optimal islanding and load shedding

7/26/2019 Optimal Islanding and Load Shedding

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Optimal islanding and load shedding

Paul Trodden, Waqquas Bukhsh,Andreas Grothey, Jacek Gondzio, Ken McKinnon

March 11, 2011

Contents

1 Introduction 32 Optimal load shedding 3

2.1 AC OLS problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 DC OLS problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Cutting lines without uncertainty . . . . . . . . . . . . . . . . . . . . . . 6

3 Optimal islanding and load shedding 143.1 Motivation and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 DC IP islanding formulation . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 DC islanding of IEEE 14-bus system 18

4.1 DC IP islanding of network . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 AC OLS on islanded network . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Comparison over different islanded networks . . . . . . . . . . . . . . . . 24

5 DC islanding of IEEE 24-bus RTS 245.1 DC IP islanding with = 0 .5 . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Effect of varying load-supply probability, d . . . . . . . . . . . . . . . . 26

6 Larger systems: computational results 29

7 Extensions to the IP formulation 327.1 Loss modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.2 Generator switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

8 Feasibility problems 35

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Notation

Notation conventions

Upper case is used for constants, (parameters in upright text and sets in calligraphictext), and lower case for variables. Indices are lower case and always subscripts. Super-scripts are part of the name.

Sets

B set of buses, indexed by bB 0 subset of buses of uncertain status (and preassigned to section 0)B 1 subset of buses preassigned to section 1G set of generators, indexed by gG b set of generators attached to bus bS = {0, 1}, set of sectionsL set of lines, indexed by lL0 subset of lines of uncertain statusD set of loads, indexed by dDb set of loads attached to bus b

Parameters

A line-bus matrix; line l goes from b = A l, 1 to b = A l, 2P Gg real power output from generator gP Dd , Q

Dd real and reactive power demands at load d (at nominal voltage)

GLl , BLl conductance and susceptance of line l

GBb , B

Bb shunt conductance and susceptance at bus b

P L ,maxl limit on real power ow in line lS L ,maxl limit on apparent power ow in line lV maxb , V

minb max and min voltages at bus b

Og set of possible values for ( pGg , q Gg )

d probability of losing load d if it is part of section 0M d reward per unit of delivered real power at load d l maximum difference in phase angle across a connected line l + maximum difference in phase angle across a disconnected line

Variables pGg , q

Gg real and reactive power output of generator g after change

pDd , q Dd real and reactive power absorbed by demand d

vb voltage at bus b b voltage phase angle at bus b pL ,frl , q

L ,frl real and reactive power ows into line l from bus b = A l, 1

pL , tol , q L ,tol real and reactive power ows into line l from bus b

= A l, 2

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pLl , q Ll real and reactive power that would ow into line l

were the line connected d proportion of load d supplied after shedding

l 01 switch to disconnect line l; line is disconnected iff l = 0 b section (0 or 1) that bus b lies in

1 Introduction

This document forms the follow-up report for the Edinburgh presentation at the Black-out project plenary meeting in Durham, 19 January 2011. The organization of thereport is as follows.

The next section describes an optimal load shedding formulation (with both AC andDC variants) and applies it to a 14-bus example system under abnormal operation. It isdemonstrated that cutting lines, without creating islands, can lead to less load shedding.

In Section 3, the motivations for islanding are described and the IP formulation ispresented. The islanding optimization is then applied to a number of test networks. InSection 4, the 14-bus network is islanded and the results compared with results from anAC model applied to the islanded network. A comparison with AC results over differentislands shows agreement between AC and DC, but also indicates a need for modellingof losses. Secondly, in Section 5, the 24-bus IEEE Reliability Test System is studied,and the effect of varying an islanding optimization parameter is investigated. Finally,the islanding optimization is applied to larger networks in Section 6, chiey to obtain ameasure of computational scaling.

In the nal two sections, modications to the islanding formulation are presentedand current difficulties with the method are discussed. For the former, modications aremade to include both loss modelling in the DC formulation and the on/off switching of generating units (as opposed to continuous variation of outputs). In the latter section,the problem of subsequent AC feasibility in a DC-islanded network is described, andideas to solve (or ameliorate) the problem are presented.

2 Optimal load shedding

The optimal load shedding model assumes the form of an optimal power ow optimiza-tion problem, but permits a proportion of the load at any bus to be shed. The nexttwo subsections briey describes the AC and DC OLS formulations, before a 14-bus

network example is used to show that cutting lines in some cases, even without failuresand uncertainty, can maximize supplied load.

2.1 AC OLS problem

The objective is to maximize the supply of real power to loads:

maxdD

M d d P Dd (1)

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subject to,

Kirchhoffs current law (KCL) for conservation of ow at each bus. For all b B :

gGb

pGg =dD b

pDd +lL :A l, 1 = b

pL ,frl lL :A l, 2 = b

pL ,tol + GBb v2b , (2a)

gGb

q Gg =dD b

q Dd +lL :A l, 1 = b

q L ,frl lL :A l, 2 = b

q L ,tol BBb v

2b , (2b)

Kirchhoffs voltage law (KVL) across each line l L. If b = A l, 1 is the from endbus and b = A l, 2 is the to end bus,

pL ,frl = G11l v

2b + vbvb G

12l cos( b b ) + B

12l sin( b b ) , (3a)

pL ,to

l = G22l v

2b + vb

vb G21l cos( b

b) + B21l sin( b

b) , (3b)q L ,frl = B

11l v

2b + vbvb G

12l sin( b b ) B

12l cos( b b ) , (3c)

q L ,tol = B22l v

2b + vb vb G

21l sin( b b) B

21l cos( b b) . (3d)

The parameters G11l , G12l , G

21l , G

22l and B

11l , B

12l , B

21l , B

22l are the real and imagi-

nary elements of the admittance matrix of line l:

Y l =Y 11l Y

12l

Y 21l Y 22l

= G11l G

12l

G21l G22l

+ j B11l B

12l

B 21l B22l

.

In most cases, when the line is not a transformer with off-nominal turns ratioand/or contains no line charging capacitance,

G11l = G22l = G

12l = G

21l = G

Ll

B 11l = B22l = B

12l = B

21l = B

Ll ,

and the more standard form of KVL is recovered. We use this formulation toretain generality; also, in each of the networks simulated in this report a numberof lines exist with l = 1 and/or non-zero line charging capacitance, in which caseG11l = G

Ll , B

11l = B

Ll , etc .

Load shedding constraints. We assume that load d D may be reduced bydisconnecting a proportion 1 d and that the ratio of real to reactive load staysthe same.

pDd = d P Dd , (4a)

q Dd = dQDd , (4b)

0 d 1. (4c)

(Currently, we assume that loads are constant power, though it is possible to useother models in the formulation).

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Generation constraints: the real and reactive power outputs of the generator g G lie in some feasible set of operation (see, for example, [1, Figure 3 .19, p93]).

pGg , q

Gg Og . (5)

Line limits. The formulation permits a number of different ways in which to imposelimits on line capacity, (depending on what is specied in the network data). Forexample, MVA ratings on apparent power for each line l L :

pL ,frl )2 + q L ,frl )

2 S L ,maxl2 , (6a)

pL ,tol )2 + q L ,tol )

2 S L ,maxl2 . (6b)

Note that this effectively corresponds to a heating constraint, applied at each endof the line.

