emtp modelling of control and power electronic devices

EMTP Modelling of Control and

Power Electronic Devices

by

BENEDITO DONIZETI BONATTO

M.A.Sc. in Electrical Engineering, State University of Campinas, Brazil, 1995.

B.A.Sc. in Electrical Engineering, Federal School of Engineering of Itajuba, Brazil, 1991.

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

in

THE FACULTY OF GRADUATE STUDIES

(Department of Electrical and Computer Engineering)

We accept this thesis as conformingto the required standard

THE UNIVERSITY OF BRITISH COLUMBIA

October 2001

c Benedito Donizeti Bonatto, 2001

Abstract

The quality of the electric power delivered to customers by utilities may not be acceptablefor some types of sensitive loads, which are typically power electronics- and computer-basedloads, particularly in the control of industrial processes. There are cases where the increas-ing use of power electronics to enhance process eÆciency and controllability creates powerquality problems. The growing application of shunt capacitors for voltage support, powerfactor correction, and system loss reduction, as well as the use of series capacitors (xedor controlled, for line reactance compensation) will increase the potential risk of transientdisturbance amplications and potential electrical and mechanical resonances in the presenceof more and more power electronic devices, and of steam and gas turbines in distributed andco-generation power plants. As the natural order of the system grows, so does its abilityto oscillate more! At the same time, new power electronic devices also oer the means foradequate \power conditioning", to meet the special requirements of electric power quality ina system.

To evaluate the promising solutions oered with the introduction of more and more powerelectronic devices in transmission and distribution systems, such as FACTS (Flexible ACTransmission Systems) Controllers and Custom Power Controllers, as well as to analyzetheir interaction and impact on either the load or the network side, computer programsbased on the EMTP (Electromagnetic Transients Program) are becoming more useful. Thedevelopment of new EMTP-based models for representation of controls and power electronicdevices has been the main subject of this Ph.D. thesis project. Its main contributions aresummarized as follows:

development of a \simultaneous solution for linear and nonlinear control and electricpower system equations" (SSCPS) in EMTP-based programs, through the compensationmethod and the Newton-Raphson iterative algorithm. This solution method eliminatesnot only the one time step delay problem at the interface between the solution of powerand control circuits, but also all the internal delays, which may exist in methods basedon the transient analysis of control systems (TACS) since 1977. A \circuit approach"was proposed in this thesis, as an innovative alternative to the solution presented byA. E. A. Araujo in 1993;

experimental implementation in MicroTranR (the UBC version of the EMTP), based

on SSCPS, of a \simultaneous solution" for: linear and nonlinear current and voltagedependent sources; independent current and voltage sources, which can also be con-nected between two ungrounded nodes; hard and soft limiters; transfer functions; math-ematical and transcendental FORTRAN functions; special control devices and somedigital logic gates; transformation of variables (such as the abc to 0 transformation

ii

ABSTRACT iii

and its inverse); voltage-controlled switches; nonlinear model of a diode semiconductor;

development of the subroutine \GATE" in MicroTran, allowing the dynamic controlof the turn-on and turn-o times of semiconductor devices (e.g., thyristors, GTO's,IGBT's, etc.), which are modeled as EMTP-based voltage-controlled switches;

development of power electronics simulation cases in MicroTran, using the simultaneoussolution approach (SSCPS) for the dynamic control of semiconductor switching devices(as in a three-phase six-pulse thyristor-controlled bridge rectier, and a three-phasePWM voltage source inverter (VSI)) and evaluation of current and voltage waveforms;

interaction with a Brazilian utility company and industries for the realization and anal-ysis of eld measurements of electromagnetic phenomena aecting the quality of power,such as voltage sags and voltage swells; harmonic current and voltage distortions; tran-sients, etc., with determination of causes, consequences and investigation of possiblesolutions for power quality problems, as for example, the application of Custom PowerControllers;

synthesis of simulation guidelines for the evaluation of the impact of power electronicdevices on the quality of power, based on realistic eld measurements and EMTP timeand frequency domain simulations.

The assessment of electric power quality, with the use of EMTP-based programs, and theevaluation of the technical impact of power electronic devices on the quality of power, canhopefully be performed with the models developed in this Ph.D. thesis project.

Contents

Abstract ii

List of Tables vi

List of Figures vii

Acknowledgements xii

Quote xiii

1 Electric Power Quality and Power Electronic Devices: An Overview 1

1.1 Introduction: Better Electricity Quality at "Possibly" Lower Prices? . . . . . 1

1.1.1 Computer Analysis and Simulation of Electric Power Quality Phenomena 3

1.1.2 Electric Power Quality Monitoring . . . . . . . . . . . . . . . . . . . 4

1.1.3 Power Quality Standards . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.4 Custom Power Related Publications . . . . . . . . . . . . . . . . . . . 9

1.2 Motivation for Thesis Research . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Contributions of this Research Project . . . . . . . . . . . . . . . . . . . . . 13

2 Simultaneous Solution of Control and Electric Power System Equations(SSCPS) in EMTP-based Programs 15

2.1 Previous Developments on Transient Analysis of Control Systems (TACS) . . 15

2.2 Current and Voltage Dependent Sources in EMTP-based Programs . . . . . 18

2.2.1 Compensation Method . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.2 Dependent Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.3 Ideal Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2.4 Independent Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.5 Newton-Raphson Algorithm . . . . . . . . . . . . . . . . . . . . . . . 36

iv

CONTENTS v

2.2.6 Possible Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3 Development of Control Transfer Functions in EMTP-based Programs . . . . 42

2.4 Development of Limiters for Control Systems in EMTP-based Programs . . . 48

2.5 Development of Intrinsic FORTRAN Functions in EMTP-based Programs . 54

2.6 Development of Control Devices in EMTP-based Programs . . . . . . . . . . 58

3 Power Electronics Modelling in EMTP-based Simulations 64

3.1 Modelling Power Electronics in Electric Power Engineering Applications . . . 65

3.2 Simultaneous Solution for Voltage-Controlled Switches in EMTP-based Pro-grams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.3 Implementation of Nonlinear Diode Model in EMTP-based Programs . . . . 78

3.4 Control Modelling Aspects of Power Electronic Devices . . . . . . . . . . . . 88

4 Evaluation of the Impact of Power Electronic Devices on the Quality ofPower 90

4.1 Dynamic Interaction between Power Electronic Devices and Power Systems . 91

4.2 Power Quality Assessment through EMTP-based Programs . . . . . . . . . . 104

4.2.1 Induction Furnace Harmonic Study . . . . . . . . . . . . . . . . . . . 104

4.2.2 Voltage Sag Analysis with EMTP-based Simulation . . . . . . . . . . 120

4.2.3 Welding Industry Voltage Fluctuation Study A Visual Flicker Case 121

4.3 EMTP-based Simulation Cases with SSCPS . . . . . . . . . . . . . . . . . . 125

4.3.1 Basic Control and Control Devices Simulation Cases . . . . . . . . . 125

4.3.2 Power Electronics Simulation Cases . . . . . . . . . . . . . . . . . . . 133

4.4 Synthesis of Simulation Guidelines for Studies with EMTP-based Programs . 150

5 Conclusions and Recommendations for Future Work 153

5.1 Conclusions and Main Contributions . . . . . . . . . . . . . . . . . . . . . . 153

5.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . . 156

Bibliography 160

List of Tables

3.1 Comparison between voltage and current in a diode as a function of its parametric values. 80

4.1 Global harmonic distortion limits for the system voltages recommended in Brazil. . . . . 110

4.2 Comparison between eld measurements and EMTP simulation results for the operating

condition with the harmonic passive lters turned OFF. . . . . . . . . . . . . . . . . 119

4.3 Comparison between eld measurements and EMTP simulation results for the operating

condition with the harmonic passive lters turned ON. . . . . . . . . . . . . . . . . . 120

vi

List of Figures

1.1 Typical Design Goals of Power-Conscious Computer Manufacturers. (Source: IEEE Std.

446-1987, \IEEE Recommended Practice for Emergency and Standby Power Systems for

Industrial and Commercial Applications.") . . . . . . . . . . . . . . . . . . . . . . 8

1.2 CBEMA curve revised by the Information Technology Industry Council (ITIC). . . . . . 9

1.3 (a) Thyristor in an industrial power converter. (b) Thyristors in a high voltage direct

current (HVDC) System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1 EMTP and TACS interface with 1 time step delay. . . . . . . . . . . . . . . . . . . . 16

2.2 M-phase Thevenin equivalent circuit. . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Representation of branch equation k as a voltage source in series with a resistance. . . . 20

2.4 Representation of branch equation k as a current source in parallel with a resistance. . . 21

2.5 Current-controlled voltage source (CCVS). . . . . . . . . . . . . . . . . . . . . . . 23

2.6 Current-controlled current source (CCCS). . . . . . . . . . . . . . . . . . . . . . . 24

2.7 Voltage-controlled voltage source (VCVS). . . . . . . . . . . . . . . . . . . . . . . . 25

2.8 Symbol for operational amplier. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.9 Inverting amplier circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.10 Non-inverting amplier circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.11 Adder circuit with operational amplier. . . . . . . . . . . . . . . . . . . . . . . . 29

2.12 Ideal integrator circuit with operational amplier. . . . . . . . . . . . . . . . . . . . 30

2.13 Generalization of inverter amplier circuit. . . . . . . . . . . . . . . . . . . . . . . 30

2.14 First-order lag circuit using ideal operational amplier. . . . . . . . . . . . . . . . . . 31

2.15 Voltage-controlled current source (VCCS). . . . . . . . . . . . . . . . . . . . . . . . 31

2.16 Ideal transformer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.17 Independent current source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.18 Independent voltage source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.19 Newton-Raphson algorithm experimentally implemented in MicroTran. . . . . . . . . . 39

2.20 Circuit with ideal operational amplier. . . . . . . . . . . . . . . . . . . . . . . . 41

vii

LIST OF FIGURES viii

2.21 Simulation results of circuit with ideal operational amplier (noninverting amplier circuit). 41

2.22 Transfer function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.23 Observer form block-diagram of transfer function in equation 2.80. . . . . . . . . . . . 44

2.24 Possible computer implementation of the transfer function block-diagram in Fig. 2.23. . . 45

2.25 Block-diagram representation of a rst-order transfer function. . . . . . . . . . . . . . 45

2.26 Observer form block-diagram of rst-order transfer function of Fig. 2.25. . . . . . . . . 46

2.27 Possible computer implementation of rst-order transfer function of Fig. 2.25. . . . . . . 46

2.28 Realistic rst-order lag circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.29 Time domain simulation of rst-order transfer function. . . . . . . . . . . . . . . . . 47

2.30 First-order transfer function with windup (static) limiter. . . . . . . . . . . . . . . . 49

2.31 First-order transfer function with non-windup (dynamic) limiter. . . . . . . . . . . . . 49

2.32 Transient response of a rst-order transfer function with windup and non-windup limiter. 50

2.33 Soft limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.34 Zero-order transfer function with soft limits. . . . . . . . . . . . . . . . . . . . . . . 53

2.35 Time domain response for a sinusoidal excitation input u(t) illustrating the eects of hard

and soft limits on the output x(t). . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.36 Open loop control system with "supplemental devices S1,S2 and S3". . . . . . . . . . . 54

2.37 Nonlinear control block-diagram with a sinusoidal intrinsic FORTRAN function. . . . . . 55

2.38 Circuit implementation for the simultaneous solution of a sinusoidal FORTRAN function. 55

2.39 Transport delay control device. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.40 Circuit implementation for the simultaneous solution of a transport delay control device. . 59

2.41 Transient simulation of a transport delay control device. . . . . . . . . . . . . . . . . 60

2.42 Transient simulation of a pulse delay control device. . . . . . . . . . . . . . . . . . . 61

2.43 Pulse delay control device with arbitrary input signal. . . . . . . . . . . . . . . . . . 62

2.44 Logic gate "NOT". . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.45 Circuit implementation of a logic gate "NOT" for simultaneous solution. . . . . . . . . 63

3.1 Power semiconductor devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2 Voltage-controlled switch in EMTP-based programs. . . . . . . . . . . . . . . . . . . 70

3.3 Test cases for transient simulation of voltage-controlled, bipolar in voltage and bidirectional

current owing switch, thyristor and GTO. . . . . . . . . . . . . . . . . . . . . . . 71

3.4 Simulation of a voltage-controlled bidirectional current owing switch. . . . . . . . . . 72

3.5 Simulation of a simplied model for thyristors. . . . . . . . . . . . . . . . . . . . . . 72

3.6 Simulation of a simplied model for GTO's. . . . . . . . . . . . . . . . . . . . . . . 73

LIST OF FIGURES ix

3.7 Circuit with \simultaneous solution" of a voltage-controlled switch. . . . . . . . . . . . 76

3.8 Simulation with simultaneous solution of a voltage-controlled switch. . . . . . . . . . . 76

3.9 One time step delay in EMTP-based switches. . . . . . . . . . . . . . . . . . . . . . 77

3.10 Diode symbol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.11 V-I diode characteristic and network Thevenin equivalent circuit equation. . . . . . . . 81

3.12 Circuit implementation for the simultaneous solution of a nonlinear diode model. . . . . 82

3.13 V-I diode characteristic and dierent network Thevenin equivalents. . . . . . . . . . . . 82

3.14 Dc-dc converter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.15 Half-wave rectier with freewheeling diode. . . . . . . . . . . . . . . . . . . . . . . 85

3.16 Electric circuit with a nonlinear diode model. . . . . . . . . . . . . . . . . . . . . . 86

3.17 Transient simulation of a nonlinear diode model in an EMTP-based program. . . . . . . 86

3.18 Detail of the transient simulation of a nonlinear diode model in an EMTP-based program. 87

3.19 V-I nonlinear characteristic of the diode resulting from the EMTP simulation. . . . . . . 87

4.1 Circuit with a single-phase diode-bridge rectier. . . . . . . . . . . . . . . . . . . . . 93

4.2 Current drawn from the source by a single-phase diode-bridge rectier. . . . . . . . . . 94

4.3 Harmonic amplitude spectrum of the current drawn from the source by a single-phase

diode-bridge rectier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.4 Current through and voltage across the total inductance, and voltage waveform distortion

at the point of common coupling (PCC). . . . . . . . . . . . . . . . . . . . . . . . 96

4.5 Harmonic amplitude spectrum of the voltage waveform distortion at the point of common

coupling (PCC). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.6 Four-wire, three-phase system with \balanced" single-phase diode-bridge rectiers. . . . . 97

4.7 Current owing through the neutral conductor. . . . . . . . . . . . . . . . . . . . . 98

4.8 Harmonic amplitude spectrum of the current owing through the neutral conductor. . . . 99

4.9 Voltage waveshape measured at the outlet of the Power Electronics Laboratory of the

Department of Electrical and Computer Engineering at UBC, Vancouver, B.C., Canada. . 100

4.10 Measured voltage waveshape, its fundamental component and its harmonic distortion. . . 101

4.11 Harmonic amplitude spectrum of the outlet waveshape voltage. . . . . . . . . . . . . . 102

4.12 Phase-angle of the harmonic components of the outlet waveshape voltage. . . . . . . . . 103

4.13 (a) Metal melting by an induction furnace. (b) Induction furnace operation. . . . . . . . 106

4.14 Current measurements in a distribution feeder supplying induction furnaces at the time of

maximum voltage distortion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.15 (a) Phase \A" current measured with harmonic passive lters turned o. (b) Phase-to-phase

\A-B" voltage measured with harmonic passive lters turned o. . . . . . . . . . . . . 108

LIST OF FIGURES x

4.16 (a) Phase \A" current measured with harmonic passive lters turned on. (b) Phase-to-phase

\A-B" voltage measured with harmonic passive lters turned on. . . . . . . . . . . . . 109

4.17 THD harmonic trend, with harmonic passive lters turned o from 12:00 midnight to 06:00am.109

4.18 THD harmonic trend, with harmonic passive lters turned on all the time. . . . . . . . 110

4.19 Distribution substation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.20 Current-source, parallel-resonant inverter for induction heating. . . . . . . . . . . . . . 114

4.21 Amplitude of the positive sequence system impedance at the PCC with harmonic lters. . 114

4.22 Phase angle of the positive sequence system impedance at the PCC with harmonic lters. 115

4.23 (a) Phase \A" current simulated with harmonic passive lters turned o. (b) Phase-to-

phase \A-B" voltage simulated with harmonic passive lters turned o. . . . . . . . . . 116

4.24 (a) Phase \A" current measured with harmonic passive lters turned o. (b) Phase-to-phase

\A-B" voltage measured with harmonic passive lters turned o. . . . . . . . . . . . . 116

4.25 (a) Phase \A" current simulated with harmonic passive lters turned on. (b) Phase-to-

phase \A-B" voltage simulated with harmonic passive lters turned on. . . . . . . . . . 117

4.26 (a) Phase \A" current measured with harmonic passive lters turned on. (b) Phase-to-phase

\A-B" voltage measured with harmonic passive lters turned on. . . . . . . . . . . . . 117

4.27 Instantaneous ideal compensation current to be \injected" by a shunt active lter. . . . . 118

4.28 Voltage sag measurements (%RMS versus time duration) with an overlay of the CBEMA

curve. For time durations less than 1 cycle the equipment seems to measure peak values. . 122

4.29 (a) Phase-to-phase \A-B" measured voltage sag. (b) Phase-to-phase \A-B" simulated volt-

age sag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.30 Instantaneous voltage uctuations causing light ickering eect. . . . . . . . . . . . . 124

4.31 Modulated voltage and respective amplitude frequency spectrum . . . . . . . . . . . . 124

4.32 Control block diagram of a second order dierential equation with poles on the imaginary

axis of the complex plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.33 Solution of system with bounded resonance oscillations. . . . . . . . . . . . . . . . . 126

4.34 Introduction of a one time step delay in the control block diagram. . . . . . . . . . . . 127

4.35 Solution of system with unstable resonance oscillations caused by the introduction of one

time step delay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.36 Classical linearized \swing equation", used in power system small-signal stability studies of

a single machine connected to an innite bus. . . . . . . . . . . . . . . . . . . . . . 130

4.37 Simulation results of the synchronous machine rotor angle deviation, in the presence of a

positive damping torque coeÆcient. . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.38 Simulation results of the synchronous machine rotor angle deviation, in the presence of

negative damping torque coeÆcient. . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.39 Canonical second order transfer function representation of the single-machine innite bus

system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

LIST OF FIGURES xi

4.40 Circuit for the dynamic control of the ring angle (\") of a thyristor. . . . . . . . . . 135

4.41 Voltages and currents in a circuit with dynamic control of the ring angle of a thyristor. . 136

4.42 Circuit for the dynamic control of the ring angles of a three-phase six-pulse thyristor-bridge

rectier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

4.43 Voltages and currents with dynamic control of the ring angles of a three-phase six-pulse

thyristor-bridge rectier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

4.44 Dynamic control of the ring angles of a three-phase six-pulse thyristor-bridge rectier. . 140

4.45 Dynamic voltage control signals at the output of the proportional-integral (PI) and the

limiter control blocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4.46 Circuit for the dynamic control of three-phase PWM voltage source inverter (VSI). . . . 144

4.47 Phase \A" modulation and triangular carrier waveforms for generation of gating signals

through sinusoidal pulse width modulation (PWM). . . . . . . . . . . . . . . . . . . 145

4.48 Node voltage \vSA" generated by a three-phase PWM voltage source inverter (VSI). . . . 146

4.49 Voltage across the load \vSANEUTR" and current supplied to the load by a three-phase

PWM voltage source inverter (VSI). . . . . . . . . . . . . . . . . . . . . . . . . . . 147

4.50 Load currents supplied by a three-phase PWM voltage source inverter (VSI). . . . . . . 148

4.51 Line-to-line voltage generated by a three-phase PWM voltage source inverter (VSI). . . . 149

Acknowledgements

I would like to thank God for the gift of learning. My most sincere thanks to my parents,Dorival and Isolina, and to all my relatives for their unconditional love and support. To mywife, Luciana, and my daughters, Alexa and Aline, my love and my heartfelt thanks for theirstrong participation in this life project altogether. I dedicate a very special note of thanksto our special friends Wany, Fernando, Fulvia, Alexandre, Martha, and Richard for theircareful and kindness personal assistance.

I owe a tremendous debt of gratitude to Dr. Hermann W. Dommel, my Ph.D. thesissupervisor, for all his personal and professional encouragement, share of wisdom and supportfor the development of this thesis. (The responsibility for any remaining errors is solelymine.) I also thank Dr. Dommel for the honor and opportunities of have being his teachingassistant.

I also thank Dr. William G. Dunford for kindly accepting to be my Ph.D. thesis co-supervisor, with Dr. Dommel becoming a Professor Emeritus at UBC. I have also learnedwith Dr. Jose R. Mart, who has excellent teaching skills. Professor Sandoval CarneiroJr. from the Federal University of Rio de Janeiro (UFRJ), Brazil, has gently been verysupportive, right from the start of this Ph.D. program in Canada.

I most specially appreciate the help, acceptance and advice of many individuals withoutwhom this opportunity would never have become fruitful. Professors, sta members, pastand present colleagues and friends at the Department of Electrical and Computer Engineeringof the University of British Columbia (UBC) have been the source of inspiration and supportto pursue scientic and personal growth. I also thank my former Brazilian professors andcolleagues at the State University of Campinas (UNICAMP), and at the Federal School ofEngineering of Itajuba (EFEI), for building and enhancing the foundation of my knowledgein science and engineering.

I would like to sincerely thank the Fundac~ao Coordenac~ao de Aperfeicoamento de Pessoalde Nvel Superior (CAPES), Braslia - Brazil, for the nancial support to this Ph.D. thesisproject. Without it, my dream would never come true.

I also thank the Brazilian utility company ELEKTRO - Eletricidade e Servicos S.A., witha special reference to Francisco Alfredo Fernandes, for providing opportunities for a practicalinteraction in power quality analysis, through a professional cooperation with the engineerErnesto A. Mertens Jr.. I acknowledge and thank students, professors and sta at the ETEProf. Armando Bayeux da Silva, a technical high school of the CEETEPS - Centro Estadualde Educac~ao Tecnologica Paula Souza, S~ao Paulo, Brazil, for all the teaching experiences Iwas able to conduct, which enriched my communication and leadership skills.

Last but not least, I thank and acknowledge the contributions of many people, notmentioned, not forgotten, who certainly have had an impact and in uence on my living andstudying at UBC, Vancouver, B.C., Canada, since August 24, 1997.

Vancouver, B.C., Canada Benedito Donizeti BonattoOctober 09, 2001.

xii

\Engineering:engineering is the application of mathematical and scientic principles to practical ends, asin the design construction, and operation of economical and eÆcient structures, equipment,and systems. It's art and communication, politics and nance, modeling and simulation,invention, approximation, measurement and estimation, and more. It's a way to think

about problems."

http:nnwww.eg3.com

xiii

Chapter 1

Electric Power Quality and PowerElectronic Devices: An Overview

THE PURPOSE of this research project was to develop reasonably accurate models for

control systems and power electronic devices to evaluate their impact on the quality

of power. These models and methods were developed for implementation in the Electromag-

netic Transients Program (EMTP) [1], [2], or in similar programs. A \circuit approach" is

used for the simultaneous solution of the control and electric power system equations, thus

eliminating any time step delay in the digital time domain simulation. Such time step delays

can cause inaccuracies or numerical instabilities. The advantages of the circuit approach is

its \generality and exibility" for modelling multi-terminal linear and nonlinear control de-

vices, which are needed in the analysis of electromagnetic phenomena aecting the quality of

power. This chapter presents an introduction to power quality problems and their relation

with power electronics, followed by a description of the motivations for this Ph.D. thesis

research and its relevant contributions.

1.1 Introduction: Better Electricity Quality at "Possi-

bly" Lower Prices?

The demand of electricity customers for increased quality of power, at possibly reduced

prices, is forcing governments, regulatory agencies, utility companies, and equipment manu-

facturers to develop new structures for the electricity market. Deregulation of the electricity

1

1.1. Introduction: Better Electricity Quality at "Possibly" Lower Prices? 2

industry has been proposed as a solution to make the present utility companies more com-

petitive in oering better services and better quality at lower prices to their customers.

However, as in any business, this may require some investments in the infrastructure of the

power system, to cope with the new demands of the modern types of loads (power electron-

ics and microcomputer based). In this scenario, traditional economic analysis, such as pay

back return or rate of interest, might show that these investments are only feasible with

concurrent increases in electricity taris. The paradox of more quality for less money still

remains a topic for discussion in forums such as government regulatory agencies.

With the growing utilization of automation and control based on the use of microproces-

sors, of power electronic devices, and of modern manufacturing techniques, industries have

been able to produce goods faster and with increasing quality. However, with such modern-

ization new issues have emerged regarding the quality of electricity. Sensitive loads tend to

shut down if there are small variations in the network voltage. Also, harmonic distortions

caused by nonlinear loads may result in wrong operation or may increase the losses in power

system components. Another problem is capacitor switching in the utility system, which

may cause problems for adjustable speed drives (ASD's), which are used more and more by

industry. All these problems point out that more attention must be paid to power qual-

ity problems. Economic losses expressed in terms of interrupted production, of damage to

equipment, and of time delays in the processing of goods and the consequent negative impact

on customers have caused a rising number of complaints about power quality problems in

many electric utility companies. Estimating the cost of poor power quality is a diÆcult task.

Nevertheless, the annual approximate value would be in the order of hundreds of million

of dollars in damage. As an example, the cost per year to U. S. A. industry in lost time

and revenue due to power related problems were estimated in 1993 as US$26,000,000,000 [3].

The costs tend to grow as the sensitivity and use of microprocessor-based devices tend to

increase. The Electric Power Research Institute (EPRI) stated that in the year 2000, 60%

to 70% of total utility power generated within the U. S. A. would be controlled by power

electronics, compared to 30% in 1995.

Power electronic devices, however, are also able to \guarantee" a certain expected level

of electricity quality to a sensitive or special load, and such devices exist today. Flexible AC


Transmission Systems (FACTS) technology, Custom Power Controllers, active lters, among

other power electronics applications, oer promising solutions for improving the quality of

power in transmission and distribution systems.

The introduction of more and more power electronic devices into the network will create

issues of compatibility of operation not only in steady state, but also under transient con-

ditions. New models for innovative equipment, as well as new philosophies for their control

and operation, will then be required. It is not enough to evaluate the electric quality condi-

tions only at the interface of power electronic systems with the electric power system. The

propagation of electromagnetic phenomena into the industrial or utility network must be

evaluated as well [4]. Therefore, software packages such as the ElectroMagnetic Transients

Program (EMTP), have become important and necessary tools to analyze the impact of large

power electronic devices on the quality of electric power. The aim of this research project

was the development of new EMTP-based models for control and power electronic devices,

thus allowing the accurate evaluation of the impact of high power electronics on the quality

of power. As part of the project, eld tests were conducted in cooperation with a Brazilian

utility company, to provide realistic power quality data measurements.

1.1.1 Computer Analysis and Simulation of Electric Power Qual-

ity Phenomena

Digital computer simulation of electromagnetic transients in single- and multiphase net-

works is well established [1]. Since the publication of [1] in 1969, signicant improvements

have been made in models for electrical power system components such as transmission lines

[5], transformers, turbine-generators and cables [6]. Of particular interest for this research

project are the necessary advances in the simulation of power electronic devices within power

systems.

