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11

Operational Defense of Power System Cascading

Outages (Project S-26)

James McCalley and Siddhartha Khaitan Department of Electrical and Computer

Engineering Iowa State University

PSERC Tele-Seminar, April 15, 2008

22

Analogiestime

Airplanes getting too close to each other Avoidance Action

by the TCAS

Normal Stage Emergency

Stage

Collision

Collision avoided

Traffic Alert and Collision Avoidance System

Without action

Emergency Response Information System

Direct detection

Electric variablesPower System

Disturbance

System Protection Scheme

INPUTACTION DECISIONPROCESS

Automation of SPS Design

33

Research Problem Statement

“to provide operators with a very fast (online) computational capability to

predict system response and identify corrective actions through analytical

modeling and fast numerical simulation studies for low probability, high-

consequence catastrophic events, by exploiting the state of the art in software

and hardware”

44

Today’s Seminar

Historical perspective of cascading events in past 40 yearsN-k contingency selectionAssessment, Detection and Control action DeterminationDFS Based Island Detection and Simulation Numerical MethodsArchrivals

55

Recent Blackouts in Power Systems

66

Location Date MW Lost Duration People affected Approximate cost

US-NE 11/9/1965 20000 13 hours 30 millionUS-NE 7/13/1977 6000 22 hours 3 million 300 millionFrance 12/19/1978 30000 10 hours

West Coast 12/22/1982 12350 5 millionSweden 12/27/1983 > 7000 5.5 hours 4.5 million Brazil 4/18/1984 15762Brazil 8/18/1985 7793

Hydro Quebec 4/18/1988 18500US-West 1/17/1994 7500

Brazil 12/13/1994 8630US-West 12/14/1994 9336 1.5 million

Brazil 3/26/1996 5746US-West 7/2/1996 11743 1.5 million US-West 7/3/1996 1200 small numberUS-West 8/10/1996 30489 7.5 million 1 billion dollars

MAPP, NW Ontario 6/25/1998 950 19 hours 0.152 millionSan Francisco 12/8/1998 1200 8 hours 1 million

Brazil 3/11/1999 25000 4 hours 75 million Brazil 5/16/1999 2000India 1/2/2001 12000 13 hours 220 million 107 millionRome 6/26/2003 2150 7.3 millionUS-NE 8/14/2003 62000 1-2 days 50 million 4-6 billion

Denmark/Sweden 9/23/2003 6300 6.5 hours 5 millionItaly 9/28/2003 27000 19.5 hours 57 million

Croatia 12/1/2003 1270 mwh 2.5 millionGreece 7/12/2004 9000 3 hours 5 million

Moscow/Russia 5/24-25/2005 2500 >6 hours 4 million European Blackout 11/412006 > 6400 1 Hour 15 million

IMP

AC

T

6

77

Location Date Generation trip Transmission trip Time between initiating and secondary, pre-collapse events

US-NE 11/9/1965 no Four 230KV lines few minutes

US-NE 7/13/1977 Yes Yesoccurred in a sequence between 20 to 45

minutes after initial eventFrance 12/19/1978 yes > 30 minutes

West Coast 12/22/1982 No Yes FastSweden 12/27/1983 No Yes 50 secondsBrazil 4/18/1984 Xmer yes 9-10 minutesBrazil 8/18/1985 No yes

Hydro Quebec 4/18/1988 Transformer yes 2-3 secondsUS-West 1/17/1994 Yes Yes FastBrazil 12/13/1994 yes yes

US-West 12/14/1994 No Yes 40-52 secondsBrazil 3/26/1996 Xmer Yes

US-West 7/2/1996 yes yes 20 secondsUS-West 7/3/1996 No yes fastUS-West 8/10/1996 yes (13 generators) yes 5-7 minutes

MAPP, NW Ontario 6/25/1998 No yes 44 minutes

San Francisco 12/8/1998 yes yes 16 secondsBrazil 3/11/1999 No Yes > 30 secondsBrazil 5/16/1999 No YesIndia 1/2/2001Rome 6/26/2003 No NoUS-NE 8/14/2003 yes yes more than 2 hours

