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1、1,Chapter 4 Modeling of Nonlinear Load,Organized byTask Force on Harmonics Modeling&SimulationAdapted and Presented by Paulo F Ribeiro AMSCMay 28-29,2008,Contributors:S.Tsai,Y.Liu,and G.W.Chang,2,Chapter outline,IntroductionNonlinear magnetic core sourcesArc furnace3-phase line commuted convertersSt
2、atic var compensatorCycloconverter,3,Introduction,The purpose of harmonic studies is to quantify the distortion in voltage and/or current waveforms at various locations in a power system.One important step in harmonic studies is to characterize and to model harmonic-generating sources.Causes of powe
3、r system harmonics Nonlinear voltage-current characteristics Non-sinusoidal winding distributionPeriodic or aperiodic switching devicesCombinations of above,4,Introduction(cont.),In the following,we will present the harmonics for each devices in the following sequence:Harmonic characteristicsHarmoni
4、c models and assumptionsDiscussion of each model,5,Chapter outline,IntroductionNonlinear magnetic core sourcesArc furnace3-phase line commuted convertersStatic var compensatorCycloconverter,6,Nonlinear Magnetic Core Sources,Harmonics characteristicsHarmonics model for transformersHarmonics model for
5、 rotating machines,7,Harmonics characteristics of iron-core reactors and transformers,Causes of harmonics generationSaturation effectsOver-excitationtemporary over-voltage caused by reactive power unbalance unbalanced transformer loadasymmetric saturation caused by low frequency magnetizing currentt
6、ransformer energizationSymmetric core saturation generates odd harmonics Asymmetric core saturation generates both odd and even harmonics The overall amount of harmonics generated depends onthe saturation level of the magnetic corethe structure and configuration of the transformer,8,Harmonic models
7、for transformers,Harmonic models for a transformer:equivalent circuit modeldifferential equation modelduality-based modelGIC(geomagnetically induced currents)saturation model,9,Equivalent circuit model(transformer),In time domain,a single phase transformer can be represented by an equivalent circuit
8、 referring all impedances to one side of the transformerThe core saturation is modeled using a piecewise linear approximation of saturationThis model is increasingly available in time domain circuit simulation packages.,10,Differential equation model(transformer),The differential equations describe
9、the relationships between winding voltageswinding currentswinding resistancewinding turnsmagneto-motive forcesmutual fluxesleakage fluxesreluctancesSaturation,hysteresis,and eddy current effects can be well modeled.The models are suitable for transient studies.They may also be used to simulate the h
10、armonic generation behavior of power transformers.,11,Duality-based model(transformer),Duality-based models are necessary to represent multi-legged transformersIts parameters may be derived from experiment data and a nonlinear inductance may be used to model the core saturationDuality-based models a
11、re suitable for simulation of power system low-frequency transients.They can also be used to study the harmonic generation behaviors,12,GIC saturation model(transformer),Geomagnetically induced currents GIC bias can cause heavy half cycle saturationthe flux paths in and between core,tank and air gap
12、s should be accountedA detailed model based on 3D finite element calculation may be necessary.Simplified equivalent magnetic circuit model of a single-phase shell-type transformer is shown.An iterative program can be used to solve the circuitry so that nonlinearity of the circuitry components is con
13、sidered.,13,Rotating machines,Harmonic models for synchronous machineHarmonic models for Induction machine,14,Synchronous machines,Harmonics origins:Non-sinusoidal flux distributionThe resulting voltage harmonics are odd and usually minimized in the machines design stage and can be negligible.Freque
14、ncy conversion processCaused under unbalanced conditions SaturationSaturation occurs in the stator and rotor core,and in the stator and rotor teeth.In large generator,this can be neglected.Harmonic modelsunder balanced condition,a single-phase inductance is sufficientunder unbalanced conditions,a im
15、pedance matrix is necessary,15,Balanced harmonic analysis,For balanced(single phase)harmonic analysis,a synchronous machine was often represented by a single approximation of inductance h:harmonic order:direct sub-transient inductance:quadrature sub-transient inductanceA more complex model a:0.5-1.5
16、(accounting for skin effect and eddy current losses)Rneg and Xneg are the negative sequence resistance and reactance at fundamental frequency,16,Unbalanced harmonic analysis,The balanced three-phase coupled matrix model can be used for unbalanced network analysisZs=(Zo+2Zneg)/3Zm=(ZoZneg)/3 Zo and Z
17、neg are zero and negative sequence impedance at hth harmonic orderIf the synchronous machine stator is not precisely balanced,the self and/or mutual impedance will be unequal.,17,Induction motors,Harmonics can be generated fromNon-sinusoidal stator winding distributionCan be minimized during the des