Although steady-state solutions at fundamental and harmonic frequencies have been pro-

posed to analyze power quality problems, the complexity of periodic switching in power elec-

tronic devices can only be studied thoroughly through time-domain simulations, e.g., with

EMTP-based programs [4], [7], [8], [9], [10], [11], [12], [13].


EMTP-type simulation is particularly useful for the analysis of the dynamic interaction

of distributed Custom Power Controllers in a power system. Despite international demon-

stration projects for some Custom Power Controllers already with some years of operating

experience, how these new devices will interact within each particular power system is still an

open question. The performance obtained from some prototypes, or from simulations with

simplied models, may not be suÆcient for real applications, where dangerous resonance

and other unforeseen problems may occur. These concerns make more realistic electromag-

netic transient program based simulations important when detailed models of Custom Power

Controllers become available. This could avoid more expensive corrective actions after in-

stallation. Measurements and simulations also become necessary for performance evaluations

under dierent network and load conditions.

All those facts have encouraged the development of new practical models, methods and

guidelines for the appropriate use of EMTP-based programs as potential tools for power

quality studies.

Nevertheless, good engineering judgment in setting up the power quality problem and

representing the physical phenomena with appropriate models, does still represent the major

challenge, despite the impressive accuracy obtained with the models available today.

1.1.2 Electric Power Quality Monitoring

In recent years, tremendous improvements have also been made in digital measuring in-

struments, because of the use of microprocessors and digital signal processing techniques

[14], [15]. This made it possible to conduct power quality surveys in many countries around

the world. The Electric Power Research Institute (EPRI) commissioned an extensive survey

of distribution system power quality in the U. S. A. [16]. In Canada, the Canadian Elec-

tricity Association (CEA) developed a guideline for the power quality that utility customers

experience, in a three-year project [17]. A quantitative measure of the deterioration of the

ideal sinusoidal waveform from the growing utilization of power electronic devices resulted

from these surveys. In many of the real-world problems, momentary voltage variations have

been the main cause for shutting down microprocessor controlled industrial processes. The


diversity of problems to cope with, and their complex interaction, has created a need for

more research on power quality issues [18]. Moreover, with the present deregulation process

in the electricity industry, power quality has become a key factor for utilities and customers

in a competitive market.

The experience in monitoring power quality phenomena has increased in the latest years,

as reported by the related literature [15], [19], [20], [21], [22], [23], [24], [25], [26], [27]. Well

known power quality problems have been summarized and at the same time new problems

have been discussed in [28], [29], [30], [31], [32], [33], [34], [35], [3], [36], [37]. The need

for more detailed information on power disturbances, and the search for new techniques to

process the amount of available measured data, have motivated research into applications of

modern theories in the power quality area, as discussed in [38], [39], [40], [41], [42].

According to the author's experience as a \power quality engineer" in a Brazilian utility

company, there are many dierent reasons why electricity customers become dissatised with

the quality of the electric power delivered by utilities, and why they complain. This happen

partly because there may indeed be technical problems. Partly, the motivation comes from

the need to reduce electricity costs in the industrial production process, looking for better

taris and better contracts, as everybody tries to survive in a competitive and aggressive

business environment.

On the technical side, the most common power quality problems are caused by a fatal

combination of sensitive electronic-based loads and a high incidence of voltage sag phenom-

ena [43], [44], [45], [46]. Most of these voltage sags are due to faults in transmission and

distribution systems, caused by lightning phenomena, which cannot be easily avoided or

minimized. It is also common to nd poor voltage regulation within the industry electric

system [45], [46]. This aggravates the impact of voltage sags, causing frequent process mal-

function or interruption with nancial losses, which are rarely presented explicitly by the

industry personnel, unless any kind of nancial compensation is legally required. There is a

wide range of alternatives for possible solutions to technical problems in the quality of the

electric power supply. Usually, the immediate most cost-eective measure is to minimize the

cause or eects of the problem close to its origin, depending on the type of electromagnetic


phenomena involved. Typically, voltage sag problems can be minimized by proper adjust-

ments in the sensitivities of the load or load control, whenever this is technically feasible.

However, in some cases compensation through the use of power electronic devices might be

a promising alternative, either for an individual sensitive load or for an entire industrial pro-

cess. Usually, the utilities comply with the standards of supply set by the regulatory agency,

but the customer process is much more sensitive and some kind of electronic compensation

would be necessary. The potential con ict is sometimes created when a possible technical

solution requires high nancial investments. A cost versus benet analysis usually leads to

a cheap compromise solution; in the absence of clear regulations one needs \to live with the

problem!"

1.1.3 Power Quality Standards

Power quality has become an important issue because of the increasing use of power

electronic devices. The Institute of Electrical and Electronics Engineers, Inc. (IEEE) has

therefore developed standards to address power quality problems, which are brie y discussed

here.

The problems related to the quality of electricity are not new, since there was never an

ideal sinusoidal waveshape, with frequency and voltage exactly at their rated values. How-

ever, with the changes in the type and sensitivity of the loads in recent years, harmonic

current and voltage distortions, short- and long-duration voltage variations, impulse and os-

cillatory transients, voltage uctuations (causing visual icker), power frequency deviations,

voltage unbalance among the phases in a three-phase system, and other electromagnetic

phenomena are increasingly causing power quality problems.

IEEE Std. 1159-95 [47] denes and characterizes electromagnetic phenomena which may

cause power quality problems. It also provides recommended practices for monitoring electric

power quality.

Most utility regulations dealing with harmonics are based on IEEE Std 519-1992, [48],

[49], [50]. This standard presents recommended practices and requirements for harmonic

control in electric power systems. It addresses most of the issues of harmonic generation


by power electronic converters, arc furnaces, static VAR compensators and power electronic

controlled drives. It also discusses:

system response characteristics;

eects of harmonics;

reactive power compensation and control;

harmonic analysis methods;

harmonic measurements;

recommended practices and harmonic limits for individual customers;

recommended practices and harmonic limits for utilities;

methodology for evaluating new harmonic sources.

In the later task, time-domain simulation can be particularly useful to predict equipment

and power system behaviour. It thus can help engineers to provide some answers in detecting

harmonic related or other power quality problems. The IEEE Std. 519-1992 [48] is currently

been revised to account for interharmonics in power systems and the possible application of

probabilistic approaches in harmonics evaluation.

IEEE Std. 446-1987 [51] covers the recommended practice for emergency and standby

power systems for industrial and commercial applications. A computer \voltage tolerance en-

velope", shown in Fig. 1.1, also known as the CBEMA curve (Computer Business Equipment

Manufacturing Association curve) is presented in this standard.

The Information Technology Industry Council (ITIC) revised the CBEMA curve, which is

presented in Fig. 1.2. It shows that computer- and power electronics-based loads, properly

designed by the manufacturers, should be able to withstand a complete interruption of

voltage supply for up to 20ms, a voltage sag of 30 percent for 0:5s, 20 percent for 10s or

10 percent in steady state. It also denes the upper limits in the input voltage that should

be tolerated. The CBEMA (ITIC) curve has been widely used as an important \reference"


0.001 0.01 0.1 1.0 10.0 100.0 1000.00%

100%

200%

300%

400%

0.5

30%

115%

87%

106%

2sTIME IN CYCLES [60Hz]

PE

RC

EN

T V

OLT

AG

E

VOLTAGEBREAKDOWN CONCERN

COMPUTER VOLTAGETOLERANCE ENVELOPE

LACK OF STORED ENERGY INSOME MANUFACTURERS'EQUIPMENT

Figure 1.1: Typical Design Goals of Power-Conscious Computer Manufacturers. (Source: IEEE Std.446-1987, \IEEE Recommended Practice for Emergency and Standby Power Systems for Industrial and

Commercial Applications.")

for the susceptibility level of computer- and power electronics-based loads. However, due

to the great variety of products and processes, and their response to transient variations in

the supply voltage, there are cases where the load sensitivity is much more strict than the

CBEMA (ITIC) curve, which has to be determined then case-by-case for an adequate power

quality assessment and proposal of solutions.

IEEE Std 1100-1992 [52] presents the recommended practice for powering and grounding

sensitive electronic equipment. It addresses the multidisciplinary area of power quality,

giving practical guidelines on load and source compatibility concerns.

Voltage uctuations causing visual icker are being studied by the Task Force IEEE

P1453 on Light Flicker, which is considering the adoption of existing standards and practices


10−4

10−3

10−2

10−1

100

101

102

103

104

0

100

200

300

400

500

600

Voltage ToleranceEnvelope

Applicable to Single−Phase120−V Equipment

1us 1ms 3ms 20ms 0.5s 10s SteadyState

11090

40

7080

120140

Duration in Cycles (c) and in seconds (s)

Per

cent

of N

omin

al V

olta

ge (

RM

S o

r P

eak

Equ

ival

ent)

Figure 1.2: CBEMA curve revised by the Information Technology Industry Council (ITIC).

of the IEC (International Electrotechnical Commission) and UIE (International Union for

Electroheat) for measuring such types of disturbances. This task force is also reviewing other

IEEE standards and recommendations on this issue. Other IEEE Standards within the IEEE

Color Series Books (http://www.ieee.org) provide useful recommendations about complex

issues on topics associated with the quality of power in utility, industrial and commercial

installations.

1.1.4 Custom Power Related Publications

This section presents a collection of publications related to Custom Power technology

for the improvement of the quality of power. Some of the papers present actual application

examples.


High voltage direct current (HVDC) and exible AC transmission systems (FACTS tech-

nology) have been used for some time to extend power transfer capability, to improve power

system stability, and for other reasons. Dr. Narain G. Hingorani introduced the acronym

FACTS (Flexible AC Transmission System) for high power electronics applications in trans-

mission systems [53], [54], [55]. HVDC, static Var compensator (SVC), thyristor controlled

series compensations (TCSC), static synchronous compensator (STATCOM), static syn-

chronous series compensation (SSSC) and unied power ow controller (UPFC) are exam-

ples of the so called FACTS devices. Reference [56] provides an annotated bibliography of

HVDC and FACTS devices. It also includes a list of FACTS installations, with data on

manufacturers, utility companies, countries, etc. It shows that, despite the high costs of

these high power electronic devices, they are gaining in acceptability around the world.

The term \Custom Power" was also introduced by Hingorani, to represent power elec-

tronics applications designed to mitigate power quality problems in industrial and distribu-

tion systems [57], [58], [59]. The distribution static condenser (D-STATCOM), the voltage

sag compensator (also known as DVR - dynamic voltage restorer), the solid-state breaker

(SSB), the solid-state transfer switch (SSTS), among others, are examples of such Custom

Power Controllers. Various manufacturers have proposed shunt, series, or shunt/series dy-

namic compensation schemes, with dierent acronyms, as solutions to specic power quality

problems. \The D-STATCOM, although based on the STATCOM, has a wider range of

applications. In fact, the D-STATCOM can be designed for reactive power control, or for

voltage control of the fundamental frequency, but it may also include higher frequencies as

in shunt active power lters." The integration of series- and shunt active lters, referred

to as unied power quality conditioner (UPQC) [60], [61], is promising to be the denite

solution for the majority of power quality problems. \However, its high cost may make it

useful only in some special cases. On the other hand, the shunt or series devices such as the

D-STATCOM or the voltage sag compensator will probably play a signicant role in future

distribution systems" 1.

The ongoing deregulation process in many countries is also fostering competition in the

1From personal communication with Dr.-Ing. Maurcio Aredes, COPPE/UFRJ, Rio de Janeiro, RJ,Brazil.


electric power industry, which accelerates the application of new technologies in the trans-

mission and distribution system. For example, there are applications being developed for

superconducting magnetic energy storage devices (SMES) for low voltage distribution sys-

tems, which will provide voltage support for a few seconds to sensitive processing equipment

during times of voltage sags.

Case studies with practical applications of Custom Power Controllers can also be down-

loaded directly from the web sites of some manufacturers, as for example:

http://www.siemenstd.com/prods/FPQD/dvr.html - case studies for the voltage sag

compensator DVR (dynamic voltage restorer of Siemens);

http://www.siemenstd.com/prods/FPQD/cp.html - general power quality information

on DVR, D-STATCOM, Solid State Breaker, Transfer Switch and Premium Power

Park;

http://sac.sandc.com/products/purewave/ups pubs.asp - for S & C UPS products;

http://www.softswitching.com - for SoftSwitching Technologies products;

and many others.

Figs. 1.3 (a) and (b) present semiconductor power devices, thyristors, used in an industrial

power electronic converter and in a HVDC system, respectively. Thyristors are considered

the \backbone" of the high power electronics revolution. Other types of semiconductors

being used are the gate turn-o thyristor (GTO), the MOS controlled thyristor (MCT), the

static induction thyristor (SITh), and the insulated gate bipolar transistor (IGBT). The so

considered, in 1997, state of the art of these devices can be found in reference [62], along with

a description of the main characteristics of HVDC, static Var compensator (SVC), thyristor

controlled series compensation (TCSC), static synchronous compensator (STATCOM), static

synchronous series compensation (SSSC) and unied power ow controller (UPFC). The

benets of the application of FACTS technology in a power system depend on the reliability

of the specic FACTS device, which in turn depends on the reliability of the semiconductor

1.2. Motivation for Thesis Research 12

(a) (b)

Figure 1.3: (a) Thyristor in an industrial power converter. (b) Thyristors in a high voltage direct current(HVDC) System.

devices used. Although the semiconductor power devices act as switches, they are not ideal

switches and many physical limitations do apply.

1.2 Motivation for Thesis Research

The motivations for this thesis research are summarized as follows:

EMTP-based simulations can handle the complexity of electromagnetic phenomena

needed for power quality analysis, once the appropriate models and methods are de-

veloped or improved. This research project evaluated the available EMTP models

for power quality studies, and developed new models where the existing ones needed

improvements, especially for the simulation of power electronics-based devices;

To analyze the in uence and interaction of dierent new power electronic devices on

the quality of power is important for electric utilities and their customers. Moreover,

with a worldwide deregulation process in the electricity industry, power quality analysis

1.3. Contributions of this Research Project 13

will rise in importance and urgency. Therefore, as part of this project eld tests were

developed in a Brazilian electric utility company, where realistic power quality cases

were analyzed and simulations were performed;

The diversity of power quality phenomena requires an interdisciplinary approach and

specialized engineering skills. With opportunities available for interaction with other

researchers at The University of British Columbia (UBC), appropriate courses were

attended, especially in the power electronics area, which was helpful for the under-

standing and development of new models for implementation in MicroTran, the UBC

version of the EMTP;

The opportunity to conduct practical eld tests in cooperation with an electric utility

company was a valuable experience, and necessary for the validation of digital computer

models.

1.3 Contributions of this Research Project

This Ph.D. thesis oers new models for the digital computer simulation of control and

power electronic devices. These models were developed for implementation in EMTP-based

programs or in similar programs. An innovative \circuit approach" was developed for the

simultaneous solution of control and power systems equations, as an alternative to the ap-

proach of A. E. A. Araujo [63] developed in 1993. The main dierences and important

advantages are summarized as follows:

With the addition of ideal operational ampliers, transfer functions can be imple-

mented with a \circuit approach", where the circuit elements R, L, C are solved by

the main code of the EMTP. If integration methods are changed in the EMTP, for

example from trapezoidal rule to backward Euler as done in some versions at instants

of discontinuities with the CDA technique, no extra coding is needed. Operational

ampliers are not aected by integration rule changes. Moreover, if ideal operational

ampliers are implemented in steady-state solution, the frequency response of linear

1.3. Contributions of this Research Project 14

control systems could be easily calculated in EMTP-based programs by just using the

frequency scan option.

A. E. A. Araujo [63] uses FORTRAN-like statements for control functions such as

Y = COS(X) in the input, which are then handled with a FORTRAN interpreter. In

this thesis, this function and similar functions are pre-dened control block types.

The \multi-terminal voltage-controlled voltage source concept" implemented in this

thesis with the compensation method and the Newton-Raphson iterative algorithm is

\general and exible", thus providing an easy EMTP-based modelling of any linear or

nonlinear control device. This is very useful for the dynamic analysis of novel power

electronic controllers, such as distributed FACTS and Custom Power Controllers in

transmission and distribution power systems.

This Ph.D. thesis is organized as follows: Chapter 2 presents the simultaneous solution

method for control and electric power system equations (SSCPS) in EMTP-based programs.

Chapter 3 discusses the developments made for power electronics models in EMTP-based

simulations. Chapter 4 presents simulation cases of power quality assessment with the use

of the existing features of MicroTran, the UBC version of the EMTP. SSCPS simulation

cases with the new models of Chapter 2 and the developments for the dynamic control of

power semiconductors presented in Chapter 3 are illustrated in practical power electronics

controllers. Simulation guidelines for the evaluation of the impact of power electronic devices

on the quality of power are summarized in Chapter 4. Finally, Chapter 5 presents the main

conclusions and contributions made in this Ph.D. thesis, and also points out the author's

recommendations for future work.

Chapter 2

Simultaneous Solution of Control andElectric Power System Equations(SSCPS) in EMTP-based Programs

2.1 Previous Developments on Transient Analysis of

Control Systems (TACS)

The computer subroutine TACS (acronym for \Transient Analysis of Control Systems")

was developed in 1977 [64] for the simulation of control systems in the EMTP (acronym

for \Electromagnetic Transients Program"). The general philosophy of the solution method

adopted at that time required a one-time-step delay at the interface between TACS and the

electric network solution 1. This non-simultaneous approach was probably used because it

was easier to write a code separated from the main program, with a simple interface. The

main program passed information to the TACS program, which then returned information

to the main program for use one time step later, as illustrated in Fig. 2.1. Moreover, control

system equation matrices in TACS are usually unsymmetric, whereas the network elements

in the EMTP result in symmetric matrices. By separating the solution into two parts, the

code for symmetric matrices in the EMTP could be maintained.

The solution in two parts, with a time delay of one t between them, was an expedient

way to implement control system equations, but it proved to be the cause of critical numerical

1Many other software programs, such as PSIM [65], also require a one-time-step delay between the solutionof control and power systems equations, which makes the solution method non-simultaneous.

15

2.1. Previous Developments on Transient Analysis of Control Systems (TACS) 16

Electric Network Solution( EMTP )

Control System Solution( TACS )

Time Delay1 ∆t

Figure 2.1: EMTP and TACS interface with 1 time step delay.

instabilities and inaccuracies in some cases in the time domain simulation of electric and

power electronic system transients [66], [67], [68], [63]. In cases where the EMTP and TACS

elements form a closed loop (or feedback system according to control theory), the eect of

the interface delay cannot always be eliminated by using a small step size t, as stated in

[69].

Besides the time delay between TACS and EMTP, even more time-step delays were

introduced by the internal solution algorithm of TACS, in order to deal with nonlinearities

in feedback control loops. The TACS internal solution is therefore non-simultaneous for

some control cases, and also sequential for its implemented devices. Improvements have

been made through the years in the TACS subroutine of some versions of the EMTP, such

as better ordering of its variables to minimize the number of delays inside TACS [70], using

the compensation method to eliminate the one-time-step delay in the EMTP-TACS interface

[71], development of a new TACS program \MODELS" [72] and its possible applications for

simultaneous solution of power electronics systems equations [73].

In 1993 A.E.A. Araujo proposed a simultaneous solution of both sets of equations, electric

network equations and control systems equations, as a way to eliminate the one-time-step

2.1. Previous Developments on Transient Analysis of Control Systems (TACS) 17

delay problem at the interface, as well as the internal control delays [67], [68], [63]. The aug-

mented matrix with the control equations becomes unsymmetric due to the structure of the

control equations. Most of the equations of both the electric network and the control systems

are usually linear, while some are nonlinear. A proper partition of the system of equations

would allow the solution to be separated into two subsystems, one linear and another non-

linear. A.E.A. Araujo chose to solve the system of linear equations inside the EMTP, and

the system of nonlinear equations (including nonlinearities from the electric network and

from the control system) with the compensation method in an iterative Newton-Raphson

algorithm as in [74]. The control equations, both linear and nonlinear, were developed inside

the subroutine \CONNEC", which is a user-dened subroutine in the MicroTran version of

the EMTP of the University of British Columbia. \Similarly to TACS, the trapezoidal rule

of integration was used to numerically integrate the rst-order dierential equations inside

CONNEC, for example, in the implementation of transfer functions. The code was written

to prove the ideas, but as far as the author knows, was not implemented in a production

version of the EMTP."

In this research project, the simultaneous solution of the electric network and control

equations in EMTP-based programs is achieved with a \circuit implementation" of the con-

trol system. With this novel approach for EMTP-based programs, elements of the control

circuit which already exist in the EMTP, such as resistances and capacitances, are solved by

the EMTP proper, while elements missing inside the EMTP, such as ideal operational am-

pliers 2 and current and voltage dependent sources, are solved in the subroutine CONNEC

with the compensation method. This circuit approach is an alternative to the mathematical

representation adopted by Araujo, and gives some important advantages, such as \generality

and exibility" for control modelling in EMTP-based programs.

The compensation method with an iterative Newton-Raphson procedure is used for the

solution of the added linear and nonlinear control system elements, such as dependent

sources, dierent types of limiters, as well as intrinsic FORTRAN functions and some special

control devices, as explained in the following sections. \Among the added elements, the de-

2The author acknowledges the help of Mr. Jesus Calvi~no-Fraga for indicating in 1998 in his technicalreport for a graduate course, the need for modeling operational ampliers in MicroTran [75].

2.2. Current and Voltage Dependent Sources in EMTP-based Programs 18

pendent sources are the most important ones for control system modelling." The FORTRAN

code for the added elements in subroutine CONNEC has approximately 5,000 lines of code,

compared to 15,000 lines of code in the main part of the MicroTran version of the EMTP.

2.2 Current and Voltage Dependent Sources in EMTP-

based Programs

Since the publication of [1] describing the rst version of the EMTP, many others have

contributed to the development of models as documented in [2] and elsewhere. As far as

the author knows, dependent sources of all possible types have not been implemented in

any EMTP-based program. Dependent sources expand the capabilities of EMTP-based

programs considerably for modelling many electric and electronic circuits and devices. With

a voltage-controlled voltage source, for example, it becomes easy to simulate operational

ampliers. These can then be used to set up control circuits with analog-computer block-

diagrams. As long as the equations of the dependent sources are linear, they could be added

directly to the network equations used in EMTP-based programs (with the modied nodal

analysis (MNA) presented in [76] and [77], but the matrix would then become unsymmetric

and a linear equation solver for unsymmetric matrices would have to be used. Another

alternative discussed here in more detail is based on the compensation method, which can

also handle nonlinear eects with a Newton-Raphson algorithm. Nonlinear eects arise with

the inclusion of saturation or limits in the dependent sources. The main motivation for the

use of the compensation method is its \generality and exibility" in modelling linear and

nonlinear devices in EMTP-based programs. This section provides then the fundamental

equations for the implementation of dependent sources in EMTP-based programs, as well as

of independent sources, which can also be connected between two ungrounded nodes.

2.2.1 Compensation Method

The compensation method has long been used in EMTP-based programs for solving

the equations of nonlinear elements with the Newton-Raphson iterative method [74]. If

the nonlinear elements are not too numerous, this approach connes the iterations to a


relatively small system of equations, compared to the nodal equations for the entire system.

This approach is used here for solving the equations of dependent sources as a special case

of nonlinear elements. Without limiters, the equations are linear, but with an unsymmetric

matrix.

When there are M nonlinear elements in a circuit, the following system of equations 2.1

to 2.6, allows the simultaneous solution of the nonlinear equations with the rest of the linear

network [2],[78], which is then represented by its M-phase Thevenin equivalent circuit, as

illustrated in Fig. 2.2:

vM

vs(M)

ZM

[ rTHEV ][ vOPEN ] [ i ]

vs(4)

Z4

vs(3)

Z3

vs(2)

Z2

vs(1)

Z1

[ v ]

... ...

...

v4 v3 v2 v1

i1i2i3i4

iM

vOPEN _ MvOPEN _ 1

Figure 2.2: M-phase Thevenin equivalent circuit.

[vOPEN ] + [rTHEV ] [i] + [v] = 0 (2.1)

where:

[vOPEN ] =

26664

vOPEN1

vOPEN2

...vOPENM

37775 (2.2)


[rTHEV ] =

26664

r11 r12 r1Mr21 r22 r2M...

.... . .

...rM1 rM2 rMM

37775 (2.3)

[i] =

26664

i1i2...iM

37775 (2.4)

[v] =

26664

v1v2...vM

37775 (2.5)

Equations 2.6 are the branch equations of the nonlinear elements:

vk = fk ([v] ; [i] ; t; etc::::) k = 1; :::M (2.6)

If the branch equations in 2.6 are linear, as in the case of dependent sources, they can be

represented in the form of a voltage source behind an impedance, as illustrated in Fig. 2.3,

or in the form of a current source in parallel with an impedance, as shown in Fig. 2.4. It is

assumed here that the branch impedances are not coupled, and that they are resistive (Rk).

For other types of impedances, the equations would have to be modied.

vk

ik

vsource(k)

Rk

ck

dk

Figure 2.3: Representation of branch equation k as a voltage source in series with a resistance.


vk

ik

Rk

ck

dk

isource(k)

isource(k) = vsource(k) / Rk

Figure 2.4: Representation of branch equation k as a current source in parallel with a resistance.

After the two systems of equations 2.1 and 2.6 have been solved in subroutine CONNEC,

the currents [i] of 2.4 are returned to the main program, which adds the eect of the M non-

linear branches to the previously calculated open-circuit solution for all nodes with unknown

voltages,

[e] = [eOPEN ] [zT ] [i] (2.7)

where:

[e] is a column vector with the nal solution for the N node voltages;

[eOPEN ] is a column vector with the previously calculated open circuit solution for all the N

nodes with unknown voltages;

[zT ] is a rectangular matrix with N rows and M columns (N = number of nodes with

unknown voltages and M = number of branches solved with the compensation method) 3;

[i] = column vector with the M compensating branch currents.

2.2.2 Dependent Sources

This section presents the necessary equations for implementing current and voltage de-

pendent sources in EMTP-based programs by using the compensation method. The following

important assumptions are made:

3For further details about the compensation method, and the calculation of matrix [zT ], please, seereference [74].


A Thevenin equivalent circuit can be calculated where the dependent source is to

be connected, and also where the controlling current or controlling voltage is to be

measured. In cases where this calculation fails, the connection of large resistors in

parallel may make a Thevenin equivalent circuit possible.

Proper precautions are taken to handle extremely large numbers and zero values.

The following models are derived: Current-Controlled Voltage Source (CCVS), Current-

Controlled Current Source (CCCS), Voltage-Controlled Voltage Source (VCVS) and Voltage-

Controlled Current Source (VCCS). In all cases, the equations from the Thevenin equivalent

circuit are the same, namely, for the controlling branch j

vOPENj + rj1i1 + ::::::+ rjjij + rjkik + ::: + rjM iM + vj = 0

(2.8)

and for the dependent source branch k

vOPENk + rk1i1 + ::::::+ rkjij + rkkik + ::: + rkM iM + vk = 0

(2.9)

where:

vOPENk = voltage vk for [i] = 0 (open circuit).

rkk = Thevenin resistance (self resistance of branch k).

rkj = Thevenin resistance (coupling or mutual resistance between branches k and j).