Denmark/Sweden 9/23/2003 yes yes 5 minutesItaly 9/28/2003 No Yes 25 minutes

Croatia 12/1/2003 Yes Yes 30 secondsGreece 7/12/2004 Yes No 10 minutes

Moscow/Russia 5/24-25/2005 No Yes >12 hours

European Blackout 11/4/2006 yes Yes 30 minutes

PR

E-C

OLL

AP

SE

EV

EN

TS

7

88

Location Date Collapse time #successive eventsUS-NE 11/9/1965 13 minutes ManyUS-NE 7/13/1977 1 hour ManyFrance 12/19/1978 > 30 minutes Many

West Coast 12/22/1982 few minutes ManySweden 12/27/1983 > 1 minute ManyBrazil 4/18/1984 > 10 minutes TopologyBrazil 8/18/1985 Topology

Hydro Quebec 4/18/1988 < 1minute ManyUS-West 1/17/1994 1 minute 3Brazil 12/13/1994 many

US-West 12/14/1994 substation topologyBrazil 3/26/1996 Topology

US-West 7/2/1996 36 seconds Several

US-West 7/3/1996 > 1 minute Prevented by fast op. action

US-West 8/10/1996 > 6 minutes ManyMAPP, NW Ontario 6/25/1998 >44 minutes substation topology

San Francisco 12/8/1998 16 seconds manyBrazil 3/11/1999 30 seconds substation topologyBrazil 5/16/1999 TopologyIndia 1/2/2001Rome 6/26/2003US-NE 8/14/2003 > 1 hour Many

Denmark/Sweden 9/23/2003 7 minutes ManyItaly 9/28/2003 27 minutes Many

Croatia 12/1/2003 few seconds manyGreece 7/12/2004 14 minutes few

Moscow/Russia 5/24-25/2005 14 hours Many

European Blackout 11/4/2006 30 minutes Many

CO

LLA

PS

E T

IME

&

N

O. O

F S

UC

CE

SS

IVE

EV

ENTS

8

99

Summary of blackout attributesImpact: • 3 of largest 4 blackouts occurred in last 10 years• # of blackouts > 1000 MW doubles every 10 years

Pre-event conditions:• Extreme weather• Extreme conditions• Weakened topology

Triggering events:• Various kinds of N-1 or• N-k (k>1) with fault + nearby protection failure

1010

Summary of blackout attributesPre-collapse events:• 50% involved generation, 95% involved

transmission• 50% had significant time between initiating &

pre-collapse events• 40% involved proper protection actionNature of collapse:• Successive tripping of components and/or• Voltage collapseCollapse time and # of events: • 50% were “slow”• 60% involved many cascaded (dependent)

events

1111

Scenario for 50% of blackouts

1. Weakened conditions: Heavy load, and/or one or more gen or ckt outage possibly followed by readjustments

2. Initiating event: One or several components trip because of fault and/or other reasons;

2. Steady-state progression (slow succession): a. System stressing: heavy loading on lines, xfmrs, unitsb. Successive events: Other components trip one by one

with fairly large inter-event time intervals3.Transient progression in fast succession:

a. Major parts of system go under-frequency and/or under-voltage.

b. Components begin tripping quicklyc. Uncontrolled islanding and/or voltage collapse

1212

Intelligent Detection and Prevention of Failures

Cond it ion A ctuated P rotec tion A ction M odeling

Adaptive Tim e Step

High Perform ance Computing

Utiliz e spars ity based coding

Multifrontal Linear Solver

Com putationa l Characte rist ics

Fast and Slow Dynam ics

Generator redispatch

Load shedding

System island ing

Sim ula tion mod el com plex ity

De cis ion sets priori ty

Simulator Attributes

12

1313

Initiating Contingency

Selection

1414

TerminologyTerminologyInitiating eventInitiating event

A disturbance consisting of firstA disturbance consisting of first--disturbance followed by outage of disturbance followed by outage of one or more components; one or more components; may include protection failuremay include protection failure

NN--k initiating eventsk initiating eventsImplied that k>1 Implied that k>1 initiating events with loss of multiple elementsinitiating events with loss of multiple elementsNN--1 events are order 1. N1 events are order 1. N--k events may be order 1 to order k.k events may be order 1 to order k.

