Optimization-based Islanding of Power Networks using Piecewise Linear AC Power Flow

Transcription

1 arxiv: v2 [math.oc] 25 Oct 2013 OPTIMIZATION-BASED ISLANDING OF POWER NETWORKS 1 Optimization-based Isanding of Power Networks using Piecewise Linear AC Power Fow P. A. Trodden, Member, IEEE, W. A. Bukhsh, Student Member, IEEE, A. Grothey, and K. I. M. McKinnon Sets Abstract In this paper, a fexibe optimization-based framework for intentiona isanding is presented. The decision is made of which transmission ines to switch in order to spit the network whie minimizing disruption, the amount of oad shed, or grouping coherent generators. The approach uses a piecewise inear mode of AC power fow, which aows the votage and reactive power to be considered directy when designing the isands. Demonstrations on standard test networks show that soution of the probem provides isands that are baanced in rea and reactive power, satisfy AC power fow aws, and have a heathy votage profie. Index Terms Controed isanding; Piecewise inear approximation; Power system modeing; Integer programming. NOMENCLATURE B Buses. L Lines. G Generators. D Loads. B Buses connected by ine. L i Lines connected to bus i. G i Generators ocated at bus i. D i Loads ocated at bus i. B 0 Buses assigned to section 0. B 1 Buses assigned to section 1. L 0 Set of uncertain ines. B G Set of generator buses. Submitted to IEEE Transactions on Power Systems This work was supported by the UK Engineering and Physica Sciences Research Counci (EPSRC) under grant EP/G060169/1. P. A. Trodden is with the Department of Automatic Contro & Systems Engineering, University of Sheffied, Mappin Street, Sheffied S1 3JD, UK (e-mai: p.trodden@shef.ac.uk). W. A. Bukhsh, A. Grothey, and K. I. M. McKinnon are with the Schoo of Mathematics, University of Edinburgh, James Cerk Maxwe Buiding, Edinburgh EH9 3JZ, UK (e-mai: w.a.bukhsh@sms.ed.ac.uk, a.grothey@ed.ac.uk, k.mckinnon@ed.ac.uk).

2 OPTIMIZATION-BASED ISLANDING OF POWER NETWORKS 2 Parameters G B i,bi B Shunt conductance, susceptance at bus i. g,b,b C Conductance, susceptance, shunt susceptance of ine. τ Off-nomina tap ratio of ine (if transformer). V i,v + i Min., max. votage magnitude at bus i. Pg G,Pg G+ Min., max. rea power outputs of generator g. Q G g,q G+ g Min., max. reactive power outputs of generator g. Pd D,QD d Rea, reactive power demands of oad d. P L+ Rea power oss imit of ine. Θ,Θ + Max. ange across if connected, disconnected. c g (p G g ) Generation cost function for generator g. β d Loss penaty for oad d. Variabes v i,δ i Votage magnitude and phase at bus i. θ ij δ i δ j, votage phase difference between bus i and j. Note θ ij = θ ji. y ij cosθ ij. Note y ij = y ji. z ij sinθ ij. Note z ij = z ij. v i,vj θ ij y ij,qij Votage magnitudes at either end of ine (which connects buses i and j). Votage phase difference across a ine. Note θ ij = θ ji. cosθ ij. Note y ij = y ji. Rea, reactive power injection at bus i into ine (which connects buses i and j). p G g,qg G Rea, reactive power outputs of generator g. p D d,qd d Rea, reactive power suppied to oad d. α d Proportion of oad d suppied. γ i Binary. Section (0 or 1) assignment of bus i. ζ g Binary. Connection status of generator g. ρ Binary. Connection status of ine. I. INTRODUCTION T HE ast decade has seen a number of notabe cases of wide-area backouts as a consequence of severe disturbances and cascading faiures [1] [3]. Athough preventive and corrective systems exist to ameiorate the effects of severe disturbances, the operation of networks coser to imits, together with increased uncertainty in oad and distributed generation, means that cascading faiures may be harder to prevent, or stop once instigated [4]. Thus, intentiona isanding is attracting attention as a corrective measure for imiting the effects of severe disturbances and preventing widearea backout. Intentiona isanding aims to spit a network, by disconnecting ines, into eectricayisoated isands. The chaenge is that, if an isand is to be feasibe, it must satisfy both static constraints oad/generation baance, network constraints, system imits and dynamic constraints, i.e., for frequency and votage stabiity. Furthermore, the act of isanding must not cause a oss of synchronism or votage coapse.