Current-Controlled Voltage Source (CCVS)

Assume that the controlling current is measured through a branch between nodes a and

b in a circuit, such that vj is its branch voltage and ij is its branch current, i.e.,

vj = va vb (2.10)

ij = iab (2.11)

and that the dependent source, CCVS, is connected between nodes c and d with branch

voltage

vk = vc vd (2.12)


and branch current

ik = icd (2.13)

The necessary equations for the implementation of a current-controlled voltage source as

illustrated in Fig. 2.5 are 2.8 and 2.9, as well as:

vj

Rin[ rTHEV j ]

ij

vOPEN jvk

Ω ij

Rout [ rTHEV k ]

ik

vOPEN k

Figure 2.5: Current-controlled voltage source (CCVS).

vj = Rinij (2.14)

vk = ij +Routik (2.15)

where:

Rin = Input resistance of branch j.

Rout = Output resistance of the dependent source in branch k.

= Gain over the controlling or measured current, applied as voltage dependent source at

branch k.

Inserting equation 2.14 into 2.8 and equation 2.15 into 2.9, results in:

vOPENj + rj1i1 + ::::::+ (rjj +Rin) ij + rjkik + :::+ rjM iM = 0

(2.16)

vOPENk + rk1i1 + :::::: + (rkj + ) ij + (rkk +Rout) ik + ::: + rkM iM = 0

(2.17)


Using the two equations 2.16 and 2.17 is preferable to using the four equations 2.8,

2.9, 2.14 and 2.15, because it reduces the number of equations which have to be solved in

subroutine CONNEC from 4 to 2. Whenever possible, the voltages should be eliminated in

this reduction from 4 to 2 equations, because the solution will then produce the currents,

which are the variables that have to be passed back to the main program.

For an ideal current-controlled voltage source, Rin = 0 and Rout = 0, from which results:

vOPENj + rj1i1 + ::::::+ rjjij + rjkik + ::: + rjM iM = 0

(2.18)

vOPENk + rk1i1 + ::::::+ (rkj + ) ij + rkkik + :::+ rkM iM = 0

(2.19)

If expressed in matrix form, one can see that the matrix becomes unsymmetric, since matrix

element j k is no longer equal to matrix element k j.

Current-Controlled Current Source (CCCS)

The necessary equations for the implementation of a current-controlled current source as


vj

Rin[ rTHEV j ]

ij

vOPEN jvk Β ij Rout

[ rTHEV k ]

ik

vOPEN k

Figure 2.6: Current-controlled current source (CCCS).

vj = Rinij (2.20)

vk = RoutBij +Routik (2.21)


where:

B = Gain over the controlling or measured current, applied as dependent current source at

branch k.

By inserting equation 2.20 into 2.8 and equation 2.21 into 2.9 one can also obtain, re-

spectively, the following equations:

vOPENj + rj1i1 + ::::::+ (rjj +Rin) ij + rjkik + :::+ rjM iM = 0

(2.22)

vOPENkRout

+ rk1Rout

i1 + :::

::: +

rkjRout

+ Bij +

rkkRout

+ 1ik + :::+ rkM

RoutiM = 0

(2.23)

Observe that the division by Rout as done in equation 2.23, allows the use of very large

numbers for Rout without numerical diÆculties.

For an ideal current-controlled current source, Rin = 0 and Rout !1, resulting in:


(2.24)

Bij + ik = 0 (2.25)

Voltage-Controlled Voltage Source (VCVS)

The necessary equations for the implementation of a voltage-controlled voltage source as


vj

Rin[ rTHEV j ]

ij

vOPEN jvk

Α vj

Rout [ rTHEV k ]

ik

vOPEN k

Figure 2.7: Voltage-controlled voltage source (VCVS).


vj = Rinij (2.26)

vk = Avj +Routik = ARinij +Routik (2.27)

where:

A = Gain over the controlling or measured voltage, applied as dependent voltage source at

branch k.

By inserting equation 2.26 into 2.8 and dividing the resulting equation by Rin to avoid

numerical diÆculties, results in equation 2.28. In order to eliminate the voltages and keep

only the currents as variables, and also to allow the use of very large numbers for the gain

A, the following calculations are done: (equation 2.26 inserted into 2.8) minus the result of

[(equation 2.27 inserted into 2.9) and divided by the gain A]. This procedure eliminates Rin

in the resulting equation 2.29:

vOPENjRin

+rj1Rin

i1 + :::

::: +rjj+RinRin

ij +

rjkRin

ik + ::: +rjMRin

iM = 0(2.28)

vOPENj +vOPENk

A+rj1

rk1A

i1 + :::

::: +rjj

rkjA

ij +

rjk

rkk+RoutA

ik + :::

::: +rjM rkM

A

iM = 0

(2.29)

Based on the equations 2.28 and 2.29 for a voltage-controlled voltage source, if

A!1,

Rin !1, and

Rout ! 0,

then equations 2.30 and 2.31 are obtained, which can be used to model \ideal operational

ampliers". Note that the use of equation 2.31 only makes sense if there are feedback paths

modelled in the network part, which create the \rjk" coupling resistance. Then equation 2.31


will produce the correct current ik, which is returned to the main program for the calculation

of voltages by compensation. Note also that the equation 2.31 is exactly stating that vj = 0.

(Please, see equation 2.8.)

ij = 0 (2.30)


(2.31)

Ideal Operational Ampliers

The commercially available operational amplier is in reality an integrated-circuit chip,

constructed essentially with many transistors and resistors in an integrated package. Oper-

ational ampliers, often called OP AMPS, are frequently used in sensor circuits to amplify

signals, in active ltering and control circuits for compensation purposes and endless ap-

plications in analog electronics [79], [80], [81], [82]. Fig. 2.8 presents the symbol used for

representation of an operational amplier. The voltage placed across the two input termi-

Figure 2.8: Symbol for operational amplier.

nals (the non-inverting terminal (+) and the inverting terminal()), is to be amplied and

to appear at the output terminals (one of which is grounded, but this grounding is usually

omitted on the symbol). Since the gain of the operational amplier is very high, it is neces-

sary to have an external feedback circuit to make it stable. \In practice the input resistance


Rin of an OP AMP is usually well in excess of 1M, the voltage gain A is at least 105, and

the output resistance Rout is a few tens of ohms " [79], and then it can usually be modelled

as a voltage-controlled voltage source (VCVS). Many other electrical properties, which are

temperature and frequency dependent, have to be considered though in realistic applications.

In the ideal operational amplier, no current would ow into the input terminals (Rin =

1 as in an open circuit), the output voltage would not be aected by the load connected

to the output terminal (Rout = 0), and the gain would be innite (A = 1 so that the

voltage at the non-inverting input terminal would be equal to the voltage at the inverting

input terminal). Therefore, the fundamental concepts for the analysis of circuits with ideal

operational ampliers are to assume that the two input terminals of the ideal operational

amplier constitute \at the same time" [77]:

\an open circuit" (equation 2.30), AND

\a virtual short-circuit" (equation 2.31).

\In this thesis, if not otherwise clearly indicated, the assumption is made that all opera-

tional ampliers are ideal"!.

There are many variations and combinations of OP AMP circuits. The two basic ones

are the inverting amplier (Fig. 2.9) and the non-inverting amplier circuit (Fig. 2.10), with

the transfer functions are given by equations 2.32 and 2.33 respectively. Fig. 2.11 illustrates

an adder, a special case of the inverting amplier, where the output is a linear sum of the

input voltages, with the transfer function given by equation 2.34. Fig. 2.12 shows an ideal

integrator, with the transfer function as of equation 2.35.

Eo(s)Ei(s)

= R2R1 (2.32)

Eo(s)Ei(s)

=1 + R2

R1

(2.33)

Eo(s)R4

= Ei1(s)R1

+ Ei2(s)R2

+ Ei3(s)R3

(2.34)


ei ( t )R1

R2

eo ( t )

Figure 2.9: Inverting amplier circuit.

ei ( t )eo ( t )

R1

R2

Figure 2.10: Non-inverting amplier circuit.

ei1 ( t )

R2

R4

eo ( t )

R1

R3

ei2 ( t )ei3 ( t )

Figure 2.11: Adder circuit with operational amplier.

Eo(s)Ei(s)

= 1RCs (2.35)

Fig. 2.13 presents a generalization of the inverting amplier circuit, which is very useful

to obtain Laplace transfer functions by using the impedance approach [81]. With the ideal

operational amplier, a \virtual ground" potential appears at the inverting input terminal,


ei ( t )

R

eo ( t )

C

Figure 2.12: Ideal integrator circuit with operational amplier.

since the non-inverting input terminal is grounded. Moreover, no current ows into the

input terminals of the ideal OP AMP. Therefore, the same current owing through the

complex impedance Z1(s) has to ow through the complex impedance Z2(s), resulting in

Ei(s) = Z1(s)I(s) and Eo(s) = Z2(s)I(s). The transfer function for this generalized

inverter circuit is given by equation 2.36.

Ei ( s ) Eo ( s )Z1(s)

Z2(s)I1 ( s )

I2 ( s )

Figure 2.13: Generalization of inverter amplier circuit.

Eo(s)Ei(s)

= Z2(s)Z1(s)

: (2.36)

For example, in the circuit shown in Fig. 2.14, the transfer function is derived with ideal

operational amplier using the impedance approach.

The complex impedances Z1(s) and Z2(s) for this circuit are:

Z1(s) = R1 (2.37)

Z2(s) =1

Cs+ 1R2

= R2

R2Cs+1 (2.38)


ei ( t )

R1

eo ( t )

C

R2

Figure 2.14: First-order lag circuit using ideal operational amplier.

The transfer function is therefore obtained as

Eo(s)Ei(s)

= R2

R1

1R2Cs+1

: (2.39)

Voltage-Controlled Current Source (VCCS)

The necessary equations for the implementation of a voltage-controlled current source as


vj

Rin[ rTHEV j ]

ij

vOPEN jvk Γ vj Rout

[ rTHEV k ]

ik

vOPEN k

Figure 2.15: Voltage-controlled current source (VCCS).

vj = Rinij (2.40)

vk = Routvj +Routik (2.41)


where:

= Gain over the controlling or measured voltage, applied as dependent current source at

branch k.

By inserting equation 2.40 into 2.8 and dividing the resulting equation by Rin to avoid

numerical diÆculties, results in equation 2.42. In order to eliminate the voltages and keep

only the currents as variables, and also to allow the use of very large numbers for Rout, the

following calculations are done: times (equation 2.40 inserted into 2.8) minus the result

of [(equation 2.41 inserted into 2.9) and divided by Rout]. This procedure eliminates Rin in

the resulting equation 2.43:

vOPENjRin

+rj1Rin

i1 + :::

::: +rjj+RinRin

ij +

rjkRin

ik + ::: +rjMRin

iM = 0(2.42)

vOPENj +vOPENkRout

+rj1

rk1Rout

i1 + :::

:::+rjj

rkjRout

ij +

rjk

rkk+RoutRout

ik + :::

:::+rjM rkM

Rout

iM = 0

(2.43)

For an ideal voltage-controlled current source, Rin !1 and Rout !1, resulting in:

ij = 0 (2.44)

vOPENj + rj1i1 + ::::::+ rjjij + (rjk 1) ik + ::: + rjM iM = 0

(2.45)

2.2.3 Ideal Transformers

Even though an ideal transformer model has already been implemented in most EMTP-

based programs, with the equations described in [2] , or with similar equations, an ideal

transformer can also be implemented as a special dependent source. The necessary equations

for the implementation of the ideal transformer as illustrated in Fig. 2.16 are 2.8 and 2.9,

as well as:


vj vk

ij ik1 : n

Figure 2.16: Ideal transformer.

vjvk

=1

n(2.46)

ijik

= n (2.47)

where:

1n= nj

nk= turns ratio of the ideal transformer.

From the equations above and from 2.8 and 2.9, one can easily obtain:

ij + nik = 0 (2.48)

vOPENj +vOPENk

n+rj1

rk1n

i1 + :::

:::+rjj

rkjn

ij +

rjk

rkkn

ik + :::

:::+rjM rkM

n

iM = 0

(2.49)

Equations 2.48 and 2.49 can be used to model an ideal transformer. It is important to

mention that, normally, better models for electric transformers, which may include saturation

eects, are used in EMTP-based simulations. For further details, please see, for example,

references [2] and [6].

2.2.4 Independent Sources

It may be useful in a circuit or device model to have an independent current or indepen-

dent voltage source connected between two ungrounded nodes. This can be accomplished


with the same technique used for dependent sources, but using only one equation in this

case.

Independent Current Source

Assuming that the independent current source is connected between nodes c and d with

branch voltage

vk = vc vd (2.50)

and branch current

ik = icd (2.51)

then the necessary equations for the implementation of an independent current source as

illustrated in Fig. 2.17 are:

vk isource Rout

[ rTHEV k ]

ik

vOPEN k

Figure 2.17: Independent current source.

vOPENk + rk1i1 + ::::::+ rkkik + :::+ rkM iM + vk = 0

(2.52)

vk = Routisource +Routik (2.53)

where:


isource = independent current source of branch k, which can be constant or a function of

time.

From the equations above, one can also obtain the following equation:

vOPENkRout

+ rk1Rout

i1 + :::

:::+rkk+RoutRout

ik + :::+ rkM

RoutiM + isource = 0

(2.54)

For the ideal current source, Rout !1, resulting in:

ik + isource = 0 (2.55)

Of course, there is a much easier way to represent an independent current source between

nodes c and d directly in the nodal equations of the EMTP: inject the current source into

node c and with a negative sign into node d [2].

Independent Voltage Source

The necessary equations for the implementation of an independent voltage source as

illustrated in Fig. 2.18 are:

vk

vsource

Rout [ rTHEV k ]

ik

vOPEN k

Figure 2.18: Independent voltage source.

vOPENk + rk1i1 + ::::::+ rkkik + :::+ rkM iM + vk = 0

(2.56)

vk = vsource +Routik (2.57)


where:

vsource = independent voltage source of branch k, which can be constant or a function of

time.

From the equations above, one can also obtain the following equation:

vOPENk + rk1i1 + :::::: + (rkk +Rout) ik + ::: + rkM iM + vsource = 0

(2.58)

For an ideal voltage source, Rout = 0, resulting in:

vOPENk + rk1i1 + ::::::+ rkkik + :::+ rkM iM + vsource = 0

(2.59)

Another approach for voltage sources between ungrounded nodes frequently used in

EMTP-based programs is the insertion of an ideal transformer between the two ungrounded

nodes, with a voltage source to ground on the other side.

2.2.5 Newton-Raphson Algorithm

The equations for current and voltage dependent sources have been presented in the

previous sections, as well as the equations of independent sources which may be connected

between two ungrounded nodes. The solution of these equations is based on the compensation

method, which is already being used to solve nonlinear equations associated with nonlinear

elements in electric or electronic circuits with Newton-Raphson (N-R) iteration schemes. The

Newton-Raphson algorithm is well known, widely used and has quadratic convergence if the

initial estimate is close to the solution. For completeness, the Newton-Raphson algorithm is

presented in this section as it is in [77] and the reader is referred to mathematical books or

numerical analysis books or network solutions books, if more detailed information is needed.

In the scalar case the N-R iteration to solve

f(x) = 0 (2.60)

is given by

xk+1 = xk +xk = xk f(xk)=f 0(xk); (2.61)


where the iteration count is denoted by the superscripts.

Consider now the system of M nonlinear equations fi in M variables xi:

f1(x1; x2; : : : ; xM) = 0f2(x1; x2; : : : ; xM) = 0...fM(x1; x2; : : : ; xM) = 0

(2.62)

Denote, for easy notation, the vector of variables by [x] and the vector of functions by

[f(x)]. Then 2.62 has a compact form:

[f (x)] = 0 (2.63)

Assume that the system has a solution; denote it by [x] and expand each function in a

Taylor series about [x]:

f1 (x) = f1(x) +

@f1@x1

(x1 x1) +@f1@x2

(x2 x2) + +@f1@xM

(xM xM) +

f2 (x) = f2(x) +

@f2@x1

(x1 x1) +@f2@x2

(x2 x2) + +@f2@xM

(xM xM) + ...

fM (x) = fM(x) + @fM@x1

(x1 x1) +@fM@x2

(x2 x2) + +@fM@xM

(xM xM) +

(2.64)

Assuming that x is close to x, higher order terms may be neglected and the system may

be written in linearized form:

[f (x)] [f (x)] + [J ] ([x] [x]) (2.65)

where

[J ] jx =

26664

@f1@x1

@f1@x2

@f1@xM

@f2@x1

@f2@x2

@f2@xM

......

. . ....

@fM@x1

@fM@x2

@fM@xM

37775jx

(2.66)

is the Jacobian matrix of the function [f(x)], which has to be calculated at each iteration

step. If equation 2.65 is set to zero and solved, the result will not be the vector [x] (because

the higher-order terms have been neglected) but some new value for [x]. Using superscripts

to indicate the iteration count results in:

fxk

+ [J ]xk+1

xk

= 0 (2.67)


Formally, the solution of 2.67 is obtained by writing

xk+1

=xk [J ]1

fxk

(2.68)

In practice, the Jacobian matrix is not inverted. Instead dene

xk

=xk+1

xk: (2.69)

Then

[J ]xk

=

fxk

(2.70)

is solved by LU factorization and the new [xk+1] is obtained from

xk+1

=xk+xk

: (2.71)

Formulae 2.70 and 2.71 represent the Newton-Raphson algorithm for systems of equa-

tions, which reduce the error norm iteratively so that

f xk+1 f xk (2.72)

This iterations scheme is repeated until the errors are lower than a specied tolerance.

For the case of a system of linear equations, as in the case of linear dependent sources,

convergence to the solution is achieved with just one iteration step. More iteration steps

are required for the solution of a system of nonlinear equations depending on how close the

initial guess is to the nal solution. For highly nonlinear functions, the standard application

of the iteration scheme of the Newton-Raphson algorithm may cause numerical problems

involving computer over ows.

The solution algorithm experimentally implemented in the MicroTran version of the

EMTP at The University of British Columbia is presented in Fig. 2.19. This method

presents \generality and exibility properties", thus looking promising for future work in

detailed modelling of circuits and devices, as will be explained later in this thesis.


INPUT DATA

INITIAL GUESSFOR CURRENTS

CALCULATION OFBRANCH VOLTAGES[from Eq. (2.6) or (2.1)]

CALCULATION OFRIGHT HAND SIDE

[negative valuesof Eq. (2.1)]

IS ITACCURATEENOUGH?

BUILD JACOBIAN MATRIX[partial derivatives

of Eq. (2.1)]

SOLVE FOR CURRENTS

CHECK FOR LIMITS

RETURN CURRENTS TO MAIN PROGRAM

IS THEREANY LIMIT

VIOLATION?Yes

Yes

UPDATECURRENTS

No

SET VARIABLETO ITS LIMIT

MAX. NO. OFITERATIONS WAS

EXCEEDED?

No

No

Yes STOP

Figure 2.19: Newton-Raphson algorithm experimentally implemented in MicroTran.


2.2.6 Possible Applications

The methodology presented in the previous sections for the implementation of dependent

sources in EMTP-based programs permits the computational development of many practical

applications, such as:

1. Current and voltage sensors;

2. Operational ampliers;

3. User-dened coupled branches in a circuit;

4. Modelling of electronic components, where the physical behavior would need to be

represented by nonlinear equations;

5. User-dened linear and nonlinear functions;

6. User-dened modelling of linear and nonlinear devices, limited only by the creativity

and ingenuity of the user.

Figure 2.20 and Fig. 2.21 illustrate the solution method with an example of a noninverting

amplier circuit, as commonly used in practical analog electronics. It consists of a sinusoidal

voltage source, an ideal operational amplier and 2 resistors (Rf and Rg). The ideal opera-

tional amplier was modelled using equations 2.30 and 2.31, whereas the sinusoidal voltage

source and the resistors are part of the network, represented through a Thevenin equivalent

circuit. If Rf = 2Rg, then voutput = 3vinput, as shown in Fig. 2.21. Alternatively, to get

an amplication of 3 in circuit simulation, one could just use a voltage-controlled voltage

source, with equations 2.28 and 2.29 with the gain A set to a value of 3. In this case a

load resistor should be connected in the output of the dependent source, to avoid numerical

oating subnetwork problems.

Indeed, in theory this noninverting amplier circuit should result in:

voutputvinput

=

1 +

Rf

Rg

(2.73)


voutput

Rf

Rg

f=60 [Hz]

vinput

vinput=1.0 0 o [V]

Figure 2.20: Circuit with ideal operational amplier.

0 5 10 15 20 25 30 35 40 45 50−4

−3

−2

−1

0

1

2

3

4

Time ( ms )

Vo

ltag

e (

V )

voutput

vinput

Figure 2.21: Simulation results of circuit with ideal operational amplier (noninverting amplier circuit).

The next sections will present a technique for the simulation of transfer functions in

EMTP-based programs, some improvements already made for the implementation of satura-

tion or limits for the elements or sources presented in this work, as well as the implementation

of some other special control devices.

2.3. Development of Control Transfer Functions in EMTP-based Programs 42

2.3 Development of Control Transfer Functions in EMTP-

based Programs

A transfer function as in Fig. 2.22, is dened in the frequency domain (Laplace transfor-

mation of a continuous time system) by the equation 2.74, which represents the output signal

X(s) as a function of the input signal U(s) for a particular linear time-invariant system.

X ( s )U ( s ) H ( s )

Figure 2.22: Transfer function.

H(s) =X(s)

U(s)= k

bms

m + bm1sm1 + ::: + b1s1 + b0

ansn + an1sn1 + ::: + a1s1 + a0

= k

B(s)

A(s)(2.74)

with n m and an 6= 0.

It is possible to reorganize the terms of equation 2.74 as follows:

(ansn + an1sn1 + :::+ a1s

1 + a0)X(s) =

k (bmsm + bm1sm1 + ::: + b1s

1 + b0)U(s)(2.75)

1 + an1

ans1 + ::: + a1

ans1n + a0

ansn

X(s) =

kbmansmn + bm1

ansm1n + ::: + b1

ans1n + b0

ansn

U(s)

(2.76)

X(s) = kbmansmn + bm1

ansm1n + :::

+ b1ans1n + b0

ansn

U(s)

an1

ans1 + ::: + a1

ans1n + a0

ansn

X(s)

(2.77)


For m = n, 4 this results in:

X(s) = kbnan

+ bn1

ans1 + :::

+ b1ans1n + b0

ansn

U(s)

an1

ans1 + ::: + a1

ans1n + a0

ansn

X(s)

(2.78)

X(s) = k bnanU(s) + s1

hk bn1

anU(s) an1

anX(s)

i+ :::

+s1nhk b1anU(s) a1

anX(s)

i+ sn

hk b0anU(s) a0

anX(s)

i (2.79)

X(s) = k bnanU(s) + s1

nk bn1

anU(s) an1

anX(s)

+ :::

::: + s1hk b1anU(s) a1

anX(s)

+ s1

k b0anU(s) a0

anX(s)

io (2.80)

A transfer function block-diagram realization of equation 2.80 is presented in Fig. 2.23,

which, according to [79], is called the observer form.

The solution technique proposed in this thesis is based on the fact that a practical

realization of a transfer function block-diagram can be accomplished with the use of circuit

components, such as operational ampliers, resistors and capacitors 5. In practice, an analog

signal processing scheme is usually designed as the rst step for a digital signal processing

derivation. Moreover, in this work, the derivation of an analog circuit model for the transfer

function implementation takes advantage of all the options already implemented in EMTP-

based programs, and redundant computational work is avoided. This way, the equations for

the digital model of a transfer function are automatically constructed inside the EMTP, which

4If m < n then bn = bn1 = ::: = bm+1 = 0:5Analog computers were commonly used in the past to solve power system control and stability dierential

equations [83].


kb1___an

a1___an

1___s

+

-

kb0___an

a0___an

1___s

+

-

kbn___an

+

kbn-1___an

an-1___an

1___s

+

-

+++

U ( s )

X ( s )

. . .

. . .

. . .

Figure 2.23: Observer form block-diagram of transfer function in equation 2.80.

uses the trapezoidal integration rule, or the backward Euler rule whenever CDA technique is

applied [84], [85]. With analog circuit modelling based on operational ampliers, the method

presented here is general and allows an arbitrary design of transfer functions (or many special

control devices) by the users of EMTP-based programs.

A simplication can be made in a computer transfer function implementation, in contrast

to a physical circuit implementation, in order to reduce the number of operational ampliers

needed: resistances and capacitances can assume negative values 6. As long as the system

eigenvalues (or poles of the transfer function) remain on the left hand side of the complex

plane, the system is stable. Negative values then may be used here for capacitances and

resistances connected on the feedback path of the \ideal operational ampliers", which re-

sults in a stable solution for transfer functions of linear systems. Such a possible computer

implementation of the transfer function block-diagram of Fig. 2.23 is presented in Fig. 2.24.

Assume for example, that the rst-order transfer function illustrated in the block-diagram

of Fig. 2.25 is to be implemented with the proposed technique. Equation 2.80 becomes in

6Proper precautions should be taken, though, whenever the equivalent digital self admittance of a nodebecomes equal to zero. The connection of large resistances to the node can easily solve this problem.


u ( t )

x ( t )

. . .

. . .

. . .

an___ M Ωkb0

an___ M Ωkb1

an___ M Ωkbn-1

an___ M Ωkbn

an_ ___ M Ω a0

an_ ___ M Ω a1

an_ ___ M Ω an-1

1MΩ 1M Ω

1M Ω1M Ω

-1µ F-1µ F

-1µ F-1M Ω

Figure 2.24: Possible computer implementation of the transfer function block-diagram in Fig. 2.23.

X ( s )U ( s )

10__________

0.01 s + 1

Figure 2.25: Block-diagram representation of a rst-order transfer function.

this case, with n = 1:

X(s) = s1kb0a1U(s)

a0a1X(s)

(2.81)

where, for illustration purposes,

kb0 = 10,

a0 = 1 and

a1 = 0:01 seconds.

The observer form block-diagram for equation 2.81 is presented in Fig. 2.26, and its

\possible computer implementation" is illustrated in Fig. 2.27. The realization of this rst-

order transfer function can also be done with a physically-based realistic rst-order lag circuit

as shown in Fig. 2.28, which requires two inverting amplier circuits 7, instead of just one

7The author acknowledges the help of Mr. Jesus Calvi~no-Fraga in a practical laboratory experiment for


required in the \more economic" computer implementation. There may be cases where the

realistic implementation is needed, which the proposed method can handle as well without

any restrictions. Fig. 2.29 presents the time domain transient response x(t) of the rst order

transfer function implemented as in Fig. 2.27 and as in Fig. 2.28, for a pulse u(t) of 1V

with a duration of 25ms.

kb0___a1

a0___a1

1___s

+

-

U ( s )

X ( s )

Figure 2.26: Observer form block-diagram of rst-order transfer function of Fig. 2.25.

u ( t )

x ( t )

a1___ M Ωkb0

a1_ ___ M Ω a0

- 1µ Fu ( t ) x ( t )1 k Ω

- 10 k Ω

- 1µ F

=

Figure 2.27: Possible computer implementation of rst-order transfer function of Fig. 2.25.

the circuit of Fig. 2.28, which validated the simulation results presented in Fig. 2.29.


u ( t )1 k Ω

10 k Ω

1µ F

x ( t )10 k Ω

10 k Ω

- x ( t )

Figure 2.28: Realistic rst-order lag circuit.

