View
3
Download
0
Category
Preview:
Citation preview
TowardsImprovingtheResilienceofPowerSystems
Devendra Shelar |shelard@mit.eduAugust30,2018
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
Subjectto• Networkconstraints• Componentconstraints• Voltageconstraints
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
Subjectto• Networkconstraints• Componentconstraints• Voltageconstraints
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
Subjectto• Networkconstraints• Componentconstraints• Voltageconstraints
Thisisamixed-integerbilevel linearprogram:NP-hard!
Islandingregime
24
max?∈𝒟
minA∈𝒰
𝐿��h.-'i. 𝑑, 𝑢
Subjectto• Networkconstraints• Componentconstraints• Voltageconstraints
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 |shelard@mit.eduAugust30,2018
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
BendersDecompositionapproach
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⋆
BendersDecompositionapproach
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
Subjectto• Networkconstraints• Componentconstraints• Voltageconstraints
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
69
DefenderResponseandAllocation:Diversification
• 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
DefenderResponseandAllocation:Diversification
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.
DefenderResponseandAllocation:Diversification
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
StageII- Adversarialnodedisruptionsa. Whichnodestocompromise(𝛿)?b. Set-pointmanipulation(𝑠𝑝3)?
StageI- AllocationofDERsoverradialnetworksa. Sizeandlocationb. Activeandreactivepowersetpoints (𝑥I)?
StageIII- Optimaldispatch/response(𝑥8)a. Maintainvoltageb. Exerciseloadcontrolornot
Goals:1. Determinethebestresourceallocation2. Identifyvulnerable/criticalnodes3. Determineoptimaldispatchpost-contingency
79
Resiliency-awareResourceAllocation
StageII- Adversarialnodedisruptionsa. Whichnodestocompromise(𝛿)?b. Set-pointmanipulation(𝑠𝑝3)?
StageIII- Optimaldispatch/response(𝑥8)a. Maintainvoltageb. Exerciseloadcontrolornot
Goals:1. Identifyvulnerable/criticalnodes2. Determineoptimaldispatchpost-contingency
80
Recommended