Alternatively, the data may provide a P L ,maxl and require limits on real powerow. Or, phase angle limits

l A l, 1 A l, 2 l .

Voltage limits at each bus b B :

V minb vb V maxb . (7)

Reference bus constraint: b0 = 0 . (8)

The AC OLS is always a nonlinear problem, owing to the non-linearity of the KVLconstraints.

2.2 DC OLS problem

The DC OLS problem makes use of the standard DC ow model assumptions toproduce a formulation with real power only and no line losses. The problem is asfollows.

maxdD

M d d P Dd (9)

subject to

KCL for all b B :

gGb

pGg =dD b

pDd +lL :A l, 1 = b

pLl lL :A l, 2 = b

pLl , (10)

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KVL for all l L. If b = A l, 1 is the to end bus and b = A l, 2 is the from endbus,

pLl =

B Ll l

b

b , (11)

Note that we include the off-nominal turns ratio l to permit the general case of aline being a transformer; usually, l = 1.

Load shedding constraints for all d D:

pDd = d P Dd , (12a)

0 d 1. (12b)

Generation constraints for all g G :

P G ,ming pGg P

G ,maxg . (13)

Usually we shall assume that generation can be decreased but not increased, giventhe long times required to do so. In that case, P G ,maxg = P Gg .

Line limits, which may only be expressed either was MW ratings on real power foreach line l L :

P L ,maxl pLl P

L ,maxl (14)

Or, phase angle limits: l b b ,l L

Reference bus constraint: b0 = 0 . (15)

The main advantage of the DC OLS is its linearity; the KVL expression is linear,and so the resulting problem is an LP.

2.3 Cutting lines without uncertainty

The following numerical simulations use the IEEE 14-bus test system (see [2]), which isshown in Figure 1. Generator outputs at buses 1 and 2 are as follows.

0 pG1 200 MW,

0 pG2 200 MW,

150 q G1 150 MVAr ,

150 q G2 150 MVAr ,

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~~

~

~

~ 3

75

14

13

6

1

10

9

4

8

11

2

12

Figure 1: IEEE 14-bus test system.

while the condensers at buses 3, 6 and 8 have zero real power output capacity, andreactive power output in the range 0 to +140 MVAr. Total real power demand in thenetwork (at nominal voltage) is 259 MW. This mirrors the settings in [3]. Line limitsare not specied as standard for this system, and so a 100 MVA limit is imposed oneach line. Phase angle limits are also not specied, but will be dealt with separately inthe examples that follow.

To simulate an abnormal operating condition, the generator at bus 2 is deleted fromthe system (as was done in [3]). Subsequently, the revised generation constraints are

0 pG1 200,

0 pG2 0.

Real power demand now exceeds supply by 59 MW. No buses are marked as uncertain; itis assumed that knowledge of the network state is accurate. The idea in this simulationis not to island an unhealthy part of the network, but to shed loads to obtain anoptimal operating point in the abnormal conditions.

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~~

~

~

~ 3

75

14

13

6

1

10

9

4

8

11

2

12

(a) No line cuts

~~

~

~

~ 3

75

14

13

6

1

10

9

4

8

11

2

12

(b) Lines (2 , 3) and (2 , 5) cut

Figure 2: IEEE 14-bus test system under abnormal conditions.

OLS solution with no line cuts

Tables 1 and 2 shows the bus and line data for the AC OLS and DC OLS respectively.The key points are summarized as follows.

Total load supplied is 147 .9 MW (AC) and 154 .6 MW (DC).

The load at bus 2 has been fully shed in both cases, while the AC OLS sheds94.9% of the load at bus 3 compared with 87 .8% for DC OLS.

Total real power generation is 153 .8 MW (AC) and 154 .6 MW (DC). In neithercase is the sole remaining generator operating at capacity (200 MW),

Line (1, 2) is operating at capacity. All others are within limits.

Bus and line values for AC and DC are of the same orders of magnitude and if thesame signs; small differences add up to a total of 5 .9 MW in losses for AC.

It is not immediately apparent from the data as to why the single generator is unableto operate nearer to capacity, and so reduce the load shed. Sensitivity analysis showsthat by far the highest shadow price is on the MVA limit constraint for line (1 , 2),indicating an extra 63 .9 MW of load could be supplied for a 1 p.u. (100 MVA) increasein capacity.

OLS solution with line cuts

Tables 3 and 4 show the results from the AC and DC OLS formulations when lines (2 , 3)and (2 , 5) are cut prior to solving the optimization problems.

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b vb b (deg) pGg q Gg p

Dd P

Dd q

Dd Q

Dd d

1 1.053 0.00 153.8 4.12 1.035 3.10 0.0 21.7 0.0 12.7 0.003 1.025 5.50 0.0 0.0 4.8 94.2 1.0 19.0 0.054 1.012 7.10 47.8 47.8 3.9 3.9 1.005 1.013 6.22 7.6 7.6 1.6 1.6 1.006 1.060 11.60 0.0 13.9 11.2 11.2 7.5 7.5 1.007 1.043 10.308 1.056 10.30 0.0 7.69 1.041 11.97 29.5 29.5 16.6 16.6 1.0010 1.037 12.20 9.0 9.0 5.8 5.8 1.0011 1.045 12.03 3.5 3.5 1.8 1.8 1.00

12 1.045

12

.45 6

.1 6

.1 1

.6 1

.6 1

.0013 1.039 12.50 13.5 13.5 5.8 5.8 1.00

14 1.022 13.23 14.9 14.9 5.0 5.0 1.00

(a) Bus data

l A l, 1 A l, 2 pL , frl p

L , tol q

L , frl q

L , tol MVA limit (deg)

1 1 2 100.0 98.2 1.7 1.3 100 3.102 1 5 53.9 52.4 5.8 5.0 100 6.223 2 3 22.5 22.3 2.0 1.7 100 2.404 2 4 41.9 41.0 0.4 0.3 100 4.005 2 5 33.8 33.2 1.1 2.9 100 3.126 3 4 17.5 17.3 0.8 1.6 100 1.607 4 5 34.9 35.1 7.9 7.4 100 0.888 4 7 28.8 28.8 3.4 5.1 100 3.209 4 9 16.6 16.6 1.3 0.1 100 4.8710 5 6 42.9 42.9 13.6 9.3 100 5.3811 6 11 6.6 6.5 5.0 4.9 100 0.4312 6 12 7.7 7.7 2.7 2.6 100 0.8513 6 13 17.4 17.2 8.0 7.5 100 0.9014 7 8 0.0 0.0 7.5 7.6 100 0.0015 7 9 28.8 28.8 2.4 1.6 100 1.6716 9 10 6.0 6.0 2.8 2.7 100 0.2217 9 14 9.9 9.8 2.7 2.4 100 1.2518 10 11 3.0 3.0 3.1 3.1 100 0.1719 12 13 1.6 1.5 1.0 1.0 100 0.0520 13 14 5.2 5.1 2.7 2.6 100 0.73

(b) Line data

Table 1: AC OLS on 14-bus network under abnormal operation with no line cuts. Unitsof real and reactive power are MW and MVAr respectively; voltages are per unit.

