Chapter 4 Modeling of Nonlinear Load

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# Chapter 4 Modeling of Nonlinear Load - PowerPoint PPT Presentation

Tutorial on Harmonics Modeling and Simulation. Chapter 4 Modeling of Nonlinear Load. 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.

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Tutorial on Harmonics Modeling and Simulation

### Chapter 4 Modeling of Nonlinear Load

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
• Causes of harmonics generation
• Saturation effects
• Over-excitation
• temporary over-voltage caused by reactive power unbalance
• 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
• 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=(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=(Zo+2Zpos)/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
• 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
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 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
• YTCR=YRS 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.