Chance-constrainedACOPFwithprobabilisticguaranteesandscalableperformance

MariaVrakopoulouMarieCuriePost-doctoralfellow,UCBerkeley

IncollaborationwithJenniferMarleyandProf.IanHiskens (UniversityofMichigan)

ControlofComplexSystems:AnIntegratedPerspectiveonModernPowerGridControlACCWorkshopMay22-23,2017

1

Operator

Decisions

Load

time

PVpower

Demand

Contingencies

Decisionsunderuncertainty

Windpower

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 1/30

time

100%

Powerflow

steadystatelimit

Decisionsunderuncertainty

Inthepast:• Notheavilyloadedsystem• LowshareofRenewableEnergy

Sources(RES)

Enoughmargintowithstanduncertainty

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 2/30

Decisionsunderuncertainty

time

100%

Powerflow

steadystatelimitBut..

demand(higherloadingonthesystem)penetrationofRES(higheruncertainty)

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 2/30

Decisionsunderuncertainty

time

100%

Powerflow

steadystatelimit

Operatingclosertothemargins

Maybenotpossibletowithstanduncertaintyorunnecessaryexpensivedesign

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 2/30

6

NeedforreliablestochasticOptimalPowerFlow formulations

q TheACOptimalPowerFlow(OPF)problem

q StochasticACOPFformulation

q Conclusions

orunnecessaryexpensivedesign

Outline

q TheACOptimalPowerFlow(OPF)problem

q StochasticACOPFformulation

q Conclusions

orunnecessaryexpensivedesign

Outline

GeneralACOPFmodel

subject to

quadratic cost of generation

powerflow equations

operationallimits

voltage magnitude and angle

windpowerforecast

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 3/30

OPFchallenges

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 4/30

• Nonconvexconstraints

• Needtobeappliedinlargenetworks

Different solution methodologies aim to find an approximate solution, a local minimum or ideally the optimal solution in a scalable manner

OPFsolutionmethodologyexamples

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 5/30

• ApproximationsDC OPF: linearized power flow equations - quadratic problem

• Finding the global optimumConvex Relaxations: Semidefinite program (SDP), Second order cone program (SOCP)

• Finding a local minimumAC-QP OPF : Quadratic program (QP) formulation - separate AC power flow based update

OPFsolutionmethodologyexamples

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 5/30

• ApproximationsDC OPF: linearized power flow equations - quadratic problem

• Finding the global optimumConvex Relaxations: Semidefinite program (SDP), Second order cone program (SOCP)

• Finding a local minimumAC-QP OPF : Quadratic program (QP) formulation - separate AC power flow based update

OPFsolutionmethodologyexamples

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 5/30

• ApproximationsDC OPF: linearized power flow equations - quadratic problem

• Finding the global optimumConvex Relaxations: Semidefinite program (SDP), Second order cone program (SOCP)

• Finding a local minimumAC-QP OPF : Quadratic program (QP) formulation - separate AC power flow based update

AC-QPOPFalgorithm

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 6/30

[A. Wood, B. Wollenberg, and G. Sheble, Power Generation, Operation and Control, 3rd ed. John Wiley and Sons, Inc., 2013]

• Initialize the AC power flow using an SCOP solution• SOCP provides a lower bound of the OPF problem

q TheACOptimalPowerFlow(OPF)problem

q StochasticACOPFformulation

q Conclusions

orunnecessaryexpensivedesign

Outline

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 8/30

Steady state operating points

Givenageneration-loadmismatchtheautomaticcontrolloopswillleadtoanewpost-disturbanceoperatingpoint

Pre-disturbanceoperating point

Post-disturbanceoperating point

source:swissgrid

Example:Frequency controlSaferegion

PrimarySecondary Tertiary

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 9/30

• AGCresponse(secondarycontrol)

Satisfytheconstraintsatthesteadystatepointafter

• aredispatch action(ortertiarycontrol)

