Towards Improving the Resilience of Power Systemsshelard/slides/2018_Aug_30_LANL.pdf · 2018. 11. 26. · J. A. Momoh, et al. -"A review of selected optimal power flow literature

TowardsImprovingtheResilienceofPowerSystems

Devendra Shelar |[email protected],2018

Collaborators:Saurabh Amin,IanHiskens

ResearchFocus

Smartgridresilience

Algorithmsforbilevel

optimizationproblems

Modelingofcyberphysical

failures

2

Outline

•Motivation:Resilience-Awareoperations

•AttackmodelsandProblemformulation

•Mainresults

3

Cyber-Physicaldisruptions

HurricaneMaria(September2017)• Customersfacing

blackoutsformonths

4

MetcalfSubstation(April2013)• Sniperattackon17

transformers• Telecommunicationcablescut• 15million$worthofdamage• 100mn $forsecurityupgrades

Ukraineattack(Dec‘15,‘16)• Firsteverblackouts

causedbyhackers• Controllersdamagedfor

months

ResearchchallengeExistingliteratureconsiders:• Physicalsecurityoftransmissionnetworks• DCpowerflowmodels

Limitedfocuson:• SmartDistributionnetworks(DNs)• Optimalattacker/defenderstrategiesbasedon:• Networktopology• Tradeoffsinresourceallocation

Myapproachcombines:• Physics-basedoptimalattack• Semantics-awaresoftwarememoryattack

5

Distributionnetworkattackscenarios

6

• Agent• Disgruntledemployee• Externalhacker• BuggySCADAimplementation

• NESCOVulnerabilities(EPRI)• Massremotedisconnectofsmartmeters• SimultaneousdisconnectofDERs• Rapidoverchargingofelectricvehicles

• Impact:supply-demanddisturbances(suddenorprolonged)

Substation

Transmission lines

Generation

Control Central

Distribution

lines

Typical communication

New communication

requirenments

Background:Security-constrainedOPF

§ EconomicDispatch problemtoensureanoperationalpowersystemdespitecontingencies

§ Accountsforappropriatecorrectiveactions forthesaidcontingency

Mainissues• OnlycapturesN-kcontingenciesforsmallk.Typicallyk=1or2• Assumesapriorifixedsetofcontingencies• Doesnotmodelstrategicattacker-inducedfailures

7

A.Monticelli,etal.- "Security-ConstrainedOptimalPowerFlowwithPost-ContingencyCorrectiveRescheduling”J.A.Momoh,etal.- "Areviewofselectedoptimalpowerflowliteratureto1993.II.Newton, linearprogrammingandinteriorpointmethods”

Ourformulation:Resilience-AwareOPF

8

Subjectto• Networkconstraints• Componentconstraints• Voltageconstraints

Minimize

Overallallocations

𝐶"##$%"&'$( + Maximize

Overalldisruptions

𝐶*$+&,%$(&'(-.(%/Minimize

Overallresponses

StageI StageII StageIII

Resilience-AwareOPF(3-Stages)

9

min3∈𝒜

𝐶36678(𝑎) + max?∈𝒟

minA∈𝒰

𝐿 𝑎, 𝑑, 𝑢


RAOPF(StagesIIandIII)

Pre-contingencystate

Worst-casepost-contingencystate

Aspecificattackscenario

10

Substation

Transmission lines

Generation

Control Central

Distribution

lines

Typical communication

New communication

requirenments

Incorrectcommands

Adversary:• HackDERSCADAanddisruptDERs• Createsupply-demanddisturbance• Causefrequencyandvoltageviolations• Inducenetworkfailures(cascades)

DistributedEnergyResources(DERS)

Distributionsubstation

𝑃HI, 𝑄HI

𝑣HI −Δv

TNleveldisturbance

Attack-inducedDNlevelsupply-demandimbalance

SOresponse

DERdisconnect-- cascade

loaddisconnect

𝑃H8, 𝑄H8

vH%

A3-regimepictureTransmissionnetwork(TN)

Microgridislanding

WhenTNandDNleveldisturbancesclear,thesystemcanreturntoitsnominalregime

11

Grid-connectedregime• Canabsorbtheimpactof

disturbances

Islandingmoderegime• Largerdisturbancesmay

forcemicrogrid islanding

Cascaderegime• Highseverityvoltage

excursions,thenmoreDERdisconnects(cascades),moreloadshedding

Ourapproach

Mostattacker-defenderinteractionscanbemodeledas• Supply-demandimbalanceinducedbyattacker• Control(reactiveandproactive)bythesystemoperator

• Abstraction:Bilevel (ormultilevel)optimizationproblems

• Supplementssimulationbasedapproaches• Forexample,co-simulationofcyberandpowersimulators

Resilience-awareOPF(StagesIIandIII)

13

StageII- Adversarialnodedisruptionsa. Whichnodestocompromise(𝛿)?

