ali pinar seminar: a combinatorial approach to vulnerability analysis of the power grid

8/9/2019 Ali Pinar Seminar: A Combinatorial Approach to Vulnerability Analysis of the Power Grid

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Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin

Corporation, for the U.S. epartment of !nergy"s National Nuclear Security #dministration under contract !-#C$%-&%#L'($$$. S#N N). *$++-

A Combinatorial Approach toVulnerability Analysis of the

Power GridAli PinarSandia National Laboratories

UCSB Institute for ner!y


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Essentially, all models are wrong, but some

are useful. George E.

P. Box


Graph models areuseful.

#$#%$&'#( &


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Power blac)outs are a !lobal problem

Au!ust &''* blac)out a+ected (' million

people in New ,or)- Pennsyl.ania- /hio-0ichi!an- Vermont- 0assachusetts-Connecticut- New 1ersey- /ntario2

3he time to reco.er from the blac)outwas as lon! as % days at an estimatedcost of 4%5#' B

Similar occurrences elsewhere6 India7&'#&8- Bra9il and Para!uay 7&'':8-;rance5Swit9erland5Italy 7&''*8

UCSB Institute for ner!y#$#%$&'#( *


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3he !riduires the system to beresilient to only one failure-because hi!her standards arenot enforceable2

Uncertainty inherent in manyrenewable resources and theincreasin! load on the system forceus to operate closer to thefeasibility boundary2

Goal6

detect .ulnerabilities of thepower networ)

Include contin!ency analysisas a constraint in systemsplannin!

Northeast blackout started with three

broken lines.


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? y s t s pro em sohard@

Combinatorial compleity

Defender-attacker-defendergames lead to problems where

an optimization problems is a

constraint for another.

Uncertainty

Power system operations

inherently involve uncertainties,which are increasing with

renewable penetration.

Compleity of power flow

Flow of power can be modeled

by differential algebraicequations, which are difficult to

solve by themselves !ven

steady state is described by

nonlinear equations.

Need for eact solutions

!ven the basic appro"imation

lead to #P-hard problems, and

heuristics lead to compromise

from security.


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?hat can we sol.e@0a=imum5ow and minimum cut

Gi.en a !raph with capacities on ed!es- a source s- anda terminal t- push as much ow as possible from s to t2

Cut is a bipartitionin! of the .ertices into S and 3- sothat s in S and t in 32

Capacity of a cut is the cumulati.e capacity of ed!esbetween S and 32

0in5cut is a cut with minimum capacity2

Volume of ma=5ow capacity of a min5cut2

ts (/0

*/1

*/*

1/1

%/2%/%

+/(

Capacity30

Capacity3+$

#$#%$&'#( UCSB Institute for ner!y


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(

k 3+, ma-flow3 0k 3$, ma-flow3 ++

k 3*, ma-flow3 (

?hat else can we sol.e@Networ) inhibition problem

Cut min2 number of lines- )- so that ma= ow is below a

speciDed bound2 Shown to be NP5complete 7Phillips #::#82

3he classical min5cut problem is a special .ersion of networ)inhibition- where ma=5ow is set to 9ero2

Can be formulated as 0ILP with EVEFEE binary .ariables2

k 31, ma-flow3+


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How useful is this model@

The Good

?e can sol.enetwor) interdictionproblems2

The Bad

;low of power is di+erent- and thusthe results are not applicable2

The Beautiful

Understandin! the combinatorial structure

can !uide heuristic techni>ues that modelthe power ow more accurately2

;urther analysis will show that the !apbetween the two models is smaller than it

loo)s2


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Power ow is !uided by potentialdi+erence between nodes

SimpliDed model

;i= .olta!es at # wor) only on acti.e power

e>uations2 Ais the node5arc incidence matri= that describes !rid

topolo!y2 p is the state of the system2

Sin(A

) is a dia!onal matri= whose entries are the an!ulardi+erence2

#cti4e power

5eacti4e power$6 4oltage

6 phase angle

76 susceptance

2

i

j

2Vl VVu

BijViVjsin(i j )

(Vi

,i

)

(Vj

,j

)

BijViVjcos(i j )

F(A,,p) =ABsin(A)p= 0


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Pictorial representation of a blac)out

)perating point corresponding to current load

and generation

+

*

7oundary of feasibility region shifts

as we add/remo4e lines

8nfeasibility of power flow e9uations lead to a blackout.

Cascading is initiated by a significant disturbance to the system.

