Digital Processing and Reconstruction of Complex AC Signals

124 Pages·2009·3.18 MB·English

by  Predrag Petrović Ph.D., Milorad Stevanović Ph.D. (auth.)

Predrag Petrovic, Ph. D.
Milorad Stevanovic, Ph. D.
Digital Processing
and Reconstruction
of Complex AC Signals
~
Springer
ACADEMIC MIND
Predrag Petrovic, Ph. D.
UniversityofKragujevac, TechnicalFaculty Cacak
MiloradStevanovic, Ph. D.
UniversityofKragujevac, TechnicalFaculty Cacak
DIGITALPROCESSINGANDRECONSTRUCTION
OFCOMPLEXACSIGNALS
Reviewers
SrdanStankovic, Ph.D.
fullprofessor, UniversityofBelgrade,FacultyofElectricalEngineering
Slavoljub Marjanovic, Ph.D.
fullprofessor, UniversityofBelgrade,FacultyofElectricalEngineering
(c)2009
ACADEMICMIND, Belgrade, Serbia
SPRINGER-VERLAG, Berlin Heidelberg, Germany
Design ofcoverpage
ZoricaMarkovic, Academic Painter
Printed inSerbiaby
Planeta print,Belgrade
Circulation
400copies
ISBN978-86-7466-363-9
ISBN978-3-642-03842-6
Library ofCongressControl Number: assigned
NOTICE:Nopartofthispublicationmaybereprodused,storedinaretreivalsystem,ortransmitted
inanyformorbyanymeans,electronic,mechanical,photocopying,recordingorotherwise,without
thepriorwrittenpermissionofthepublishers.Allrightsreservedbythepublishers.
Digital Processing
and Reconstruction
of Complex AC Signals
CONTENTS
1.INTRODUCTION.....................................................................................................................•....•........5
1.1Basic PrinciplesoftheSuggestedMeasuringMethod 6
1.2MathematicalProofofJustificationoftheSuggestedMeasuringMethod 11
1.3MathematicalProofofCorrectnessinActivePowerProcessing 14
1.3.1.AnalysisoftheNumericalProceduresSuggestedin theProcessingofPeriodicSignals 18
1.4AdaptabilityoftheSuggestedAlgorithmUsed for MeasuringofElectricalValuesinElectric
Utilities.........•..........•.................................................•..•.....•...••...................................................................19
1.5AnalysisofPossibleSourcesofErrorsinDigitalProcessing
WithaSuggestedMeasurementConcept ..................•....•............................................................21
1.5.1SimulationResults.....•.......•...•..............................................•......•.•...•...•..•..•..............................•26
1.5.2AnalysisoftheErrorCausedbyImprecisioninDeterminingSamplingInterval............•...27
1.6SimulationoftheSuggestedMeasuringMethodintheMatlabProgramPackage 33
1.6.1SoftwareTestingoftheSuggestedMeasuringConceptofElectricalValuesBasedon
MeasurementResultsinRealElectricUtilities........••....•.....•..•...................................................•...•..38
1.7PracticalRealizationoftheSuggestedDigitalMeasuringSystem 41
1.8ResultsofPracticalMeasurementswithRealizedDigitalWattmeter 49
References 55
2.DIGITALPROCESSINGOFSYNCHRONOUSLYSAMPLEDAC SIGNALSIN
PRESENCEOFINTERHARMONICSANDSUBHARMONICS 58
2.1SynchronousSamplingin thePresenceofSubharmonicsandInterharmonics 59
2.1.1DerivedConditionsforPreciselyProcessing..............•......•........•..•.............•....................•...••...•.67
2.1.2AsynchronousSampling.••..•..•...•..•.••..•.••..•..•..•..•..•.••....•..••..•..•.•..•..•.••.•..•..•.•.••......•..........•..••..•••..69
2.2SimulationResults 70
2.3CalculationoftheTruncationErrorsinCaseof
AsynchronousSamplingofComplexAC Signals 73
2.3.1AnalysisofWorstCaseErrors 73
2.3.1.1AverageMethod............•............................................................................................................75
2.3.1.2TrapezoidalMethod...................................................................................................•..•..•....••...76
2.3.1.3Stenbakken'sCompensation..............................•.•..•...•.................•........•...•.••..•.....••..................77
2.3.1.4Zu-LiangCompensation...•............•.....................................................................•..•..................77
2.3.1.5AverageMethod-approximateexpression...•........................................•...•..••.......••.•..•..••........•78
2.3.1.6TrapezoidalMethod-approximateexpression...................................................................•....•.78
2.3.1.7Stenbakken'sCompensation-approximateexpression....................................................•......•78