…canincludeotherattackmodels

StageIII- Optimaldispatch/response(𝑥8)a. Exerciseloadcontrolornotb. Disconnectsloads/DGs?c. Maintainvoltageregulation

…possibletoconsiderfrequencyregulationGoals:1. Identifycriticalnodes2. Determineoptimalresponse

ModelingofGrid-connected/Cascaderegimes

14

max?∈𝒟

minA∈𝒰

𝐿 𝑑, 𝑢


Networkmodel

15

𝒢 = (𝒩, ℰ)

0 𝑖 𝑗

𝑘

𝑙

𝑝𝑐\ + 𝐣𝑞𝑐\

𝑝𝑐\ + 𝐣𝑞𝑐\

𝑝𝑔6 + 𝐣𝑞𝑔6

𝑝𝑔6 + 𝐣𝑞𝑔6

𝐫ab + 𝐣𝐱ab

𝑃ab + 𝐣𝑄ab

va vb

v6

v\

vH Impedance

Powerflow

Voltages

Nominalload

Actualload

Nominalgeneration

Actualgeneration

DefendermodelinGrid-connectedregime

• Defenderresponse:onlyloadcontrol• 𝑢 = 𝛽

• 𝛽a ∈ 𝛽a , 1 :loadcontrolparameteratnode𝑖

𝑝𝑐a = 𝛽a𝑝𝑐a, 𝑞𝑐a = 𝛽a𝑞𝑐a

Defenderresponse:Howmuchloadcontrolshouldbeexercised?

16

LossesinGrid-connectedregime

17

𝐿fgh.-'i. =

Wheret ≥ max

'∈𝒩 vH($i − va

Wmg𝑃H

Costofactivepowersupply

Wno𝑡

Costoflossofvoltageregulation

qWrg,a(1 − 𝛽a)a∈s

Costofloadcontrol+ +

DefendermodelinCascaderegime

Defenderresponse:loadcontrol,connectivitycontrol𝑢 = 𝛽, 𝑘𝑔, 𝑘𝑐

𝑘𝑔a = t1, ifDG𝑖isdisconnected0, otherwise.

𝑘𝑐a = t1, ifload𝑖isdisconnected0, otherwise.Connectivityconstraintsaremixed-integerlinear:• Connectedimpliesnoviolations• Violationimpliesnotconnected

Defenderresponse:WhichloadsandDGstodisconnect?

18

Similarlyforloads!

VoltageboundsforDG

𝑘𝑔a = 0 ⟹ va ∈ vga, vga

va ∉ vga, vga ⟹ 𝑘𝑔a = 1

LossesinCascaderegime

19

𝐿g�h.-'i. ≡ 𝐿fgh.-'i. +

qW��,a𝑘𝑐aa∈s

Costofloaddisconnection

Attackermodel

Attackerstrategy:𝑑 = 𝛿, ΔvH𝛿a = t1, ifnode𝑖isattacked

0, otherwise.

q𝛿aa

≤ k

• ΔvH: amountbywhichsubstationvoltagedrops• DuetophysicaldisturbanceortemporaryfaultintheTN

Attackerstrategy:• Whichnodestocompromise? 20

Attacker’sresourcebudget

Effectofattackeractions

• DERdisruptionmakesitsoutputzero.

𝑘𝑔a ≥ 𝛿a𝑝𝑔a = 1− 𝑘𝑔a 𝑝𝑔a𝑞𝑔a = 1− 𝑘𝑔a 𝑞𝑔a

• TN-sidedisturbanceimpactssubstationvoltage

vH = vH($i − ΔvH

21

Linearpowerflows

22

Powerconservation

vH = vH($i − Δv

𝑃ab = q 𝑃b\\:b→\

+ 𝑝𝑐b − 𝑝𝑔b

vb = va − 2(𝐫ab𝑃ab + 𝐱ab𝑄ab)

𝑥 = (𝑝𝑐,𝑞𝑐, 𝑝𝑔, 𝑞𝑔, v)Systemstate

Voltagedrop

𝑄ab = q 𝑄b\\:b→\

+ 𝑞𝑐b − 𝑞𝑔b

Cascaderegime

23

ℒ ∶= max?∈𝒟

minA∈𝒰

𝐿g�h.-'i. 𝑑, 𝑢


Thisisamixed-integerbilevel linearprogram:NP-hard!

