Optimal islanding and load shedding

Transcription

1 Optima isanding and oad shedding Pau Trodden, Waqquas Bukhsh, Andreas Grothey, Jacek Gondzio, Ken McKinnon March 11, 2011 Contents 1 Introduction 3 2 Optima oad shedding AC OLS probem DC OLS probem Cutting ines without uncertainty Optima isanding and oad shedding Motivation and assumptions DC IP isanding formuation DC isanding of IEEE 14-bus system DC IP isanding of network AC OLS on isanded network Comparison over different isanded networks DC isanding of IEEE 24-bus RTS DC IP isanding with β = Effect of varying oad-suppy probabiity, β d Larger systems: computationa resuts 29 7 Extensions to the IP formuation Loss modeing Generator switching Feasibiity probems 35 1

2 Notation Notation conventions Upper case is used for constants, (parameters in upright text and sets in caigraphic text), and ower case for variabes. Indices are ower case and aways subscripts. Superscripts are part of the name. Sets B set of buses, indexed by b B 0 subset of buses of uncertain status (and preassigned to section 0) B 1 subset of buses preassigned to section 1 G set of generators, indexed by g G b set of generators attached to bus b S = {0, 1}, set of sections L set of ines, indexed by L 0 subset of ines of uncertain status D set of oads, indexed by d set of oads attached to bus b D b Parameters A ine-bus matrix; ine goes from b = A,1 to b = A,2 Pg G rea power output from generator g Pd D,QD d rea and reactive power demands at oad d (at nomina votage) G L,BL conductance and susceptance of ine G B b,bb b shunt conductance and susceptance at bus b P L,max imit on rea power fow in ine S L,max imit on apparent power fow in ine Vb max,vb min max and min votages at bus b O g set of possibe vaues for (p G g,qg G ) β d probabiity of osing oad d if it is part of section 0 M d reward per unit of deivered rea power at oad d maximum difference in phase ange across a connected ine + maximum difference in phase ange across a disconnected ine Variabes p G g,qg g p D d,qd d v b δ b p L,fr,q L,fr p L,to,q L,to rea and reactive power output of generator g after change rea and reactive power absorbed by demand d votage at bus b votage phase ange at bus b rea and reactive power fows into ine from bus b = A,1 rea and reactive power fows into ine from bus b = A,2 2

3 ˆp L, ˆqL rea and reactive power that woud fow into ine were the ine connected α d proportion of oad d suppied after shedding ρ 0 1 switch to disconnect ine ; ine is disconnected iff ρ = 0 γ b section (0 or 1) that bus b ies in 1 Introduction This document forms the foow-up report for the Edinburgh presentation at the Backout project penary meeting in Durham, 19 January The organization of the report is as foows. The next section describes an optima oad shedding formuation (with both AC and DC variants) and appies it to a 14-bus exampe system under abnorma operation. It is demonstrated that cutting ines, without creating isands, can ead to ess oad shedding. In Section 3, the motivations for isanding are described and the IP formuation is presented. The isanding optimization is then appied to a number of test networks. In Section 4, the 14-bus network is isanded and the resuts compared with resuts from an AC mode appied to the isanded network. A comparison with AC resuts over different isands shows agreement between AC and DC, but aso indicates a need for modeing of osses. Secondy, in Section 5, the 24-bus IEEE Reiabiity Test System is studied, and the effect of varying an isanding optimization parameter is investigated. Finay, the isanding optimization is appied to arger networks in Section 6, chiefy to obtain a measure of computationa scaing. In the fina two sections, modifications to the isanding formuation are presented and current difficuties with the method are discussed. For the former, modifications are made to incude both oss modeing in the DC formuation and the on/off switching of generating units (as opposed to continuous variation of outputs). In the atter section, the probem of subsequent AC feasibiity in a DC-isanded network is described, and ideas to sove (or ameiorate) the probem are presented. 2 Optima oad shedding The optima oad shedding mode assumes the form of an optima power fow optimization probem, but permits a proportion of the oad at any bus to be shed. The next two subsections briefy describes the AC and DC OLS formuations, before a 14-bus network exampe is used to show that cutting ines in some cases, even without faiures and uncertainty, can maximize suppied oad. 2.1 AC OLS probem The objective is to maximize the suppy of rea power to oads: max d D M d α d P D d (1) 3