2.3.1.8Zu-LiangCompensation-approximateexpression 79
2.3.2 SimulationResults 79
References.............................................................................•.•..•.......................................•.•.................83
AppendixA 84
3.RECONSTRUCTIONOFNONUNIFORMLYSAMPLEDAC SIGNALS 86
3.2.ProposedMethodofProcessing 88
3.2.1 TheDeterminantsofthe VanderMondeMatrix 89
3.2.2ReconstructionofBandLimitedSignalsinForm ofFourierSeries 90
3.3.SimulationResultandErrorAnalysis 95
2
3.4.PossibleHardwareRealizationofthe ProposedMethodofProcessing 99
References..•..••......••..••..........•.......•...••.•...•...••.••................•.•.•..........••..•..•••.••.•••..••..•••...•...••...........•.••..102
AppendixB...•...•..•...•...•...•.•..............••.•..•..••..••..•...•...•..•..•.......•..........••.••..........••........•.....••.••..•..•.••....104
AppendixC•..••..••••••..•..•........•.....•.•....•.•..••.••...••.•••••..••••..•••••...........•.•..••.•..•••..••.•••..••......••••••••••••..•.••.••104
AppendixD•...•...••.....•...•.....••.•.•............•..••..••..••.•...........•.•....•......•.•....•.............•.....••...••••....•....•.•.••.••.108
AppendixE...........................................................................................................................•..•......••...110
4.NEW METHODFORPROCESSINGOF BASIC ELECTRICALVALUES BASEDON
DEFINITIONFORMULAINTIMEDOMAIN...•..•.........................................................•.....•....••.112
4.1.SuggestedMethodofProcessing 112
4.1.1EstimationofMeasuringUncertainty 115
4.2.Simulationofthe SuggestedMeasuringMethod 115
4.3.Practicalrealizationofthe proposedalgorithm 117
4.4.ExperimentalResults 119
References..•.••...•................................•...•....................•.........•..•..........•......•..••....••..••............•....•..•.••.•.121
3
PREFACE
Thismonographisthe result ofalong-term work ofthe authors ontheproblems ofdigital processing
ofcomplex periodic signals ofvoltage and current which can be found inthe distribution network. This
scientific fieldhasbeengiven special attention intheliterature abroad through agreat number ofarticles
publishedintheleadingjournals,textbooks andtheotherpublishingforms.
Over the recent period, the authors ofthe monograph have published a large number ofpapers in a
number ofjournalsandhave presentedattheleading internationalconferences,which verifies theresults
theyhave achieved inthis scientific field. Theirpatents whichhavebeenregisteredintheirhome country
aretheresult ofthework.
The problem ofthe reconstructionofcomplex periodic signals, which isinthe focus ofChapter three
of this monograph, has been given special attention. A completely new protocol, which enables the
development ofmuch superior and more efficacious algorithms, has been developed, and the obtained
results areunique inglobalpractice.
Chapter two deals with the processing ofnonharmonic components ofac signals, i.e. interharmonics
and subharmonics based on the principles ofsynchronous sampling. The conditions, required for the
performance of such a processing, have necessarily been derived. Chapter four looks at a newly
developed method for the calculation ofbasic parameters ofthe processed voltage and current signals.
Having beingpracticallyverified, thismethodhasdemonstratedexceptionallyfavourable performance.
The authors believe that the monograph will be beneficial to specialists and experts involved in
problems ofthis scientific field, and hope that it will serve as a useful guideline to follow in further
investigations.
Theauthors wishtoexpresstheirgratitude totheirlanguage editorLidijaPalurovic, M.A.Philo!'
4
1.INTRODUCTION
The first chapter of this book is dedicated to the problem of measuring basic electric quantities in
electric utilities (voltage, current, power, frequency), both from the aspect ofaccuracy of this type of
measurementsandthepossibilities ofsimpleandpractical realization.Theconventional algorithmsused
tothispurpose(calculatingtheactivepowerandtheRMSvalueofthevoltageandcurrentsignalsthatare
theobjectofprocessing) arebasedontheuseofintegrationorsummationprocess onlimittimeinterval.
Acharacteristicofthisapproach isthatitoffersthecorrectresultiftheinputsignalisperiodical intime.
