This is a repository copy of MILP formulation for controlled islanding of power networks.

Transcription

1 This is a repository copy of MILP formuation for controed isanding of power networks. White Rose Research Onine URL for this paper: Version: Accepted Version Artice: Trodden, P., Bukhsh, W., Grothey, A. et a. (1 more author) (2013) MILP formuation for controed isanding of power networks. Internationa Journa of Eectrica Power and Energy Systems, 45 (1) ISSN Reuse Uness indicated otherwise, futext items are protected by copyright with a rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 aows the making of a singe copy soey for the purpose of non-commercia research or private study within the imits of fair deaing. The pubisher or other rights-hoder may aow further reproduction and re-use of this version - refer to the White Rose Research Onine record for this item. Where records identify the pubisher as the copyright hoder, users can verify any specific terms of use on the pubisher s website. Takedown If you consider content in White Rose Research Onine to be in breach of UK aw, pease notify us by emaiing eprints@whiterose.ac.uk incuding the URL of the record and the reason for the withdrawa request. eprints@whiterose.ac.uk

2 MILP formuation for controed isanding of power networks P. A. Trodden a,, W. A. Bukhsh a, A. Grothey a, K. I. M. McKinnon a a Schoo of Mathematics, University of Edinburgh, James Cerk Maxwe Buiding, King s Buidings, Mayfied Road, Edinburgh EH9 3JZ, United Kingdom Abstract This paper presents a fexibe optimization approach to the probem of intentionay forming isands in a power network. A mixed integer inear programming (MILP) formuation is given for the probem of deciding simutaneousy on the boundaries of the isands and adjustments to generators, so as to minimize the expected oad shed whie ensuring no system constraints are vioated. The soution of this probem is, within each isand, baanced in oad and generation and satisfies steady-state DC power fow equations and operating imits. Numerica tests on test networks up to 300 buses show the method is computationay efficient. A subsequent AC optima oad shedding optimization on the isanded network mode provides a soution that satisfies AC power fow. Time-domain simuations using second-order modes of system dynamics show that if penaties were incuded in the MILP to discourage disconnecting ines and generators with arge fows or outputs, the actions of network spitting and oad shedding did not ead to a oss of stabiity. Keywords: optimization, integer programming, controed isanding, backouts 1. Introduction In recent years, there has been an increase in the occurrence of wide-area backouts of power networks. In 2003, separate backouts in Itay [1], Sweden/Denmark [2] and USA/Canada [3] affected miions of customers. The wide-area disturbance in 2006 to the European system caused This work is supported by the UK Engineering and Physica Sciences Research Counci (EPSRC) under grant EP/G060169/1. Corresponding author. Te Fax Emai addresses: Ô ÙкØÖÓ Ò º ºÙ (P. A. Trodden), Ûº º Ù Ñ º º ºÙ (W. A. Bukhsh), º ÖÓØ Ý º ºÙ (A. Grothey), ºÑ ÒÒÓÒ º ºÙ (K. I. M. McKinnon) Preprint submitted to Esevier September 21, 2012

3 the system to spit in an uncontroabe way [4], forming three isands. More recenty, the UK network experienced a system-wide disturbance caused by an unexpected oss of generation; backout was avoided by oca oad shedding [5]. Whie the exact causes of wide-area backouts differ from case to case, some common driving factors emerge. Modern power systems are being operated coser to imits: iberaization of the markets, and the subsequent increased commercia pressures and change in expenditure priorities, has ed to a reduction in security margins [6, 7, 8]. A more recenty occurring factor is increased penetration of variabe distributed generation, notaby from wind power, which brings significant chaenges to secure system operation [9]. For severa arge disturbance events, e.g., [3], studies have shown that a wide-area backout coud have been prevented by intentionay spitting the system into isands [10]. By isoating the fauty part of the network, the tota oad disconnected in the event of a cascading faiure is reduced. Controed isanding or system spitting is therefore attracting an increasing amount of attention. The probem is how to spit the network into isands that are as cosey baanced as possibe in oad and generation, have stabe steady-state operating points within votage and ine imits, and so that the action of spitting does not cause dynamic instabiity. This is a considerabe chaenge, since the search space of ine cutsets grows exponentiay with network size, and is exacerbated by the requirement for strategies that obey non-inear power fow equations and satisfy operating constraints. It is not computationay practica to tacke a these aspects of the probem simutaneousy within a singe optimization, and approaches in the iterature differ according to which aspect is treated as the primary objective. Additionay different search methods have been proposed for defining the isand boundaries. An exampe where the primary objective is to produce oad baanced isands is [11]. This proposes a three-phase ordered binary decision diagram (OBDD) to generate a set of isanding strategies. The approach uses a reduced graph-theoretica mode of the network to minimize the search space for isanding; power fow anayses are subsequenty executed on isands to excude strategies that vioate operating constraints, e.g., ine imits. In other approaches the primary objective is to spit the network into eectromechanicay stabe isands, commony by spitting so that generators with coherent osciatory modes are grouped. 2

