Harmonic Balance Finite Element Method E-Book

Junwei Lu

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley - IEEE
• Sprache: Englisch

Seitenzahl: 331

Veröffentlichungsjahr: 2016

Beschreibung

The first book applying HBFEM to practical electronic nonlinear field and circuit problems 

• Examines and solves wide aspects of practical electrical and electronic nonlinear field and circuit problems presented by HBFEM
• Combines the latest research work with essential background knowledge, providing an all-encompassing reference for researchers, power engineers and students of applied electromagnetics analysis
• There are very few books dealing with the solution of nonlinear electric- power-related problems
• The contents are based on the authors’ many years’ research and industry experience; they approach the subject in a well-designed and logical way
• It is expected that HBFEM will become a more useful and practical technique over the next 5 years due to the HVDC power system, renewable energy system and Smart Grid, HF magnetic used in DC/DC converter, and Multi-pulse transformer for HVDC power supply
• HBFEM can provide effective and economic solutions to R&D product development
• Includes Matlab exercises

Leseprobe

Table of Contents

Cover

Title Page

Preface

About the Companion Website

1 Introduction to Harmonic Balance Finite Element Method (HBFEM)

1.1 Harmonic Problems in Power Systems

1.2 Definitions of Computational Electromagnetics and IEEE Standards 1597.1 and 1597.2

1.3 HBFEM Used in Nonlinear EM Field Problems and Power Systems

References

2 Nonlinear Electromagnetic Field and Its Harmonic Problems

2.1 Harmonic Problems in Power Systems and Power Supply Transformers

2.2 DC‐Biased Transformer in High‐Voltage DC Power Transmission System

2.3 Geomagnetic Disturbance and Geomagnetic Induced Currents (GIC)

2.4 Harmonic Problems in Renewable Energy and Microgrid Systems

References

3 Harmonic Balance Methods Used in Computational Electromagnetics

3.1 Harmonic Balance Methods Used in Nonlinear Circuit Problems

3.2 CEM for Harmonic Problem Solving in Frequency, Time and Harmonic Domains

3.3 The Basic Concept of Harmonic Balance in EM Fields

3.4 HBFEM for Electromagnetic Field and Electric Circuit Coupled Problems

3.5 HBFEM for a DC‐Biased Problem in High‐Voltage Power Transformers

References

4 HBFEM for Nonlinear Magnetic Field Problems

4.1 HBFEM for a Nonlinear Magnetic Field with Current‐Driven Source

4.2 Harmonic Analysis of Switching Mode Transformer Using Voltage‐Driven Source

4.3 Three‐Phase Magnetic Frequency Tripler Analysis

4.4 Design of High‐Speed and Hybrid Induction Machine using HBFEM

4.5 Three‐Dimensional Axi‐Symmetrical Transformer with DC‐Biased Excitation

References

5 Advanced Numerical Approach using HBFEM

5.1 HBFEM for DC‐Biased Problems in HVDC Power Transformers

5.2 Decomposed Algorithm of HBFEM

5.3 HBFEM with Fixed‐Point Technique

5.4 Hysteresis Model Based on Neural Network and Consuming Function

5.5 Analysis of Hysteretic Characteristics Under Sinusoidal and DC‐Biased Excitation

5.6 Parallel Computing of HBFEM in Multi‐Frequency Domain

References

6 HBFEM and Its Future Applications

6.1 HBFEM Model of Three‐Phase Power Transformer

6.2 Magnetic Model of a Single‐Phase Transformer and a Magnetically Controlled Shunt Reactor

6.3 Computation Taking Account of Hysteresis Effects Based on Fixed‐Point Reluctance

6.4 HBFEM Modeling of the DC‐Biased Transformer in GIC Event

6.5 HBFEM Used in Renewable Energy Systems and Microgrids

References

Appendix

Appendix I & II

Appendix III

Index

End User License Agreement

List of Tables

Chapter 01

Table 1.1 Sources and problems of harmonics

Chapter 02

Table 2.1 Current distortion due to nonlinear load and power electronics

Table 2.2 Harmonic components in 6‐pulse and 12‐pulse rectifier transformers

Chapter 03

Table 3.1 Comparison of time‐periodic steady‐state nonlinear EM field analysis method

Chapter 05

Table 5.1 Different DC bias conditions specified by quantity in the magnetic field

Table 5.2 Errors between calculated and measured results in exciting current

Table 5.3 Each harmonic component of magnetic induction in one element in the silicon steel region under DC bias conditions

Table 5.4 Scheme of setting the relaxation factor for the convergence of the harmonic solution