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b b (deg) pGg pDd P Dd d

1 0.00 154.62 3.75 0.0 21.7 0.003 6.82 0.0 11.5 94.2 0.124 8.41 47.8 47.8 1.005 7.38 7.6 7.6 1.006 12.93 0.0 11.2 11.2 1.007 11.858 11.85 0.09 13.70 29.5 29.5 1.0010 14.06 9.0 9.0 1.0011 13.75 3.5 3.5 1.00

12

14.24 6

.1 6

.1 1

.0013 14.52 13.5 13.5 1.00

14 15.65 14.9 14.9 1.00

(a) Bus data

l A l, 1 Al, 2 pLl P L , maxl (deg)

1 1 2 100.0 100 3.752 1 5 54.6 100 7.383 2 3 25.6 100 3.064 2 4 41.6 100 4.655 2 5 32.9 100 3.636 3 4 14.1 100 1.597 4 5 38.6 100 1.038 4 7 29.3 100 3.449 4 9 17.1 100 5.2910 5 6 41.2 100 5.5511 6 11 5.9 100 0.8212 6 12 7.2 100 1.3013 6 13 16.9 100 1.5914 7 8 0.0 100 0.0015 7 9 29.3 100 1.8516 9 10 6.6 100 0.3717 9 14 10.3 100 1.9618 10 11 2.4 100 0.3119 12 13 1.1 100 0.2920 13 14 4.6 100 1.13

(b) Line data

Table 2: DC OLS of 14-bus system under abnormal conditions with no line cuts. Unitsof real power are MW.

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Having made these line cuts, the generator at bus 1 is now operating at or near tomaximum capacity, providing 200 MW (DC) and 198 .1 MW (AC) to the rest of thenetwork. For AC, the same buses (2 and 3) as before have shed or part-shed their loads;

bus 2 has fully shed but as a consequence of the increased generation bus 3 hasshed only 56.2% compared with 94 .9% previously.

The situation for DC at rst appears to be rather different: bus 2 has fully shed,while, of the rest, the only load that has not shed is at bus 3 all other buses havepart-shed between 3 .6% (bus 5) and 99 .6% (bus 2). However, it should be noted thatthe DC is almost indifferent to which loads are shed, since (i) no losses are modelled(ii) reactive power and voltage is neglected, and (iii) all rewards per unit supply, M din the objective, are unity. The former point indicates that some loss modelling isrequired if the IP islanding formulation is to shed the correct loads. The second pointhas implications for how line limits are modelled: if a 100 MVA limit is assumed tobe equivalent to a 100 MW limit for DC, then DC will be able to squeeze more realpower down a line. Despite this, what is important here is the overall load supply: infact, by cutting lines (2 , 3) and (2 , 5) an extra 36 MW of load has been supplied for ACand an additional 46 for DC.

Disconnecting lines (2 , 3) and (2 , 5) has increased the real power carried over lines(1, 5) and (2 , 4) to near-capacity levels. Previously these lines carried 54 .6 and 41.6 MWrespectively. Meanwhile, line (1 , 2) remains at capacity. The net effect is a higher owof real power from the buses 1 and 2 to the rest of the network. This better solutionis not available without the line cuts in place; with the lines present, any phase angledifferences between bus 2 and buses 3 and 5 implies a non-zero ow of power. Vieweddifferently, with all lines intact the network is unable to establish the conditions requiredto move enough generated power from buses 1 and 2 to the rest of the buses.

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Dd P

Dd q

Dd Q

Dd d

1 1.060 0.00 198.1 13.02 1.041 3.04 0.0 21.7 0.0 12.7 0.003 0.984 17.14 0.0 18.9 41.2 94.2 8.3 19.0 0.444 0.996 12.57 47.8 47.8 3.9 3.9 1.005 0.997 11.48 7.6 7.6 1.6 1.6 1.006 1.060 17.06 0.0 22.9 11.2 11.2 7.5 7.5 1.007 1.038 15.788 1.060 15.78 0.0 13.59 1.035 17.44 29.5 29.5 16.6 16.6 1.0010 1.032 17.67 9.0 9.0 5.8 5.8 1.0011 1.042 17.49 3.5 3.5 1.8 1.8 1.00

12 1.044 17.91 6.1 6.1 1.6 1.6 1.0013 1.039 17.96 13.5 13.5 5.8 5.8 1.0014 1.018 18.70 14.9 14.9 5.0 5.0 1.00

(a) Bus data


L , tol q

L , frl q


1 1 2 100.0 98.3 0.2 0.8 100 3.042 1 5 98.1 93.3 12.8 1.6 100 11.483 2 4 98.3 93.1 0.8 11.4 100 9.534 3 4 41.2 42.5 10.6 8.6 100 4.575 4 5 41.7 42.0 10.4 9.6 100 1.096 4 7 28.3 28.3 8.6 10.4 100 3.21

7 4 9 16.2 16.2 0.6 2.0 100 4.878 5 6 43.8 43.8 6.4 2.1 100 5.589 6 11 7.1 7.0 6.1 5.9 100 0.4310 6 12 7.8 7.8 2.8 2.7 100 0.8611 6 13 17.7 17.4 8.5 8.1 100 0.9012 7 8 0.0 0.0 13.2 13.5 100 0.0013 7 9 28.3 28.3 2.9 2.0 100 1.6614 9 10 5.5 5.5 1.7 1.7 100 0.2215 9 14 9.5 9.4 2.0 1.8 100 1.2616 10 11 3.5 3.5 4.1 4.1 100 0.1817 12 13 1.7 1.6 1.1 1.1 100 0.0518 13 14 5.6 5.5 3.4 3.2 100 0.74

(b) Line data

Table 3: AC OLS of 14-bus system under abnormal conditions with cuts to lines (2 , 3)and (2 , 5). Units of real and reactive power are MW and MVAr respectively; voltagesare per unit.

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b b (deg) pGg pDd P

Dd d

1 0.00 200.02 3.75 0.1 21.7 0.003 25.59 0.0 94.2 94.2 1.004 14.94 17.0 47.8 0.365 13.53 7.3 7.6 0.966 18.83 0.0 10.7 11.2 0.957 18.058 18.05 0.09 19.73 27.8 29.5 0.9410 20.00 7.8 9.0 0.86

11

19.63 2

.9 3

.5 0

.8312 20.05 5.5 6.1 0.91

13 20.35 12.9 13.5 0.9514 21.48 13.8 14.9 0.93

(a) Bus data

l A l, 1 Al, 2 pLl P L , maxl (deg)

1 1 2 100.0 100 3.752 1 5 100.0 100 13.533 2 4 99.9 100 11.194 3 4 94.2 100 10.655 4 5 53.3 100 1.426 4 7 26.5 100 3.117 4 9 15.5 100 4.788 5 6 39.4 100 5.309 6 11 5.7 100 0.8010 6 12 6.7 100 1.2211 6 13 16.2 100 1.5212 7 8 0.0 100 0.0013 7 9 26.5 100 1.6714 9 10 5.0 100 0.2715 9 14 9.3 100 1.7516 10 11 2.8 100 0.3717 12 13 1.2 100 0.3118 13 14 4.5 100 1.13

(b) Line data

Table 4: DC OLS of 14-bus system under abnormal conditions with cuts to lines (2 , 3)and (2 .5). Units of real power are MW.