4 If the system can be spit aong boundaries of coherent generator groups whie not causing excessive imbaance between oad and generation, then the system is ess ikey to ose stabiity. Determining the required cutset of ines invoves, as a secondary objective, considerations of oadgeneration baance and other constraints; agorithms incude exhaustive search [12], minima-fow minima-cutset determination using breadth-/depth-first search [13], graph simpification and partitioning [14], and metaheuristics [15, 16]. The authors of [17] propose a framework that, iterativey, identifies the controing group of machines and the contingencies that most severey impact system stabiity, and uses a heuristic method 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 generator. Subsequenty, the network is coarsey spit aong domain intersections before refinement of boundaries to minimize imbaances. The current paper presents an optimization framework for controed isanding. The method s primary objective is to minimize the expected amount of oad that has to be disconnected whie eaving the isanded network in a baanced steady state. The post-isanding dynamics are not modeed expicity in the optimization, as this greaty increases the computationa difficuty of the probem. Instead penaties are used to discourage arge changes to power fows, and it is shown by simuation that this resuts in the isanding soutions being dynamicay stabe. The proposed approach has two stages: first, a mixed-integer inear programming (MILP) isanding probem, which incudes the inear DC fow equations and fow imits, is soved to determine a DC-feasibe soution; secondy, an AC optima oad shedding optimization is soved to provide an AC-feasibe steady-state post-isanding operating point. Integer programming has many appications in power systems, but its use in network spitting and backout prevention is imited. Bienstock and Mattia [19] proposed an IP-based approach to the probem of designing networks that are robust to sets of cascading faiures and thus avoid backouts; whether to upgrade a ine s capacity is a binary decision. Fisher et a. [20], Khodaei and Shahidehpour [21] propose methods for optima transmission switching for the probem of minimizing the cost of generation dispatch by seecting a network topoogy to suit a particuar oad. In common with the formuation presented here, binary variabes represent switches that open or cose each ine and the DC 3

5 power fow mode is used, resuting in a MILP probem. However, in the current paper sectioning constraints are present, and the probem is to design baanced isands whie minimizing oad shed. The organization of the paper is as foows. The next section outines the motivation and assumptions that underpin the approach. The DC MILP isanding formuation is deveoped in Section 3. The AC optima oad shedding probem is described in Section 4. In Section 5 computationa resuts are presented. In Section 6, the dynamic stabiity of the networks in response to isanding is investigated. Finay, concusions are drawn in Section Motivation An appication of isanding which has received itte attention is isanding in response to particuar contingencies so as to isoate vunerabe parts of the network. For exampe after some faiure, part of the network may be vunerabe to further faiure, or a suspected faiure of monitoring equipment may have resuted in the exact state of part of the network being uncertain. In such a case an action that woud prevent cascading faiures throughout the network is to form an isand surrounding the uncertain part of the network so isoating it from the rest. A method that does not take into account the ocation of the troube when designing isands may eave the uncertain equipment within a arge section of the network, a of which may become insecure as a resut. Figure 1(a) iustrates the situation: uncertain ines and buses are indicated by a?. Figure 1(b) shows a possibe isanding soution for this network: a uncertain buses have been paced in Section 0 and a uncertain ines with at east one end in Section 1 are disconnected. The foowing distinction is made between sections and isands. The spit network consists of two sections, an unheathy Section 0 and a heathy Section 1 with no ines connecting the two sections, and a uncertain equipment in Section 0. However, neither section is required to be connected so may contain more than one isand: in Figure 1(b), Section 1 comprises isands 1, 3 and 4, and Section 0 is a singe isand. The optimization wi determine the boundaries of the sections, the number and boundaries of the isands, the generator adjustments, and the amount of each oad that is panned to be shed. A baance has to be found between the oad that is panned to be shed and the residua oad that is eft in Section 0, which may be ost because that section is vunerabe. This can be achieved by taking as objective the sum of the vaue of the oads remaining in both sections after the panned 4