3 OPTIMIZATION-BASED ISLANDING OF POWER NETWORKS 3 The majority of approaches to isanding aim to find, as a primary objective, eectromechanicay stabe isands. A popuar approach first uses sow coherency anaysis to determine groupings of machines with coherent osciatory modes, and then aims to spit the network aong the boundaries of these groups [5], [6]. Determining the required cutset of ines invoves considerations of oad-generation baance, power fows, and other constraints: agorithms incude pre-specification of boundaries [7], exhaustive search [5], [6], minima-fow minima-cutset determination using a combination of breadth- and depth-first search [8], graph simpification and partitioning [9], spectra custering [10], and meta-heuristics [11], [12]. A key attraction of the sow-coherencebased approach is that generator groupings are dependent on machine properties and argey independent of faut ocation and to a esser extent operating point [6]. If the network can be spit aong the boundaries of these groups, whie not causing excessive oad/generation imbaance or disruption, the system is ess ikey to ose stabiity. Moreover, groupings and ine cutsets can be determined offine. As a consequence, the on-ine action of isanding is fast, and the approach has been demonstrated effectivey by simuations of rea scenarios [13], [14]. Another approach uses ordered binary decision diagrams (OBDDs) to determine baanced isands [15]. Subsequenty, power fow and transient stabiity anayses can be used to iterate unti feasibe, stabe isands are found [16]. In [17], a framework is proposed that iterativey identifies the controing group of machines and the contingencies that most severey impact system stabiity. A heuristic method is used to search for a spitting strategy that maintains a desired stabiity margin. Wang et a. [18] empoyed a power fow tracing agorithm to first determine the domain of each generator, i.e., the set of oad buses that beong to each. Subsequenty, the network is coarsey spit aong domain intersections before refinement of boundaries to minimize imbaances. Whie it is known that the sensitivity of coherent machine groupings to faut ocation is ow, it is true that spitting the network aong the boundaries of a-priori determined coherent groups is not, in genera, the ony isanding soution that maintains stabiity. Moreover, such isands may be undesirabe in terms of other criteria, such as the amount of oad shed, the votage profie or the possibiity that the impacted region may be contained within a arger than necessary isand. For exampe, in [10], the sowcoherence-based isanding of the 39-bus New Engand system isoates the network s argest generator in an isand with no oad. In [19], an optimization-based approach to isanding and oad shedding was proposed. A key feature is that, unike many other methods, it can take into account a part of the network that is desired to be isoated a troubesome area when determining isands, and isoate this whie minimizing the expected amount of oad shed or ost. The probem is formuated as a singe mixed integer inear programming (MILP) probem, meaning that power baances, fows, and operating imits may be handed expicity when designing isands, and satisfied in each isand in a feasibe soution. The isanding MILP probem has simiarities with the transmission switching probem [20], in that the decision variabes incude which ines to disconnect, whie power fow constraints must be satisfied foowing any disconnection. Both approaches isanding and transmission switching may be seen as network topoogy optimization probems with added power fow constraints. In both cases, incusion of AC power fow aws in the constraints resuts in a mixed integer noninear program (MINLP), which

4 OPTIMIZATION-BASED ISLANDING OF POWER NETWORKS 4 is difficut to sove. Hence, inear DC power fow has been used to date, resuting in a more computationay favourabe MILP or MIQP probem. A disadvantage of the DC power fow mode is that the effect of ine disconnections on network votages is not considered. This is not excusive to MILP-based isanding and transmission switching; a number of isanding approaches consider rea power ony, and assume that reactive power may be compensated ocay after spitting. In [19], however, cases were reported where a soution coud not be found to satisfy AC power fow and votage constraints when the isands were designed considering DC power fow, even when sufficient reactive power generation capacity was present in each isand. Investigation found that oca shortages or surpuses of reactive power ed to abnorma votages in certain areas of the network. This paper presents a new method for controed isanding that respects votage and reactive power constraints. A piecewise inear approximation to AC power fow is deveoped and then used in a MILP-based approach to isanding: decisions are which ines to disconnect, which oads to shed and how to adjust generators. Resuts on test networks show this eiminates the AC-infeasibiities reported in [19]. The method is fexibe and abe to dea with different reasons for isanding. For exampe, to minimize the oad shed whie spitting the network so that coherent synchronous machines remain in the same isand. Or, to spit the network in two so as to ensure that the most of it is eft in a known safe state, isoated from a troubed region that has been identified as a possibe trigger for cascading faiures. The objective woud be to minimize the oad that is panned to be shed, pus the expected extra oad that might be ost due to faiures in the sma isand surrounding the troubed region. There can be many reasons for suspecting troube from a region e.g., incompete or inconsistent measurements, estimates of system stress such as coseness to instabiity or equipment operating imits, indications of component faiures, or other behaviour patterns that simuations have shown to be correated with cascading faiure [21] but the precise definition of what evidence woud ead to isanding being initiated is compex and is beyond the scope of this paper. The organization of this paper is as foows. In the foowing section, the piecewise inear AC power fow mode is presented, and its use is demonstrated in an Optima Power Fow (OPF) probem. In Section III, the isanding formuation is described. Section IV presents computationa resuts for test networks. Concusions are made in Section V. II. PIECEWISE LINEAR AC POWER FLOW A. A inear-pus-cosine mode of AC power fow The inear DC mode is a widey accepted approximation to AC power fow, whose benefits (inearity, simpicity) often outweigh its shortcomings. Recenty, however, there has been renewed research interest in the DC mode itsef [22] and more accurate aternative inearizations [23]. Recent work [19] by the authors found that a DC-based approach to controed isanding sometimes ed to infeasibe isands being created, mainy owing to out-of-bound votages and oca shortages or surpuses of reactive power. Motivated by this, this section presents a piecewise inear approximation to AC power fow, in which votage and reactive power are modeed.