Table 5.5 Calculation results under different DC bias excitations

Table 5.6 Comparison of the computational cost between the two methods

Table 5.7 Harmonic components of

B

x

and reluctivity in point B

Table 5.8 DC flux density (

B

x0

) under different DC bias and AC excitations

Table 5.9 AC flux density (

B

x1

) under different DC bias and AC excitations

Table 5.10 Data of magnetization curve under DC‐biased magnetization (

H

dc

 = 107 A/m)

Table 5.11 Comparison of the two methods

Table 5.12 Comparison of the exciting current

Table 5.13 Comparison of the flux density

Table 5.14 Comparison of the two methods (

I

dc

 = 0.426 A)

Table 5.15 Comparison of the two methods (

I

dc

 = 0.847 A)

Table 5.16 Simulated results of hysteresis loops under sinusoidal excitation (

I

dc

 = 0 A)

Table 5.17 Simulated results of hysteresis loops under sinusoidal excitation (

I

dc

 = 0.426 A)

Table 5.18 Simulated results of hysteresis loops under sinusoidal excitation (

I

dc

 = 0.847 A)

Table 5.19 Iron loss under sinusoidal magnetization

Table 5.20 Iron loss in different areas under sinusoidal magnetization

Table 5.21 Iron loss in different areas under DC‐biased magnetization (

I

dc

 = 0.426 A)

Table 5.22 Comparison between measured and predicted results

Chapter 06

Table 6.1 Comparison of the RMS value of the magnetizing currents under different DC‐biased conditions

Table 6.2 Comparison of the peak value of magnetizing currents under different DC‐biased conditions

List of Illustrations

Chapter 01

Figure 1.1 (a) Sine wave. (b) Distorted waveform or non‐sinusoidal

Figure 1.2 Distorted waveform and number of harmonics by Fourier series

Figure 1.3 Harmonic distortion of the electrical current waveform, where the distorted waveform is composed of fundamental and 3rd harmonics

Figure 1.4 Creation of distorted current

Chapter 02

Figure 2.1 (a) Characteristics of magnetic impedance associated with a B‐H curve and permeability. (b) Excitation current corresponding to a sinusoidal voltage excitation associated with a hysteresis B‐H curve

Figure 2.2 Excitation current corresponding to a sinusoidal voltage excitation

Figure 2.3 Nonlinear magnetic and nonlinear dielectric materials, (a) the B‐H hysteresis loop of the magnetic material, and (b) the direction of the polarization (D‐E) hysteresis loop of ferroelectric material

Figure 2.4 Harmonics in time domain presentation

Figure 2.5 Frequency domain graphs – frequency spectrums

Figure 2.6 Examples of load current waveforms with harmonics

Figure 2.7 (a) Waveforms of flux; (b) voltage for sinusoidal magnetizing current in nonlinear magnetics

Figure 2.8 (a) Magnetic flux waveforms; (b) current waveform with a square excitation voltage

Figure 2.9 The full bridge transformer‐isolated buck converter

Figure 2.10 A two‐transistor version of the forward converter

Figure 2.11 The flyback converter, a single‐transistor isolated buck‐boost converter

Figure 2.12 A half‐bridge LLC converter

Figure 2.13 Magnetic triplers with three‐phase input at 50Hz and single phase output at 150Hz. (a) magnetic frequency tripler with three input magnetizing coils and two secondary coils connected in a series as an output. (b) magnetic frequency tripler using three secondary coils connected in a series

Figure 2.14 Three phase rectifier transformer (6‐pulse transformer). (a) three phase transformer with Y connection at secondary side. (b) waveforms of 6‐pulse output voltage for three phase rectifier transformer with delta connection at the secondary side

Figure 2.15 Waveforms of 6‐pulse input voltage and current

Figure 2.16 Three phase with 12‐pulse rectifier transformer with Y and delta connections at the secondary side

Figure 2.17 Waveforms of input voltage and current for 12‐pulse rectifier transformer

Figure 2.18 Schematic diagram of the DC bias phenomenon

Figure 2.19 Inpouring reverse current for compensation

Figure 2.20 The changing geomagnetic field (

φ

gm

) induces a geoelectric field (

E

ge

) that drives currents in conductor loops

Figure 2.21 A small DC content in the AC may cause a power transformer to enter half‐cycle saturation

Figure 2.22 Magnetizing current and hysteresis loop under different direct magnetizing current (single phase transformer)

Figure 2.23 Three‐phase DC/AC inverter connected to a building transformer

Figure 2.24 Output voltage waveform

Figure 2.25 Frequency spectrum

Figure 2.26 Microgrid coordinator/coordinated control system, conceptual commercial‐level microgrid architecture

Figure 2.27 Basic concept of the microgrid connected through a PCC to the grid

Figure 2.28 (a) Voltage waveform at the PCC; (b) VSC current waveforms from the inverter

