Post on 19-Nov-2020
ApplicationsofBilevelMixed-IntegerProgrammingtoPowerSystemsResilience
Devendra ShelarJointworkwithSaurabh AminandIanHiskens
January19,2018
Outline
•Motivation•Modeling• Networkmodel• Generalizeddisruptionmodel• Multi-regimeSystemOperator(defender)model
• Grid-connected,cascade,islanding
• Bilevel formulation• Bendersdecomposition
• Resourcedispatch• ControllableDGs,islandingcapabilities• Trilevelformulation– solutionapproach
2
Cyberphysical disruptions
HurricaneMaria(September2017)• Customersfacing
blackoutsformonths
3
MetcalfSubstation(April2013)• Sniperattackon17
transformers• Telecommunicationcablescut• 15million$worthofdamage• 100mn $forsecurityupgrades
Ukraineattack(Dec2015-2016)• Firsteverblackouts
causedbyhackers• Controllersdamagedfor
months
Attackscenarios
8
=>supply-demandimbalance(sudden/prolonged)
ThreeregimesofSOoperation
Distributionsubstation
𝑃"#, 𝑄"#
𝑉"#
𝑃"', 𝑄"'
𝑉"'
Transmissionnetwork
−Δ𝑉
TNleveldisturbance
Attack-inducedDNlevelsupply-demandimbalance
SOresponse
DERdisconnect-- cascade
loaddisconnect
Microgridislanding
WhenTNandDNleveldisturbancesclear,thesystemcanreturntoitsnominalregime
5
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• Flexibletoallowforbothcontinuousanddiscretevariables• Goodsolutionapproaches:Duality,KKTconditions,Benderscut,MILP• Providepracticallyusefulinsightstodeterminecriticalscenarios
• Supplementssimulationbasedapproaches• Forexample,co-simulationofcyberandpowersimulators
Ourcontributions
Bilevel problem
Regime?
7
[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
RelatedWork(partial)(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)Cyber-physicalsecurityofnetworkedcontrolsystems• E.Bitar,K.Poolla,AGiani:Dataintegrity,Observability• H.Sandberg,K.Johansson:Securecontrol,networkedcontrol• B.Sinopoli,J.Hespanha:Secureestimationanddiagnosis• T.Basar,C.Langbort:Networksecuritygames
8
NetworkmodelPowerflowontreenetworks- Baran andWumodel(1989):• 𝒢 = (𝒩, ℰ)– treenetworkofnodesandedges• 𝑝𝑐2 , 𝑞𝑐2 - realandreactivenominalpowerdemandatnode𝑖• 𝑝𝑔2, 𝑞𝑔2 - realandreactivenominalpowerfromuncontrollablegenerationatnode𝑖
• 𝑉2- voltagemagnitudeatnode𝑖• z28 = r28 + 𝐣x28 - impedanceonline(𝑖, 𝑗)• 𝑃28, 𝑄28 - realandreactivepowerfromnode𝑖 to node𝑗• 𝑝2, 𝑞2 - netrealandreactivepowerconsumedatnode𝑖
9
GeneralizeddisruptionmodelAttackerstrategy:𝑎 = 𝛿, 𝑝𝑑A,𝑞𝑑A, Δ𝑉"• 𝛿:attackvector,with𝛿2 = 1 ifnode𝑖 isattackedand0otherwise• Satisfy∑ 𝛿22 ≤ 𝑀 (attacker’sresourcebudget)
• 𝑝𝑑2A, 𝑞𝑑2A - attacker’sactive/reactivepowerdisturbanceatnode𝑖(generalmodel:capturesvariousattackscenarios)
• Δ𝑉": voltagedropatsubstationnode• DuetophysicaldisturbanceortemporaryfaultintheTN
Attackerstrategy:• Whichnodestocompromise?• Whatset-pointstochoose?