6 ???? (a) Network prior to isanding Section 1 Section 0 Section 1???? Isand 1 Isand 2 (b) Network post isanding Isand 3 Isand 4 Figure 1: (a) Iustration of a network with uncertain buses and ines, and (b) the isanding of that network by disconnecting ines. oad shedding minus a proportion of the the vaue of the oad remaining in Section 0 after the panned oad is shed. 3. MILP isanding formuation This section presents a MILP formuation for the probem of finding a steady state isanded soution in a stressed network, whie minimizing the expected oad ost. Consider a network that comprises a set of buses B={1, 2,..., n B } and a set of ines L. The two vectors F and T describe the connection topoogy of the network: a ine L connects bus F to bus T. There exists a set of generators G and a set of oads D. A subset G b of generators is attached to bus b B; simiary, D b contains the subset of oads present at bus b B Sectioning constraints Motivated by the previous section, the intention is to partition the buses and ines between Sections 0 and 1. It is suspected that some subset B 0 B of buses and some subset L 0 L of 5

7 ines are fauty or at risk. No uncertain components are aow in Section 1. A binary variabe γ b is defined for each bus b B; γ b is set equa to 0 if b is paced in section 0 and γ b = 1 otherwise. A binary variabe ρ is defined for each L; ρ = 0 if ine is disconnected and ρ = 1 otherwise. Constraints (1a) and (1b) appy to ines in L \ L 0. A ine is cut if its two end buses are in different sections (i.e. γ F = 0 and γ T = 1, or γ F = 1 and γ T = 0). Otherwise, if the two end buses are in the same section then ρ 1, and the ine may or may not be disconnected. Thus, these constraints enforce the requirement that any certain ine between sections 0 and 1 sha be disconnected. ρ 1+γ F γ T, L \ L 0, ρ 1 γ F + γ T, L \ L 0. (1a) (1b) Constraints (1c) and (1d) appy to ines assigned to L 0. A ine L 0 is disconnected if at east one of the ends is in Section 1. Thus, an uncertain ine either (i) sha be disconnected if entirey in Section 1, (ii) sha be disconnected if between sections 0 and 1, or (iii) may remain connected if entirey in Section 0. ρ 1 γ F, L 0, ρ 1 γ T, L 0, (1c) (1d) Constraints (1e) and (1f) set the vaue of γ b for a bus b depending on what section that bus was assigned to. B 1 is defined as the set of buses that are required to remain in Section 1. It may be desirabe to excude buses from the unheathy section, and such an assignment wi usuay reduce computation time. γ b = 0, b B 0, γ b = 1, b B 1. (1e) (1f) Given some assignments to B 0, B 1 and L 0, the optimization wi disconnect ines and pace buses in Sections 0 or 1, hence partitioning the network into Sections 0 and 1. What ese is paced in Section 0, what other ines are cut, and which oads and generators are adjusted, are degrees of freedom for the optimization, and wi depend on the objective function. 6

8 3.2. DC power fow mode with ine osses The power fow mode empoyed is a variant of the DC mode. As in the standard DC mode it assumes unit votage at each bus and uses a inearization of Kirchhoff s votage aw (KVL), but unike the standard DC mode the variant aso accounts for ine osses. Kirchhoff s current aw is appied at each bus b B: p G g= p D d + p L (p L h L ), (2) g G b d D b L:F =b L:T =b where p G g is the rea power output of generator g G b at bus b, p D d is the rea power demand from oad d D b. The variabe p L is the rea power fow from bus F into the first end of ine, and p L h L is the fow out of the second end of ine into bus T, the difference in the fows being the oss h L. The standard DC mode has no ine oss, i.e., h L = 0, but this mode resuts in the oad oss being underestimated. Actua ine osses are non-inear functions of votages and phase ange differences, and these can be approximated in the DC mode by a piecewise inear function. However investigations have shown that this offers itte or no improvement in the objective over a simpe constant-oss approximation, but adversey affects computation [22]. In this paper, therefore, a constant oss mode is empoyed. The oss for ine is given by h L = ρ h L0, (3) where h L0 is the oss immediatey before isanding. The incusion of ρ drives the oss to zero if the isanding optimization cuts the ine. The inearized version of Kirchhoff s votage aw (KVL) has the form ˆp L = B L ( ) δf δ T, (4) where B L is the susceptance of ine and ˆp L an auxiiary variabe for the rea power fow. When the ine is connected then it is required that p L = ˆp L, but when it is disconnected then pl = 0 and ˆp L is free. This is modeed as foows. ρ P Lmax p L P Lmax ρ, (5a) (1 ρ ) ˆP Lmax ˆp L p L ˆP Lmax (1 ρ ), (5b) 7

