1 Chapter 4 Modeling of Nonlinear Load Organized by Task Force on Harmonics Modeling & Simulation Adapted and Presented by Paulo F Ribeiro AMSC May 28-29,

1

Chapter 4 Modeling of Nonlinear Load

Chapter 4 Modeling of Nonlinear Load

Organized by

Task Force on Harmonics Modeling & Simulation

Adapted and Presented by Paulo F Ribeiro

AMSC

May 28-29, 2008

Contributors: S. Tsai, Y. Liu, and G. W. Chang

2

Chapter outlineChapter outline

• Introduction

• Nonlinear magnetic core sources

• Arc furnace

• 3-phase line commuted converters

• Static var compensator

• Cycloconverter

3

IntroductionIntroduction

• 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 power system harmonics – Nonlinear voltage-current characteristics

– Non-sinusoidal winding distribution

– Periodic or aperiodic switching devices

– Combinations of above

4

Introduction (cont.)Introduction (cont.)

• In the following, we will present the harmonics for each devices in the following sequence:

1. Harmonic characteristics

2. Harmonic models and assumptions

3. Discussion of each model

5


• Introduction


• Arc furnace



• Cycloconverter

6

Nonlinear Magnetic Core Sources Nonlinear Magnetic Core Sources

• Harmonics characteristics

• Harmonics model for transformers

• Harmonics model for rotating machines

7

Harmonics characteristics of iron-core reactors and transformers

Harmonics characteristics of iron-core reactors and transformers

• Causes of harmonics generation– Saturation effects– Over-excitation

temporary over-voltage caused by reactive power unbalance unbalanced transformer load asymmetric saturation caused by low frequency magnetizing current transformer energization

• Symmetric core saturation generates odd harmonics • Asymmetric core saturation generates both odd and even

harmonics • The overall amount of harmonics generated depends on

– the saturation level of the magnetic core– the structure and configuration of the transformer

8

Harmonic models for transformersHarmonic models for transformers

• Harmonic models for a transformer:– equivalent circuit model

– differential equation model

– duality-based model

– GIC (geomagnetically induced currents) saturation model

9

Equivalent circuit model (transformer)Equivalent circuit model (transformer)

• In time domain, a single phase transformer can be represented by an equivalent circuit referring all impedances to one side of the transformer

• The core saturation is modeled using a piecewise linear approximation of saturation

• This model is increasingly available in time domain circuit simulation packages.

10

Differential equation model (transformer)Differential equation model (transformer)

• The differential equations describe the relationships between – winding voltages– winding currents– winding resistance– winding turns– magneto-motive forces– mutual fluxes– leakage fluxes– reluctances

• Saturation, hysteresis, and eddy current effects can be well modeled.

• The models are suitable for transient studies. They may also be used to simulate the harmonic generation behavior of power transformers.

NNNNN

N

N

NNNNN

N

N

N

i

i

i

dt

d

LLL

LLL

LLL

i

i

i

RRR

RRR

RRR

v

v

v

2

1

21

22221

11211

2

1

21

22221

11211

2

1

11

Duality-based model (transformer)Duality-based model (transformer)

• Duality-based models are necessary to represent multi-legged transformers

• Its parameters may be derived from experiment data and a nonlinear inductance may be used to model the core saturation

• Duality-based models are suitable for simulation of power system low-frequency transients. They can also be used to study the harmonic generation behaviors

Magnetic circuit Electric circuit

Magnetomotive 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

/1 R/1

AB / AIJ /

12

GIC saturation model (transformer)GIC saturation model (transformer)

• Geomagnetically induced currents GIC bias can cause heavy half cycle saturation– the flux paths in and between core,

tank and air gaps should be accounted

• A 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 considered.

F

~AC

DC

Rc1 Ra1

Ra4

Ra4’

Rt4

Rc3

Rc2

Rc2

Ra3

Rt3

13

Rotating machinesRotating machines

• Harmonic models for synchronous machine

• Harmonic models for Induction machine

14

Synchronous machinesSynchronous machines

• Harmonics origins:– Non-sinusoidal flux distribution

The resulting voltage harmonics are odd and usually minimized in the machine’s design stage and can be negligible.

– Frequency conversion process Caused under unbalanced conditions

– Saturation Saturation occurs in the stator and rotor core, and in the

stator and rotor teeth. In large generator, this can be neglected.