18、ign stageTransientsHarmonics are induced during cold-start or load changingThe above-mentioned phenomenon can generally be neglectedThe primary contribution of induction motors is to act as impedances to harmonic excitationThe motor can be modeled asimpedance for balanced systems,ora three-phase cou
19、pled matrix for unbalanced systems,18,Harmonic models for induction motor,Balanced ConditionGeneralized Double Cage ModelEquivalent T ModelUnbalanced Condition,19,Generalized Double Cage Model for Induction Motor,Stator,Excitation branch,At the h-th harmonic order,the equivalent circuit can be obtai
20、ned by multiplying h with each of the reactance.,mutual reactance of the 2 rotor cages,2 rotor cages,20,Equivalent T model for Induction Motor,s is the full load slip at fundamental frequency,and h is the harmonic order-is taken for positive sequence models+is taken for negative sequence models.,21,
21、Unbalanced model for Induction Motor,The balanced three-phase coupled matrix model can be used for unbalanced network analysisZs=(Zo+2Zpos)/3 Zm=(ZoZpos)/3 Zo and Zpos are zero and positive sequence impedance at hth harmonic orderZ0 can be determined from,22,Chapter outline,IntroductionNonlinear mag
22、netic core sourcesArc furnace3-phase line commuted convertersStatic var compensatorCycloconverter,23,Arc furnace harmonic sources,Types:AC furnaceDC furnaceDC arc furnace are mostly determined by its AC/DC converter and the characteristic is more predictable,here we only focus on AC arc furnaces,24,
23、Characteristics of Harmonics Generated by Arc Furnaces,The nature of the steel melting process is uncontrollable,current harmonics generated by arc furnaces are unpredictable and random.Current chopping and igniting in each half cycle of the supply voltage,arc furnaces generate a wide range of harmo
24、nic frequencies,25,Harmonics Models for Arc Furnace,Nonlinear resistance modelCurrent source modelVoltage source modelNonlinear time varying voltage source modelNonlinear time varying resistance modelsFrequency domain modelsPower balance model,26,Nonlinear resistance model,simplified to,R1 is a posi
25、tive resistorR2 is a negative resistorAC clamper is a current-controlled switchIt is a primitive model and does not consider the time-varying characteristic of arc furnaces.,modeled as,27,Current source model,Typically,an EAF is modeled as a current source for harmonic studies.The source current can
26、 be represented by its Fourier seriesan and bn can be selected as a function ofmeasurement probability distributionsproportion of the reactive power fluctuations to the active power fluctuations.This model can be used to size filter components and evaluate the voltage distortions resulting from the
27、harmonic current injected into the system.,28,Voltage source model,The voltage source model for arc furnaces is a Thevenin equivalent circuit.The equivalent impedance is the furnace load impedance(including the electrodes)The voltage source is modeled in different ways:form it by major harmonic comp
28、onents that are known empiricallyaccount for stochastic characteristics of the arc furnace and model the voltage source as square waves with modulated amplitude.A new value for the voltage amplitude is generated after every zero-crossings of the arc current when the arc reignites,29,Nonlinear time v
29、arying voltage source model,This model is actually a voltage source modelThe arc voltage is defined as a function of the arc lengthVao:arc voltage corresponding to the reference arc length lo,k(t):arc length time variationsThe time variation of the arc length is modeled with deterministic or stochas
30、tic laws.Deterministic:Stochastic:,30,Nonlinear time varying resistance models,During normal operation,the arc resistance can be modeled to follow an approximate Gaussian distribution is the variance which is determined by short-term perceptibility flicker index PstAnother time varying resistance mo
31、del:R1:arc furnace positive resistance and R2 negative resistanceP:short-term power consumed by the arc furnaceVig and Vex are arc ignition and extinction voltages,31,Power balance model,r is the arc radiusexponent n is selected according to the arc cooling environment,n=0,1,or 2recommended values f
32、or exponent m are 0,1 and 2K1,K2 and K3 are constants,32,Chapter outline,IntroductionNonlinear magnetic core sourcesArc furnace3-phase line commuted convertersStatic var compensatorCycloconverter,33,Three-phase line commuted converters,Line-commutated converter is mostly usual operated as a six-puls
33、e converter or configured in parallel arrangements for high-pulse operations Typical applications of converters can be found in AC motor drive,DC motor drive and HVDC link,34,Harmonics Characteristics,Under balanced condition with constant output current and assuming zero firing angle and no commuta
