atps - statistical estimation s8

8/6/2019 ATPS - Statistical Estimation S8

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Statistical EstimationStatistical Estimation

Maximum Likelihood WeightedMaximum Likelihood Weighted

LeastLeast--Squares EstimationSquares Estimation


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Statistical EstimationStatistical Estimation

RefersRefers toto aa procedureprocedure wherewhere samplessamples

(measurements)(measurements) areare usedused toto calculatecalculate valuevalue of of

unknownunknown parametersparameters

SinceSince samplessamples areare inexact,inexact, estimatesestimates areare alsoalso

inexactinexact

PROBLEM:PROBLEM:HowHow toto formulateformulate bestbest estimateestimate ofof unknownunknown

parametersparameters givengiven thethe availableavailable measurementsmeasurements


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Most commonly encounteredMost commonly encountered

criteriacriteria ((Three types)Three types)

Maximum likelihood criterionMaximum likelihood criterion

Weighted least squares criterionWeighted least squares criterion Minimum variance criterionMinimum variance criterion

WhenWhen normallynormally distributed,distributed, unbiasedunbiased metermetererrorerror distributionsdistributions areare assumed,assumed, eacheach ofof themthem

resultsresults inin identicalidentical estimatorsestimators


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Maximum likelihood criterionMaximum likelihood criterion

WhereWhere thethe objectiveobjective isis toto maximizemaximize

probabilityprobability (likelihood)(likelihood) thatthat thethe

estimateestimate ofof thethe statestate variablevariable ,is,isthethe truetrue valuevalue of of thethe statestate variablevariable

vector,vector, xx

Maximize P( )= xMaximize P( )= x

^

X

^

X


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Weighted least squares criterionWeighted least squares criterion

WhereWhere thethe objectiveobjective isis toto minimizeminimize

thethe sumsum of of thethe squaressquares of of thetheweightedweighted deviationsdeviations ofof thethe estimatedestimated

measurementsmeasurements ,, fromfrom thethe actualactual

measurements,measurements, ZZ

^

Z


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Minimum variance criterionMinimum variance criterion

WhereWhere thethe objectiveobjective isis toto minimizeminimize thethe

expectedexpected valuevalue ofof thethe sumsum of of thethe

squaressquares of of thethe deviationsdeviations ofof thetheestimatedestimated componentscomponents ofof thethe statestate

variablevariable vectorvector fromfrom thethe correspondingcorresponding

componentscomponents ofof thethe truetrue statestate variablevariable

vectorvector


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MaximumMaximum likelihoodlikelihood approachapproach isis utilizedutilized

herehere becausebecause itit introducesintroduces thethe

measurementmeasurement errorerror weightingweighting matrixmatrix [R][R] inin

aa straightforwardstraightforward mannermanner

MaximumMaximum likelihoodlikelihood estimatorestimator assumesassumes

thatthat wewe knowknow thethe ProbabilityProbability DensityDensity

FunctionFunction (PDF)(PDF) ofof randomrandom errorserrors inin thethemeasurementmeasurement


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LeastLeast squaressquares estimatorestimator doesdoes notnot requirerequirethethe PDFPDF forfor measurementmeasurement errorserrors

IfIf PDFPDF ofof measurementmeasurement errorerror isis assumedassumed toto

bebe normalnormal (Gaussian)(Gaussian) distributiondistribution,, samesameestimationestimation formulaformula isis obtainedobtained

PreciselyPrecisely speakingspeaking WeightedWeighted leastleast--squaressquaresestimationestimation formulaformula isis thethe resultresult


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Random measurement errorRandom measurement error

TheThe valuevalue obtainedobtained fromfrom thethe measurementmeasurement

devicedevice isis closeclose toto thethe truetrue valuevalue of of thethe

parameterparameter beingbeing measuredmeasured butbut differsdiffers byby anan

unknownunknown errorerror

RandomRandom numbernumber servesserves toto modelmodel

uncertaintyuncertainty inin thethe measurementmeasurement


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Mathematical modelMathematical model

LetLet ZZmm bebe thethe valuevalue ofof aa measurementmeasurement asas receivedreceivedfromfrom aa measurementmeasurement devicedevice

LetLet ZZtt bebe thethe truetrue valuevalue ofof thethe quantityquantity beingbeing

measuredmeasured

LetLet bebe thethe randomrandom measurementmeasurement errorerror

ThenThen measurementmeasurement valuevalue cancan bebe representedrepresented byby