• Harmonic models– under balanced condition, a single-phase inductance is

sufficient– under unbalanced conditions, a impedance matrix is

necessary

15

Balanced harmonic analysisBalanced 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 inductance

• A 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/""qdh LLhL

"dL

"qL

negnega

h jhXRhZ

16

Unbalanced harmonic analysisUnbalanced harmonic analysis

• The 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 order

• If the synchronous machine stator is not precisely balanced, the self and/or mutual impedance will be unequal.

smm

msm

mms

h

ZZZ

ZZZ

ZZZ

Z

17

Induction motorsInduction motors

• Harmonics can be generated from– Non-sinusoidal stator winding distribution

Can be minimized during the design stage

– Transients Harmonics are induced during cold-start or load changing

– The above-mentioned phenomenon can generally be neglected

• The primary contribution of induction motors is to act as impedances to harmonic excitation

• The motor can be modeled as– impedance for balanced systems, or– a three-phase coupled matrix for unbalanced systems

18

Harmonic models for induction motorHarmonic models for induction motor

• Balanced Condition– Generalized Double Cage Model

– Equivalent T Model

• Unbalanced Condition

19

Generalized Double Cage Model for Induction MotorGeneralized Double Cage Model for Induction Motor

Rs jXs

RcjXm

jXr

R1(s)

jX1

R2(s)

jX2

Stator

Excitation branch

At 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

20

Equivalent T model for Induction MotorEquivalent 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.

Rs jhXs

Rc jhXm

jhXr

Rrsh

h

shsh

1

21

Unbalanced model for Induction MotorUnbalanced model for Induction Motor

• The balanced three-phase coupled matrix model can 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 order• Z0 can be determined from Rs0 jXs0

Rm0

20.5Rr0

(-2+3s)

jXm0

2jXr0

2

Rm0

20.5Rr0

(4-3s)

jXm0

2jXr0

2

smm

msm

mms

h

ZZZ

ZZZ

ZZZ

Z

22


• Introduction


• Arc furnace



• Cycloconverter

23

Arc furnace harmonic sourcesArc furnace harmonic sources

• Types:– AC furnace

– DC furnace

• DC 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

Characteristics of Harmonics Generated by Arc FurnacesCharacteristics 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 harmonic frequencies

(a)

25

Harmonics Models for Arc Furnace Harmonics Models for Arc Furnace

• Nonlinear resistance model

• Current source model

• Voltage source model

• Nonlinear time varying voltage source model

• Nonlinear time varying resistance models

• Frequency domain models

• Power balance model

26

Nonlinear resistance modelNonlinear resistance model

(a)

simplified to

• R1 is a positive resistor

• R2 is a negative resistor

• AC clamper is a current-controlled switch

• It is a primitive model and does not consider the time-varying characteristic of arc furnaces.

modeled as

27

Current source modelCurrent source model

• Typically, an EAF is modeled as a current source for harmonic studies. The source current can be represented by its Fourier series

• an and bn can be selected as a function of– measurement

– probability distributions

– proportion 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 harmonic current injected into the system.

1 0cossin

n nnnL tnbtnati

28

Voltage source modelVoltage 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 components that are known

empirically account 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 varying voltage source modelNonlinear time varying voltage source model

• This model is actually a voltage source model

• The 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 variations

• The time variation of the arc length is modeled with deterministic or stochastic laws.– Deterministic:

– Stochastic:

00 laoVtklaV

tDlltl o sin12

tRltl o

30

Nonlinear time varying resistance modelsNonlinear 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 Pst

• Another 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 RRarc

2

2

2

2

2

1

R

V

R

VP

VR

exig

ig

31

Power balance modelPower balance model

• r is the arc radius

• exponent n is selected according to the arc cooling environment, n=0, 1, or 2

• recommended values for exponent m are 0, 1 and 2

• K1, K2 and K3 are constants

22

321 i

r

K

dt

drrKrK

mn

32


• Introduction


• Arc furnace



• Cycloconverter

33

Three-phase line commuted convertersThree-phase line commuted converters

• Line-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 link

34

Harmonics CharacteristicsHarmonics Characteristics

• Under 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 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 determined by

– F(,) is an overlap function

)]}cos([cos/{),(6 hFII dh

h

ha thhIti )sin()/2()( 11

1pnh

35

Harmonic Models for the Three-Phase Line-Commutated ConverterHarmonic Models for the Three-Phase Line-Commutated Converter