9 where P Lmax is the maximum possibe magnitude of rea power fow through a ine, and ˆP Lmax shoud be arge enough to aow two buses across a disconnected ine to maintain sufficienty different phase anges. (Note that at the very minimum ˆP Lmax ˆp L P Lmax.) If ρ = 0, then p L = 0 but may take whatever vaue is necessary to satisfy the KVL constraint (4), whie if ρ = 1 then p L = ˆp L. Line imits P Lmax may be expressed either directy as MW ratings on rea power for each ine, or as a imit on the phase ange difference across a ine. Since in the mode the rea power through a ine is just a simpe scaing of the phase difference across it, then any phase ange imit may be expressed as a corresponding MW imit Generation constraints In the short time avaiabe when isanding in response to a contingency it is not possibe to start up generators. Generators that are operating can either have their input power disconnected, in which case their rea output power drops to zero, or their output can be changed to a vaue within a sma interva, [ Pg G, Pg G+ ] say for generator g, around their 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. This aternative operating regime is modeed by the constraint ζ g P G g p G g ζ gp G+ g, where ζ g is a binary variabe. If ζ g = 0 then generator g is switched off and p G g ζ g = 1 and its output is p G g [ Pg G, Pg G+ ]. (6) = 0; otherwise 3.4. Load shedding Because of the imits on generator power outputs and network constraints it may not be possibe after isanding to fuy suppy a oads. It is therefore necessary to permit some shedding of oads. Note that this is intentiona oad 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. 8

10 Suppose that a oad d D has a constant rea power demand P D d. It is assumed that this oad may be reduced by disconnecting a proportion 1 α d, where 0 α d 1, so the oad deivered is p D d = α dp D d. (7) 3.5. Objective function The overa goa in isanding is to spit the network and eave it in a secure steady state whie maximizing the expected vaue of the oad suppied. Suppose a reward M d per unit is associated with the suppy of oad d. However if this oad is part of Section 0, then because this section is vunerabe, it is assumed there is a risk of not being abe to suppy power to that oad. Accordingy, a oad oss penaty 0 β d < 1 is defined, which may be interpreted as the probabiity of being abe to suppy a oad d if paced in section 0. If d is paced in Section 1, a reward M d is reaized per unit suppy, but if d is paced in Section 0 a ower reward of β d M d < M d is reaized. The expected vaue of the oad suppied is J DC : ( ) J DC = M d P d βd α 0d + α 1d, where, d D α d = α 0d + α 1d, d D, 0 α 0d 1, d D, (8a) (8b) 0 α 1d γ b, b B, d D b. (8c) 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 M d P d α d. On the other hand, if γ b = 1 then α 1d = α d and α 0d = 0, giving a higher reward M d P d α d. Thus maximizing J DC gives a preference for γ b = 1 and a smaer section 0. The DC optima isanding probem with objective of maximizing J DC usuay has mutipe feasibe soutions with objectives cose to the optima vaue. This fexibiity is expoited by introducing two penaty terms to the objective which are sma enough not to affect significanty the primary objective, but improve the computationa performance and provide the fexibiity to guide 9

11 the search towards soutions with good dynamic behaviour. The modified objective is to maximize J DC ε 1 W (1 ρ ) ε 2 W g (1 ζ g ) (9) L\L 0 g G where the W, ε 1, W g and ε 2 are non-negative weights. The vaue of J DC in the optima is denoted by J DC. The penaties discourage the disconnection of heathy ines and generators, i.e. they encourage the binary variabes ρ and ζ g to take the integer vaues 1 in the LP reaxations. This improves computationa efficiency by reducing the size of the branch and bound tree. A uniform weight, e.g., W = 1,, wi discourage equay a ine cuts, whie cuts to highfow ines may be more heaviy discouraged with W = s L0, where s L0 is the pre-isanding apparent power fow through the ine. Generation disconnection is uniformy penaized by setting W g equa to the generator s capacity P Gmax g Overa formuation The overa formuation for isanding is to maximize (9) subject to constraints (1) to (8). The resuting probem is a MILP. 4. Post-isanding AC optima oad shedding The soution of the DC isanding optimization incudes vaues for oads shed and generator rea outputs. In genera, however, because these vaues come from a inearized mode that ignores votage and reactive power, these vaues wi not be exacty optima or feasibe for the true AC probem. Therefore, to determine a good feasibe AC soution for the isanded network, an AC optima oad shedding (OLS) probem is soved after the isanding optimization using the isands and generator disconnections determined by the DC isanding. The AC-OLS optimization probem has the same form a standard OPF probem except that the objective is to maximize the expected vaue of oad suppied. The AC-OLS is soved for the network in its isanded state. That is, the set L is modified by removing ines for which ρ = 0. Furthermore, any generator for which ζ g = 0 has its upper and ower bounds on rea power set to zero; others are free to vary rea power output within a restricted region, as described previousy. 10