However, inreal systems,voltage andcurrentsignalsarenotnecessarilyofaperiodical quantity, dueto
the presence of nonharmonic components or/and possible stochastic variation. It is for this reason that,
very often, the result of processing is not an instantaneous value of processing signal - it is rather a
processed (observed)valueinsometimeinterval,bothwithconventionalandcertainnewalgorithmsfor
processing[1-19].
Themeasuringequipmentusedtothispurposeworldwide(aswellasinourelectricutilities)canbeof
differentprecision class(fromclass0.1toclass2),andmeasuringusuallypresupposes thattherecording
onthe net takes place from a specified momentto another moment, specified with aparticular protocol
(these two moments can be some minutes or somehours away). This further implies that we deal with
measurements not conducted on-line (continuously in time), and for which we can use a realized
measurementsystemduetoitsaccuracyandprice.
Manyapplicationsinvolvedigitalprocessingofperiodicsignals[1-19].Forexample,bothvoltageand
current inelectric utilities areperiodic signals containing harmonic components. The measuring method
proposedinthischapterisbasedonselectingsamplesoftheinputvariableinalargenumberofperiodsin
which the system (electric utilities, inthis case) is considered to be stationary. The stationary condition
canbeprovedbythevaluesobtainedfrommeasuringoftheRMS(rootmeansquareoreffective)values
ofthevoltagesystem.Stationarityofthesystemsuggeststhatslowlychangingquantities,suchascurrent
andvoltage, andtheir harmonic content, are constant within the measuring interval. Inthis case, unlike
the Nyquistcriteria, undersampling is possible. The current makes this system nonlinear as the type of
load which will be used and the time ofits connection to the investigated system cannot be predicted.
However,afteracertainnumber ofperiods, thecurrentcanbeconsideredasaslowlychanging variable
duringprocessing.
Theutilitiesareinertsystemsincharacter,sothatthesamplingduringseveralperiods oftheobserved
systemvariables canbeperformed. It is forthis reason that very slow,low-cost, but very accurate AID
converters, suchasadual-slope type, wereused intheproposed measuring system. Voltage andcurrent
from real electric utilities were used as input variables. The sampling procedure is initiated arbitrarily.
Thedistancebetweentwoconsecutivesamplesisgivenby:
tdelay = N ·T +L\t (1.1)
whereN isthe number of periods between sampling, T isthe period ofthe inputvoltage, and.<1tisthe
delaydeterminedbythedelayofelements intheprocessing circuit..<1tdependsontheharmonic content
of the input signal. All conditions, which have to satisfy both Nand .<1t to get an accurate result of
measuring,canbederived.Forthatreasontheycannotbearbitrary.
The suggested measuring method is classified as a synchronous sampling method which does not
introduce any error when measuring sine and spectrally limited complex periodic voltage and current
signals. Compared to other methods, it is the simplest method from the viewpoint of realizing a
microcomputer block using the simplest algorithm with a relatively small number ofsamples Win the
measuring cycle. However, itisnon-ideal synchronization of samplingfrequency with the frequency of
themeasuringsignalthatintroducesasignificanterror.Specialattentionhasbeenpaidtothisproblem.
Taking into account the presented facts, it is obvious that the time interval necessary for the correct
processing of the observed quantities becomes very short from the viewpoint of inertia of such huge
systemsaselectricutilities. Thishasbeen absolutely confirmedbyexperimental measurementswehave
performed.Themeasuringtimefortheproposedmethodisabout 1second.
5
Theproblem of noisethatcanoccurin a realsystem, as averaging isperformed to determine the
average power, isnotconsidered important astheaverage noisevalueiszero.Thepossible nonlinear
distortions intransitionprocesses donotlastverylong,sotheycanbeavoidedwhenthefunctioning of
thiswattmeterdesignedformeasuringperiodicvariablesisconsidered.
Mathematicalanalysisoftheproposedmeasurementmethodandthedefinedconditionsinwhichthey
giveanabsolutely(mathematically)correctresultwilldependontheharmoniccontentoftheinputsignal
whichistheobjectofprocessing.Allofthisismadeforthenetsthatoperateatfrequencies of50and60
Hz.Thuswecreateapossibilitytodefineanadaptive algorithmforprocessingwhichcantrackchanges
inslowly-changingsystemssuchaselectricutilities. Consequently, itispossibletomakeacorrectionof
thesamplingfrequencybasedontheestablishedchange.
