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### Chapter 4 Modeling of Nonlinear Load

Chapter outlineChapter outlineChapter outline

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 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

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=(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.

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=(Zo+2Zpos)/3
- Zm=(ZoZpos)/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

- ris 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

- 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.

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

- 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.

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