B D Bonatto PhD Thesis

EMTP Modelling of Control andPower 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 conforming to 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 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 increasing 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 disturbance 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 does 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 transmission and distribution systems, such as FACTS (Flexible AC Transmission Systems) Controllers and Custom Power Controllers, 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 representation of controls and power electronic devices has been the main subject of this Ph.D. thesis project. Its main contributions 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 compensation method and the Newton-Raphson iterative algorithm. This solution method eliminates not only the one time step delay problem at the interface between the solution 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. A \circuit approach" was proposed in this thesis, as an innovative alternative to the solution presented by A. E. A. Araujo in 1993; R (the UBC version of the EMTP), based experimental implementation in MicroTran on SSCPS, of a \simultaneous solution" for: linear and nonlinear current and voltage dependent sources; independent current and voltage sources, which can also be connected between two ungrounded nodes; hard and soft limiters; transfer functions; mathematical and transcendental FORTRAN functions; special control devices and some digital 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 control of 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 simultaneous solution approach (SSCPS) for the dynamic control of semiconductor switching devices (as in a three-phase six-pulse thyristor-controlled bridge recti  interaction with a Brazilian utility company and industries for the realization and analysis of . and a three-phase PWM voltage source inverter (VSI)) and evaluation of current and voltage waveforms.er.  synthesis of simulation guidelines for the evaluation of the impact of power electronic devices on the quality of power. such as voltage sags and voltage swells. as for example. transients. based on realistic . with determination of causes.eld measurements of electromagnetic phenomena aecting the quality of power. harmonic current and voltage distortions. the application of Custom Power Controllers.. etc. consequences and investigation of possible solutions for power quality problems. The assessment of electric power quality.D. can hopefully be performed with the models developed in this Ph. . with the use of EMTP-based programs. and the evaluation of the technical impact of power electronic devices on the quality of power. thesis project.eld measurements and EMTP time and frequency domain simulations. . . . . . . . . . . . . . . .2. 2. . . . . . . . . . . .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. .3 Contributions of this Research Project . . . . 15 18 18 21 32 33 36 . . . . .1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1. . . . . . . . . . . . .2.1. . .5 Newton-Raphson Algorithm . . . . . . . . . . . . . . . . . . . . . . . . iv . . . . . . . . . . . 1 3 4 6 9 12 13 2 Simultaneous Solution of Control and Electric Power System Equations (SSCPS) in EMTP-based Programs 15 2. . . . . . . .2.1 Introduction: Better Electricity Quality at "Possibly" Lower Prices? .1 Compensation Method . . . . . . . . . 1. . . 2. . . . . . . . .2 Dependent Sources . .4 Independent Sources . . . . . . . . 1. . . . . . 2. . . 2. . 1. . . . . . . . .1. . . . . . 1. . . . . . . .3 Power Quality Standards . . . . . .3 Ideal Transformers . . . . .2. .2 Motivation for Thesis Research . . . . . . . . . . . . . . . 2. . .2 Current and Voltage Dependent Sources in EMTP-based Programs . . 2. . . . . . . .1 Computer Analysis and Simulation of Electric Power Quality Phenomena 1. . . . . .1 Previous Developments on Transient Analysis of Control Systems (TACS) . . . . . . . . . . . . . . . .2 Electric Power Quality Monitoring . .4 Custom Power Related Publications . .2. . . . . . . . . . . 1. . . . . . . . . . . . .5 2. .CONTENTS 2. . .2. .1 Conclusions and Main Contributions . . . . . . . . . . 3 Power Electronics Modelling in EMTP-based Simulations 3. v . .2 Simultaneous Solution for Voltage-Controlled Switches in EMTP-based Programs . . . 4. . . 4. . . . .3 EMTP-based Simulation Cases with SSCPS . .1 Dynamic Interaction between Power Electronic Devices and Power Systems . . . . .6 Possible Applications .3 2. 4. . . 