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8

9

10

Time ( ms )

Vo

ltag

e (

V )

x(t)

u(t)

Figure 2.29: Time domain simulation of rst-order transfer function.

This technique can easily be used with the solution for the \ideal operational ampliers"

implemented in subroutine CONNEC. There is no time delay between the electric network

and control equations as in TACS, and the solution of both systems of equations is simulta-

neous. The modied nodal analysis method (MNA) [76] could also be used for the solution

of operational ampliers and other \linear branch equations" as presented in [77] . However,

2.4. Development of Limiters for Control Systems in EMTP-based Programs 48

with the modied nodal analysis (MNA), the network and control system equation matrix

becomes unsymmetric, and zero diagonal elements may appear, which requires pivoting tech-

niques. Also, as the number of added branch equations increases, requiring extra columns

and extra rows, the dimension of the matrix may become very large, which may eventually

decrease the computational eÆciency, if proper techniques are not used. Possibily, if the sim-

ulation time becomes a very important issue, as in the case of digital real-time simulators,

a combined solution could be investigated, such that all linear control and system equations

could be solved using the MNA with appropriate techniques, and the remaining nonlinear

equations could be solved with the compensation method. The main motivation here in this

thesis for using the compensation method with a Newton-Raphson iterative algorithm is its

\generality and exibility" to model nonlinear (and linear as a special case) control devices

in EMTP-based programs, particularly because the branch Thevenin equivalent circuit is

readily available, as in the CONNEC subroutine of MicroTran.

The implementation of limits associated with rst order transfer functions, as well as

special devices and intrinsic FORTRAN functions, are presented in the following sections.

2.4 Development of Limiters for Control Systems in

EMTP-based Programs

The use of limiters in control loops may introduce extra time delays in the solution

method implemented in TACS. This can result in inaccuracies and instabilities [66], [67].

The technique proposed in this thesis for the solution of limiters overcomes this diÆculty.

With the Newton-Raphson iterative algorithm in the compensation method, a simultaneous

solution for limiters can be found.

There are two types of limiters associated with rst-order transfer functions : windup

(also referred to as static limiter) and non-windup (dynamic limiter) [66], [2]. "Non-windup

limiters should only be used with rst-order transfer functions. For second and higher-order

transfer functions it is no longer clear which variables should be limited. ... Even for the rst-

order transfer function, the meaning of the limiting function is confused if it has any zeros"


[2]. Reference [66] presents an appropriate model for a proportional-integral (PI) controller

(which can be represented as a transfer function with one zero) with a non-windup limiter.

In a lead-lag control function block, for example, the way in which a non-windup limiter can

be realized is not unique; the interpretation of the limiting action should therefore be based

on the electronic implementation of the physical device [86].

The main dierence between windup and non-windup limiters is the way in which the

limited variable comes o its limit. To illustrate that, the rst-order transfer function pre-

sented earlier in Fig. 2.25 is assumed to have a windup limiter as in Fig. 2.30, and a

non-windup limiter as in Fig. 2.31. The time domain simulation of the output variable x(t)

for both cases, for a pulse input excitation u(t) of 1V, is presented in Fig. 2.32.

X ( s )U ( s )

10__________

0.01 s + 1

+ 5

slope=1

Figure 2.30: First-order transfer function with windup (static) limiter.

X ( s )U ( s )

10__________

0.01 s + 1

+ 5

slope=1

Figure 2.31: First-order transfer function with non-windup (dynamic) limiter.

Note that the output variable x(t) reaches its limit at the same time for both cases, but

x(t) backs o the limit rst for the non-windup (dynamic) limiter. The reason is that for the

windup limiter the output variable is just clipped at the limit, whereas in the non-windup

limiter the dierential equation is actually modied [66], [87], [2].


0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8

9

10

Time ( ms )

Vo

ltag

e (

V )

x(t) without limiter

x(t) with windup limiter(static)

x(t) with non−windup limiter (dynamic)

u(t)

Figure 2.32: Transient response of a rst-order transfer function with windup and non-windup limiter.

The implemented solution for limiters uses the methodology proposed in Section 2.2.1.

As indicated in the algorithm illustrated in Fig. 2.19 of Section 2.2.5, a simultaneous system

solution is rst found without considering any of the limits. Then, each limit violation is

veried in the sequence of the input data given by the user. If a particular limit has been

reached, all previous indications of limit violations are cleared, and a solution is found for this

particular limiter and all of its consequences on the other limiters. This cause-consequence

iterative process has been found to be a very \robust method" in all cases tested, and has

given the correct solution for all limiters, \independent of the ordering of the input data given

by the user". Also, because the compensation method is properly applied 8, the solution for

limiters is simultaneous without time delays.

The maximum (xmax) and minimum (xmin) limiting values are part of the input data. For

8TACS seems to use a pseudo-compensation method to solve limiters [2].


example, in the case of the rst order transfer function with a non-windup limiter illustrated

in Fig. 2.31, with a computer model as in Fig. 2.27, it is possible to represent the non-windup

hard limiting action with a simple change in the equations for the ideal operational amplier,

such that equations 2.30 and 2.31 are replaced by equations 2.82 and 2.83, respectively:


(2.82)

vOPENk + rk1i1 + :::::: + rkjij + rkkik + :::+ rkM iM + vklimit = 0

(2.83)

where vklimit = xmax, or vklimit = xmin.

By using these equations it becomes easy to observe the limits accurately. In practice,

in a realistic rst order lag circuit, as in Fig. 2.28, the clamping action is done with the use

of Zener diodes connected in parallel with the capacitor in the feedback loop of the rst OP

AMP for a non-windup (dynamic) limiter, or with Zener diodes connected in parallel with

the resistor in the feedback loop of the \second" OP AMP for a windup (static) limiter 9.

Another example is the simple limiter control block. In this case, one could use the

equations for an \ideal voltage-controlled voltage source" including the limiting values in

the output voltage, as follows:

ij = 0 (2.84)

vOPENk + rk1i1 + :::::: + rkjij + rkkik + :::+ rkM iM + vklimit = 0

(2.85)

where vklimit = xmax, or vklimit = xmin.

Implementation of soft limits

The limiters presented in the previous section assume xed values (hard limits) for the

maximum and minimum of the output variable. It may be useful to allow soft limits as

well, as recommended in [2]. With soft limits, the slopes in the limited region are nonzero.

9The author acknowledges the help of Mr. Jesus Calvi~no-Fraga in a practical laboratory experiment,which validated the simulation results presented in Fig. 2.32.


Hard limits are then just a special case of soft limits when the slopes are set to zero. The

equations for soft limits, with the notation from Fig. 2.33, are:

x(t) =

8<:

Ku(t); if xmin < Ku(t) < xmax,xmin +Kmin[u(t) umin]; if Ku(t) xmin,xmax +Kmax[u(t) umax]; if Ku(t) xmax.

(2.86)

x ( t )

u ( t )

x max

x min

slope=K max

slope=K min

u max

u minSLOPE = GAIN "K"

Figure 2.33: Soft limits.

Consider, for example, the zero-order transfer function (constant gain) in Fig. 2.34. The

time domain response for a sinusoidal excitation input u(t) of 1V is presented in Fig. 2.35,

illustrating the eects of hard and soft limits on the output x(t).

In this thesis project, soft limits (and hard limits as a special case) have been implemented

for all the current and voltage dependent sources presented in Section 2.2. Limits can also

be easily implemented for all the FORTRAN functions and special devices which will be

discussed in the following sections.


X ( s )U ( s ) K=4

+ 2

- 2

K max= + 0.1 ( 10% )

K min= + 0.1 ( 10% )

Figure 2.34: Zero-order transfer function with soft limits.

0 5 10 15 20 25 30 35 40 45 50−4

−3

−2

−1

0

1

2

3

4

Time ( ms )

Vo

ltag

e (

V )

x(t) with soft limit

x(t) with hard limit u(t)

Figure 2.35: Time domain response for a sinusoidal excitation input u(t) illustrating the eects of hardand soft limits on the output x(t).

2.5. Development of Intrinsic FORTRAN Functions in EMTP-based Programs 54

2.5 Development of Intrinsic FORTRAN Functions in

EMTP-based Programs

\Supplemental variables and devices", as dened by the EMTP Rule Book, dier in

TACS from the transfer function blocks, as follows:

1. they are not solved with the matrix of the set of linear equations in TACS.

2. they are calculated sequentially, instead of simultaneously (so that the data cards must

be ordered accordingly).

Supplemental variables, such as intrinsic FORTRAN functions and special devices, are

solved in TACS in a sequential way. \In Fig. 2.36, the special device S1 would be solved

after G2 has been solved, and S2 would be solved after G3 has been solved. The solution

would still be simultaneous in this case. In general, the sequence of calculations is more

complicated, with non-simultaneous solutions through time delays. For details, the reader

should consult the EMTP Rule Book" [2]. The sequential solution requires a denition by

the user of \input (e.g. S1 in Fig. 2.36", \output (e.g. S3 in Fig. 2.36)" and \inside (e.g.

S2 in Fig. 2.36)" groups of devices. A special ordering of these device blocks is necessary

to minimize the time delays introduced by this sequential solution method. \To make the

solution as much simultaneous as possible, the user should keep the number of internal

devices as low as possible, and use input and output devices instead whenever possible" [2].

S1G1 ( s ) G2 ( s ) S2

G3 ( s ) S3

Figure 2.36: Open loop control system with "supplemental devices S1,S2 and S3".

In this research project a \truly simultaneous solution" is achieved for intrinsic FOR-

TRAN functions, and no special ordering is necessary, i.e., the input of data by the user is

arbitrary. The solution technique applies the compensation method in a similar way as done

in Section 2.2 for the implementation of current and voltage dependent sources. Assume, for


example, the control block-diagram of Fig. 2.37, with a nonlinear relationship between the

output voltage vk(t) and the input voltage vj(t).

vkvj K2 SIN ( K1 vj )

Figure 2.37: Nonlinear control block-diagram with a sinusoidal intrinsic FORTRAN function.

A simultaneous solution can be obtained for this nonlinear function by representing the

block-diagram of Fig. 2.37 in the form of an electric circuit as shown in Fig. 2.38. The nec-

essary equations are 2.87 and 2.88 and the branch equations 2.89 and 2.90. These equations

resemble those of a voltage-controlled voltage source presented in Section 2.2.

vj

Rin[ rTHEV j ]

ij

vOPEN jvk

K2 sin ( K1 vj )

Rout [ rTHEV k ]

ik

vOPEN k

Figure 2.38: Circuit implementation for the simultaneous solution of a sinusoidal FORTRAN function.

For the controlling branch, the equation is


(2.87)

and for the dependent source branch it is


(2.88)

where:




vj = Rinij (2.89)


vk = K2 (sin (K1vj)) +Routik = K2 (sin (K1Rinij)) +Routik (2.90)

where:

K1 = Gain over the controlling or measured voltage.

K2 = Gain applied to the dependent source in branch k.

From the equations above, one can also obtain the following equations:

vOPENjRin

+rj1Rin

i1 + :::

::: +rjj+RinRin

ij +

rjkRin

ik + ::: +rjMRin

iM = 0(2.91)

vOPENk + rk1i1 + :::::: + [rkjij +K2 (sin (K1Rinij))] + (rkk +Rout) ik + ::: + rkM iM = 0

(2.92)

If a large number is used for Rin10, then the solution for the current ij will be very small

but not exactly equal to zero. This allows the convergence to a non trivial solution for the

current ik, and of course, for the input and output voltages of this nonlinear control block

of a sine function.

These equations can then be solved with the implemented Newton-Raphson algorithm

illustrated in Fig. 2.19, of Section 2.2. Note that for the proper application of the compen-

sation method there must always be possible a Thevenin equivalent circuit. Therefore, in

cases where there is a oating subnetwork, i.e., a node without connection to ground (as for

example if the output of an intrinsic FORTRAN function has no circuit elements connected

to ground), then the insertion of a big resistance between this node and ground easily over-

comes this problem, as is usually done in some versions of the EMTP for the solution of

nonlinear elements [2].

Applying this technique, the following nonlinear intrinsic FORTRAN functions were im-

plemented,

SIN

10In theory Rin ! 1, making ij = 0, which would result in a trivial solution for equation 2.92. Rout isobviously assumed to be equal zero.


COS

TAN

COTAN

SINH

COSH

TANH

ASIN

ACOS

ATAN

EXP

LOG

LOG10

SQRT,

as well as the mathematical operations 11:

multiplication ()

division (=)

exponentiation ().

Proper precautions were taken to handle mathematical and computational problems such

as division by zero, square root of negative number, exponentiation of negative number

with non-integer exponent, logarithm of zero or of negative values, etc.. The evaluation of

trigonometric functions accepts the argument in degrees, which is then converted to radians

internally. The inverse of trigonometric functions gives the answer already converted to

degrees.11Addition (+) and subtraction () can be implemented with the use of just one ideal operational amplier.

2.6. Development of Control Devices in EMTP-based Programs 58

2.6 Development of Control Devices in EMTP-based

Programs

A simultaneous solution is also obtained for special control devices, by using the same

approach presented in the previous section. Their input can be in arbitrary order dened

by the user. To illustrate the potential of this technique, the detailed development of some

useful special devices will be presented.

Transport Delay

Assume, for example, the time delay control block-diagram of Fig. 2.39, (also called

"transport delay" in TACS), where the input voltage vj(t) only aects the output after the

elapsed time t+ , or conversely, the output voltage vk(t) only depends on the past history

value of the input voltage, i.e., vj(t ).

vkvj DELAY

Figure 2.39: Transport delay control device.

The transport delay of Fig 2.39 can be represented in the form of an electric circuit 12

as in Fig. 2.40, with circuit equations 2.93 and 2.94, and branch equations 2.95 and 2.96.

If not otherwise indicated, all the current and voltage variables are the instantaneous values

at the present time t.

For the controlling branch, the equation is


(2.93)

and for the dependent source branch


(2.94)

12For generality reasons, the equations are derived including Rin and Rout, however the ideal equationsare used in the computer implementation.


vj

Rin[ rTHEV j ]

ij

vOPEN jvk

K2 vj ( t - τ )

Rout [ rTHEV k ]

ik

vOPEN k

Figure 2.40: Circuit implementation for the simultaneous solution of a transport delay control device.

where:




vj = Rinij (2.95)

vk = K2vj (t ) +Routik (2.96)

where:

K2 = Gain applied to the controlling or measured past history voltage vj(t ), to create

an independent source at time t, in branch k.

From the equations above, one can also obtain the following equations:

vOPENJRin

+rj1Rin

i1 + :::

::: +rjj+RinRin

ij +

rjkRin

ik + ::: +rjMRin

iM = 0(2.97)

vOPENk + rk1i1 + :::::: + rkjij + (rkk +Rout) ik + ::: + rkM iM +K2vj (t ) = 0

(2.98)

If Rin !1 and Rout ! 0, then

ij = 0 (2.99)

vOPENk + rk1i1 + ::::::+ rkjij + rkkik + :::+ rkM iM +K2vj (t ) = 0

(2.100)


which can be solved with the implemented Newton-Raphson algorithm illustrated in Fig.

2.19, of Section 2.2.

Considering that the delay time is not usually an integer multiple of the simulation

time step t, some type of interpolation must be used. Linear interpolation has been chosen

for that purpose, in a way similar to the transient time domain simulation of a transmission

line model [1], [2]. Fig. 2.41 illustrates the time response of a transport delay control device

where = 4:1667ms, t = 166:6667s, K2 = 1; the input voltage signal vj(t) is a sinusoidal

source of 1V. Note in Fig. 2.41 that the output voltage signal vk(t) is actually equal to the

input voltage signal vj(t), but delayed in time in 4.1667ms.

0 5 10 15 20 25 30 35 40 45 50−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time ( ms )

Vo

ltag

e (

V )

delayed signal

delay 4.1667ms

Figure 2.41: Transient simulation of a transport delay control device.

Pulse Delay

Applying the same technique used for the implementation for a transport delay, it is

possible to develop a model for a pulse delay control device. In a pulse delay, the negative to


positive and positive to negative zero crossings of the input signal are detected and a pulse is

created with the specied delay and with the width of respective time between the two zero

crossings of the input signal. This way there is no need to store all the past history values of

the input signal, just the respective times of zero crossing, which is a \more computational

economic" delay if the output signal will always have to be a pulse, irrespective of the shape

of the input signal. Fig. 2.42 illustrates the time response of a pulse delay control device

where = 20ms, K2 = 1, and the input voltage signal vj(t) is a 1V pulse source. Fig. 2.43

shows the time response of a pulse delay control device where = 20ms, K2 = 1, and the

input voltage signal vj(t) is an arbitrary signal source.

0 5 10 15 20 25 30 35 40 45 50−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time ( ms )

Vo

ltag

e (

V )

20ms delay

delayed pulse

Figure 2.42: Transient simulation of a pulse delay control device.


0 5 10 15 20 25 30 35 40 45 50−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time ( ms )

Vo

ltag

e (

V )

20ms delay

delayed pulse

Figure 2.43: Pulse delay control device with arbitrary input signal.

Logic Gate "NOT"

The necessary equations for the simultaneous solution of a logic gate NOT, as illustrated

in Fig. 2.44 and Fig. 2.45, are 2.93, 2.94, 2.95 and

vk = K2(1 vj) +Routik (2.101)

where:

K2 = 1 is the gain over the controlling or measured voltage (with either vj = 0 or vj = 1),

being applied as a dependent source in branch k.

vkvj

Figure 2.44: Logic gate "NOT".


vj

Rin[ rTHEV j ]

ij

vOPEN jvk

K2 ( 1 - vj )

Rout [ rTHEV k ]

ik

vOPEN k

Figure 2.45: Circuit implementation of a logic gate "NOT" for simultaneous solution.

From equations 2.93, 2.94, 2.95 and equation 2.101, one can also obtain the following

equations:

vOPENJRin

+rj1Rin

i1 + :::

::: +rjj+RinRin

ij +

rjkRin

ik + ::: +rjMRin

iM = 0(2.102)

vOPENj vOPENK

K2+ 1 +

rj1 +

rk1K2

i1 + :::

::: +rjj +

rkjK2

ij +

rjk +

rkk+RoutK2

ik + :::

::: +rjM + rkM

K2

iM = 0

(2.103)

If Rin !1, Rout ! 0 and K2 = 1, then the following equations are obtained:

ij = 0 (2.104)

vOPENj vOPENK + 1 + (rj1 + rk1) i1 + :::::: + (rjj + rkj) ij + (rjk + rkk) ik + :::::: + (rjM + rkM) iM = 0

(2.105)

Other logic gates, such as "AND", "NAND", "OR", "NOR", etc. can be implemented

in a similar way.

Chapter 3

Power Electronics Modelling inEMTP-based Simulations

\The application of semiconductor devices in the electric power eld has been steadily

increasing, and a study of power electronics (as it is commonly called) is now a feature

of most electrical and electronics engineering courses. The power semiconductor devices,

such as the diode, thyristor, triac, and power transistor, are used in power applications as

switching devices. The development of theory and application relies heavily on waveforms

and transient responses, which distinguishes the subject of power electronics from many

other engineering studies" [88].

\Computer simulation can greatly aid in the analysis, design and education of Power

Electronics. However, simulation of power electronics systems is made challenging by the

following factors: 1) extreme nonlinearity presented by the switches, 2) time constants within

the system may dier by several orders of magnitude, and 3) a lack of models. Therefore, it is

important that the objective of the computer analysis be evaluated carefully and appropriate

simulation packages be chosen" [89].

\In system level investigation, it is often adequate to represent semiconductor switches

within converters by ideal switches. This is important in order to minimize the overall

simulation time. But, it is very desirable if the same simulation package has the detailed

device models to design snubbers and gate drives. The simulation package should also be

able to represent the controller portion of the converter system by its functional features

64

3.1. Modelling Power Electronics in Electric Power Engineering Applications 65

in as simplied a manner as possible. Yet, it should be able to model the controller on a

component level if needed" [89].

The preceding quotations show the need for the development and implementation of

simplied as well as detailed nonlinear models of semiconductor devices in EMTP-based

programs. Such models can be used for the transient simulation of electromagnetic phe-

nomena in low and high power circuit networks. They are especially useful for the detailed

evaluation of the impact of power electronic devices on electric power quality, which is the

main emphasis of this thesis. For power quality studies, transient phenomena have to be

evaluated not only at the interface between power electronic devices and the power system,

but the propagation of transients through the supply network and neighboring systems, and

the impact of transients coming from the supply power system on the electronic devices have

to be analyzed as well. The following sections discuss some of the IEEE recommendations

for power electronics modelling, the simultaneous solution of voltage-controlled switches in

EMTP-based programs, the implementation of a nonlinear diode model, and some aspects

of control of power electronic devices.

3.1 Modelling Power Electronics in Electric Power En-

gineering Applications

Fig. 3.1 presents the major power semiconductor devices: Diode, Thyristor, Gate

Turn-O (GTO) Thyristor and Gate-Controlled Thyristor (GCT), MOS Turn-O Thyristor

(MTO), Emitter Turn-O Thyristor (ETO), MOS Controlled Thyristor (MCT), Transistor,

Insulated Gate Bipolar Transistor (IGBT), and Metal Oxide Semiconductor Field Eect

Transistor (MOSFET) [90]. Some of these semiconductor devices are already well known.

High-power semiconductor devices with increasing switching and power capabilities appear

on the market every year (such as the Integrated Gate-Commutated Thyristors (IGCTs)),

which extend the potential applications of modern power electronics techniques to basically

all voltage levels in utility companies and industrial sites. Models for the existing and new

power electronics devices are therefore necessary for analyzing existing or future applications.


Anode

Cathode

Anode

Cathode

Gate( turn-on )

Anode

Cathode

Gate( turn-on &turn-off )

DIODE THYRISTOR GTO and GCT

Anode

Cathode

Turn-offGate

MTO

Turn-onGate

Anode

Cathode

Turn-on Gate

ETO

Turn-off Gate

Anode

Cathode

Gate( turn-on &turn-off )

MCT

Collector

Emitter

Base

TRANSISTOR

Collector

Emitter

Gate

IGBT

Drain

Source

Gate

MOSFET

Figure 3.1: Power semiconductor devices.


Guidelines for modeling power electronics in electric power engineering applications, es-

pecially for use in EMTP-based programs, can be found in [91], [92]. These guidelines can

also be useful when using other digital simulation tools. The approximately 60 test cases in

the computer exercise collection of Dr. N. Mohan [93], [94], [89] which are available for both

EMTP and PSpice R simulations, are also useful for power electronics digital simulations.

In particular, the dierences between existing EMTP and PSpice models may give hints for

the improvement of EMTP models. Dr. Ned Mohan also presents in [89] a power electronics

library with special sub-circuits, which are used to represent switching electronic devices

and some control devices. According to [89] \in case of extreme nonlinearity, PSpice uses

extremely small time steps and is also prone to problems of voltage convergence. To avoid

this problem of extreme nonlinearity such as that associated with diodes, R-C snubbers are

connected across them. The values of R and C in these snubbers are not optimized, rather

these are based on speeding up the simulation without distorting the system voltage and

current waveforms signicantly."

As stated in [91], [92], power electronics modelling depends on the objectives of the

study. Depending on the type of study, dierent software tools and solution techniques can

be applied. In steady-state or harmonic analysis, the main concern is the injection and

propagation of harmonic currents into the transmission and distribution system, which may

cause unacceptable voltage distortions and dangerous resonances. The power electronic sub-

system is then modelled as \known" harmonic current sources. These currents are assumed

to be independent of voltage variations at the point of common coupling (PCC), where the

power electronics load is connected. However, in many applications such as adjustable speed

drives, active power conditioning, FACTS and Custom Power Controllers, etc., the operation

of a power electronics device closely depends on and can aect the dynamics and the electric

transient behavior of the connected system. Variations of the system parameters, such as

voltage and current amplitude, frequency, phase-angle displacement among the phases in

a three-phase system, instantaneous or average power, etc., need to be used by the power

electronics control, to properly adjust the ring time of the semiconductors, which in turn

might have a feedback eect on the system. Therefore, for transient analysis, a more complex

and detailed representation of the power electronics devices as well as of the supplying power


system is required. Some model simplication and system reduction [92] might be necessary

for practical reasons. This can be acceptable, provided that the equivalent model is validated

against practical measurements, and that it ts the study investigation needs.

The representation of semiconductor switching devices is commonly simplied in power

level application studies. Therefore, the nonlinear characteristic of a diode is usually repre-

sented in a simplied form as a two-terminal uncontrollable unidirectional current owing

switch, or in some programs as a voltage-controlled switch. Series on-state and parallel o-

state resistances can be added to represent the semiconductor losses. In some EMTP-based

programs, such as MicroTran, series resistances must be included to allow multiple switch

connections on the same circuit node. Parallel resistances can provide a resistive connection

between the DC sides of rectiers and inverters and the AC local ground, thus avoiding

oating sub-network problems [95].

The use of simplied switch models for power electronics devices may be justied to

speed up the simulation time for system level studies, but it may also give wrong and

misleading results, especially related to semiconductor commutation phenomena. Also, the

EMTP solution at xed discrete time intervals t may result in inaccurate turn-on or turn-

o switching times, causing unrealistic high frequency transients in the simulation of power

electronic devices. Backtracking techniques [67] and/or resynchronization techniques ([96]

pages 185, 204, 207), or even the Clock Synchronized Structure Changing Concept (CSSC)

[97] can be used to minimize the problem. Interpolation and/or extrapolation as well as

resynchronization techniques seem to be more and more applied even in the EMTP-based

solution of modern control for power electronics systems, as in the software PSCAD/EMTDC

[98], [99]. Therefore, for EMTP-based simulations of power electronics, it is much more

important to use such techniques than to reduce the time step size.

Numerical oscillations caused by the trapezoidal rule of integration in solving the system

of equations may also be a problem for EMTP-based simulations. The use of techniques

such as CDA (\Critical Damping Adjustment" [84], [85]) is eective in the elimination of

numerical oscillations. MicroTran has CDA implemented, but other EMTP versions may

not, or may use dierent approaches.


If the gating circuit is not considered in the study, three-terminal, controllable, unidi-

rectional current owing semiconductor devices can be represented by simplied switches

with gate turn-on and turn-o controls. Dierent ring controls can be applied to represent

thyristors, GTO's, IGBT's, etc. However, \in many actual power electronics applications, in

order to provide a continuous current ow path for an inductive load, a reversal diode (free

wheeling diode) is used in parallel with a controllable switching device to form the basic

power electronic unit" [91]. The implementation of a \basic power electronic unit" in digi-

tal programs requires special care with respect to \instantaneous commutation phenomena"

[95], [96], [88], [97].

