The factored approach to solving nonlinear power system...
Transcript of The factored approach to solving nonlinear power system...
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
© Gómez-Expósito
• 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
© Gómez-Expósito
• 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
© Gómez-Expósito
• 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
© Gómez-Expósito
• 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
© Gómez-Expósito
• Letybeanauxiliaryvectorcomposedofthemdis:nctaddi:vetermsinh(.),eachwithtrivialinverse.Then,thefollowingfactoredformarises:
Factoredsolu:onapproach
Under-determinedlinearsystem
Over-determinedlinearsystem
mone-to-onemappingswithexplicitinverse:
© Gómez-Expósito
• Conven:onalNR(forcomparison):
• Two-stepfactoredprocedure:
Factoredsolu:onapproach
Step0:Ini:aliza:onx0 ày0
Step1:
Step2:
<ε NO
YESENDFactoredJacobian:
© Gómez-Expósito
• Conven:onalNR:
ComputaGonalissues:
Factoredsolu:onapproach
Step0:Ini:aliza:onx0 ày0
Step1:
Step2:
<ε NO
YESEND
Least-distanceproblem:Choleskyfactorscomputedonlyonce
© Gómez-Expósito
• Conven:onalNR:
ComputaGonalissues:
Factoredsolu:onapproach
Step0:Ini:aliza:onx0 ày0
Step1:
Step2:
<ε NO
YESEND
Trivialone-to-onenonlinearmappingf()withdiagonalJacobian
© Gómez-Expósito
• Conven:onalNR:
ComputaGonalissues:
Factoredsolu:onapproach
Step0:Ini:aliza:onx0 ày0
Step1:
Step2:
<ε NO
YESEND
Least-squaresproblem:FactoredJacobiansimplertocompute
© Gómez-Expósito
• Conven:onalNR:
ComputaGonalissues:
Factoredsolu:onapproach
Step0:Ini:aliza:onx0 ày0
Step1:
Step2:
<ε NO
YESEND
LinearmismatchvectorusedinnextiteraGon
© Gómez-Expósito
1.Sumsofsingle-variablenonlinearelementaryfuncGons:
Canonicalforms
Example:
© Gómez-Expósito
Canonicalforms2.Sumsofproductsofsingle-variablepowerfuncGons:
Preliminarylogarithmicchangeofvariables:
© Gómez-Expósito
CanonicalformsExample:
© Gómez-Expósito
CanonicalformsRemark:Ifm=n,thenthereisnoneedtoiterate
© Gómez-Expósito
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:
© Gómez-Expósito
Extendingtherangeofreachablesolu:ons
MulGplicityofsoluGons:Whichsolu:onisitera:velyreached?
• NewtonRaphson:dependsonthebasinofaWrac:oninwhichtheini:alguessx0lies
• Factoredmethod:canbecontrolledbyselec:ngthecomputedrangeforeachindividualnonlinearfunc:on
Examples:⇒ ℜ(u)> 0
⇒ ℜ(u)< 0
Periodicfunc:ons
⇒
⇒
© Gómez-Expósito
Extendingtherangeofreachablesolu:ons
Example:2x2P.Boggs’system
Onlythreesolu:ons:
NewtonRaphson’sbasinsofaYracGon:unpredictableirregularbehavior
Whiteareas:divergence
A
B
C
ABC
© Gómez-Expósito
Extendingtherangeofreachablesolu:ons
Example:2x2P.Boggs’system
Foranyx0 !!Withthebasicdefini:onsconvergestoAWithconvergestoBWithconvergestoCWithyieldscomplexsolu:on
Factoredmethod:newvariables
A
B
C
© Gómez-Expósito
Infeasibilityandcomplexsolu:ons
• Feasiblecase:pissuchthatthereexistsatleastarealxsa:sfyingh(x)=p
• Infeasiblecase:onlycomplexvaluesofxsa:sfyh(x)=p
Empiricalevidence:• Newton’smethodcannotgenerallyconvergetocomplexsolu:ons
• Thefactoredmethodisflexibleenoughtoconvergetocomplexsolu:onswheneverrealsolu:onscannotbefound
• ChoosinganiniGalguesswithacertainimaginarycomponentishelpfultoachieveconvergencetocomplexsolu:ons
• Quadra:cconvergenceratealsotocomplexsolu:ons
© Gómez-Expósito
Infeasibilityandcomplexsolu:ons
Example: Canonicalform:
© Gómez-Expósito
FactoredLoadFlowSolu:on
Factoredmodel:
p: Specifiedquan::es(PQandPVbuses)x: Statevariables(2N-1)
(N+2b)
© Gómez-Expósito
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
© Gómez-Expósito
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
© Gómez-Expósito
FactoredLoadFlowSolu:onConvergencebasins:2-busexample
© Gómez-Expósito
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
© Gómez-Expósito
FactoredLoadFlow
ConvergenceratesfortheIEEE118-bussystem[PVbusesconvertedtoPQbuses]
NR
Factored
© Gómez-Expósito
FactoredLoadFlowSolu:onIEEE14-bussystem:evoluGonofvoltagesforincreasingloadings
Voltagemagnitudes(p.u.)Voltagephaseangle(deg.)
Infeasibleregion
Infeasibleregion
© Gómez-Expósito
Maximumloadingpointdetermina:onProblem:Findingthesaddle-nodeorlimit-inducedbifurcaGonpoint
© Gómez-Expósito
Maximizethevalueofλ:
• Op:miza:onproblem• Con:nua:onmethods
Maximumloadingpointdetermina:on
© Gómez-Expósito
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
© Gómez-Expósito
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
© Gómez-Expósito
ComparisonofsimulaGonresults
Maximumloadingpointdetermina:on
© Gómez-Expósito
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
© Gómez-Expósito
FactoredWLSStateEs:ma:on
LinearWLSSE
Non-linearexplicittransforma:on
LinearWLSSE
Factoriza:onofnonlinearWLSproblems
© Gómez-Expósito
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{ }
© Gómez-Expósito
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
PP
ε
ε
+=
+=
∑∑
z = By+ e y , cov−1( y) =GB = BTWB
© Gómez-Expósito
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
© Gómez-Expósito
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
© Gómez-Expósito
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
© Gómez-Expósito
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
© Gómez-Expósito
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
© Gómez-Expósito
FactoredWLSStateEs:ma:on
Tests
QualityIndices
1000simula:ons(toobtainpdf’s)2redundancylevels
Benchmarknetworks:118-,298-,2948-bus
• Accuracy(withasinglerun)• Computa:on:meandspeed-ups
© Gómez-Expósito
FactoredWLSStateEs:ma:on
Voltageerrors
Phaseangleerrors
Accuracy(singlerun)
Factored(singlerun)
© Gómez-Expósito
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)
© Gómez-Expósito
FactoredWLSStateEs:ma:on
Advantages
• EnhancedconvergencerateFirst(linear)stepalwaysgivesanesGmate
• Earlydetec:onofbaddataortopologyerror• LesscomputaGonaleffort(constantorsimplerJacobians)
Approx.3Gmesspeedup(foraccurateenoughmeasurements)
• Moreadvantageouswithaccuratemeasurements
• Moreadvantageousunderpeakloading
© Gómez-Expósito
On-goingresearchefforts
Modelenhancement&otherpromisingapplicaGons
• Incorpora:onofregula:ngdevices• Distribu:onnetworks(b=N-1)• Op:malPowerFlows
• Nonlinearcontrol• Eigensystemanalysis
• Nonlinearalgebraic-differen:alequa:ons
© Gómez-Expósito
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!