Islandingregime

24

max?∈𝒟

minA∈𝒰

𝐿��h.-'i. 𝑑, 𝑢


𝐿��h.-'i. ≡ 𝐿g�h.-'i. + Costofislanding

q W�f,ab𝑘𝑚ab(a,b)∈�

Systemresilience

• ℒ�3� = ∑ W��,aa∈s ∶maximumloss• Costofdisconnectionofallloads

• Systemresilience• Percentagedecreaseinsystemperformancerelativetomaximumloss• =100 1 − ℒ

ℒ��

25

BendersDecompositionvs.Optimal

26

Grid-connected,cascade,andIslandingregime

Grid-connectedandCascaderegime

Uncontrolled(multi-round)cascade

Inreality,defendermaynotbeabletoinstantaneouslydetectandidentifyattack,andoptimallyrespondtoit

Noresponsecascadealgorithm• Initialcontingency• Forr=1,2,…• Computepowerflows• Determinethenodesthatviolatethevoltagebounds• Disconnecttheloadsornon-controllableDGsaccordingly

27

UncontrolledvsCascaderegime

28N=36

PerformanceofBendersDecomposition

29

Res�$h+&,%"+.

= 1−𝐿

𝐿�3� 100%

Summary(sofar)

• ResourceallocationanddispatchinelectricityDNs• understrategiccyber-physicalfailures•Multi-regimedefenderresponse

• Bendersdecompositionapproachforsolvingbilevel MILPs

• Structuralresultsonworst-caseattacksanddefenderresponse

30

LearningofPowerTransmissionDynamicsfrompartialPMUobservations

Devendra Shelar |[email protected],2018

Collaborators:AndreyLokhov,NathanLemons,SidhantMisra,MarcVuffray

Motivation

• Stateestimation• Optimalresourceallocationforimprovedresiliency• Secureandefficientoperations

• Dynamicmodelestimation• Detectionoffaults/attacks• Promptandaccurateresponse

• Data-drivenapproach

32

Preliminaries

• Dynamicalequation:𝑥�� = 𝐴𝑥� + 𝐹𝑣�• 𝐴 ∈ 𝑅s×s :dynamicmatrix:,• 𝑥� ∈ 𝑅s ∶statevector• 𝑣� ∈ Rs:Noisevector• 𝐹 :Noise-scalingmatrix

33

Assumptions• Temporalindependenceofnoisevectors• 𝑣a and𝑣b areindependentforall𝑖 ≠ 𝑗

• Spatialindependenceofnoisevectors• 𝐹 isadiagonalmatrix(thereisnospatialmixingofnoise)

Learningunderfullobservability

Given:observations𝑥�for𝑡 = 1,2,⋯ , 𝑛 + 1Result:• MaximumlikelihoodestimatorofA[1]

𝐴¦ = Σ�,�ΣHWhere

ΣH =1𝑛q

𝑥�𝑥�¨I

�©�

andΣ� =1𝑛q

𝑥��𝑥�¨I

�©�

• Alsothesolutionofleastsquaresregression[1]A.Lokhov etal.OnlineLearningofPowerTransmissionDynamics 35

LinearSwingDynamicsmodel

• Network 𝒱, ℰ• 𝒱 setofnodes,𝑁 = |𝒱| numberofnodes• ℰ setofedges

Swingequation𝑀a𝜃a + 𝐷a 𝜃a − 𝜔H = 𝑃a

� − 𝑃a¸

• 𝑃a� : mechanicalpowerinput

• -𝑃a¸ : electricalpoweroutput

36

Powersystemmodel

Usingchangeofvariables• 𝛿a : phasedeviationsfromsteadystatevalues• 𝜔a :relativegeneratorrotorspeedrelativenominalfrequency

𝑀a 𝜔a + 𝐷a𝜔a = − q 𝛽ab 𝛿a − 𝛿ba,b ∈ℰ

+ 𝛿𝑃a

��

= 0s×s 𝐼s×s−𝑀,�𝐿 −𝑀,�𝐷

½¾

𝛿𝜔 + 0 0

0 𝑀,�0s𝛿𝑃

37

Discretedynamicalmodel

• Usingdiscretizationwithtimestep T• 𝐴 = (𝐼 + 𝐴?𝑇)