Chapter 03

Figure 3.1 Distorted waveform and spectrum. (a) Distorted waveforms; (b) Spectrum

Figure 3.2 The concept of harmonic balance for a non‐linear circuit

Figure 3.3 The circuit diagram for using harmonic balance in a non‐linear circuit

Figure 3.4 Magnetic core for a 2D transformer structure, and its typical B‐H cure. (a) Transformer with nonlinear magnetic core. (b) B‐H curve and permeability

Figure 3.5 B‐H curve with hysteresis characteristics (a), and without hysteresis characteristics (b)

Figure 3.6 B‐H curve with hysteresis characteristics and DC‐biased condition. (a) H‐B curve with hysteresis; (b) H‐B curve with hysteresis and DC‐biased case

Figure 3.7 Size of the system matrix. (a) Static FEM; (b) HBFEM

Figure 3.8 Simulation model of LLC converter with resonant tank with idea transformer

Figure 3.9 Coupling between the electric circuit and the magnetic field

Figure 3.10 HVDC power transmission system

Figure 3.11 B‐H curve with hysteresis characteristics and a DC biased condition. (a) DC‐biased hysteresis loop; (b) Magnetizing current

Figure 3.12 The block diagram of the three phase HVDC transformer including neutral points

Chapter 04

Figure 4.1 Current‐source excitation to magnetic field. (a) Switch‐mode push‐pull converter; (b) Zero‐voltage switched resonant converter

Figure 4.2 Magnetic system with current‐source excitation. (a) Magnetic configuration; (b) Hysteresis characteristic

Figure 4.3 The magnetic flux distribution for fundamental and third harmonic components. (a) Fundamental component; (b) Third harmonic component

Figure 4.4 The analysis model with an air gap and two slots at the central leg, and B‐H curve

Figure 4.5 The experimental results compared with numerical computation results in the case of current source excitation, where … indicates an experimental result and — indicates a numerical result. (a) Magnetic density B

1

; (b) Magnetic density B

2

; (c) Magnetic density B

3

; (d) Excitation current density

J

Figure 4.6 The magnetic flux distribution for fundamental and third harmonic components. (a) Fundamental harmonic component; (b) Third harmonic component

Figure 4.7 Voltage‐source to the magnetic system used for switch mode transformers

Figure 4.8 Generalized model of voltage‐source to the magnetic system

Figure 4.9 Magnetic core for a 2‐D transformer structure and its B‐H cure. (a) Magnetic core (b) B‐H curve

Figure 4.10 Magnetic flux distribution for their harmonic components at phase of zero degree. (a) fundamental component; (b) third harmonic component; (c) fifth harmonic component

Figure 4.11 Comparison between computation and measurement. (a) Input voltage source; (b) Current caused by voltage source

Figure 4.12 A circuit diagram of a magnetic frequency tripler

Figure 4.13 B‐H curve with (a) hysteresis; and (b) without hysteresis characteristics

Figure 4.14 A configuration of the magnetic frequency tripler with a voltage driven source connected to the magnetic system

Figure 4.15 A three‐phase magnetic tripler problem as a voltage‐driven source connected to the magnetic system

Figure 4.16 Geometric size ½, a configuration of the magnetic frequency tripler ½ for numerical computation model

Figure 4.17 Output voltage waveform: (a) experimental result, (b) simulation result

Figure 4.18 Flux distribution of fundamental, third and fifth harmonics.

Figure 4.19 Comparison between computation and experiment results of waveforms of input and output currents, and input voltage (phase U) and neutral voltage

V

NN″

Figure 4.20 The waveforms of the magnetic flux density distribution for each phase of magnetic leg and output side of magnetic legs

Figure 4.21 Characteristics of output current against load and phase U input current against input voltage. (a) Output current against load; (b) Phase U input current against input voltage

Figure 4.22 High‐speed hybrid induction motor consists of three‐phase to two‐phase magnetic frequency tripler

Figure 4.23 The compact structure of high‐speed and hybrid induction motor and three‐phase magnetizing windings and four additional coils connected with the capacitors

Figure 4.24 Numerical model of electric machine taking account of motion

Figure 4.25 The distributions of magnetic flux in the core at excitation voltage 213 V. (a) Magnetic flux distributions of different harmonic component; (b) The impact of slip on the distributions of magnetic flux

Figure 4.26 The waveform of the flux density at different legs of the magnetic tripler

Figure 4.27 The fundamental and third harmonic components of magnetic flux distributions at different rotating angles

Figure 4.28 Magnetic flux distributions of high‐speed and hybrid induction motor: (a) at normal rotating case; (b) at stopping case

Figure 4.29 Input voltage vs output voltage for both fundamental and third harmonics