)ur focus is detecting initiating e4ents and analy:ing the network

for 4ulnerabilities.

UCSB Institute for ner!y#$#%$&'#(

#'


11/37

Contin!ency analysis as a bi5le.eloptimi9ation problem

Add inte!er 7binary8 line parameters- - to identify bro)en lines

0easure the blac)out se.erity as the distance to feasibility boundary 3he load sheddin! problem

Bile.el50INLP problem

cut minimumnumber of lines so that

shortestdistance to feasibility 7i2e2 se.erity8 is at least as lar!e as aspeciDed tar!et

0an!asarian ;romowit9 constraint >ualiDcation conditions are satisDed fora sli!htly modiDed system2

*

+

:

7oundary of feasibility region shiftsas we add/remo4e lines

)perating point corresponding to

current load and generation

UCSB Institute for ner!y#$#%$&'#( ##


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3his approach leads to a 0i=edInte!er Nonlinear Pro!ram 70INLP8

se4erity abo4e threshold

satisfy the ;;< optimalityconditions

feasible power flow

feasible load shedding

minimi:e number oflines cut

UCSB Institute for ner!y#$#%$&'#( #&

min,!,,1 ,2 ,3 ,4,5,6 | |s.t. F(A"(1 ),,p+!) = 0

/2 A"(1 ) /2e!g #0 pg +!g pg

pl pl +!l 0e0

gl

+

4 32 1

= 0

F

+ A"(1 )(6 5) = 0

1!l = 0; 2(pl +!l ) = 04!g = 0; 3(pg +!g) = 0;5(/2+ A"(1 )) = 0;6(/2A"(1 )) = 0;1,...,6 0 {0,1}


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Jela=ation wor)s on smallproblems

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8!!! 1$-7us System

;our candidate lines identiDed2

3wo are su"cient to cause ablac)out2

;ailure of these lines can causea blac)out with K%* 0? loss outof a total load of #(( 0?82

Solutions found usin! SN/P32

onde, et al. Se4ere Multiple Contingency Screening in !lectric ower Systems,

8!!!


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8!!! ++' 7us System

#* candidate lines are

identiDed ;ailure of these lines can

cause a blac)out with #(0? loss of load 7total load is%&'' 0?8

Better solutions e=ist


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Power5ow 1acobian corresponds to the !raphLaplacian

Mey obser.ation6he $a%obian matrix,whi%h %hara%teri!es the feasibility boundary,has the same stru%ture as the &apla%ian inspe%tral graph theory.

;easibility of the optimi9ation problem pushesthe 1acobian towards a second 9ero ei!en.alue2

Second 9ero ei!en.alue corresponds to

disconnected re!ions in the !raph2

Node-arc

incidence matri

Node-arc

incidence matriDiagonal matrices with non-

negative weights

. et al., )ptimi:ation Strategies for the =ulnerability #nalysis of the

ower >rid, S8#M ?. )ptimi:ation, =ol. *$, No. %, pages +0'2-+'+$, *$+$@

UCSB Institute for ner!y#$#%$&'#( #(

F

=$ =AB"((1 )cos(A)A


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;easible solutions ha.ea combinatorial structure

3heoretical analysis of the bile.el0INLP formulation shows6

System is split into load5richand !eneration5rich re!ions2

3here is at least onesaturated line from the!eneration rich re!ion to theload rich re!ion2

Blac)out si9e can beappro=imated by the!eneration$load mismatchand capacity of ed!es inbetween2

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ractical application6 !ploit the combinatorial structure to find a loosely

coupled decomposition with a high generation/load mismatch

UCSB Institute for ner!y#$#%$&'#( #


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18/37

Can we wor) on a nonlinear modelwithout sol.in! nonlinear e>uations@

It is not free lunch- but a !ood deal2?hy does it wor)@?e are not proposin! a power ow model-we only Dnd why power is not owin!2

3his is a ow problem2

3he !oal of the load sheddin! problem is

to ma)e this model wor)2?hat is missin!@?e can document the underlyin!assumptions that lead to the !raph model2

But we cannot >uantify the !ap betweenthe two models- yet2

?e need a representation of a typicalcases2

UCSB Institute for ner!y#$#%$&'#( #K


19/37

-*

-*-+

-1

-1

(

2

(

%2

*

*

1

*

+$k 3+, ma-flow3 0

k 3$, ma-flow3 ++

k 3*, ma-flow3 (

Networ) inhibition problem

Cut min2 number of lines so that ma= ow is below a speciDed

bound2 Shown to be NP5complete 7Phillips #::#82

3he classical min5cut problem is a special .ersion of networ)inhibition- where ma=5ow is set to 9ero2