4 subject to, Kirchhoff s current aw (KCL) for conservation of fow at each bus. For a b B : p G g = p D d + p L,fr p L,to + G B b v2 b, (2a) g G b d D b L:A,1 =b L:A,2 =b qg G = qd D + q L,fr q L,to Bb B v2 b, (2b) g G b d D b L:A,1 =b L:A,2 =b Kirchhoff s votage aw (KVL) across each ine L. If b = A,1 is the from end bus and b = A,2 is the to end bus, [ ] p L,fr = G 11 vb 2 + v bv b G 12 cos(δ b δ b ) + B 12 sin(δ b δ b ), [ ] (3a) p L,to = G 22 vb 2 + v b v b G 21 cos(δ b δ b ) + B 21 sin(δ b δ b ), [ ] (3b) q L,fr = B 11 vb 2 + v bv b G 12 sin(δ b δ b ) B 12 cos(δ b δ b ), [ ] (3c) q L,to = B 22 vb 2 + v b v b G 21 sin(δ b δ b ) B 21 cos(δ b δ b ). (3d) are the rea and imagi- The parameters G 11,G 12,G 21,G 22 and B 11 nary eements of the admittance matrix of ine : [ Y 11 Y = Y 12 ] [ G 11 Y 21 Y 22 = G 12 G 21 G 22,B 12,B 21,B 22 ] + j [ B 11 B 12 B 21 B 22 In most cases, when the ine is not a transformer with off-nomina turns ratio and/or contains no ine charging capacitance, G 11 = G 22 = G 12 = G 21 = G L B 11 = B 22 = B 12 = B 21 = B L, and the more standard form of KVL is recovered. We use this formuation to retain generaity; aso, in each of the networks simuated in this report a number of ines exist with τ 1 and/or non-zero ine charging capacitance, in which case G 11 G L,B11 B L, etc. Load shedding constraints. We assume that oad d D may be reduced by disconnecting a proportion 1 α d and that the ratio of rea to reactive oad stays the same. p D d = α dp D d, q D d = α dq D d, ]. (4a) (4b) 0 α d 1. (4c) (Currenty, we assume that oads are constant power, though it is possibe to use other modes in the formuation). 4

5 Generation constraints: the rea and reactive power outputs of the generator g G ie in some feasibe set of operation (see, for exampe, [1, Figure 3.19, p93]). ( p G g,qg G ) Og. (5) Line imits. The formuation permits a number of different ways in which to impose imits on ine capacity, (depending on what is specified in the network data). For exampe, MVA ratings on apparent power for each ine L : ( p L,fr ( p L,to ) 2 + ( q L,fr ) 2 + ( q L,to ) 2 ( S L,max ) 2, (6a) ) 2 ( S L,max ) 2. (6b) Note that this effectivey corresponds to a heating constraint, appied at each end of the ine. Aternativey, the data may provide a P L,max and require imits on rea power fow. Or, phase ange imits Votage imits at each bus b B: δ A,1 δ A,2. V min b v b V max b. (7) Reference bus constraint: δ b0 = 0. (8) The AC OLS is aways a noninear probem, owing to the non-inearity of the KVL constraints. 2.2 DC OLS probem The DC OLS probem makes use of the standard DC fow mode assumptions to produce a formuation with rea power ony and no ine osses. The probem is as foows. max d D M d α d P D d (9) subject to KCL for a b B: g G b p G g = d D b p D d + L:A,1 =b p L L:A,2 =b p L, (10) 5

6 KVL for a L. If b = A,1 is the to end bus and b = A,2 is the from end bus, p L = BL ( ) δb δ b, (11) τ Note that we incude the off-nomina turns ratio τ to permit the genera case of a ine being a transformer; usuay, τ = 1. Load shedding constraints for a d D: p D d = α dp D d, (12a) 0 α d 1. (12b) Generation constraints for a g G: P G,min g p G g P G,max g. (13) Usuay we sha assume that generation can be decreased but not increased, given the ong times required to do so. In that case, P G,max g = P G g. Line imits, which may ony be expressed either was MW ratings on rea power for each ine L : p L P L,max (14) Or, phase ange imits: P L,max δ b δ b, L Reference bus constraint: δ b0 = 0. (15) The main advantage of the DC OLS is its inearity; the KVL expression is inear, and so the resuting probem is an LP. 2.3 Cutting ines without uncertainty The foowing numerica simuations use the IEEE 14-bus test system (see [2]), which is shown in Figure 1. Generator outputs at buses 1 and 2 are as foows. 0 p G MW, 0 p G MW, 150 q1 G 150 MVAr, 150 q2 G 150 MVAr, 6

7 Figure 1: IEEE 14-bus test system. whie the condensers at buses 3, 6 and 8 have zero rea power output capacity, and reactive power output in the range 0 to +140 MVAr. Tota rea power demand in the network (at nomina votage) is 259 MW. This mirrors the settings in [3]. Line imits are not specified as standard for this system, and so a 100 MVA imit is imposed on each ine. Phase ange imits are aso not specified, but wi be deat with separatey in the exampes that foow. To simuate an abnorma operating condition, the generator at bus 2 is deeted from the system (as was done in [3]). Subsequenty, the revised generation constraints are 0 p G 1 200, 0 p G 2 0. Rea power demand now exceeds suppy by 59 MW. No buses are marked as uncertain; it is assumed that knowedge of the network state is accurate. The idea in this simuation is not to isand an unheathy part of the network, but to shed oads to obtain an optima operating point in the abnorma conditions. 7

8 (a) No ine cuts (b) Lines (2,3) and (2,5) cut Figure 2: IEEE 14-bus test system under abnorma conditions. OLS soution with no ine cuts Tabes 1 and 2 shows the bus and ine data for the AC OLS and DC OLS respectivey. The key points are summarized as foows. Tota oad suppied is MW (AC) and MW (DC). The oad at bus 2 has been fuy shed in both cases, whie the AC OLS sheds 94.9% of the oad at bus 3 compared with 87.8% for DC OLS. Tota rea power generation is MW (AC) and MW (DC). In neither case is the soe remaining generator operating at capacity (200 MW), Line (1,2) is operating at capacity. A others are within imits. Bus and ine vaues for AC and DC are of the same orders of magnitude and if the same signs; sma differences add up to a tota of 5.9 MW in osses for AC. It is not immediatey apparent from the data as to why the singe generator is unabe to operate nearer to capacity, and so reduce the oad shed. Sensitivity anaysis shows that by far the highest shadow price is on the MVA imit constraint for ine (1,2), indicating an extra 63.9 MW of oad coud be suppied for a 1 p.u. (100 MVA) increase in capacity. OLS soution with ine cuts Tabes 3 and 4 show the resuts from the AC and DC OLS formuations when ines (2,3) and (2,5) are cut prior to soving the optimization probems. 8

