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
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
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
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
62
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
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)
Initiating Event G102 trips at 290 seconds
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
Initiating Event G102 trips at 290 seconds
Without Generator Protection
First Control Action (reconfiguration) to avoid generator trip at 595.142 seconds
Second Control Action (Load Shedding) to avoid generator trip
Circuit loading with and without generator protection
Voltage response of system with and without generator protection
Predicting Response and Identifying Control Actions
Without Generator Protection
First Control Action
Second Control Action
3030
ONE-LINE DIAGRAM OF THE TEST SYSTEM
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
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
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
62
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
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
Without Generator Protection
First Generator Trip at 619.71s
Second Generator Trip at 925.37
Third Generator Trip at 1260.17
Initiating contingency: L215
Without Generator Protection
First Generator Trip at 619.71s
Second Generator Trip at 925.37
Third Generator Trip at 1260.17
Initiating contingency: L215
WithoutGenerator Protection
First Generator
Trip at 619.71s
Second Generator
Trip at 925.37
Initiating contingency: L215
WithoutGenerator Protection
First Generator
Trip at 619.71s
Second Generator
Trip at 925.37
Initiating contingency: L215
Voltage response of system with and without generator protection
Circuit loading with and without generator protection
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
ONE-LINE DIAGRAM OF THE TEST SYSTEM
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
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
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
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44
40
41
42
33
34
35
65
39
37
36
66
62
58
101
102
103
104
105
106
107
108
109
110
111
112
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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
Load ramping 20%from t=900s to
t=2700s
L215 (initiating event at t=300s)
L401 (line trip at t=619.71s)
L402 (line trip at t=619.71s)
L302 (line trip at t=619.71s)
L301 (line trip at t=619.71s)
G101 (generator
trip at t=619.71s)
3535
ONE-LINE DIAGRAM OF THE TEST SYSTEM
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
Load ramping 20%from t=900s to
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
Without Generator Protection
Generator Trip and four tie-line trip
at 619.71s leading to island formation
Generator Trip in bottom island
at 945.27
Initiating contingency: L215
Without Generator Protection
Generator Trip and four tie-line trip
at 619.71s leading to island formation
Generator Trip in bottom island
at 945.27
Initiating contingency: L215
Circuit loading with and without generator protection
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