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3 Optimal islanding and load shedding

The previous section showed that it is possible to reduce the load shed by allowing

lines to be cut, though it is uncommon for the best solution to split the network intoislands. However, so far we have assumed perfect knowledge of the post-fault state of the network. More realistically, it may be known that there is a problem in some partof the network, but the form and extent of the problem is not known. In such a case,a robust solution to prevent cascading failures would be to isolate the uncertain part of the network from the certain part, by forming one or more islands.

In this section, we present a DC Integer Programming (IP) formulation for minimiz-ing the load shed in an electricity network under stress. The goal is to maximise theexpected load that remains connected. Initially we will formulate a single stage problem.

3.1 Motivation and assumptions

Following some failure in the network, we wish to disconnect lines, vary loads andestablish stable islands to avoid cascading failures and eventual blackout. We assumethat, post-fault, there are parts of the network that are suspected of having a fault andsome where we are reasonably sure have no faults. Figure 3 depicts such a situation fora ctional network.

??

? ?

(a) Network prior to islanding

?

??

?

Island 1 Island 2

Section 0 Section 1Section 1

Island 3 Island 4(b) Network post islanding

Figure 3: (a) Fictional network with uncertain buses and lines, and (b) the islanding of that network by disconnecting lines.

Our aim is to split the network into disconnected sections so that the possible faultsare all in one section. It is desirable that this section be small, since it may be prone

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to failure, and that the other section is able to operate with little load shedding. Wewould also like the problem section to shed as little load as possible. Figure 3 thereforealso shows a possible islanding solution for this network, where all of the uncertain

buses and lines have been placed in a section 0. At this point we make the followingdistinction between sections and islands .

The optimized network shall consist of two sections, an unhealthy section 0 anda healthy section 1. No lines shall connect the two sections. On the other hand,neither section is required to be a single, connected component.

An island describes a connected component of the network.

Thus, either section may contain a number of islands as is exemplied by in Figure 3,where section 1 comprises islands 1, 3 and 4. Section 0 is a single island, but is notalways necessarily so.

We will assume that the state of the system immediately after the initial fault isknown, and that the state at the end of the calculation can be quickly and accuratelypredicted. We have central control of generation, load shedding and line breakers; weassume that we can instantaneously and simultaneously reduce the demand, reduce realpower generation levels 1 , vary the reactive power generation levels within bounds, andalso disconnect lines. It is assumed that such instantaneous variations do not causetransient instability.

We require that after the adjustments the system is in a feasible stable steady state.If it is optimal to do so, the model will split the network into islands but without furtherconstraints it will not necessarily do this.

3.2 DC IP islanding formulationThe formulation is obtained by adding sectioning constraints to the DC OLS problem.The resulting problem is a Mixed Integer Linear Problem (MILP), describing a loss-less p system with voltage-independent loads. It is, of course, possible to develop anequivalent formulation for the AC model, but this results in a MINLP, which is difficultto solve. (The basic graph partitioning formulation is a standard one, but linking thebinary line variables to the electricity network variables appears to be new.)

Sectioning constraints

We dene a set S = {0, 1} that numbers the sections of the network: section 0 shallbe the unhealthy section, while section 1 shall be healthy. We suspect that somesubset B 0 of buses and some subset L0 of lines have a possible fault; it is these we wishto conne to section 0.

1 Note that in more recent work and following feedback from the January meeting in Durham we have assumed that real power generation cannot be instantaneously varied; a generators mechanicalinput may be held at its current level or removed completely, while the machine remains electricallyconnected to the network and able to supply reactive power. This modication is described in Section 7.For the results presented here, however, we assume that real power can be varied.

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We introduce a binary decision variable b with each bus b B ; b shall be set equalto 0 iff b is placed in section 0 and b = 1 otherwise. To partition the network in sucha way, we need to disconnect lines. Accordingly, we dene a binary decision variable lfor each l L; l = 0 iff line l is disconnected and l = 1 otherwise.

The rst pair of constraints operates on each line not pre-assigned to L0 . The valueof l for the line is zero and the line is cut if the two end buses are in differentsections ( i.e. A l, 1 = 0 and A l, 2 = 1, or A l, 1 = 1 and A l, 2 = 0). Otherwise, if thetwo end buses are in the same section, be it section 0 or 1, l 1; that is, the line mayor may not be disconnected. Thus, these constraints enforce the requirement that anycertain line between sections 0 and 1 shall be disconnected.

l 1 + A l, 1 A l, 2 ,l L \ L0 , (16a)

l 1 A l, 1 + A l, 2 ,l L \ L0 . (16b)

The second pair of constraints examines lines pre-assigned to L0

and sets l = 0 disconnects the line if either or both of the ends are in healthy section 1. Thus, anuncertain line (i) shall be disconnected if entirely in section 1, (ii) shall be disconnectedif between sections 0 and 1, (iii) may remain connected if entirely in section 0.

l 1 A l, 1 ,l L0 , (16c)

l 1 A l, 2 ,l L0 , (16d)

The nal pair of constraints simply sets the value of b for a bus b depending onwhat set that bus was pre-assigned to. So if b B 0 then b = 0. In addition, we denea set B 1 , to which any buses that are desired to remain in section 1 may be assigned; if b B 1 then b = 1.

b = 0 ,b B 0 , (16e) b = 1 ,b B 1 . (16f)

This completes the description of the sectioning constraints. The IP optimizationwill disconnect lines and place buses in section 0 or 1, as directed by these constraints,depending on the pre-assignments to B 0 , B 1 and L1 . What else is placed in section 0 andwhat other lines are cut are degrees of freedom for the optimization, and will depend onthe objective function.

Objective function

The overall objective of islanding is to minimize the risk of system failure. In ourmotivation we assumed that there is some uncertainty associated with a particular subsetof buses and/or lines; we suspect there may be a fault and so we wish to isolate thesecomponents from the rest of the network.

Owing to uncertainty, we assume that there is a chance that section 0 will eitherblack out or lose a proportion of its load beyond what we expect from our plan. We

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therefore wish to reduce the value attributed to load supplied to section 0 in our model.In placing any load in section 0, we shall assume a lowered probability of being ableto supply power to that load, since the probability of that section failing is higher. In

particular, we assume that we have a probability of 1 of being able to supply a load dplaced in section 1, but a probability of only d 1 of being able to supply the sameload if placed in section 0 with the uncertain components. We wish to maximize thevalue of supplied demand:

maxdD

M d P d d 0d + 1d , (17)

and

d = 0d + 1d ,d D, (18a)0 0d 1,d D, (18b)

0 1d b,b B , d Db. (18c)

Here we have introduced a new variable sd for the load d delivered in section s S .The above constraints permit it to be non-zero in only one section. The interpretationof this is that a load d at a bus b will be worth d M d d P d if b is in section 0, or M d d P dif b is in section 1. Thus the objective has a preference for a smaller section 0.