9 b v b δ b (deg) p G g q G g p D d P D d q D d Q D d α d A,1 A,2 p L,fr (a) Bus data p L,to q L,fr q L,to MVA imit δ (deg) (b) Line data Tabe 1: AC OLS on 14-bus network under abnorma operation with no ine cuts. Units of rea and reactive power are MW and MVAr respectivey; votages are per unit. 9

10 b δ b (deg) p G g p D d P D d α d (a) Bus data A,1 A,2 p L P L,max δ (deg) (b) Line data Tabe 2: DC OLS of 14-bus system under abnorma conditions with no ine cuts. Units of rea power are MW. 10

11 Having made these ine cuts, the generator at bus 1 is now operating at or near to maximum capacity, providing 200 MW (DC) and MW (AC) to the rest of the network. For AC, the same buses (2 and 3) as before have shed or part-shed their oads; bus 2 has fuy shed but as a consequence of the increased generation bus 3 has shed ony 56.2% compared with 94.9% previousy. The situation for DC at first appears to be rather different: bus 2 has fuy shed, whie, of the rest, the ony oad that has not shed is at bus 3 a other buses have part-shed between 3.6% (bus 5) and 99.6% (bus 2). However, it shoud be noted that the DC is amost indifferent to which oads are shed, since (i) no osses are modeed (ii) reactive power and votage is negected, and (iii) a rewards per unit suppy, M d in the objective, are unity. The former point indicates that some oss modeing is required if the IP isanding formuation is to shed the correct oads. The second point has impications for how ine imits are modeed: if a 100 MVA imit is assumed to be equivaent to a 100 MW imit for DC, then DC wi be abe to squeeze more rea power down a ine. Despite this, what is important here is the overa oad suppy: in fact, by cutting ines (2,3) and (2,5) an extra 36 MW of oad has been suppied for AC and an additiona 46 for DC. Disconnecting ines (2, 3) and (2, 5) has increased the rea power carried over ines (1, 5) and (2, 4) to near-capacity eves. Previousy these ines carried 54.6 and 41.6 MW respectivey. Meanwhie, ine (1, 2) remains at capacity. The net effect is a higher fow of rea power from the buses 1 and 2 to the rest of the network. This better soution is not avaiabe without the ine cuts in pace; with the ines present, any phase ange differences between bus 2 and buses 3 and 5 impies a non-zero fow of power. Viewed differenty, with a ines intact the network is unabe to estabish the conditions required to move enough generated power from buses 1 and 2 to the rest of the buses. 11

12 b v b δ b (deg) p G g q G g p D d P D d q D d Q D d α d A,1 A,2 p L,fr p L,to (a) Bus data q L,fr q L,to MVA imit δ (deg) (b) Line data Tabe 3: AC OLS of 14-bus system under abnorma conditions with cuts to ines (2, 3) and (2,5). Units of rea and reactive power are MW and MVAr respectivey; votages are per unit. 12

13 b δ b (deg) p G g p D d P D d α d (a) Bus data A,1 A,2 p L P L,max δ (deg) (b) Line data Tabe 4: DC OLS of 14-bus system under abnorma conditions with cuts to ines (2,3) and (2.5). Units of rea power are MW. 13

14 3 Optima isanding and oad shedding The previous section showed that it is possibe to reduce the oad shed by aowing ines to be cut, though it is uncommon for the best soution to spit the network into isands. However, so far we have assumed perfect knowedge of the post-faut state of the network. More reaisticay, it may be known that there is a probem in some part of the network, but the form and extent of the probem is not known. In such a case, a robust soution to prevent cascading faiures woud be to isoate the uncertain part of the network from the certain part, by forming one or more isands. In this section, we present a DC Integer Programming (IP) formuation for minimizing the oad shed in an eectricity network under stress. The goa is to maximise the expected oad that remains connected. Initiay we wi formuate a singe stage probem. 3.1 Motivation and assumptions Foowing some faiure in the network, we wish to disconnect ines, vary oads and estabish stabe isands to avoid cascading faiures and eventua backout. We assume that, post-faut, there are parts of the network that are suspected of having a faut and some where we are reasonaby sure have no fauts. Figure 3 depicts such a situation for a fictiona network.???? (a) Network prior to isanding Section 1 Section 0 Section 1???? Isand 1 Isand 2 (b) Network post isanding Isand 3 Isand 4 Figure 3: (a) Fictiona network with uncertain buses and ines, and (b) the isanding of that network by disconnecting ines. Our aim is to spit the network into disconnected sections so that the possibe fauts are a in one section. It is desirabe that this section be sma, since it may be prone 14