12 The probem is to maximize the tota vaue of rea power suppied to the oads: J AC = max R d α d P d, d D (10) subject to, f (x)=0, g(x) 0, (p G g, qg g ) O g, g G, (11a) (11b) (11c) (p D d, qd d )=α d(p D d, QD d ), d D. (11d) Here, R d is the reward for suppying oad d, and is equa to M d if the oad has been paced in Section 1 and β d M d if paced in Section 0. The equaity constraint (11a) captures Kirchhoff s current and votage aws in a compact form; x denotes the coection of bus votages, anges, and rea/reactive power injections across the isanded network. The inequaity constraint (11b) captures ine imits and bus votage imits. The set O g is the post-isanding region of operation for generator g, and depends on the soution of the isanding optimization and pre-isanded outputs of the generator. If ζ d = 1 the unit remains fuy operationa, and its output may vary within some region around the pre-isanded operating point; most generay (p G g, qg g ) O g( p G0 g, ) ( qg0 g, where p G0 ) is the pre-isanding operating point and O g is defined by the output capabiities of the generating unit. If rea and reactive power are independent, p G g [ Pg G, Pg G+ ] and q G g [ Q G g, Q G+ ] g. If, conversey, ζg = 0, then rea power output is set to zero: p G g = 0. In that case, the unit may remain eectricay connected to the network, with reactive power output free to vary within some specified interva [ Q G g, Q G+ ] g. g, qg0 g Each oad is assumed to be homogeneous, i.e. rea and reactive components are shed in equa proportions. The AC-OLS is a noninear programming (NLP) probem and may be soved efficienty by interior point methods. 11

13 Tabe 1: Pre- and post-isanding generator outputs for the 24-bus test exampe. Bus p G (MW) q G (MVAr) Pre Post Pre Post Computationa resuts This section presents computationa resuts using the above isanding formuation. First, a demonstration is given of the isanding approach on a 24-bus network. Foowing that, the construction of the further test probems is described, then computation times for different convergence criteria for the MILP isanding cacuation is given, and finay the accuracy of the DC soutions are assessed by comparing them with the AC soutions bus network case study The IEEE Reiabiity Test System [23] network comprises 38 ines and 24 buses, 17 of which have oads attached. Tota generation capacity is 3405 MW from 32 synchronous generators. The tota oad demand is 2850 MW. The isanding scenario is described as foows. With the network operating initiay at a state determined from an AC OPF, it is suspected that bus 9 has a faut, and it is decided to isand this bus to avoid further faiures; hence, bus 9 is assigned to B 0. It is assumed that β d = 0.75, d D. In obtaining a new steady-state soution for the isanded network, each generator is permitted to varying rea power output by up to 5% of its pre-isanding vaue, or switch off. In the objective, a unity reward, R d = 1, is assumed for each oad, and sma penaties are paced on ine cuts and generator disconnections (ε 1 = 0.001, W = 1, ε 2 = 0.01, W g = P Gmax g in (9)). Figure 2 shows the optima isanding soution, obtained by soving the DC MILP isanding probem. Tabe 1 shows the rea and reactive power outputs at each generator bus, both prior to, and after, isanding. A individua unit outputs are within imits. Tabe 2 shows the objective 12

14 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 7 Figure 2: Isanding of the 24-bus network. vaue the expected oad suppy and tota vaues of generation and oad for the DC isanding soution, and compares these vaues with those obtained from the post-isanding AC OLS. Buses 9, 11, 14, 16, 17 and 22 have been paced in section 0. No generators have been switched off. Of the origina 2850 MW demand, 469 MW has been paced in the risky section 0, and MW of oad has been shed (as determined by the AC OLS). The returned AC OLS soution is feasibe with respect to system ine fow imits and a votages are between 0.95 and 1.05 p.u. Note that isanding bus 9 aone woud have resuting in the oss of the entire 175 MW oad at that bus, pus further possibe osses in section 1 in order to baance the system. The optimized soution paces more buses and oads in section 0 than is stricty necessary, but aows baanced, feasibe isands to be obtained with minimum expected oad shed Further isanding test cases A set of isanding test cases was buit based on test networks with between 9 and 300 buses. For a network with n B buses, n B scenarios were generated by assigning in turn each singe bus to B 0. No assignments were made to B 1. The possibe post-isanding range of outputs for generator g when it is generating were defined 13

