Iterative McCormick Relaxation for Joint Impedance Control and Network Topology Optimization

Iterativ e McCormic k Relaxation for Join t Imp edance Con trol and Net w ork T op ology Optimization Junseon P ark ∗ , § , Hy eongon P ark ∗ , Rah ul K. Gupta § ∗ Puky ong National Univ ersit y , Busan, South K orea, § W ashington State Univ ersit y , Pullman, W A, USA. Abstract—P ow er system op erators are increasingly deplo y- ing V ariable Impedance Devices (VIDs), e.g., Smart Wires, and Net work T op ology Optimization (NTO) sc hemes for mitigating op erational challenges such as line and transformer congestion, and voltage violations. This work aims to optimize and co or- dinate the op eration of distributed VIDs considering xed and optimized topologies. This problem is inheren tly non-linear due to p ow er o w equations as well as bilinear terms in tro duced due to v ariable line imp edance of VIDs. F urthermore, the top ology optimization scheme mak es it a mixed integer nonlinear prob- lem. T o tac kle this, w e introduce using McCormick relaxation sc heme, which conv erts the bilinear constraints into a linear set of constraints along with the DC pow er ow equations. W e prop ose an iterative correction of the McCormick relaxation to enhance its accuracy . The prop osed framew ork is v alidated on standard IEEE b enchmark test systems, and we present a p erformance comparison of the iterative McCormick metho d against the non-linear, SOS2 piecewise linear approximation, and original McCormick relaxation. Index T erms—McCormick relaxation, Imp edance con troller, Net work top ology optimization, Optimal pow er o w. I. In tro duction The pow er system is facing op erational challenges suc h as p o wer ow congestion and voltage quality issues [ 1 ], [ 2 ] due to massiv e in terconnection of in termitten t renew able energy resources (RERs), in v erter-based resources (IBRs) [ 3 ] as w ell as data cen ter loads [ 4 ]. On the one hand, utilities are lo oking for dierent long-term solutions such as net work expansion, transformer upgrade [ 5 ], etc. On the other hand, several Grid Enhancing T ec hnologies (GET s) ha ve emerged as promising alternativ es to improv e net w ork exibility and mitigate c ongestion issues [ 6 ]. These include Netw ork T op ology Optimization (NTO) [ 7 ], V ariable Imp edance Devices (VIDs), also referred to as Smart Wire Devices (SWDs) [ 8 ], etc. Prior w ork has demonstrated that NTO can signi- can tly reduce op erational costs relative to xed-topology systems [ 7 ], and improv e op erational exibility . In con- trast, VIDs, or SWDs, dynamically adjust line impedance to redistribute pow er ows from congested to lightly loaded transmission lines [ 8 ]. VIDs are a type of exible A C transmission system (F A CTS), suc h as Th yristor Con- trolled Series Comp ensator (TCSC), Static Synchronous Series Comp ensator (SSSC), Unied Po wer Flow Con- troller (UPFC), etc., and can b e mo deled b y v ariable line impedance. The v ariable imp edance/reactance allows exibilit y in the transmission net w ork to re-route the p o w er ows and help a v oid congestion and reduce costs. In the existing literature, the optimization of distributed VIDs across has b een widely inv estigated. Incorp orat- ing VIDs in to op erational planning introduces additional nonlinearities in b oth the optimal p ow er ow (OPF) and switching problems, as VIDs are typically modeled through v ariable line imp edance [ 8 ], [ 9 ]. Dieren t methods ha v e b een prop osed to tac kle the nonlinearities. F or exam- ple, the w ork in [ 8 ] proposes a mixed in teger reform ulation; ho w ev er, the con trols are limited to the maxim um and minim um op eration setp oints of the VIDs. A similar strategy was used in [ 9 ] for the placement of the SWDs to facilitate integration of large-scale wind p ow er plants. Ov erall, these methods are fo cused on the planning of the SWDs. The work in [10] developed a scheme for the op eration of SWDs using a sensitivity