控制電纜燒損事故之分析改善與預(yù)防對策課件

1、1Chapter 4 Modeling of Nonlinear Load Organized byTask Force on Harmonics Modeling & SimulationAdapted and Presented by Paulo F Ribeiro AMSCMay 28-29, 2008Contributors: S. Tsai, Y. Liu, and G. W. Chang2Chapter outlineIntroductionNonlinear magnetic core sourcesArc furnace3-phase line commuted con

2、vertersStatic var compensatorCycloconverter3IntroductionThe 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

3、of power system harmonics Nonlinear voltage-current characteristics Non-sinusoidal winding distribution Periodic or aperiodic switching devices Combinations of above4Introduction (cont.)In the following, we will present the harmonics for each devices in the following sequence:Harmonic characteristic

4、sHarmonic models and assumptions1. Discussion of each model5Chapter outlineIntroductionNonlinear magnetic core sourcesArc furnace3-phase line commuted convertersStatic var compensatorCycloconverter6Nonlinear Magnetic Core Sources Harmonics characteristicsHarmonics model for transformersHarmonics mod

5、el for rotating machines7Harmonics characteristics of iron-core reactors and transformersCauses of harmonics generationSaturation effectsOver-excitationntemporary over-voltage caused by reactive power unbalance nunbalanced transformer loadnasymmetric saturation caused by low frequency magnetizing cu

6、rrentntransformer 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 transformer8Harmonic mo

7、dels for transformersHarmonic models for a transformer: equivalent circuit model differential equation model duality-based model GIC (geomagnetically induced currents) saturation model 9Equivalent circuit model (transformer)In time domain, a single phase transformer can be represented by an equivale

8、nt circuit 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. IpRpLpLsRsIexLmImRm+-Vm+-VinVout+-Is10Differential equation model (tra

9、nsformer)The differential equations describe the relationships between winding voltages winding currents winding resistance winding turns magneto-motive forces mutual fluxes leakage fluxes reluctancesSaturation, hysteresis, and eddy current effects can be well modeled. The models are suitable for tr

10、ansient studies. They may also be used to simulate the harmonic generation behavior of power transformers. NNNNNNNNNNNNNNNiiidtdLLLLLLLLLiiiRRRRRRRRRvvv212122221112112121222211121121 11Duality-based model (transformer)Duality-based models are necessary to represent multi-legged transformersIts param

11、eters may be derived from experiment data and a nonlinear inductance may be used to model the core saturationDuality-based models are suitable for simulation of power system low-frequency transients. They can also be used to study the harmonic generation behaviorsMagnetic circuitElectric circuitMagn

12、etomotive Force (FMM) Ni Electromotive Force (FEM) E Flux Current I Reluctance Resistance R Permeance Conductance Flux density Current density Magnetizing force H Potential difference V /1R/1AB/AIJ/12GIC saturation model (transformer)Geomagnetically induced currents GIC bias can cause heavy half cyc

13、le saturation the flux paths in and between core, tank and air gaps 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

14、 circuitry so that nonlinearity of the circuitry components is considered.FACDCRc1Ra1Ra4Ra4Rt4Rc3Rc2Rc2Ra3Rt313Rotating machinesHarmonic models for synchronous machineHarmonic models for Induction machine14Synchronous machinesHarmonics origins: Non-sinusoidal flux distributionnThe resulting voltage

15、harmonics are odd and usually minimized in the machines design stage and can be negligible. Frequency conversion processnCaused under unbalanced conditions SaturationnSaturation occurs in the stator and rotor core, and in the stator and rotor teeth. In large generator, this can be neglected.Harmonic

16、 models under balanced condition, a single-phase inductance is sufficient under unbalanced conditions, a impedance matrix is necessary15Balanced harmonic analysisFor balanced (single phase) harmonic analysis, a synchronous machine was often represented by a single approximation of inductance h: harm

17、onic order : direct sub-transient inductance : quadrature sub-transient inductanceA more complex model a: 0.5-1.5 (accounting for skin effect and eddy current losses) Rneg and Xneg are the negative sequence resistance and reactance at fundamental frequency 2/qdhLLhLdLqLnegnegahjhXRhZ16Unbalanced har

18、monic analysisThe balanced three-phase coupled matrix model can be used for unbalanced network analysis Zs=(Zo+2Zneg)/3 Zm=(ZoZneg)/3 Zo and Zneg are zero and negative sequence impedance at hth harmonic orderIf the synchronous machine stator is not precisely balanced, the self and/or mutual impedanc

19、e will be unequal.smmmsmmmshZZZZZZZZZZ17Induction motorsHarmonics can be generated from Non-sinusoidal stator winding distributionnCan be minimized during the design stage TransientsnHarmonics are induced during cold-start or load changing The above-mentioned phenomenon can generally be neglectedThe

20、 primary contribution of induction motors is to act as impedances to harmonic excitationThe motor can be modeled as impedance for balanced systems, or a three-phase coupled matrix for unbalanced systems18Harmonic models for induction motorBalanced ConditionGeneralized Double Cage ModelEquivalent T M

21、odelUnbalanced Condition19Generalized Double Cage Model for Induction MotorRsjXsRcjXmjXrR1(s)jX1R2(s)jX2StatorExcitation branchAt the h-th harmonic order, the equivalent circuit can be obtained by multiplying h with each of the reactance. mutual reactance of the 2 rotor cages 2 rotor cages 20Equival

22、ent T model for Induction Motors 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. RsjhXsRcjhXmjhXrRrshhshsh121Unbalanced model for Induction MotorThe balanced three-phase coupled matrix model ca