Successive eventsSuccessive eventsSignificant changes in configuration/conditions after initiatingSignificant changes in configuration/conditions after initiating event.event.Assumed to be predictable with advanced simulator modelingAssumed to be predictable with advanced simulator modelingIncludes operation of protection Includes operation of protection asas--designeddesigned, to trip element, to trip element•• Generator: field winding Generator: field winding overexcitationoverexcitation, loss of field, loss of , loss of field, loss of

synchronism, synchronism, overfluxoverflux, , overvoltageovervoltage, , underfrequency,undervoltageunderfrequency,undervoltage•• Transmission: impedance, Transmission: impedance, overcurrentovercurrent backup, outbackup, out--ofof--stepstep

1515

EventEvent--probability order and spaceprobability order and spaceProbability orderProbability order

Order 1: Probability of single element outage in next hr: P=10Order 1: Probability of single element outage in next hr: P=10--55

Order 2: Probability of 2 independent outages: POrder 2: Probability of 2 independent outages: P22=10=10--1010

Order 3: Probability of 3 independent outages: POrder 3: Probability of 3 independent outages: P33=10=10--1515

Order=number of independent eventsOrder=number of independent eventsProvides probability scale in considering initiating event likelProvides probability scale in considering initiating event likelihoodihood

PP2

P3

∑

Pi=1

P4

Construct response plans for each initiating event so that operaConstruct response plans for each initiating event so that operator is tor is ““readyready”” in case initiating event occurs. in case initiating event occurs. Approach: Prepare response plans for initiating events resultingApproach: Prepare response plans for initiating events resulting in in failure, according to decreasing probability of initiating eventfailure, according to decreasing probability of initiating event, to , to maximize maximize ““operator readiness.operator readiness.””Assume successive events are predictable given initiating eventAssume successive events are predictable given initiating event

Probability space: Probability space: the space of all the space of all possible initiating possible initiating eventsevents

1616

Preventive/corrective action paradigmPreventive/corrective action paradigm

ProbabilityProbability<<PP

Initiating event identification

and probability calculation

High-probability

events

Low-probability

events

Assessment Assessment

Violation detection Failure detection

Probability>PProbability>P

Determine preventive actionand implement it

Determine corrective actionand store it

1717

Component model Component model

1.1. Functional group tripping , ~ PFunctional group tripping , ~ P•• Proper relay tripping, may trip multiple componentsProper relay tripping, may trip multiple components

2.2. Fault plus breaker failure to trip, ~ PFault plus breaker failure to trip, ~ P22

•• Breaker stuck or protection fail to send the signal to openBreaker stuck or protection fail to send the signal to open•• Two neighboring functional groups trippedTwo neighboring functional groups tripped

3.3. Inadvertent tripping of two or more components, ~ PInadvertent tripping of two or more components, ~ P22

•• Inadvertent tripping of backup breaker to a primary faultInadvertent tripping of backup breaker to a primary fault4.4. Any of the above together with independent outage of any Any of the above together with independent outage of any

other single component in a selected set, ~ Pother single component in a selected set, ~ P22 ~ P~ P33

Definition: A functional group is a group of components that operate & fail together as a result of breaker locations within the topology that interconnects them.

We admit four types of initiating events:

1818G-1

BR-1

CAP-1

BS-4 SW-1

BS-6

BR-2

BR-3

BR-4

SW-2 SW-3 SW-4

BS-1

BS-2

BS-3

BS-7 BS-8 BS-9

LN-1

LN-3LN-4

LN-5

LN-2

GROUND

BS-5

BS-10

TR-1 BS: BR: G: CAP: SW: LN: TR: FG:

Bus SectionBreaker Generator Capacitor Switch Line Transformer Functional Group

Legend

FG-6

FG-1

FG-2

FG-3

FG-4

FG-5

FG-7

Functional group decomposition

FG-7

FG-3

FG-4

FG-2

FG-1

FG-5 FG-6

BR-1

BR-2

BR-3

SW-2 SW-3

BR-4

FG : SW : BR :

Functional GroupOpen Switch Breaker

1919

HighHigh--Risk Initiating EventsRisk Initiating Events

ji

j i

CFGremoval failure

C FGP P

∈

= ∑

( ) = ij iji iB BFG FGfault stuck fault fault per demandP P P P+ ×

Functional Groups provides for efficient Functional Groups provides for efficient initiating event identification and probability initiating event identification and probability computation.computation.