• Harmonic models can be categorized as– frequency-domain based models

current source model transfer function model Norton-equivalent circuit model harmonic-domain model three-pulse model

– time-domain based models models by differential equations state-space model

36

Current source modelCurrent 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 typical 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 fundamental frequency phase angle, 1

spsphratedh IIII 1/

)( 11 spsphh h

37

Transfer Function Model Transfer Function Model

• The simplified schematic circuit can be used to describe the transfer function model of a converter

• G: the ideal transfer function without considering firing angle variation and commutation overlap

• G,dc and G,ac, relate the dc and ac sides of the converter

• Transfer functions can include the deviation terms of the firing angle and commutation overlap

• The effects of converter input voltage distortion or unbalance and harmonic contents in the output dc current can be modeled as well

cbaVGV dcdc ,,, ,

a,b,ciGi dcac , ,

38

Norton-Equivalent Circuit Model Norton-Equivalent Circuit Model

• The nonlinear relationship between converter input currents and its terminal voltages is

– I & V are harmonic vectors

• If 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 variations in its terminal voltage harmonics or unbalance

– IN = Ib - YJVb (Norton equivalent)

)(VI f

39

Harmonic-Domain ModelHarmonic-Domain Model

• Under normal operation, the overall state of the converter is specified by the angles of the state transition– These angles are the switching instants 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 domain

• The overall dc voltage

– Vk,p: 12 voltage samples p: square pulse sampling functions

– H: the highest harmonic order under consideration

• The converter input currents are obtained in the same manner using the same sampling functions.

H

h

H

n

np

hpk

pppkd VVV

1

2

1,

12

1,

40


• Introduction


• Arc furnace



• Cycloconverter

41

Harmonics characteristics of TCRHarmonics characteristics of TCR

• Harmonic currents are generated for any conduction intervals within the two firing angles

• With 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)(

21

hh

hhh

L

VI

Rh

42

Harmonics characteristics of TCR (cont.)Harmonics characteristics of TCR (cont.)

• Three single-phase 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 current

tt

tthhLh

V

ti hhh

and 0 ,0

)],cos()[cos()(

1

h

hhs thVtv )sin()(

43

Harmonic models for TCRHarmonic models for TCR

• Harmonic models for TCR can be categorized as– frequency-domain based models

current source model transfer function model Norton-equivalent circuit model

– time-domain based models models by differential equations state-space model

44

Current Source ModelCurrent Source Model

h

hhh thIti )sin()(

tt

tthhLh

V

ti hhh

and 0 ,0

)],cos()[cos()(

1

by discrete Fourier analysis

45

Norton equivalence for the harmonic power flow analysis of the TCR for the h-th harmonic

Norton-Equivalent ModelNorton-Equivalent Model

• The input voltage is unbalanced and no coupling between different harmonics are assumed

1)( eqeqh LjhY heqheqh Ljh IVΙ )/(

hhh V V hhh I I )sin/( Req LL

46

Transfer Function ModelTransfer Function Model

• Assume the power system is balanced and is represented by a harmonic Thévenin equivalent

• The voltage across the reactor and the TCR current can be expressed as

• YTCR=YRS can be thought of TCR harmonic admittance matrix or transfer function

R SV s V

STCRRRR V Y V Y I

47

Time-Domain ModelTime-Domain Model

sSc

c

RSc

c

VLi

v

L

s

Ldt

didt

dv

10

0)1

(

10Model 1

Model 2i

L

RR

L

v

dt

di T

48


• Introduction


• Arc furnace



• Cycloconverter

49

Harmonics Characteristics of CycloconverterHarmonics 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-pulse

– p = 6 or p= 12, and 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.

oih kffpmf 2)1(

oih kffpmf 6)1(

50

Harmonic Models for the Cycloconverter Harmonic Models for the Cycloconverter

• The harmonic frequencies generated by a cycloconverter depend on its changed output frequency, it is very difficult to eliminate them completely

• To date, the time-domain and current source models are commonly used for modeling harmonics

• The harmonic currents injected into a power system by cycloconverters still present a challenge to both researchers and industrial engineers.

Documents

1 Chapter 4 Modeling of Nonlinear Load Organized by Task Force on Harmonics Modeling & Simulation Adapted and Presented by Paulo F Ribeiro AMSC May 28-29,