𝛿��𝜔��ÂÃÄ

=𝐼s×s 𝑇𝐼s×s

−𝑇𝑀,�𝐿 𝐼s×Å − 𝑇𝑀,�𝐷½

𝛿𝜔Æ�Â

+ 0 00 𝑇𝑀,�

Ç

0s𝛿𝑃ÈÉÂ

𝑥�� = 𝐴𝑥� + 𝐹𝑣�

38

Learningunderpartialobservability

• ℋ ⊆ 𝒱 setofhiddennodes(withoutPMUs)• 𝒪 = 𝒱 ∖ℋ setofobservablenodes(withPMUs)

39

1 2

4 3

56ℋ

Rearrangementofdynamicmatrix𝛿��𝒪

𝜔��𝒪

𝛿��ℋ

𝜔��ℋ

= 𝐴𝒪𝒪 𝐴𝒪ℋ𝐴ℋ𝒪 𝐴ℋℋ

𝛿�𝒪

𝜔�𝒪

𝛿�ℋ

𝜔�ℋ

+ 𝐺 00 𝐻

0𝑣�𝒪0𝑣�ℋ

Bychangeofnotation,

𝑦��𝑧�� = 𝐵 𝐶

𝐷 𝐸𝑦�𝑧� + 𝐺 0

0 𝐻𝑢�𝑤�

Problemstatemement• Givenmeasurementsfromobservablenodes𝑦�for𝑡=1,2,⋯ , n• Goal:TorecoverdynamicmatrixA• Orequivalently,recoversub-matricesB,C,D,E

Somesimpleobservations• Stablesystemimplies

|𝜆�3� 𝐸 | ≤ |𝜆�3� 𝐴 | < 1• Thus,𝐸\ ≈ 0 forsufficientlylarge𝑘

• Largesusceptance valuesimplymoreunstablesystem

41

𝑘 = 250

42

Eliminatinghiddennodemeasurements

∴ 𝑦��\��¨ = 𝑦��\¨ 𝑦��\,�¨ ⋯ 𝑦�′

𝐵′(𝐶𝐷)′⋮

(𝐶𝐸\,�𝐷)′

+ 𝐺𝐶𝐻⋯𝐶𝐸\,�𝐻

𝑢��\𝑤��\,�

⋮𝑤�

¨

𝑦��\¨ = 𝑌�¨𝑋 + 𝜂�

Connectivityrestrictions

• Eachobservablenodeisconnectedtoatmostonehiddennode• { 𝑜, ℎ ∈ ℰ: ℎ ∈ ℋ | ≤ 1∀𝑜 ∈ 𝒪

• Eachhiddennodeisconnectedtoexactlyoneobservablenode• { 𝑜, ℎ ∈ ℰ: 𝑜 ∈ 𝒪 | ≤ 1∀ℎ ∈ ℋ

43

1 2

4 3

56ℋ

44

Somesimpleproperties

𝑦��\��¨ = 𝑦��\¨ 𝑦��\,�¨ ⋯𝑦� ′

𝐵′(𝐶𝐷)′⋮

(𝐶𝐸\,�𝐷)′

+ 𝐺𝐶𝐻⋯𝐶𝐸\,�𝐻

𝑢��\𝑤��\,�

⋮𝑤�

¨

𝑦��\¨ = 𝑌�¨𝑋 + 𝜂�

Properties• 𝐺 isdiagonalbyassumption• Underconnectivityrestriction,forall𝑚 = 0,1,⋯ , 𝑘 − 1, 𝐶𝐸�𝐻 isoftheform

0 0x 0 ,where

• x ∈ 𝑅𝒪×ℋ with• exactly1non-zeroentrypercolumn,and• atmost1non-zeroentryperrow.

Implications

Fortimesteps 𝑡 = 𝑖, 𝑖 + 𝑘, 𝑖 + 2𝑘,⋯ • Thenoisevectors𝜂� satisfybothtemporalandspatialindependence• Thus,wecanuseleastsquaresestimator

𝑦��\��¨

𝑦��å\��¨

⋮𝑦��8\��¨

=

𝑌�¨𝑌��\′⋮

𝑌��8\ ′

𝑋 + 𝜂�

𝑟 ≈ SX

45

Leastsquaresestimator

• 𝑋è = 𝑆¨𝑆 ,� 𝑆¨𝑟 ,orequivalently,

𝐵′(𝐶𝐷)′⋮

(𝐶𝐸\,�𝐷)′

=

ΣH Σ� ⋯Σ\Σ,� ΣH ⋯Σ\,�⋮Σ,\

⋮Σ,\��

⋮⋯ ΣH

,� Σ\Σ\,�⋮ΣH

Where

Σa =1

𝑙 − 𝑗 + 1q𝑦b\�a

6

b©�

𝑦b\¨

• Allows,recoveryofBmatrixinastraightforwardmanner.