DC ow-phase angle relations in lines

When a line l is connected, a ow of real power is established depending on the differencesin phase angle at each end of the line. However, if l = 0 and a line is disconnected, wemust still permit a difference in phase at each end of the line, only with zero ow throughthe line. To achieve this, we replace real power pLl in the DC KVL constraint (11) with pLl .

pLl = B Ll

l A l, 1 A l, 2 . (19)

Then, when line l is connected we will set pLl = pLl , and when l is disconnected p

Ll = 0.

We model this as follows.Assume the minimum and maximum possible values of pLl are P

L ,minl , P

L ,maxl re-

spectively, and of pLl are P L ,minl , P

L ,maxl

2 . Then,

pLl = l, 1 (P L ,minl ) + l, 2( P

L ,minl ) + l, 3P

L ,maxl + l, 4 P

L ,maxl , (20a)

pLl = l, 1P L ,minl + l, 3P

L ,maxl , (20b)

l = l, 1 + l, 3 , (20c)1 l = l, 2 + l, 4 , (20d)

l,i 0,i {1, 2, 3, 4}. (20e)2 In practice, P L , minl , P

L , maxl may be set according to line limits, i.e. P

L , minl = P

L , maxl . On the

other hand, P L , minl , P L , maxl should be sufficiently large to not constrain the solution.

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When the sectioning constraints set a particular l = 0, then l, 1 = l, 2 = 0 andso pLl = 0. However, because l, 2 + l, 4 = 1, p

Ll may take whatever value necessary to

satisfy the KVL constraint (19).

Phase angle constraints

If phase angle constraints are present in the DC OLS formulation, they must be modiedto take into account line disconnections. If a line l has been disconnected, the formulationshould relax the phase angle difference restriction between the two end buses. For alll L :

+ + ( l + )l A l, 1 A l, 2 + + ( l + )l , (21)

Overall formulation

The overall IP formulation for islanding is maximize the objective (17) subject to

sectioning constraints (16);

KCL (10);

KVL (19);

ow-phase angle line constraints (20);

load model and shedding, (12) and (18);

generation limits (13), with P G ,maxg = P Gg ;

line limits: real power (14) or phase angle (21).

The reference bus angle constraint ( b1 = 0) is not necessary; in fact, since theoptimization may create two or more islands a constraint may be required for eachisland created, which is not straightforward to implement. If we do desire to remove theredundancy in absolute values of phase angle, most likely for computational reasons, wecan add a small penalty to the objective.

4 DC islanding of IEEE 14-bus system

The 14-bus network is initially operating under nominal conditions; all loads are fullysupplied (259 MW in total), with outputs of 131 .2 MW and 127 .8 MW for the generatorsat buses 1 and 2 respectively.

Bus 2 is marked uncertain and assigned to set B 0 . The probability of being able tosupply any load placed in Section 0 is d = 0 .5,d D. No lines are uncertain, and nobuses are pre-assigned to Section 1.

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Constraint limits for the DC IP optimization are set as follows. The generation con-straints assume that real power output can only be decreased from the current operatingpoint.

0 pG1 131.2 MW,

0 pG2 127.8 MW.

This still, however, assumes a continuous decrease is possible instantaneously, which isnot realistic; this assumption is tightened in Section 7.

In the absence of real data, line limits are again set to 100 MVA for all lines. Becausethe DC model does not include reactive power, we assume a line limit of 100 MW. Phaseangle differences across lines are limited to 15 degrees. Linearization errors in the DCow model increase rapidly from this point onwards: the error in cos x 1 is 3.5% atx = 15 degrees.

4.1 DC IP islanding of network

The results of the islanding optimization applied to the network are shown in Table 5,and the islanded network is depicted in Figure 4(a).

Lines 1, 5, 7, 9 and 15 have been cut, islanding buses 2, 3, 4, 7 and 8 thelower-right of the network into Section 0. Consequently, the generator at bus 1serves the healthy upper-left part of the network, while the generator at 2 servesthe unhealthy part.

Real power delivery to loads is 100 % at all load buses except for buses 2 and 3,where 0% and 84 .8 % of the respective demanded levels are supplied.

A total of 223 .1 MW of real power is generated. The generator at bus 1 hasbeen re-scheduled to output 95 .3 MW enough to fully-supply each load in itssection while the output of the generator at bus 2 is unchanged at 127 .8 MW.Despite operating at maximum output, this generator is unable to supply all loadwithin its section.

The expected total load supplied using the probability d for those loads inSection 0 is 159.2 MW.

No line is at its limit, though line (1 , 5) carries 95 .3 MW against an assumed limitof 100 MW. Similarly, no phase angle limits are active.

4.2 AC OLS on islanded network

An AC OLS was solved for the islanded network in Figure 4. The objective function isagain to maximize the load supplied, but now includes the probability measure for theany buses placed in section 0. The real power generation limits are the same as for the

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b b (deg) pGg pDd P Dd d b

1 0.00 95.3 12 32.72 127.8 0.0 21.7 0.00 03 40.83 0.0 80.0 94.2 0.85 04 39.45 47.8 47.8 1.00 05 12.89 7.6 7.6 1.00 16 24.70 0.0 11.2 11.2 1.00 17 39.45 08 39.45 0.0 09 34.72 29.5 29.5 1.00 110 33.51 9.0 9.0 1.00 111 29.50 3.5 3.5 1.00 1

12

26.58 6

.1 6

.1 1

.00 113 27.68 13.5 13.5 1.00 1

14 33.26 14.9 14.9 1.00 1

(a) Bus data

l A l, 1 A l, 2 pLl P L , maxl (deg) l

1 1 2 0.0 100 32.72 02 1 5 95.3 100 12.89 13 2 3 67.7 100 8.11 14 2 4 60.1 100 6.73 15 2 5 0.0 100 19.83 06 3 4 12.3 100 1.39 17 4 5 0.0 100 26.55 08 4 7 0.0 100 0.00 19 4 9 0.0 100 4.73 010 5 6 87.7 100 11.80 111 6 11 34.3 100 4.80 112 6 12 10.4 100 1.88 113 6 13 31.8 100 2.98 114 7 8 0.0 100 0.00 115 7 9 0.0 100 4.73 016 9 10 21.8 100 1.21 117 9 14 7.7 100 1.45 118 10 11 30.8 100 4.01 119 12 13 4.3 100 1.10 120 13 14 22.6 100 5.59 1

(b) Line data

Table 5: DC islanding of 14-bus system in response to uncertainty at bus 2. Units of real power are in MW.