15 Tabe 2: Comparison of DC isanding and post-isanding AC OLS soutions for the 24-bus network. DC MILP AC OLS Objective (9) Penaties Exp. oad suppy, JDC or J AC (MW) Generation (MW) Load suppied (MW) Load shed (MW) as where P min g [ P G g, Pg G+ ] [ = P min, P max ] [ p G0 and P max g the pre-isanding generation and R G g g g g R G g, pg0 g + R G ] g, are the minimum and maximum steady-state imits when generating, p G0 g is the imit on change of output owing to ramp rate imits and/or generator reserve. For the 24-bus network, for which ramp rates are given, R G g maximum change over 2 minutes. For a other networks R G g pre-isanding generation eves are those obtained by soving an AC OPF. is is set to the is set to equa to 5% of p G0 g. The Where no ine imits are present for a network, a maximum phase ange difference of 0.4 rad is imposed for each ine. In the objective function, a vaue of 0.75 is used for the oad oss penaty β d, whie the vaues of ε 1 and ε 2 in (9) the penaties on ine cuts and generator disconnection respectivey are 0.1 and , with W = 1 and W g = P G+ g Computation times and optimaity 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. Especiay in the second case it is important to be abe to produce feasibe soutions within short time periods even if these are not necessariy optima. Resuts are therefore presented for a range of optimaity toerances: feasibe i.e. first integer feasibe soution found, and reative MIP gaps of 5%, 1% and 0%. 14

16 tcomp (s) Feasibe 5% MIP gap 1% MIP gap 0% MIP gap Figure 3: Mean, max and min times for finding, to different eves of optimaity, isanding soutions for different test networks. n B Probems were soved on a dua quad-core 64-bit Linux machine with 8 GiB RAM, using AMPL 11.0 with parae CPLEX 12.3 to sove MILP probems. Computation times quoted incude ony the time taken to sove the isanding optimization to the required eve of optimaity, and not the AC-OLS, and are obtained as the eapsed (wa) time used by CPLEX during the ÓÐÚ command. A time imit of 5000 seconds is imposed. Figure 3 shows the times required to find obtain feasibe isanding soutions to varying proven eves of optimaity. Minimum, mean and maximum times are obtained for each network by soving each of the n B scenarios once. The first set of times show that a probems are soved to feasibiity we within 1 s. In a cases, a feasibe soution was found at the root note, without requiring branching. For a MILP probem soved by branch and bound, the optima integer soution is bounded from beow (for maximization) by the highest integer objective vaue found so far during the soution process, and from above by the maximum objective of the reaxed soution among a eaf nodes of the tree. The reative MIP gap is the reative error between these two bounds. Figure 3 indicates the progress made by the CPLEX sover, in terms of the times required to reach reative MIP gaps of 5%, 1% and 0% (i.e., optimaity) respectivey. Performance is very promising for soving to 5%, with a probems soved to this toerance within five seconds. Times to 1% and 0% gaps are of the same order for the smaer networks (up to 39 buses), but the 57-, 118- and 300-bus networks can taken significanty onger to sove to these toerances. 15

17 Tabe 3: Reative errors (%) between optima and returned soutions. Feasibe 5% gap 1% gap Min Mean Max Tabe 3 shows the means of the reative errors between the soution vaue returned at termination of the sover and the actua optimum, where this has been obtained by soving the probem to fu optimaity (0%). The actua gaps between eary termination soutions and the true optima are nearer zero than the 5% or 1% bounds. Therefore, good isanding soutions with respect to the DC mode can be provided even when the sover is terminated eary and moreover these soutions can be found quicky. Moreover, because the DC mode is an approximation of the AC mode, there is itte advantage in soving it to proven optimaity Feasibiity and accuracy of DC isanding soution The post-isanding AC-OLS showed that some of the isanding soutions were AC infeasibe, i.e., there was no soution to the AC-OLS ying within norma votage bounds. Reaxing the norma votage bounds by an extra 0.06 p.u. gave a soution in a cases; however, this is not aways possibe for practica networks. It was noted that in many of these AC-infeasibe cases, there was sufficient goba reactive power capacity in each isand. However, reactive power and votage is a oca probem, and hence achieving a goba reactive power baance is not sufficient to ensure a norma votage profie. This is an issue that is overooked by most controed isanding schemes, and instead it is assumed that reactive power can be compensated ocay. This is not aways a justifiabe assumption, however, and further research is needed on methods for obtaining, directy, isands with a heathy votage profie. Tabe 4 gives the number of these AC-infeasibe cases as we as the number that did not sove to 0% optimaity gap within 5000 seconds. For a the remaining cases the differences between the objectives as predicted by the DC isanding optimization and the actua vaue from post-isanding 16