15 to faiure, and that the other section is abe to operate with itte oad shedding. We woud aso ike the probem section to shed as itte oad as possibe. Figure 3 therefore aso shows a possibe isanding soution for this network, where a of the uncertain buses and ines have been paced in a section 0. At this point we make the foowing distinction between sections and isands. The optimized network sha consist of two sections, an unheathy section 0 and a heathy section 1. No ines sha connect the two sections. On the other hand, neither section is required to be a singe, connected component. An isand describes a connected component of the network. Thus, either section may contain a number of isands as is exempified by in Figure 3, where section 1 comprises isands 1, 3 and 4. Section 0 is a singe isand, but is not aways necessariy so. We wi assume that the state of the system immediatey after the initia faut is known, and that the state at the end of the cacuation can be quicky and accuratey predicted. We have centra contro of generation, oad shedding and ine breakers; we assume that we can instantaneousy and simutaneousy reduce the demand, reduce rea power generation eves 1, vary the reactive power generation eves within bounds, and aso disconnect ines. It is assumed that such instantaneous variations do not cause transient instabiity. We require that after the adjustments the system is in a feasibe stabe steady state. If it is optima to do so, the mode wi spit the network into isands but without further constraints it wi not necessariy do this. 3.2 DC IP isanding formuation The formuation is obtained by adding sectioning constraints to the DC OLS probem. The resuting probem is a Mixed Integer Linear Probem (MILP), describing a ossess p δ system with votage-independent oads. It is, of course, possibe to deveop an equivaent formuation for the AC mode, but this resuts in a MINLP, which is difficut to sove. (The basic graph partitioning formuation is a standard one, but inking the binary ine variabes to the eectricity network variabes appears to be new.) Sectioning constraints We define a set S = {0,1} that numbers the sections of the network: section 0 sha be the unheathy section, whie section 1 sha be heathy. We suspect that some subset B 0 of buses and some subset L 0 of ines have a possibe faut; it is these we wish to confine to section 0. 1 Note that in more recent work and foowing feedback from the January meeting in Durham we have assumed that rea power generation cannot be instantaneousy varied; a generator s mechanica input may be hed at its current eve or removed competey, whie the machine remains eectricay connected to the network and abe to suppy reactive power. This modification is described in Section 7. For the resuts presented here, however, we assume that rea power can be varied. 15

16 We introduce a binary decision variabe γ b with each bus b B; γ b sha be set equa to 0 iff b is paced in section 0 and γ b = 1 otherwise. To partition the network in such a way, we need to disconnect ines. Accordingy, we define a binary decision variabe ρ for each L; ρ = 0 iff ine is disconnected and ρ = 1 otherwise. The first pair of constraints operates on each ine not pre-assigned to L 0. The vaue of ρ for the ine is zero and the ine is cut if the two end buses are in different sections (i.e. γ A,1 = 0 and γ A,2 = 1, or γ A,1 = 1 and γ A,2 = 0). Otherwise, if the two end buses are in the same section, be it section 0 or 1, ρ 1; that is, 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 A,1 γ A,2, L \ L 0, ρ 1 γ A,1 + γ A,2, L \ L 0. (16a) (16b) The second pair of constraints examines ines pre-assigned to L 0 and sets ρ = 0 disconnects the ine if either or both of the ends are in heathy section 1. Thus, an uncertain ine (i) sha be disconnected if entirey in section 1, (ii) sha be disconnected if between sections 0 and 1, (iii) may remain connected if entirey in section 0. ρ 1 γ A,1, L 0, ρ 1 γ A,2, L 0, (16c) (16d) The fina pair of constraints simpy sets the vaue of γ b for a bus b depending on what set that bus was pre-assigned to. So if b B 0 then γ b = 0. In addition, we define a set B 1, to which any buses that are desired to remain in section 1 may be assigned; if b B 1 then γ b = 1. γ b = 0, b B 0, γ b = 1, b B 1. (16e) (16f) This competes the description of the sectioning constraints. The IP optimization wi disconnect ines and pace buses in section 0 or 1, as directed by these constraints, depending on the pre-assignments to B 0, B 1 and L 1. What ese is paced in section 0 and what other ines are cut are degrees of freedom for the optimization, and wi depend on the objective function. Objective function The overa objective of isanding is to minimize the risk of system faiure. In our motivation we assumed that there is some uncertainty associated with a particuar subset of buses and/or ines; we suspect there may be a faut and so we wish to isoate these components from the rest of the network. Owing to uncertainty, we assume that there is a chance that section 0 wi either back out or ose a proportion of its oad beyond what we expect from our pan. We 16

17 therefore wish to reduce the vaue attributed to oad suppied to section 0 in our mode. In pacing any oad in section 0, we sha assume a owered probabiity of being abe to suppy power to that oad, since the probabiity of that section faiing is higher. In particuar, we assume that we have a probabiity of 1 of being abe to suppy a oad d paced in section 1, but a probabiity of ony β d 1 of being abe to suppy the same oad if paced in section 0 with the uncertain components. We wish to maximize the vaue of suppied demand: max d D M d P d ( βd α 0d + α 1d ), (17) and α d = α 0d + α 1d, d D, (18a) 0 α 0d 1, d D, (18b) 0 α 1d γ b, b B,d D b. (18c) Here we have introduced a new variabe α sd for the oad d deivered in section s S. The above constraints permit it to be non-zero in ony one section. The interpretation of this is that a oad d at a bus b wi be worth β d M d α d P d if b is in section 0, or M d α d P d if b is in section 1. Thus the objective has a preference for a smaer section 0. DC fow-phase ange reations in ines When a ine is connected, a fow of rea power is estabished depending on the differences in phase ange at each end of the ine. However, if ρ = 0 and a ine is disconnected, we must sti permit a difference in phase at each end of the ine, ony with zero fow through the ine. To achieve this, we repace rea power p L in the DC KVL constraint (11) with ˆp L. ˆp L = BL τ ( δa,1 δ A,2 ). (19) Then, when ine is connected we wi set p L = ˆp L, and when is disconnected pl = 0. We mode this as foows. Assume the minimum and maximum possibe vaues of ˆp L L,min L,max are ˆP, ˆP respectivey, and of p L are P L,min, P L,max2. Then, ˆp L = λ,1 (P L,min ) + λ,2 ( L,min ˆP ) + λ,3 P L,max L,max + λ,4 ˆP, (20a) p L = λ,1 P L,min + λ,3 P L,max, (20b) ρ = λ,1 + λ,3, (20c) 1 ρ = λ,2 + λ,4, (20d) λ,i 0, i {1,2,3,4}. 2 In practice, P L,min, P L,max may be set according to ine imits, i.e. P L,min L,max other hand,, ˆP shoud be sufficienty arge to not constrain the soution. ˆP L,min = P L,max (20e). On the 17