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22/37


Dd P

Dd q

Dd Q

Dd d

1 0.992 0.00 99.7 8.12 1.060 3.80 127.8 11.5 21.7 21.7 12.7 12.7 1.003 1.033 1.54 0.0 8.1 55.6 94.2 11.2 19.0 0.594 1.037 1.17 47.8 47.8 3.9 3.9 1.005 0.940 13.43 7.6 7.6 1.6 1.6 1.006 1.060 25.20 0.0 61.3 11.2 11.2 7.5 7.5 1.007 1.060 1.178 1.060 1.17 0.0 0.0

9 0.976

32.06 24.9 29.5 14.0 16.6 0.8410 0.981 31.08 9.0 9.0 5.8 5.8 1.0011 1.014 28.21 3.5 3.5 1.8 1.8 1.0012 1.040 26.53 6.1 6.1 1.6 1.6 1.0013 1.028 26.96 13.5 13.5 5.8 5.8 1.0014 0.978 30.88 14.9 14.9 5.0 5.0 1.00

(a) Bus data


L , tol q

L , frl q


1 1 5 99.7 94.2 8.1 10.1 100 13.432 2 3 52.5 51.3 2.0 1.9 100 5.343 2 4 53.6 52.1 3.2 4.0 100 4.97

4 3 4

4.3 4

.3

1

.3

0

.1 100

0

.375 4 7 0.0 0.0 0.0 0.0 100 0.00

6 5 6 86.6 86.6 11.7 30.6 100 11.777 6 11 32.9 31.9 9.7 7.6 100 3.008 6 12 11.3 11.2 2.8 2.5 100 1.329 6 13 31.1 30.5 10.7 9.5 100 1.7610 7 8 0.0 0.0 0.0 0.0 100 0.0011 9 10 18.7 18.8 1.9 1.6 100 0.9812 9 14 6.2 6.3 2.3 2.1 100 1.1813 10 11 27.8 28.4 4.2 5.8 100 2.8714 12 13 5.1 5.0 0.9 0.9 100 0.4315 13 14 22.0 21.2 4.5 2.9 100 3.92

(b) Line data

Table 6: AC OLS on islanded network.

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~

~

~~

~3

75

14

13

6

1

10

9

4

8

1112

2?

(a)

~

~

~~

~3

75

14

13

6

1

10

9

4

8

1112

2?

(b)

~

~

~~

~3

75

14

13

6

1

10

9

4

8

1112

2?

(c)

~

~

~~

~3

75

14

13

6

1

10

9

4

8

1112

2?

(d)

Figure 4: Islanded networks: network schematic diagrams indicating islanded sections.Red line indicates boundary of unhealthy Section 0. Network (a) is obtained bysolution of the DC islanding optimization. Networks (b), (c) and (d) are manuallycreated by making small variations to (a).

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4.3 Comparison over different islanded networks

To obtain a clearer picture, we now compare use of the DC model with the AC model for

a number of different islanded networks. Four different congurations are dened, shownas (a) to (d) in Figure 4. Network (a) is that formed by the DC islanding optimization,and then (b), (c), and (d) are derived from A as follows:

(b) additionally has the load bus 9 in Section 0.

(c) moves buses 4 , 7 and 8 into Section 1.

(d) retains load bus 4 in Section 0, but moves 7 and 8 into Section 1.

In Table 7, we compare for each network the expected total load delivered as pre-dicted by DC and AC OLS.

Network P d P D

d ` d 0

d + 1

d P g pG

g

DC AC DC AC

(a) 159 .1800 153.2096 223.0600 227.4307(b) 129 .6800 128.3228 193.5600 197.3711(c) 157 .9500 148.7439 215.9000 219.6065(d) 159 .1800 153.3485 223.0600 227.4307

Table 7: Comparison of DC and AC results for the islanded networks AD (shown inFigure 4). Units of real power are MW.

These results indicate the shortcomings of the DC formulation as a decision makerof which lines to disconnect; the DC optimization problem provides multiple optima,equally ranking networks (a) and (d). The AC optimization shows that (d) is a (marginally;0.1% difference) higher-value solution when losses are taken into account. This resultis intuitive: the line (7 , 9), which is cut in network A, is actually of higher conductivitythan line (4 , 7). Consequently, cutting (4 , 7) instead as is done in network (d) leadsto lower total line losses. This issue could be worse than it rst appears; network (d)is not necessarily a global optimum and different cuts may lead to an even higher-valuesolution. In the worst case, the loss-less DC IP formulation could in theory choose a so-lution that is signicantly different in objective value and network topology to thetrue AC optimum. This reinforces the case for modelling losses in the IP formulation.

5 DC islanding of IEEE 24-bus RTSThe IEEE RTS [4] comprises 24 buses and 38 lines. Of the buses, 17 have loads attached,and the total demand real power demand is 2850 MW. Generation capacity is 3405 MWfrom 32 synchronous generators; in addition, there is one synchronous condenser at bus14. The network is depicted in Figure 5.

Under nominal operation, total load demand is 2850 MW, while total generationcapacity is 3405 MW.

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~ ~~

~

~~ ~ ~

~ ~~

2118 22

16 19

17

12

15 14 13

3 9 10

1 2 7

4

24 11

8

23

20

65

Figure 5: Schematic of the IEEE RTS. Generating units are grouped at buses.

In [5], a number of most-likely contingency scenarios are determined for the RTS,under a set of specied assumptions. The most probable collapse sequence is found to bethe consecutive tripping of the transmission line between bus 15 and bus 24 and the linebetween bus 3 and bus 9. The authors show that load ow computations subsequentlyfail, since the system fails to supply the load at bus 3.

In this section, we simulate this collapse sequence, and show that further failure maybe prevented by a combination of islanding and load shedding. The simulation proceedsas follows. We begin with the network operating under nominal conditions. We thenassume that the line (15 , 24) has tripped, and study the network immediately after thisrst failure. Line (3 , 9) is marked as uncertain (and assigned to set L0), 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 real power output may be instan-taneously decreased but not increased. The maximum generation limits for the DCislanding optimization are set to the operating points prior to the rst fault, obtainedfrom solution of a DC OPF; these will be different to the real (AC) outputs, since lossesare neglected. Table 8 shows the generator outputs at nominal operation; the total costof this generation is $61k per hour.

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g b P G , maxg (MW)

1 1 162 1 163 1 764 1 765 2 166 2 167 2 768 2 769 7 57.110 7 57.111 7 57.112 13 76.313 13 76.314 13 76.3

15 14 0(a) Buses 114

g b P G , maxg (MW)

16 15 2.417 15 2.4

18 15 2.419 15 2.420 15 2.421 15 15522 16 15523 18 40024 21 40025 22 5026 22 5027 22 5028 22 5029 22 5030 22 5031 23 15532 23 15533 23 350

(b) Buses 1524

Table 8: Generator outputs at nominal operation

5.1 DC IP islanding with = 0 .5

With the probability d set to 0 .5 for all demands, the islanding optimization producesthe network shown in Figure 6(a). Buses 3 and 24 have been contained in section 0by disconnecting lines (1 , 3) and (3 , 9), and the entire load at 3 has been shed. Noloads have been shed in section 1, but to account for the reduced demand the followinggenerators 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 load-supply probability, d

How should the probability d be set? For the islanding case presented in the previoussubsection, neither bus 3 nor bus 24 is a generation bus, so the islanding of only 3 and24 requires full load shedding at these buses. The probability of being able to supplyany load placed in Section 0 will depend on the loading and status of lines within thatpart, and also the generation capacity in that island.