10
Defendermodel:Grid-connectedregime
Defenderresponse:𝑑 = 𝛽
• 𝛽2 ∈ 𝛽2,1 :loadcontrolparameteratnode𝑖• 𝑝𝑐2 = 𝛽2𝑝𝑐2, 𝑞𝑐2 = 𝛽2𝑞𝑐2
Defenderresponse:Howmuchloadcontrolshouldbeexercised? 11
𝑝𝑐2 , 𝑞𝑐2 - nominalpowerdemandatnode𝑖
Defendermodel:Cascaderegime
Defenderresponse:𝑑 = 𝛽, 𝑘𝑐, 𝑘𝑔
• 𝑘𝑐2 =0ifloadisconnected,1otherwise.• 𝑘𝑔2 =0ifuncontrolledDGisconnected,1otherwise.
• Voltageconstraintsforconnectivity:𝑘𝑐2 = 0 ⟹ 𝑉2 ∈ 𝑉'2,𝑉'
L2
𝑘𝑔2 = 0 ⟹ 𝑉2 ∈ 𝑉M2, 𝑉ML 2
Defenderresponse:
WhichloadsandDGstodisconnect? 12
voltageboundsforload(resp.generation)connectivity
Defendermodel:Islandingregime
Defenderresponse:𝑑 = 𝛽, 𝑘𝑐, 𝑘𝑔, 𝑝𝑟, 𝑞𝑟, 𝑘𝑚
• 𝑝𝑟, 𝑞𝑟 - dispatchofresources(DERs)• 𝑘𝑚28 =1,ifline 𝑖, 𝑗 ∈ 𝜒isopen,0otherwise.
• Microgrid formationaffectspowerflowsandvoltages:
𝑘𝑚28 = 1 ⟹ Q𝑃28 = 𝑄28 = 0𝑉8 = 𝑉RST
𝑘𝑚28 = 0 ⟹ 𝑝𝑟8 = 0, 𝑞𝑟8 = 0
Defenderresponse:
Whichlinestodisconnect? 13
𝜒 - setoflineswhichcanbedisconnectedtoformmicrogrids
Powerflowconstraintsbeforedisruption
• Netpowerconsumedatanode
• LinearPowerflows(LPF)
• Voltagedropequation
14
𝑃28 = U 𝑃8VV:8→V
+ 𝑝2
𝑄28 = U 𝑄8VV:8→V
+ 𝑞2
𝑉8 = 𝑉2 − (r28𝑃28 + x28𝑄28)
𝑝2 = 𝑝𝑐2 − 𝑝𝑔2𝑞2 = 𝑞𝑐2 − 𝑞𝑔2
𝑉" = 𝑉"RST
Powerflowconstraintsafterdisruption
• Netpowerconsumedatanode
• LinearPowerflows(LPF)
• Voltagedropequation
15
𝑃28 = U 𝑃8VV:8→V
+ 𝑝2
𝑄28 = U 𝑄8VV:8→V
+ 𝑞2
𝑉8 = 𝑉2 − (r28𝑃28 + x28𝑄28)
𝑝2 = 𝑝𝑐2 − 𝑝𝑔2 + 𝛿2𝑝𝑑A2⋆
𝑞2 = 𝑞𝑐2 − 𝑞𝑔2 + 𝛿2𝑞𝑑A2⋆
𝑉" = 𝑉"RST − Δ𝑉"
PowerflowconstraintsafterSOdispatch
• Netpowerconsumedatanode
• LinearPowerflows(LPF)
• Voltagedropequation
16
𝑃28 = U 𝑃8VV:8→V
+ 𝑝2
𝑄28 = U 𝑄8VV:8→V
+ 𝑞2
𝑉8 = 𝑉2 − (r28𝑃28 + x28𝑄28)
𝑝2 = 𝑝𝑐2 − 𝑝𝑔2 + 𝛿2𝑝𝑑A2⋆ − 𝑝𝑟2
𝑞2 = 𝑞𝑐2 − 𝑞𝑔2 + 𝛿2𝑞𝑑A2⋆ − 𝑞𝑟2
𝑉" = 𝑉"#YZ − Δ𝑉"
Losses
Costofactivepowersupply:
Lossofvoltageregulation:where𝑡2 ≥ 𝑉2 − 𝑉RST
Costincurredduetoloadcontrol:
LossinGrid-Connectedregime:
17
𝐿^_ 𝑥 ≡ 𝑊cd𝑃"
𝐿ef 𝑥 ≡ 𝑊efU𝑡22∈g
,
𝐿h_ 𝑥 ≡U𝑊h_,2(1 − 𝛽2)2∈g
𝐿idjkM2Zk 𝑥 = 𝐿^_ 𝑥 + 𝐿ef 𝑥 + 𝐿h_(𝑥)
Attacker-Defenderproblem[AD] - Bilevel formulation
AD ℒ ∶= maxA∈𝒜
minw∈𝒟
𝐿y_z{|}T{ 𝑥 𝑎, 𝑑