5 OPTIMIZATION-BASED ISLANDING OF POWER NETWORKS 5 The AC power fow equations are described as foows. Rea and reactive power baances at each bus i B give p G g = p D d + +G B i v2 i, g G i d D i L i,j B :j i qg G = qd D + q ij Bi B v2 i, g G i d D i L i,j B :j i A ine L connects bus i B to bus j B,j i. The power fows from i to j are = vi 2 Gii +G ij v iv j y ij +B ij v i v j z ij, q ij = vib 2 ii B ij v i v j y ij +G ij v iv j z ij, with a simiar expression from j to i, where τ 2 G ii = G jj = τ G ij = τ G ji = g, τ 2 B ii = B jj = τ B ij = τ B ji = b +0.5b C. The convention is for a transformer to be ocated at the from end (bus i) of a branch. The standard DC approximation to AC power fow inearizes these equations by using the approximations v i = v j = 1, z ij = θ ij, y ij = 1, and b g 0 yieding = B ij θ ij. The reactive power variabes and equations are dropped. In the mode in this paper, votage and reactive power are retained. Expanding the ine fows about v i = 1, v j = 1 and θ ij = 0 (hence y ij = 1,z ij = 0): ( vi +v j +y ij 2 ) +B ij G ii (2v i 1)+G ij z ij, q ij B ii (1 2v i) B ij ( vi +v j +y ij 2 ) +G ij z ij. In a standard inearization, the sma-ange approximations woud then be used: y ij = cosθ ij 1 and z ij = sinθ ij θ ij. Tab. I gives the maximum absoute errors for each of the constituent terms in the inearized fows, over a typica range of operating votages and anges, i.e., 0.95 v i 1.05 at each end of the ine, and θ ij 40. The cosine approximation incurs the argest error. Fig. 1 shows maximum and minimum power fows and errors over this range of votages and anges for a ine with g = 1,b = 5,b C = 1. Approximation errors are obtained for when the y ij = cosθ ij term is approximated as 1 (a inear mode) and modeed exacty (inear pus cosine). In both cases, z ij = θ ij sinθ ij is used. Athough itte reduction in errors is apparent in the rea fows, the importance of modeing the cosine term is cear for reactive fows. A simiar anaysis shows that incuding the sine term (instead of its inearization) in addition to the cosine term reduces the error in the rea fows sighty, but makes no significant difference to the reactive power. Since the infeasibiities that occur using the DC approach to isanding are mainy owing to the reactive power and votage imits [19], the appropriate approximation to use is the inear-pus-cosine one. And athough cosine terms cannot be used directy in an MILP mode, they can be modeed to arbitrary eves of accuracy by piecewise inear functions. The next section demonstrates the use of the mode in an OPF formuation.

6 OPTIMIZATION-BASED ISLANDING OF POWER NETWORKS 6 Rea power (p.u.) Lin. error Lin.+cos error Reactive power (p.u.) 1 0 Lin. error Lin.+cos error q ij Fig θ ij (deg) θ ij (deg) Maxima and minima of power fows, and of approximation errors, as a function of phase ange difference. B. Piecewise inear AC OPF The piecewise inear (PWL) AC OPF probem is defined as min ( ) c g p G g g G subject to, i B, the inearized power baances: p G g = Pd D + +G B i (2v i 1), g G i d D i L i,j B :j i qg G = Q D d + q ij Bi B (2v i 1), g G i d D i L i,j B :j i Line fows for a L,i,j B : i j: = G ii (2v ( i 1)+G ij vi +v j +y ij 2 ) +B ij θ ij, q ij = B ii (1 2v i) B ij ( vi +v j +y ij 2 ) +G ij θ ij. The N-piece PWL approximation to cosθ ij. For a L,i,j B : i j. (1a) (1b) y ij = h ij,k θ ij +d ij,k, θ ij [x ij,k,x ij,k+1 ],k = 0...N 1, (2) TABLE I APPROXIMATION ERRORS IN LINE FLOW TERMS Term Approximation Max abs error vi 2 2v i v iv jy ij v i +v i +y ij v iv jz ij z ij y ij z ij θ ij

7 OPTIMIZATION-BASED ISLANDING OF POWER NETWORKS 7 Generation cost ($K/hour) AC SOS Reaxed DC 3 Fig Load (% of peak) Generation costs as a function of oad for the 9-bus network. where h ij,k and d ij,k are chosen so that the approximation coincides with cosx at breakpoints {x ij,0,...,x ij,n }. System imits are appied: V i v i V + i, i B, Pg G p G g P g G+, g G, Q G g qg G Q G+ g, g G, +p ji P L+, L. (3) Note that ine fow imits are imits on rea power (I 2 R) oss. If an MVA imit ( S L+ ) is given, this may be converted by assuming nomina votage, i.e., P L+ S L+ 2. = g g 2 +b2 The impementation of the PWL mode of cosθ ij (2) requires either binary variabes or specia ordered sets of type 2 (SOS-2) [24]. The overa probem is then, depending on c g, a mixed integer inear or quadratic program (MILP or MIQP). If (2) is repaced by its reaxation y ij h ij,k θ ij +d ij,k, then the probem becomes a convex optimization probem and no binary variabes or SOS sets are needed. Since rea and reactive ine osses decrease as y ij increases, it is tempting to assume that equaity wi hod for one of the PWL sections, and this reaxation wi yied a tight resut. However, as Fig. 2 shows, situations exist where the SOS formuation is necessary. This shows optima generation costs against oad eve, as obtained by OPFs using AC, PWL with SOS, reaxed PWL, and DC power fow modes. The network is the WSCC 9-bus network modified to set votage imits to ±5% and the ower reactive power imit for each generator is raised from 300 to 5 Mvar. This means that at ow oad eves the generators find it increasingy difficut to baance the reactive power, as more ines become sources rather than sinks of reactive power, and the generation cost rises with faing oad. Whie the SOS PWL is abe to capture this effect, the reaxed PWL and DC-based modes are not; the former cheats by having some ines continue to store reactive power irrespective of their end votages and anges aowed because y ij < h ij,k θ ij +d ij,k is permitted and this aows more of the rea power to