Figure 7 shows the effect of varying d on the expected total load supplied. It isassumed that d = for all d, so that any load placed in section 0 will have a supplyprobability of . The results indicate three regimes of operation:

1. 0 0.55, where buses 3 and 24 have been islanded. This is the network shownin Figure 6(a).

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~ ~~

~

~~ ~ ~

~ ~~

2118 22

16 19

17

12

15 14 13

9 10

1 2 7

4

11

8

23

20

65

24

3?

?

?

(a)

~ ~~

~

~~ ~ ~

~

2118 22

16 19

17

12

15 14 13

9 10

7

4

11

8

23

20

65

24

3?

?

?

~~

1 2

(b)

Figure 6: Schematic diagrams of the IEEE RTS, islanded under different values of d .Generating units are grouped at buses.

2. 0.55 < 0.98, where buses 1 , 2, 3 and 24 have been islanded. In this cong-uration, 9 .4% of the load at bus 3 is shed, while all other loads are fully served.However, because buses 1 and 2 are now in the unhealthy section 0, the prob-ability of being able to supply these loads is now rather than 1. This is thenetwork shown in Figure 6(b).

3. 0.98 < 1, where no lines have been cut and consequently no islands formed.In this conguration, the probability of being able to supply any load placed inSection 0 is sufficiently high and so all buses are assigned to Section 0.

Thus, an increasing probability leads to a growing section 0, as it becomes less risky toplace loads there.

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d

E x p e c t e d t o t a l l o a d s u p p l i e d ( % )

0 0.2 0.4 0.6 0.8 193

94

95

96

97

98

99

100

Figure 7: Sweep of d and resulting expected total load supplied (solid line). The dashedline shows the expected supply with no line cuts permitted.

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6 Larger systems: computational results

The islanding formulation was applied to a number of different networks, ranging from

a 9-bus to the 2383-bus model of the Polish transmission network during the winterpeak. For each network, 100 instances of the islanding optimization were solved, eachto 1% optimality; in each instance, a single bus was randomly selected as uncertain andassigned to B 0 . All operations were executed on an Intel Core 2 Quad 2 .66 GHz Linuxmachine with 4 GiB RAM, using ILOG AMPL CPLEX 11 .1 as the solver.

Figure 8 shows the computational results, including times, solver iterations, branchand bound nodes, and iterations per node. To each node count was added unity, sincethese are logarithmic plots, and where no branch and bound nodes were required (thesolver found an integer solution to the initial LP relaxation) a node count of zero isreturned.

The computation times plot indicates very approximately a linear relationship

between log( N b) and log( tcomp ). The longest solution time recorded is 80 minutes foran instance of the 300-bus system, though the average times for that and the 2383-busnetwork are 100 and 160 seconds respectively. That the longest time observed was forthe 300-bus system is indicative of a tendency for that network (and also the 57-busnetwork) to require a large amount of solution time. The number of iterations followsa similar pattern. This may imply that there is something particularly hard aboutforming islands in these networks.

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N b

t c o m p

( s )

9 14 2430 39 57 118 300 238310 4

10 2

100

102

104

(a) Computation times

N b

N i t e r s

9 14 2430 39 57 118 300 2383100

102

104

106

108

(b) Iterations

Figure 8: Computational results for different networks. Mean, max and min valuesindicated.

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N b

N n o d e s

9 14 2430 39 57 118 300 2383100

102

104

106

(c) Branch and bound nodes

N b

N i t e r s /

N n o d e s

9 14 2430 39 57 118 300 2383100

101

102

103

104

105

(d) Iterations per node

Figure 8: (continued) Computational results for different networks. Mean, max and minvalues indicated.

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7 Extensions to the IP formulation

This section describes modications to the basic DC IP formulation.

7.1 Loss modelling

The real power losses in a line l are determined directly from the KVL equations (3).Suppose, with some abuse of notation, that l = A l, 1 A l, 2 and also b = A l, 1 , b

= A l, 2 .Then,

hLl = pL ,frl + p

L ,tol

= G11l v2b + G

22l v

2b + vbvb G

12l + G

21l cos l + B

12l B

21l sin l .

We proceed in the same way as for deriving the DC ow equations from the AC model.Assume that voltages are at nominal levels at each end of the line, i.e., vb = vb = 1.

hLl = G11l + G

22l + G

12l + G

21l cos l + B

12l B

21l sin l .

We also assume that B 12l = B21l , so that,

hLl = G11l + G

22l + G

12l + G

21l cos l .

In a general case, if l is the off-nominal turns ratio of a transformer line l, then G11l =GLl /

2l , G

22l = G

Ll and G

12l = G

21l = G

Ll / l . This gives the line loss as

hLl = GLl

l 1 l

+ l 2cos l .

Of course, for the usual case when a transmission line is not a transformer with off-nominal turns ratio, l = 1 and

hLl = 2 GLl 1 cos l .

In either case, we may model the loss while maintaining a linear formulation by usinga piecewise-linear (PWL) approximation to cos l (or 1 cos l ). Having done so, theKCL constraint (10) is modied to include the loss of power over a line. For all b B ,

gGb

pGg =dD b

pDd +lL :A l, 1 = b

pLl lL :A l, 2 = b

pLl hLl , (22)

That is, it is implicitly assumed that pLl is the power injected at the from end of a

line, while pLl h

Ll is the power injected at the to end of a line.Figure 9 shows PWL-modelled line losses 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, line losses are always overestimated. Therefore, a second resultis shown for a PWL approximation with a small corrective offset (( x)2 / 16) to makethe errors either side of the curve equal. The plot shows that as N p becomes large, lossmodelling becomes more accurate. However, an offset of around 5% is present in thelimit; it is thought that this is due to off-nominal voltages in the AC solution.

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N p

P g

p G g

P d

p D d ( M W )

0 5 10 15 20 25 30 35 40 45 500

50

100

150

Figure 9: Line losses as a function of N p, the number of pieces in the PWL approxima-tion. The dashed line shows the true AC losses; losses are shown for PWL loss modellingwith (green) and without (blue) the corrective offset. N p = 0 means no loss modelling.

Loss modelling in 14-bus network

Revisiting the different islanding congurations of the 14-bus network ((a)(d) in Fig-ure 4), Table 9 compares objective value, total generation and total line losses for DC,AC and DC with loss modelling. The latter formulation employed 20 pieces in the PWLapproximation of the loss term.

P d P Dd ` d 0 d + 1 d P g p

Gg P l h

Ll

DC AC DC+ DC AC DC+ DC AC DC+

(a) 159 .1800 153.2096 152.2476 223.0600 227.4307 227.7600 0 11.6696 13.2414(b) 129 .6800 128.3228 128.0748 193.5600 197.3711 197.9766 0 6.5255 7.6270(c) 157 .9500 148.7439 147.9266 215.9000 219.6065 220.7488 0 12.9126 14.8722(d) 159 .1800 153.3485 152.2476 223.0600 227.4307 227.7600 0 11.5276 13.2414

Table 9: Comparison of DC, AC and DC with losses (DC+) results for the islandednetworks (a)(d) (shown in Figure 4). Units of real power are MW.