Recoveringsubmatrices C,E,andD

• Undertheconnectivityrestrictions,𝐶 and𝐷 aresparsematricessuchthat𝐶 = 0 0

x 0 , 𝐷 = 0 0z 0 and𝐸a = 𝑅�a 𝑅åa

Rê' 𝑅ëa• x ∈ 𝑅𝒪×ℋ withexactly1non-zeroentrypercolumnandatmost1non-zeroentryperrow.

• z ∈ 𝑅ℋ×𝒪 withexactly1non-zeroentryperrowandatmost1non-zeroentrypercolumn.

• Rba ∈ 𝑅ℋ×ℋ isadiagonalmatrixforj = 1,2,3,4and𝑖 = 1,2,⋯

• Hence,givenvaluesof𝐶𝐸�𝐷, 𝐶𝐸å𝐷and𝐶𝐸ê𝐷,arerelativelysimplernon-linearexpressionsofentriesinC,EandD.

Concludingremarks

Summary• Connectivityrestrictioncanbeleveragedtolearnthedynamicalmodelwithpartialobservability.• Thesepropertiesmaybeapplicabletootherdomains

• Identifyingpropertiesofnon-linearoptimizationmodel

Futurework• Relaxingassumptionssuchasconnectivityrestrictionandusingsmallervaluesof𝑘.

48

Questions?

Thankyou

49

BendersDecompositionapproach

• Reformulatebudget-k-max-loss problemastarget-loss-min-cardinality problem.Let𝐿�3ïð¸� beminimumtargetloss.

50

AttackerMasterproblem• Initializewithnocuts

min q𝛿aa

s. t. Bendercuts𝛿a ∈ {0,1}

Defenderproblem(SameasStageIII)

minA∈𝒰

𝐿(𝛿, 𝑢)s.t.• Networkconstraints• Componentconstraints• Voltagebounds


51

AttackerMIP

min q𝛿aa

s. t. Benderscuts𝛿a ∈ {0,1}

DefenderMIP

minA∈𝒰

𝐿(𝛿⋆, 𝑢)

𝐿𝑃(𝛿⋆, 𝑢ô⋆)

𝑢⋆ = (𝑢ô⋆, 𝑢õ⋆ )

𝛿⋆

𝛿⋆

Benderscut

𝐿 𝛿⋆, 𝑢⋆≥ 𝐿�3ïð �

yes

no

Exit𝑢⋆


52

min 𝑐ö𝑦𝑠. 𝑡. 𝐴𝑦 ≥ 𝑏 + 𝑄𝛿a�¸ï

𝐿𝑃 𝛿a�¸ï,𝑢ô ≡

𝜆⋆ö 𝑏 + 𝑄𝛿 ≥ 𝐿�3ïð¸� + 𝜖

52

Fixedattackerstrategyforcurrentiteration

Responsewithfixedintegervalues

Benderscut

OptimaldualvectorsolutiontoLP RighthandsideofLP

Smallnumber≈ 10,ú

TechnicalDetail

• BadBenderscutsmayarise• IfnoStageIIIconstraintshavenon-zerocoefficientsforbothattackvariablesandcontinuousinnervariables• Whichindeedisthecaseinourproblem!• Mayperformasbadlyasbruteforce!

• Suggestion!Approximatereformulation?• Ensurepositivecoefficientsofattackvariablesinconstraintshavingcontinuousinnervariables• Significantcomputationalspeed-up• Solutionsfor118nodenetworkobtainedinlessthan2minutes

• Approximationerrorproducessub-optimalmin-cardinalityattacks53

Resilience-AwareOPF- Trilevelformulation

54

min3∈𝒜

𝐶36678(𝑎) + max?∈𝒟

minA∈𝒰

𝐿 𝑎, 𝑑, 𝑢


pre-contingencystate𝑥7

post-contingencystate𝑥8

Resiliency-awareResourceAllocation(StageI)

StageI- AllocationofDERsoverradialnetworksa. Sizeandlocationb. Activeandreactivepowersetpoints (𝑥I)?

Resource

allocation

BG supply

Supply-demand

Balance

Flexible

Loads

Supply

Reserves

Total

capacity

DERs

Supply

Reserves

55

Suppose,somecontrollableDERsarenotvulnerabletoattack.