18 When the sectioning constraints set a particuar ρ = 0, then λ,1 = λ,2 = 0 and so p L = 0. However, because λ,2 + λ,4 = 1, ˆp L may take whatever vaue necessary to satisfy the KVL constraint (19). Phase ange constraints If phase ange constraints are present in the DC OLS formuation, they must be modified to take into account ine disconnections. If a ine has been disconnected, the formuation shoud reax the phase ange difference restriction between the two end buses. For a L : ( + + ( + )ρ ) δa,1 δ A,2 + + ( + )ρ, (21) Overa formuation The overa IP formuation for isanding is maximize the objective (17) subject to sectioning constraints (16); KCL (10); KVL (19); fow-phase ange ine constraints (20); oad mode and shedding, (12) and (18); generation imits (13), with P G,max g = P G g ; ine imits: rea power (14) or phase ange (21). The reference bus ange constraint (δ b1 = 0) is not necessary; in fact, since the optimization may create two or more isands a constraint may be required for each isand created, which is not straightforward to impement. If we do desire to remove the redundancy in absoute vaues of phase ange, most ikey for computationa reasons, we can add a sma penaty to the objective. 4 DC isanding of IEEE 14-bus system The 14-bus network is initiay operating under nomina conditions; a oads are fuy suppied (259 MW in tota), with outputs of MW and MW for the generators at buses 1 and 2 respectivey. Bus 2 is marked uncertain and assigned to set B 0. The probabiity of being abe to suppy any oad paced in Section 0 is β d = 0.5, d D. No ines are uncertain, and no buses are pre-assigned to Section 1. 18

19 Constraint imits for the DC IP optimization are set as foows. The generation constraints assume that rea power output can ony be decreased from the current operating point. 0 p G 1 0 p G MW, MW. This sti, however, assumes a continuous decrease is possibe instantaneousy, which is not reaistic; this assumption is tightened in Section 7. In the absence of rea data, ine imits are again set to 100 MVA for a ines. Because the DC mode does not incude reactive power, we assume a ine imit of 100 MW. Phase ange differences across ines are imited to 15 degrees. Linearization errors in the DC fow mode increase rapidy from this point onwards: the error in cos x 1 is 3.5% at x = 15 degrees. 4.1 DC IP isanding of network The resuts of the isanding optimization appied to the network are shown in Tabe 5, and the isanded network is depicted in Figure 4(a). Lines 1, 5, 7, 9 and 15 have been cut, isanding buses 2, 3, 4, 7 and 8 the ower-right of the network into Section 0. Consequenty, the generator at bus 1 serves the heathy upper-eft part of the network, whie the generator at 2 serves the unheathy part. Rea power deivery to oads is 100 % at a oad buses except for buses 2 and 3, where 0% and 84.8 % of the respective demanded eves are suppied. A tota of MW of rea power is generated. The generator at bus 1 has been re-schedued to output 95.3 MW enough to fuy-suppy each oad in its section whie the output of the generator at bus 2 is unchanged at MW. Despite operating at maximum output, this generator is unabe to suppy a oad within its section. The expected tota oad suppied using the probabiity β d for those oads in Section 0 is MW. No ine is at its imit, though ine (1,5) carries 95.3 MW against an assumed imit of 100 MW. Simiary, no phase ange imits are active. 4.2 AC OLS on isanded network An AC OLS was soved for the isanded network in Figure 4. The objective function is again to maximize the oad suppied, but now incudes the probabiity measure for the any buses paced in section 0. The rea power generation imits are the same as for the 19

20 b δ b (deg) p G g p D d P D d α d γ b (a) Bus data A,1 A,2 p L P L,max δ (deg) ρ (b) Line data Tabe 5: DC isanding of 14-bus system in response to uncertainty at bus 2. Units of rea power are in MW. 20

