equas 1, then the ine is in service and needs to satisfy the physica constraint f ij = B ij (θ i θ j ). On the other hand, if x ij equas 0, then (8) (and hence (7f)) is redundant as it is dominated by (7d). The term arge enough for M ij is ambiguous, and indeed the design of an effective M ij is a chaenging task that wi be studied beow. Definition 1. For every (i, j) S, it is said that M ij is feasibe for the OTS if it preserves the equivaence between (8) and (7f) in the OTS. The smaest feasibe M ij is denoted by ij. Remark 1. Note that the vaue of ij is independent of the vaues of M r, for (r, ) S\(i, j), in the inearized OTS formuation, as ong as they are chosen to be feasibe. In other words, given an instance of the OTS, the vaue of ij is the same if M r satisfies M r r for every (r, ) S\(i, j). The probem under investigation in this section is the foowing: Given an instance of OTS, is there an efficient agorithm to compute ij or a good approximation of that for every (i, j) S? It is desirabe to find the smaest feasibe vaues for every M ij, (i, j) S, in (8) because of two reasons: 1. Commony used methods for soving MILP or MIQP probems, such as cutting-pane and branch-and-bound agorithms, are based on iterative convex reaxations of the constraints. Therefore, whie a sufficienty arge vaue for M ij does not change the feasibe region of the OTS after repacing (7f) with (8), it may have a significant impact on the feasibe region of its convex reaxation. Sma vaues for M ij yied stronger convex reaxations with smaer feasibe sets. 2. Large vaues for M ij may cause numerica issues for convex reaxation sovers. For every (i, j) S, define F ij as the set of a points {f, x, Θ, p} F such that x ij = 0. Lemma 1. The equation ij = B ij max {f,x,θ,p} F ij { θ i θ j } (9) hods for every fexibe ine (i, j) S. Proof. Consider a number M ij such that M ij B ij max Fij { θ i θ j }. Every {f, x, Θ, p} F satisfies (8) with the chosen M ij and hence, M ij is feasibe. Now, assume that M ij < B ij max Fij { θ i θ j }. It is straightforward to observe that there exists {f, x, Θ, p} F for which (8) is vioated and, hence, M ij is not feasibe. This impies that ij = B ij max Fij { θ i θ j }. Due to Lemma 1, the probem of finding ij for every (i, j) S reduces to finding the max Fij { θ i θ j }. Remark 2. Note that, for a given (i, j) S, the term max Fij { θ i θ j } is finite if and ony if the buses i and j are connected for every feasibe point in F ij. This means that the inearization of the OTS is we-defined if and ony if the

4 Fig. 1: This depicts the topoogy of the network in Exampe 1. The soid and dashed edges denote the ines with ON and OFF switches, respectivey. power network remains connected at every feasibe soution in F ij for a (i, j) S. The next exampe iustrates a scenario where the max Fij { θ i θ j } is not finite. Exampe 1. Consider the network with 6 buses and 8 ines in Figure 1. Assume that the network is decomposed into two disjoint components (known as isands) with the buses {1, 2, 3} and {4, 5, 6} at a feasibe point {f, x, Θ, p} F 16. Define Θ as θ i = θ i for i {1, 2, 3} and θ i = θ i + τ for i {4, 5, 6}, where τ is an arbitrary scaar. It can be verified that {f, x, Θ, p} F 16 for every τ. Furthermore, θ 6 θ 1 = θ 6 θ 1 + τ, which impies that max F16 { θ 6 θ 1 } + as τ +. To avoid unbounded vaues for ij, the existence of a connected spanning subnetwork connecting a the nodes in the network with fixed ines wi be assumed in the next section. In what foows, it wi be shown that, even if max Fij { θ i θ j } is bounded for every (i, j) S, one cannot devise an agorithm that efficienty finds max Fij { θ i θ j } because it amounts to an NP-hard probem. Furthermore, the impossibiity of any constant factor approximation of max Fij { θ i θ j } in the inearized OTS is proven. Theorem 1. Consider an instance of the OTS and seect a fexibe ine (i, j) S. Uness P = NP, it hods that: - (NP-hardness) there is no poynomia-time agorithm for finding max Fij { θ i θ j }; - (Inapproximabiity) there is no poynomia-time constantfactor approximation agorithm for finding max Fij { θ i θ j }. Proof. The proof is provided in the Appendix. Theorem 1 together with Lemma 1 impies that finding ij is both NP-hard and inapproximabe