/ Pr( )/Pr( )

j iFG FGinadvertentP FG j trips FG i trips

FG i trips= − −

−

I

Functional Group Contingency

Stuck Breaker Contingency

Inadvertent Tripping Contingency

Forms ~59% of the protection related category classification of major disturbances (Source: NERC’s DWAG)

2020

Topology Variation can lead to N-k outage

Variation in substation topology results from: Forced Outages &Operator action for

1. Facility maintenance2. Mitigating undesirable operating condition as high circuit loading or out- of-limit voltages.

Change in system topology exposes the system to N-k contingency with probability order equivalent to N-1 contingencies and thus high risk.

2121

Contingency SelectionContingency Selection

BUSBAR-1

L1 L2 L3

S2 (off)

S3 (off)

B1 (on)

L1 L2 L3

B3 (off)

S1 (off)

B2 (on)

B2 (off)

B1 (off)

S3 (on)

S2 (on)

S1 (on)

BUSBAR-2 backup BUSBAR-2

BUSBAR-1

B3 (on)

N−3 exposure increases P2

to 3P when maintaining busbar

1 in

double breaker-double bus, for fault on any line. An N-1 or N-2 contingency cannot occur.

B 1

B 4

B 2

B 3

L in e -1 L in e -3

L in e -2

L in e -4

R in g B u s

B 1

B 4

B 2

B 3

L in e -3

L in e -2

L in e -4

R in g B u s L in e -1

N−2

exposure increases P2

to P when maintaining B4 in ring bus, for fault on Line 3. An N-1 contingency for Line 3 cannot occur.

2222

Number of contingencies of type N-k resulting from a single fault (Order P) for IEEE RTS96

k 1 2

No. 63 4

Number of contingencies of type N-k resulting from a fault/breaker failure (order P2) for

IEEE RTS96

k 1 2 3 4

No. 82 88 1 3

Number of contingencies of type N-k resulting from ITC (Order P2) for IEEE RTS96

k 2 3 4

No. 137 5 5

Test Results on IEEE RTS 96 Test System

2323

Testing on Large EMS Model

k 1 2 3 4 5 6 7 8 9 10 11No. 2022 468 49 14 5 3 2 1 0 0 1

k 1 2 3 4 5 6 7 9 10 11 12 13 14 15 17No. 3011 1248 356 134 63 31 23 0 1 1 7 1 0 0 1

Type Bus Line Xfmr Gen ShuntNo. 1549 1830 697 353 357

Number of contingencies of type N-k resulting from a fault/breaker failure (order P2)

Number of contingencies of type N-k resulting from a single fault (Order P)

Number of components in the system

Use graph-search to identify functional groups, and order P and P2 contingencies

2424

Assessment, Assessment, detection, & detection, &

corrective action corrective action determinationdetermination ProbabilityProbability<<PP

Initiating event identification

and probability calculation

Low-probability

events

Assessment

Failure detectionDetermine corrective action

and store it

2525

Failure detection & preventionFailure detection:• Activation of any protection that would

trip an additional component• Detection of an overload,

underfrequency, or undervoltage condition exceeding tolerable thresholds

Failure prevention: • Expert system with multiplicity of

possible actions taken based on failure type detected

• Includes a simple/robust optimization to identify action for relieving overloads.