It is clear that the modied DC model conservatively estimates losses, as expected.The ranking of the different islands is unchanged; all are in agreement that network(d) is the most optimal, yet even with losses modelled the DC formulation continues to

33


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place equal objective value on network (a). The AC results conrm that line losses aremarginally lower and load delivery higher for network (d).

Thus initially the introduction of loss modelling appears not to remedy the problem

of selecting the optimal islands. However, further investigation shows that neither (a)nor (d) is actually the optimal island selected by the IP islanding optimization modiedfor losses. The optimal islanded network is similar to network (d) but with an additionalline, (3, 4), disconnected.

P d P Dd ` d 0 d + 1 d P g p

Gg P l h

Ll

DC AC DC+ DC AC DC+ DC AC DC+

(e) 159 .1800 153.3723 152.3702 223.0600 227.4307 227.7600 0 11.4800 13.0590

Table 10: Comparison of DC, AC and DC with losses (DC+) results for the islandednetwork (e). Units of real power are MW.

We nd for this network, (e), that the AC OLS also returns a higher objective valueand lower losses, while the DC without losses returns the same objective value as it didfor (a) and (d).

Further research of loss modelling in the DC formulation is in progress. In particular,we are are investigating the application of the modied model to larger networks andthe effect on computation times.

7.2 Generator switching

So far, it has been assumed that real power generation can be instantaneously decreasedfrom current levels; accordingly, in the IP formulation pGg is constrained by lower and up-per bounds. However, such an assumption is not realistic, particularly for the timescalesassumed for islanding, since ramp up/down rates for turbines are orders of magnitudeslower.

A more accurate scenario is that real power output of each generator obeys a binaryconstraint: either the generator may continue to output at its current level, or theturbine valves may be opened, reducing the mechanical input power to zero. In thislatter case, the generator remains electrically connected to the network thus able toact as a source/sink of reactive power but real power output falls to zero. It is notclear what timescales are involved for the step-/ramp-down of mechanical input power,but we will assume this to be instantaneous. We may then model this as follows.

The generation constraint (13) in the IP formulation is replaced by

pGg = gP Gg , (23a)

g {0, 1}, (23b)

for all g G . If g = 0 then generator g is switched off; otherwise it outputs P Gg . Fromthe DC models point of view, the switched off generating unit contributes no power to

34


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the network. However, in a subsequent AC OLS, we note that P G ,maxg = P Gg = 0, andthe generating constraints become

0 pGg 0,

QG ,ming q Gg Q

G ,maxg ,

so that the unit is able to supply reactive power ( e.g. for voltage control).Initial investigations on the 14-bus network show this approach to be restrictive,

since only two real power generating units are present. For the 24-bus network, 32generating units are spread across the network and the approach is more successful. (Itis anticipated that the same will be true for larger networks; as more generating unitsare present the degrees of freedom are increased.)

8 Feasibility problemsThe greatest problem to date is that of obtaining islanding solutions that are subse-quently always feasible AC solutions. We describe such an instance for the 24-busnetwork here.

We begin with the network in the same post-fault state as in Section 5. To sum-marize, line (15 , 24) has tripped, and line (3 , 9) and buses 3 and 24 are uncertain. AnIP islanding optimization is executed, with d = 0 .75, using both the modications forswitched generation and loss modelling. In the islanding solution,

buses 1, 2, 3 and 24 have been placed in section 0, similar to the network shownin Figure 6;

in addition, line (17 , 18) and one of the lines from 18 to 21 (there are two inparallel) have been cut;

the two 16 MW generators at bus 1 and one of the two 16 MW generators at bus2 have been switched off;

40% of the load at bus 3 has shed, and 1% of the load at bus 4. No shedding takesplace in section 1.

Following islanding, total real power generation in section 0 is 320 MW and totaldemand is 312 .1 MW. Total reactive power demand is 64 MVAr while the total reactive

power capability limit of the generators in that island is 160 MVAr. Similarly, generationcapability in section 1 is sufficient to supply the load. Ostensibly, therefore, the islandhas a feasible steady-state operating point.

Attempting to solve an AC OLS (with generators either remaining at constant realpower output or switching off, as appropriate) on the post-islanded network results ina reported infeasibility. This is despite the fact that disconnected generators remainfree to vary reactive power output, and there was sufficient reactive power generationavailable in each island to meet reactive power demands. Examining the constraints,

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it is found that the relaxing upper voltage limits at buses 1 and 2, and lower voltagelimits at buses 4 and 9, recovers a feasible solution. In this solution, however, voltagesat buses 1 and 2 are high (1 .1642 and 1.1658 p.u. respectively), while voltages are very

low at buses 4 and 9 (0 .8730 and 0.9161 p.u. respectively).Note that this loss of feasibility also occurs for DC islanding of the network without

the modications for losses and generator switching, thus it is not a consequence of these modications. It should also be stressed that not all islanding congurations of the 24-bus network lead to infeasibility. However, this is clearly a problem, and raisesthe question of what we can infer from a feasible DC solution about feasibility in AC.Is this a problem in general for islanding, unless AC power ow is explicitly consideredin deriving the islanding solution?

The loss of feasibility in our case is exacerbated by constraining real and reactiveloads to be shed in equal proportions. In fact, if real and reactive loads may be shedindependently, then no feasibility problems occur. However, this is not likely to be arealistic assumption. It may be that feasibility is less of a problem on larger networks,as more degrees of freedom are available. Other ideas we have, in increasing order of complexity, are

in AC OLS (following DC islanding), modelling each load in two parts: a high P /low Q part and a low P / high Q part, and freeing the optimization to shed theseloads independently. This would break the restriction of shedding P / Q in equalproportions without going as far as fully independent shedding.

modelling reactive power ows and voltages in a decoupled AC islanding formu-lation by including a linearized q v model. The p and q v systems are indepen-dent (voltage is xed in p , is xed in q v).

PWL-modelled AC islanding, using special ordered sets and a MILP solver (CPLEX)that can exploit these.

full AC islanding, by use of MINLP.

References

[1] J. Machowski, J. W. Bialek, and J. R. Bumby, Power System Dynamics: Stability and Control , 2nd ed. John Wiley & Sons, Ltd, 2008.

[2] R. D. Christie. (1999, August) Power systems test case archive. University of Washington. [Online]. Available: http://www.ee.washington.edu/research/pstca/

[3] M. A. Mostafa, M. E. El-Hawary, G. A. N. Mbamalu, M. M. Mansour, K. M. El-Nagar, and A. N. El-Arabaty, Steady-state load shedding schemes: a performancecomp, Electric Power Systems Research , vol. 38, pp. 105112, 1996.

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[4] Reliability Test System Task Force of the Application of Probability Methods Sub-committee, IEEE reliability test system, IEEE Transactions on Power Apparatus and Systems , vol. PAS-98, no. 6, pp. 20472054, 1979.

[5] J. Hazra and A. K. Sinha, Prognosis of catastrophic failures in electric power sys-tems, in IEEE International Conference on Industrial Technology , December 2006,pp. 13491354.

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optimal islanding and load shedding

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