within any constant factor, hence providing a negative answer to the question raised in this section. Remark 3. The decision version of the OTS is known to be NP-compete [17]. One may specuate that the NP-hardness of finding the best M ij for every (i, j) S may foow directy from that resut. However, notice that there are some we-known probems with disjunctive constraints, such as the minimization of tota tardiness on a singe machine, which are known to be NP-hard [31] and yet there are efficient methods to find the optima parameters of their big-m reformuation [32]. Theorem 1 shows that not ony is finding the best M ij for the OTS NP-hard, but one cannot hope for obtaining a strong inearized reformuation of the probem based on the big-m method. 5 Note that one may choose to use McCormick inequaities [25] instead of the big-m method to obtain a inear reformuation of the biinear constraint (7f). In what foows, it wi be shown that the compexity of finding the optima parameters of McCormick inequaities is the same as those in the big- M method for the OTS. The McCormick inequaities can be written in the foowing form for a fexibe ine (i, j): f ij u ij xij=1x ij, f ij ij xij=1x ij, f ij B ij (θ i θ j ) ij xij=0x ij, f ij B ij (θ i θ j ) u ij xij=0x ij, (10a) (10b) (10c) (10d) where u ij xij=1 and ij xij=1 are the respective upper and ower bounds for B ij (θ i θ j ) in the case where the ine (i, j) is in service. Simiary, u ij xij=0 and ij xij=0 are the respective upper and ower bounds for B ij (θ i θ j ) when the switch for the fexibe ine (i, j) is off. It can be verified that the foowing equaities hod: u ij xij=1 = fij max, (11a) ij xij=1 = fij max, (11b) u ij xij=0 = B ij max{θ i θ j }, F ij (11c) ij xij=0 = B ij min F ij {θ i θ j }. (11d) Therefore, Theorem 1 immediatey resuts in the NP-hardness and inapproximabiity of the pair ( ij xij=0, u ij xij=0). Athough finding a good approximation of ij is a difficut task in genera, one can find a non-conservative vaue for M ij in poynomia time for power systems whose fixed ines form a connected spanning subgraph. IV. OTS WITH A FIXED CONNECTED SPANNING SUBGRAPH In this section, we consider a power system with the property that the set of fixed ines contains a connected spanning tree of the power system. The objective is to show that a non-trivia upper bound on M ij can be efficienty derived by soving a shortest path probem. Furthermore, it wi be proven that this upper bound is tight in the sense that there exist instances of the OTS with a fixed connected spanning subgraph for which this upper bound equas ij. Before presenting this resut, it is desirabe to state that the OTS is hard to sove even under the assumption of a fixed connected spanning subgraph. Theorem 2. The OTS with a fixed connected spanning subgraph is NP-hard. Proof. The proof foows from a sight modification of the argument made in the proof of Theorem 3.1 in [17]. Consider a feasibe point {f, x, Θ, p} F. For any ine (i, j) L, we have B ij (θ i θ j ) = B ij (θ r θ ), (12) (r,) P ij

5 5 where P ij is an arbitrary path from node i to node j in the fixed spanning connected subgraph of G. This impies that ij = B ij (θr opt θ opt r ) B ij, (13) B r (r,) P ij (r,) P ij f max where {f opt, Θ opt, x opt, p opt } arg max F { θ i θ j }. Note that (13) hods for every path P ij in the fixed connected spanning subgraph of the network. We wi use this observation in Theorem 3 to derive strong upper bounds for ij. Denote the undirected weighted subgraph induced by the fixed ines in the power system as G I (B I, W I ), where B I = B and W I is the set of a tupes (i, j, w ij ) such that (i, j) L\S and w ij is the weight corresponding to (i, j) defined as f max ij /B ij. Let P I;ij and p I;ij be the set of edges in a shortest simpe path between nodes i and j in G I and its ength, respectivey. Theorem 3. For every fexibe ine (i, j) S, the inequaity ij B ij p I,ij (14) hods. Moreover, there exists an instance of the OTS for which this inequaity is tight. Proof. Based on (13), we have max f r ij B ij = B ij w r = B ij p I,ij. B r (r,) P I,ij (r,) P I,ij (15) Furthermore, a simiar feasibe soution that is derived in the second part of the proof of Theorem 1 (provided in the Appendix) can be used to show