21 DC isanding optimization. Thus, because the AC system modes osses, ess power wi be avaiabe to suppy oads and hence a greater proportion of oad wi be shed. Tabe 6 shows the resuts. The saient points are summarized as foows. 41% of the oad at bus 2 (the uncertain bus) is shed, but no shedding of other oads occurs in the unheathy part of the network. Conversey, the DC optimization predicted that the oad at bus 2 woud be fuy shed, whie the oad at bus 3 woud be suppied at 85 %. The generator at bus 1 is operating at 99.7 MW compared with the 95.3 MW predicted by the DC optimization. However, whie DC predicted that 95.3 MW woud be sufficient to fuy suppy oads in Section 1, for AC the 100 MVA imit on ine (1, 5) acts as a botteneck for Section 1. Once osses are accounted for, the generator at 1 is unabe to deiver sufficient power into Section 1, and 16% of the oad at bus 9 is shed. The tota expected oad suppied is MW versus the predicted for DC, a shortfa accounted for by ine osses. Line fows are again, in genera, in the same direction as, and reasonaby cose to, those predicted by DC. The most significant resut is that the DC IP isanding has produced a network that is steady-state stabe for AC.Again the AC and DC modes shed different oads, suggesting that some consideration of the osses shoud be incuded in the IP mode. 21

22 b v b δ b (deg) p G g q G g p D d P D d q D d Q D d α d A,1 A,2 p L,fr p L,to (a) Bus data q L,fr q L,to MVA imit δ (deg) (b) Line data Tabe 6: AC OLS on isanded network. 22

23 ? 2 8? (a) (b) ? 2 8? (c) (d) Figure 4: Isanded networks: network schematic diagrams indicating isanded sections. Red ine indicates boundary of unheathy Section 0. Network (a) is obtained by soution of the DC isanding optimization. Networks (b), (c) and (d) are manuay created by making sma variations to (a). 23

24 4.3 Comparison over different isanded networks To obtain a cearer picture, we now compare use of the DC mode with the AC mode for a number of different isanded networks. Four different configurations are defined, shown as (a) to (d) in Figure 4. Network (a) is that formed by the DC isanding optimization, and then (b), (c), and (d) are derived from A as foows: (b) additionay has the oad bus 9 in Section 0. (c) moves buses 4,7 and 8 into Section 1. (d) retains oad bus 4 in Section 0, but moves 7 and 8 into Section 1. In Tabe 7, we compare for each network the expected tota oad deivered as predicted by DC and AC OLS. Network P d P d D `βd α 0d + α 1d P g pg g DC AC DC AC (a) (b) (c) (d) Tabe 7: Comparison of DC and AC resuts for the isanded networks A D (shown in Figure 4). Units of rea power are MW. These resuts indicate the shortcomings of the DC formuation as a decision maker of which ines to disconnect; the DC optimization probem provides mutipe optima, equay ranking networks (a) and (d). The AC optimization shows that (d) is a (marginay; 0.1% difference) higher-vaue soution when osses are taken into account. This resut is intuitive: the ine (7,9), which is cut in network A, is actuay of higher conductivity than ine (4,7). Consequenty, cutting (4,7) instead as is done in network (d) eads to ower tota ine osses. This issue coud be worse than it first appears; network (d) is not necessariy a goba optimum and different cuts may ead to an even higher-vaue soution. In the worst case, the oss-ess DC IP formuation coud in theory choose a soution that is significanty different in objective vaue and network topoogy to the true AC optimum. This reinforces the case for modeing osses in the IP formuation. 5 DC isanding of IEEE 24-bus RTS The IEEE RTS [4] comprises 24 buses and 38 ines. Of the buses, 17 have oads attached, and the tota demand rea power demand is 2850 MW. Generation capacity is 3405 MW from 32 synchronous generators; in addition, there is one synchronous condenser at bus 14. The network is depicted in Figure 5. Under nomina operation, tota oad demand is 2850 MW, whie tota generation capacity is 3405 MW. 24

25 Figure 5: Schematic of the IEEE RTS. Generating units are grouped at buses. In [5], a number of most-ikey contingency scenarios are determined for the RTS, under a set of specified assumptions. The most probabe coapse sequence is found to be the consecutive tripping of the transmission ine between bus 15 and bus 24 and the ine between bus 3 and bus 9. The authors show that oad fow computations subsequenty fai, since the system fais to suppy the oad at bus 3. In this section, we simuate this coapse sequence, and show that further faiure may be prevented by a combination of isanding and oad shedding. The simuation proceeds as foows. We begin with the network operating under nomina conditions. We then assume that the ine (15, 24) has tripped, and study the network immediatey after this first faiure. Line (3,9) is marked as uncertain (and assigned to set L 0 ), as are bus 3 and bus 24, which are assigned to B 0. No buses are assigned to B 1. As before, we constrain the generators so that rea power output may be instantaneousy decreased but not increased. The maximum generation imits for the DC isanding optimization are set to the operating points prior to the first faut, obtained from soution of a DC OPF; these wi be different to the rea (AC) outputs, since osses are negected. Tabe 8 shows the generator outputs at nomina operation; the tota cost of this generation is $61k per hour. 25