Can be formulated as 0ILP with EVEFEE binary .ariables2

k 31, ma-flow3+

UCSB Institute for ner!y#$#%$&'#( #:


20/37

0ILP formulation for networ) inhibition


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problemcan be sol.ed e"ciently

3he inte!rality !ap is pro.ably small2

/nly one fractional .ariable after each solution2

=perimented on a simpliDed model for ?estern stateswith #*-*% nodes and #-(&' lines- used PIC/ for sol.in!the 0ILPs2

.en the lar!est instances can be sol.ed in small time-moti.atin! us for more hi!her obecti.es2

UCSB Institute for ner!y#$#%$&'#(

3a)e * Inhibitin! bisection


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3a)e *6 Inhibitin! bisectionproblem

i4ide graph into two parts bisectionD so that load/generation mismatch is maimum. cutsi:e is minimum.

imbalance3+$@ cutsi:e31imbalance3++@ cutsi:e3(

imbalance3 2@ cutsi:e3*

-*-+

-1

-1

(

2

(

%

2 *

*

1

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+$

-*


Lesieutre et al., ower system etreme e4ent analysis

using graph partitioning, N#S *$$2.

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&&


23/37

Sol.in! the inhibitin! cutproblem

Constrained problem is NP5complete2

Goal6 minimi9e 7cutsi9e( )7#5 8 imbalance

is the relati.e importance of cutsi9e compared to imbalance2

Solution6 use a standard min5cut al!orithm2

0in5cut !i.es an optimalsolution to the lineari9ed inhibitin! bisectionproblem2

/ther .ersions are sol.able 0inimi9e cutsi9e$imbalance

0inimi9e capacityO7cutsi9e5#8$cutsi9e

s

t

(

-*-+

-1

-12

(

%2

*

*

1

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+$

-*

., Aogel, and Lesieutre,


24/37

In t n! sect on ena esfast analysis of lar!e systems

=perimented on aSimpliDed model for?estern states with#*-*% nodes and #-(&'lines2

Complete analysis usin!

Goldber!


25/37

;indin! .ulnerabilities is !ood D=in! them is

better6 N5) sur.i.able networ) desi!n problem Impro.e a networ) e"ciently to ma)e

it resilient to contin!encies 0inimi9ethe impro.ement cost such that

the minimumnumber of lines for thema=imumow to be below a threshold B isabo.e a threshold C2

Solution approaches6 A sin!le problem with a separate set of

constraints for each contin!ency forms a !iant problem

Benderuirements

the number of subproblems is still .ery lar!e-prohibiti.ely e=pensi.e for lar!e N and )2

Proposed 0ethod6 elayed Contin!encyGeneration


Infrastructure/pera ons

Infrastructure

Pisrup ons

Infrastructure

Au!menta on#st Le.el

&nd Le.el

*rd Le.el

#$#%$&'#( &(

elayed contin!ency


26/37

Jestricted0aster Problem

Sub5Problem*

Sub5Problem

Sub5Problem+

xt

/JACL@/P3I0AL,S7QI38

/P3I0ALN/7AN?C/N3INGNC,8

Optcut(s)

xt#

&

*

elayed contin!ency!eneration

/utline of the al!orithm

Sol.e a restricted master problemto identify candidate lines to add2

Sol.e the networ) inhibitionproblem

If we cannot brea) the networ)-current solution is optimal

If not- add a constraint to themaster problem for the identiDed.ulnerability2

New constraints e=clude re!ionsof the solution space- as opposedto indi.idual solutions2


"cient solution of the interdiction problem is the )ey enabler2

3he same framewor) can be applied to plannin! and operationalproblems

?e ha.e applied this approach to Unit commitment

3ransmission and !eneration e=pansion

Chen et al. Contingency-Constrained Unit Commitment

with ost-Contingency Correcti4e 5ecourse,

to appear in #nnals of )perations 5esearch

#$#%$&'#( &


27/37

elayed contin!ency analysis limitscontin!encies bein! considered2

Problem6 3ransmission and Generation =pansion


Chen et al., Contingency-5isk 8nformed ower System esign,

to appear in 8!!!