factor approach, but they are not applicable to dieren t top ologies. In this w ork, we fo cus on the join t optimal opera- tion of the VIDs with base and optimized topology . First, the net w ork top ology and bus-bar splitting are mo deled using the no de-breaker representation, which enables detailed switc hing decisions while main taining net w ork connectivity and op erational feasibility . Then, VIDs are incorp orated to provide additional exibility by allo wing controlled adjustments of line reactances within predened op erating limits, thereby redistributing p ow er o ws and alleviating congestion. The DC p ow er ow model is adopted to represent p ow er ow physics, extended to accoun t for the v ariable line reactances introduced b y the VIDs. As this problem has bilinear constraints, w e prop ose a McCormick relaxation-based approach to linearize the bilinear constraints in the DC-OPF and NTO problems. Ho w ev er, this scheme usually results in large errors due to the McCormic k relaxation, esp ecially when the allow able c hange in the net w ork imp edance is large, for example, 30-50% of the rated v alue. F or enhancing the accuracy of the McCormick relaxation approach, w e prop ose an iterativ e McCormick metho d, where the b ounds on the maxim um allow able range in the imp edance is changed in steps. The key contribution of this w ork lies in lineariza- tion techniques using the iterative McCormick relaxation approac h and its p erformance comparison with resp ect to the nonlinear and mixed integer schemes. The pap er is organized as follo ws. Section II presen ts the problem formulation. Section I I I presents the relaxation using the McCormick metho d and SOS2 approximation. Section IV presen ts the simulation setup and results on dieren t IEEE testcases, and nally , Section V concludes the main contribution of the presented work. I I. Problem F orm ulation W e consider a p ow er transmission netw ork consisting of N b buses, N g generators and connected with several VIDs. The ob jective is to optimize the op eration of gen- erators, load-shedding, distributed VIDs and the netw ork top ology . The optimization ob jectives and the constrain ts are dened b elow. A. Ob jectiv e function The prop osed optimization aims to minimize the total system op erating cost, which consists of (i) generation cost ( C Gen ) and (ii) load-shedding cost ( C LS ) , dened as follo ws. 1) Generation cost: They are mo deled as a quadratic function, as shown in Equation (1a), where the net generation is the sum of the generation pro duced at the t w o busbars. C Gen = N g ∑ g =1 ( c g, 2 ( P g, 1 + P g, 2 ) 2 + c g, 1 ( P g, 1 + P g, 2 ) + c g, 0 ) (1a) where, c g, 0 , c g, 1 and c g, 2 are the cost parameters of the generators and P g, 1 and P g, 2 refer to the generator units at busbar 1 and 2, resp ectively . 2) Load shedding cost: When transmission capacity constrain ts fail to meet the total load demand, load shedding may o ccur. Multiplying the load shedding p ow er b y a high-cost V O LL minimizes the load shedding cost. The load shedding cost can b e calculated as shown in Equation (1b). C LS = V OLL N b X b =1 ( X d ∈ D b ( P max b,d − ( P b,d, 1 + P b,d, 2 ))) (1b) where, VOLL refers to the v alue of loss of load, P b,d, 1 and P b,d, 2 are the demand at busbars 1 and 2 of bus b . The sym b ol P max b,d refer to the maximum demand at bus b . D b represen ts a set con taining the indices of the demand per bus b . B. Constrain ts The constraint set consists of a p ow er ow mo del, constrain ts on the netw ork top ology , constraints on the generators and load-shedding. These are describ ed below. 