• Primarycontrol

Steady state operating points

Optimaldecision making under uncertainty

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 10/30

source:swissgrid

timeScheduling point

• Optimaldecisions

• Uncertain forecast

• Historicaldata ofthe uncertainty

Deployment point

• Apply the optimaldecisions

• Realization of theuncertainty

Dayahead market:one day beforeIntra-day markethours before

Uncertaintymodel

Optimaldecisionmaking

Apply inrealtime

Schedulingpoint Deploymentpoint

ChanceConstrainedOPF

subject to

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 11/30

subject to

Satisfytheconstraintsinaprobabilisticsense

ChanceConstrainedOPF

subject to

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 12/30

subject to

Solvethechanceconstraintusingdata-basedoptimization

Thescenarioapproach[1]• Noassumptionsontheunderlyingdistributionoftheuncertainty• Guaranteesa-priorithatthechanceconstraintwillbesatisfiedwithacertain

confidence• Requiresconvexitywithrespecttodecisionsvariables

Probabilisticallyrobustdesign[2]• Mixtureofrandomizedandrobustoptimization• Noassumptionsontheunderlyingdistributionoftheuncertainty• Guaranteesa-priorithatthechanceconstraintwillbesatisfiedwithacertain

confidence• Noparticularstructurerequiredwithrespecttothedecisionvariable

[1] G. Calafiore and M. Campi, TAC, 2006[2] K. Margellos, P. Goulart and J. Lygeros, TAC, 2014[3] M. C. Campi, S. Garatti, F. A. Ramponi, CDC, 2015

Data-basedoptimization

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 13/30

Thenon-convexscenariooptimization[3]• Noassumptionsontheunderlyingdistributionoftheuncertainty• Applicabletocon-convexproblems• Providesa-posterioriguarantees

• Substitutethechanceconstraintwithafinitenumberofhardconstraintsbasedonscenariosoftheuncertainty

• Howmanyscenariosdoweneedtoprovideprobabilisticguarantees?

Thescenario approach [1]

number ofdecision elementsconfidence

violationlevel

numberof scenarios

𝑁 ≥ #$

%%&#

ln #)+ 𝑁+ − 1

𝛿#

𝛿/

𝑖 = 1…𝑁for

Data-basedoptimization

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 14/30

𝑖

• Usethescenarioapproach tofind‘bounds‘oftheuncertaintyelements

• Howmanyscenariosdoweneedtoprovideprobabilisticguarantees?

Step1

number ofuncertaintyelements

confidenceviolationlevel

numberof scenarios

Probabilistically robustdesign(two-step approach)[2]

𝑁 ≥ #$

%%&#

ln #)+ 2𝑁5 − 1

𝛿#

𝛿/

Data-basedoptimization

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 15/30

• Usethescenarioapproach tofind‘bounds‘oftheuncertaintyelements

• Howmanyscenariosdoweneedtoprovideprobabilisticguarantees?

Step1

number ofuncertaintyelements

confidenceviolationlevel

numberof scenarios

Probabilistically robustdesign(two-step approach)[2]

𝑁 ≥ #$

%%&#

ln #)+ 2𝑁5 − 1

𝛿#

𝛿/

Data-basedoptimization

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 15/30

Step2

• Solvearobustreformulationoftheinitialchanceconstrainedproblem

• Anysolutionoftherobustproblemisfeasibleforthechanceconstrainedwithconfidenceatleast

forall 𝛿 ∈ ∆8

𝛿#

𝛿/

∆8

1 − 𝛽

Dealingwiththechanceconstraint

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 15/30

Probabilistically robustdesign(two-step approach)[2]

Priorwork

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 16/30

[M.Vrakopoulou,K.Margellos,J.Lygeros,andG.Andersson,“ProbabilisticguaranteesfortheN-1securityofsystemswithwindpowergeneration,”inPMAPS2012]

DCpowerflow• Usingactivepowerpolicies

allowtotriviallysatisfytheequalityconstraintsandtheproblemcanbetractable

• Methods[1]and[2]wereefficientlyused

• ButsolutionnotaccuratefortheACconstraints

Failstosatisfythedesiredviolationlevel

Satisfiesthedesiredviolationlevel

§ MonteCarloevaluationfor10000windpowerrealizations

IEEE30-busnetwork§ 4loadprofiles§ Desiredviolationlevel10%

Priorwork

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 16/30

[M.Vrakopoulou,M.Katsampani,K.Margellos,J.Lygeros,andG.Andersson “Probabilisticsecurity-constrainedACoptimalpowerflow,”inIEEEPowerTech Conference,2013]

• SDPformulation,activepowerandgeneratorvoltagepoliciesneedtobeusedtomaketheproblemtractable.