2626

ONE-LINE DIAGRAM OF THE TEST SYSTEM

56

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L115

L114

L113

38

53

51

45

BUS-107 BUS-108

BUS-109

L105

L106

L118

L117

L116

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34

29 30

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28 1

TG103

52

61 60

63

57

58

17

T301BUS-301

L402

L216

3

G102 G103

G101

G203G202

L302

L302

L302

L402

L301

L301

L402

L107

L108

L109

L103

L111

L112

L110

L102

L101

T302T402

TG102

TG202

TG101

L207

L208

L209

L210

L211

L212

L203

L202

L201

L206

L204

L205

L218

L217

L213

L215

L214

LOAD103

LOAD101

LOAD102

LOAD202

LOAD201

LOAD203

L104

TG203

TG201G201

BUS-101

BUS-103

BUS-104

BUS-105 BUS-106

BUS-102

BUS-401

BUS-205

BUS-209

BUS-208

BUS-207

BUS-203BUS-202

BUS-201

BUS-204

L401

L401

T4014

5

1 2

6 7

8

9 10

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BUS-302 BUS-402

BUS-206

Load ramping 20% from t=900s to

t=2700s

G102 (initiating generator

trip at t=290s)

2727

Initiating Event G102 trips at 290 seconds

First Control Action (reconfiguration)

Second Control Action (Load Shedding)


First Control Action (reconfigur ation)

Second Control Action (Load Shedding)

No control action

Voltage response of systemCircuit loading

Tracking and Avoidance Decision Support Tool

No Generator Protection No Generator Protection

2828

With and Without Generator Protection

Voltage response of system with and without generator protection

Circuit loading with and without generator protection

2929


Without Generator Protection

First Control Action (reconfiguration) to avoid generator trip at 595.142 seconds

Second Control Action (Load Shedding) to avoid generator trip



Predicting Response and Identifying Control Actions


First Control Action

Second Control Action

3030


56

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L115

L114

L113

38

53

51

45

BUS-107 BUS-108

BUS-109

L105

L106

L118

L117

L116

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34

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31

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TG103

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T301BUS-301

L402

L216

3

G102 G103

G101

G203G202

L302

L302

L302

L402

L301

L301

L402

L107

L108

L109

L103

L111

L112

L110

L102

L101

T302T402

TG102

TG202

TG101

L207

L208

L209

L210

L211

L212

L203

L202

L201

L206

L204

L205

L218

L217

L213

L215

L214

LOAD103

LOAD101

LOAD102

LOAD202

LOAD201

LOAD203

L104

TG203

TG201G201

BUS-101

BUS-103

BUS-104

BUS-105 BUS-106

BUS-102

BUS-401

BUS-205

BUS-209

BUS-208

BUS-207

BUS-203BUS-202

BUS-201

BUS-204

L401

L401

T4014

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1 2

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154

BUS-302 BUS-402

BUS-206

L215 (initiating event at t=300s)

G101 (generator trip at t=619.71s

G102 (generator trip at t=925.37s)

Load ramping 20%from t=900s to

t=2700s

Generator trip initiated by

overexcitation(V/Hz relay)

G201 (generator trip at

t=1260.17s)

A Cascading Scenario

3131


First Generator Trip at 619.71s

Second Generator Trip at 925.37

Third Generator Trip at 1260.17

Initiating contingency: L215


First Generator Trip at 619.71s

Second Generator Trip at 925.37

Third Generator Trip at 1260.17


WithoutGenerator Protection

First Generator

Trip at 619.71s

Second Generator

Trip at 925.37


WithoutGenerator Protection

First Generator

Trip at 619.71s

Second Generator

Trip at 925.37




A Cascading Scenario

3232

C

G

A

B

D F

E

Automatic Island Detection and Simulation

Depth First Search (Recursive Algorithm)Graph Search Algorithm for component detection within each Island

F

Traversing order: A, B, D, F, E, C, G

•The figure shown is a representative structure of the cascading phenomena starting at root node A and progressing on any of the three branches initially, which can further branch depending on the trajectory of the system. •DFS is ideally suited for simulating the cascading phenomena in power system.

