AntonioGómez-ExpósitoDpt.ofElectricalEng.–EndesaRedChair

UniversityofSevilla

The factored approach to solving nonlinear power system problems

UCDublin,February23,2017

•  Introduc:onandmo:va:on•  Factoredsolu:onapproach•  Canonicalformsofnonlinearsystems•  Extendingtherangeofreachablesolu:ons•  Complexsolu:onsforinfeasiblecases•  Applica:ontopowersystemproblems

–  Loadflowsolu:on– Maximumloadabilitydetermina:on– WLSstatees:ma:on

Contents

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•  JohnRice(fatherofmathema<calso=wareconcept,1969):

“solvingsystemsofnonlinearequa<onsisperhapsthemostdifficultprobleminallofnumericalcomputa<ons”

Introduc:on

•  Ingeneral,nonlinearequa:onsystemsmustbesolveditera:vely–  Sometrivialcasesallowforclosed-formsolu:ons

•  Newton-Raphsonitera:vealgorithmisthereferencemethodinthegeneralcase(sincetheXVIIcentury):–  First-orderlocalapproxima:onssuccessivelyperformed– Quadra:cconvergenceratenearthesolu:on–  SamedivergencerateoutsidethebasinsofaWrac:on–  Exploita:onofJacobiansparsityforlargesystems:keyforitsterrificsuccessinpowersystemsapplica:ons

Tinneyet.al.lateinthe60’s:sparseLUfactoriza:onofJacobian

Introduc:on

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•  ManyimprovementssofartovanillaNR:– Quasi-Newtonmethods:approximate(constant)Jacobian–  InexactNewtonmethods:approximatecomputa:onofNewton’sstep(precondi:oners)

– Higher-ordermethods:super-quadra:cconvergence(morecostly)

– GloballyconvergentNewtonmethods:linesearch,con:nua:on/homotopy,trustregion

– Others:polynomialapproxima:on(Chebyshev’s),solu:onofdifferen:alequa:ons(Davidenko’s),etc.

Introduc:on

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•  Keyidea:“Lookforaglobalnonlineartransforma:onthatcreatesanalgebraicallyequivalentsystemonwhichNewton’smethoddoesbeWerbecausethenewsystemismorelinear”.

Sofar,“nogeneralwaytoapplythisideahasbeenfound;itsapplica:onisproblem-specific”.[JuddK.L.,“Numericalmethodsineconomics”,MITPress,NewYork,1998,pp.174-176]

•  Factoredapproach:firstsystema:caWempttoachievethisgoalonabroadrangeofnonlinearsystemsofprac:calinterest

Mo:va:on

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•  Thecustomaryultra-compactexpression:h(x)=p(pisgiven,xtobefound)

hidesthetypicalstructureofh(.):sumsofnonlinearexpressionsofdifferentcomplexity

•  Large-scalenonlinearsystemsarealwayssparse:– Numberofaddi:vetermsinasetofnnonlineareqs.innvariablesisroughlyO(n),notO(n2)

–  Someofthoselinear/nonlineartermsmayappearseveral:mes:

Letm>nbethenumberofdis:nctterms

Mo:va:on

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•  Letybeanauxiliaryvectorcomposedofthemdis:nctaddi:vetermsinh(.),eachwithtrivialinverse.Then,thefollowingfactoredformarises:

Factoredsolu:onapproach

Under-determinedlinearsystem

Over-determinedlinearsystem

mone-to-onemappingswithexplicitinverse:

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•  Conven:onalNR(forcomparison):

•  Two-stepfactoredprocedure:

Factoredsolu:onapproach

Step0:Ini:aliza:onx0 ày0

Step1:

Step2:

<ε NO

YESENDFactoredJacobian:

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•  Conven:onalNR:

ComputaGonalissues:

Factoredsolu:onapproach

Step0:Ini:aliza:onx0 ày0

Step1:

Step2:

<ε NO

YESEND

Least-distanceproblem:Choleskyfactorscomputedonlyonce

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•  Conven:onalNR:

ComputaGonalissues:

Factoredsolu:onapproach

Step0:Ini:aliza:onx0 ày0

Step1:

Step2:

<ε NO

YESEND

Trivialone-to-onenonlinearmappingf()withdiagonalJacobian

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•  Conven:onalNR:

ComputaGonalissues:

Factoredsolu:onapproach

Step0:Ini:aliza:onx0 ày0

Step1:

Step2:

<ε NO

YESEND

Least-squaresproblem:FactoredJacobiansimplertocompute

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•  Conven:onalNR:

ComputaGonalissues:

Factoredsolu:onapproach

Step0:Ini:aliza:onx0 ày0

Step1:

Step2:

<ε NO

YESEND

LinearmismatchvectorusedinnextiteraGon

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1.Sumsofsingle-variablenonlinearelementaryfuncGons:

Canonicalforms

Example:

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Canonicalforms2.Sumsofproductsofsingle-variablepowerfuncGons:

Preliminarylogarithmicchangeofvariables:

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CanonicalformsExample:

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CanonicalformsRemark:Ifm=n,thenthereisnoneedtoiterate

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p1 = x1x2 + x12

p2 = 2x1x2 − 4x12

m=n=2

p1p2

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥= 1 1

2 −4

⎡

⎣⎢

⎤

⎦⎥y1y2

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

u1 = ln y1

u2 = ln y2

u1

u2

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥= 1 1

2 0

⎡

⎣⎢

⎤

⎦⎥

ln x1

ln x2

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

x1x2

UnlikeinNewton’smethod,itera:onsarisebecausem>n

Canonicalforms3.Conversiontocanonicalforms:systemaugmentaGonAddvariables/equa:onsasneeded

Example:

Newvariableintroduced:

Theequivalent“canonical”systembecomes:

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Extendingtherangeofreachablesolu:ons

MulGplicityofsoluGons:Whichsolu:onisitera:velyreached?

•  NewtonRaphson:dependsonthebasinofaWrac:oninwhichtheini:alguessx0lies

•  Factoredmethod:canbecontrolledbyselec:ngthecomputedrangeforeachindividualnonlinearfunc:on

Examples:⇒ ℜ(u)> 0

⇒ ℜ(u)< 0

Periodicfunc:ons

⇒

⇒

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Extendingtherangeofreachablesolu:ons

Example:2x2P.Boggs’system

Onlythreesolu:ons:

NewtonRaphson’sbasinsofaYracGon:unpredictableirregularbehavior

Whiteareas:divergence

A

B

C

ABC

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Extendingtherangeofreachablesolu:ons

Example:2x2P.Boggs’system

Foranyx0 !!Withthebasicdefini:onsconvergestoAWithconvergestoBWithconvergestoCWithyieldscomplexsolu:on

Factoredmethod:newvariables

A

B

C

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Infeasibilityandcomplexsolu:ons

•  Feasiblecase:pissuchthatthereexistsatleastarealxsa:sfyingh(x)=p

•  Infeasiblecase:onlycomplexvaluesofxsa:sfyh(x)=p

Empiricalevidence:•  Newton’smethodcannotgenerallyconvergetocomplexsolu:ons

•  Thefactoredmethodisflexibleenoughtoconvergetocomplexsolu:onswheneverrealsolu:onscannotbefound

•  ChoosinganiniGalguesswithacertainimaginarycomponentishelpfultoachieveconvergencetocomplexsolu:ons

•  Quadra:cconvergenceratealsotocomplexsolu:ons

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Infeasibilityandcomplexsolu:ons

Example: Canonicalform:

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FactoredLoadFlowSolu:on

Factoredmodel:

p: Specifiedquan::es(PQandPVbuses)x: Statevariables(2N-1)

(N+2b)

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FactoredLoadFlowSolu:on

Pij = (gsh,i + gij )Ui − gijKij − bijLijQij = −(bsh,i + bij )Ui + bijKij − gijLij

Linear(underdetermined)powerflowmodel:Powerinjec:ons

with•  Voltagemagnitudes

Pisp = Pij∑

Qisp = Qij∑

[Vi2 ]sp =Ui

Ey = p

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F−1 =

U1

U2

U3

12K12 −L12

12L12 K12

12K13 −L13

12L13 K13

12K23 −L23

12L23 K23

"

#

$$$$$$$$$$$$$$$$$$$$$$$$

%

&

''''''''''''''''''''''''

FactoredLoadFlowSolu:on

TheJacobiananditsinversearetriviallyobtained

Jacobianelements:

InvertedJacobianelements:

F-1 becomessingulariffUi=0foranyi

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FactoredLoadFlowSolu:onConvergencebasins:2-busexample

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v1=1.0 + j0 p.u. s2 = 0.9 + j0.6 p.u.

z=0.01 + j0.1 p.u.1 2

Θ (deg)

V (p

.u.)

−150 −100 −50 0 50 100 1500

2

4

6

8

10

8

9

67

54 3

NC

Θ (deg)

V (p

.u.)

−150 −100 −50 0 50 100 1500

2

4

6

8

10

6

3

14 5

NC2

Newton-Raphson Factoredmethod

FactoredLoadFlowSolu:on

ConvergenceratesfortheIEEE300-bussystem

NR

Factored

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FactoredLoadFlow

ConvergenceratesfortheIEEE118-bussystem[PVbusesconvertedtoPQbuses]

NR

Factored

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FactoredLoadFlowSolu:onIEEE14-bussystem:evoluGonofvoltagesforincreasingloadings

Voltagemagnitudes(p.u.)Voltagephaseangle(deg.)