• Powerflows,DERcapabilities,voltagebounds• Defendermodel(resourcesandcapabilities)• Attackermodel(resourcesandcapabilities)
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SystemState𝑥 = (𝑝, 𝑞, 𝑃, 𝑄, 𝑉)
Attacker-Defenderproblem[AD] – Cascaderegime
AD ℒ ∶= maxA∈𝒜
minw∈𝒟
𝐿_~z{|}T{ 𝑥 𝑎, 𝑑
• Powerflows,DERcapabilities,voltagebounds• Defendermodel(resourcesandcapabilities)• Attackermodel(resourcesandcapabilities)
19
Where 𝐿_~z{|}T{ 𝑥 ≡ 𝐿y_z{|}T{ 𝑥 + 𝐿~� 𝑥• Costofloadshedding
𝐿~� 𝑥 ≡ U𝑊~�,2𝑘𝑐22∈𝒩
• 𝑊~�,2 :costofunitloadshedding
Attacker-Defenderproblem[AD] – Islandingregime
AD ℒ ∶= maxA∈𝒜
minw∈𝒟
𝐿��z{|}T{ 𝑥 𝑎, 𝑑
• Powerflows,DERcapabilities,voltagebounds• Defendermodel(resourcesandcapabilities)• Attackermodel(resourcesandcapabilities)
20
Where𝐿��z{|}T{ 𝑥 ≡ 𝐿y_z{|}T{ 𝑥 + 𝐿�y 𝑥• Costofmicrogrid islanding
𝐿�y 𝑥 ≡ U 𝑊�y,28𝑘𝑚28(2,8)∈�
• 𝑊�y,28 :costofasinglemicrogrid islandformationatnode𝑗
Benderscutapproach
2
5
6
7
8 12
11
9
1
0
3
4
10
21
ComputationalresultsforCascaderegime
22
𝑝𝑑A⋆
𝑠𝑟L
Loadsheddingvs����
23
Noresponse- (multi-round)cascade
Worst-caselossundernodefenderresponse
Analgorithm• Initialcontingency• Forr=1,2,…• Computenewpowerflows• DetermineasingleloadsorDGthatmaximallyviolatesitsvoltagebounds• Disconnectthatdeviceaccordingly
24
OnlinevsSequentialvsIslanding
25
ValueoftimelydisconnectionsValueoftimelyIslanding
DefenderResponseandAllocation:Diversification
• SomeDERscontributeto𝐿efmorethan𝐿^_,andviceversa
26
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
nodes
Specialcaseof𝜒 = 0,1
DefenderResponseandAllocation:Diversification
• Diversificationholdsfor“heterogeneousallocation”withdownstreamDERswithmorereactivepower
27
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
nodes
• Post-contingencylossesarethesameforuniformvs.heterogeneousresourceallocations
• Pre-contingencyvoltageprofileisbetterforheterogeneousresourceallocation
Heterogeneousresourceallocationcansupportmoreloadsthanuniformone.
DefenderResponseandAllocation:Diversification
28
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
nodes
Bigpicture:Wheredoesitallfit?