3333

Contingencyoccurred

Contingencyresults in two sub-systems

Contingencyresults in a single

sub-system

Stable system

Cascaded system

Note: Contingencies at first level are deliberate

3434


56

55

L115

L114

L113

38

53

51

45

BUS-107 BUS-108

BUS-109

L105

L106

L118

L117

L116

33

34

29 30

31

32

44

45

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35

43

41

42

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40

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18

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67

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22

25

26 27

28 1

TG103

52

61 60

63

57

58

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T301BUS-301

L402

L216

3

G102 G103

G101

G203G202

L302

L302

L302

L402

L301

L301

L402

L107

L108

L109

L103

L111

L112

L110

L102

L101

T302T402

TG102

TG202

TG101

L207

L208

L209

L210

L211

L212

L203

L202

L201

L206

L204

L205

L218

L217

L213

L215

L214

LOAD103

LOAD101

LOAD102

LOAD202

LOAD201

LOAD203

L104

TG203

TG201G201

BUS-101

BUS-103

BUS-104

BUS-105 BUS-106

BUS-102

BUS-401

BUS-205

BUS-209

BUS-208

BUS-207

BUS-203BUS-202

BUS-201

BUS-204

L401

L401

T401

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1 2

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107

108

109

110 112

111

115

116

117

118 119

120

121

122

123

124

125

126

127 128

146

151

150

149 148

147

153

135

136

138

137 139

140

141

142

143

144

145 133

134

129 130

131

132

155

152

154

BUS-302 BUS-402

BUS-206


t=2700s

L215 (initiating event at t=300s)

L401 (line trip at t=619.71s)




G101 (generator

trip at t=619.71s)

3535


56 55

L115

L114

L113

38

53

51

45

BUS-107 BUS-108

BUS-109

L105

L106

L118

L117

L116

33

34

29 30

31

32

44

45

36

35

43

41

42

39

40

37

38

48

47

53

62 46

51

50

49

64

4

3

2

5

6

7

8

9

10

11 14

15 12

16

65 59

13

18

54

66

67

20

19

23

24

21

22

25

26 27

28 1

TG103

52

61 60

63

57

17

T301 BUS-301

L402

L216

3

G102 G103

G203G202

L302

L302

L302

L402

L301

L3

01

L107

L108

L109

L103

L111

L112

L110

L102

L101

T302T402

TG102

TG202 L207

L208

L209

L210

L211

L212

L203

L202

L201

L206

L204

L205

L218

L217

L213

L214

LOAD103

LOAD101

LOAD202

LOAD201

LOAD203

L104

TG203

TG201G201

BUS-103

BUS-104

BUS-105 BUS-106

BUS-102

BUS-401

BUS-205

BUS-209

BUS-208

BUS-207

BUS-203BUS-202

BUS-201

BUS-204

L401

T401

4 5

1 2

6 7

8

9 10

11

12

13

14

67

16

17

63

19

20

15

18 21

6864

22

23

24

25

26

27

28

29

30

31

32

54

55

56

59

61

60

57

52

49

50

46

47

48

43

44

40

41

42

33

34

35

65

39

37

36

66

58

101

102

103

104

105

106

107

108

109

110

111

112

113

114

116

117

119

120

122

123

124

125

126

127

128

129

130

131

132

115

118 121

151

152

153

149

150

146

147

148

143

144

145

140

141

142

154

155

156

160

159

157

137

138

139

135

134

133

136

158

103

114 113

104

101

102

105

106

107

108

109

110 112

111

115

116

117

118 119

120

121

122

123

124

125

126

127 128

146

151

150 149 148

147

153

135

136

138

137 139

140

141

142

143

144

145 133

134

129 130

131

132 155

152

154

BUS-302 BUS-402

BUS-206

Top Island

Bottom Island


t=2700s

G201 (generator

trip at t=945.27s)

under review IEEE Transactions on Power Systems under review IEEE Transactions on Power Systems under review IEEE Transactions on Power Systems