• Method[2]wasapplied• Butscalabilityissues!!

ACpowerflow

IEEE14-busnetwork§ Desiredviolationlevel10%§ MonteCarloevaluationfor10000windpower

realizations

Data-basedoptimization

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 17/30

Thenon-convexscenariooptimization[3]

• UseNscenariosasaninputtothenon-convexoptimizationalgorithm• Calculatethesupportscenariosk oftheobtainedsolution• Providesa-posterioriguaranteesfortheupperboundofchanceconstraint

violationlevel

[3] M. C. Campi, S. Garatti, F. A. Ramponi, CDC, 2015

29

Non-convexalgorithm

Data-basedoptimization

Supportscenarios

Non-convexalgorithm

k

CanwefindasmallnumberofscenarioskthatmustbeincludedintheproblemsothatthesolutionwouldbethesameastheoneusingalltheNscenarios?

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 18/30

30ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 19/30

StochasticAC-QPOPF

[J.F.Marley,M.VrakopoulouandI.A.Hiskens,AnAC-QPoptimalpowerflowalgorithmconsideringwindforecastuncertainty,IEEEPESInnovativeSmartGridTechnologies,ISGT-Asia2016]

31ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 20/30

ScenariosintheOPF

32

Casestudies

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 21/30

Test the stochastic AC-QP algorithm for the following IEEE benchmarknetworks:

• 14-Bus: 5 Generators, 20 Branches, 2 wind buses

• 30-Bus: 6 Generators, 41 Branches, 5 wind buses

• 57-Bus: 7 Generators, 80 Branches, 7 wind buses

• 118-Bus: 54 Generators, 186 Branches, 10 wind buses

N=1500scenariosβ=0.001

33

Increasingthenetworksize

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 22/30

2

3

4

5

6

7

8

9

Num

ber o

f Sup

port

Scen

ario

s in

AC-

QP

pOPF

118 Bus w/ 10 Wind: Number of Support Scenarios

14 30 57 11810

20

30

40

50

60

70

Tota

l Exe

cutio

n Ti

me

(Sec

.)

118 Bus w/ 10 Wind: AC-QP Time

14 30 57 118

34

Increasingthenetworksize

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 23/30

0

0.1

0.2

0.3

0.4

0.5

% o

f 10k

Sce

nario

s w

ith A

C V

iola

tions

118 Bus w/ 10 Wind: AC-QP Violation Probability (Empirical) - AC Power Flow

14 30 57 1182

2.5

3

3.5

4

4.5

Theo

retic

al P

roba

bilit

y of

Vio

latio

n

118 Bus w/ 10 Wind: Theoretical Probability of Violation

14 30 57 118

35

IncreasingthenumberofscenariosN

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 24/30

36

IncreasingthenumberofscenariosN

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 25/30

37

IncreasingthenumberofscenariosN

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 26/30

q TheACOptimalPowerFlow(OPF)problem

q StochasticACOPFformulation

q Conclusions

orunnecessaryexpensivedesign

Outline

39

Conclusions

TheAC-QPOPFmethodhasbeenextendedtoincludewindpoweruncertainty,throughtheadditionofafinitenumberofwindscenarios.

TheresultingstochasticAC-QPOPFalgorithmoffersseveraladvantages:

• ItdoesnotrelyuponmodelapproximationsandproducesanACfeasiblesolution.

• Itprovidesaprobabilisticallyrobustsolutionwitha-posterioriprobabilisticviolationguarantees.

• Thenumberofsupportscenariosremainsmallastheproblemsizeincreaseshighlightingpromisingscalabilityproperties

ACC 2017 Chance-constrained AC OPF with probabilistic guarantees 28/30

Thankyouforyourattention!

mariavr@berkeley.edu