the tightness of this upper bound. Theorem 3 proposes a bound strengthening scheme for every fexibe ine in the OTS probem that can be carried out as a simpe preprocessing step before soving the OTS using any branch-and-bound method. The agorithm for the proposed bound strengthening method is described beow: Data: G I (B I, W I ) and B = {B ij (i, j) S} Resut: M ij for every (i, j) S for (i, j) S do find p I;ij using Dijkstra s agorithm; M ij B ij p I;ij ; end Agorithm 1: Bound strengthening method for inearized OTS with fixed connected spanning subgraph The worst-case compexity of performing this preprocessing step is O(n s n 2 b ) since it is equivaent to performing n s rounds of Dijkstra s agorithm on the weighted graph G I (it can aso be reduced to O(n s (n n s + n b og n b )) if the agorithm is impemented using a Fibonacci heap) [33]. This preprocessing step can be processed in an offine fashion before reaizing the demand in the system. The impact of this preprocessing step on the runtime of the sover wi be demonstrated on different cases in Section V. As mentioned in the Introduction, the existence of a fixed connected spanning subgraph in power systems is a practica assumption since power operators shoud guarantee the reiabiity of the system by ensuring the connectivity of the power network. Therefore, due to Theorem 3, one can design reativey sma vaues for M ij s in order to strengthen the convex reaxation of OTS. Consider the cost function for the OTS. In practice, for the production panning probems of power systems, it is often the case that a quadratic objective function is used in order to better imitate the actua cost of production, speciay for therma generators [34]. However, the noninearity introduced by a quadratic cost function makes the OTS hard to sove in genera. Due to this inherent difficuty in soving this MIQP, most of the existing iterature on the OTS assumes a inear cost function [16], [17] for the production of eectricity in a generators. The main chaenge of soving the MIQP is the fact that the optima soution of its continuous reaxation often ies in the interior or on the boundary of its reaxed feasibe region which may be infeasibe for the origina MIQP (as opposed to the extreme point soutions in MILP). More precisey, even obtaining the convex hu of the feasibe region is not enough to guarantee the exactness of such continuous reaxations, since the optima soution of the reaxed probem does not aways correspond to an extreme point in the convex hu if the objective function is quadratic. On the other hand, branch-andbound methods heaviy rey on iterative continuous reaxations of the origina probem. This woud introduce non-fractiona soutions for the binary variabes of the probem in most of the iterations which often eads to a high number of iterations. One way to partiay remedy this probem is to reformuate the probem by introducing auxiiary variabes such that a new inear function is minimized and the od quadratic objective function is moved to the constraints. This guarantees that the continuous reaxation of the reformuated probem wi obtain an optima soution that is an extreme point of the reaxed feasibe region. This is a key reason behind the success of different conic reaxation and strengthening methods in MIQP [35], [36]. In other words, different vaid inequaities and bound strengthening methods wi be more effective since they can partiay describe the convex hu of the reformuated probem with a inear objective function. Assume that the objective function is quadratic in the form of n g i=1 g i(p i ), where g i (p i ) is defined as (5). Upon defining a new set of variabes t i for i G, one can reformuate the objective function as n g i=1 g i(p i, t i ) where g i (p i, t i ) = a i t i + b i p i + c i. (16) subject to the additiona convex constraints p 2 i t i, i G (17) One can easiy verify that the aforementioned modifications ead to an exact reformuation of the origina OTS. However, it wi be shown through different case studies that our proposed bound strengthening method is notaby more effective when appied to the OTS with the inear objective function (16) and the additiona set of constraints (17). This probem is referred to as modified formuation of OTS henceforth.