3636


Generator Trip and four tie-line trip

at 619.71s leading to island formation

Generator Trip in bottom island

at 945.27



Generator Trip and four tie-line trip

at 619.71s leading to island formation

Generator Trip in bottom island

at 945.27



3737

Numerical Methods

3838Ax b=

( ),dx f x ydt

= ( )0 ,g x y=

[ ] ( ) ( )1 1 11 , 0n n n n nx x h x hf x yθ θ+ + +− − − − =&

( )1 1, 0n ng x y+ + =

( )1 1, 0n nF x y+ + =

( ) 11 1i in nx x xγ−+ += − Δ

( ) 11 1i in ny x yγ−+ += − Δ

x

y

RxJ

Ry⎡ ⎤Δ⎡ ⎤

= ⎢ ⎥⎢ ⎥Δ⎣ ⎦ ⎣ ⎦

The power system dynamic equations can be summarized as

which, are discretized using Theta-method giving

which is summarized as a set of non-linear equations given by:

which are solved at each time step using Newton-Raphson method:

where and are solved as:

equivalent to a linear system:

xΔ yΔ

Integration algorithm

3939

Computational speed is the major goal…

Intelligent Jacobian BuildingVariable Time StepChoice of Integration AlgorithmVery fast Sparse Linear Solvers

Direct Methods• Intelligent Symbolic and Numeric Factorization

Iterative Methods• Development of robust preconditioner

Parallelization

4040

Ingredients in the solution of sparse linear systems

Ordering step:

Reorders rows and columns to reduce fill in L & U.

Symbolic factorization step:

Determine locations of nonzeros in L & U.

Set up data structures for storing nonzeros of L & U.

Allocate memory for the nonzeros.

Numerical factorization step:

Input numerical values.

Compute L & U via intelligent factorization on multiple small, but dense “fronts”, with pivoting to maintain numerical stability.

Triangular solution step:

Use L & U to perform forward and backward substitutions to solve linear system.

4141

MultifrontalMultifrontal Methods for Linear SolversMethods for Linear Solvers

6789

3789

467

569

239

137

1 X X X 2 XX X3 XXX *X*4 X XX 5 XX X6 XXX*X*7 X *X *XX* 8 X XXXX9 X* X**XX

•Not in open literature for power system time domain simulation applications

•Not Used in Commercial Grade Simulators

•Can be seen as an enabling technology for

fast simulation

4242

MultifrontalMultifrontal factorizationfactorizationAll arithmetic happens on dense square or rectangular matrices.All arithmetic happens on dense square or rectangular matrices.

Needs extra memory for a stack of pending update matricesNeeds extra memory for a stack of pending update matrices

Potential parallelism:Potential parallelism:

1.1. between independent tree branchesbetween independent tree branches

2.2. parallel dense ops on frontal matrixparallel dense ops on frontal matrix

42

4343

Performance Comparison for a 6 bus Test System

0

50

100

150

200

250

300

350

1 2 3 4 5 6

Contingency Number

Tim

e (s

econ

ds)

Gaussian

Multifrontal

43

4444

Performance Comparison for a 32-generator IEEE Test System

0

200

400

600

800

1000

1200

1400

1600

1 2 3

Contingency number

Tim

e (s

econ

ds)

GaussianMultifrontal

44

4545

SummarySummary1.1. ProbabilityProbability--based contingency selection, using based contingency selection, using

substation topology breakersubstation topology breaker--switch data, is a switch data, is a must for todaymust for today’’s EMS. No reason not to.s EMS. No reason not to.

2.2. Continuous, anticipatory computing to provide Continuous, anticipatory computing to provide decisiondecision--support corrective action for lowersupport corrective action for lower-- probability events makes sense.probability events makes sense.

•• Preparing operators for rare events is fundamental Preparing operators for rare events is fundamental to operating engineering systems having to operating engineering systems having catastrophic potential; has precedent in air traffic catastrophic potential; has precedent in air traffic control, nuclear, & process control.control, nuclear, & process control.

•• It motivates significant improvements in modeling It motivates significant improvements in modeling and computation. and computation. MultifrontalMultifrontal solvers for linear solvers for linear systems should be widely implemented in EMS systems should be widely implemented in EMS software.software.

4646

Publications1. Multifrontal Solver for Online Power System

Time Domain Simulation: under review IEEE Transactions on Power Systems

2. Probability Estimation of High Risk N-k Inadvertent Contingencies for Online Security Assessment: under review IEEE Transactions on Power Systems

3. DFS based algorithm for simulating cascading: under preparation

4. Time Domain simulator with Generation Protection: under preparation

5. A study of Blackouts in past 40 years: under preparation

operational defense of power system cascading outages ... › ecow › get › generalinf ›...

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