1.1BasicPrinciplesoftheSuggestedMeasuringMethod
Voltageandcurrentoccurringinrealelectricutilities(therealcircuitofalternatingcurrent)wereused
asinputvariables. Practically, measuring isbasedonthedirectmeasurement method, e.g.voltageand
currentmeasurementduringwhichtheproblemofphasecorrelationbetweenvoltageandcurrentsignals
isnotevidencedsinceinstantaneoussignalvaluesarefirstsampled,andthenprocessed.It isaknownfact
that the phase anglerepresents a seriousdefectin measurement methods realizedwith an analogue
voltmeter and ampermeter. In this case,it is dueto the factthat measuring is restricted onlyto the
resistiveload.Inthemeasurementsystemwearesuggesting,thesamplesofvoltageandcurrentaretaken
inarbitrarymoments,withthedistancebetweentwoconsecutivesamplesgivenby(1.1).
Basedon the obtained samples, seriesv(k), i(k) (k = 1,2,...U') of voltage and currentsamples are
formed. Thesamplingprocedure isinitiatedarbitrarilyanditenablesthereconstruction ofthemeasured
valuesinaccordancewiththefollowingdiagram(Figure 1.1).
v[v]
Idelay
310
I[ms]
10 20 30 40 50
-310
Figure 1.1Proposedmethodofsampling
Theaveragepowerofsignals sampledinthiswayiscalculatedwhenthesumofallWinstantaneous
valuesofpowerisdividedbythewholenumberofsamples W,accordingtothefollowingequation:
1W-I 1W-I T
p!::'e= W ~V(ti~(t;)= W ~P(ti}, ti=to+iW (1.2)
1=0 1=0
whereWisanarbitrarynumberdeterminedbythenumberofsamplesneededforaprecisereconstruction
ofthemeasured value,to- theinitialmoment ofmeasurements, whereO<to<T/W. ThenumberWis a
natural number, obtained as the number of samples necessary for a precise reconstruction of the
measurement quantities. Theobtained valuefortheaverage poweris compared duringthe simulation
withtheaveragepowerdeterminedusingthedefinitionformeanaveragepower:
1fT
P=- v(t).i(t)lit (1.3)
To
whereTistheperiodoftheinputvoltage.
Aftermeasuringaseriesofsamplesthewattmeterrepeatsthesameprocedureonthenextseriesofthe
samelengthuntilitisswitchedoff.
6
Measuring theactivepowerina systemwithcomplex periodical signalsandwitharbitrary loadis
basedontheprincipleappliedinthecaseofastemwithsimpleperiodicalsignals.
The measurement method proposed in this chapter is based on selecting samples (the original
assumption)oftheinputvariableinalargenumberofperiodsinwhichthesystem(theelectricutilityin
this case) is considered to be stationary. Stationarity of the system provides consistency in slowly
changing quantities, such as currentand voltage, and their harmonic contentwithin the measuring
interval. Thisiswhyveryslow,low-cost, butveryaccurate dual-slope AIDconverterswereusedinthe
proposedmeasuringsystem. Forthistypeofconverter,theaccuracyofconversionwillonlybegoverned
bythereferencevoltage. BydeterminingthetransferfunctionAOw)(amplitudetransferfunctionofdual
slopeADC), wenoticethatthisis alinearsystemwithconst. amplitude characteristic, andthisisthe
reasonwhyitdoesnotincludeanadditionaldistortioninthemeasurementvalue[20].
ThebasicstructureofsuchADCwith8-bitsresolutionispresentedinFigure1.2.
sw;
r - --,
I SAMPLE I-Vul
-V-u.ljABNODLD I~ ~ --"\M~~
IL __ J+Vre
STRATL
eLK
BUSY OW
Q, Q, QsQ4 QJ Q2 QI QI
Figure1.2Dual-slopeADCwith8-bitsresolution
ForthistypeofADCthefollowingequationisvalid,anditistheonewebaseontheconversionof
themeasuredvalue:
1 Jtl
VI(t1)=_R·C. V';nput •dt (1.4)
to
whereT]= t]-to.
v. v.
V (t)= Rm.Cput.T.=Rm.pCut •2n•t (1.5)
I 1 1 C
whereT] = t:- torepresentstheperiodoftimeinwhichthecountercountedZ"clockimpulsesofperiod
t..Inequations(1.4)and(1.5)weassumedthatVinputistheconstantvalueintheperiodofintegration.Ifit
isnotthecase,wemustplacethesampleinfrontofADCandholdthecircuit.
Anthemomentdeterminedast2,thevoltageattheoutputoftheintegratorisvo= 0,i.e.