The UBC version MicroTran of the EMTP, up to now, only allows pre-dened timing

for the closing and opening of switches representing semiconductor devices. This is done

through the denition of a modied ring angle (\ ") and of a \switching frequency" for

each semiconductor. The frequency is needed to calculate the switching period and the time

of switching from , which assumes a xed reference at time t = 0. This implementation with

xed opening and closing times has limitations, because it does not use the real ring angle

(\") dened from a zero crossing detection related to the voltages at the semiconductor

terminals, and because it ignores the dynamics of the control circuits. MicroTran users could

write their own ring control subroutine based on an available ALPHA subroutine, but only

a few users have used this option. Also, sensing voltages and currents from the main program

would introduce a one time step delay with this approach, and users must be aware of that.

For PWM control techniques, the auxiliary program \PWM" can be useful to calculate and

dene the closing and opening times, which would then be read in as a switching table. The

dynamics of the control circuit would be ignored, which limits the application of \PWM" to

steady-state behavior.

As part of this thesis project, a subroutine \GATE" was developed to simulate power

electronics dynamic control schemes with more accuracy and exibility. As its name indi-

cates, the subroutine GATE allows a simplied gate ring control of a semiconductor, i.e,

it can control its turn-on time, and, for some devices, also its turn-o time. The control

signal is assumed to be a gate voltage signal, dened for simplicity, between the gate node

and ground. This subroutine was derived from the subroutine ALPHA (which can be user


dened), to implement new four-terminal, dynamically controlled semiconductor devices as

voltage-controlled switches, as illustrated in Fig. 3.2.

i

vgate

Voltage-Controlled

BidirectionalSwitch

Figure 3.2: Voltage-controlled switch in EMTP-based programs.

With the subroutine GATE, most of the three-terminal controllable power semiconductor

devices can be represented. Fig. 3.3 presents some test cases for the transient simulation of:

voltage-controlled, \bipolar in voltage" (i.e., may conduct irrespective if it is forward

or reverse biased) and bidirectional current owing switch;

Thyristor (simplied model, as a voltage-controlled, \unipolar in voltage" (i.e., may

only conduct when forward biased), and unidirectional current owing switch);

GTO (simplied model, as a voltage-controlled, \unipolar in voltage", unidirectional

current owing switch, with turn-o capabilities).

Figs. 3.4, 3.5 and 3.6 illustrate the controlling properties of the bidirectional switch, thyris-

tor and GTO, respectively. Since the solution for switches follows the algorithm already

implemented in most EMTP-based programs [1], including MicroTran, the change of switch

position (\status" on or o) only happens one time step after the enabling gate signal [100].

If it becomes necessary to avoid this delay problem for certain types of simulations, an

alternative implementation for a \simultaneous" solution for voltage-controlled switches is

presented in the next section of this chapter.


1 Ω

i

vsource = 10 sin ( ω t ) [ V ]

vgate

f = 60 [ Hz ]

vsource

vgate = 1 [ V ]

1 Ω

i

vgate

vsource

1 Ω

i

vgate

vsource

Voltage-Controlled

BidirectionalSwitch

THYRISTOR

GTO

Figure 3.3: Test cases for transient simulation of voltage-controlled, bipolar in voltage and bidirectionalcurrent owing switch, thyristor and GTO.


0 2 4 6 8 10 12 14 16

−10

−5

0

5

10

Time ( ms )

Vo

ltag

e (

V )

Cu

rre

nt

( A

)

Gate Voltage

Source Voltage

Switch Current

Figure 3.4: Simulation of a voltage-controlled bidirectional current owing switch.

0 2 4 6 8 10 12 14 16

−10

−5

0

5

10

Time ( ms )

Vo

ltag

e (

V )

C

urr

en

t (

A )

Gate Pulse

Source Voltage Thyristor Current

Figure 3.5: Simulation of a simplied model for thyristors.


0 2 4 6 8 10 12 14 16

−10

−5

0

5

10

Time ( ms )

Vo

ltag

e (

V )

C

urr

en

t (

A )

Gate Pulse

Source Voltage GTO Current

Figure 3.6: Simulation of a simplied model for GTO's.

3.2. Simultaneous Solution for Voltage-Controlled Switches in EMTP-based Programs 74

3.2 Simultaneous Solution for Voltage-Controlled Switches

in EMTP-based Programs

The implementation of voltage-controlled bidirectional current owing switches in EMTP-

based programs, in such a way that a simultaneous solution is found at each time step for

the controlling gate voltage and the voltage-controlled switch, can be done with the com-

pensation method presented in Section 2.

The necessary equations are, for the controlling branch with the gate voltage,


(3.1)

and for the voltage-controlled switch branch


(3.2)

where:




It is possible to assume as branch equations:

vj = Rinij (3.3)

and

vk = [vj (Ron Roff ) +Roff )] ik (3.4)

where:

Ron = On-state resistance, and

Roff = O-state resistance, for the voltage-controlled switch connected at branch k.

From equation 3.4 it is easy to verify that:

if vj = 1, then vk = Ronik;


else if vj = 0, then vk = Roff ik.

Moreover,

if Ron = 0, then vk = 0;

else if Roff !1, then ik = 0.

In order to sense the gate control voltage, assume Rin !1, which results in

ij = 0 (3.5)

vOPENk + rk1i1 + ::: + rkjij + rkkik + :::+ rkM iM++ [vj (Ron Roff ) +Roff )] ik = 0

(3.6)

which then can be solved with the implemented Newton-Raphson algorithm illustrated in

Fig. 2.19. Observe, however, that the voltage signal vj is calculated using equation 3.1 (i.e.,

vj = vOPENj rj1i1 ::: rjjij rjkik ::: rjM iM) in the solution algorithm, which is

always correct and less prone to numerical problems.

Assume, for example, the simple circuit with a simultaneous solution for a voltage-

controlled switch presented in Fig. 3.7. Applying an enabling control voltage signal (1V) at

the node gate, the switch turns on at the same time the enabling signal is received. When

the gating signal becomes zero (0V), the switch turns o at the corresponding same time,

as illustrated in Fig. 3.8. In the case of a conventional switch, as it is implemented in most

EMTP-based programs, the turn-on and turn-o would occur 1 time step later, as illustrated

in Fig. 3.9.

A similar approach has been proposed in [100] and [73] for the simultaneous solution

(also called "synchronized solution" in [73]) of voltage-controlled switches in EMTP-based

programs. Every time a switch changes its status, special computer techniques, such as, for

example, the Critical Damping Adjustment (CDA) [84], [85], would have to be triggered to

avoid numerical oscillation problems in the simulation of power electronic devices.


vsource = 10 sin( ω t ) [ V ]

f = 60 [ Hz ]vgate = 1 [ V ]

1 Ω

i

vgate

vsource

"Simultaneous"Voltage-

ControlledBidirectional

Switch

Figure 3.7: Circuit with \simultaneous solution" of a voltage-controlled switch.

0 2 4 6 8 10 12 14 16

−10

−5

0

5

10

Time ( ms )

Vo

ltag

e (

V )

C

urr

en

t (

A )

Gate Pulse

Source Voltage "Simultaneous" Switch Current

Figure 3.8: Simulation with simultaneous solution of a voltage-controlled switch.


0 1 2 3 4 5 6 7 8−2

0

2

4

6

8

10

12

Time ( ms )

Vo

ltag

e (

V )

C

urr

en

t (

A )

Gate Pulse

Source Voltage

"Simultaneous" Switch Current

EMTP Switch Current

Figure 3.9: One time step delay in EMTP-based switches.

3.3. Implementation of Nonlinear Diode Model in EMTP-based Programs 78

3.3 Implementation of Nonlinear Diode Model in EMTP-

based Programs

The use of simplied models for diodes, as for example, voltage-controlled switches and

piecewise linear representation, usually gives simulation results with acceptable accuracy for

most of the power system studies. However, according to [89], detailed nonlinear modelling

of semiconductors is needed to design snubbers and gate drive circuits. A detailed nonlinear

model for a diode is also needed in the synthesis of equivalent networks to represent, for

example, a bipolar transistor with the Ebers-Moll model [77], [101]. The semiconductor

diode, with its symbol shown in Fig. 3.10, is therefore, the most common nonlinear element

in power electronics.

i (t)

v (t)

ANODE

CATHODE

Figure 3.10: Diode symbol.

The terminal behavior of a diode [77], with respect to current and voltage as shown in

3.10, is described by

i(t) = Is

he(

qv(t)kT ) 1

i(3.7)

where:

i(t) is the current through the diode, from anode to cathode,

v(t) is the voltage across the diode, i.e., the potential dierence between the anode and

cathode terminals,


Is is a constant which depends on the physical properties of the diode, and is usually in the

range of 106[A] to 109[A] 1,

q = 1:6022 1019C is the charge of an electron,

k = 1:3806 1023J=ÆK is the Boltzmann's constant, and

T is the temperature in degrees Kelvin (273:16ÆK = 0ÆC).

When the polarity of v(t) is as shown in Fig. 3.10, the diode is in the conducting region.

At 17ÆC 290ÆK, the constant VT kT=q 25mV . For v(t) < 3VT ( 75mV ),

i(t) Is. The value Is is usually referred to as the saturation current. If the diode

is forward biased with v(t) > 4VT (over 100mV ), equation 3.7 may be approximated by

i(t) = Ise(qv(t)=kT ). Table 3.1 expresses the relationship between v(t)=VT and i(t)=Is, which

are derived from

i(t) = Is

e

v(t)VT

1

(3.8)

When a constant voltage V0 is applied to the diode, a constant current I0 ows through

it. The pair of values (V0; I0) is called the operating point of the diode. For each operating

point along the characteristic curve of the diode, one could dene a dynamic resistance of the

diode, which relates increments of the voltage to the increments of the current (dv(t)=di(t)

for v(t) = V0 and i(t) = I0). For higher frequencies, additional physical eects come into

play and the diode may no longer be treated as a simple nonlinear resistor. Charges stored

in the semiconductor material will also require the inclusion of \dynamic capacitive eects"

in the nonlinear model of a diode. The value of the capacitance is, in general, a function of

the voltage across the diode. The reader is referred to [77], [101] and other references for

further details. Reference [89], (pp. 71-1 to 71-7), discuss the PN junction diode switching

characteristics, including a sub-circuit for a diode model with reverse recovery.

In this thesis project, the compensation method presented in Section 2.2, with a Newton-

Rhapson solution algorithm, is used for the solution of the nonlinear model of a diode. In-

version of the diode characteristic curve from equation 3.8 and inclusion of a series resistance

Rout results in branch equation 3.10, which is solved together with the linear network equa-1Actually, Is is temperature dependent and may assume default values of 1014[A] at 27ÆC [101].


Table 3.1: Comparison between voltage and current in a diode as a function of its parametric values.

v(t)VT

i(t)Is

......

7 1095.6336 402.4295 147.4134 53.8983 19.0862 6.3891 1.7180 0-1 -0.632-2 -0.865-3 -0.950-4 -0.982-5 -0.983-6 -0.998-7 -0.999...

...

tion 3.9 to determine the operating point of the diode for a particular network condition, as

illustrated in Fig. 3.11:

vOPENk + rk1i1 + : : :+ rkkik + : : :+ rkM iM + vk = 0 (3.9)

vk = VT lnh

ikIs+ 1

i+Routik (3.10)

where:



vk = voltage of branch k, i.e., voltage across the diode (the potential dierence vanode

vcathode) plus the voltage drop across the series resistance Rout.

ik = current through the diode, i.e., the current owing from the anode to the cathode

terminal.

Rout = series resistance for the diode model (which may be assumed to be equal zero).


−0.5 0 0.5 1 1.5 2 2.5−3

−2

−1

0

1

2

3

Current ( A )

Vo

ltag

e (

V )

vOPEN

(t)

iSC

(t) = vOPEN

(t) / (rkk

+ Rout

)

iSC

(t)

V0

I0

Linear Network Thevenin Equivalent Circuit

Nonlinear Diode Characteristic

Resulting nonlinear equation to be solvedwith the Newton−Raphson method

Figure 3.11: V-I diode characteristic and network Thevenin equivalent circuit equation.

Inserting equation 3.10 into equation 3.9 results in:

vOPENk + rk1i1 + : : :

: : :+ [rkk +Rout] ik + VT lnh

ikIs+ 1

i+ : : :+ rkM iM = 0

(3.11)

A general circuit representation of equation 3.11 is illustrated in Fig. 3.12.

Equation 3.11 is solved iteratively at each time step. Note that the Thevenin equivalent

circuit equation changes from step to step, with a change in vOPEN(t), and in general with a

change in the Thevenin equivalent resistance (slope dv=di). Fig. 3.13 illustrates this, where

it is assumed, for simplicity, that the Thevenin equivalent resistance does not change along

the simulation time.

The iterative solution with the Newton-Raphson algorithm requires an initial guess. De-

pending on how close the initial guess is to the nal solution, convergence can be very fast or


vk

vT ln ( ik / Is + 1 )

Rout[ rTHEV k ]

ik

vOPEN k

Figure 3.12: Circuit implementation for the simultaneous solution of a nonlinear diode model.

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

Current ( A )

Vo

ltag

e (

V )

vOPEN

( t )

iSC

( t )

V−I Diode nonlinearcharacteristic

Network Thevenin equivalent at time "t"

−Is

Figure 3.13: V-I diode characteristic and dierent network Thevenin equivalents.


on the contrary, convergence can be very slow for some cases, or numerical problems such as

computer over ow may even arise. Therefore, it is important to derive a heuristic computer

technique for initial guesses, to accommodate the highly nonlinear exponential characteristic

of a diode, and speed up the convergence of the solution. A robust rule should be general

and insensitive to the network or diode parameters. The detailed development of such a

technique for implementation in EMTP-based programs is presented in the following.

A simple inspection of Fig. 3.11 reveals some \physical" candidates for the initial guess of

the nonlinear diode current, associated with the network conditions, i.e., with the particular

linear network Thevenin equivalent circuit:

ik =vOPENk

(rkk+Rout)(3.12)

ik = Is

e

vOPENkVT

1

(3.13)

ik = Is (3.14)

Equation 3.12 is the short circuit current ikSC , which becomes the initial guess for the

diode current ik in the conduction mode. Then,

if vOPENk 0 and vOPENk > VT lnh

ikSCIs

+ 1i, the initial current is assumed to be

equal to ikSC , as calculated with equation 3.12.

If vOPENk 0 and vOPENk VT lnh

ikSCIs

+ 1i, then the voltage across the diode is

rst estimated to be equal to vOPENk, and a better estimate for the current is calculated

with equation 3.13 (Another simple alternative would be just to assume ik = 0);

On the other hand, if vOPENk < 0 and vOPENk 6:0VT , then again the voltage

across the diode is rst estimated to be equal to vOPENk, and the initial current is then

calculated with equation 3.13;

If vOPENk < 0 and vOPENk < 6:0VT , then the diode current is assumed to be as

calculated with equation 3.14. Alternatively, one could use a linear function instead of

using equation 3.14.


There are other possibilities for the initialization of the variables when a Newton-Raphson

method is being used [77]. For example, piecewise linear approximations could be used rst

to determine the initial guess (either for the current or for the voltage across the diode) and

then, the linear characteristic would be replaced by the detailed nonlinear equation. This

approach might be useful for speeding up the solution of networks with multiple nonlinear

elements, such as diodes, transistors, etc.

A two-slope piecewise linear approximation of the diode characteristic is accurate enough

in many cases. Even though the equations are linear in that case it is not known a priori

on which segment the solution ends up. This is particularly true if there are many diodes

to be solved. The production code of MicroTran still uses the Newton-Raphson method for

the two-slope piecewise linear representation, which will then decide, iteratively, the solution

points for the N diodes. Theoretically, there is no guarantee that the iterations will converge

in such cases, but by limiting the new voltages and currents to technically reasonable values,

the method has so far converged in all tested cases, with up to 6 diodes.

The use of linear piecewise or detailed nonlinear models of electronics semiconductors

might also be a requirement for the correct solution of commutation phenomena in power

electronics circuits. \In circuits with gate turn-o thyristors (GTO's), commutation of the

current into a diode must often be instantaneous, without any current interruption. An

example for such situation is the buck-boost (step-down/up dc-dc) converter shown in `Fig.

3.14 [93]' and the half-wave rectier circuit with a freewheeling diode in `Fig. 3.15'. ... If

the diode is represented as a switch which closes when the voltage from anode to cathode

becomes positive (either built into the code as in MicroTran, or controlled in this way

through TACS), then the positive voltage `across the diode' will only be seen in the next

time step immediately following the time step in which the GTO turned o. This is one time

step too late, because the current in the GTO would already have dropped to zero." [95],

[78]. Therefore, EMTP-type simulation of \instantaneous commutation" with simple switch

models might give wrong results in the case of switching converters. User knowledge of the

commutation process can be used to pre-dene the switches which will have simultaneous

commutation [78], but it may not work for all cases. A better option is the compensation

method used in this work, which assures a simultaneous solution of all equations [95].


GTO DIODE

i

Figure 3.14: Dc-dc converter.

DIODE 1

DIODE 2

i

v = 155 sin( ω t ) [ V ]

- 50 [ V ]

Figure 3.15: Half-wave rectier with freewheeling diode.

To illustrate the solution method presented here, assume, for example, the simple electric

circuit with a nonlinear diode as in Fig. 3.16, where the basic parameters for the diode are:

Is = 1012[A] and VT = 0:026[V ]. The time response solution for this circuit is presented

in Fig. 3.17. Fig. 3.18 shows the detail of the diode current at the zero crossing 2, whereas

Fig. 3.19 presents the V-I nonlinear characteristic of the diode resulting from the EMTP

simulation. The average number of iterations with the Newton-Raphson algorithm, and

using the proposed heuristic technique for initial guess, was 1 for this case, assuming a

convergence tolerance error of less than 1010 for the resulting nonlinear equation, and also

forcing at least 1 iteration in the algorithm.

Detailed modelling of other semiconductor devices, such as bipolar transistors, eld ef-

2The modelling of the reverse recovery current would require improvements in the nonlinear diode model,as for example the inclusion of the junction capacitance, which is also nonlinear and voltage dependent [77],[101].


vDIODE

1 Ω

iDIODE

Vmax = 2 [V]

f = 60 [Hz]

Figure 3.16: Electric circuit with a nonlinear diode model.

0 2 4 6 8 10 12 14 16−3

−2

−1

0

1

2

3

Time ( ms )

Vo

llta

ge

( V

)

Cu

rre

nt

( A

)

Diode Current

Diode Voltage

Figure 3.17: Transient simulation of a nonlinear diode model in an EMTP-based program.

fect transistors, etc. can be accomplished with the \synthesis of equivalent networks" (sub-

circuits) using nonlinear diodes, dependent sources, voltage-dependent capacitances and re-

sistances [101], [77]. The Ebers-Moll model of a bipolar transistor, or more complex models

(Gummel-Poon) could be used as well, but their description is beyond the scope of this

thesis project. The use of macromodels, based on functional terminal conditions, can also

be derived with the interconnection of circuit elements, especially with the use of the de-

pendent sources presented in Section 2.2. Piecewise linear approximations can also be used

to model device characteristics, but one should be aware of the fact that even though the

3.3.Im

plem

entation

ofNonlin

earDiodeModelin

EMTP-based

Program

s87

7.58

8.59

9.5−

2

−1.5

−1

−0.5 0

0.5 1

1.5 2x 10

−12

Tim

e ( m

s )

Current ( A )

Is=1

.0E

−1

2 [A

]

Dio

de

Cu

rren

t

Figu

re3.18:

Deta

ilofthetra

nsien

tsim

ulatio

nofanonlin

eardiodemodelin

anEMTP-based

program.

−0.2

00.2

0.40.6

0.81

1.21.4

−2

−1.5

−1

−0.5 0

0.5 1

Cu

rren

t ( A )

Voltage ( V )

Figu

re3.19:

V-Inonlin

earcharacteristic

ofthedioderesu

ltingfro

mtheEMTPsim

ulatio

n.

3.4. Control Modelling Aspects of Power Electronic Devices 88

approximations are continuous from segment to segment, the derivatives are discontinuous.

Spline tting techniques can then be used to approximate the device characteristic by sepa-

rate low-order polynomials between adjacent segments, with the cubic spline being the most

popular one [77].

3.4 Control Modelling Aspects of Power Electronic De-

vices

The digital simulation of control devices for power electronics applications, such as recti-

ers, inverters, DC-DC converters, AC-AC converters, motor drives, etc. is made challenging

because of the more frequent use of mixed analog and (real-time) digital signal processing

(DSP) control techniques.

The derivation of reliable system references requires a growing number of digital and

analog components. Their modelling can, in general, not be simplied without compromis-

ing the simulation accuracy and stability. For example, signal sensoring and zero crossing

prediction or detection usually requires an appropriate use of signal ltering, either through

analog design (operational ampliers, resistances and capacitances) or its respective digital

implementation. However, in the simulation of DSP controls it is important to pay attention

to the digital sampling frequency with respect to the selection of the time step size (t).

System frequency tracking requires the use of appropriate phase-locked loop (PLL) con-

trols, which are of fundamental importance for the control of high power electronics appli-

cations, such as FACTS and Custom Power Controllers [90].

Pulse width modulation (PWM) techniques, which are commonly used in voltage-sourced

and current-sourced converters, require particular attention in digital implementations, due

to the discrete nature of computer simulation programs, where the time step size may aect

the results of discrete binary comparators, thus in uencing the simulation results.

Special transformation of variables into other reference systems (e.g., abc to 0 transfor-

mation, which is used in active lter control), requires a simultaneous solution of control and

system equations. Digital simulation of hysteresis eects, limits (windup and non-windup)

3.4. Control Modelling Aspects of Power Electronic Devices 89

and various non-linearities in the control system also requires appropriate models for a suc-

cessful computer simulation.

Moreover, with variations in the supply voltage caused by power system disturbances,

the control, ideally, should be able to actively withstand and support its primary regulation

functions without disrupting the electric supply to the controlled load. However, in most

practical power quality cases, typically related to voltage sag problems, power electronics-

based loads (i.e., their control) are either extremely sensitive to momentary voltage variations

causing frequent shut-downs in industrial process operation, or improper designs may become

the cause of many power quality problems.

As part of this thesis project, \basic control devices" were experimentally implemented

in MicroTran. The main advantage of this development, compared to TACS (Transient

Analysis of Control System) used by many other EMTP versions, is that a \true simultane-

ous solution" is found through the compensation method using a Newton-Raphson iteration

scheme. Therefore, provided that appropriate computer techniques are used to allow conver-

gence in the solution, this method has been shown to be very robust for the cases simulated

up to now, and does not have the 1 time step delay present in the interface of EMTP and

TACS and also does not have any internal delays in the linear and non-linear control solution.

With graphical user-interfaces for MicroTran, it would become easy to dene libraries of

control devices as well as of power components (and sub-circuits, similarly as in [65], [102],

[103]), in such a way that no dierentiation between power and control circuit will then

be necessary, since a unique simultaneous solution approach would be used, based on the

method and algorithm presented in this Ph.D. thesis.

Chapter 4

Evaluation of the Impact of PowerElectronic Devices on the Quality ofPower

THE analysis of the dynamic interaction between power electronic devices and power

systems and the assessment of electric power quality phenomena can be thoroughly

done with EMTP-based programs. The objective of this Ph.D. thesis research project was

to develop reasonably accurate models for EMTP-based programs, with which one could

evaluate the impact of high power electronic devices on the quality of power. To make the

thesis results more valuable to utilities and industries, the following practical tasks were

carried out:

Cooperation with a utility company in Brazil, ELEKTRO - Eletricidade e Servicos

S. A. 1, which kindly agreed to provide data and real power quality problem cases

to validate simulation results. Financial support for this cooperation was provided

by CAPES 2, a federal agency of the Brazilian Government, for a eld activity at

ELEKTRO with the duration of three months.

Appropriate models for time-domain simulations using EMTP-based programs were

applied to simulate some of the real power quality problems experienced by ELEKTRO.

1ELEKTRO - Eletricidade e Servicos S. A. , Rua Ary Antenor de Souza, 321 - Jardim Nova America,CEP 13053-024, Campinas-S. P. , BRAZIL.

2CAPES - Fundac~ao Coordenac~ao de Aperfeicoamento de Pessoal de Nvel Superior, Esplanada dosMinisterios, Anexo I, Sala 215, Caixa Postal 00365, CEP 70047-900, Braslia - D. F. , BRAZIL.

90

4.1. Dynamic Interaction between Power Electronic Devices and Power Systems 91

Based on the power quality monitoring and simulations, a synthesis of simulation

guidelines for power quality evaluation was developed, with emphasis to determine the

dynamic interaction between power electronic devices and the power systems, through

the use of time and frequency-domain techniques with EMTP-based programs.

This chapter presents some simulation cases of power quality assessment with the use of

the existing features of MicroTran, the UBC version of the EMTP. The simultaneous solu-

tion of control and electric power system equations (SSCPS), with the new circuit approach

presented in Chapter 2 and with the models developed for the dynamic control of power

semiconductor devices presented in Chapter 3, are illustrated through practical control and

power electronics controllers simulation cases. Important simulation guidelines for the eval-

uation of the impact of power electronic devices on the quality of power are also summarized

in this Chapter.

4.1 Dynamic Interaction between Power Electronic De-

vices and Power Systems

The interaction between the utility supply system and power electronics-based loads,

(such as electronic converter controlled electric motor drives), depends on a variety of factors

[104], as for example:

1. the type of \front-end" electronic converter, which converts line-frequency AC into DC

(diode-bridge rectiers (which are unidirectional in power ow), switch mode converters

(in which, power ow can be reversed), thyristors converters (which can be made

bidirectional in power ow);

2. the number of phases (single-phase, three-phase) from the supply system used by the

converter, and the converter conguration (e.g., 6, 12, 24, 48, etc. pulses converters),

which also aects the waveform current distortion;

3. the \strength" (or \stiness") of the utility system, determined by its \short-circuit

power";


4. the number of power electronics based loads, the point of their connection in the

network, and the electric power of electronic converters, which will in uence their

impact on the electric supply system and vice-versa;

5. the design and control of the electronic converter. The choice of parameters and compo-

nents, particularly in cases where low-cost choices are made, can cause a deterioration

on the quality of power of the utility system. On the other hand, the converter opera-

tion (i.e., its control) can easily be disrupted by power system disturbances travelling

through the utility network, as is very common in the case of sensitive loads.

The most common lower-power electronics based load uses a single-phase diode-bridge

rectier [94], [104] which draws highly distorted waveshape current from the utility system,

and may cause harmonic associated problems, such as:

overheating in neutral conductors due to the ow of high third harmonic currents;

voltage distortion along the distribution circuits;

increase in power losses;

risk of resonances with utility or industry power factor correction capacitor banks, etc.

This type of rectier only draws current close to the maximum peak of the \assumed sinu-

soidal" utility voltage source, in order to recharge the capacitor lter on the DC side of the

rectier. To illustrate that, Fig. 4.1 presents a typical single-phase diode bridge rectier [93],

[94], [95], [104] 3, where Ls1 represents the system Thevenin equivalent inductance (resistance

ignored here) and Ls2 represents any series inductance added in the AC side of the rectier.