28/37

;ast solutions enable analyses ofthe decision space

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#%

#

#K&'

&&

' & % K #' #& #% # #K &' && &% & &K *' *&

M#

M&M*M%

Rloadloss

Cost7

Billion48

UCSB Institute for ner!y#$#%$&'#( &K


29/37

Cost of perfectness

'

&

%

K

#'

#&

#%

#

#K

&'

&&

' & % K #' #& #% # #K &' && &% & &K *' *&

M#

M&M*M%

Rloadloss

Cost

7Billion48

Infrastructurein.estment costcanbereducedby('Rif #Rlossof loadisacceptable

UCSB Institute for ner!y#$#%$&'#( &:


30/37

BeneDts of humbleness

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K#'

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#%

#

#K

&'

&&

' & % K #' #& #% # #K &' && &% & &K *' *&

M#M&M*M%

Rloadloss

Cost7Billion48

Lossof loadT%RatM%

Lossof loadT&RatM*

Lossof loadT#RatM&

?ithabud!et of4Billion

UCSB Institute for ner!y#$#%$&'#( *'

N t St U t i ti f bl


31/37

Ne=t Step6 Uncertainties of renewablesposea crucial challen!e for !rid operations

Source6 http6//saferen4ironment.wordpress.com

Aull link

Source6 http6//www.thesierraleonetelegraph.com/Gp3(1&1

Most renewable resources cannot be

controlled and in4ol4e significant

uncertainties.

Figh penetration of renewables lead to

a significant change in operations due

to uncertainty.

Storage technologies are not ade9uate

enough, yet.

)perations re9uire decision making

under uncertainty.

Stochastic optimi:ation is essential.

7etter models for handling uncertainty

are needed.

UCSB Institute for ner!y#$#%$&'#( *#
http://saferenvironment.wordpress.com/http://saferenvironment.wordpress.com/2008/11/03/wind-energy-renewable-energy-harnesses-natural-wind-power-%E2%80%93-effective-answer-for-emission-problem-towards-cleaner-safer-and-greener-environment/http://saferenvironment.wordpress.com/2008/11/03/wind-energy-renewable-energy-harnesses-natural-wind-power-%E2%80%93-effective-answer-for-emission-problem-towards-cleaner-safer-and-greener-environment/http://saferenvironment.wordpress.com/


32/37

/perational problems re>uirestochastic optimi9ation

Sample Problem6 UnitCommitment

;undamental problem inoperations

3wo sta!e problem ecide on the state of bi! andslow !enerators under aprediction of demand$renewables

/perate on a reali9ation ofuncertainties to minimi9e!eneration costs

Standard approach6 0onte Carlosamplin!

Unit

commitment

5eali:ation

of demand

rediction

of demand

!conomic

dispatch

Setup

costs

>eneration

cost

Commitment

decisions

UCSB Institute for ner!y#$#%$&'#( *&

"cient 0odel for


33/37

"cient 0odel forUncertainty6

Polynomial Chaos =pansion rror for 0onte CarloVar7f8$s>rt7S8

Accurate estimations renderoptimi9ation problems impractical2

Proposed Solution6 Polynomial chaos

e=pansion Commonly used for uncertainty

>uantiDcation in CS applications

Core idea6 preprocess the random.ariables to build a surro!ate thatrepresents random .ariables

compactly Promisin! Initial results6

Currently wor)in! on addin! thisto the optimi9ation loop

(hiam and De)arco* +imply put, when

uncertainty is credibly accounted for such

methods yield solutions for economic

benefit of a transmission e"pansion in

which the +error bars are often larger than

the nominal predicted benefit.


Safta et al., rid )perations roblems,roc. N#S *$+%

#$#%$&'#( **


34/37

Concludin! remar)s ssentially all models are wron!- but some are useful2

Geor!e 2 P2 Bo=

Graph models are useful2 ?e ha.e made si!niDcant ad.ances in .ulnerability

analysis2 Special structure of a feasible solution to our 0INLP formulation can

be e=ploited for a simpler approach for .ulnerability detection2

/ur combinatorial techni>ues can analy9e .ulnerabilities of lar!esystems in a short amount of time2

Inte!ration of system dynamics remain as a challen!e2

?e can incorporate .ulnerability analysis as a constraintto !rid operations and plannin! problems2

elayed contin!ency !eneration approach shows promisin! results forN5)5e contin!ency constrained problems2

?e need no.el approaches for handlin! uncertainties2 0onte Carlo al!orithms cannot pro.ide the desired accuracy in

estimations2

UCSB Institute for ner!y#$#%$&'#( *%


35/37

Ac)nowled!ements

Jichard Chen- Amy Cohn- Vaibha. onde- Nen!;an- ,onatan ;o!el- Bernard Lesieutre- VanessaLope9- 1uan 0e9a- Habib Nam- Adam Jeichert-Sandip Joy- Cosmin Safta- 1ean5paul ?atson- andChao ,an! contributed to this wor)2