153 5. . . . . . . .6 2. Development of Control Transfer Functions in EMTP-based Programs . . . . .1 Induction Furnace Harmonic Study . . . . . . . . . . . . . .3 Implementation of Nonlinear Diode Model in EMTP-based Programs . . . . .4 Control Modelling Aspects of Power Electronic Devices . . . . . . . . . . . .2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . Development of Intrinsic FORTRAN Functions in EMTP-based Programs Development of Control Devices in EMTP-based Programs .2. . . . . 3. 5 Conclusions and Recommendations for Future Work 91 104 104 120 121 125 125 133 150 153 5. . . . . . . . . 3. . 40 42 48 54 58 64 65 74 78 88 4 Evaluation of the Impact of Power Electronic Devices on the Quality of Power 90 4. . . . Development of Limiters for Control Systems in EMTP-based Programs . . . . . . . . . . . . . . . . . . . . . .4 2. . . . . 4. .2.3. . 3. . 4. . .2 Voltage Sag Analysis with EMTP-based Simulation .3 Welding Industry Voltage Fluctuation Study { A Visual Flicker Case 4. . . . . . . . . . . .2 Power Electronics Simulation Cases . 4. . . . . .3.2 Power Quality Assessment through EMTP-based Programs . . . . . . . 4. . .2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Basic Control and Control Devices Simulation Cases . .1 Modelling Power Electronics in Electric Power Engineering Applications . . . . . . . . . . .4 Synthesis of Simulation Guidelines for Studies with EMTP-based Programs . . . . 156 Bibliography 160 . . . . . . . 2 Comparison between . 80 4.1 Global harmonic distortion limits for the system voltages recommended in Brazil.List of Tables 3. .1 Comparison between voltage and current in a diode as a function of its parametric values. 110 4. . . . eld measurements and EMTP simulation results for the operating condition with the harmonic passive . 3 Comparison between . . . . . . . .lters turned OFF. 119 4. . . . . . . . . . eld measurements and EMTP simulation results for the operating condition with the harmonic passive . . .lters turned ON. . . . . . 120 vi . . . . . . . . . . . . . . . . .15 2. . 1. . . . . . . . . . . . .List of Figures 1.2 CBEMA curve revised by the Information Technology Industry Council (ITIC). . . . .17 2. . . . .11 2. . . .8 2. . . . .20 16 19 20 21 23 24 25 27 29 29 29 30 30 31 31 33 34 35 39 41 EMTP and TACS interface with 1 time step delay. . . . . . . . . . . . . . . . . . . Current-controlled current source (CCCS).4 2. Voltage-controlled voltage source (VCVS). . Symbol for operational ampli.1 Typical Design Goals of Power-Conscious Computer Manufacturers. . . . . . .14 2. . 12 2.7 2. . . . \IEEE Recommended Practice for Emergency and Standby Power Systems for Industrial and Commercial Applications. . . . . . 1.10 2. . . . . . . . .6 2.2 2. . .13 2. . . . . . . . . . . . . . . . . . . . . . 446-1987. Representation of branch equation k as a voltage source in series with a resistance. . ." ) .16 2. . . . . . . (b) Thyristors in a high voltage direct current (HVDC) System. . . .19 2. Representation of branch equation k as a current source in parallel with a resistance. . . Current-controlled voltage source (CCVS). . . . M-phase Thevenin equivalent circuit. . . . . . . . . . .3 2. . . . . . . . .3 (a) Thyristor in an industrial power converter. . .12 2. . . . . . .5 2. .18 2. .1 2. . . .9 2. . . . . . . . (Source: IEEE Std. . . . . . . . . . . . . . . . . . . . . . . . Inverting ampli. . . . . .er. . . . . .er circuit. . . . . . . . . . . . . . . . . . . . . Non-inverting ampli. . . . . . . . . Adder circuit with operational ampli. . . . . . . . . . . . . . . .er circuit. . . . . . . . . . Ideal integrator circuit with operational ampli. . .er. . . . . . . . . . . . . . . . . . . . . . . . . Generalization of inverter ampli.er. . . . . . . . . . . . . . . . . . . . . First-order lag circuit using ideal operational ampli. . . . . . . . . . .er circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Circuit with ideal operational ampli. . . . . . . . . . . . . . . . . . Ideal transformer. . . . . . . . . . . . . . . . Independent current source. . . . . . . . . . . . . Newton-Raphson algorithm experimentally implemented in MicroTran. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Voltage-controlled current source (VCCS). . . . . . . . . .er. . . . . . . . . . . . . . . . . Independent voltage source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 9 . . . . . . . . . . . . . . . . . . vii .er. 32 2.29 2.LIST OF FIGURES 2.22 2.45 Simulation results of circuit with ideal operational ampli.36 2.30 2.41 2.34 2.43 2.40 2.27 2.42 2.44 2.28 2.33 2.39 2.24 2.35 2.21 2.26 2.25 2.31 2.38 2.23 2.37 2. er (noninverting ampli. . . .80. . . . Block-diagram representation of a . . . . .23. . . . .er circuit). . . Transfer function. . . . . . . . . . . 2. . . . . . . . . Possible computer implementation of the transfer function block-diagram in Fig. Observer form block-diagram of transfer function in equation 2. . . . . . . . . . . . . . . .rst-order transfer function. . . Observer form block-diagram of . . . . . . . . . . rst-order transfer function of Fig. . . . . . . . . 