ZZmm = Z= Ztt ++


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Probability Density FunctionProbability Density Function

If measurement error is unbiased, PDF ofIf measurement error is unbiased, PDF of isis

usually chosen asusually chosen as normal distribution withnormal distribution with

zero meanzero mean

standard deviationstandard deviation of random numberof random number

22 variancevariance of random numberof random number

)2/exp(2

1)(

22 WLTW

L !PDF


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Normal distributionNormal distribution

PDF()

2 3


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IfIf isis largelarge,, measurementmeasurement isis relativelyrelativelyinaccurateinaccurate..

ii..ee..,, PoorPoor--qualityquality measurementmeasurement devicedevice

SmallSmall valuevalue ofof denotesdenotes smallsmall errorerror spreadspread

ii..ee..,, higherhigher--qualityquality measurementmeasurement devicedevice

NormalNormal distributiondistribution isis commonlycommonly usedused for for

modelingmodeling measurementmeasurement errorserrors sincesince itit isis thethedistributiondistribution thatthat willwill resultresult whenwhen manymany factorsfactors

contributecontribute toto thethe overalloverall errorerror


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Maximum likelihood ConceptsMaximum likelihood Concepts

A simple DC circuit is considered.A simple DC circuit is considered.

EstimateEstimate thethe valuevalue ofof thethe voltagevoltage sourcesource xxtt

usingusing anan ammeterammeter withwith anan error error havinghaving aaknownknown SDSD

AmmeterAmmeter givesgives aa readingreading ofofzz11mm

,, whichwhich isis equalequaltoto thethe sumsum ofof zz11tt (( truetrue currentcurrent flowingflowing inin thethe

circuit)circuit) andand 11 (error(error presentpresent inin thethe ammeter)ammeter)


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Simple DC circuitSimple DC circuit

Xt (volts)

Ammeter

z1

m

r1


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i.e.,i.e., ZZ11mm = Z= Z11

tt ++ 11

Since mean value ofSince mean value of11 is zero,is zero,

mean value ofmean value ofZZ11mm is equal tois equal to ZZ11

tt

ThusThus PDF of ZPDF of Z11mm can be written ascan be written as

11 SD for random errorSD for random error11

]2

)(exp[2

1)(21

2

11

1

1WTW

tm

m zzzPDF !


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If value of resistance, rIf value of resistance, r11 is known, thenis known, then

FindFind anan estimateestimate ofof xx (called(called xxestest)) thatthatmaximizesmaximizes thethe probabilityprobability thatthat thethe observedobservedmeasurementmeasurement ZZ11

mm wouldwould occuroccur

]

2

)1

(

exp[

2

1)(

21

2

11

1

1

WTW

xr

z

zPDF

m

m

!


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Probability of zProbability of z11mm can be written ascan be written as

As dzAs dz11mm 00

!

mm

m

dzz

z

mmmdzzPDFzP

11

1

111 )()(

mmm dzzPDFzP 111 )()( !


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MaximumMaximum likelihoodlikelihood procedureprocedure requiresrequires

maximizationmaximization ofof probprob.. ofof zz11mm ,, whichwhich isis aa

functionfunction ofof xx

ii..e,e,

ForFor convenienceconvenience naturalnatural logarithmlogarithm of of

PDF(zPDF(z11mm)) isis usedused

{Maximizing{Maximizing LnLn ofof PDF(zPDF(z11mm)) alsoalso maximizemaximize PDF(zPDF(z11

mm)})}

mm

x

m

x dzzPDF

zp 111 )(max)(max !


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i.e,i.e,

oror

Since first term is a constant, it is ignoredSince first term is a constant, it is ignored

)]([max 1m

xzPDFLn

]2

)1

(

)2([max21

2

11

1W

TW

xr

z

n

m

x


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Maximize the function by minimizing secondMaximize the function by minimizing second

term since it has a negative coefficientterm since it has a negative coefficient

i.e,i.e,

same assame as

]2

)1

(

)2([max 21

2

11

1 WTW

xr

z

n

m

x

]2

)1

([min

21

2

11

W

xrzm

x


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Value of x that minimizes RHT is found byValue of x that minimizes RHT is found by

taking first derivative and equating to zerotaking first derivative and equating to zero

i.e.,i.e.,

0

)1

(

]2

)1

(

[ 211

1

1

21

2

1

1

!