26 g b Pg G,max (MW) (a) Buses 1 14 g b P G,max g (MW) (b) Buses Tabe 8: Generator outputs at nomina operation 5.1 DC IP isanding with β = 0.5 With the probabiity β d set to 0.5 for a demands, the isanding optimization produces the network shown in Figure 6(a). Buses 3 and 24 have been contained in section 0 by disconnecting ines (1,3) and (3,9), and the entire oad at 3 has been shed. No oads have been shed in section 1, but to account for the reduced demand the foowing generators have decreased their outputs. The three 69 MW generators at bus 7 have decreased output to 25 MW each. The 155 MW unit at bus 16 has decreased its output to 93 MW. 5.2 Effect of varying oad-suppy probabiity, β d How shoud the probabiity β d be set? For the isanding case presented in the previous subsection, neither bus 3 nor bus 24 is a generation bus, so the isanding of ony 3 and 24 requires fu oad shedding at these buses. The probabiity of being abe to suppy any oad paced in Section 0 wi depend on the oading and status of ines within that part, and aso the generation capacity in that isand. Figure 7 shows the effect of varying β d on the expected tota oad suppied. It is assumed that β d = β for a d, so that any oad paced in section 0 wi have a suppy probabiity of β. The resuts indicate three regimes of operation: 1. 0 β 0.55, where buses 3 and 24 have been isanded. This is the network shown in Figure 6(a). 26

27 ? ? ? 3? ? 3? (a) (b) Figure 6: Schematic diagrams of the IEEE RTS, isanded under different vaues of β d. Generating units are grouped at buses < β 0.98, where buses 1,2,3 and 24 have been isanded. In this configuration, 9.4% of the oad at bus 3 is shed, whie a other oads are fuy served. However, because buses 1 and 2 are now in the unheathy section 0, the probabiity of being abe to suppy these oads is now β rather than 1. This is the network shown in Figure 6(b) < β 1, where no ines have been cut and consequenty no isands formed. In this configuration, the probabiity of being abe to suppy any oad paced in Section 0 is sufficienty high and so a buses are assigned to Section 0. Thus, an increasing probabiity eads to a growing section 0, as it becomes ess risky to pace oads there. 27

28 100 Expected tota oad suppied (%) β d Figure 7: Sweep of β d and resuting expected tota oad suppied (soid ine). The dashed ine shows the expected suppy with no ine cuts permitted. 28

29 6 Larger systems: computationa resuts The isanding formuation was appied to a number of different networks, ranging from a 9-bus to the 2383-bus mode of the Poish transmission network during the winter peak. For each network, 100 instances of the isanding optimization were soved, each to 1% optimaity; in each instance, a singe bus was randomy seected as uncertain and assigned to B 0. A operations were executed on an Inte Core 2 Quad 2.66 GHz Linux machine with 4 GiB RAM, using ILOG AMPL CPLEX 11.1 as the sover. Figure 8 shows the computationa resuts, incuding times, sover iterations, branch and bound nodes, and iterations per node. To each node count was added unity, since these are ogarithmic pots, and where no branch and bound nodes were required (the sover found an integer soution to the initia LP reaxation) a node count of zero is returned. The computation times pot indicates very approximatey a inear reationship between og(n b ) and og(t comp ). The ongest soution time recorded is 80 minutes for an instance of the 300-bus system, though the average times for that and the 2383-bus network are 100 and 160 seconds respectivey. That the ongest time observed was for the 300-bus system is indicative of a tendency for that network (and aso the 57-bus network) to require a arge amount of soution time. The number of iterations foows a simiar pattern. This may impy that there is something particuary hard about forming isands in these networks. 29

30 tcomp (s) N b (a) Computation times Niters N b (b) Iterations Figure 8: Computationa resuts for different networks. Mean, max and min vaues indicated. 30

31 Nnodes N b (c) Branch and bound nodes Niters/Nnodes N b (d) Iterations per node Figure 8: (continued) Computationa resuts for different networks. Mean, max and min vaues indicated. 31

32 7 Extensions to the IP formuation This section describes modifications to the basic DC IP formuation. 7.1 Loss modeing The rea power osses in a ine are determined directy from the KVL equations (3). Suppose, with some abuse of notation, that δ = δ A,1 δ A,2 and aso b = A,1,b = A,2. Then, h L = p L,fr + p L,to = G 11 v 2 b + G22 v 2 b + v bv b [ (G 12 + G 21 ) cos δ + ( B 12 B 21 ) ] sin δ. We proceed in the same way as for deriving the DC fow equations from the AC mode. Assume that votages are at nomina eves at each end of the ine, i.e., v b = v b = 1. h L = G 11 + G 22 + ( G 12 + G 21 ) cos δ + ( B 12 B 21 ) sin δ. We aso assume that B 12 = B 21, so that, h L = G 11 + G 22 + ( G 12 + G 21 ) cos δ. In a genera case, if τ is the off-nomina turns ratio of a transformer ine, then G 11 G L /τ2, G22 = G L and G 12 = G 21 = G L /τ. This gives the ine oss as h L = GL ( 1 ) + τ 2cos δ. τ τ Of course, for the usua case when a transmission ine is not a transformer with offnomina turns ratio, τ = 1 and h L = 2G L ( ) 1 cos δ. In either case, we may mode the oss whie maintaining a inear formuation by using a piecewise-inear (PWL) approximation to cos δ (or 1 cos δ ). Having done so, the KCL constraint (10) is modified to incude the oss of power over a ine. For a b B, p G g = p D d + p L ( p L h L ), (22) g G b d D b L:A,1 =b L:A,2 =b That is, it is impicity assumed that p L is the power injected at the from end of a ine, whie p L h L is the power injected at the to end of a ine. Figure 9 shows PWL-modeed ine osses for a DC OLS as the number of pieces in the approximation is increased. Because a standard PWL approximation to 1 cos(x) sits above the curve, ine osses are aways overestimated. Therefore, a second resut is shown for a PWL approximation with a sma corrective offset (( x) 2 /16) to make the errors either side of the curve equa. The pot shows that as N p becomes arge, oss modeing becomes more accurate. However, an offset of around 5% is present in the imit; it is thought that this is due to off-nomina votages in the AC soution. 32 =

