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Chapter 4 Modeling of Nonlinear Load

Tutorial on Harmonics Modeling and Simulation

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

Chapter outline

- Introduction
- Nonlinear magnetic core sources
- Arc furnace
- 3-phase line commuted converters
- Static var compensator
- Cycloconverter

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 power system harmonics
- Nonlinear voltage-current characteristics
- Non-sinusoidal winding distribution
- Periodic or aperiodic switching devices
- Combinations of above

Introduction (cont.)

- In the following, we will present the harmonics

for each devices in the following sequence - Harmonic characteristics
- Harmonic models and assumptions
- Discussion of each model

Chapter outline

- Introduction
- Nonlinear magnetic core sources
- Arc furnace
- 3-phase line commuted converters
- Static var compensator
- Cycloconverter

Nonlinear Magnetic Core Sources

- Harmonics characteristics
- Harmonics model for transformers
- Harmonics model for rotating machines

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

Harmonic models for transformers

- Harmonic models for a transformer
- equivalent circuit model
- differential equation model
- duality-based model
- GIC (geomagnetically induced currents) saturation

model

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.

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.

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

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.

Rotating machines

- Harmonic models for synchronous machine
- Harmonic models for Induction machine

Synchronous machines

- Harmonics origins
- Non-sinusoidal flux distribution
- The resulting voltage harmonics are odd and

usually minimized in the machines 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

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 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

Unbalanced harmonic analysis

- The balanced three-phase coupled matrix model can

be used for unbalanced network analysis - Zs(Zo2Zneg)/3
- Zm(Zo?Zneg)/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.

Induction 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

Harmonic models for induction motor

- Balanced Condition
- Generalized Double Cage Model
- Equivalent T Model
- Unbalanced Condition

Generalized Double Cage Model for Induction Motor

Stator

mutual reactance of the 2 rotor cages

Excitation branch

2 rotor cages

At the h-th harmonic order, the equivalent

circuit can be obtained by multiplying h with

each of the reactance.

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.

Unbalanced model for Induction Motor

- The balanced three-phase coupled matrix model can

be used for unbalanced network analysis - Zs(Zo2Zpos)/3
- Zm(Zo?Zpos)/3
- Zo and Zpos are zero and positive sequence

impedance at hth harmonic order - Z0 can be determined from

Chapter outline

- Introduction
- Nonlinear magnetic core sources
- Arc furnace
- 3-phase line commuted converters
- Static var compensator
- Cycloconverter

Arc 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

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 harmonic frequencies

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

Nonlinear resistance model

simplified to

modeled as

- 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.

Current 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.

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 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

Nonlinear 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

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 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

Power balance model

- r is the arc radius
- exponent n is selected according to the arc

cooling environment, n0, 1, or 2 - recommended values for exponent m are 0, 1 and 2
- K1, K2 and K3 are constants

Chapter outline

- Introduction
- Nonlinear magnetic core sources
- Arc furnace
- 3-phase line commuted converters
- Static var compensator
- Cycloconverter

Three-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

Harmonics 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

Harmonic 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

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 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

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

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)

Harmonic-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.

Chapter outline

- Introduction
- Nonlinear magnetic core sources
- Arc furnace
- 3-phase line commuted converters
- Static var compensator
- Cycloconverter

Harmonics 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

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

Harmonic 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

Current Source Model

by discrete Fourier analysis

Norton-Equivalent Model

- The input voltage is unbalanced and no coupling

between different harmonics are assumed

Norton equivalence for the harmonic power flow

analysis of the TCR for the h-th harmonic

Transfer 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 - YTCRYRS can be thought of TCR harmonic

admittance matrix or transfer function

Time-Domain Model

Model 1

Model 2

Chapter outline

- Introduction
- Nonlinear magnetic core sources
- Arc furnace
- 3-phase line commuted converters
- Static var compensator
- Cycloconverter

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-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.

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.