!

WW r

xr

zxr

z

dx

d

mm

mest zrx 11!


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TheThe maximummaximum likelihoodlikelihood estimateestimate ofof

voltagevoltage isis thethe measuredmeasured currentcurrent timestimesthethe knownknown resistance,resistance, whichwhich isis obviousobvious

ConsiderConsider anotheranother situationsituation byby addingadding aasecondsecond measurementmeasurement circuitcircuit wherewhere thethebestbest estimateestimate isis notnot soso obviousobvious

SecondSecond ammeterammeter andand resistanceresistance isisaddedadded asas shownshown


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DC circuit with two currentDC circuit with two current

measurementsmeasurements

Xt (volts)Ammeter

z1m

r1 r2

z2m


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AssumeAssume rr11 & r& r22 are knownare known

Model each ammeter reading asModel each ammeter reading as sum ofsum of

true value & random errortrue value & random error

111 L!tm zz

222 L!tm zz


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ErrorsErrors areare representedrepresented byby normallynormallydistributeddistributed randomrandom variablevariable withwith PDFPDF

)2/exp(2

1)( 21211

1 WLTW

L !PDF

)2/exp(2

1

)(

2

2

2

22

2 WLTW

L !PDF


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As before we can writeAs before we can write

]

2

)1

(

exp[

2

1)(

21

2

11

1

1

WTW

xr

z

zPDF

m

m

!

]2

)

1

(exp[

2

1)(

22

2

22

2

2WTW

xrzzPDF

m

m

!


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LikelihoodLikelihood functionfunction mustmust bebe ProbabilityProbability ofof

obtainingobtaining measurementsmeasurements ZZ11mm andand ZZ22

mm

SinceSince randomrandom errorserrors 11 andand 22 areare

independentindependent randomrandom variablesvariables

)()()( 2121mmmm zPzPandzzP !

mmmmdzdzzPDFzPDF 2121 )()(!


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mm

m

dzdz

xr

z

2122

2

22

2

]}2

)1

(

exp[2

1{*

WTW

]}2

)1(

exp[2

1{)(

21

2

11

1

21WTW

xr

z

andzzP

m

mm

!


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To maximize take itsTo maximize take its natural logarithmnatural logarithm

)(max 21mm

xandzzp

]2

)

1

()2(

2

)

1

()2([max

22

2

22

221

2

11

1W

TW

W

TWxrz

Lnxrz

Ln

mm

x

!

]2

)1(

2

)1(

[min22

2

22

21

2

11

WW

xr

zxr

z mm

x

!


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ValueValue of of x,x, thatthat minimizesminimizes RHTRHT isis foundfound byby

takingtaking firstfirst derivativederivative andand equatingequating toto zerozero

ii..ee..,,

0]

2

)1

(

2

)1

(

[22

2

22

21

2

11

!

WW

xr

zxr

z

dx

d

mm

0

)1

()1

(

222

22

211

11

!

WW r

x

r

z

r

x

r

z mm


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oror

IfIf oneone ofof thethe ammeterammeter isis of ofsuperiorsuperior qualityquality,, itsits

variancevariance willwill bebe muchmuch smallersmallerthanthan thethe otherother

IfIf 2222


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MaximumMaximum likelihoodlikelihood methodmethod of of estimatingestimating

unknownunknown parameterparametergivesgives aa wayway toto weightweight thethemeasurementsmeasurements properlyproperly accordingaccording toto their their

qualityquality

MaximumMaximum likelihoodlikelihood estimateestimate ofof unknownunknown

parameterparameter isis alwaysalways expressedexpressed asas thatthat valuevalue

whichwhich givesgives thethe minimumminimum ofof thethe sumsum of of thethe

squaressquares ofof thethe differencedifference betweenbetween eacheachmeasuredmeasured valuevalue andand truetrue valuevalue withwith weightedweighted

byby variancevariance ofof thethe metermeter errorerror


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EstimatingEstimating aa singlesingle parameterparameter xx usingusing NNmm

measurements,measurements, expressionexpression becomesbecomes

wherewhere

ffii == functionfunction usedused toto calculatecalculate thethe valuevalue

beingbeing measuredmeasured byby iithth measurementmeasurement

ii22

== variancevariance forfor iithth

measurementmeasurementJ(x)J(x) == measurementmeasurement residualresidual

NNmm == numbernumberofof independentindependent measurementsmeasurements

ZZiimm == iithth measuredmeasured quantityquantity

!