Frequencydeviationmodel𝑓($i − 𝑓8 = −𝑓h.- 𝑃H7 − 𝑃H8

Voltagedeviationmodelv($i − vH8 = −vh.- 𝑄H7 − 𝑄H8

Pre-contingencyresourceallocation𝑎 = (𝑝𝑔7, 𝑞𝑔7)

56

Resiliency-AwareOPF- Trilevelformulation

DefenderResponseandAllocation:Diversification

57

• SomeDERscontributeto𝐿nomorethan𝐿mg,andviceversa

• Diversificationholdsfor“heterogeneousallocation”withdownstreamDERswithmorereactivepower

• Post-contingencylossesarethesameforuniformvs.heterogeneousresourceallocations

• Pre-contingencyvoltageprofileisbetterforheterogeneousresourceallocation

2

5

6

7

8 12

11

9

1

0

3

4

10

Left lateral (l)

AC > V R

Right lateral (r)

V R > AC

Attacked

EV nodes

GoingfromLPFtoNPF

Lowerandupperboundtheoptimallossfornon-linearpowerflowswithoptimallossescomputedusinglinearpowerflows.

Theorem:Letℒ,ℒ¦ , andℒü denotetheoptimallossesusingNPF,LPF,andϵ-LPFrespectively.Then,

ℒ¦ ≤ ℒ ≤ ℒü +𝜇𝑁

2𝜇 + 4 .

Remarks• For𝜇 = 0.5, 𝑁 = 37,

!s

å!�ë=3.7.Withtypicalϵ (max.ratiooflinelossto

powerflows),thegapbetweentheboundsissmall(3-5%).

58

Ourcontributions

Bilevel problem

Regime?

59

[1]Shelar D.andAmin.S- "SecurityassessmentofelectricitydistributionnetworksunderDERnodecompromises”[2]Shelar D.,Amin.SandHiskens I.– “TowardsResilience-AwareResourceAllocationandDispatchinElectricityDistributionNetworks”[3]Shelar D.,SunP.,Amin.SandZonouz S.- “CompromisingSecurityofEconomicDispatchsoftware”

Attackermodel

Regulationobjectives

Defendermodel

Grid-Connected regime Cascade/Islanding regimes

DERdisruptions• GreedyApproach• IEEETCNS2016[1]

DNvulnerability tosimultaneousEVovercharging [2]

SecurityofEconomicDispatch• KKTbasedreformulation• DSN2017[3]

Multiple regimes• Innerproblem:mixed-integervars• Bendersdecomposition

UncontrolledvsCascadevsIslanding

60

ValueoftimelyIslandingValueoftimelydisconnections

N=24

StrategicdeploymentofportableDERsforpost-hurricanepowerrestorationefforts

61

SN1SN2

SN3 SN4

• Asimplerproblem• Given

• setofsubnetworks• repairtimesoflines• inventoryofportableDERswithvaryingcapabilities

• Question• WhatisoptimaldeploymentofportableDERssuchthatlostdemandisminimized?

PortableDERsforpowerrestoration

• Morechallengingproblem• WhatistheoptimaldeploymentofportableDERsbeforethehurricanetominimizeexpectedlostdemand?

62

Powercomponent

failuresmodel

Stormwindfield

simulation

Networksimulation,outage

prediction

Optimalresourceallocation

Technicaldetail

63

𝑘𝑔a ≥ 𝛿a𝑝𝑔a = 1 − 𝑘𝑔a 𝑝𝑔a𝑞𝑔a = 1− 𝑘𝑔a 𝑞𝑔a

0 ≥ 01 ≥ 01 ≥ 1

Originalconstraints LPconstraints

𝑘𝑔a ≥ 𝛿a𝑝𝑔a = 1− 1 − 𝜂 𝑘𝑔a − 𝜂𝛿a 𝑝𝑔a𝑞𝑔a = 1 − 1− 𝜂 𝑘𝑔a − 𝜂𝛿a 𝑞𝑔a

Reformulatedconstraints:Choose𝜂 = 10𝜖 Cases𝛿a = 1, 𝑘𝑔a = 1✔𝑘𝑔a = 0, 𝛿a = 0✔𝑘𝑔a = 1, 𝛿a = 0❓

GoingfromLPFtoNPFTheorem:Letℒ, ℒ¦, andℒübeoptimalsolutionstoattacker-defendergameunderNPF,LPF,andϵ-LPFrespectively;anddenotetheoptimallossesby,respectively.Then,

ℒ¦ ≤ ℒ ≤ ℒü +𝜇𝑁

2𝜇 + 4.

