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**Chapter 4 Modeling of Nonlinear Load**

Contributors: S. Tsai, Y. Liu, and G. W. Chang Organized by Task Force on Harmonics Modeling & Simulation Adapted and Presented by Paulo F Ribeiro AMSC May 28-29, 2008 Certain devices exhibit nonlinear input and output characteristics which make the prediction of its behavior difficult. We are familiar with linear system, for nonlinear devices, we can still understand its characteristics through linearized modeling. In this chapter, we will look at different nonlinear devices and see how to model them under harmonic studies. 中正--電力品質實驗室

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**Chapter outline Introduction Nonlinear magnetic core sources**

Arc furnace 3-phase line commuted converters Static var compensator Cycloconverter In today’s talk, we will briefly introduce some common models of different devices for harmonic studies. Among the harmonics generating devices, we select the most significant ones listed on this slide. There are other devices such as PCs, TVs, energy efficient lights, and battery chargers also generate currents rich in odd harmonics, but the harmonic magnitudes are usually small and the phase angles are diversified. Harmonic modeling for their group effects generally requires statistical or probabilistic approaches. These topics will not be discussed here. 中正--電力品質實驗室

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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 中正--電力品質實驗室

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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 中正--電力品質實驗室

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**Chapter outline Introduction Nonlinear magnetic core sources**

Arc furnace 3-phase line commuted converters Static var compensator Cycloconverter 中正--電力品質實驗室

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**Nonlinear Magnetic Core Sources**

Harmonics characteristics Harmonics model for transformers Harmonics model for rotating machines First, we will introduce the harmonic models for non-linear magnetic core sources. We will briefly introduce its general characteristics, and then we will look into different ways to model transformers and rotating machines 中正--電力品質實驗室

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**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 中正--電力品質實驗室

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**Harmonic models for transformers**

Harmonic models for a transformer: equivalent circuit model differential equation model duality-based model GIC (geomagnetically induced currents) saturation model 中正--電力品質實驗室

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**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. The equivalent model for transformer in time domain, a single phase transformer can be represented on the right. 中正--電力品質實驗室

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**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. 中正--電力品質實驗室

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**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 中正--電力品質實驗室

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**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. 中正--電力品質實驗室

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**Rotating machines Harmonic models for synchronous machine**

Harmonic models for Induction machine Since synchronous machines and induction motors have different harmonic characteristics, we will illustrate the harmonic models for synchronous and induction machines. 中正--電力品質實驗室

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**Synchronous machines Harmonics origins: Harmonic models**

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 中正--電力品質實驗室

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**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: (accounting for skin effect and eddy current losses) Rneg and Xneg are the negative sequence resistance and reactance at fundamental frequency 中正--電力品質實驗室

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**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. 中正--電力品質實驗室

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**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 The model is similar to that of the synchronous machines as we previously talked about. 中正--電力品質實驗室

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**Harmonic models for induction motor**

Balanced Condition Generalized Double Cage Model Equivalent T Model Unbalanced Condition For induction motor, we can generally model it from 2 aspects: under balanced condition and unbalanced condition 中正--電力品質實驗室

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**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. 中正--電力品質實驗室

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**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. 中正--電力品質實驗室

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**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 中正--電力品質實驗室

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**Chapter outline Introduction Nonlinear magnetic core sources**

Arc furnace 3-phase line commuted converters Static var compensator Cycloconverter 中正--電力品質實驗室

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**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 中正--電力品質實驗室

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**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 Figure (a) shows the typical voltage-current characteristics of the furnace arc Figure (b) shows the resulting arc current and voltage when a sinusoidal source supplies the furnace The last figure shows the typical spectrum of arc furnace current. It can be seen that besides integer harmonics, the arc furnace currents are also rich in inter-harmonics. 中正--電力品質實驗室

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**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 Although an arc furnace can be modeled simply as an inductor in series with a resistor, many complex models were proposed to more precisely represent its characteristics and to study its impacts on power systems. Here is the list of 7 models to be covered here. Next, we will briefly introduce each one of them. 中正--電力品質實驗室

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**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. The simplified V-I characteristics can be modeled into the circuit as shown. Several characteristics are listed. 中正--電力品質實驗室

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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. 中正--電力品質實驗室

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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 中正--電力品質實驗室

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**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: Deterministic: Dl is the maximum variation of arc length, and is an angular frequency in the range of 0.5 to 25 Hz, which is the popular flicker frequency range of an arc furnace. Stochastic: R(t) is a band-limited white noise. Its frequency band is in the range of the flicker frequency of the arc furnace. 中正--電力品質實驗室

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**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 中正--電力品質實驗室

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**Power 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 中正--電力品質實驗室

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**Chapter outline Introduction Nonlinear magnetic core sources**

Arc furnace 3-phase line commuted converters Static var compensator Cycloconverter 中正--電力品質實驗室

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**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 中正--電力品質實驗室

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**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 the ac harmonic currents generated by a six-pulse converter include all odd harmonics except triplens 中正--電力品質實驗室

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**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 中正--電力品質實驗室

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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 For harmonic current magnitude, the subscript 'sp' indicates the typical harmonic current spectrum of the source 中正--電力品質實驗室

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**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 the dc voltage is directly computed by summing each input phase voltage, V, multiplied by its corresponding transfer function. I is the input current of the converter of each phase 中正--電力品質實驗室

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**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) In the iterative harmonic analysis, the converter is usually represented by a fixed harmonic current source at each iteration. For better convergence, the Norton equivalent can be used for the converter. 中正--電力品質實驗室

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**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. 中正--電力品質實驗室

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**Chapter outline Introduction Nonlinear magnetic core sources**

Arc furnace 3-phase line commuted converters Static var compensator Cycloconverter 中正--電力品質實驗室

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**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 (b) shows a typical current waveform generated by a TCR branch, where the dash line represents the supply voltage. All odd harmonics are present in the input current for the TCR 中正--電力品質實驗室

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**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 中正--電力品質實驗室

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**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 中正--電力品質實驗室

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Current Source Model by discrete Fourier analysis 中正--電力品質實驗室

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**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 中正--電力品質實驗室

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**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 中正--電力品質實驗室

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**Time-Domain Model Model 1 Model 2**

Model 1: a single-phase SVC that includes a fixed capacitor in parallel with the TCR. The power system is represented by its Thévenin equivalent. s is the switching function Model 2: two anti-parallel thyristors are replaced by two time-varying resistors, Rt1 and Rt2. An equivalent resistance, RT, is then used to represent these resistors. RT becomes zero when any of the thyristors is conducted 中正--電力品質實驗室

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**Chapter outline Introduction Nonlinear magnetic core sources**

Arc furnace 3-phase line commuted converters Static var compensator Cycloconverter 中正--電力品質實驗室

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**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. the first term specifies the characteristic harmonics from a p-pulse converter and the second term represents sidebands of each of the dominant characteristic harmonics, which may not be integers 中正--電力品質實驗室

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**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. 中正--電力品質實驗室

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