Figure 4.30 Various waveforms, including input voltages, output voltage and voltage accrued at neutral point

Figure 4.31 (a) Simulation model of a switching mode transformer. (b) B‐H curve of the magnetic core

Figure 4.32 Magnetic flux distribution [4]. (a) DC flux; (b) First harmonic flux; (c) Second harmonic flux

Figure 4.33 Flux density of the transformer with DC biased excitation. (a) Experimental result; (b) Simulation result

Figure 4.34 The 2‐D axi‐symmetrical numerical model of voltage excitation and distributions of eddy current in the windings. (a) Axi‐symmetrical winding configuration; (b) Eddy current at

Z

 = 0 mm, (c) Eddy current at

Z

 = 2.25 mm

Chapter 05

Figure 5.1 Operation modes of the high‐voltage direct current transmission system

Figure 5.2 Epstein frame‐like core model

Figure 5.3 Magnetizing curve of the silicon‐steel sheet

Figure 5.4 Exciting current under different DC bias (

U

m

 = 

U

m,2

 = 133 V;

B

m

 = 

B

m,2

 = 0.49 T)

Figure 5.5 Exciting current under different DC bias (

U

m

 = 

U

m,3

 = 240 V;

B

m

 = 

B

m,3

 = 0.88 T)

Figure 5.6 Exciting current under different DC bias (

U

m

 = 

U

m,4

 = 370 V;

B

m

 = 

B

m,4

 = 1.37 T)

Figure 5.7 Exciting current under different DC bias (

U

m

 = 

U

m,6

 = 495 V;

B

m

 = 

B

m,6

 = 1.82 T)

Figure 5.8 Each harmonic component of exciting current under different DC bias (

U

m

 = 

U

m,3

 = 240 V;

B

m

 = 

B

m,3

 = 0.88 T)

Figure 5.9 DC bias effect on different harmonics (

U

m

 = 

U

m,3

 = 240 V;

B

m

 = 

B

m,3

 = 0.88 T)

Figure 5.10 Each harmonic component of exciting current under different DC bias (

U

m

 = 

U

m,6

 = 495 V;

B

m

 = 

B

m,6

 = 1.82 T)

Figure 5.11 AC voltage (peak value) effect on each harmonic component under 50% DC bias (

I

dc

 = 

I

dc,2

 = 0.847 A;

H

dc

 = 

H

dc,2

 = 213.12 A/m)

Figure 5.12 DC bias effect on different harmonics (

U

m

 = 

U

m,6

 = 495 V;

B

m

 = 

B

m,6

 = 1.82 T)

Figure 5.13 One quarter of the computational model

Figure 5.14 DC component of flux density (

B

x

,0

) under different DC bias conditions

Figure 5.15 Magnetic flux density (

B

x

) under different DC bias when the AC excitation is constant (

U

m

 = 

U

m,3

 = 240 V)

Figure 5.16 AC voltage effect on waveforms of magnetic flux density (

B

x

) under 50% DC bias (

I

dc

 = 

I

dc,2

 = 0.847 A)

Figure 5.17 AC voltage effect on DC and AC components of the magnetic flux density (

B

x

,0

and

B

x

,1

) under 25% DC bias

Figure 5.18 AC voltage effect on DC and AC components of the magnetic flux density (

B

x

,0

and

B

x

,1

) under 50% DC bias

Figure 5.19 AC voltage effect on DC and AC components of the magnetic flux density (

B

x

,0

and

B

x

,1

) under 75% DC bias

Figure 5.20 DC bias effect on DC and AC components of the magnetic flux density (

B

x

,0

and

B

x

,1

) with alternating voltage of 133 V

Figure 5.21 DC bias effect on DC and AC components of the magnetic flux density (

B

x

,0

and

B

x

,1

) with alternating voltage of 240 V

Figure 5.22 DC bias effect on DC and AC components of the magnetic flux density (

B

x

,0

and

B

x

,1

) with alternating voltage of 370 V

Figure 5.23 The total flux distribution

Figure 5.24 Flux distribution of harmonic components (

ωt

 = 

π

/3). a. DC component; b. Fundamental component; c. Second harmonic component; d. Third harmonic component; e. Fourth harmonic component; f. Fifth harmonic component; g. Sixth harmonic component; h. Seventh harmonic component; i. Eighth harmonic component; j. Ninth harmonic component

Figure 5.25 An HVDC transmission system

Figure 5.26 LCM and the B‐H curve. (a) Laminated core model (

σ

steel

 = 2.22 × 10

6

 S/m); (b) Exciting coil and search coil (

σ

coil

 = 5.71 × 10

7

 S/m); (c) B‐H curve of the LCM

Figure 5.27 Schematic circuit diagram of the core model in experiment

Figure 5.28 Exciting currents under different DC bias conditions (

I

dc

 = 0.847 A;