1 ~ V
VI(t2)=VI(t1)--·JVREF .dt=VI(t1)- REF .T2=0 , (1.6)
R·C ~ R·C
whereT2 = t:-t] = it;isthetimeinwhichthecounterregisterediclockimpulses.Thelasttwoequations
infer:
7
~.2n.t
- VREF -i-t =0 (1.7)
R.C R.C
C C ,
inotherwords:
2n
i=--·V (1.8)
V ul
REF
The lastformula revealsthatthenumber i,i.e.thenumber registeredatthecounter atthemoment t2is
proportionalto the absolute value ofthe input voltage. Similarly, it suggests that it isdependentneither
ontheresistance Rnorcapacity Cnortheamplitude oftheclockimpulsest.:
A standard dual slope AID converter operates with a sampling frequency between 4 and 96 Hz,
depending on the input amplitude. This type ofconverter with a resolution of16bits was used for the
developmentofthedescribed digitalmeasuringsystem.
Another important advantage of this method (integrating ADC) is that the input signal becomes
averagedasitdrives the integratorduring the fixed-time portion ofthe cycle. Any changes inthe analog
signal during that period oftime have a cumulative effect on the digital output at the end ofthat cycle.
Other ADC strategies merely "capture" theanalog signal levelatasinglepoint intimeevery cycle. Ifthe
analog signal is"noisy", i.e.containingsignificantlevels ofspurious voltage spikes/dips oneofthe other
ADC converter technologies may occasionally convert a spike or dip because it captures the signal
repeatedly at a signal point time. A dual-slope ADC, on the other hand, averages togetherall the spikes
anddipswithin theintegrationperiod, thusprovidinganoutputwithgreater noise immunity. Delta-sigma
(or more accurately, sigma-delta) ADCs isthe newest architecture, and it isused in systems demanding
high-resolution data acquisition; but when they are used in instrumentation, their filter delays prevent
multiplexingandloopstability. Otherwise, theyhavepoorstepresponse contrarytodual-slope ADCs.
The influencethat ADC architecture(over adequate model ofADC) has onthe error inconversion is
very well-known for three most important converter types: integrating, with successive approximations
and flash converter [21]. The different sources of error were analyzed in the form of integral and
differential nonlinearity, in order to define unique model oferror. Inthis way we can generate the table
withcorrectionfactors andadiagnostic model forerrordetection. Numericalsimulations were carried out
in order to valorise the theoretical results. Inthis way, apossibilityis opened to model an ADfeedback
block on the above-mentioned converters, in a unique way [22]. The error in the processing section for
the analogue signal atthe integrationAD will have apolynomial form, sothat itcan be compensatedin
the simplest way. Taking into considerationthe confirmedlinearity ofthe suggested methods in[22],we
canrightly assumethattheapplied dual-slope ADCrepresents anoptimal solutionunderthiscriterion.
The suggested concept ofmeasuringthe RMS oreffective value ofcurrent, voltage andpower can be
classifiedasasynchronoussampling method through which, intheory, inequaltimeintervals, Wsamples
are taken. However, non-ideal synchronization of sampling frequency with the frequency of the
measuringsignal introduces asignificanterror. Specialattention hasbeenpaidtothisproblem.
Theory ofsynchronous sampling was developed for periodic alternating signals (sine and complex
periodic signals). In the classification ofmeasurement signals, these are the first and second type of
measurementsignals. Synchronoussampling with anowndefinition isperformedwhen acertainperiodT
is divided into Wequidistant intervals, after which Wsamples ofcurrent and voltage are taken in equal
time intervals TIW. Accordingtotheexpandeddefinition ofsynchronoussampling, wetake Wsamples of
the measurement signal and form Wpower samples in equal time intervals over M periods ofsignal,
provided Mand Wdo not have a common factor. It was proved mathematically (the sample set theory)
[4], that if we take Wsamples ofa periodic function equidistantly over M periods, we will obtain the
samevalue ofobservedperiodic function asinthecaseofuniformly performedsampling with Wsamples
over a period, if the starting time moment to is the same, and provided that M and Wdo not have a
common factor. A broadened definition of synchronous sampling enables us to increase the sampling
interval from TIWtoMTIW, which canbeimportant fromtheaspectofpracticalrealization.
Stocton and Clarke [3] derived the theory of synchronous sampling based on a representation of
periodic signals v(t) and i(t) in the form ofa Fourier series. The instantaneous power ofsuch signals
p(t)=v(t)i(t)canbepresentedas:
8