Remarks• Voltagesforℒ¦ 𝑟𝑒𝑠𝑝. ℒü upper(resp.lower)boundvoltagesforℒ• Powerflowsforℒ¦ 𝑟𝑒𝑠𝑝. ℒü lower(resp.upper)boundpowerflowsforℒ

• For𝜇 = 0.5, 𝑁 = 37,!så!�ë =3.7.Withtypicalϵ (max.ratiooflinelosstopowerflows),

thegapbetweentheboundsissmall(3-5%).• Betterboundscanbederived

64

Twosimplerproblems

≡max#min$𝐿 𝑥 𝛿,𝜙

s. t. constraints,linearpowerflow LPF or(ϵ − LPF)

65

ℒ( (LPFmodel)ℒ) (ϵ-LPFmodel)

ϵ-LPFstate:𝑥* = v*,ℓ, , 𝑠𝑐, 𝑠𝑔, 𝑆ü ∈ 𝒳)

𝑆üab = ∑ 𝑆üb\\ + (1 + ϵ)𝑠bvb. = va. − 2𝐑𝐞 𝑧ab𝑆¦ab

LPFstate:𝑥2 = v2, ℓè, 𝑠𝑐, 𝑠𝑔, 𝑆¦ ∈ 𝒳(

𝑆¦ab = ∑ 𝑆¦b\\ + 𝑠b + 𝑧abℓabvb3 = v2a − 2𝐑𝐞 𝑧ab𝑆¦ab + 𝑧ab

åℓab

ϵ chosenbasedonthesizeofthetreenetworkandthemaxratiooflinelossestopowerflows

Structureofattacks

0 5 10

M

0

500

1000

1500

LLC(in$)

W

C= 2

W

C= 10

W

C= 18

BF

GA

BC NPF

BC LPF

• Downstreamnodesaremorecriticalforvoltageregulation• Greedyapproachcomputes“near-optimal”solutions• Loadcontrolisnoteffectiveforhigherintensityattacks• LoadcontrolreacheshighersaturationlevelsforhigherweightageforLVR

0 5 10M

0

200

400

600

800

1000

1200

LVR(in$)

W

C= 2

W

C= 10

W

C= 18

BFGABC NPFBC LPF

Defendermodel(Cascaderegime)

Defenderresponse:𝑢 = 𝛽, 𝑘𝑔, 𝑘𝑐

𝑘𝑔a = t1, ifDG𝑖isdisconnected0, otherwise.

𝑘𝑐a = t1, ifload𝑖isdisconnected0, otherwise.Connectivitycondition:

𝑘𝑔a = 0 ⟹ va ∈ vga , vga

va ∉ vga , vga ⟹ 𝑘𝑔a = 1

Defenderresponse:WhichloadsandDGstodisconnect?

67

Similarlyforloads!

VoltageboundsforDG

StrategicdeploymentofportableDERsforpost-hurricanepowerrestorationefforts• Damagetolinesresultinsubnetworks (SNs)

• Usualrestorationstepsare:• Repairthedamagedlines• Connecttomaingrid• Restorethepowersupply

• HowcanportableDERshelp?

68

SN1SN2

SN3 SN4

Literaturesurvey

(T1)Interdictionandcascadingfailureanalysisofpowergrids• R.Baldick,K.Wood,D.Bienstock:NetworkInterdiction,Cascades• A.Verma,D.Bienstock:N-kvulnerabilityproblem• D.Papageorgiou,R.Alvarez,etal.:Powernetworkdefense• X.Wu,A.Conejo:GridDefensePlanning

(T2)Data-integrityattacks• E.Bitar,K.Poolla,AGiani:Dataintegrity,Observability• H.Sandberg,K.Johansson:Securecontrol,networkedcontrol• B.Sinopoli,J.Hespanha:Secureestimationanddiagnosis

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• SomeDERscontributeto𝐿nomorethan𝐿mg,andviceversa

2

5

6

7

8 12

11

9

1

0

3

4

10

Left lateral (l)

AC > V R

Right lateral (r)

V R > AC

Attacked

EV nodes

70


Amin

• Diversificationholdsfor“heterogeneousallocation”withdownstreamDERswithmorereactivepower

2

5

6

7

8 12

11

9

1

0

3

4

10

Left lateral (l)

AC > V R

Right lateral (r)

V R > AC

Attacked

EV nodes

71

• Post-contingencylossesarethesameforuniformvs.heterogeneousresourceallocations

• Pre-contingencyvoltageprofileisbetterforheterogeneousresourceallocation

Heterogeneousresourceallocationcansupportmoreloadsthanuniformone.