U

ac

 = 320 V)

Figure 5.29 Computational region: one quarter of the square laminated core model

Figure 5.30 Calculated flux density, exciting current and reluctivity (

I

dc

 = 1.27 A,

U

ac

 = 370 V). (a) Waveform of flux density

B

x

in point B; (b) Exciting current; (c) Waveform of reluctivity

Figure 5.31 Harmonic flux distributions (

I

dc

 = 1.27A,

U

ac

 = 370V). (a) DC flux; (b) Fundamental (

ωt

 = π/2); (c) Second order (2

ωt

 = π/2); (d) Third order (3

ωt

 = π/2); (e) Fourth order (4

ωt

 = π/2); (f) Fifth order (5

ωt

 = π/2)

Figure 5.32 Hysteresis loops under DC‐biased and sinusoidal excitations

Figure 5.33 The SLC and computational region

Figure 5.34 Measured hysteresis loops (solid line) and magnetization curve (dashed line) under DC‐biased magnetization (

H

dc

 = 107 A/m)

Figure 5.35 Calculated and measured exciting current (

H

dc

 = 213 A/m,

U

ac

 = 320 V)

Figure 5.36 Waveforms of calculated

ν

FP

(dashed line) and magnetic flux density (solid line) in point

E

in SLC (

H

dc

 = 107 A/m,

U

ac

 = 368 V)

Figure 5.37 Evolution of the relative error of the harmonic solution in two methods (

H

dc

 = 107 A/m,

U

ac

 = 368 V,

N

c

 = 11)

Figure 5.38 Comparison of calculated exciting currents (

H

dc

 = 107 A/m,

U

ac

 = 495 V);

i

1

and

i

2

are exciting currents calculated by basic and DC‐biasing magnetization curves respectively, and

i

m

is the measured result

Figure 5.39 Comparison of calculated magnetic flux density (

U

ac

 = 240 V, 368 V, 495 V;

H

dc

 = 107 A/m).

B

1

and

B

2

are computed by the basic and DC‐biasing magnetization curves respectively,

B

av

is the average value obtained in (5‐47)

Figure 5.40 Magnetizing current under sinusoidal excitation

Figure 5.41 Hysteresis model based on consuming function

Figure 5.42 Laminated core model and electric schematic diagram

Figure 5.43 Simulated and measured hysteresis loop under sinusoidal magnetization. (a) Symmetrical hysteresis loops. (b) Simulated and measured results

Figure 5.44 Computed and measured magnetizing currents under different DC‐biased magnetizations. (a)

U

ac

 = 80 V,

I

dc

 = 0.425 A; (b)

U

ac

 = 370 V,

I

dc

 = 0.425 A; (c)

U

ac

 = 452 V,

I

dc

 = 0.425 A

Figure 5.45 Simulated and measured hysteresis loops under DC‐biased magnetization. (a) DC‐biasing hysteresis loops (

H

dc

 = 107 A/m); (b) DC‐biasing hysteresis loops (

H

dc

 = 213 A/m); (c) Simulated and measured results

Figure 5.46 Hyperbolic tangent transfer function

Figure 5.47 Training performance (three hidden layers with ten hidden neurons in each layer)

Figure 5.48 Training performance (four hidden layers with 20 hidden neurons in each layer)

Figure 5.49 Comparison between the measured and simulated results (

I

dc

 = 0 A)

Figure 5.50 Comparison between the measured and simulated results (

I

dc

 = 0.426 A)

Figure 5.51 Comparison between the measured and simulated results (

I

dc

 = 0.847 A)

Figure 5.52 Geometric model of the laminated core for computation

Figure 5.53 Magnetizing current under sinusoidal magnetization (

U

ac

 = 420 V,

I

dc

 = 0 A)

Figure 5.54 Magnetizing current under DC‐biased magnetization (

U

ac

 = 420 V,

I

dc

 = 0.426 A)

Figure 5.55

H

y

‐

B

y

hysteresis loop on point B under sinusoidal magnetization (

U

ac

 = 420 V,

I

dc

 = 0 A)

Figure 5.56 Hysteresis loop on point A under sinusoidal magnetization (

U

ac

 = 420 V,

I

dc

 = 0 A).(a)

H

x

‐

B

x

hysteresis loop. (b)

H

y

‐

B

y

hysteresis loop

Figure 5.57

H

y

‐

B

y

hysteresis loop on point B under DC‐biased magnetization (

U

ac

 = 420 V,

I

dc

 = 0.426 A)

Figure 5.58 Hysteresis loop on point A under DC‐biased excitation (

U

ac

 = 420 V,

I

dc

 = 0.426 A). (a)