2. Possible computer implementation of .25. rst-order transfer function of Fig. . 2. . . Realistic . . . .25. . . . Time domain simulation of . . . . . . . . . . .rst-order lag circuit. . . . . . . . . . . . . . . . . . . Transient response of a . . . . . . . . . . . . . . . . First-order transfer function with non-windup (dynamic) limiter. . . . . . . . . . . . . . . . . . . . . .rst-order transfer function. First-order transfer function with windup (static) limiter. . . . . . . . . 3. . . . . . . . Pulse delay control device with arbitrary input signal. .S 2 and S 3". . Zero-order transfer function with soft limits. . . . . . . . . . . . Nonlinear control block-diagram with a sinusoidal intrinsic FORTRAN function. . . 3. . . . Circuit implementation for the simultaneous solution of a sinusoidal FORTRAN function. . . . . . . . . . . . . 3. Soft limits. . . . . . . . . . . . . . . . . . . . . . . . . 3. . . .3 Test cases for transient simulation of voltage-controlled. . . . .5 Simulation of a simpli. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Circuit implementation of a logic gate "NOT" for simultaneous solution. . . . . . . Circuit implementation for the simultaneous solution of a transport delay control device. . . . . . . . . . . . . . . . . . . . . 3. Transport delay control device. thyristor and GTO. . . . . .2 Voltage-controlled switch in EMTP-based programs. . . . . . . . . . . . . . . Transient simulation of a transport delay control device. . . . . . . . . . . . . . .rst-order transfer function with windup and non-windup limiter. . . . . . . . bipolar in voltage and bidirectional current owing switch. . . . . . . . . . . . . . Logic gate "NOT". . . . Open loop control system with "supplemental devices S 1. . . . . . . . . Transient simulation of a pulse delay control device. . . . . . . . . . . . . . . . . .1 Power semiconductor devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time domain response for a sinusoidal excitation input u(t) illustrating the eects of hard and soft limits on the output x(t). . . . . . . . . . . . . . . . . .4 Simulation of a voltage-controlled bidirectional current owing switch. . . . . . . . . . . . . . 3. .ed model for thyristors. . . . . . . . . . . . . . . . .6 Simulation of a simpli. . . . . . . . . . . viii 41 42 44 45 45 46 46 47 47 49 49 50 52 53 53 54 55 55 58 59 60 61 62 62 63 66 70 71 72 72 73 .ed model for GTO's. . . . . . . . . . . . . . . . . . Half-wave recti.10 3. Simulation with simultaneous solution of a voltage-controlled switch. . . . . . . . . .18 3. . . . . . Diode symbol. . . . . . Circuit implementation for the simultaneous solution of a nonlinear diode model. . . . . .12 3. . . .LIST OF FIGURES 3. . . . . . . . . . . .11 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 3. . . . . V-I diode characteristic and dierent network Thevenin equivalents. .17 3. . . . . . . . . . . . . . . . . . .9 3. . . . . . .15 3. . . Dc-dc converter. . . . . . . . .19 ix Circuit with \simultaneous solution" of a voltage-controlled switch. . . . . . . . V-I diode characteristic and network Thevenin equivalent circuit equation.8 3. . .14 3.13 3. . . . . . . . . . .7 3. One time step delay in EMTP-based switches. . . . . . . 1 Circuit with a single-phase diode-bridge recti. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transient simulation of a nonlinear diode model in an EMTP-based program. . 76 76 77 78 81 82 82 85 85 86 86 87 87 4. .er with freewheeling diode. . . Detail of the transient simulation of a nonlinear diode model in an EMTP-based program. . . . . . . V-I nonlinear characteristic of the diode resulting from the EMTP simulation. . . . . . . . Electric circuit with a nonlinear diode model. . . . . . . . . . . . . . . . . 4. . . . .er. . . . . .2 Current drawn from the source by a single-phase diode-bridge recti. . . . . . . . . .3 Harmonic amplitude spectrum of the current drawn from the source by a single-phase diode-bridge recti. . 4.er. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three-phase system with \balanced" single-phase diode-bridge recti. . . . . 4. . . . . . . . 4. . . . . . . . . .6 Four-wire.5 Harmonic amplitude spectrum of the voltage waveform distortion at the point of common coupling (PCC). . . . . . . . . . .er. . . . 4. . . . . . . . . . . . . . . . . . . . . . . . . and voltage waveform distortion at the point of common coupling (PCC).4 Current through and voltage across the total inductance. . . . . . . . . . . . . . . Canada. .14 Current measurements in a distribution feeder supplying induction furnaces at the time of maximum voltage distortion. . . . . . .. . . . . . . . . . . 