Assume that the diodes are ideal and the circuit has been energized a long time ago, such

that the DC capacitor lter has already been charged (i.e., the transient energization has

gone and \stead state waveforms" are present). Note, from Fig. 4.2 , that the current is

(and consequently id) only starts to ow during the positive semicycle of the source voltage,

when vSA is greater than the voltage vdc (i.e., when diodes 1 and 2 are forward biased at

3Reference [95] presents useful guidelines on power electronics applications using the EMTP.


time t1). Conversely, during the negative subcycle, when vSA is less than vdc, the current is

starts to ow (i.e., when diodes 3 and 4 are forward biased at time t4), with id = is owing

in the same direction, corresponding then to the electronic current rectication process).

Fig.4.3 presents the harmonic amplitude spectrum of the current drawn from the source by

a single-phase diode-bridge rectier.

D1

20 Ω

id

Vmax = 169.7 [V]

f = 60 [Hz]

Ls2Ls1 vPCCvSA vA

D3

D4 D2

1000 µ F

is

vdc

0.5 mH 0.5 mH

+

-

Figure 4.1: Circuit with a single-phase diode-bridge rectier.

To better understand the waveform voltage distortions caused by a single-phase diode-

bridge rectier, one can apply the Kirchho's second law to the circuit of Fig. 4.1, and

derive the following equations,

vSA vL vA = 0 (4.1)

vL = vL1 + vL2 = Ls1di1dt

+ Ls2di2dt

= (Ls1 + Ls2)disdt

= Ldisdt

(4.2)

where:

vL = voltage across the total inductance L = Ls1 + Ls2,

i1 = i2 = is.

4.1.Dynam

icInteraction

betw

eenPow

erElectron

icDevices

andPow

erSystem

s94

05

1015

2025

3035

4045

50−

200

−150

−100

−50 0 50

100

150

200

Tim

e ( m

s )

Voltage ( V ) Current ( A )

vd

c

is

vS

A

vA

t1 t4

Figu

re4.2:

Curren

tdraw

nfro

mthesource

byasin

gle-p

hase

diode-b

ridgerecti

er.

Therefore,

thefollow

ingvoltages

canbedened:

vA=vSALdisdt

(4.3)

vPCC=vSALs1 d

isdt

(4.4)

From

Fig.

4.4itcan

beseen

that

theinductan

cesin uence

therate

ofchange

(andcon

-

sequently

thewavesh

ape)

ofthecurren

t,such

thatisstarts

togrow

exponentially,

reachinga

maximum

peak

valuewhen

thevoltage

acrosstheinductan

ceiszero,

afterwhich

vLchanges

itspolarity

(becau

sethederivative

ofthecurren

tchanges

itssign

al4)

physically

tryingto

keepthecurren

t ow

inguntil

thecurren

tnally

becom

eszero.

Follow

ingtheequation

s4.3,

4.4andFig.

4.4onecan

easilyunderstan

dthevoltage

waveform

sdistortion

sin

therecti

er

input,vA,andin

thepoin

tof

common

couplin

g(PCC),vPCC,where

manyoth

erload

smay

4Asamatter

offact,

manyDC-DCelectro

nicconverters

relyontheinducto

rphysica

lproperties

durin

gsw

itchingtra

nsien

ts,to

either

stepvolta

ges

up(boost)

orstep

volta

ges

dow

n(buck).


0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

Harmonic Order

RM

S C

urr

en

t (

A )

Third Harmonic Component

Figure 4.3: Harmonic amplitude spectrum of the current drawn from the source by a single-phase diode-bridge rectier.

be connected and may be aected by the waveform voltage distortion. Fig.4.5 presents the

harmonic amplitude spectrum of the voltage at the PCC.

In a four-wire, three-phase system with \well balanced" loads, composed by single-phase

diode bridge rectiers, as illustrated in Fig. 4.6, the current owing through the neutral

conductor has mainly third harmonic components (i.e, 180Hz, for a 60Hz fundamental) as

shown in Fig. 4.7 and Fig. 4.8. Observe that the third harmonic component of the neutral

current in Fig.4.8 is approximately 3 times the third harmonic component of the current of

a single-phase diode-bridge rectier as illustrated in Fig.4.3. These high currents may cause

overheating in the neutral conductor (usually designed with a smaller cross section than the

phase conductors), thereby creating a potential hazardous risk, such as re.

The widespread use of this type of inexpensive rectier, usually as the front end of low

power appliances, such as television sets, computers and compact uorescent lamps (CFL)

4.1.Dynam

icInteraction

betw

eenPow

erElectron

icDevices

andPow

erSystem

s96

02

46

810

1214

16−

200

−150

−100

−50 0 50

100

150

200

Tim

e ( m

s )


vP

CC

is

vS

A v

A

vL

s1

vL

Figu

re4.4:

Curren

tthroughandvolta

geacro

ssthetotalinducta

nce,

andvolta

gewavefo

rmdisto

rtionat

thepointofcommoncouplin

g(PCC).

(which

areused

more

andmore

becau

seofits

high

erenergy

eÆcien

cy),can

have

acumulative

negative

eect

onthequality

ofpow

ersupplied

toresid

ential,

commercial

andindustrial

custom

ers[105]

.For

exam

ple,

Fig.

4.9andFig.

4.10presen

tthevoltage

wavesh

ape

measu

redwith

adigital

oscilloscopeat

theoutlet

ofthePow

erElectron

icsLaboratory

of

theDepartm

entof

Electrical

andCom

puter

Engin

eeringat

UBC,Vancou

ver,B.C.,Canada

5.Thewavesh

apedistortion

isprob

ably

caused

bythelarge

number

ofcom

puters

inthe

build

ing,as

well

asintheentire

university.

Italso

aects,

presen

tlywith

minor

conseq

uences,

alltheoth

erload

sin

thebuild

ingsupplied

fromthesam

ecom

mon

bus,andeven

tually,

also

prop

agatesthrou

ghtheBC

Hydro

electricsupply

system

.Fig.

4.11andFig.

4.12show

theresu

ltsof

aFourier

analy

sisof

theoutlet

voltagecurve

(theDCcom

ponentpresen

tin

theharm

onicam

plitu

despectru

mat

Fig.

4.11migh

tbecau

sedbyinaccu

raciesor

dcosets

inthemeasu

ringequipment).

Alth

ough,accord

ingto

thepresen

tstan

dard

s[48],

[49],such

5Thehelp

ofMr.

Kenneth

Wick

sin

doingthismeasurem

entisgratefu

llyacknow

ledged.


0 5 10 15 20 25 30 35 40 45 500

10

20

30

40

50

60

70

80

90

100

Harmonic Order

Pe

rce

nta

ge

of

Fu

nd

am

en

tal (

% )

Fund = 119.6844 VoltsRMS = 119.9200 VoltsCreast Factor = 1.3851Min = −166.1000 VoltsMax = 166.1000 VoltsTHD = 6.2778 %HRMS = 7.5136 VoltsTIF / IT = 121.8154

Figure 4.5: Harmonic amplitude spectrum of the voltage waveform distortion at the point of commoncoupling (PCC).

10 -6 Ω

iNEUTRAL

vSA

isa

vSB

vSC

isb

isc

diod

ebr

idge

diod

ebr

idge

diod

ebr

idge

Vmax = 169.7 [V]

f = 60 [Hz]

Figure 4.6: Four-wire, three-phase system with \balanced" single-phase diode-bridge rectiers.

harmonic voltage distortions are usually within acceptable limits, other sensitive equipment

may be aected, and better alternatives for power conversion are actually available.

4.1.Dynam

icInteraction

betw

eenPow

erElectron

icDevices

andPow

erSystem

s98

05

1015

2025

3035

4045

50−

200

−150

−100

−50 0 50

100

150

200

Tim

e ( m

s )


vS

A

iNeutral

Figu

re4.7:

Curren

t ow

ingthroughtheneutra

lconducto

r.

Byusin

g,for

exam

ple,

step-up(boost)

DC-D

Ccon

verters(con

sistingof

IGBT'ssw

itch-

ingat

high

frequency

with

PWM

control

techniques,

freewheelin

gdiodes

andinductors),

incon

nection

with

diode-b

ridge

rectiers,

pow

erfactor

corrected(PFC)interfaces

canbe

design

ed.Such

PFC

circuits

areable

todraw

almost

sinusoid

alcurren

tsat

closeto

unity

pow

erfactor.

Single-

orthree-p

hase

controlled

thyristor

converters

canalso

adversely

aect

thequal-

ityof

pow

er,dueto

their

distorted

curren

twaveform

s,thenotch

ingof

theinputvoltage

waveform

caused

bythecom

mutation

amongthethyristors,

andthepoor

pow

erfactor.

Thechoice

ofapow

erelectron

iccon

verterisbased

onits

inten

ded

application

andon

the

price.

With

theintro

duction

ofmore

strictstan

dard

sfor

pow

erquality,

andwith

growing

concern

saboutthedynam

icinteraction

betw

eenpow

erelectron

icdevices

andthepow

er

system

,new

technologies

with

lessim

pact

and/or

lesssuscep

tibility

topow

erdistu

rbances

aregain

ingmarket

acceptan

ce.Exam

ples

areactive

lters

andoth

erdynam

iccom

pensatin

g


0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

Harmonic Order

RM

S C

urr

en

t (

A )

Third Harmonic Component

Figure 4.8: Harmonic amplitude spectrum of the current owing through the neutral conductor.

devices.

For a proper identication and solution of power quality problems, transient and steady-

state analysis are needed, which include models not only for the power electronics but also

the power system part, because of the dynamic interaction between them. Power electronics

based loads can either be the cause of problems in the power system, or they can be adversely

aected by electromagnetic transient phenomena coming from the power system. Reference

[24] presents the fundamental denition of electromagnetic phenomena aecting the electric

power quality, with realistic cases and practical monitoring results and examples of power

disturbances, such as:

Voltage sags and interruptions;

Transient overvoltages;

Harmonics;


0 2 4 6 8 10 12 14 16−200

−150

−100

−50

0

50

100

150

200Phase Voltage

time ( ms )

Vo

ltag

e (

V )

Figure 4.9: Voltage waveshape measured at the outlet of the Power Electronics Laboratory of the Depart-ment of Electrical and Computer Engineering at UBC, Vancouver, B.C., Canada.

Long duration voltage variation;

Wiring and grounding practices.

Voltage sags are by far the most common cause of disruption of operation of power elec-

tronics based loads, such as electronically controlled motor drives. Many industrial processes

(e.g. pulp and paper, textile, automotive, etc.) rely on accurate speed and torque control

through the use of power electronics, and thus become more or less vulnerable and suscepti-

ble to power quality problems depending on the sensitivity of these devices. This is actually

at the heart of many power quality problems!

It is also common that utility capacitor switching creates high frequency transients,

which may propagate through the distribution system and cause amplied transient voltage

oscillations in low voltage power factor capacitor banks in industry [106]. The commutation

of thyristors in current source inverter (CSI) adjustable speed drives (ASD) in industrial


0 2 4 6 8 10 12 14 16−200

−150

−100

−50

0

50

100

150

200Phase Voltage

time ( ms )

Vo

ltag

e (

V )

Measured Phase Voltage

Fundamental Voltage

Harmonic Distortion

Figure 4.10: Measured voltage waveshape, its fundamental component and its harmonic distortion.

plants may excite natural resonance modes of weak distribution and other industrial systems,

causing high frequency oscillations in the voltages and, consequently, \nuisance tripping" of

sensitive loads. [107].

Resonances tend to occur more frequently as more power factor and voltage support

capacitor banks are used in the system, mainly to control voltage on transmission or distri-

bution lines. Typically, the 5th order harmonic current commonly \injected" into the power

system by traditional power electronics converters, has the potential to cause problems, such

as capacitors failures. The importance of power quality then depends on its economic im-

pact on the industry, the utility, the society, and the country. Appropriate means to predict

problems in the early design stage or to diagnose and mitigate problems in existing systems

becomes a very important task of \power quality engineers".

EMTP-based programs, because of many available computer models for power systems

and power electronics, have become a necessary engineering tool for the evaluation of the


0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5Harmonic Amplitude Spectrum

Harmonic Order

Pe

rce

nt o

f F

un

da

me

nta

l ( %

)

Fundamental = 100%

Figure 4.11: Harmonic amplitude spectrum of the outlet waveshape voltage.

impact of power electronic devices on the quality of power. An extensive literature survey

about EMTP-based models for time and frequency domain analysis of electric and electronic

power systems can be found in [92], [6], [2], and elsewhere. Specialized conferences, such as

the International Power System Transients Conference (IPST) held every two years, provide

opportunities to exchange information about new techniques and practical experiences in

the use of EMTP-based programs.


0 5 10 15 20 25 30 35 40 45 50

−150

−100

−50

0

50

100

150

Harmonic Phase Angle

Harmonic Order

de

gre

es

( º

)

Figure 4.12: Phase-angle of the harmonic components of the outlet waveshape voltage.

4.2. Power Quality Assessment through EMTP-based Programs 104

4.2 Power Quality Assessment through EMTP-based

Programs

This section presents some real cases of power quality assessment through EMTP-based

simulations. Field test measurements made by the author and comparisons with time and

frequency domain computer analysis are also shown for some of the cases. Section 4.2.1

presents a harmonics case study. Section 4.2.2 presents a voltage sag case study. Section 4.2.3

is concerned with visual light icker caused by voltage uctuations.

4.2.1 Induction Furnace Harmonic Study

The problem of harmonic analysis in power systems is usually studied with steady-state

solution techniques, which use linear solutions at the harmonic frequencies. The charac-

teristic harmonic spectra of non-linear loads are assumed to be known, and are modeled

as current sources at the respective harmonic frequency. In reality, the harmonic current

sources are not exactly known, because they depend on the behaviour of the power system

as well. For example, harmonics from transformer saturation clearly depend on the voltage

magnitude and waveform at the transformer terminal. Only time domain simulations of

the EMTP type can address the interaction between the system and the harmonic sources,

which can result in non-characteristic harmonics as well. Time domain simulations can also

be useful to develop other types of power quality studies, such as fault analysis, transient

impulses caused by switching utility capacitor banks, diagnosing the eects of special loads

into the system, troubleshooting the failure of sensitive loads, evaluating the application of

\Custom Power" devices as solutions to power quality problems, etc.

This section presents an application of the EMTP in a distribution system study, where

the harmonics injected into a distribution feeder by induction furnaces were the prime con-

cern for power quality. It is taken from a harmonic problem experienced by ELEKTRO -

Eletricidade e Servicos S. A. 6, an electric utility in the southeast of Brazil. The three-phase

power system with a digital model of the induction furnace as a power electronic load was

6ELEKTRO - Eletricidade e Servicos S. A. , Rua Ary Antenor de Souza, 321 - Jardim Nova America,CEP 13053-024, Campinas, S. P. , BRAZIL.


simulated with the EMTP over a time span which was long enough to reach steady state.

Voltage and current waveforms were then analyzed with a Fourier analysis program to obtain

the harmonic content of the distorted waveforms. Field measurements at the point of com-

mon coupling between the utility and the industry are presented as well. Electromagnetic

transients programs are more accurate in representing nonlinear eects of the supply system,

and allow more detailed modelling of power electronic loads and devices, than steady-state

harmonic programs. Power electronics and power quality have such a strong correlation that

can only be fully described and analyzed with the use of time domain simulation techniques.

Induction heating has gained wide acceptance in industry because this type of heat-

ing process is considered clean, quick and eÆcient. On the other hand, the use of power

electronic devices for induction heating introduces harmonic currents, causing voltage dis-

tortions in the electric supply network. The dynamic interaction of these harmonics with the

electric system in terms of system conguration, loading and other conditions, may result

in linear resonances, or even in undesirable steady-state conditions, which all can result in

misoperation, failure and life reduction of equipment, with consequent economical losses.

Induction furnaces are power electronics-based loads, where the heat in the electrically

conducting workpiece to be melted is produced by circulating currents through electromag-

netic induction. Series or parallel-resonant inverters are typical congurations used to supply

energy to the induction coil, at a selected frequency, which can be in a range varying from

the power system frequency to a few hundred kilohertz [94]. Figs. 4.13 (a) and (b) illustrate

an induction furnace in operation.

The operation of these induction furnaces has produced distortions in the current and

voltage waveforms, and has created incompatibility problems between these special loads and

other sensitive loads connected on the same distribution feeder. Changes in the conguration

of the power supply system as well as application of passive lters, have minimized the

eects, but have not completely eliminated the harmonics power quality problem. Field

measurements have been made for dierent operating conditions to evaluate the eectiveness

of the already installed harmonic passive lters.


(a) (b)

Figure 4.13: (a) Metal melting by an induction furnace. (b) Induction furnace operation.

(a) History of the Induction Furnace Harmonic Problem

This section summarizes the actions taken by ELEKTRO - Eletricidade e Servicos S. A. ,

in assisting an industrial customer to deal with the harmonic problem created by the opera-

tion of induction furnaces in a distribution system. The problem emerged after a customer

complained that its sensitive loads (computer-based loads) were suering from the poor qual-

ity of the utility-supplied power. Coincidentally, this customer was connected to the same

feeder where one of the industrial plants of the same customer had been operating induc-

tion furnaces. At that time, a harmonic problem was detected and the sensitive load was

connected to a feeder from another substation. It was also recommended to the customer

to provide corrective ltering of the injected harmonic currents from the induction furnaces.

Since there was no clear regulation about this issue at that time, and because of the economic

costs involved, the solution was postponed by the customer.

Two years later, the utility installed a 9.0 MVAr capacitor bank in the same substation

supplying this customer. Resonance eects caused overcurrents in this capacitor bank, which

led the utility to change the capacitor digital overcurrent protection relays to electromechan-

ical relays. After that, the protection fuses of the customer's capacitor banks started to blow.

The utility's capacitor bank was turned o until another 25/30 MVA-138/13.8kV transformer

was later installed in the substation. Then, measurements were made after the utility capac-


itor bank was connected to a dierent bus bar from the feeder that supplies the induction

furnaces customer.

In August 1995, new measurements were made at the substation, especially on the feeder

supplying the induction furnaces [108]. Fig. 4.14 shows the feeder current at the moment

of maximum harmonic distortion. Sometimes later, the customer installed 4th and 5th order

harmonic tuned passive lters, with an economic motivation imposed by a new Brazilian

power factor legislation, which essentially changed from a minimum monthly average power

factor of 0.85 to a minimum hourly average of 0.92 (inductive limit from 06:00am to 12:00

midnight and capacitive limit from 12:00 midnight to 06:00am). Measurements were made in

1998 to verify the system overall performance on this feeder and in the respective distribution

substation.

August 29, 1995 at 20:40:49 GMTLIM_IV

Phase B Current

SS Wave

0 20 40 60 80 100 120 140

-600

-400

-200

0

200

400

600

Time (mSeconds)

Am

ps

0 5 10 15 20 25 30

0

20

40

60

80

100

120

Harmonic

Am

ps

Fund 192.7

RMS 235.6

CF 2.091

Min -492.6

Max 487.2

THD 70.41

HRMS 135.7

TIF/IT 101821

BMI/Electrotek

Figure 4.14: Current measurements in a distribution feeder supplying induction furnaces at the time ofmaximum voltage distortion.

(b) Harmonic Measurements

Harmonic measurements were made at the point of common coupling (PCC) between

the utility and the induction furnace customer, with modern monitoring equipment of the

type \Dranetz/BMI-PQNode 8010 and 8020", which acquires 256 samples per cycle (60Hz)

in each of the voltage channels, and 128 samples per cycle (60Hz) in each of the current


channels. With this equipment, it is possible to monitor power quality phenomena such as

voltage sags, voltage swells, outages, cold load pick-up, transient impulses, waveshape faults,

harmonic distortions and frequency deviations [15], [108], [43].

The following gures illustrate some of the harmonic measurements made in March/April

1998. Fig. 4.15 (a) shows the phase \A" measured current and Fig. 4.15 (b) shows the mea-

sured voltage between phases A and B, with their respective harmonic amplitude spectrum.

Both measurements were taken at a particular time when the induction furnace operation

was with the 4th and 5th order harmonic passive lters turned o (see Fig. 4.17).

April 01, 1998 at 03:58:49 LocalM_VPhase A CurrentSS Wave

0 10 20 30 40 50 60 70

-400-300-200-100

0100200300400

Time (mSeconds)

Am

ps

0 5 10 15 20 25 30

05

101520253035

Harmonic

Am

ps

Fund 198.9

RMS 204.8

CF 1.531Min -308.5

Max 313.4

THD 18.89

HRMS 37.58

TIF/IT 52660

BMI/Electrotek

(a)

April 01, 1998 at 03:58:49 LocalM_VPhase A-B VoltageSS Wave

0 10 20 30 40 50 60 70

-30000

-20000

-10000

0

10000

20000

Time (mSeconds)

Vo

lts

0 5 10 15 20 25 30

00.5

11.5

22.5

33.5

4

Harmonic

% F

un

d

Fund 13557

RMS 13592

CF 1.489

Min -20237Max 19805THD 7.172HRMS 972.3TIF/IT 234.9

BMI/Electrotek

(b)

Figure 4.15: (a) Phase \A" current measured with harmonic passive lters turned o. (b) Phase-to-phase\A-B" voltage measured with harmonic passive lters turned o.

Figs. 4.16 (a) and (b) illustrate the same measurements, but during an operating condi-

tion when the 4th and 5th harmonic passive lters were turned on (see Fig. 4.18).

The melting process inside the induction furnaces aects the total harmonic distortion

(THD) of the measured voltage at the point of common coupling very strongly. Fig. 4.17

shows the THD historic trend during one regular day of work, with the 4th and 5th harmonic

passive lters turned o from about 12:00 midnight to 06:00am. This THD trend presents a

characteristic variation along the day, depending on the load cycle at the industry. During

the time when the harmonic lters were turned o (from about 12:00 midnight to 06:00am)

4.2.Pow

erQuality

Assessm

entthrou

ghEMTP-based

Program

s109

Ap

ril 09

, 19

98

at 0

3:3

0:4

0 L

oca

lM

_V

Ph

ase

A C

urre

nt

SS

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ve01

02

03

04

05

06

07

0

-40

0-3

00

-20

0-1

00 0

10

02

00

30

0

Tim

e (m

Se

con

ds)

Amps

05

10

15

20

25

30

0 1 2 3 4 5 6 7

Ha

rmo

nic

Amps

Fu

nd

19

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RM

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ax

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ctrote

k

(a)

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at 0

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oca

lM

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Ph

ase

A-B

Vo

ltag

eS

S W

ave0

10

20

30

40

50

60

70

-25

00

0-2

00

00

-15

00

0-1

00

00

-50

00 0

50

00

10

00

01

50

00

20

00

0

Tim

e (m

Se

con

ds)

Volts

05

10

15

20

25

30

00

.20

.40

.60

.8 11

.21

.4

Ha

rmo

nic

% Fund

Fu

nd

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TH

D2

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9H

RM

S3

62

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IF/IT

11

5.1

BM

I/Ele

ctrote

k

(b)

Figu

re4.16:

(a)Phase

\A"curren

tmeasured

with

harm

onicpassiv

elters

turned

on.(b)Phase-to

-phase

\A-B"volta

gemeasured

with

harm

onicpassiv

elters

turned

on.

itiseasy

tosee

asubstan

tialincrease

intheTHDvalu

es.Fig.

4.18presen

tstheTHDhistoric

trendfor

anoth

eroperatin

gcon

dition

,with

the4thand5thharm

oniclters

turned

ondurin

g

allthetim

e.

TH

D (%

)

0 1 2 3 4 5 6 7 8

12:09:10

13:19:03

14:29:04

15:39:04

16:49:05

17:58:59

19:08:59

20:19:00

21:29:01

22:38:53

23:48:54

00:58:55

02:08:56

03:18:48

04:28:49

05:38:50

06:48:51

07:58:51

09:08:45

10:18:46

11:28:46

Figu

re4.17:

THDharm

onictren

d,with

harm

onicpassiv

elters

turned

ofro

m12:00midnightto

06:00am.

From

these

charts

itispossib

leto

conclu

dethat

passive

lters

canminim

izetheharm

onic

voltagedistortion

s,butnot

eliminate

them

completely.

New

pow

erelectron

icdevices

such

4.2.Pow

erQuality

Assessm

entthrou

ghEMTP-based

Program

s110

TH

D (%

)

0 1 2 3 4 5 6 7 8 12:00:53

13:10:54

14:20:54

15:30:47

16:40:48

17:50:49

19:00:50

20:10:42

21:20:43

22:30:44

23:40:44

00:50:38

02:00:39

03:10:39

04:20:40

05:30:41

06:40:33

07:50:34

09:00:35

10:10:36

11:20:28

Figu

re4.18:

THDharm

onictren

d,with

harm

onicpassiv

elters

turned

onallthetim

e.

asactive

lters,

migh

tbeableto

dynam

icallycom

pensate

thedistortion

sandim

prove

the

quality

ofpow

erat

theinterface

ofindustrial

andpow

ersystem

s.

Asfar

astheauthor

know

s,new

Brazilian

legislationaboutharm

onics

andoth

erpow

er

quality

phenom

enaare

curren

tlyunderdevelop

ment.Som

erecom

mendation

shave

been

used

toguideutility

system

plan

ners

andoperators

insupplyingpow

erto

special

loads.

Refer-

ence

[50]for

exam

ple,

presen

tssom

ecriteria

andglob

alvoltage

harm

oniclim

its(Table4.1),

based

ontheexperien

ceof

Brazilian

pow

erutilities,

aswell

ason

standard

sfrom

CIG

RE

(Conferen

ceIntern

ationale

des

Gran

dsReseau

xElectriq

ues

aHaute

Tension

-Intern

ational

Conferen

ceon

Large

High

Voltage

Electric

System

s),IEC

(Intern

ational

Electrotech

nical

Com

mission

)andIEEE(T

heInstitu

teof

Electrical

andElectron

icsEngin

eers).

Table4.1:

Globalharm

onicdisto

rtionlim

itsforthesystem

volta

ges

recommended

inBrazil.

Voltage

<69kV:THDmax=6%

Voltage

69kV:THDmax=3%

Odd

Even

Odd

Even

Order

Valu

e%

Order

Valu

e%

Order

Valu

e%

Order

Valu

e%

3,5,

75

2,4,

62

3,5,

72

2,4,

61

9,11,

133

8

19,

11,13

1.58

0.5

15to

252

15to

251

27

127

0.5


(c)Digital Modelling of Induction Furnaces in Distribution Networks

A three-phase detailed modelling of the distribution system, including the linear and non-

linear loads, transformer saturation eects, unbalanced conditions, power electronic loads,

automatic control devices, frequency dependent characteristics of the system and of the loads,

and so on, would be the ideal and recommended database for an electromagnetic transient

simulation, in order to analyze power quality phenomena. This would require a complete and

well-organized database of the system and load parameters, which is rarely available in prac-

tice. For distribution system planning and operation, such details are usually not required,

unless some specic power quality problem emerges as urgent and important. Typically, nei-

ther some important data is available, nor appropriate models exist to represent the physical

behaviour by digital simulation. One must therefore use simplications, which may make

the simulations unrealistic. Therefore, the development of more accurate models is needed

for power quality studies.