H

x

‐

B

x

hysteresis loop; (b)

H

y

‐

B

y

hysteresis loop

Figure 5.59 The NN DC‐biasing hysteresis model in the limb‐yoke region

Figure 5.60 The training performance based on NN

Figure 5.61 The predicted and measured DC‐biasing hysteresis loops (

H

dc

 = 213 A/m,

B

acm

 = 0.5065 T)

Figure 5.62 Comparison of exciting current between measurement and calculation (

H

dc

 = 107 A/m,

U

ac

 = 470 V)

Figure 5.63 The calculated hysteresis loops in point

A

(

H

dc

 = 107 A/m,

U

ac

 = 470 V)

Figure 5.64 The calculated hysteresis loops in point

A

(

H

dc

 = 107 A/m,

U

ac

 = 470 V)

Figure 5.65 The calculated hysteresis loops in point

B

(

H

dc

 = 107 A/m,

U

ac

 = 470 V)

Figure 5.66 Size of the equation matrices for HBFEM method (b) compared with the traditional FEM (a)

Figure 5.67 Domain decomposition approach for the FEM mesh

Figure 5.68 Domain Decomposition approach for (a) the finite difference mesh and (b) by block

Figure 5.69 A sample of (a) Red‐black ordering of a 6 × 4 grid and (b) Associated matrix

Figure 5.70 Domain decomposition and frequency domain divided for the system matrix

Chapter 06

Figure 6.1 A circuit diagram of a three‐phase transformer. (a) Δ/Y; (b) Δ/Δ

Figure 6.2 B‐H curve with hysteresis

Figure 6.3 Three‐phase transformer with three input magnetizing coils and three secondary coils as an output winding in Y/Y connection. (a) transformer circuit diagram; (b) 2D cross section of transformer

Figure 6.4 A simplified power system wiring diagram under no‐load conditions

Figure 6.5 Equivalent magnetic circuit model structure

Figure 6.6 Scheme circuit for iron core structure and coil arrangement

Figure 6.7 Scheme circuit for iron core structure and coil arrangement

Figure 6.8 The laminated core model

Figure 6.9 Experimental circuit of the DC‐biased test

Figure 6.10 Exciting current under 25%

I

dc

(U = 293 V)

Figure 6.11 Exciting current under 50%

I

dc

(U = 293 V)

Figure 6.12 The magnetic flux density under different DC‐biased magnetization with an alternating voltage of 293 V

Figure 6.13 The reluctance under different DC bias (U = 159 V)

Figure 6.14 Magnetic circuit model of a single‐phase three limb transformer

Figure 6.15

U

ac

 = 95 V,

I

dc

 = 0

Figure 6.16

U

ac

 = 95 V,

I

dc

 = 0.2708 A

Figure 6.17

U

ac

 = 95 V,

I

dc

 = 0.5091 A

Figure 6.18

U

ac

 = 95 V,

I

dc

 = 1.5214 A

Figure 6.19 Waveforms of magnetic flux in the central core

Figure 6.20 Waveforms of magnetic flux in the side yoke

Figure 6.21 (a) DC‐biased voltage and current waveform against magnetizing curve; (b) the problems caused by combined DC and AC excitation

Figure 6.22 GIC or HVDC DC‐biased transformer, including internal equivalent circuit (leakage inductance and winding resistance) and external circuits (transmission line impedance)

Figure 6.23 (a) B‐H curve with hysteresis characteristics and DC‐biased condition; (b) excitation current during DC bias

Figure 6.24 Block diagram of harmonics current flowing through a transformer in the microgrid with nonlinear load and DG and DS

Figure 6.25 Transformer with Y/Δ connected to the grid

Figure 6.26 Waveforms of voltage and current in the LV side due to power electronic devices

Appendix

Figure A1.1 (a) The laminated core model; (b) geometric model for numerical computation

Figure A1.2 The flowchart of the Matlab program for 2‐D numerical computation

Figure A3.1 (a) Simulation model of a switching mode transformer; (b) B‐H curve of the magnetic core

Figure A3.2 Axi‐symmetrical structure in the

z

and

r

domain

Figure A3.3 Example program flowchart of a three‐dimensional axi‐symmetrical transformer with DC‐biased excitation

Guide

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HARMONIC BALANCE FINITE ELEMENT METHOD

APPLICATIONS IN NONLINEAR ELECTROMAGNETICS AND POWER SYSTEMS

Junwei Lu, Xiaojun Zhao and Sotoshi Yamada

This edition first published 2016© 2016 John Wiley & Sons Singapore Pte. Ltd

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Library of Congress Cataloging‐in‐Publication Data