6 6 V. NUMERICAL RESULTS In this section, case studies on different test cases are conducted to evauate the effectiveness of the proposed preprocessing method in soving the OTS. A of the test cases are chosen from the pubicy avaiabe MATPOWER package [37], [38]. The simuations are run on a aptop computer with an Inte Core i7 quad-core 2.50 GHz CPU and 16GB RAM. The resuts reported in this section are for a seria impementation in MATLAB using the CVX framework and the GUROBI 6.00 sover with the defaut settings. The reative optimaity gap threshod is defined as z UB z LB z UB 100, where z UB and z LB are the objective vaue corresponding to the best found feasibe soution and the best found ower bound, respectivey. If the sover obtains a feasibe soution for the OTS with the reative optimaity gap of at most 0.1% within a time imit (to be defined ater), it is said that an optima soution is found. A. Data Generation First, we study the IEEE 118-bus system. There are 185 ines in this test case. In a of the considered instances, a randomy generated connected spanning subgraph of the network with 120 fixed ines is chosen and the remaining ines are considered fexibe. We compare the proposed preprocessing method with the case where M ij is chosen as (i,j) L f ij max /B ij for every (i, j) S. This conservative vaue does not expoit the underying structure of the network. To generate mutipe instances of the OTS, the oads are mutipied by a oad factor α chosen from the set {α 1, α 2,..., α k }. Furthermore, a uniform ine rating is considered for a ines in the system. We examine both inear and quadratic cost functions and perform the foowing comparisons: For the instances with a inear cost function, the tota runtime is computed for the cases with and without the proposed bound strengthening method (denoted by L-T and L-C, respectivey) for different oad factors and cardinaity ower bounds. For the instances with a quadratic cost function, the runtime is computed for four different formuations: 1) the modified formuation with the proposed bound strengthening method (denoted by Q-ET), 2) the modified formuation without the proposed bound strengthening method (denoted by Q-EC), 3) the origina formuation with the proposed bound strengthening method (denoted by Q-OT), and 4) the origina formuation without the proposed bound strengthening method (denoted by Q-OC). We aso study six different arge-scae Poish networks that are equipped with hundreds of switches. For each test case, a singe oad factor is considered for the OTS with inear and quadratic cost functions and the effect of the proposed bound strengthening method on the runtime is investigated. Simiar to the IEEE 118-bus case, we fix a randomy chosen connected spanning subgraph of the network with fixed ines is considered. B. IEEE 118-bus System In this subsection, the OTS probem is studied for the IEEE 118-bus system with 65 switches. Two types of cost functions are considered for this system: Linear cost function: Figure 2a shows the runtime with respect to the various oad factors. For a of these experiments, the ower bound on the cardinaity of the ON switches is set to 45, i.e. r = 45 in (7i). It can be observed that, for sma vaues of the oad factor, the OTS is reativey easy to sove with a inear cost function and the sover can easiy find the optima soution within a fraction of second with or without the bound strengthening method. On the other hand, as the oad factor increases, the OTS becomes harder to sove and the proposed bound strengthening method has a significant impact on the runtime. In particuar, when the oad factor equas 0.8, the strengthened formuation of the OTS is soved 8.73 faster. In the second experiment, the performance of the sover is evauated as a function of the ower bound on the number of the ON switches. As pointed out in [17], the OTS becomes computationay hard to sove with a reativey arge ower bound. This behavior is observed in Figure 2b. However, note that the negative effect of increased ower bound diminishes when the bound strengthening step is performed. Specificay, the strengthened formuation is soved 2.66 times faster on average for the first two cardinaity ower bounds (10 and 20) and 6.53 times faster on average for the ast two cardinaity ower bounds (40 and 50). Quadratic cost function: When the cost function is quadratic, the runtime of the sover is drasticay increased. Nevertheess, the modified formuation of the OTS combined with the proposed bound strengthening method reduces the runtime significanty. For a experiments, a time imit of 3, 000 seconds is imposed. For those instances that are not soved within the time imit, the reative optimaity gap that is achieved by the sover at termination is reported. The runtime for different formuations of the OTS with respect to various oad factors is depicted in Figure 3a. Simiar to the previous case, the ower bound on the cardinaity of