18 Tabe 4: Number of unique probems incuded in the comparisons. n B MIP gap > 0% Votage infeasibe Cases compared (%) J DC J AC J AC n B Figure 4: Mean, max and min reative errors between DC and AC objective vaues. AC-OLS was cacuated as is shown in Figure 4. The adopted isanding soution in each case is that from soving the probem to fu optimaity. The comparison in Figure 4 shows how we the DC mode predicts the AC objective. There are a few cases where the DC objective is a significant over-estimate, however on average the objective vaues are within 0.3%. 6. Dynamic stabiity Soving the MILP isanding probem and, subsequenty, the AC-OLS provides a feasibe steadystate operating point for the network in its post-isanding configuration. Since the objective minimizes the oad shed and the constraints imit the changes to generator outputs, the proposed soution wi naturay imit, to some extent, the disruption to the system power fows. Nevertheess, since neither the transient response is modeed when designing isands nor the generators are necessariy grouped according to coherent modes, it is possibe that the isanding actions may ead 17

19 Tabe 5: Resuts of time-domain simuations with origina ine-cut and generator penaties. n B Cases compared Unstabe Stabe to dynamic instabiity. This section therefore describes the use of a time-domain simuation to investigate this issue. Resuts are presented for the previous 9- to 300-bus isanding cases to test whether or not the act of isanding induces dynamic instabiity. Time-domain simuations for a of the isanding scenarios described in Sec. 5.3 were performed using PSAT [24]. Second-order non-inear modes of synchronous machine dynamics were used with machine parameters taken from [25], with a damping coefficient D = 0.5. The oads were assume to have no dynamics. In practice the damper windings and the contro systems (turbine governor, AVR) woud act to dampen the osciations more than in the simuation making the rea system more stabe than in the simuations. Each simuation is started from an undisturbed pre-isanding operating point, at which a generators have an anguar frequency of 1 p.u. and the network is baanced. The resuts are given in Tabe 5. In tota, 14 out of 452 isanding soutions were found to ead to instabiity. Investigation of the individua cases found that, in a cases, severe transients were caused by cuts to high-fow ines. However, re-soving the isanding optimizations with increased penaties on high-fow ines (W (W g = P G+ g and ε 2 = 1) resuted in a cases being stabe. = s L0 and ε 1 = 10 4 d P D d ) and switching-off of generators As expected these arger penaty coefficients caused a drop in the primary DC objective vaue, J DC. However Tabe 6 shows that this degradation is sma, and is an acceptabe trade-off for removing a of the unstabe cases. As an added advantage, as Figure 5 shows, sove times for the arger networks are shorter with the heavier penaties. 18

20 Tabe 6: Decrease in objective J DC (%) for ine-cut and generator penaties increased versus previous penaties. n B Min Mean Max origina penaties new penaties tcomp (s) Figure 5: Mean, max and min sove times to optimaity with origina penaties (ε 1 = 0.1, W = 1, ε 2 = 10 4, W g = Pg G+ ) and new penaties (ε 1 = 10 4 d P D d, W = s L0, ε 2 = 1, W g = Pg G+ ). n B 7. Concusions In this paper, an optimization-based approach to controed isanding and intentiona oad shedding has been presented. The proposed method uses MILP to determine which ines to cut, oads to shed, and generators to switch or adjust in order to isoate an uncertain or faiure-prone region of the network. The optimization framework aows inear network constraints a oss-modified DC power fow mode, ine imits, generator outputs to be expicity incuded in decision making, and produces baanced, steady-state feasibe DC isands. AC isanding soutions are found via the subsequent soving of an AC optima oad shedding probem. The dynamic stabiity of resuting isanding soutions is assessed via time-domain simuation. The approach has been demonstrated through exampes on a range of test networks, and the practicaity of the method in terms of computation time has been demonstrated. Good feasibe 19

21 isanding soutions can be found very quicky. Whie the dynamic response is not expicity modeed in the optimizations, time-domain simuations of the isanding soutions have indicated that instabiity is avoided by appropriate choices of penaties on cuts to high-fow ines and disconnections of generating units, both of which discourage disruption to the network. Furthermore, it was shown that these penaties had a sma effect on the amount of oad required to be shed. This paper has aso served to raise the issue of ensuring adequate reactive power and a heathy votage profie after controed isanding, and issue overooked by most approaches, which instead assume that reactive power may be compensated ocay. Further research is needed in this area. Future research wi investigate the incusion of constraints for dynamic stabiity in the probem, the generaization of the method to partitioning the system foowing sow coherency anaysis, and the modeing of reactive power in the optimization to ensure a heathy votage profie. Acknowedgements The authors woud ike to thank Professor Janusz Biaek and Dr Patrick McNabb of the Schoo of Engineering and Computing Sciences, Durham University for hepfu discussions on power system dynamics. References [1] Fina Report of the Investigation Committee on the 28 September 2003 Backout in Itay, Fina Report, Union for the Coordination of the Transmission of Eectricity (UCTE), [2] S. Larsson, A. Dane, The back-out in southern Sweden and eastern Denmark, September 23, 2003, in: IEEE Power Systems Conference and Exposition, [3] U.S.-Canada Power System Outage Task Force, Fina Report on the August 14, 2003 Backout in the United States and Canada: Causes and Recommendations, Fina Report, URL ØØÔ»»Ö ÔÓÖØ º Ò Ö Ýº ÓÚ», [4] Fina Report System Disturbance on 4 November 2006, Fina Report, Union for the Coordination of the Transmission of Eectricity (UCTE), [5] Report of the Nationa Grid Investigation into the Frequency Deviation and Automatic Demand Disconnection that occurred on the 27th May 2008, Fina Report, Nationa Grid, [6] J. W. Biaek, Are backouts contagious?, IEE Power Engineer 17 (6) (2003)