Names: Lu, Junwei, author. | Zhao, Xiaojun, (Electrical engineer), author. | Yamada, Sotoshi, author.Title: Harmonic balance finite element method : applications in nonlinear electromagnetics and power systems / Junwei Lu, Xiaojun Zhao, and Sotoshi Yamada.Description: Solaris South Tower, Singapore : John Wiley & Sons, Inc., [2016] | Includes bibliographical references and index.Identifiers: LCCN 2016009676| ISBN 9781118975763 (cloth) | ISBN 9781118975787 (epub)Subjects: LCSH: Electric power systems--Mathematical models. | Harmonics (Electric waves)--Mathematics. | Finite element method.Classification: LCC TK3226 .L757 2016 | DDC 621.3101/51825--dc23 LC record available at https://lccn.loc.gov/2016009676

This book is dedicated to my wife Michelle, without her support I would never complete this book, and in memory to my parents.

– Junwei Lu

This book is dedicated to my wife Weichun Cui, since she has helped me a lot during the writing of this book. I also would like to express my gratitude to my beloved parents, who have always supported me.

– Xiaojun Zhao

Preface

In writing this book on the Harmonic Balance Finite Element Method (HBFEM): Applications in Nonlinear Electromagnetics and Power Systems, two major objectives were borne in my mind. Firstly, the book intends to teach postgraduate students and design engineers how to define quasi‐static nonlinear electromagnetic (EM) field and harmonic problems, build EM simulation models, and solve EM problems by using the HBFEM. Secondly, this book will delve into a field of challenging innovations pertinent to a large readership, ranging from students and academics to engineers and seasoned professionals.

The art of HBFEM is to use Computational Electromagnetics (CEMs) with harmonic balance theories, and CEM technologies (with IEEE Standard 1597.1 and IEEE Standard 1597.2) to analyze or investigate nonlinear EM field and harmonic problems in electrical and electronic engineering and electrical power systems. CEM technologies have been significantly developed in the last three decades, and many commercially available software packages are widely used by students, academics and professional engineers for research and product design. However, it takes untrained engineers or users several months to understand how to use those packages properly, due to a lack of knowledge on CEMs and EM modeling, and computer simulation techniques. This is particularly true for the harmonic analysis technique, which has not been fully presented in any CEM textbook or used in any commercially available packages. Although a number of CEM‐related books are available, these books are normally written for experts rather than students and design engineers. Some of these books only cover one or a few areas of CEMs, and many common CEM techniques and real‐world harmonic problems are not introduced. This book attempts to combine the fundamental elements of nonlinear EM, harmonic balance theories, CEM techniques and HBFEM approaches, rather than providing a comprehensive treatment of each area.

This book covers broad areas of harmonic problems in electrical and electronic engineering and power systems, and includes the basic concepts of CEMs, nonlinear EM field and harmonic problems, IEEE Standards 1597.1 and 1597.2, and various numerical analysis methods. In particular, it covers some of the methods that are very useful in solving harmonic‐related problems – such as the HBFEM – that are not mentioned in any other numerical calculation books or commercial software packages. In relation to computational technology, this book introduces high‐performance parallel computation, cloud computing, and visualization techniques. It covers application problems from component level to system level, from low‐frequency to high‐frequency, and from electronics to power systems.

This book is divided into six chapters and three appendices. Chapter 1 provides a short introduction to the HBFEM used for solving various harmonic problems in nonlinear electromagnetic field and power systems. This chapter will also discuss definitions of CEM techniques and the various methods used for nonlinear EM problem solving. It also describes high‐performance computation, visualization and optimization techniques for EMs, and CEM standards and validation (IEEE Standard 1597.1 and IEEE Standard 1597.2, 2010).

Chapter 2 highlights some fundamental EM theory used in nonlinear EM fields, harmonic problems in transformer power supplies, DC‐biased phenomenon in High Voltage Direct Current (HVDC) power transformers, harmonic problems in geomagnetic disturbances (GMDS), geomagnetic induced current (GIC), harmonic problems in distributed energy resource (DER) systems and microgrids, and future smart grids with electric vehicles (EV) and vehicle to grid (V2G).

Chapter 3 covers: the fundamental theory of harmonic balance methods used in nonlinear circuit problems; CEM for nonlinear EM field and harmonic problems; basic concepts of HBFEM used in nonlinear magnetic field analysis; HBFEM for electric circuits and magnetic field coupled problems; HBFEM for three‐phase electric circuits coupled with magnetic field; and HBFEM for DC‐biased HVDC power transformers.

Chapter 4 investigates HBFEM and its applications in nonlinear magnetic fields and harmonic problems. Several case study problems are presented, such as: HBFEM for a nonlinear magnetic field with current driven (inductor and single phase transformer); HBFEM for a nonlinear magnetic field with voltage‐driven (switch mode power supply transformer); three‐phase magnetic tripler transformer (electric circuit and magnetic field coupled problems); three‐phase high speed motor based on frequency tripler using HBFEM; DC‐biased 3D asymmetrical magnetic structure transformer using HBFEM.