1) T ransmission Grid Mo del: W e mo del the transmis- sion netw ork using the DC p o wer o w approximation [11], whic h assumes that (i) the angle of the voltage dierence is small, (ii) the net w ork is reactive, and (iii) the voltage magnitudes are close to 1 p er unit. Let the symbol b l represen t the susceptance of line l , θ l,f r and θ l,to denote the voltage angle for line l at ”from” and ”to” ends. Then, the ow in line l , P l can b e expressed using DC p ow er o w mo del as P l = b l ( θ l,f r − θ l,to ) . (2) 2) Net w ork T op ology Optimization Mo del: NTO iden- ties optimal netw ork top ology by utilizing con trollable switc hing devices, including transmission switches and substation circuit breakers (CBs). Substation CBs en- able reconguration of internal bus arrangements and their in terfaces with the rest of the netw ork, thereby aecting grid exibility , reliability , and securit y . V arious bus congurations are used in practice, each requiring dieren t num b ers of breakers and oering distinct reli- abilit y c haracteristics; among them, the break er-and-a- half sc heme is common in high-voltage substations due to its fav orable balance of reliability and op erational exibilit y [12]. In this work, we mo deled the substations using the no de-breaker (NB) representation. Figure 1 sho ws a generalized break er-and-a-half la y out with line- switc hing capability . This NB mo del provides additional reconguration options, allo wing the “from’’ and “to’’ busbars to be op ened or closed indep enden tly and enabling loads and generators to connect to either busbar. 𝑮𝒆𝒏𝒆𝒓𝒂𝒕𝒐𝒓 𝒉 𝒈 𝑭𝒓𝒐𝒎 * 𝑩𝒖𝒔 𝑩𝒖𝒔𝒃𝒂𝒓 '𝟏 𝑩𝒖𝒔𝒃𝒂𝒓 '𝟐 𝒉 𝒃 𝑻𝒐 * 𝑩𝒖𝒔 𝑩𝒖𝒔𝒃𝒂𝒓 '𝟏 𝑩𝒖𝒔𝒃𝒂𝒓 '𝟐 𝒍𝒐𝒂𝒅 𝒉 𝒈 𝒉 𝒃 𝒉 𝒍, 𝒇𝒓 𝒉 𝒍, 𝒕𝒐 𝒉 𝒍 Fig. 1. Generalized breaker-and-half model. Considering the switching capability of the busbars and the lines, the NTO mo del can b e mathematically mo deled using busbar binary v ariables h b [ 7 ], [13]: − θ max (1 − h b ) ≤ θ b, 1 − θ b, 2 ≤ θ max (1 − h b ) ∀ b, (3a) where θ b, 1 and θ b, 2 are the voltage angles at busbars 1 and 2 and θ max is the maxim um dierence of angle betw een the busbars. In (3a), when busbars are merged, the dierence in v oltage angle phase b etw een busbars must b e iden tical, and when split, it must not exceed the maxim um angle phase. The generators connected at eac h busbar can b e mo d- eled using binary v ariables h g , and can be expressed as (1 − h g ) P min g ≤ P g, 1 ≤ (1 − h g ) P max g ∀ g (3b) h g P min g ≤ P g, 2 ≤ h g P max g ∀ g . (3c) The symbols P min g and P max g denote the minim um and maxim um limits. Eqs. (3b) and (3c) mo del when a generator is connected to busbar 1 or at busbar 2, and vice versa. Similarly , the connection and controllabilit y of the demand at eac h of the busbars are mo deled using binary v ariable h d using (3d) and boun ded by (3e) 0 ≤ P b,d, 1 ≤ (1 − h d ) P max b,d ∀ b (3d) 0 ≤ P b,d, 2 ≤ h d P max b,d ∀ b. (3e) The limit on the pow er-o w for each line and the switc hing of the lines are expressed as − (1 − h l,e ) P max l ≤ P l,e, 1 ≤ (1 − h l,e ) P max l ∀ l, e (3f ) − h l,e P max l ≤ P l,e, 2 ≤ h l,e P max l ∀ l, e (3g) − h l P max l ≤ P l,e, 1 ≤ h l P max l ∀ l, e (3h) h l,e ≤ h l ∀ l, e (3i) P l = P l,e, 1 + P l,e, 2 ∀ l, e (3j) where h l,e refer to binary v ariable represen ting whether line l is on or o. e = { “ from ” , “ to ” } refer to the ends of the line. P max l denes the maxim um capacity of line l . P l,e, 1 and P l,e, 2 refer to p ow er ow of line l at the end side of e through busbar 1 and busbar 2 resp ectively . Eqs. (3f) and (3g) dene the transmission line constraints: when a line is connected to busbar 1, the p ow er ow to the other busbars must b e zero, and vice versa. Eqs. (3h) and (3i) specify the line status, ensuring that no p ow er o ws when the line is open. Constraint (3j) enforces pow er conserv ation by requiring that the total p ow er o wing out of a busbar equals the p ow er owing from the ‘from’ bus to the ‘to’ bus. The line switching constraints are expressed using the big- M metho d, giv en b y (3k), (3l), and (3m) − M (1 − h l ) ≤ b l ( θ l,f r − θ l,to ) − P l ≤ M (1 − h l ) ∀ l (3k) − h l,e θ max ≤ θ l,e − θ l,e, 1 ≤ h l,e θ max ∀ l, e (3l) − (1 − h l,e θ max ) ≤ θ l,e − θ l,e, 2 ≤ h l,e θ max ∀ l, e. (3m) The binary v ariables for busbars, generators, demand