the switches is set to 45 for different oad factors. It can be observed that when the oad factor equas 0.5, the sover can find the optima soution within the time imit ony for Q-ET. As the oad factor increases, the average runtime decreases for a formuations. As it is cear from Figure 3a, Q-ET significanty outperforms other formuations for a oad factors. Specificay, the runtime for Q-ET is at east 5.95, 2.96, and times faster than Q-OT, Q-EC, and Q-OC on average, respectivey. Notice that these vaues are the under-estimators of the actua speedups since the sover was terminated before finding the optima soution in many cases. Next, consider the runtime for different formuations with respect to the change in the cardinaity ower bound of ON switches. It can be observed that the soution times for Q-OT, Q-EC, and Q-OC increase as the ower bound increases. This observation supports the argument made in [17] suggesting that a arge ower bound on the cardinaity of the ON switches woud make the OTS harder to sove in genera. However,

7 7 Running Time (seconds) L-T L-C Running Time (seconds) L-T L-C Load Factor Cardinaity (a) Running time vs. oad factor (b) Running time vs. cardinaity ower bound Fig. 2: These pot show the runtime of different formuations of OTS with a inear cost function with respect to different oad factors and cardinaity ower bounds. L-T and L-C correspond to the origina formuation with and without the bound strengthening method, respectivey. Running Time (second) % 16.3% 5.59% 17.3% Load Factor 0.21% Q-ET Q-OT Q-EC Q-OC Running Time (second) % Cardinaity 14.5% 3.58% Q-ET Q-OT Q-EC Q-OC 32.4% (a) Running time vs. oad factor (b) Running time vs. cardinaity ower bound Fig. 3: These pot show the runtime of different formuations of OTS with a quadratic cost function with respect to different oad factors and cardinaity ower bounds. Q-ET, Q-EC, Q-OT, and Q-OC correspond to the modified formuation with the proposed bound strengthening method, the modified formuation without the proposed bound strengthening method, the origina formuation with the proposed bound strengthening method, and the origina formuation without the proposed bound strengthening method, respectivey. notice that the cardinaity constraint has a minor effect on the runtime of Q-ET. Notice that Q-OC has the worst runtime on average among different settings of the oad factor and cardinaity ower bound. This impies that the proposed reformuation of the objective function together with the bound strengthening step is crucia to efficienty sove the OTS with a quadratic objective function. C. Poish Networks In this part, the proposed bound strengthening method is appied to sove the OTS for Poish networks. As for the 118-bus system, the runtime is evauated for both inear and quadratic cost functions. In a of the simuations, the cardinaity ower bound on the number of ON switches is set to 0. The number of fexibe ines varies from 70 to 400. The time imit is chosen as 14, 400 seconds (4 hours) for the sover. If the time imit is reached, the optimaity gap of the best found feasibe soution (if one exists) is reported. For the test cases with a quadratic cost function, ony the modified formuation of the probem is considered because it significanty outperforms the origina formuation. Tabe I reports the computationa improvements when the bound strengthening method is incorporated into the formuation as a preprocessing step. This tabe incudes the foowing coumns: # continuous: The number of continuous variabes in the system; # binary: The number of binary variabes corresponding to the fexibe ines in the system; Time: The runtime (in seconds) for soving the OTS with and without bound strengthening method within the time imit; Optgap: The reative optimaity gap within the time imit. Note that the sover is terminated when this gap is ess than 0.1%; Speedup: The speedup in the runtime when the proposed bound strengthening method is used as a preprocessing step. It can be observed from Tabe I that the presented bound strengthening method can notaby reduce the computation time. In particuar, the sover can be up to times faster if the bound strengthening method is used to strengthen the formuation. Moreover, on average (excuding the case 3375wp with a quadratic cost function), the soution time is at east 6.55 times faster if the bound strengthening method is performed prior to soving the probem. For the case 3375wp with a quadratic cost function, the sover cannot obtain a fea-