For this case study, some realistic data was available, and some simplications had to

be made for other data. The actual system under study is shown in Fig. 4.19. A Thevenin

equivalent circuit with a series connection of coupled resistances and inductances was used

to represent the 138kV transmission system, based on the given three-phase short-circuit

power (2881.3 MVA, angle of - 78 degrees) and single-line-to-ground short-circuit power

(1734.1 MVA, angle of - 77.7 degrees) from which the positive and zero sequence impedances

at the frequency of 60Hz can be calculated (assuming 100MVA as a base power, 138kV

as base voltage, and zero fault resistance for a single-line-to ground fault, one obtains:

Zposp:u: = 1=Ssc3php:u:, Zzerop:u: = 3=Ssc1phgp:u: 2 Zposp:u ). In reality, the resistance

and inductance derived from these impedances are frequency-dependent, which was ignored

in this study.

The transformer model used was based on three single-phase coupled impedances (\IN-

VERSE" option in MicroTran). The distribution line was modelled as a three-phase cou-

pled -circuit, with positive and zero sequence parameters at 60Hz. For the frequencies of

interest here, a -circuit representation is reasonably accurate. For higher frequencies, a

distributed-parameter line model would have to be used, either with constant parameters or


with frequency-dependent parameters.

The data for all the distribution feeders with their respective loads, and for the capaci-

tor banks for power factor correction were available. Not enough information was available

though for the induction furnaces. A digital model based on [93], [94] was therefore used:

a current-source, parallel-resonant inverter for induction heating, as shown in Fig. 4.20.

The resonant inverter is used to create variable frequency at the induction coil. The six-

pulse controlled rectier on the AC side of each induction furnace was supplied through a

13.8/0.48kV - 3.0MVA - 5.9% three-phase unit transformer in delta/wye-grounded connec-

tion. Saturation eects were not considered in this simulation. Realistic parameter values

of resistance, inductance and capacitance were available for the 4th and 5th order harmonic

passive lters.

EMTP-based programs can perform time-domain transient analysis or frequency-domain

analysis. A frequency-domain analysis was done for this case to nd the system impedance

as a function of frequency, seen from the point of common coupling (PCC) of the induction

furnace, as shown in Fig. 4.21 and Fig. 4.22. Not only the impedance at multiples of the

fundamental frequency was evaluated, but over the continuous frequency range as well, by

using a small step increment (f) in the frequency variation input parameter. It was small

enough to allow linear interpolation between calculated points. The two minimum impedance

values at 240Hz and 300Hz shown in Fig. 4.21 correspond to the eects of the 4th and 5th

order harmonic lters. Fig. 4.22 also shows the zero crossings of the phase angle of the system

impedance, from inductive to capacitive and vice-versa, thus indicating parallel (maximum

impedance) or series (minimum impedance) resonant conditions, respectively.

Next, the three-phase distribution system with the digital model of the induction furnace

as a power electronics load was simulated as a transients case (t = 16:6667s), using the

MicroTran version of the EMTP, until a time when steady state was reached. Voltage and

current waveforms were then processed through a Fourier analysis program to obtain the

harmonic content of the distorted waveforms. For this simulation case, all the distribution

feeders were represented. The induction furnace operating condition selected for this case

corresponds to the time of maximum total harmonic distortion (THD) measured at the point


138 kV

Bus I - 13.8kV Bus II - 13.8kV

InductionFurnaces

HarmonicPassive Filters

PCC

477.0 ACSR - 2km

25/30MVA138/13.8kV

9.72% (25MVA)

25/30MVA138/13.8kV

10.2% (25MVA)

Ssc 3ph = 2,881.3 / -78.0 o MVASsc 1phg = 1,734.1 / -77.4 o MVA

9.0MVAr13.8kV

3.0MVA13.8/0.48kV

5.9%

336.4 ACSR -0.5km

0.48kV

4th 5th

Figure 4.19: Distribution substation.

of commom coupling (PCC), without the harmonic lters (see Fig. 4.17).

Figs. 4.23 (a) and (b) show the current and voltage waveforms respectively, with their

harmonic contents, at the point of common coupling (PCC) between the utility electric

system and the customer facilities, for the induction furnaces operation with the 4th and 5th

order harmonic passive lters turned o. Fig. 4.15 is repeated here as Fig. 4.24 to facilitate


Lc

Cr Lr Rload

VA

VB

VC

INDUCTIONFURNACE

INVA

INVB

1 3 5

4 26

POS

NEG

Ld

Figure 4.20: Current-source, parallel-resonant inverter for induction heating.

0 50 100 150 200 250 300 350 400 450 5000

5

10

15

20

25

30

35

40

45

50

frequency ( Hz )

Imp

ed

an

ce M

ag

nitu

de

( O

hm

s )

Figure 4.21: Amplitude of the positive sequence system impedance at the PCC with harmonic lters.

the comparison between the simulated and the measured results.

Figs. 4.25 (a) and (b) show the current and voltage waveforms, but for induction furnace


0 50 100 150 200 250 300 350 400 450 500−100

−80

−60

−40

−20

0

20

40

60

80

100

frequency ( Hz )

Ph

ase

An

gle

( D

eg

ree

s )

Figure 4.22: Phase angle of the positive sequence system impedance at the PCC with harmonic lters.

operation with the 4th and 5th order harmonic passive lters turned on. This operating

condition corresponds to the time of minimum total harmonic distortion (THD) measured

at the point of common coupling (PCC), but in this case with the harmonic passive lters

turned on (see Fig. 4.18). Both operating conditions have approximately the same value

of fundamental current (199A RMS at the phase-to-phase RMS voltage of 13.8kV). Again,

Fig. 4.16 is repeated here as Fig. 4.26 to facilitate the comparison between the simulated

and the measured results.


0 10 20 30 40 50 60 70−400

−200

0

200

400Phase "A" Currrent

Time ( ms )

Cu

rre

nt (

A )

0 5 10 15 20 25 300

5

10

15

20

25

30

35

40

Harmonic Order

Pe

ak

Am

plit

ud

e (

A )

(a)

0 10 20 30 40 50 60 70−3

−2

−1

0

1

2

3x 10

4 Phase−to−phase "A−B" Voltage

Time ( ms )

Vo

ltag

e (

V )

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

Harmonic Order

Pe

ak

Am

plit

ud

e (

% F

un

d )

(b)

Figure 4.23: (a) Phase \A" current simulated with harmonic passive lters turned o. (b) Phase-to-phase\A-B" voltage simulated with harmonic passive lters turned o.


0 10 20 30 40 50 60 70

-400-300-200-100

0100200300400

Time (mSeconds)

Am

ps

0 5 10 15 20 25 30

05

101520253035

Harmonic

Am

ps

Fund 198.9

RMS 204.8

CF 1.531Min -308.5

Max 313.4

THD 18.89

HRMS 37.58

TIF/IT 52660

BMI/Electrotek

(a)


0 10 20 30 40 50 60 70

-30000

-20000

-10000

0

10000

20000

Time (mSeconds)

Vo

lts

0 5 10 15 20 25 30

00.5

11.5

22.5

33.5

4

Harmonic

% F

un

d

Fund 13557

RMS 13592

CF 1.489

Min -20237Max 19805THD 7.172HRMS 972.3TIF/IT 234.9

BMI/Electrotek

(b)

Figure 4.24: (a) Phase \A" current measured with harmonic passive lters turned o. (b) Phase-to-phase\A-B" voltage measured with harmonic passive lters turned o.


0 10 20 30 40 50 60 70−300

−200

−100

0

100

200


Time ( ms )

Cu

rre

nt (

A )

0 5 10 15 20 25 300

1

2

3

4

5

6

7

Harmonic Order

Pe

ak

Am

plit

ud

e (

A )

(a)

0 10 20 30 40 50 60 70−3

−2

−1

0

1

2

3x 10

4 Phase−to−phase "A−B" Voltage

Time ( ms )

Vo

ltag

e (

V )

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Harmonic OrderPe

ak

Am

plit

ud

e (

% F

un

d )

(b)

Figure 4.25: (a) Phase \A" current simulated with harmonic passive lters turned on. (b) Phase-to-phase\A-B" voltage simulated with harmonic passive lters turned on.

Some care must be taken when dening the time step size (which corresponds to the

inverse of the sampling frequency), for the transient simulation. Relatively large step sizes

introduce errors into the results of a time-domain simulation and consequently into the post-


0 10 20 30 40 50 60 70

-400-300-200-100

0100200300

Time (mSeconds)

Am

ps

0 5 10 15 20 25 30

01234567

Harmonic

Am

ps

Fund 198.7RMS 199.0CF 1.515Min -301.5Max 294.2THD 5.585HRMS 11.10TIF/IT24637

BMI/Electrotek

(a)


0 10 20 30 40 50 60 70

-25000-20000-15000-10000-5000

05000

100001500020000

Time (mSeconds)

Vo

lts

0 5 10 15 20 25 30

00.20.40.60.8

11.21.4

Harmonic

% F

un

d

Fund 14011RMS 14037CF 1.431Min -20083Max 19896THD 2.589HRMS 362.7TIF/IT 115.1

BMI/Electrotek

(b)

Figure 4.26: (a) Phase \A" current measured with harmonic passive lters turned on. (b) Phase-to-phase\A-B" voltage measured with harmonic passive lters turned on.


processing frequency-domain analysis. The explanation is that the dierential equations

of inductances and capacitances in EMTP-based programs are solved with the trapezoidal

integration rule, thus producing errors which are a function of the frequency and the time

step size (t) [2], [12]. Moreover, It is also recommended that the time step size be such

that the period of the fundamental frequency is an integer multiple of t, in order to avoid

the generation of non-characteristic harmonics in the post processing Fourier analysis. For

example, if the system nominal frequency is 60:0Hz and the maximum frequency expected in

the transient simulation is in the order of fmax = 6kHz, then the step size can be calculated

as t = 1=(10 fmax) = 16:66666s;

New solutions to this harmonic problem could be investigated using EMTP-based sim-

ulations, as for example the possible use of active lters [60], [61] to minimize harmonic

distortions. Such power electronic device should be able to inject a shunt compensated cur-

rent, as shown in Fig. 4.27, at the point of common coupling, thus improving the quality of

power at the interface of industrial and utility power systems.

0.05 0.052 0.054 0.056 0.058 0.06 0.062 0.064 0.066−400

−200

0

200


Time ( ms )

Cu

rre

nt (

A )

0.05 0.052 0.054 0.056 0.058 0.06 0.062 0.064 0.066−40

−30

−20

−10

0

10

20

30

40Ideal Compensation Current for Phase "A"

Time ( ms )

Cu

rre

nt (

A )

Figure 4.27: Instantaneous ideal compensation current to be \injected" by a shunt active lter.


(d) Conclusions

This section presented an application of the electromagnetic transients program (EMTP)

to the analysis of a power quality issue in a distribution system, where the voltage and current

harmonic distortions were produced by induction furnaces. The history of this harmonic res-

onance problem was described. A digital model of the induction furnace as a current-source,

parallel-resonant inverter load, together with three-phase representations of the supplying

distribution system were used in a time-domain simulation. After reaching a time considered

as steady state, the simulated results were processed through a Fourier analysis program to

obtain the harmonic contents of the voltage and current waveforms at the point of com-

mon coupling. Field measurements were also presented. A comparison between the actual

measurements and the EMTP simulations is presented in Table 4.2 and Table 4.3. The dier-

ences in the values of the total harmonic distortion (THD) and the telephone in uence factor

(TIF) are possibly due to simplications or lack of realistic data in the digital modelling.

The knowledge of detailed manufacturer data of the induction furnaces and their operat-

ing conditions, such as the natural resonant frequency, would allow the improvement of the

EMTP simulation. Considering unbalanced conditions could also improve the simulation

results. Both were unfortunately not available.

Table 4.2: Comparison between eld measurements and EMTP simulation results for the operating con-dition with the harmonic passive lters turned OFF.

Phase \A" Current [A] Phase-to-phase \A-B" Voltage [V]

Parameters Measured Simulated Error % Measured Simulated Error %

Fund 198.9 201.8 1.5 13,557 13,776 1.6

RMS 204.8 204.2 0.3 13,592 13,814 1.6

CF 1.531 1.491 2.6 1.489 1.444 3.0

Min -308.5 -304.4 1.4 -20,237 -19,950 1.4

Max 313.4 304.4 2.9 19,805 19,950 0.7

THD [%] 18.89 15.42 18.4 7.172 7.487 4.4

HRMS 37.58 31.11 17.2 972.3 1,031.4 6.1

TIF/IT 52,660 64,416 22.3 234.9 520.1 121.4

EMTP-based simulations can be useful tools for harmonic analysis, based on the fact that

very detailed eects can be taken into account. Once the system is modelled for an EMTP-


Table 4.3: Comparison between eld measurements and EMTP simulation results for the operating con-dition with the harmonic passive lters turned ON.

Phase \A" Current [A] Phase-to-phase \A-B" Voltage [V]

Parameters Measured Simulated Error % Measured Simulated Error %

Fund 198.7 190.2 4.3 14,011 14,211 1.4

RMS 199.0 190.4 4.3 14,037 14,219 1.3

CF 1.515 1.474 2.7 1.431 1.422 0.6

Min -301.5 -279.5 7.3 -20,083 -20,130 0.2

Max 294.2 280.7 4.6 19,896 20,220 1.6

THD [%] 5.585 5.110 8.5 2.589 3.224 24.5

HRMS 11.10 9.72 12.4 362.7 458.2 26.3

TIF/IT 24,637 36,910 49.8 115.1 219.9 91.1

based software, any type of studies can be performed. The new models developed in this

thesis project, hopefully will contribute to make EMTP-based programs more valuable tools

for electric utility companies and industrial customers in evaluating power quality problems.

4.2.2 Voltage Sag Analysis with EMTP-based Simulation

Voltage sags or voltage dips are short duration variations in the supply voltage, caused

by faults in transmission lines or in parallel distribution feeders, or caused by the start-up

of large induction motors or other types of sudden load variations. The majority of power

quality problems are associated with voltage sags, which are very common in today's electric-

ity industry. The ride-through characteristics of modern electronic and computer-controlled

loads are very sensitive to short duration variations in the supply voltage. An entire process

may be shut down when the voltage sags momentarily. The equipment tolerance charac-

teristics to voltage sags vary very much among equipment manufacturers. Moreover, in

most cases the equipment ride-through characteristics are not known. For these reasons,

the CBEMA curve [51] (or the ITIC curve) has been widely used as a rst reference for

power quality studies related to short duration voltage variations. Fig. 4.28 presents actual

measurements of voltage sag phenomena with an overlay of the CBEMA curve 7.

7\We typically employ the curve only from 0.1 cycles and higher due to limitations in power qualitymonitoring instruments and dierences in opinion over dening the magnitude values in the subcycle timeframe." From [24], pp. 37-38.


The power system characteristics at the point where the sensitive load is connected is

another important issue for possible mitigation of voltage sag problems. Power system

protection and operating practices may also aect the success or failure of loads which are

sensitive to voltage disturbances.

Fig. 4.29 (a) shows a voltage sag phenomenon caused by a single-line-to-ground fault in a

distribution feeder, which is in parallel to a feeder supplying a \PVC" pipe (and other plastics

derived products) manufacturer with sensitive loads. The industrial process, controlled by

DC drives, is stopped if the voltage sags at the point of common coupling to less than 90%

of the nominal voltage during a time greater than 18 cycles (300ms).

Fig. 4.29 (b) shows the simulation results using MicroTran. The simulation does not

match the measurements exactly, because the dynamic behaviour of industrial loads (typ-

ically induction motors) were not included in the simulation model. Nevertheless, EMTP-

based programs have the exibility to include aggregated load models [109]. More research,

however, seems to be needed for the accurate representation in EMTP-based programs of

the dynamic behavior of loads, which is beyond the scope of this thesis project. Custom

Power Controllers such as the dynamic voltage restorer (DVR), is a promising solution for

the mitigation of voltage sag phenomena.

4.2.3 Welding Industry Voltage Fluctuation Study A Visual

Flicker Case

Flicker is historically considered a problem of perception, because the human ability to

visually sense light changes caused by voltage uctuations does vary. However, there are

certain levels of light icker which can be easily detected, though their impact on possible

human brain disorders or any other potential health damage is diÆcult to quantify. This

section presents a recent case of light icker and some simplied EMTP-based simulations.

The operation of a welding machine, to produce meshed wires for construction, connected

to a distribution system was simulated with MicroTran. If the duty cycle of the welding pro-

cess is near 8.8 Hz, the human eye would perceive the maximum visual ickering eect,

even when the RMS voltage uctuations are in a range of very small percentage deviations.


March 15, 1996 at 18:56:21 PQNode LocalPQNode Group

Triggering Phase,Max Depth 16/04/96 06:03:38 PQNode Local

0.001 0.01 0.1 1 10 100 1000 10000

0

50

100

150

200

250

300

Time (Cycles)0.001 0.01 0.1 1 10 100 1000 10000

0

50

100

150

200

250

300

Time (Cycles)

% V

olts

Max 106.0

Min 73.50

BMI/Electrotek

Figure 4.28: Voltage sag measurements (%RMS versus time duration) with an overlay of the CBEMAcurve. For time durations less than 1 cycle the equipment seems to measure peak values.

IEC Standard 1000-4-15 provides the specications for a ickermeter with \lamp-eye-brain"

frequency response to light ickering eects. Fig. 4.30 shows the simulated instantaneous

voltage for a power electronics controlled welding machine . The operating duty cycle, con-

trolled by the semiconductor ring angle, results in a modulating frequency of approximately

7Hz for this case, as shown in Fig. 4.31, which causes the visual light icker. Cost-eective

solutions for voltage uctuation problems are usually related to changes in the load duty

cycle, when this does not aect the industry productivity or the quality of the manufac-

tured product. The application of reactive dynamic compensation through power electronic

devices, such as the distribution static synchronous compensator (D-STATCOM) can eec-

tively mitigate this type of power quality problem [110].


March 25, 1996 at 08:25:40 PQNode LocalTIGRE2

Phase A-B Voltage

RMS Variation

Trigger

0 0.1 0.2 0.3 0.4 0.5

707580859095

100105110

Time (Seconds)

% V

olts

0 25 50 75 100 125 150 175 200

-100-75-50-25

0255075

100

Time (mSeconds)

% V

olts

Duration

0.233 Sec

Min 73.50

Ave 89.70

Max 100.8

Ref Cycle

30664

Uncalibrated Data

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

4 Instantaneous Phase−to−phase"A−B" Voltage at the PCC

Time ( s )

Vo

ltag

e (

Vo

lts )

(b)

Figure 4.29: (a) Phase-to-phase \A-B" measured voltage sag. (b) Phase-to-phase \A-B" simulated voltagesag.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1600

650

700

750

800

850

900

950

1000Instantaneous Voltage at the Point of Common Coupling

Time ( s )

Vo

ltag

e (

Vo

lts )

Figure 4.30: Instantaneous voltage uctuations causing light ickering eect.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1000

−500

0

500

1000Phase "A" Voltage

Time ( s )

Vo

ltag

e (

Vo

lts )

40 45 50 55 60 65 70 75 800

0.5

1

1.5

2Harmonic Amplitude Spectrum

Frequency ( Hz )

% V

olts

Figure 4.31: Modulated voltage and respective amplitude frequency spectrum

4.3. EMTP-based Simulation Cases with SSCPS 125

4.3 EMTP-based Simulation Cases with SSCPS

This section presents a collection of test cases to validate the EMTP-based models de-

veloped in this thesis. Where an interesting and already published benchmark simulation

case with all the necessary data was readily available, it was selected because it made com-

parisons easy. In other cases, simple circuits were assembled by the author to test and prove

the ideas.

4.3.1 Basic Control and Control Devices Simulation Cases

This section provides more explanations about the theory presented in the previous chap-

ters, with some applications to test cases.

(a) The Eect of a One Time Step Delay in the Solution of EMTP-based Simu-lations

Reference [66] presents some interesting cases where the one time step delay in the solution

of control and system equations, or the internal delays inside TACS, give wrong or inaccurate

simulation results. For example, the simulation of a transfer function with poles on the

imaginary axis of the complex plane is very sensitive to time delays, which can \move" the

poles to the right half of the complex plane, resulting in instability. To illustrate this, Fig.

4.32 presents a control block diagram of a transfer function with a second order dierential

equation. The simultaneous solution is shown in Fig. 4.33. The intentional introduction of

a one time step delay in the control system, as illustrated in Fig. 4.34, leads to an unstable

resonance condition, as shown in Fig. 4.35. These results emphasize the \importance of a

simultaneous solution approach", which is critical in some cases for the correct simulation

of control and power system equations with EMTP-based programs.


1_____

s-

+

1_____

s

1.0 [V]

v IN v OUT

Figure 4.32: Control block diagram of a second order dierential equation with poles on the imaginaryaxis of the complex plane.

0 10 20 30 40 50 60 70 80−8

−6

−4

−2

0

2

4

6

8

Time ( s )

Vo

ltag

e (

V )

vout

vin

Figure 4.33: Solution of system with bounded resonance oscillations.


1_____

s-

+

1_____

s

1.0 [V]

v IN v OUT

delay1 ∆t

Figure 4.34: Introduction of a one time step delay in the control block diagram.

0 10 20 30 40 50 60 70 80−8

−6

−4

−2

0

2

4

6

8

Time ( s )

Vo

ltag

e (

V )

vout

vin

Figure 4.35: Solution of system with unstable resonance oscillations caused by the introduction of onetime step delay.


(b) Basic Control Blocks, Transfer Functions and Filters

Laplace transfer functions are essential for control design and simulation. Generally,

classical control blocks such as proportional (P), integral (I), derivative (D), (and their com-

binations PI, PD, PID), lead-lag's for phase compensation, washout lters, etc., are present

in any analog or digital control scheme. Many if not most of the rst, second or higher

order dierential equations used to model the physical behavior of electrical, mechanical,

chemical, and any other systems use such transfer functions for their mathematical repre-

sentation. Transfer functions can also represent passive and active lters (i.e., connections of

circuits with operational ampliers, resistors and capacitors). Depending on their properties

and frequency response, these lters are referred to with special names, such as low-pass,

Butterworth, Chebyshev, Legendre-Papoulis, Bessel, elliptic, high-pass, band-pass, notch,

band-elimination, all-pass (magnitude), all-pass (phase), etc. [80]). Therefore, the method-

ology proposed in this thesis for the simultaneous solution of transfer functions (Chapter

2) expands considerably the potential applications of EMTP-based programs in time and

frequency domain simulation studies.

It is important to mention that other general purpose and powerful computer tools,

such as MATLAB [103], can also be very useful for engineers and scientists, especially for

the design of control systems. The computer program SPICE seems to be more used for

electronics and power electronics simulations. EMTP-based programs, however, still seem

to have more detailed and proven models, especially for the power system part.

This section then presents some test cases with classical control blocks, which are simu-

lated with the simultaneous solution method implemented experimentally in MicroTran, the

UBC version of the EMTP. The rst control block diagram case is illustrated in Fig. 4.36,

which represents the classical linearized \swing equation", which is used in power system

small-signal stability studies of a single machine connected to an innite bus, as extracted

from page 731 of reference [86], where:

KS = 0:757pu torque/rad = synchronizing torque coeÆcient in pu torque/rad;

KD = 10 or -10 pu/pu = damping torque coeÆcient in pu torque/pu speed deviation;

H = 3:5 MWs/MVA = inertia constant in MWs/MVA;


!r = speed deviation in pu = (!r !0)=!0;

Æ = rotor angle deviation in electrical rad;

!0 = rated speed in electrical rad/s = 2f0 = 377 rad/s for a 60Hz system;

Tm = 0:1pu = mechanical torque deviation in pu;

Tes = synchronizing torque component in pu;

Ted = damping torque component;

Ta = TmTeS Ted = accelerating torque deviation.

Fig. 4.37 illustrates the simulation results for a -0.1pu disturbance in the per unit me-

chanical torque deviation (Tm). With a positive damping torque coeÆcient (KD = 10)

the rotor angle deviation (Æ) presents damped natural oscillations and reaches a new stable

operation point in steady state. As shown in Fig. 4.38 with a negative damping torque co-

eÆcient (KD = 10) the rotor angle deviation (Æ) presents amplied natural oscillations,

as expected, causing small-signal instability. Alternatively, the control block diagram pre-

sented in Fig. 4.36 could also be represented by a canonical second order transfer function,

as presented in Fig. 4.39, where:

K = 1KS

(4.5)

!n =pKS

!02H (4.6)

= 12

KD

2H!n= 1

2KDp

KS2H!0(4.7)


KS

KD

ω0_____

s

1_____2H s

-

-

+ ∆δ∆ωr

∆Tes

∆Ted

∆Ta∆Tm

- 0.1 p.u.

Figure 4.36: Classical linearized \swing equation", used in power system small-signal stability studies ofa single machine connected to an innite bus.

0 1 2 3 4 5 6 7 8 9 10−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

Time ( s )

Va

ria

ble

s in

pe

r u

nit

( p

.u.

)

∆

5*

Tes

Ted

Ta

∆ω

Tm

∆δ

∆

∆

∆

Figure 4.37: Simulation results of the synchronous machine rotor angle deviation, in the presence of apositive damping torque coeÆcient.


0 1 2 3 4 5 6 7 8 9 10−150

−100

−50

0

50

100

150

200

Time ( s )

Ro

tor

an

gle

de

via

tion

( p

.u.

)

∆δ

Figure 4.38: Simulation results of the synchronous machine rotor angle deviation, in the presence ofnegative damping torque coeÆcient.

K ωn2

_________________s2 + 2 ξωn s + ωn

2

∆δ∆Tm

- 0.1 p.u.

Figure 4.39: Canonical second order transfer function representation of the single-machine innite bussystem.


(c) Voltage and Current Sensors, Control Devices, Function Blocks and DigitalLogic Gates

The solution method for voltage and current sensors, control devices, function blocks and

digital logic gates follows the procedures for dependent sources presented in Chapter 2. For

example, the simultaneous solution through the compensation method for a generic linear

or nonlinear voltage-controlled voltage source can be used

to sense a voltage signal, with the compensating current at the controlling branch j

set to zero, i.e., ij = 0;

then to dene any linear or nonlinear voltage function at the controlled branch k, i.e.,

vk = f(vj);

and nally to use the Newton-Raphson algorithm presented in Section 2.2.5, (Fig.

2.19) to calculate a solution for the compensating currents ij and ik.

If more than one control voltage signal needs to be sensed, their respective compensating

branch currents are \simply set to zero". On the other hand, if currents have to be sensed,

their respective branch voltages can be set to zero. Since control signals are usually voltage

signals (or current signals converted to voltage signals), a current sensor can be represented

by a current-controlled voltage source.

The implementation of control devices (such as transport or time delay blocks, com-

parators, zero crossing detection and generation of gate ring signals, Clark transformation,

Park transformation, etc.), FORTRAN function blocks (such as SIN, COS, TAN, ASIN,

ACOS, LOG, EXP, SQRT, MULTIPLICATION, DIVISION, etc.) and digital logic gates

(AND, OR, NOT, NAND, NOR, etc.) uses the same concept of the simultaneous solution

for voltage-controlled voltage sources presented before.

The implementation of a simultaneous solution for limiters requires special attention, as

for example in the case of the widely used PI controllers with non-windup limiters. Reference

[66] clearly explains the problem and the solution for a correct EMTP-based simulation.


Special functions, as for example for the evaluation of average values (e.g. \rolling average

power" presented in [66]), root mean square (RMS) values, instantaneous total harmonic

distortion (THD) values, etc. can be easily modelled with control blocks, and with sub-

circuit implementation.