33 150 P g pg g P d pd d (MW) N p Figure 9: Line osses as a function of N p, the number of pieces in the PWL approximation. The dashed ine shows the true AC osses; osses are shown for PWL oss modeing with (green) and without (bue) the corrective offset. N p = 0 means no oss modeing. Loss modeing in 14-bus network Revisiting the different isanding configurations of the 14-bus network ((a) (d) in Figure 4), Tabe 9 compares objective vaue, tota generation and tota ine osses for DC, AC and DC with oss modeing. The atter formuation empoyed 20 pieces in the PWL approximation of the oss term. P d P d D `βd α 0d + α 1d P g pg g P hl DC AC DC+ DC AC DC+ DC AC DC+ (a) (b) (c) (d) Tabe 9: Comparison of DC, AC and DC with osses ( DC+ ) resuts for the isanded networks (a) (d) (shown in Figure 4). Units of rea power are MW. It is cear that the modified DC mode conservativey estimates osses, as expected. The ranking of the different isands is unchanged; a are in agreement that network (d) is the most optima, yet even with osses modeed the DC formuation continues to 33

34 pace equa objective vaue on network (a). The AC resuts confirm that ine osses are marginay ower and oad deivery higher for network (d). Thus initiay the introduction of oss modeing appears not to remedy the probem of seecting the optima isands. However, further investigation shows that neither (a) nor (d) is actuay the optima isand seected by the IP isanding optimization modified for osses. The optima isanded network is simiar to network (d) but with an additiona ine, (3, 4), disconnected. P d P d D `βd α 0d + α 1d P g pg g P hl DC AC DC+ DC AC DC+ DC AC DC+ (e) Tabe 10: Comparison of DC, AC and DC with osses ( DC+ ) resuts for the isanded network (e). Units of rea power are MW. We find for this network, (e), that the AC OLS aso returns a higher objective vaue and ower osses, whie the DC without osses returns the same objective vaue as it did for (a) and (d). Further research of oss modeing in the DC formuation is in progress. In particuar, we are are investigating the appication of the modified mode to arger networks and the effect on computation times. 7.2 Generator switching So far, it has been assumed that rea power generation can be instantaneousy decreased from current eves; accordingy, in the IP formuation p G g is constrained by ower and upper bounds. However, such an assumption is not reaistic, particuary for the timescaes assumed for isanding, since ramp up/down rates for turbines are orders of magnitude sower. A more accurate scenario is that rea power output of each generator obeys a binary constraint: either the generator may continue to output at its current eve, or the turbine vaves may be opened, reducing the mechanica input power to zero. In this atter case, the generator remains eectricay connected to the network thus abe to act as a source/sink of reactive power but rea power output fas to zero. It is not cear what timescaes are invoved for the step-/ramp-down of mechanica input power, but we wi assume this to be instantaneous. We may then mode this as foows. The generation constraint (13) in the IP formuation is repaced by p G g = ζ gp G g, ζ g {0,1}, (23a) (23b) for a g G. If ζ g = 0 then generator g is switched off; otherwise it outputs P G g. From the DC mode s point of view, the switched off generating unit contributes no power to 34

35 the network. However, in a subsequent AC OLS, we note that Pg G,max the generating constraints become = P G g = 0, and Q G,min g 0 p G g 0, q G g Q G,max g, so that the unit is abe to suppy reactive power (e.g. for votage contro). Initia investigations on the 14-bus network show this approach to be restrictive, since ony two rea power generating units are present. For the 24-bus network, 32 generating units are spread across the network and the approach is more successfu. (It is anticipated that the same wi be true for arger networks; as more generating units are present the degrees of freedom are increased.) 8 Feasibiity probems The greatest probem to date is that of obtaining isanding soutions that are subsequenty aways feasibe AC soutions. We describe such an instance for the 24-bus network here. We begin with the network in the same post-faut state as in Section 5. To summarize, ine (15,24) has tripped, and ine (3,9) and buses 3 and 24 are uncertain. An IP isanding optimization is executed, with β d = 0.75, using both the modifications for switched generation and oss modeing. In the isanding soution, buses 1, 2, 3 and 24 have been paced in section 0, simiar to the network shown in Figure 6; in addition, ine (17,18) and one of the ines from 18 to 21 (there are two in parae) have been cut; the two 16 MW generators at bus 1 and one of the two 16 MW generators at bus 2 have been switched off; 40% of the oad at bus 3 has shed, and 1% of the oad at bus 4. No shedding takes pace in section 1. Foowing isanding, tota rea power generation in section 0 is 320 MW and tota demand is MW. Tota reactive power demand is 64 MVAr whie the tota reactive power capabiity imit of the generators in that isand is 160 MVAr. Simiary, generation capabiity in section 1 is sufficient to suppy the oad. Ostensiby, therefore, the isand has a feasibe steady-state operating point. Attempting to sove an AC OLS (with generators either remaining at constant rea power output or switching off, as appropriate) on the post-isanded network resuts in a reported infeasibiity. This is despite the fact that disconnected generators remain free to vary reactive power output, and there was sufficient reactive power generation avaiabe in each isand to meet reactive power demands. Examining the constraints, 35