8 8 TABLE I: This tabe shows the runtime of the sover with and without bound strengthening for Poish networks with inear and quadratic cost functions. The superscript corresponds to the cases where the sover is terminated before finding the optima soution due to the time imit. With Bound Tightening Without Bound Tightening Cases Cost Function # Continuous # Binary Time Optgap Time Optgap Speedup 3120sp Linear < 0.1% 3, 623 < 0.1% 7.60 Quadratic , 900 < 0.1% 14, % wp Linear < 0.1% 931 < 0.1% 2.23 Quadratic < 0.1% 3, 960 < 0.1% sp Linear , 942 < 0.1% 2, 508 < 0.1% 1.29 Quadratic , 060 < 0.1% 3, 417 < 0.1% wp Linear , 447 < 0.1% 14, % 5.88 Quadratic , 570 < 0.1% 14, % wp Linear < 0.1% 77 < 0.1% 0.79 Quadratic , 301 < 0.1% 14, wop Linear < 0.1% 118 < 0.1% 6.94 Quadratic < 0.1% 3, 523 < 0.1% Average 1, 472 6, sibe soution in 14, 400 seconds without bound strengthening. However, the sover can find an optima soution within 4, 301 seconds after performing the proposed preprocessing step. VI. CONCLUSION Finding an optima topoogy of a power system subject to operationa and security constraints is a daunting task. In this probem, certain ines are fixed/uncontroabe, whereas the remaining ones coud be controed via on/off switches. The objective is to co-optimize the topoogy of the grid and the parameters of the system (e.g., generator outputs). Common techniques for soving this probem are mosty based on mixed-integer inear or quadratic reformuations using the big- M or McCormick inequaities foowed by iterative methods, such as branch-and-bound or cutting-pane agorithms. The performance of these methods party reies on the strength of the convex reaxation of these reformuations. In this paper, it is shown that finding the optima parameters of a inear or convex reformuation based on big-m or McCormick inequaities is NP-hard. Furthermore, the inapproximabiity of these parameters up to any constant factor is proven. 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The ongest path probem is defined as foows: Given an undirected graph G(V, E), where V and E stand for the sets of vertices and edges, respectivey, what is the ongest simpe path between two particuar vertices i and j in V? Let the ength of the ongest path be denoted as p opt. We construct an instance of the OTS probem in the foowing way: Consider V buses and, for every (r, ) E, connect buses r and through a ine with an arbitrary orientation that is equipped with a switch (note that S = E in this case). For each ine (r, ) E, its susceptance and fow capacity are set to 1. For every bus s {i, j} in the system, we set d s = p min s = p max s = 0, which impies that there is no oad or generator connected to bus s. Connect a generator with p min i = p max i = 1 to bus i. Furthermore, connect a oad d j = 1 to bus j. Finay, set r = 0. The instance designed above is feasibe if and ony if there is a simpe path between buses i and j in G. Furthermore, the size of the constructed instance of the OTS probem is i v 1 v 2 v k j Fig. 4: This graph shows the visuaization of the path P in the proof of Theorem 1. The soid edges denote the ines in P (with ON switches) and the dashed edges correspond to the remaining ines. poynomia in the size of the instance of the ongest path probem. Denote the feasibe region of the designed instance of the OTS probem as F. Note that ij = max Fij { θ i θ j } due to Lemma (1). Without oss of generaity, we drop the absoute vaue in the remainder of the proof. According to the defined characteristics of the oads and generators in the system, for any feasibe soution of the OTS probem, there shoud be at east one simpe path from bus i to bus j consisting of ony ines that are switched on. Therefore, for every (f, Θ, x, p ) arg max Fij {θ i θ j }, there exists a path P = {(i, v 1 ), (v 1, v 2 ),..., (v k, j)} with x rk = 1 for a (r, k) P. This simpe path is visuaized in Figure 4. With no oss of generaity, assume that the direction of the fow on the ines respect the directions in P. Based on Figure 4, one can verify that θ i θ j = (θr θ ) = fr fr max p opt (r,) P (r,) P (r,) P (18) Now, it is desirabe to construct a feasibe soution ( f, Θ, x, p) F that incudes a simpe path with ines that are switched on from buses i to j whose ength is p opt. To this end, consider the instance of the ongest path probem and suppose that P opt = {(i, u 1 ), (u 1, u 2 ),..., (u, j)} defines the ongest simpe path in G between nodes i and j. For every fexibe ine (i, j) in the corresponding instance of the OTS probem, we set x ij to 1 if this ine beongs to P opt and set to 0 otherwise. Moreover, we set θ j to 0 and define θ k = p opt kj for every bus k in P opt, where p opt kj is the ength of the unique path between buses k and j in P opt. This yieds that fr is equa to 1 for every ine (r, ) in P opt. Furthermore, for every fexibe ine (t, s) that does not beong to P opt, we set fts to 0. To satisfy (7h), set p i = 1. Therefore, a feasibe soution ( f, Θ, x, p) is constructed that satisfies the foowing property: θ i θ j θ i θ j = θ i = p opt (19) Inequaity (19) together with (18) estabishes the proof of the NP-hardness of finding max Fij { θ i θ j }. The inapproximabiity of the probem foows from the fact that, uness P = N P, there is no poynomia-time constant-factor approximation agorithm for determining the ongest path between nodes i and j in G.