Chapter 4 Modeling of Nonlinear Load

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

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=(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

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

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.

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

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.

Chapter 4 Modeling of Nonlinear Load