8 OPTIMIZATION-BASED ISLANDING OF POWER NETWORKS 8 be generated by the cheaper generators. III. A FORMULATION FOR SYSTEM ISLANDING USING PIECEWISE LINEAR AC POWER FLOW In [19], the probem of determining how to spit a transmission network into isands is considered. The aim is to imit the effects of possibe cascading faiures and prevent the onset of wide-area backouts by re-configuring the network via ine switching so that probem areas are isoated. The MILP-based method defines two sections of the network. A of the buses that must be isoated are pre-assigned to section 0, and the optimization determines which other buses and ines to pace in section 0. A the remaining components are in section 1. This creates at east two isands. The optimization wi aso determine the best strategy to adjust generation and shed oad so as to estabish a oad-generation baance in each isand whie respecting a network equations and operating constraints after the spit. A. Motivation: effect of topoogy changes on votage profie Soution of the MILP isanding probem provides a set of ines to switch, oads to shed and generators to adjust. However, if ony the DC power fow equations are incuded in the constraints, the effects of changing the network topoogy on votages and reactive power fows is not considered. Thus, in [19], an AC optima oad shedding (OLS) probem is soved after the MILP isanding probem, using the isanded network topoogy. If a soution to this can be found, the isanded network is feasibe with respect to AC power fow and operating constraints. The soution provides the correct generator output and oad adjustments to make, now having considered votage and reactive power. However a number of the isanding soutions in [19] were AC infeasibe, primariy due to vioation of votage bounds; soutions coud be recovered by reaxing the norma imits. One such exampe, for the 24-bus IEEE Reiabiity Test System (RTS) [25], is described as foows. Given the probem of isoating bus 6 whie minimizing the expected oad shed or ost, the optima soution isands buses 1, 2 and 6, as indicated in Fig. 3. There remains sufficient rea power capacity in both isands to meet demand, and no oad is shed. Moreover, but not by design, there is sufficient reactive power capacity in each isand to meet the tota reactive power demand. Despite this, a feasibe soution to the AC-OLS cannot be found. Softening the votage bounds recovers a soution, but with an abnormay ow votage of p.u. at bus 6 and an over-imit fow on ine (2, 6). Further inspection reveas that this situation has arisen because of the disconnection of ine(6,10), a cabe with high shunt capacitance. The passive shunt reactor at bus 6 woud, in norma circumstances, baance ocay the excess reactive power and maintain a satisfactory votage profie. This probem coud be avoided by inking together the disconnection of ine(6,10) and the shunt reactor at 6. The optima soution when these actions are inked is shown in the right-hand diagram of Fig. 3, and it yieds a better feasibe soution than when the reactor is not disconnected. Rues ike this are easy to incorporate in the mode; however, it is difficut a-priori to define a possibe rues. A better approach is to aow the mode to decide the combination of equipment to disconnect, and when this is done the optima soution disconnects

9 OPTIMIZATION-BASED ISLANDING OF POWER NETWORKS Fig. 3. IEEE 24-bus RTS with bus 6 isoated. On the eft, DC method (soid) and PWL method without shunt switching (dashed). On the right, PWL method with shunt switching. both ine (6,10) and the reactor at bus 6, giving the right-hand Fig. 3 soution. The modes and resut for this exampe are given in Sections III-B4 and IV-A1. This is just one exampe of where an isanding soution formed by considering ony rea power even if network constraints are incuded is unsatisfactory. It aso shows that even if reactive power baance is achieved within each isand, oca shortages or surpuses can ead to an abnorma votage profie. Many test networks are prone to this probem [19]. Moreover, it is not just system isanding that is susceptibe; DCbased transmission switching aso does not consider the consequences on votage of disconnecting ines. Thus, there is a need for network topoogy optimization methods that can determine AC-feasibe soutions, but without having to resort to soving the fu MINLP probem. The focus of this paper is topoogy optimization for the purpose of isanding, and in the next section, a formuation is presented that uses the PWL mode of AC power fow. B. Formuation of constraints for isanding The probem is to decide which ines to switch in order to isoate a part of the network. Separation of sections is enforced by sectioning constraints. The isanded network must satisfy power baance and fow equations and operating imits, and so these are incuded as constraints in the probem. 1) Sectioning constraints: Define B 0 and B 1, where B 0 B 1 =, as the subsets of buses that are desired to be separated. For now, the motivation for this separation is eft open, but it may be that these buses in, say, B 0 represent a faiing area of the network, or are associated with a coherent group of synchronous machines that wi be separated from other groups. The proposed approach wi spit the network into two sections: section 0 wi contain a buses in B 0 and section 1 a buses in B 1. For a bus i B, γ i denotes the section (0 or 1) to which that bus is assigned. That is, if