Amin

2

5

6

7

8 12

11

9

1

0

3

4

10

Left lateral (l)

AC > V R

Right lateral (r)

V R > AC

Attacked

EV nodes

72

Effectofpowerfactoronlosses

73N=36N=12

Optimalattackerset-pointsTypically,

• Smalllinelosses:incomparisontopowerflows

• Smallimpedances:sufficientlysmalllineresistances

Assumeforsimplicity:

• Noreversepowerflows:powerflowsfromsubstationtodownstream

74

Whatareoptimalattackerset-points?

Proposition:Foradefenderaction𝜙,andgivenattackerchoiceof𝛿,theoptimalattackerset-pointisgivenby:

𝑝𝑑3⋆ = 0, 𝑞𝑑3⋆ = −𝐣𝒔𝒈𝒊

GreedyApproach[Mm8 ]max

?minA𝐿è 𝑥2 𝑑,𝑢 Mm8 − A 𝑑¦⋆ = argmax

?𝐿è 𝑥2 𝑑,𝑢;

[Mm]max?

minA𝐿 𝑥 𝑑, 𝑢

[Mm< ]max?minA

𝐿, 𝑥* 𝑑, 𝑢 Mm< − A 𝑑ü⋆ = argmax?𝐿, 𝑥* 𝑑, 𝑢

Mm= − D 𝑢⋆ = argminA𝐿 𝑥 𝑑; , 𝑢 𝒟)\⋆ 𝑢; ≡ 𝒟(\⋆ 𝑢;convergence

𝑢;

𝑢;𝑑;

75

Forfixeddefenderaction:• Forafixedattackeraction,theorderingofnodeswithrespecttotheirvoltagesremainthesamebetweenℒ¦ andℒü

• Foranyfixednode,theorderingofoptimalattackeractionswithrespecttotheirimpactonthisnoderemainsthesamebetweenℒ¦ andℒü

Defendermodel• Defenderresponse:𝑢 = 𝑝𝑟,𝑞𝑟, 𝛽

• 𝑝𝑟a, 𝑞𝑟a :activeandreactivepoweroutputofreserves(controllableDGs)atnode𝑖• 0 ≤ 𝑝𝑟a ≤ 𝑝𝑟a , 𝑝𝑟aå + 𝑞𝑟aå ≤ 𝑠𝑟>aå

• 𝛽a ∈ 𝛽a,1 :loadcontrolparameteratnode𝑖• 𝑝𝑐a = 𝛽a𝑝𝑐a, 𝑞𝑐a = 𝛽a𝑞𝑐a

Defenderresponse:Howtooptimallydispatchreserves?Howmuchloadcontrolshouldbeexercised?

76

Optimalinterdictionplan:fixeddefenderchoicesPropositionForatreenetwork,givennodes𝑖 (pivot),𝑗, 𝑘 ∈ 𝒩:• IfDGsat𝑗, 𝑘 arehomogeneousand𝑗 isbefore𝑘 w.r.t.𝑖,thenDGdisruptionat𝑘 willhavesmallereffecton𝜈a (relativetodisruptionat𝑗)• IfDGsat𝑗, 𝑘 arehomogeneousand𝑗 isatthesamelevelas𝑘 w.r.t.𝑖,thenDGdisruptionsat𝑗 and𝑘 willhavethesameeffecton𝜈a

77

SusbstationC.C.esg esga

Attack strategy

esga

Resiliency-awareResourceAllocation(StageII)

StageII- Adversarialnodedisruptionsa. Whichnodestocompromise(𝛿)?b. Set-pointmanipulation(𝑠𝑝3)?

78

…canincludeotherattackmodels

Resiliency-awareResourceAllocation


StageI- AllocationofDERsoverradialnetworksa. Sizeandlocationb. Activeandreactivepowersetpoints (𝑥I)?

StageIII- Optimaldispatch/response(𝑥8)a. Maintainvoltageb. Exerciseloadcontrolornot

Goals:1. Determinethebestresourceallocation2. Identifyvulnerable/criticalnodes3. Determineoptimaldispatchpost-contingency

79

Resiliency-awareResourceAllocation


StageIII- Optimaldispatch/response(𝑥8)a. Maintainvoltageb. Exerciseloadcontrolornot

Goals:1. Identifyvulnerable/criticalnodes2. Determineoptimaldispatchpost-contingency

80

Towards Improving the Resilience of Power Systemsshelard/slides/2018_Aug_30_LANL.pdf · 2018. 11. 26. · J. A. Momoh, et al. -"A review of selected optimal power flow literature

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