!

mN

i i

i

m

i

x

xfzxJ

12

2)]([

)(minW


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To estimate NTo estimate Nss unknown parameters using Nunknown parameters using Nmm

measurementsmeasurements

These two estimation calculation equations areThese two estimation calculation equations areknown asknown as weighted leastweighted least--square estimatorsquare estimator

!

!m

s

s

sN

N

i i

Ni

m

i

N

xxx

xxxfzxxxJ

1 2

221

21},...,{

)],.....,([),......,(min

21 W


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Matrix FormulationMatrix Formulation

IfIf functionsfunctions f fii(x(x11,x,x22,,.. xxNsNs)) areare linear linear

functions,functions, wewe cancan writewrite

sss NiNiiiNixhxhxhxfxxxf !! ...............)(),.....,( 221121


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If all fIf all fii functions are placed in a vectorfunctions are placed in a vector

wherewhere

[H] =[H] = NNmm

X NX Nss

matrix containingmatrix containing coefficientscoefficients of linearof linear

functionsfunctions

NNmm = No. of measurements= No. of measurements

NNss = No. of= No. ofunknown parametersunknown parameters being estimatedbeing estimated

? AxH

xf

xf

xf

xf

mN

!

!

)(

.

.

.

)(

)(

)(

2

1


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PlacingPlacing measurementsmeasurements in ain a vectorvector

!

mN

m

m

m

mz

z

z

z

.

.

.2

1


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Weighted leastWeighted least--squares estimator in verysquares estimator in very

compact formcompact form

wherewhere

calledcalled covariance matrix of measurement errorscovariance matrix of measurement errors

)](][[)]([)(min 1 xfzRxfzxJ mTm

x!

!

2

22

21

.

.

.][

mN

R

W

W

W


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Expand the expression & substitute for f(x)Expand the expression & substitute for f(x)

Minimum of J(x)Minimum of J(x) is found whenis found when

i.e., Gradient of J(x),i.e., Gradient of J(x), J(x) is exactly zeroJ(x) is exactly zero

mTTmm

xzRHxzRzxJ

T

][][][)(min 11 !

xHRHxxHRz TTmT

]][[][]][[11

si NforixxJ ,........2,1,0/)( !!xx


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Gradient of J(x),Gradient of J(x),

ThenThen J(x) = 0 givesJ(x) = 0 gives

This holds whenThis holds whenNNss < N< Nmm

i.e., No. of parameters being estimatedi.e., No. of parameters being estimated

< No. of measurements being made< No. of measurements being made

xHRHzRHxJ TmT ]][[][2][][2)( 11 !(

mTTest zRHHRHx ][][]]][[][[ 111 !


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whenwhenNNss = N= Nmm

whenwhenNNss > N> Nm,m,

xx isis notnot estimatedestimated toto maximizemaximize likelihoodlikelihood

functionfunction sincesince thethe conditioncondition impliesimplies thatthat manymany

valuesvalues ofof xxestest cancan bebe foundfound thatthat causecause ffii(x(xestest)) equalequaltoto zzii

mm exactlyexactly forfor allall i=i=11,,22,,..NNmm..

mest zHx1

][!


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TheThe objectiveobjective isis toto findfind xxestest suchsuch thatthat sumsum

ofof squaressquares ofof xxiiestest isis minimizedminimized

ii..ee..,,

subjectsubject toto thethe conditioncondition zzmm == [H][H] xx

SolutionSolution

xxx TN

i

i

x

s

!!1

2min

mTTestzHHHx

1]]][[[][

!


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Power system state estimationPower system state estimation

UnderUnder determineddetermined problemsproblems ((NNss >> NNmm)) isis notnotsolvedsolved usingusing thethe aboveabove equationequation

PseudoPseudo--measurementsmeasurements areare addedadded toto thethe

measurementmeasurement setset toto givegive completelycompletely determineddetermined

((NNss == NNmm )) oror overover determineddetermined ((NNss


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3 bus system3 bus system


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Meter placementMeter placement


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Suppose readings obtained areSuppose readings obtained are

puMWM 62.06212 !!

puMWM 06.0613 !!

puMWM 37.03732 !!