10 OPTIMIZATION-BASED ISLANDING OF POWER NETWORKS 10 i is to be paced in section 0, then γ i = 0. Separation between sections is achieved by switching ines: ρ denotes the connection status of a ine, and the convention foowed is for ρ = 0 when is disconnected. The exact boundaries of each section wi depend on the objective, defined ater, and the optimization wi determine how to assign to sections those buses not in B 0 or B 1, in order achieve baance and optimize the objective. However, the foowing constraints enforce the separation of sections 0 and 1, without defining precisey their boundaries. ρ 1+γ i γ j, L,i,j B : i j, γ i = s, i B s,s {0,1}. (4a) (4b) 2) Power fow: The remainder of the constraints are concerned with achieving a baanced, steady state for the isanded network. It is assumed that generators are permitted to make ony sma-scae changes to output or be switched off, and oads may be fuy or party shed in order to maintain a baance. As a consequence of these changes and the topoogica changes, bus votages, anges and ine fows wi change, and so must be modeed to ensure satisfaction of network constraints and operating imits. First, the power baances, (1a) and (1b), are incuded without modification. Next, the ine fow equations are modified so that when a ine is disconnected, power fows across it are zero irrespective of its end bus votages and anges. To assist this, we introduce ine variabes v i and v j as end votages and θ ij as the ange difference that are distinct from bus variabes v i, v j and θ ij. The foowing constraints contro the reationship between ine variabes and bus variabes. For a ine L with end buses i and j, i B : Θ ρ θ ij Θ ρ, Θ + (1 ρ ) θ ij θ ij Θ + (1 ρ ), (5a) (5b) 0 v i v i (V i + Vi )(1 ρ ), V i v i V i +(V + i V i )ρ, (5c) (5d) and i B, V i v i V + i, where Θ + Θ is a big-m constant. Of these, (5a) and (5b) force equaity of θ ij and θ ij = δ i δ ij for a connected ine, but set θ ij = 0 for a disconnected ine whie aowing the bus anges δ i and δ j to vary independenty. Likewise, if ρ = 1 then, by (5c), v i = v i and v j = v j. However, if ρ = 0 then the ine votages are set to minimum vaues v i = V i and v j = V j independent of the bus votages v i and v j. (5e)

11 OPTIMIZATION-BASED ISLANDING OF POWER NETWORKS 11 This switching between ine and bus variabes is made use of in modified ine fow equations. For a ine, = G ii (2v i 1)+G ij ( v i +v j +yij 2 ) +B ij θ ij ( G ii (2V i 1)+G ij i +V j 1) ) (6a) (1 ρ ), q ij = B ii (V ( v i +v j +yij 2 ) +G ij θij (1 2vi ) Bij ( B ii (1 2V i ) B ij (Vi +Vj 1) ) (1 ρ ), and y ij is given by (2), using θ ij. Note that since θij = 0 if ρ = 0, then y ij = 1 for a disconnected ine. Hence, if ρ = 0 then = 0, irrespective of v i, v j and θ ij = δ i δ j. If ρ = 1, the norma power fow equations are recovered. 3) Operating constraints: In the short time avaiabe when isanding in response to a contingency, any extra generation that is needed wi be achieved by a combination of the ramping-up of on-ine units and the commitment of fast-start units. For simpicity, fast-start units are not considered in the exampes in this paper. We assume that a generator that is operating can either have its input mechanica power disconnected, in which case rea output power drops to zero in steady state, or its output can be set to a new vaue within a sma interva, [ ] Pg G,Pg G+, say, for generator g, around the pre-isanded vaue. The imits wi depend on the ramp and output imits of the generator, and the amount of immediate or short-term reserve capacity avaiabe to the generator. For the test scenarios in Section IV, a time imit of 2 minutes is assumed for ramping, but the formuation permits any choice. This choice shoud be informed by existing post-contingency response protocos. For reactive power, it is assumed that a new output can be set in some range Q G g to Q G+ g. The set of possibe rea and reactive power outputs of a generator is usuay convex. For the test scenarios in Section IV, the bounds on the rea and reactive power are independent. In the more genera case, since the range of vaues for the rea power output is sma, the feasibe region for the probem is a narrow sice through a convex set, and except when the rea power output is cose to its upper imit it is a good approximation to treat the rea and reactive power bounds as independent. If this is not the case, it is straightforward to add constraints that coupe p G g and qg G. The operating regime is modeed by the constraints ζ g Pg G p G g ζ g Pg G+, g G, qg G QG+ g, g G, (7b) ζ g = 1, g { G : Pg G = 0 } G 1. (7c) Here, ζ g is a binary variabe and denotes the on/off setting of the rea power output, and G 1 is a subset of generators which are required to remain on. For oads, because of the imits on generator outputs and network constraints, it may not be possibe after isanding to fuy suppy a oads. It is therefore assumed that some shedding of oads is permissibe. Note that this is intentiona shedding, not automatic shedding as a resut of ow votages or frequency. To impement this in the rea network there has to be centra contro over equipment. For a d D, Q G g p D d = α dp D d, q D d = α dq D d, (6b) (7a) (8a) (8b)