minj∈ℛ
𝐶A��Y' 𝑥Y 𝑟 + maxA∈𝒜
minw∈𝒟
𝐿 𝑥' 𝑟, 𝑎, 𝑑
• Powerflows,DERcapabilities,voltagebounds• Defendermodel(resourcesandcapabilities)• Attackermodel(resourcesandcapabilities)
Resilience-AwareOptimalPowerFlow(RAOPF)
29
Voltagedeviationmodel𝑉#YZ − 𝑉"' = −𝑉jkM 𝑃"Y − 𝑃"'
Frequencydeviationmodel𝑓#YZ − 𝑓' = −𝑓jkM 𝑄"Y − 𝑄"'
Pre-contingencyresourceallocation𝑟 = (𝑝𝑟Y, 𝑞𝑟Y)
30
Resiliency-AwareOPF- Trilevelformulation
Finalexample:DNresiliencyisindeedimportant
31
DN1
DN2
DN3
DN4
60MW
60MW
60MW
60MW
𝐺�
𝐺� 240MW
120MW
DN1
DN2
DN3
DN4
60MW
60MW
60MW
60MW
𝐺�
𝐺�
120MW
120MW
0MW
𝑃� = 80MW
𝑃� = 80MW
DN1
DN2
DN3
DN4
30MW
30MW
30MW
30MW
𝐺�
𝐺�
40MW
80MW
• Normaloperatingscenario
• Lightningstrikes- recloser openstemporarily
• VoltagedropsattheDNsubstations
• Microgrid islandingreducesnetload
• InfeasiblepowerflowinTN
Summary• ResourceallocationanddispatchinelectricityDNs
• understrategiccyber-physicalfailures• trilevelmixed-integerformulation
• Multi-regimedefenderresponse• ApplicationofBenderscutapproachforsolvingbilevel MILPs• Structuralresultsonworst-caseattacksandtradeoffsfordefenderresponse
Futurework• Designofdecentralizeddefenderresponseusingmessagepassing• Powerrestorationovermultipletimeperiods
32
Optimalattackerset-pointsTypically,
• Smalllinelosses:incomparisontopowerflows
• Smallimpedances:sufficientlysmalllineresistances
Assumeforsimplicity:
• Noreversepowerflows:powerflowsfromsubstationtodownstream
33
Whatareoptimalattackerset-points?
Proposition:Foradefenderaction𝜙,andgivenattackerchoiceof𝛿,theoptimalattackerdisturbanceisgivenby:
𝑝𝑑A⋆ = 𝑝𝑔2Y, 𝑞𝑑A⋆ = 𝑞𝑔2Y + 𝒔𝒈𝒊 (incaseofattackonDERs)
𝑝𝑑A⋆ = 𝑝𝑐𝑒2Y, 𝑞𝑑A⋆ = 𝑞𝑐𝑒2Y (incaseofattackonEVs)
Benderscutapproach
Proposition(Bienstock 2009)Optimalvalueattackproblemforafixedattack cardinality isequivalenttoaminimumcardinalityattackproblemforafixedtargetlossvalue.
34
Benderscutapproach
AttackerMasterproblem• Initializewithnocuts
min U𝛿22
s. t. cuts𝛿2 ∈ {0,1}
Defenderproblem
minw∈𝒟
𝐿(𝑥)s.t.• Powerflows,DERcapabilities,voltagebounds• Defendermodel(resourcesandcapabilities)
Optimalvalueattack problemforafixedattackcardinality isequivalenttoaminimumcardinalityattack problemforafixedtargetloss value.
𝐿�AjMk� :minimumlossthattheattackerwantstoinflictuponthedefender
35
Benderscut• Let𝛿2�kj befixedattackerstrategyforcurrentiteration• Let𝜙�(resp.𝜙d)denoteadefenderresponsewithfixedintegervariables• Thentheinnerproblembecomesalinearprogram(LP)