connections, and lines are connected through Eqs. (3n), (3o), and (3p). When the busbars at eac h substation are connected, it is not necessary to consider the binary v ariable determining the connection of the generator, load, and end of the line. h b + h g ≤ 1 ∀ b g ∈ G b (3n) h b + h d ≤ 1 ∀ b d ∈ D b (3o) h b + h l,e ≤ 1 ∀ b, e l ∈ LF b or l ∈ LT b . (3p) Finally , the constraints (3q) and (3r) express the p ow er balance constraints, i.e., the dierence betw een generation and demand is equal to the sum of pow er o wing through the lines. The symbol G b refers to a set containing the indices of the generator p er bus b . LF b and LT b denote set of lines o w ”from” bus b and ”to” bus b , respectively . X g ∈ G b P g, 1 − X d ∈ D b P b,d, 1 − X l ∈ LF b P l + X l ∈ LT b P l = 0 ∀ b (3q) X g ∈ G b P g, 2 − X d ∈ D b P b,d, 2 − X l ∈ LF b P l + X l ∈ LT b P l = 0 ∀ b (3r) 3) VID Mo del: VIDs can b e mathematically mo deled as v ariable susceptance devices [ 8 ], [ 9 ], [14] as b l = ¯ b l + ∆ b l (4a) − r ¯ b l ≤ ∆ b l ≤ r ¯ b l ∀ l (4b) where, ¯ b l denotes the nominal susceptance and ∆ b l is a v ariable allowing deviations in the line susceptance is b ounded by a factor r ≤ 1 of nominal susceptance ¯ b l . C. Final Optimization Problem The nal optimization can b e formulated as min C Obj = C Gen + C LS , sub ject to ( 3 ) , ( 4 ) . (5) Note that the optimization problem in ( 5 ) is non-linear b ecause of the bilinear v ariables ∆ b l ( θ l,f r − θ l,to ) due to VIDs when ( 4 ) is substituted in (3k) and ( 2 ) with ∆ b l and θ l,f r , θ l,to b eing v ariables. Therefore, w e prop ose a linearization approach using McCormick Relaxation as describ ed b elo w. I I I. McCormick Relaxation and SOS2 Appro ximation A. McCormic k Relaxation Mo del 1) McCormic k Relaxation (Non-iterative): W e relax bilinear terms, ∆ b l ( θ l,f r − θ l,to ) , by dening auxiliary v ariables, then applying McCormick env elop es [15]. These McCormic k en v elop es transform the bilinear constraints to a set of linear constrain ts by dening upp er and lo w er b ounds on each of the v ariables in the bilinear term. W e dene upp er and low er b ounds on the v ariables in v olv ed in the bilinear constraint, i.e., [ b min l , b max l ] and [∆ θ min l , ∆ θ max l ] , then the McCormic k constrain t set is dened as follows. W e dene a new v ariable w l = ∆ b l ∆ θ l where ∆ θ l = θ l,f r − θ l,to . Then, the McCormic k relaxation is given as (6a)-(6d): w l ≥ b min l ∆ θ l + ∆ b l ∆ θ min l − b min l ∆ θ min l ∀ l (6a) w l ≥ b max l ∆ θ l + ∆ b l ∆ θ max l − b max l ∆ θ max l ∀ l (6b) w l ≤ b min l ∆ θ l + ∆ b l ∆ θ max l − b min l ∆ θ max l ∀ l (6c) w l ≤ b max l ∆ θ l + ∆ b l ∆ θ min l − b max l ∆ θ min l ∀ l. (6d) 2) Iterativ e McCormick Relaxation: The accuracy of the McCormick relaxation scheme is strongly inuenced b y the range of bounds associated with the v ariables in the McCormick equations. As it will b e demonstrated in the Results section, the ob jectives obtained using the McCormic k relaxation metho d exhibit a deviation from the costs computed by a nonlinear solver. T o address this discrepancy , we prop ose an iterative McCormick relaxation technique, formally describ ed in Algorithm 1 . This algorithm applies McCormick relaxation in small steps, so that the accuracy is not compromised. Algorithm 1 Iterativ e McCormic k Relaxation with Incre- men tal r Up dates Require: T arget relaxation level r nal , step size ∆ r , initial susceptances { b l } 1: Initialize ˆ b l ← b l for all lines l 2: while r < r nal do 3: Solv e ( 5 ) using McCormick relaxation ( 6 ) for r = ∆ r 4: Compute susceptance up date ∆ b l 5: Up date susceptance: ˆ b l ← b l + ∆ b l for all l 6: r ← r + ∆ r 7: end while 8: return nal solution at r nal B. SOS2 Mo del The bilinear term b l ∆ θ l in ( 2 ) and (3k) can also b e appro ximated by a tw o-dimensional piecewise-linear mo del using special ordered sets of t yp e 2 (SOS2). F or eac h line l , the range [ b min l , b max l ] and [∆ θ min l , ∆ θ max l ] are divided in to N grid b and N grid θ n um b er of interpolation p oints, denoted b y ˆ b l,i and ∆ b θ l,j . SOS2 constraints are constructed by dening contin uous v ariables λ l,i,j , α l,i , β l,j suc h that N grid b X i =1 N grid θ X j =1 λ l,i,j = 1; α l,i = X j λ l,i,j ; β l,j = X i λ l,i,j . (7a) The vectors ( α l, 1 , . . . , α l,N grid b ) and ( β l, 1 , . . . , β l,N grid θ ) are mo deled as SOS2 sets, ensuring that only tw o adjacent breakp oin ts are activ e along each axis. In our implemen- tation, λ l,i,j , α l,i , and β l,j are mo deled as contin uous v ariables, while the adjacency structure of the piecewise– linear mo del is imp osed through the SOS2 constrain ts; Y ALMIP 1 creates an SOS2 set and automatically adds the binary v ariables and linear constraints required to enforce the adjacency structure. Using the interpolation weigh ts, the line susceptance, angle dierence, and an auxiliary v ariable ˜ w l that ap- pro ximates the pro duct b l ∆ θ l are dened as b l = X i,j λ l,i,j ˆ b l,i , ∆ θ l = X i,j λ l,i,j c ∆ θ l,j , (8a) ˜ w l = X i,j λ l,i,j ˆ b l,i c ∆ θ l,j . (8b) Finally , the SOS2 approximation replaces the bilinear o w relation in (3k) through − M (1 − h l ) ≤ ˜ w l − P l ≤ M (1 − h l ) , (9a) − P max l ≤ P l ≤ P max l , (9b) where M is a suciently large constant. When h l = 1 , (9a) enforces P l = ˜ w l , yielding a piecewise–linear approx- imation of b l ∆ θ l . When h l = 0 , the constraint is relaxed, and the line o w is handled through the NTO constrain ts in (3). 1 https://yalmip.github.io/command/sos2/ IV. Numerical Simulations A. Sim ulation Setup W e sim ulate our framew ork on four dierent IEEE b enc hmark test systems, whic h are Case300, Case588_sdet, Case1354pegase, and Case1888rte [16], and will b e describ ed in the results section. W e used the dataset from the MA TPO WER testcases for the simu- lations. F or each testcase, we set the minimum output of all generators to zero in order to av oid situations where a generator’s minim um output w ould exceed the load at its corresp onding bus. T o sim ulate the congestion scenario, where the con trol of VIDs and NTO would be b enecial, w e articially reduce the capacit y of the transmission lines. This is done by rst solve the OPF for the nominal case, then we set the capacity of the lines to be 80% of the p ow er-ows obtained in the nominal case. F or the sim ulations with VIDs,w e allow the susceptance c hange factor (4b), r is set from 0 to 0.5, allo wing eac h line susceptance to v ary up to 50% of their nominal v alues. The cost of load shedding (VOLL) is set to 2,000 $ /MWh. B. Results The results of the prop osed iterative-McCormic k re- laxation is presented and compared against three dier- en t formulations: Nonlinear, McCormick, and SOS2. The p erformance are compared with resp ect to the achiev ed cost, computation time, and accuracy with resp ect to the nonlinear case. W e use Gurobi 12.0 2 as the optimization solv er for all the cases. 0 0.1 0.2 0.3 0.4 0.5 Susceptance change factor (r) 6.4 6.5 6.6 6.7 6.8 6.9 Objective Value 10 6 Nonlinear McCormick(Original) Iterative McCormick SOS2 Fig. 2. Performance comparison of ob jective function costs under dierent metho ds for Case300 using the nominal top ology . T ABLE I Error and computation time of relaxation/approx metho ds for dierent r (Case300, Nominal T opolog y) r (%) Nonlinear McCormick (Original) Iterative McCormick SOS2 Time (s) Error (%) Time (s) Error (%) Time (s) Error (%) Time (s) 0 1.18 0.000 1.05 0.000 1.06 0.000 1.80 5 1.76 0.300 1.11 0.300 1.07 0.001 6.78 10 1.91 0.681 1.10 0.117 2.13 0.000 12.42 15 1.99 0.550 1.11 0.021 3.19 0.000 8.40 20 2.07 0.697 1.10 0.099 4.21 0.000 31.03 25 2.16 0.677 1.11 0.173 5.24 0.000 31.82 30 2.07 0.680 1.10 0.172 6.34 0.047 6.98 50 1.95 0.338 1.11 0.234 10.59 0.000 9.53 2 https://www.gurobi.com 0 0.1 0.2 0.3 0.4 0.5 Susceptance change factor (r) 6.44 6.45 6.46 6.47 6.48 6.49 Objective Value 10 6 Nonlinear