12 OPTIMIZATION-BASED ISLANDING OF POWER NETWORKS 12 where 0 α d 1. Finay, ine imits are appied via constraint (3). 4) Rues for other component switching: As motivated by Section III-A, sometimes it is necessary to have rues for switching components or adjusting contros in different situations. Such rues can easiy be incuded in the formuation using standard techniques for deriving constraints from ogica rues [26]. For exampe, the switching of a shunt component at a bus i can be modeed by introducing binary and continuous variabes, ξ i and u i respectivey, constraints ξ i (2V i 1) u i ξ i (2V + i 1), (1 ξ i )(2V i 1) u i (2v i 1) (1 ξ i )(2V + i 1), and repacing the G B i (2v i 1), B B i (2v i 1) terms in (1a) and (1b) with G B i u i and B B i u i, respectivey. In Section IV-A1 this is expored further for the 24-bus exampe. C. Objective functions for isanding The genera aim is to spit the network, separating the two sections 0 and 1, yet eaving it in a feasibe state of operation. The specific motivations and objectives for isanding are discussed in this section. Ceary, if a network can be partitioned with minima disruption to oad, and with minima disturbances to generators, then its chances of viabe operation unti future restoration are increased. 1) Isoating uncertain regions and maximizing expected oad suppy: We assume that there is an identifiabe ocaized area of the network that is beieved coud be a trigger for cascading faiure. Simiar to the approach in [19], the goa is to incude this area of potentia troube in an isand, eaving the rest of the network in a known, secure steady state. The sets B 0 and L 0 consist of a buses and ines in the troubed area and, additionay, any buses and ines whose status is uncertain. To ensure section 1 contains no uncertain components, a ines L 0 remaining in this section are disconnected by repacing (4a) by ρ 1 γ i, i B. (9) Because section 0 may contain faiing components or be in an uncertain state, it is assumed there is a risk of not being abe to suppy any oad paced in that section. Accordingy, a oad oss penaty 0 β d < 1 is defined for a oad d, which may be interpreted as the probabiity of being abe to suppy a oad if paced in section 0. Suppose a reward R d is obtained per unit suppy of oad d. If d is paced in section 1 a reward R d is reaized per unit suppy; however, if d is paced in section 0, a ower reward of β d R d < R d is reaized. The objective is then to maximize the expected tota vaue of oad suppied: J exp oad = d DR d P D d (β d α 0d +α 1d ), (10) where α d = α 0d + α 1d, and 0 α 1d γ b, b B,d D b. Here a new variabe α sd is introduced for the oad d deivered in section s {0,1}. If γ b = 0 (and so the oad at bus b is in section 0), then α 1d = 0, α 0d = α d, and the reward is β d R d P D d α d. Conversey, if γ b = 1 then α 1d = α d and α 0d = 0, giving a arger reward R d P D d α d. Thus, maximizing (10) gives a preference for γ b = 1 and a smaer section 0, so that the impacted area is imited.

13 OPTIMIZATION-BASED ISLANDING OF POWER NETWORKS 13 2) Promoting generator coherency: Another aim is to ensure the synchronicity of generators within isands. Large disturbances in the network cause eectro-mechanica osciations, which can ead to a oss of synchronism. A popuar approach is to spit the system aong boundaries of near-coherent generator groups, as determined by sowcoherency anaysis [27]. Thus, weak connections between machines which give rise to sow, ighty-damped osciations are cut, eaving separate networks of tightycouped, coherent machines. Consider those buses in the network with generators attached, the set of which is defined as B G, and define B GG { (i,j) B G B G : j > i } as the set of a pairs of such buses. For what foows, it may be assumed that mutipe units at a bus are tighty couped and are aggregated to a singe unit. The dynamic couping,w ij, between a pair of machines at buses (i,j) B GG may be determined from sow-coherency anaysis. For exampe, assuming as in [10] the undamped second order swing equation, W ij = ( ω i ω j ) (δ i δ j ) = ( 1 M i + 1 M j ) Pij δ ij, where M i, ω i, δ i are the inertia constant, anguar frequency and rotor ange of the machine at bus i, and P ij is the synchronizing power coefficient or stiffness between δ ij machines at i and j. To favour, in the objective, separating oosey-couped generators, introduce a new variabe 0 η ij 1 for a (i,j) B GG. Then the constraint η ij γ i γ j η ij, (11) sets η ij to 1 if generator buses i and j are in different sections of the network (and hence eectricay isoated), but otherwise may be zero. Minimizing the function J coh = (i,j) B GG W ij η ij (12) gives a preference for machines in different sections having sma W ij, i.e., being weaky couped, and those within the same section have stronger couping. This may be used in conjunction with (10), i.e., maxj exp oad kj coh, with weighting k > 0, so that section 0 is the unheathy section, and the expected oad suppy is maximized whie keeping together strongy-couped machines. Minimizing (12) aone wi favour keeping a machines in the same section, and to force the machines apart additiona constraints may be needed. Aternativey, the foowing impementation spits the network directy into coherent groups, making different use of the sets B 0 and B 1. 3) Spitting into coherent groups: Suppose that coherent groups of generators have been determined, and that assigned tob 0 andb 1 are those buses inb G corresponding to machines in different groups. For exampe, B 0 may contain the critica coherent group of machines, and B 1 a others. The sectioning constraints wi ensure that the machines are separated, but which other buses are assigned to each section is determined by the optimization. The soution that minimizes the amount of oad shed can be found by maximizing the function J oad = d Dα d P D d. (13)