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SolutionSolution

oo NNss = 2= 2 ((11,, 22))

oo NNmm = 3= 3 (M(M1212, M, M1313, M, M3232))

oo StatesStates 11,, 22 cancan bebe estimatedestimated byby minimizingminimizing

residualresidual J(J(11,, 22)) whichwhich isis thethe sumsum ofof squaressquares ofof

individualindividual measurementmeasurement residualsresiduals divideddivided bybyvariancevariance forfor eacheach measurementmeasurement


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oo Given Meter characteristicsGiven Meter characteristics

Meter fullMeter full--scale value: 100 MWscale value: 100 MW

Meter accuracy:Meter accuracy: ++ 3 MW3 MW

TheThe interpretationinterpretation isis thatthat metersmeters willwill

givegive aa readingreading withinwithin ++ 33 MWMW ofof thethe truetrue

valuevalue beingbeing measuredmeasured forfor approxapprox.. 9999%% ofof

thethe timetime


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ErrorsErrors areare distributeddistributed accordingaccording toto normalnormalPDFPDF withwith SDSD asas shownshown

ProbabilityProbability ofof anan errorerror decreasesdecreases asas thethe

errorerror magnitudemagnitude increasesincreases

IntegratingIntegrating PDFPDF betweenbetween --33 && ++33,, thethe

valuevalue approxapprox.. toto 00..9999


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Normal distribution of meter errorsNormal distribution of meter errors


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WeWe assumeassume metersmeters accuracyaccuracy isis equalequal

toto thethe 33 pointspoints onon thethe PDFPDF

Then 3 MW corresponds to meteringThen 3 MW corresponds to metering

SD ofSD of = 1MW = 0.01 pu= 1MW = 0.01 pu


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we knowwe know

WhereWhere

xxestest = vector of= vector ofestimated stateestimated state variablesvariables

[H][H] = measurement function= measurement function coefficient matrixcoefficient matrix

[R][R] == measurementmeasurement covariance matrixcovariance matrixZZmm = vector containing measured values= vector containing measured values

mTTest zRHHRHx ][][]]][[][[ 111 !


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we havewe have

Derive [H] matrixDerive [H] matrix

write measurements as a function of statewrite measurements as a function of statevariablesvariables 11,, 22 in puin pu

!est

estest

x2

1

U

U


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puxf 06.05.2)(

113131

1313 !!!! UUU

pu

x

f 62.055)(1

122121

12

12 !!!! UUUU

puMxf 37.04)(

132223

3232 !!!! UUU


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Reference bus phase angleReference bus phase angle 33 is assumed tois assumed to

zerozero, then, then

Covariance matrixCovariance matrix for measurements,for measurements,[R][R] isis

? A

!

40

05.2

55

H

? A

!2

2

2

32

13

12

M

M

M

R

W

WW


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All quantities in puAll quantities in pu

Solve for state variablesSolve for state variables

? A

!0001.0

0001.0

0001.0

R

!

!

094286.0

028571.0

2

1

est

estestxUU


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FromFrom thethe estimatedestimated valuesvalues calculatecalculate powerpower

flowsflows && netnet generation/generation/ loadload atat eacheach busbus

CalculateCalculate residualresidual J(J(11,, 22))

232

2213232

213

22,11313

212

2211212

21

)],([)]([)],([),(

W

UU

W

UU

W

UUUU

fzfzfzJ

!

14.2),( 21 !UUJ


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33 -- bus systembus system


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Three bus with best estimatesThree bus with best estimates 11&& 22


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Suppose meterSuppose meterMM1313 is superior in qualityis superior in quality

Check how this affect the estimates of statesCheck how this affect the estimates of states

Data:Data:

MetersMeters MM1212 & M& M3232


Meter accuracy:Meter accuracy: ++ 3 MW3 MW (( = 1MW = 0.01 pu)= 1MW = 0.01 pu)

MeterMeterMM1313


Meter accuracy:Meter accuracy: ++ 0.3MW0.3MW (( = 0.1MW = 0.001 pu)= 0.1MW = 0.001 pu)


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Repeat as before & solve for statesRepeat as before & solve for states

Compare the resultsCompare the results

EstimatedEstimated flowflow onon lineline 11--33 isis closercloser totometermeter readingreading

!

!

097003.0

024115.0

2

1

est

estestx

U

U


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Three bus with better meterThree bus with better meteratat MM1313

atps - statistical estimation s8

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