min 𝑐�𝑦
𝐶𝑦 = 𝑑 + 𝑄𝛿2�kj𝑠. 𝑡. 𝐴𝑦 ≥ 𝑏
𝐿𝑃 𝛿2�kj, 𝜙� ≡
• Let(𝜆⋆, 𝛼⋆)betheoptimaldualvariablesolutiontothisLP.Benderscutisgivenby:𝜆⋆�𝑏 + 𝛼⋆� 𝑑 + 𝑄𝛿 ≥ 𝐿�AjMk�
• Thiscuteliminates𝛿2�kj fromfeasiblespaceofattackerstrategies 36
Controllabledistributedgenerationmodel
pr
qr
qr
Reactive power
Real
power
Reactive power
pr
qr
qr
Real
power
sr
37
0 ≤ 𝑝𝑟2 ≤ 𝑝𝑟2,𝑝𝑟2� + 𝑞𝑟2� ≤ 𝑠𝑟L2�𝑝𝑟2 - maximumactivepowercapacity𝑠𝑟L2 - apparentpowercapabilityofinverter
Polytopicmodelusedforcomputationalsimplicity
UncontrolledcascadevsSequential
38
Valueoftimelyresponse
N=37nodes
Microgrid islandformation
2
5
6
7
8 12
11
9
1
0
3
4
10
2
5
6
7
8 12
11
9
1
0
3
4
10
2
5
6
7
8 12
11
9
1
0
3
4
10
• 𝜒 = 0,1 , 4,5 , 4,9• 3outof8= 2 � possibleconfigurations– 13nodenetwork
39
Linearpowerflowsafterdispatch
𝑝2 = 𝑝𝑐2 − 𝑝𝑔2 − 𝑝𝑟2 + 𝛿2𝑝𝑑A2⋆
𝑞2 = 𝑞𝑐2 − 𝑞𝑔2 − 𝑞𝑟2 + 𝛿2𝑞𝑑A2⋆Netpowerconsumedatanode𝑖
Powerflowonline𝑖 → 𝑗
Voltagedropequations
40
𝑉" = 𝑉"Y − Δ𝜈
𝑃28 = U 𝑃8VV:8→V
+ 𝑝2
𝑄28 = U 𝑄8VV:8→V
+ 𝑞2
𝑉8 = 𝑉2 −(𝑟28𝑃28 + 𝑥28𝑄28)
Islandingregime(cont’d)Updatedconstraints
• An(emergency)distributedgeneratorisstartedatnode𝑗 inamicrogrid island𝑝𝑟8 ≤ 𝑠𝑟L8𝑘𝑚28𝑞𝑟 ≤ 𝑠𝑟L8𝑘𝑚28
Where𝑝𝑟8 ,𝑞𝑟8 isactiveandreactivepoweroutput;𝑠𝑟L8 istheapparentpowercapabilityoftheemergencygeneratoratnode𝑗
• Thenetpowerflowintothenode𝑗 fromthesubstationis0,i.e.𝑘𝑚28 = 1 ⟹ 𝑃28 = 𝑄28 = 0
• Thenodalvoltageatnode𝑗 isthenominalvoltage,
𝑉8 = Q𝑉2 − 𝑟28𝑃28 + 𝑥28𝑄28 , if𝑘𝑚28 = 0𝑉RST, otherwise.
41
What’snext?
42
•Whatisagoodresiliencymetric?• AllowableΔ 𝑉, 𝑝, 𝑞 withoutexceedingtarget20%𝐿µ¶
•Generalcase 𝜒 > 1• Diversification?• SolutionapproachforRAOPF(trilevel)?
Resiliency-awareResourceAllocation
StageII- Adversarialnodedisruptionsa. Whichnodestocompromise(𝛿)?b. Set-pointmanipulation(𝑠𝑝A)?
StageI- AllocationofDERsoverradialnetworksa. Sizeandlocationb. Activeandreactivepowersetpoints (𝑥#)?
StageIII- Optimaldispatch/response(𝑥')a. Maintainvoltageb. Exerciseloadcontrolornot
Goals:1. Determinethebestresourceallocation2. Identifyvulnerable/criticalnodes3. Determineoptimaldispatchpost-contingency 43
Microgrid formation(cont’d)Updatedconstraints
𝑝2 = 𝑝𝑐2 − 𝑝𝑔2 − 𝑝𝑟2 + 𝛿2𝑝𝑑A2⋆ − 𝑝𝑒2
𝑞2 = 𝑞𝑐2 − 𝑞𝑔2 − 𝑞𝑟2 + 𝛿2𝑞𝑑A2⋆ − 𝑞𝑒2
|𝑃28| ≤ 𝐶𝑎𝑝28 1− 𝑘𝑚28|𝑄28 | ≤ 𝐶𝑎𝑝28 1− 𝑘𝑚28
|𝜈8 − 𝜈#YZ | ≤ 1 − 𝑘𝑚28
|𝜈8 − 𝜈2 − 2 𝑟28𝑃28 + 𝑥28𝑄28 | ≤ 𝑘𝑚28
• Anemergencygeneratorofmicrogrid isononlyifitisinislandedstate𝑝𝑒8 ≤ 𝑠𝑒8 𝑘𝑚28𝑞𝑒8 ≤ 𝑠𝑒8 𝑘𝑚28
44