14 OPTIMIZATION-BASED ISLANDING OF POWER NETWORKS 14 Aternativey, to seek a soution that changes the generator outputs the minimay from their initia vaues P G0 g, minimize J gen = g G t g (14) where t g 0, t g p G g Pg G0, and t g p G g +Pg G0, g G. The sectioning constraints ensure that the machines are spit into two sections. If further separation is required, the optimization can be re-run on each isand of the network. 4) Penaties: Often there may be mutipe feasibe soutions with objective vaues cose to the optimum. Incuding additiona penaty terms in the objective sma enough to not significanty affect the primary objective improves computation by encouraging binary variabes to take integra vaues in the reaxations, and aso guides the soution process towards particuar soutions. For exampe, consider the penaty terms (for a minimization probem) W y (1 y )+ W L (1 ρ )+ Wg G (1 ζ ) (15) L L g G where W y, W L, W g G are weights to be chosen appropriatey. The first term penaizes 1 cosθ, and hence heps ine oss. The second penaizes cuts to ines. For exampe, setting W L equa to the some sma mutipe of the pre-isanding power fow through the ine wi penaize most heaviy disconnections of high-fow ines; in [19] it was shown that this eads more often to soutions that retain dynamic stabiity. The third term penaizes the switching-off of generators. If Wg G = ǫpg G+ then units are given uniform weighting. If, say, Wg G = ǫ ( ) 2, Pg G+ then the disconnection of arge units is discouraged. D. Overa formuation The overa probem is to optimize the chosen isanding objective (e.g., (10), (12), (13), or (14)), subject to sectioning constraints (4); ine switching constraints (5); power baance ((1a) and (1b)) and fow (6) constraints; the PWL approximation (2); generation imits (7); ine fow imits (3); oad shedding constraints (8). IV. COMPUTATIONAL RESULTS A. Isanding to minimize expected oad oss A set of scenarios was buit based on the 9-, 14-, 24-, 30-, 39-, 57-, 118- and 300-bus test systems from MATPOWER [28]. For a network with n B buses, n B scenarios were generated by assigning in turn each singe bus to B 0. No buses were incuded in B 1 and no ines in L 0. For each scenario, the isanding soution was obtained by soving the previousy described MILP probem. The feasibiity of an isanding soution was

15 OPTIMIZATION-BASED ISLANDING OF POWER NETWORKS 15 checked by soving an AC optima oad shedding (OLS) probem on the isanded network, which incudes a AC power baance, fow and operating constraints, but permits oad shedding as per (8a) and (8b). Data for the isanding probems are described as foows. In the objective function, J exp oad, a vaue of 0.75 is used for the oad oss penaty β d. The generator coherency objective, J coh, was not incuded initiay. The penaties are Wg G G+ = 0.01Pg, W y = 0.1 and W L = d P d D, so that the ine-cut penaty is scaed by the tota oad in the system. Our investigations show that these penaties have a negigibe effect (0.2%) on the quaity of the soutions, but reduce computation time by an order of magnitude. For the PWL approximation for a ine, first the ange difference prior to isanding, θ, is determined from the base-case AC OPF soution, and then 12 pieces are used over ± ( θ +10 ). Operating imits, incuding votage and ine imits, were obtained from each network s data fie [28]. Generator rea power output imits (Pg G and Pg G+ ) were set, as expained in Section III-B3, to aow a 2-minute ramp change from the current output Pg G0, where ramp rates were avaiabe in the network data, or a 5% change where they were not. In either case, the output imits were imited by capacity imits. Pg G0 was obtained by soving an AC OPF on the intact network prior to isanding. Then in the isanding probem, the ower imit was raised by 5% of (Pg G Pg G+ ). The postisanding AC OLS, however, was permitted to use the fu range, [ ] Pg G,Pg G+. This avoids those soutions where an isand is infeasibe because of too much generated rea power. 1) AC-feasibe isanding of 24-bus network: Returning to the exampe of Section III-A, the PWL AC isanding approach is appied to the probem of isanding bus 6. The isanding probem was soved both with and without the option (as part of the optimization) of switching the shunt reactor at bus 6. The optima soutions are shown in Fig. 3. Without shunt switching (PWL-AC-1), the cabe (6,10) is eft intact and the fina network topoogy is significanty different from before. With shunt switching permitted (PWL-AC-2), the cabe is again switched, but fewer buses are isanded than for the DC soution. The feasibiity of each soution was checked by soving the AC OLS probem on the isanded network, and both PWL AC soutions satisfied a AC constraints. Tab. II compares the DC, PWL-AC-1 and PWL-AC-2 soutions, using vaues obtained from both the MILP soutions and the post-isanding AC soutions. The PWL AC isanding soutions are cose to the fina AC OLS soutions. Note that the PWL AC soutions achieve AC feasibiity at the cost of a ower expected oad suppy (hence higher expected oad shed). 2) Computation time: The speed with which isanding decisions have to be made depends on whether the decision is being made before a faut has occurred, as part of contingency panning within secure OPF, or after, in which case the time scae depends on the cause of the contingency. Finding soutions that are optima, or to within a prespecified percentage of optimaity, can take an unpredictabe amount of time. Hence, especiay in the atter case of reacting after a faut has occurred, it is important to be abe to produce good feasibe soutions within short time periods even if these are not necessariy optima. To iustrate how the quaity of the soution depends on the soution time, tests were run for a set of fixed times of between 5 and 45 seconds, returning the best found integer feasibe soution. Tab. III summarizes these resuts for the 57-, 118-