4.3.2 Power Electronics Simulation Cases

The cases presented in this section are based on references [65], [93], [89], [94], [104], [95].

(a) Dynamic Control of the Firing Angle (\") of a Thyristor

Fig. 4.40 illustrates a simple circuit to demonstrate three important contributions of this

thesis, for the dynamic simulation of power electronic devices in EMTP-based programs:

simultaneous solution for voltage sensors;

zero crossing detection and generation of ring pulses with an instantaneous updating

of the controlling ring angle . The controller is enabled by a voltage signal of \1

volt" and disabled by a voltage signal of \0 volt";

EMTP-based voltage-controlled unidirectional current owing switch, to represent the

thyristor, which receives the gate signal directly from the controller.

The voltage sensor in Fig. 4.40 is modelled with the equations for an \ideal" voltage-

controlled voltage source (VCVS) presented in Section 2.2, i.e.:

ij = 0 (4.8)

vOPENj +vOPENk

A+rj1

rk1A

i1 + :::

:::+rjj

rkjA

ij +

rjk

rkkA

ik + :::

:::+rjM rkM

A

iM = 0

(4.9)

where A = 1. For this case, the synchronizing signal for the gate ring controller could have

been sensed directly from the excitation source vSA, without the need for a voltage sensor,

which was included here for completeness of the test case.


The gate ring controller is modelled as a \multi-terminal voltage-controlled voltage

source", where three controlled voltages are sensed and used to determine the voltage source

at the gate, vGATE (a pulse of amplitude equal to 1 with a specied width, or zero volts). The

output vGATE depends on the values of vSY NCHR (which is used to detect, with interpolation,

the time of zero crossings) , vALPHA (which is the ring angle in degrees, converted to time,

based on the given input frequency) and venable=disable (whose value of 1 or zero is multiplied

by vGATE to enable or disable the control ring). The following equations are used to solve

for the gate ring controller using the compensation method:

ij = 0 (4.10)

ik = 0 (4.11)

il = 0 (4.12)

vOPENm + rm1i1 + :::::: + rmmim + :::+ rmM iM + vGATE = 0

(4.13)

In addition to equations 4.10 to 4.13, conditional IF-statements are used to implement the

logic described above equation 4.10.

The thyristor was modelled as an EMTP-based voltage-controlled unidirectional current

owing switch, which receives the gate signal directly from the controller.

The solution with the compensation method using an iterative Newton-Raphson algo-

rithm requires the calculation of a Thevenin equivalent for each branch, as mentioned before

in the fundamental assumptions of Section 2.2.2. In cases where this calculation fails, the

connection of large resistors in parallel with the branch may make a Thevenin equivalent

circuit possible 8. Eventually, in an EMTP production code, the use of internal variables

for the control could result in a more economic implementation. Internal control variables

would actually not have any physical connection with the power network part of the circuit.

Fig. 4.41 shows the resulting voltages and currents simulated with MicroTran and the

method proposed in this thesis, which is a truly simultaneous solution of the power and

control circuit. The time step size used was t = 16:6667s.

8In MicroTran, the connection of a large resistance of 109 and with the near zero tolerance parameterfor checking matrix singularities, EPSILON=1012, this problem can easily be solved.


2.5 ΩVmax = 5.0 [V]

f = 60 [Hz]

vLOAD

vSA

iLOAD

5 mH

vGATE

venable /

disable

vALPHAvSYNCHR

α

1.0 [V]tstop = 25.0 [ms]

45.0 [V]Voltage Sensor

Gate Firing Controller

Firing Angle(degrees)

f=60Hzpulse width=10

degreees

Figure 4.40: Circuit for the dynamic control of the ring angle (\") of a thyristor.

4.3.EMTP-based

Simulation

Cases

with

SSCPS

136

05

1015

2025

3035

4045

50−

5

−4

−3

−2

−1 0 1 2 3 4 5

Tim

e ( m

s )


vL

OA

D

iLO

AD

vsyn

chr

vG

AT

E v

en

ab

le / d

isab

le

Figu

re4.41:

Volta

ges

andcurren

tsin

acircu

itwith

dynamiccontro

loftherin

gangleofathyristo

r.


(b) Dynamic Control of the Firing Angles of a Three-Phase Six-Pulse Thyristor-Bridge Rectier

Fig. 4.42 illustrates a phase controlled rectier with a feedback control system based on

the manual of PSIM [65]. \It should be noted that, in PSIM, the power and the control

circuit are solved separately. There is one time step delay between the power and the control

circuit solutions" [65].

The six-pulse ring controller implemented with the models developed in this thesis

project uses a simultaneous solution for the power and control circuit equations. Similarly to

the simple ring controller \" of the previous test case, a multi-terminal voltage-controlled

voltage source is used to model it. The inputs to the six-pulse ring controller are the

synchronizing signal (voltage vAC sensed from the supply system), the dynamic ring angle

(resulting from the ACOS control block, which receives the signal from the limited PI

controller, after comparison with the desired reference voltage for the DC load). From

the gating signal generated to the thyristor with identication number \1", all the other

gating signals are derived sequentially by adding a time delay corresponding to 60 degrees

at the 60Hz frequency. For starting purposes, whenever a ring signal is sent to a particular

thyristor, another \isolated" ring signal is sent to the previous thyristor as recommended

in [88].

Fig. 4.43 presents the EMTP simulated voltages and currents, using a t = 16:6667s

and with a dynamic control of the ring angles of the three-phase six-pulse thyristor-bridge

rectier. Fig. 4.44 shows the dynamic behavior of the control variables, emphasizing the

ring control signal , and Fig. 4.45 illustrates the dynamics of the voltage control signals

at the output of the proportional-integral (PI) control block and the limiter control block.


G1

2 Ω

iLOAD

Vmax = 81.65 [V]

f = 60 [Hz] vLOAD

vSYNCHR

vA

400 µ F

vDC

5 mH +

-

vB

vC

PI ACOS

G1G2G3G4G5G6

G3 G5

G4 G6G2

vALPHA

vENABLE /

DISABLE

vLIMvPIvERRvREF

vSENS = vLOAD

-

+

100 [V]

1 [V]

KP=0.01KI= 1.0

1

-1

vAC

Figure 4.42: Circuit for the dynamic control of the ring angles of a three-phase six-pulse thyristor-bridgerectier.

4.3.EMTP-based

Simulation

Cases

with

SSCPS

139

05

1015

2025

−150

−100

−50 0 50

100

150

Tim

e ( m

s )


iLO

AD

vsyn

chr

vL

OA

D

vD

C

vA

Figu

re4.43:

Volta

ges

andcurren

tswith

dynamic

contro

loftherin

gangles

ofathree-p

hase

six-pulse

thyristo

r-brid

gerecti

er.

4.3.EMTP-based

Simulation

Cases

with

SSCPS

140

05

1015

2025

3035

4045

50−

20 0 20 40 60 80

100

120

Tim

e ( m

s )


vA

LP

HA

vR

EF

vE

RR

vS

EN

S =V

LO

AD

Figu

re4.44:

Dynamiccontro

loftherin

gangles

ofathree-p

hase

six-pulse

thyristo

r-brid

gerecti

er.

4.3.EMTP-based

Simulation

Cases

with

SSCPS

141

05

1015

2025

3035

4045

500

0.2

0.4

0.6

0.8 1

1.2

1.4

Tim

e ( m

s )


vL

IM

vP

I

Figu

re4.45:

Dynamicvolta

gecontro

lsig

nalsattheoutputoftheproportio

nal-in

tegral(P

I)andthelim

itercontro

lblocks.


(c) Dynamic Control of Three-Phase PWM Voltage Source Inverter

Fig.4.46 presents a circuit for the dynamic control of a three-phase PWM voltage source

inverter (VSI) [65]. Again, the power and the control circuit equations are solved simultane-

ously with the methods proposed in this thesis. For this EMTP-type simulation it was used

a t = 16:66667s.

The phase \A" modulation and triangular carrier waveforms for generation of gating

signals through sinusoidal pulse width modulation (PWM) are presented in Fig. 4.47.

With the use of comparators and NOT logic gates the ring signals are dynamically

generated, in a simultaneous solution with the network equations through the compensation

method.

The IGBT's with anti-parallel diodes were represented, for simplicity and without much

loss of accuracy in this simulation, by the EMTP-based voltage-controlled switches, which

were implemented in this thesis through the GATE subroutine. Here it is opportune to

discuss the issue of simultaneous commutation: Let us assume that the current is owing

through the IGBT with identication number \1" from the DC source to the load. When

IGBT number 1 is turned o, the voltage vSANEUTR reverses polarity almost instanta-

neously (due to the behavior of the inductor, which forces the current to keep owing in

the same direction), thereby forward biasing the anti-parallel diode of the IGBT with iden-

tication number \4", which then starts conducting. In digital simulation programs this

means that there is simultaneous commutation between IGBT 1 and the anti-parallel diode

at IGBT 4. This could be modelled as it is, i.e., with an IGBT and diode in anti-parallel

(which would have to be modelled with piecewise linear or nonlinear model), or as a voltage-

controlled bidirectional current owing switch, where the control signals play the role of the

commutation.

Fig. 4.48 presents the node voltage \vSA" generated by the 3-Phase PWM voltage source

inverter (VSI), whereas Fig. 4.49 shows the voltage across the load \vSANEUTR" and the

current in phase \A" supplied to the load.

The dynamically generated 3-phase load currents are illustrated in Fig. 4.50. The line-


to-line voltage generated by the three-phase PWM voltage source inverter (VSI) is shown in

Fig. 4.51.

Most of the advanced Custom Power Controllers [60] (and active lters [111]) apply

this type of converter to synthesize voltages or current waveforms according to the desired

\dynamic reference modulating signal". When a current is to be synthesized, dynamic

hysteresis current-band PWM converters are used [60] 9. Therefore, the models developed in

this thesis, will hopefully be useful for the accurate EMTP-simulation of a variety of existing

and new power electronic devices, especially those aimed at improving the quality of power

in utility and industrial systems.

9The Ph.D thesis \Active Power Line Conditioners" of Dr.-Ing. Mauricio Aredes, can be downloaded bythe reader from the the web site http://www.dee.ufrj.br.


IGBT1 IGBT3 IGBT5

IGBT4 IGBT6 IGBT2

vTRI

vA

vB

vC

vSAvSB

vSC

vNEUTRiSA-NEUTR

iSB-NEUTR

iSC-NEUTR

0.8 [V]

0.8 [V]

0.8 [V]

1 [V]

10 9 Ω

3.87 Ω 7.7 mH

450 [V]

f = 60 [Hz]

f = 1500 [Hz]phase = -180 [degrees]

COMPARATOR

NOT

Figure 4.46: Circuit for the dynamic control of three-phase PWM voltage source inverter (VSI).

4.3.EMTP-based

Simulation

Cases

with

SSCPS

145

05

1015

2025

−1.5

−1

−0.5 0

0.5 1

1.5

Tim

e ( m

s )


vA

vT

RI

Figu

re4.47:

Phase

\A"modulatio

nandtria

ngularcarrier

wavefo

rmsforgenera

tionofgatin

gsig

nals

throughsin

usoidalpulse

width

modulatio

n(PWM).

4.3.EMTP-based

Simulation

Cases

with

SSCPS

146

05

1015

2025

−600

−400

−200 0

200

400

600

Tim

e ( m

s )


vS

A

Figu

re4.48:

Nodevolta

ge\vSA"genera

tedbyathree-p

hase

PWM

volta

gesource

inverter

(VSI).

4.3.EMTP-based

Simulation

Cases

with

SSCPS

147

05

1015

2025

−600

−400

−200 0

200

400

600

Tim

e ( m

s )


vS

A − v

NE

UT

R 5 * iS

A−

NE

UT

R

Figu

re4.49:

Volta

geacro

sstheload\vSANEUTR"andcurren

tsupplied

totheloadbyathree-p

hase

PWM

volta

gesource

inverter

(VSI).

4.3.EMTP-based

Simulation

Cases

with

SSCPS

148

05

1015

2025

−60

−40

−20 0 20 40 60

Tim

e ( m

s )


iSA

−N

EU

TR

iSB

−N

EU

TR

iSC

−N

EU

TR

Figu

re4.50:

Loadcurren

tssupplied

byathree-p

hase

PWM

volta

gesource

inverter

(VSI).


0 5 10 15 20 25−600

−400

−200

0

200

400

600

Time ( ms )

Vo

ltag

e (

V )

vSA

− vSB

Figure 4.51: Line-to-line voltage generated by a three-phase PWM voltage source inverter (VSI).

4.4. Synthesis of Simulation Guidelines for Studies with EMTP-based Programs 150

4.4 Synthesis of Simulation Guidelines for Studies with

EMTP-based Programs

This section emphasizes the basic issues which are critical for the successful evaluation of

the impact of power electronic devices on the quality of power. Important factors regarding

power quality monitoring, modelling and simulation in EMTP-based programs of power

system components and power electronics devices are pointed out, with the main objective

of analyzing their dynamic interaction and of evaluating their impact on electric power

quality.

For power quality monitoring the important factors are:

Evaluation based on accurate measurements of power quality phenomena, through the

use of instruments with appropriate voltage and current sensors and adequate digital

sampling frequency [15];

Use of statistical and other advanced data analysis methods to produce meaningful

information;

Comparison against national and international power quality standards, taking into

consideration system dierences and similarities;

In the analysis of power quality phenomena through time and frequency domain EMTP-

based simulations, special attention must be paid to:

the simulation step size t, which has to be chosen as a function of the maximum

frequency expected (or of concern) in the simulation. Usually, the time step size is

set to a value at least equal to one tenth of the period of the maximum frequency,

which will result in a 3% error with the trapezoidal integration rule [2], [112]. It is also

recommended that the step size be such that the period of the fundamental frequency

is an integer multiple of t, in order to avoid the generation of non-characteristic

harmonics in the post processing Fourier analysis. For example, if the system nominal

frequency is 60:0Hz and the maximum frequency expected in the transient simulation


is in the order of fmax = 6kHz, then the time step size can be calculated as t =

1=(10 fmax) = 16:66666s;

the selection of appropriate models to represent power system and load components,

especially if frequency dependence has to be taken into consideration in the simulated

phenomena [92];

nonlinearities, which are usually disregarded in many simulations. They can aect the

accuracy of the simulation results, particularly in the case of transformer saturation;

the use of simplied switch models for power electronics devices. This may be justied

to speed up the simulation time for system level studies, but it may also give wrong and

misleading results, especially related to semiconductor commutation phenomena. Also,

the EMTP solution at discrete time intervals t may result in inaccurate turn-on or

turn-o switching times, causing unrealistic high frequency transients in the simulation

of power electronic devices. Backtracking techniques [67], [68] and/or resynchroniza-

tion techniques ([96] pages 185, 204, 207) or even the Clock Synchronized Structure

Changing Concept (CSSC) [97] can be used to minimize the problem. Interpolation

and/or extrapolation as well as resynchronization techniques seem to be more and more

applied even in the EMTP-based solution of modern control for power electronics sys-

tems [98], [99]. Therefore, for better accuracy in EMTP-based simulations of power

electronics, it is much more important to use such techniques than to reduce the time

step size;

numerical oscillations caused by the trapezoidal rule of integration in solving the system

of equations. The use of techniques such as CDA (\Critical Damping Adjustment"

[84], [85]) is eective in the elimination of numerical oscillations. MicroTran has CDA

implemented, but other EMTP versions may not, or may use dierent approaches.

the one time step delay at the interface between the control and power systems solu-

tion, as well as other internal time step delays, which may exist in TACS (\Transient

Analysis of Control Systems") and in other software packages. The method SSCPS

(\Simultaneous Solution of Control and electric Power System equations") proposed in


this thesis overcomes this problem, and the user only introduces time delays if needed

to represent system physical behaviour.

Chapter 5

Conclusions and Recommendationsfor Future Work

THE MAIN GOAL of this Ph.D. thesis project was the development of EMTP-based

models for control and power electronic devices for electric power quality assessment.

EMTP simulations can oer theoretical and practical insights into the evaluation of power

quality, both by time-domain simulation techniques and by frequency-domain simulation

techniques.

5.1 Conclusions and Main Contributions

The increasing demand for electricity and other forms of energy in modern society will

create issues of con ict and interest. The simple absence of enough power generation or

of available transfer capacity may become the cause of scheduled load shedding or more

frequent blackouts, with obvious catastrophic consequences.

The growing regulatory, environmental, nancial and time constraints in building new

power plants (typically hydroelectric) and transmission lines has been forcing emergency

solutions in the electricity industry all over the world, such as the increasing use of "small"

distributed generation (mainly thermal power plants with steam or gas turbines), the use of

FACTS devices to enhance power system stability and control, and the adoption of programs

to \save energy" with the promotion for the use of more electricity eÆcient light, heat, and

motor equipment. This, in turn, might create a deteriorating impact on the quality of power,

153

5.1. Conclusions and Main Contributions 154

due to the manufacturing of usually inexpensive power electronic converters. It is a challeng-

ing environment for engineering, politics, economics, etc. since in modern human society,

electricity has become a basic commodity, which almost everybody and almost everything

depends on.

The quality of the electric power delivered to customers by utilities may not be acceptable

for some types of sensitive loads, which are typically power electronics and computer-based

loads, particularly in the control of industrial processes. There are cases where the increas-

ing use of power electronics to enhance process eÆciency and controllability creates power

quality problems. The growing application of shunt capacitors for voltage support, power

factor correction, and system loss reduction, as well as the use of series capacitors (xed or

controlled, for line reactance compensation) will increase the potential risk of transient dis-

turbance amplications and potential electrical and mechanical resonances in the presence

of more and more power electronic devices, and of steam and gas turbines in distributed and

co-generation power plants. As the natural order of the system grows, so grows its ability

to oscillate more! At the same time, new power electronic devices also oer the means for

adequate \power conditioning", to meet the special requirements of electric power quality in

a system.

To evaluate the promising solutions oered with the introduction of more and more power

electronic devices in the transmission and distribution systems, as well as to analyze their

interaction and impact on either the load or the network side, computer programs based on

the EMTP (Electromagnetic Transients Program) are becoming more useful.

The development of new EMTP-based models for more accurate representation of controls

and power electronic devices has been the main subject of this thesis project. The assessment

of electric power quality and the technical impact of power electronic devices on the quality

of power, can hopefully be performed with the models developed in this work.

The main contributions of this Ph.D. thesis project are summarized as follows:

development of a \simultaneous solution for linear and nonlinear control and electric

power system equations" (SSCPS) in EMTP-based programs, through the compen-

5.1. Conclusions and Main Contributions 155

sation method and the Newton-Raphson iterative algorithm. This solution method

eliminates not only the one time step delay problem at the interface between the so-

lution of power and control circuits, but also all the internal delays, which may exist

in methods based on the transient analysis of control systems (TACS) since 1977 [64].

A \circuit approach" was proposed in this thesis, as an innovative alternative to the

solution presented by A. E. A. Araujo in 1993 [67];

experimental implementation in Microtran, the UBC version of the EMTP, based on

SSCPS, of a simultaneous solution for:

linear and nonlinear current and voltage dependent sources (which allow, for

example the modelling of operational ampliers, ideal current and voltage sensors,

etc.);

independent current and voltage sources, which can also be connected between

two ungrounded nodes;

hard and soft limiters (which can be used to represent nonlinear eects such as

saturation);

transfer functions (which allows the simulation modelling of all types of analog

lters and classical control blocks);

mathematical and transcendental FORTRAN functions (such as +, -, * , /, SIN,

COS, TAN, ASIN, ACOS, LOG, EXP, etc.);

special control devices (such as time delays, comparators, etc.) and some digital

logic gates (NOT);

transformation of variables (such as the abc to 0 transformation and its inverse);

voltage-controlled switches;

nonlinear model of a diode semiconductor;

development of the subroutine \GATE" in MicroTran, allowing the dynamic control

of the turn-on and turn-o times of semiconductor devices (e.g., thyristors, GTO's,

IGBT's, etc.), which are modelled as EMTP-based voltage-controlled switches;

5.2. Recommendations for Future Work 156

development of power electronics simulation cases in MicroTran, using the simultaneous

solution approach (SSCPS) for the dynamic control of semiconductor switching devices

(as in a three-phase six-pulse thyristor controlled bridge rectier, and in a three-phase

PWM voltage source inverter (VSI)) and evaluation of current and voltage waveforms;

interaction with a Brazilian utility company and industries for the realization and

analysis of eld measurements of electromagnetic phenomena aecting the quality of

power, such as:

voltage sags and voltage swells;

harmonic current and voltage distortions;

transients, etc.

with determination of causes, consequences and investigation of possible solutions for

power quality problems, as for example, the application of \Custom Power Controllers";

synthesis of simulation guidelines for the evaluation of the impact of power electronic

devices on the quality of power, based on realistic eld measurements and EMTP time

and frequency domain simulations.

5.2 Recommendations for Future Work

The author's main recommendations for future work related to his Ph.D. thesis project

are listed below:

development and implementation in EMTP-based programs of methods for the \au-

tomatic calculation of initial conditions" in the simultaneous solution of control and

power electronic circuits;

development and implementation in EMTP-based programs of techniques for the cal-

culation of the \frequency response of integrated control and power system equations",

allowing for example, the determination of transfer functions between a generic input

and a generic output;


development and implementation in EMTP-based programs of techniques for the ac-

curate solution of \digital control systems";

development and implementation in EMTP-based programs of \detailed nonlinear

models of semiconductor devices" (such as transistors, etc.);

development and implementation in EMTP-based programs of algorithms for the si-

multaneous solution of \generic nonlinear dependent sources" [101];

development and implementation in EMTP-based programs of \voltage and frequency

dependent aggregated load models" [109], but suitable for time and frequency domain

simulations in power quality studies;

development and implementation in EMTP-based programs of models for FACTS and

Custom Power Controllers, such as STATCOM, UPFC, DVR, UPQC, active lters,

etc, and denition, if not yet available in the technical literature, of benchmark test

cases, possibly with eld validation of the models. The evaluation of \power system

stability and control, based on long term EMTP-type simulations", will require detailed

and complex modelling of power system components, FACTS Controllers and special

loads, with all their associated control equipment such as turbine-governor controllers,

exciter controllers, power system stabilizers, power electronics controllers, etc.. Com-

plex and very important electromagnetic and electromechanical phenomena could then

be investigated thoroughly, as for example, the damping and control of subsynchronous

resonance [86], [90].

Some of the author's publications in areas related to the thesis topic are listed here for

easy reference:

B. D. Bonatto, E. A. Mertens Jr., E. S. da Silva, and L. F. S. Dias, \Power Quality

Assessment at Sensitive Loads", submitted to the IEEE/PES Transmission and Dis-

tribution Latin America Conference (IEEE/PES T&D 2002), S~ao Paulo-SP, Brazil,

March 18-22, 2002.


B. D. Bonatto, E. H. Watanabe, E. A. Mertens Jr., H. W. Dommel, L. F. S. Dias,

M. Aredes, S. Carneiro Jr., and S. Nosaki, \Power Electronics and Electric Power

Quality: Cooperative Research at ELEKTRO, COPPE/UFRJ, and UBC", submitted

to the I Brazilian Congress on Electric Energy Technological Innovation (I CITENEL),

Braslia-DF, Brazil, November 6-7, 2001 (in Portuguese).

B. D. Bonatto, H. W. Dommel, E. H. Watanabe, M. Aredes, S. Carneiro Jr., E.

A. Mertens Jr., S. Nosaki, and L. F. S. Dias, \Custom Power Applications for the

Improvement of the Quality of Power - Literature Review", IV Brazilian Seminar

about the Quality of Power (SBQEE001), Porto Alegre-RS, Brazil, August 12-17, 2001.

B. D. Bonatto, and H. W. Dommel, \Current and Voltage Dependent Sources in

EMTP-based Programs", International Conference on Power Systems Transients -

(IPST001), Rio de Janeiro-RJ, Brazil, Volume I, pp. 299-304, June 24-28, 2001.

B. D. Bonatto, E. A. Mertens Jr., F. A. Fernandes, and L. F. S. Dias, \The Quality

of Electric Power in Coordination with the Industrial Safety", XIV National Seminar

on Electric Energy Distribution (XIV SENDI), Foz do Iguassu-PR, Brazil, November

19-23, 2000 (in Portuguese).

B. D. Bonatto, H. W. Dommel, E. A. Mertens Jr., and F. A. Fernandes, \Power

Quality Analysis based on EMTP Simulations Harmonics Case Study", 5th. Brazilian

Power Electronics Conference (COBEP099), Foz do IguassuPR, Brazil, Volume 1, pp.

135-140, September 19-23, 1999.

B. D. Bonatto, E. A. Mertens Jr., and F. A. Fernandes, \Power Quality Diagnosis

in Industrial Customers - Case Study", III Brazilian Seminar about the Quality of

Electric Power (SBQEE099), Braslia-DF, Brazil, pp. 108-113, August 08-12, 1999 (in

Portuguese).

B. D. Bonatto, T. Niimura and H. W. Dommel, \A Fuzzy Logic Application to Repre-

sent Load Sensitivity to Voltage Sags", in 8th International Conference on Harmonics

and Quality of Power (8th ICHQP), IEEE/PES, Ed., Athens, Greece, Vol. I, pp.

60-64, October 14-16 1998.


B. D. Bonatto, E. A. Mertens Jr., and F. A. Fernandes, \Power Quality Diagnosis in

Distribution System", III Latin-American Congress in Electric Energy Distribution (III

CONLADIS), S~ao Paulo-SP, Brazil, pp. 37-41, September 8-10, 1998 (in Portuguese).

L. E. O. Pinheiro, B. D. Bonatto, R. Torrezan, and F. A. Fernandes, \Power Quality

Monitoring - Practical Cases, Solutions and the Planning Perspective", XIII National

Seminar in Electrical Energy Distribution (XIII SENDI), S~ao Paulo-SP, Brazil, May

11-16, 1997 (in Portuguese).

L. E. O. Pinheiro, O. S. I. Komukai, B. D. Bonatto, and E. Yoshida, \Measurements

for Power Quality Monitoring in Distribution System", I Brazilian Seminar on Power

Quality (SBQEE096), Uberlandia-M.G., Brazil, pp. 92-97, June 10-13, 1996 (in Por-

tuguese).

S. M. Deckmann, and B. D. Bonatto, \Damping Introduced by Control Means Consid-

ering Generator Capability Limits", IFAC Control of Power Plants and Power Systems

(SIPOWER095), Cancun, Mexico, 1995.

B. D. Bonatto, and S. M. Deckmann, \Damping Introduced by SVC, CSC, and PSS

for Operation on the Synchronous Machine Capability Curve", XIII National Seminar

on Production and Transmission of Electric Energy (XIII SNPTE), Florianopolis-SC,

Brazil, 1995.

B. D. Bonatto, Damping of Electromechanical Oscillations in Electric Systems through

Reactive Dynamic Compensation, Master Thesis, The State University of Campinas -

UNICAMP, Campinas-SP, Brazil, 1995 (in Portuguese).

S. M. Deckmann, and B. D. Bonatto, Laboratory of Electrical Engineeering I, UNI-

CAMP/FEEC, Faculty of Electrical and Computer Engineering, December 1992, reed-

ited in December 1993, and in 1998. Campinas-SP, Brazil.

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