22 [7] J. W. Biaek, Backouts in the US/Canada and continenta Europe in 2003: is iberaisation to bame?, in: IEEE PowerTech Conference, Russia, [8] G. Andersson, P. Donaek, R. Farmer, N. Hatziargyriou, I. Kamwa, P. Kundur, N. Martins, J. Paserba, P. Pourbeik, J. Sanchez-Gasca, R. Schuz, A. Stankovic, C. Tayor, V. Vitta, Causes of the 2003 Major Grid Backouts in North America and Europe, and Recommended Means to Improve System Dynamic Performance, IEEE Transactions on Power Systems 20 (4) (2005) [9] D. E. Newman, B. A. Carreras, M. Kirchner, I. Dobson, The Impact of Distributed Generation on Power Transmission Grid Dynamics, in: 44th Hawaii Internationa Conference on System Science, [10] B. Yang, V. Vitta, G. T. Heydt, Sow-Coherency-based Controed Isanding A Demonstration of the Approach on the August 14, 2003 Backout Scenario, IEEE Transactions on Power Systems 21 (2006) [11] K. Sun, D.-Z. Zheng, Q. Lu, Spitting Strategies for Isanding Operation of Large-Scae Power Systems Using OBDD-Based Methods, IEEE Transactions on Power Systems 18 (2003) [12] H. You, V. Vitta, X. Wang, Sow Coherency-Based Isanding, IEEE Transactions on Power Systems 19 (1) (2004) [13] X. Wang, V. Vitta, System Isanding using Minima Cutsets with Minimum Net Fow, in: IEEE Power Systems Conference and Exposition, [14] G. Xu, V. Vitta, Sow Coherency Based Cutset Determination Agorithm for Large Power Systems, IEEE Transactions on Power Systems 25 (2) (2010) [15] L. Liu, W. Liu, D. A. Cartes, I.-Y. Chung, Sow coherency and Ange Moduated Partice Swarm Optimization based isanding of arge-scae power systems, Advanced Engineering Informatics 23 (1) (2009) [16] M. R. Aghamohammadi, A. Shahmohammadi, Intentiona isanding using a new agorithm based on ant search mechanism, Internationa Journa of Eectrica Power and Energy Systems 35 (2012) [17] M. Jin, T. S. Sidhu, K. Sun, A New System Spitting Scheme Based on the Unified Stabiity Contro Framework, IEEE Transactions on Power Systems 22 (1) (2007) [18] C. G. Wang, B. H. Zhang, Z. G. Hao, J. Shu, P. Li, Z. Q. Bo, A Nove Rea-Time Searching Method for Power System Spitting Boundary, IEEE Transactions on Power Systems 25 (4) (2010) [19] D. Bienstock, S. Mattia, Using mixed-integer programming to sove power grid backout probems, Discrete Optimization 4 (2007) [20] E. B. Fisher, R. P. O Nei, M. C. Ferris, Optima Transmission Switching, IEEE Transactions on Power Systems 23 (3) (2008) [21] A. Khodaei, M. Shahidehpour, Security-constrained transmission switching with votage constraints, Internationa Journa of Eectrica Power and Energy Systems 35 (2012) [22] P. A. Trodden, W. A. Bukhsh, A. Grothey, K. I. M. McKinnon, MILP formuation for isanding of power networks, Tech. Rep. ERGO , Edinburgh Research Group in Optimization, Schoo of Mathematics, Univer- 21

23 sity of Edinburgh, [23] Reiabiity Test System Task Force of the Appication of Probabiity Methods Subcommittee, IEEE Reiabiity Test System, IEEE Transactions on Power Apparatus and Systems PAS-98 (6) (1979) [24] F. Miano, Documentation for PSAT version 2.0.0, University of Castia La Mancha, [25] Reiabiity Test System Task Force of the Appication of Probabiity Methods Subcommittee, IEEE Reiabiity Test System 1996, IEEE Transactions on Power Systems 14 (3) (1999)