23、n be used for unbalanced network analysis Zs=(Zo+2Zpos)/3 Zm=(ZoZpos)/3 Zo and Zpos are zero and positive sequence impedance at hth harmonic orderZ0 can be determined fromRs0jXs0Rm020.5Rr0(-2+3s)jXm02jXr02Rm020.5Rr0(4-3s)jXm02jXr02smmmsmmmshZZZZZZZZZZ22Chapter outlineIntroductionNonlinear magnetic c

24、ore sourcesArc furnace3-phase line commuted convertersStatic var compensatorCycloconverter23Arc furnace harmonic sourcesTypes: AC furnace DC 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 24Characte

25、ristics of Harmonics Generated by Arc FurnacesThe 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 harmonic fre

26、quencies (a)(b)25Harmonics Models for Arc Furnace Nonlinear resistance modelCurrent source modelVoltage source modelNonlinear time varying voltage source modelNonlinear time varying resistance modelsFrequency domain modelsPower balance model26Nonlinear resistance model(a)simplified toR1 is a positiv

27、e 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 as27Current source modelTypically, an EAF is modeled as a current source for harmonic studies. The source current can be

28、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 har

29、monic current injected into the system. 10cossinnnnnLtnbtnati28Voltage source modelThe 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:nform it by

30、 major harmonic components that are known empiricallynaccount 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 reignit

31、es29Nonlinear time varying voltage source modelThis model is actually a voltage source modelThe arc voltage is defined as a function of the arc length Vao :arc voltage corresponding to the reference arc length lo, k(t): arc length time variationsThe time variation of the arc length is modeled with d

32、eterministic or stochastic laws. Deterministic: Stochastic: 00laoVtklaV tDlltlosin12 tRltlo30Nonlinear time varying resistance modelsDuring normal operation, the arc resistance can be modeled to follow an approximate Gaussian distribution is the variance which is determined by short-term perceptibil

33、ity flicker index PstAnother time varying resistance model: R1: arc furnace positive resistance and R2 negative resistance P: short-term power consumed by the arc furnace Vig and Vex are arc ignition and extinction voltages RAND22cosRAND1ln2 RRarc222221RVRVPVRexigig31Power balance modelr is the arc

34、radiusexponent n is selected according to the arc cooling environment, n=0, 1, or 2recommended values for exponent m are 0, 1 and 2K1, K2 and K3 are constants 22321irKdtdrrKrKmn32Chapter outlineIntroductionNonlinear magnetic core sourcesArc furnace3-phase line commuted convertersStatic var compensat

35、orCycloconverter33Three-phase line commuted convertersLine-commutated converter is mostly usual operated as a six-pulse 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 link34Harmonic

36、s CharacteristicsUnder balanced condition with constant output current and assuming zero firing angle and no commutation 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 nn = 1, 2, and p is the pulse number of

37、the converter For non-zero firing angle and non-zero commutation overlap, rms value of each characteristic harmonic current can be determined by F(,) is an overlap function )cos(cos/),(6hFIIdhhhathhIti)sin()/2()(111 pnh35Harmonic Models for the Three-Phase Line-Commutated ConverterHarmonic models ca

38、n be categorized as frequency-domain based modelsncurrent source modelntransfer function modelnNorton-equivalent circuit modelnharmonic-domain modelnthree-pulse model time-domain based modelsnmodels by differential equations nstate-space model36Current source modelThe most commonly used model for co

39、nverter 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 typical measured spectrum and rated load current for the harmonic source (Irated)Harmonic phase angles need to be included

40、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 fundamental frequency phase angle, 1spsphratedhIIII1/)(11spsphhh37Transfer Function Model The simplified schematic circui

41、t 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 the converterTransfer functions can include the deviation terms of the firing angle and co

42、mmutation overlapThe effects of converter input voltage distortion or unbalance and harmonic contents in the output dc current can be modeled as wellcbaVGVdcdc, ,a,b,ciGidcac , ,38Norton-Equivalent Circuit Model The nonlinear relationship between converter input currents and its terminal voltages is

43、 I & V are harmonic vectorsIf the harmonic contents are small, one may linearize the dynamic relations about the base operating point and obtain: I = YJV + IN YJ is the Norton admittance matrix representing the linearization. It also represents an approximation of the converter response to varia

44、tions in its terminal voltage harmonics or unbalance IN = Ib - YJVb (Norton equivalent)(VIf39Harmonic-Domain ModelUnder normal operation, the overall state of the converter is specified by the angles of the state transitionThese angles are the switching instants corresponding to the 6 firing angles

45、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 considerationThe converter input curr

46、ents are obtained in the same manner using the same sampling functions. HhHnnphpkpppkdVVV121,121,40Chapter outlineIntroductionNonlinear magnetic core sourcesArc furnace3-phase line commuted convertersStatic var compensatorCycloconverter41Harmonics characteristics of TCRHarmonic currents are generate

47、d 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 ) 1(sin)cos()sin(cos4)(21hhhhhLVIRh42Harmonics characteristics of TCR (cont.)Three single-phas

48、e TCRs are usually in delta 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 voltage the current through the compensator is proved to be the discontinuous curre

49、nt tttthhLhVtihhh and 0 , 0 ),cos()cos()(1hhhsthVtv)sin()(43Harmonic models for TCRHarmonic models for TCR can be categorized as frequency-domain based modelsncurrent source modelntransfer function modelnNorton-equivalent circuit model time-domain based modelsnmodels by differential equations nstate-space model44Current Source ModelhhhhthIti)sin()(tttthhLhVtihhh and 0 , 0 ),cos()cos()(1by discrete Fourier analysis45Norton equivalenc