UCSB Institute for ner!y#$#%$&'#( *(


36/37

Announcements

?e always try to create positions for talentedresearchers2 Please contact me

Networ) Science under SIA0 *rd SIA0 ?or)shop on Networ) Science- Snowbird- U3-

0ay #5#:- &'#(2 Co5located with the SIA0 C2 ynamical Systems

Abstract due by 1an #:th2

0ailin! list for networ) science 3o subscribe- send an email to siam5ns5subscribesiam2or!

UCSB Institute for ner!y#$#%$&'#( *
http://www.siam.org/meetings/ns14/mailto:[email protected]:[email protected]:[email protected]://www.siam.org/meetings/ns14/


37/37

Jelated Publications Chen- Cohn- ;an- and P2- Contin!ency5Jis) Informed Power System esi!n-W I 32 Power

Systems Vol2 &:- No2 (- pa!es6 &'K5&': preprint a.ailable as arQi.6#*'(2'K' Chen- ;an- P2- and ?atson- Contin!ency5Constrained Unit Commitment with Post5

Contin!ency Correcti.e Jecourse-X to appear in Annals of /perations Jesearch- preprinta.ailable as arQi.6#%'%2&:% Safta- Chen- Nam- P2- ?atson- 3oward Usin! Surro!ates to Accelerate Solution of Stochastic

lectricity Grid /perations Problems-W Proc2 NAPS &'#% preprint a.ailable asarQi.6#%'%2&:%

Chen- Cohn- ;an- and P2- N5)5epsilon Sur.i.able Power System esi!n-W in #&thInternationalConference on Probabilistic 0ethods Applied to Power Systems 7P0APS#&8 arQi.6#&'#2#(*'

Chen- Cohn- and P2- An Implicit /ptimi9ation Approach for Sur.i.able Networ) esi!n-W Proc2

&'## I #stInternational Networ) Science ?or)shop 7NS? &'##8 arQi.6##':2#K'# P2- 0e9a- onde- and Lesieutre- Y

/ptimi9ation Strate!ies for the Vulnerability Analysis of the Power Grid-W #A- ptimi!ation-&' 7%8- pp2 #K5#K#'- &'#'2

onde- Lope9- Lesieutre- P2- ,an!- and 0e9a- YSe.ere 0ultiple Contin!ency Screenin! in lectric Power Systems-W EEE . Power #ystems/-&*7&8- pp2 %'5%#- &''K2

Lesieutre- P2- and Joy- YPower System =treme .ent etection6 3he Vulnerability ;rontier-X inPro%. 0+st 1awaii nt. 2onf. on #ystem #%ien%es- p2 #K%- HI- &''K2 P2- Jeichert- and Lesieutre- YComputin! Criticality of Lines in a Power System-W Pro%. 3445

EEE nt. #ymp. 2ir%uits and #ystems- pp2 (ZK- New /rleans- LA- 0ay &''2 Lesieutre- Joy- onde- and P2- YPower system e=treme e.ent analysis usin! !raph partitionin!

-W Pro%. the 67th 8orth Ameri%an Power #ymp.- Carbondale- IL- &''2 onde- Lope9- Lesieutre- P- ,an!- 0e9a-

IdentiDcation of se.ere multiple contin!encies in electric power networ)s-W Pro%. the 69th

8orth Ameri%an Power #ymp Ames IA /ctober &''(
http://arxiv.org/abs/1305.0780http://arxiv.org/abs/1404.2964http://arxiv.org/abs/1404.2964http://arxiv.org/abs/1201.1530http://arxiv.org/abs/1109.1801http://epubs.siam.org/doi/abs/10.1137/070708275http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=4483749http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=4438888http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=4252572http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=4201361http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=1560502http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=1560502http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=4201361http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=4252572http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=4438888http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=4483749http://epubs.siam.org/doi/abs/10.1137/070708275http://arxiv.org/abs/1109.1801http://arxiv.org/abs/1201.1530http://arxiv.org/abs/1404.2964http://arxiv.org/abs/1404.2964http://arxiv.org/abs/1305.0780http://arxiv.org/abs/1305.0780http://arxiv.org/abs/1305.0780

ali pinar seminar: a combinatorial approach to vulnerability analysis of the power grid

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