34、tion overlap,phase a current is h=1,5,7,11,13,.Characteristic harmonics generated by converters of any pulse number are in the order of n=1,2,and p is the pulse number of the converter For non-zero firing angle and non-zero commutation overlap,rms value of each characteristic harmonic current can be
35、 determined by F(,)is an overlap function,35,Harmonic Models for the Three-Phase Line-Commutated Converter,Harmonic models can be categorized asfrequency-domain based modelscurrent source modeltransfer function modelNorton-equivalent circuit modelharmonic-domain modelthree-pulse model time-domain ba
36、sed modelsmodels by differential equations state-space model,36,Current source model,The most commonly used model for converter is to treat it as known sources of harmonic currents with or without phase angle information Magnitudes of current harmonics injected into a bus are determined from the typ
37、ical measured spectrum and rated load current for the harmonic source(Irated)Harmonic phase angles need to be included when multiple sources are considered simultaneously for taking the harmonic cancellation effect into account.h,and a conventional load flow solution is needed for providing the fund
38、amental frequency phase angle,1,37,Transfer Function Model,The simplified schematic circuit can be used to describe the transfer function model of a converterG:the ideal transfer function without considering firing angle variation and commutation overlapG,dc and G,ac,relate the dc and ac sides of th
39、e converterTransfer functions can include the deviation terms of the firing angle and commutation overlapThe effects of converter input voltage distortion or unbalance and harmonic contents in the output dc current can be modeled as well,38,Norton-Equivalent Circuit Model,The nonlinear relationship
40、between converter input currents and its terminal voltages isI&V are harmonic vectorsIf the harmonic contents are small,one may linearize the dynamic relations about the base operating point and obtain:I=YJV+INYJ is the Norton admittance matrix representing the linearization.It also represents an ap
41、proximation of the converter response to variations in its terminal voltage harmonics or unbalance IN=Ib-YJVb(Norton equivalent),39,Harmonic-Domain Model,Under normal operation,the overall state of the converter is specified by the angles of the state transitionThese angles are the switching instant
42、s corresponding to the 6 firing angles and the 6 ends of commutation angles The converter response to an applied terminal voltage is characterized via convolutions in the harmonic domainThe overall dc voltageVk,p:12 voltage samplesp:square pulse sampling functionsH:the highest harmonic order under c
43、onsiderationThe converter input currents are obtained in the same manner using the same sampling functions.,40,Chapter outline,IntroductionNonlinear magnetic core sourcesArc furnace3-phase line commuted convertersStatic var compensatorCycloconverter,41,Harmonics characteristics of TCR,Harmonic curre
44、nts are generated for any conduction intervals within the two firing anglesWith the ideal supply voltage,the generated rms harmonic currents h=3,5,7,is the conduction angle,and LR is the inductance of the reactor,42,Harmonics characteristics of TCR(cont.),Three single-phase TCRs are usually in delta
45、 connection,the triplen currents circulate within the delta circuit and do not enter the power system that supplies the TCRs.When the single-phase TCR is supplied by a non-sinusoidal input voltagethe current through the compensator is proved to be the discontinuous current,43,Harmonic models for TCR
46、,Harmonic models for TCR can be categorized asfrequency-domain based modelscurrent source modeltransfer function modelNorton-equivalent circuit modeltime-domain based modelsmodels by differential equations state-space model,44,Current Source Model,by discrete Fourier analysis,45,Norton equivalence f
47、or the harmonic power flow analysis of the TCR for the h-th harmonic,Norton-Equivalent Model,The input voltage is unbalanced and no coupling between different harmonics are assumed,46,Transfer Function Model,Assume the power system is balanced and is represented by a harmonic Thvenin equivalentThe v
48、oltage across the reactor and the TCR current can be expressed as YTCR=YRS can be thought of TCR harmonic admittance matrix or transfer function,47,Time-Domain Model,Model 1,Model 2,48,Chapter outline,IntroductionNonlinear magnetic core sourcesArc furnace3-phase line commuted convertersStatic var co
49、mpensatorCycloconverter,49,Harmonics Characteristics of Cycloconverter,A cycloconverter generates very complex frequency spectrum that includes sidebands of the characteristic harmonics Balanced three-phase outputs,the dominant harmonic frequencies in input current for 6-pulse 12-pulsep=6 or p=12,an
50、d m=1,2,.In general,the currents associated with the sideband frequencies are relatively small and harmless to the power system unless a sharply tuned resonance occurs at that frequency.,50,Harmonic Models for the Cycloconverter,The harmonic frequencies generated by a cycloconverter depend on its ch