Chapter 5 is devoted to the advanced numerical approaches of HBFEM. These include: the decomposed algorithm of HBFEM; HBFEM with a fixed‐point technique; hysteresis model based on a neural network and consuming function; and analysis of hysteretic characteristics under sinusoidal and DC bias excitation, parallel computing techniques for multi‐frequency domain problem.

Chapter 6 discusses: three‐phase power supply transformer model; magnetically controlled shunt reactors (MCSR); computation taking account of hysteresis effects based on fixed‐point reluctance; harmonics analysis in HVDC transformers (three phase model) with geo‐magnetics and geomagnetic induced current (GIC); HBFEM used for low‐voltage network transformers in renewable energy and microgrid grid systems with distributed energy resource (DER); and electric vehicle (EV) charging systems and vehicle to grid (V2G).

There are three appendices included in this book: MATLAB Program 1 (magnetic circuit analysis of a single phase transformer) and MATLAB Program 2 (main program for 2D magnetic field analysis in current driven); and Fortran program 3 (3D Asymmetrical magnetic structure transformer using HBFEM).

Junwei Lu

About the Companion Website

Don’t forget to visit the companion website for this book:

www.wiley.com/go/lu/HBFEM

There you will find valuable material designed to enhance your learning, including:

HBFEM program codes

Explanations

Scan this QR code to visit the companion website

1Introduction to Harmonic Balance Finite Element Method (HBFEM)

1.1 Harmonic Problems in Power Systems

The harmonics problem in power systems is not a new problem. It has existed since the early 1900s – as long as AC power itself has been available. The earliest harmonic distortion issues were associated with third harmonic currents produced by saturated iron in machines and transformers, or so‐called ferromagnetic loads. Later, arcing loads, like lighting and electric arc furnaces, were also shown to produce harmonic distortion. The final type, electronic loads, burst onto the power scene in the 1970s and 1980s, and has represented the fastest growing category ever since [1].

Since power system harmonic distortion is mainly caused by non‐linear loads and power electronics used in the electrical power system [2, 3], the presence of non‐linear loads and the increasing number of distributed generation power systems in electrical grids contributes to changing the characteristics of voltage and current waveforms in power systems (which differ from pure sinusoidal constant amplitude signals). The impact of non‐linear loads and power electronics used in electrical power systems has been increasing during the last decade.

Such electrical loads, which introduce non‐sinusoidal current consumption patterns (current harmonics), can be found in power electronics [4], such as: DC/AC inverters; switch mode power supplies; rectification front‐ends in motor drives; electronic ballasts for discharge lamps; personal computers or electrical appliances; high‐voltage DC (HVDC) power systems; impulse transformers; magnetic induction devices; and various electric machines. In addition, the harmonics can be generated in distributed renewable energy systems, geomagnetic disturbances (GMDs) and geomagnetic induced currents (GICs) [5, 6].

Harmonics in power systems means the existence of signals, superimposed on the fundamental signal, whose frequencies are integer numbers of the fundamental frequency. The presence of harmonics in the voltage or current waveform leads to a distorted signal for the voltage or current, and the signal becomes non‐sinusoidal. Thus, the study of power system harmonics is an important subject for electrical engineers. Electricity supply authorities normally abrogate responsibility on harmonic matters by introducing standards or recommendations for the limitation of voltage harmonic levels at the points of common coupling between consumers.

1.1.1 Harmonic Phenomena in Power Systems

A better understanding of power system harmonic phenomena can be achieved by consideration of some fundamental concepts, especially the nature of non‐linear loads, and the interaction of harmonic currents and voltages within the power system. By definition, harmonic (or non‐linear) loads are those devices that naturally produce a non‐sinusoidal current when energized by a sinusoidal voltage source. As shown in Figure 1.1, each “waveform” represents the variation in instantaneous current over time for two different loads each energized from a sinusoidal voltage source. This pattern is repeated continuously, as long as the device is energized, creating a set of largely‐identical waveforms that adhere to a common time period. Both current waveforms were produced by turning on some type of load device. In the case of the current on the left, this device was probably an electric motor or resistance heater. The current on the right could have been produced by an electronic variable‐speed drive, for example. The devices could be single‐ or three‐phase, but only one phase current waveform is shown for illustration. The other phases would be similar.

Figure 1.1 (a) Sine wave. (b) Distorted waveform or non‐sinusoidal

A French mathematician, Jean Fourier, discovered a special characteristic of periodic waveforms in the early 19th century. The method describing the non‐sinusoidal