16 OPTIMIZATION-BASED ISLANDING OF POWER NETWORKS 16 and 300-bus scenarios, quoting the average reative MIP gap of returned soutions. A the test cases with 39 or fewer buses soved to negigibe % gaps within 5 seconds, and are not shown. Tab. III aso shows the average gaps between the returned and best-known AC soutions for each scenario, where an AC soution was obtained from a returned PWL isanding soution by soving the AC-OLS on the isanded network. The mean error between the objectives of the returned PWL-AC and AC soutions was ess than 0.02%. For each network and scenario, the best-known AC soution was the best from those found from the different termination times, pus onger 1000-second runs. In the second and third sections of Tab. III, the mean vaues are over a cases that were feasibe within the time imit. The patform was a 64-bit Dua Inte Xeon processor and 128 GiB RAM with up to 12 threads and using CPLEX 12.5 as the MILP sover. The resuts show that good isanding soutions were found within 30 s and usuay sooner for a networks. Moreover, the isanding topoogy usuay changes itte, or not at a, between the soutions returned at 5s and 45s. 3) AC feasibiity: Using the DC mode 20% of cases ed to AC-infeasibe isands [19], whereas none of the isands found using the PWL AC mode were infeasibe. 4) Promoting generator coherency: The generator coherency objective, J coh, may be incuded for the24-bus network exampe by taking second-order dynamic data taken from [25]. For exampe, when B 0 = 3, maximizing just J exp oad eads to an optima soution that paces bus 1 in section 0 aong with bus 3, and an expected oad suppy of 2699 MW. In doing this, the ine between buses 1 and 2 is switched, separating the arge generator sets at these buses (which woud incur a cost of J coh = 2.26). However, when maximizing the joint objective with k = 100, the optima soution does not incude bus 1 in section 0, opting instead to eave the ine (1,2) intact and pacing just buses 3 and 9 in section 0. With k = 100, the expected oad suppy is sighty smaer (2670 MW), but the strongy-couped generators at buses 1 and 2 remain connected (J coh = 0.00). B. Coherency-based isanding The coherency-based spitting approach was appied to the 10-machine, 39-bus New Engand test network. Sow coherency anaysis, assuming second-order dynamics, TABLE II 24-BUS SYSTEM: COMPARISON OF SOLUTIONS. Soution DC PWL-AC-1 PWL-AC-2 MILP isanding soution J exp oad (MW) Generation (MW) Exp. oad shed (MW) Post-isanding AC-OLS J exp oad (MW) Generation (MW) Exp. oad shed (MW)

17 OPTIMIZATION-BASED ISLANDING OF POWER NETWORKS 17 TABLE III SOLUTIONS TO ISLANDING PROBLEMS FOR DIFFERENT TIME LIMITS. Time (s) Percentage with no isanding soution found within time 57-bus bus bus Mean % between best MIP soution and the MIP bound 57-bus bus bus Mean % between best AC soution found in time and best known 57-bus bus bus shows that the machines may be divided into two groups: those at buses 30, 31 and 39 in one group, and then a others. With B 0 = {30,31,39} and B 1 = B G \ B 0, the optima soution spits the system as shown in Tab. IV. Note that athough buses 1 3 and 5 9 are incuded in the same section as 30, 31 and 39, no generators are present at these buses. The objective was to minimize the movement of generator rea power outputs, i.e., (14). To achieve this spit and eave the isands baanced, the generator at bus 32 has to ower its output from 671 to 373 MW, whie 311 MW is shed. It is worth stating that no other soution exists that spits these two groups but requires ess tota change in generator outputs. V. CONCLUSIONS AND FUTURE WORK An optimization-based framework for the intentiona or controed isanding of power networks has been presented. The approach is fexibe with respect to the aims and objectives of isanding, and finds isands that are baanced and satisfy rea and reactive power fow and operating constraints. It has been shown that the incusion of a piecewise inear mode of AC power fow aows AC-feasibe isands to be found, where previousy a DC-based approach ed to isands with out-of-bound votages. The use of objectives that promote generator coherency has been demonstrated. Future work wi investigate the wider practica aspects of the approach by performing detaied simuations on representative networks and backout scenarios, considering TABLE IV COHERENCY-BASED ISLANDING OF 39-BUS NETWORK. Section 0 Section 1 Buses 1 3, 5 9, 30, 31, 39 4, 10 29, Generation (MW) Load suppied (MW) Load shed (MW)

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