McCormick(Original) Iterative McCormick SOS2 Fig. 3. Performance comparison of ob jective function costs under dierent metho ds for Case300 using the optimal top ology via NTO. T ABLE I I Error and computation time of relaxation/approx metho ds for dierent r (Case300, Optimal T op ology using NTO) r (%) Nonlinear McCormick (Original) Iterativ e McCormick SOS2 Time (s) Error (%) Time (s) Error (%) Time (s) Error (%) Time (s) 0 1205.93 † 0.000 1205.33 † 0.000 1205.69 † 0.000 1205.54 † 5 1206.95 † 0.419 81.97 0.419 78.55 0.388 179.80 10 920.84 0.036 41.42 0.050 128.78 0.016 47.61 15 564.82 0.028 15.53 0.017 185.17 0.003 46.16 20 375.47 0.006 22.66 0.002 246.85 0.034 135.91 25 435.44 0.012 22.39 0.082 312.15 0.005 112.66 30 342.35 0.093 20.10 0.005 365.00 0.001 75.62 50 151.38 0.051 32.76 0.006 544.09 0.000 44.16 † Indicates instances that reached the 1200 s time limit. At r = 0% , all metho ds reached the limit; at r = 5% , only the nonlinear formulation did. 1) Case300 System: The optimized ob jectiv es using the nonlinear, McCormick, iterative-McCormic k, and SOS2 are shown in Fig. 2. F rom the plot, w e can make tw o observ ations. First, the cost go es down with an increase in r i.e., the exibilit y in changing the line susceptance thanks to VIDs. Second, the iterativ e McCormick and SOS2 costs are quite close to the nonlinear case, whereas the original McCormick deviates from the nonlinear cost as r increases. This happ ens b ecause an increase in r leads to a larger McCormic k relaxation and therefore it is more inexact. W e also quan tify the dierence b etw een the costs ob- tained b y relaxed and appro ximated metho ds with resp ect to the nonlinear metho d in T able I . It also shows the computation time for each metho d. It sho ws that the iterativ e McCormic k metho d improv es up on the error, but it takes more time due to the iterations. F or this case, SOS2 p erforms the b est b oth in terms of time and accuracy . The results with optimized top ology , i.e., solving the NTO problem with VIDs, are shown in Fig 3 . Here, we see that the cost do es not change muc h with an increase in r , this is b ecause NTO already reduced the cost by a large margin, and there is not m uch impro vemen t that can b e achiev ed using VIDs for this testcase. In some cases, the v alue of the cost function increases as the r increases. This is b ecause the Gurobi solver’s Gap was set to 0.1%, leading to errors. The error is esp ecially large when r is 5% b et w een the Nonlinear mo del and the other models. This is caused by the solver not conv erging to a solution within the time limit (1200 seconds). The error and computation time are presented in T able II . W e again observed that SOS2 is the best-p erforming approximation. 2) Larger testcases: T o ev aluate whether the prop osed sc heme is scalable on larger netw orks, we sim ulate it for Case588_sdet, Case1354p egase, and Case1888rte. F or brevit y , w e present the result for r = 10% considering the nominal top ology . The results are presented in T able I I I comparing the cost, computation time, and accuracy . W e observ e that the prop osed iterative McCormick achiev es the b est p erformance in accuracy and computation time for all the larger systems. V. Conclusions This work considered the problem of optimizing the op eration of distributed v ariable imp edance devices with net w ork top ology optimization in p ow er transmission net w orks. Due to the v arying nature of the line reactances, this problem contained bilinear constraints. The w ork prop osed using McCormick env elop es to relax the bilinear constrain ts, which results in a large error compared to the ob jective obtained b y the nonlinear mo del. Therefore, w e prop osed an iterative approach to apply McCormic k en v elop es in small steps. The proposed method was compared against the nonlinear metho d, the original McCormic k relaxation, and the SOS2 appro ximation. 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