Infeasibleregion

Infeasibleregion

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Maximumloadingpointdetermina:onProblem:Findingthesaddle-nodeorlimit-inducedbifurcaGonpoint

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Maximizethevalueofλ:

•  Op:miza:onproblem•  Con:nua:onmethods

Maximumloadingpointdetermina:on

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Factoredloadflow-basedprocedure:Phase1:Increaseregularlyλun:ltwoconsecu:vesolu:onscorrespondtofeasibleandinfeasiblepoints

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

λ

Real

(V2)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Imag

l (V

2)

Imag (V2)

Real (V2)

3

2

6

1

5

4

Δλ=0.5

Maximumloadingpointdetermina:on

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Factoredloadflow-basedprocedure:Phase1:Increaseregularlyλun:ltwoconsecu:vesolu:onscorrespondtofeasibleandinfeasiblepointsPhase2:Sequen:allyperformbisec:onsearchesun:lΔλissmall

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

λ

Real

(V2)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Imag

l (V

2)

Imag (V2)

Real (V2)

3

2

6

1

5

4

Δλ=0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

2.6 2.65 2.7 2.75 2.8 2.85 2.9 2.95 30

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Real (V2)

Imag (V2)

9

14

10

7

5

6

81312

11

l m

l m

u

u

l m u

m ul

umlΔλ=0.25

Maximumloadingpointdetermina:on

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ComparisonofsimulaGonresults

Maximumloadingpointdetermina:on

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EnhancedsoluGonprocedure:parabolicapproxima:onsintheinfeasibledomain

Conven:onalsolu:onofnonlinearWLS-SE

Measurementmodel:

Maximumlikelihoodes:ma:on:

Normalequa:ons(Gauss-Newton):Covariance(dense)matrices:

FactoredWLSStateEs:ma:on

MinJ

cov(z) = H ⋅ cov(x) ⋅HT

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FactoredWLSStateEs:ma:on

LinearWLSSE

Non-linearexplicittransforma:on

LinearWLSSE

Factoriza:onofnonlinearWLSproblems

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FactoredWLSStateEs:ma:on

LinearWLSSE

Non-linearexplicittransforma:on

LinearWLSSE

Factoriza:onofnonlinearWLSproblems

z = By+ e

ijjiij

ijjiij

VVLVVK

θ

θ

sin

cos

=

=

2ii VU =

Branch

Bus(2b+Nvariables)

y = Ui,Kij,Lij{ }

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y , cov−1( y) =GB = BTWB

FactoredWLSStateEs:ma:on

LinearWLSSE

Non-linearexplicittransforma:on

LinearWLSSE

Factoriza:onofnonlinearWLSproblems

•  Noneedtochooseini:alvalues•  “Warmstart”forrepeatedruns(constantBmatrix)

qijijijijiijishijm

pijijijijiijishijm

LgKbUbbQ

LbKgUggP

ε

ε

+−++−=

+−−+=

)(

)(

,

,

[Vi2 ]m =Ui +εV

Linearmeasurementmodel:

qIijmi

pIijmi

QQ

PP

ε

ε

+=

+=

∑∑

z = By+ e y , cov−1( y) =GB = BTWB

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FactoredWLSStateEs:ma:on

LinearWLSSE

Non-linearexplicittransformaGon

LinearWLSSE

Factoriza:onofnonlinearWLSproblems

( )( )ijijij

ijijij

ii

KLLK

U

/arctan

ln

ln22

=

+=

=

θ

α

α trivialinverse

z = By+ e

u = fu( y) u , cov−1( u) = Wu = Fu−TGB

Fu−1

y , cov−1( y) =GB = BTWB

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FactoredWLSStateEs:ma:on

LinearWLSSE

Non-linearexplicittransforma:on

LinearWLSSE

Factoriza:onofnonlinearWLSproblems

αi

θi

u =Cx + eu x , cov−1(x) =GC =CT WuC

z = By+ e

u = fu( y) u , cov−1( u) = Wu = Fu−TGB

Fu−1

y , cov−1( y) =GB = BTWB

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FactoredWLSStateEs:ma:on

LinearWLSSE

Non-linearexplicittransforma:on

LinearWLSSE

Factoriza:onofnonlinearWLSproblems

Isitanon-itera:veprocedure?Notreally!!dependsontheopera:ngpoint:

Wu = Fu−TGB

Fu−1

x , cov−1(x) =GC =CT WuCu =Cx + eu

u

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FactoredWLSStateEs:ma:on

LinearWLSSE

Non-linearexplicittransforma:on

LinearWLSSE

Factoriza:onofnonlinearWLSproblems

unew =Cx Repeatun:l xk+1 ≈ xkWnew

u

u =Cx + eu x , cov−1(x) =GC =CT WuC

z = By+ e

u = fu( y) u , cov−1( u) = Wu = Fu−TGB

Fu−1

y , cov−1( y) =GB = BTWB

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FactoredWLSStateEs:ma:on

u  = { α i, α ij, θ ij }

u = Cx + eu

Stage1:LinearWLSSE

z = By + e y  = { U i, K ij, L ij }

GB=BTWB

Stage2:Non-linearone-to-onemapping

u = fu (y)Wu = cov-1(u  ) =(Fu  )-TGB( Fu  )-1

Stage3:LinearWLSSEx  = { V i, θ i }

GC=CTWuC

trivial inverse( )

( )ijijij

ijijij

ii

KLLK

U

/arctan

ln

ln22

=

+=

=

θ

α

α

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FactoredWLSStateEs:ma:on

Tests

QualityIndices

1000simula:ons(toobtainpdf’s)2redundancylevels

Benchmarknetworks:118-,298-,2948-bus

•  Accuracy(withasinglerun)•  Computa:on:meandspeed-ups

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FactoredWLSStateEs:ma:on

Voltageerrors

Phaseangleerrors

Accuracy(singlerun)

Factored(singlerun)

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FactoredWLSStateEs:ma:on

ComputaGonGmes

0

0.1

0.2

0.3

0.4

0.5

0.6

118 300

t(s)

01234567

2948

t(s)

Numberofbuses

factored(singlerun)

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FactoredWLSStateEs:ma:on

Advantages

•  EnhancedconvergencerateFirst(linear)stepalwaysgivesanesGmate

•  Earlydetec:onofbaddataortopologyerror•  LesscomputaGonaleffort(constantorsimplerJacobians)

Approx.3Gmesspeedup(foraccurateenoughmeasurements)

•  Moreadvantageouswithaccuratemeasurements

•  Moreadvantageousunderpeakloading

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On-goingresearchefforts

Modelenhancement&otherpromisingapplicaGons

•  Incorpora:onofregula:ngdevices•  Distribu:onnetworks(b=N-1)•  Op:malPowerFlows

•  Nonlinearcontrol•  Eigensystemanalysis

•  Nonlinearalgebraic-differen:alequa:ons

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References•  A.Gómez-Expósito;A.Abur;A.delaVillaJaén;C.Gómez-Quiles,“AMul:levelStateEs:ma:onParadigmforSmart

Grids”,Proc.oftheIEEE,vol.99,no.6,pp.952-976,June2011.•  C.Gómez-Quiles,A.delaVillaJaén,A.Gómez-Expósito,“AFactorizedApproachtoWLSStateEs:ma:on”,IEEE

TransacGonsonPowerSystems,vol.26,no.3,pp.1724-1732,Aug.2011.•  A.Gómez-Expósito,C.Gómez-Quiles,A.delaVillaJaén,“BilinearPowerSystemStateEs:ma:on”,IEEETransacGons

onPowerSystems,vol.27(1),pp.493-501,Feb.2012.•  C.Gómez-Quiles,H.Gil,A.delaVillaJaén,A.Gómez-Expósito,“Equality-ConstrainedBilinearStateEs:ma:on”,IEEE

TransacGonsonPowerSystems,vol.28(2),pp.902-910,May2013.•  A.Gómez-Expósito,C.Gómez-Quiles,“FactorizedLoadFlow”,IEEETransacGonsonPowerSystems,vol.28(4),pp.

4607-4614,November2013.•  A.Gómez-Expósito,“Factoredsolu:onofnonlinearequa:onsystems”,ProceedingsoftheRoyalSociety–A,2014

470,20140236,July2014.•  A.Gómez-Expósito,C.Gómez-Quiles,W.Vargas,“FactoredSolu:onofInfeasibleLoadFlowCases”,18-thPower

SystemsComputa<onConference,Wroclaw,August2014.•  C.Gómez-Quiles,A.Gómez-Expósito,W.Vargas,“Computa:onofMaximumLoadingPointsviatheFactoredLoad

Flow”,IEEETransacGonsonPowerSystems,vol.31(5),pp.4128-4134,September2016.•  C.Gómez-Quiles,A.Gómez-Expósito“FastDetermina:onofSaddle-NodeBifurca:onsviaParabolicApproxima:onsin

theInfeasibleRegion”,IEEETransac:onsonPowerSystems,inpress,2017.

Thankyou!