4. . . . . . .8 Harmonic amplitude spectrum of the current owing through the neutral conductor.10 Measured voltage waveshape. . . . 4. . . . Vancouver. . . . . .C. . . . . 4. . .11 Harmonic amplitude spectrum of the outlet waveshape voltage. . . . . . . . . . . . . . . . . 4. . . 4. . . . . . .7 Current owing through the neutral conductor. 4. . . . . 4. . B. . . . . (b) Induction furnace operation. .13 (a) Metal melting by an induction furnace. . 4.ers. . 4.9 Voltage waveshape measured at the outlet of the Power Electronics Laboratory of the Department of Electrical and Computer Engineering at UBC. .12 Phase-angle of the harmonic components of the outlet waveshape voltage. . its fundamental component and its harmonic distortion. . .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. . . . 93 94 95 96 97 97 98 99 100 101 102 103 106 107 108 . . . . . . 16 (a) Phase \A" current measured with harmonic passive .LIST OF FIGURES x 4. lters turned on. (b) Phase-to-phase \A-B" voltage measured with harmonic passive . .17 THD harmonic trend. . . with harmonic passive . .lters turned on. . . . . . . . . 109 4. 18 THD harmonic trend. 109 4.lters turned o from 12:00 midnight to 06:00am. with harmonic passive . . . . . 113 4. .21 Amplitude of the positive sequence system impedance at the PCC with harmonic . . . . . . . . . .lters turned on all the time. . . . . . . . . . . . . . . . . parallel-resonant inverter for induction heating.19 Distribution substation. . . . 110 4. . . .20 Current-source. . . . . . . . 114 4. . . . . . . . . 22 Phase angle of the positive sequence system impedance at the PCC with harmonic . 114 4. .lters. 23 (a) Phase \A" current simulated with harmonic passive . 115 4.lters. (b) Phase-tophase \A-B" voltage simulated with harmonic passive .lters turned o. .lters turned o. 116 4. . . . .24 (a) Phase \A" current measured with harmonic passive . . . . . (b) Phase-to-phase \A-B" voltage measured with harmonic passive .lters turned o. . . . . . .lters turned o. 116 4. . . . . .25 (a) Phase \A" current simulated with harmonic passive . . lters turned on. (b) Phase-tophase \A-B" voltage simulated with harmonic passive . .lters turned on. . 117 4. . . . . . .26 (a) Phase \A" current measured with harmonic passive . . (b) Phase-to-phase \A-B" voltage measured with harmonic passive .lters turned on. . 117 4. . . . . . . . . .lters turned on.27 Instantaneous ideal compensation current to be \injected" by a shunt active . . . . . . . .32 Control block diagram of a second order dierential equation with poles on the imaginary axis of the complex plane. . . .34 Introduction of a one time step delay in the control block diagram. . . . (b) Phase-to-phase \A-B" simulated voltage sag. 127 4. . . . . . . .28 Voltage sag measurements (%RMS versus time duration) with an overlay of the CBEMA curve. . . . used in power system small-signal stability studies of a single machine connected to an in. . . . . . . . . . 123 4. . . . . . . . . . . . . . . . .29 (a) Phase-to-phase \A-B" measured voltage sag. . . . For time durations less than 1 cycle the equipment seems to measure peak values. . . . . . . . . . . . . . . . . . . . 124 4. . . . . . . . . . . .36 Classical linearized \swing equation". . . . . . . . . . . 126 4. . . . . . . . . . . . . . 127 4.30 Instantaneous voltage uctuations causing light ickering eect. . .31 Modulated voltage and respective amplitude frequency spectrum . . . . . . . 118 4. . . . . . . . . . . . . . . . . . . 122 4. .33 Solution of system with bounded resonance oscillations. . . . . . . . . . . . . . . . .lter. 126 4. . . . . . . .35 Solution of system with unstable resonance oscillations caused by the introduction of one time step delay. . . . . 124 4. . . . . . . . . . . . . . . . . . . 130 4. . . .nite bus. . . . . . . . . . . . . in the presence of a positive damping torque coeÆcient. . . .37 Simulation results of the synchronous machine rotor angle deviation. . . in the presence of negative damping torque coeÆcient. . . . . . . . . . . . . . . 130 4.38 Simulation results of the synchronous machine rotor angle deviation. . . 131 4. . . . . . . . . . . . . . . . . .39 Canonical second order transfer function representation of the single-machine in. . . . . . nite bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 . . . . . . . LIST OF FIGURES 4.40 Circuit for the dynamic control of the . . . . . . . 4. . .ring angle (\") of a thyristor.41 Voltages and currents in a circuit with dynamic control of the . . ring angle of a thyristor. 4.42 Circuit for the dynamic control of the . . ring angles of a three-phase six-pulse thyristor-bridge recti. . . . . .er. . . . . . . .43 Voltages and currents with dynamic control of the . . . . . . . . . . . . . . . . . . . . 4. . . . . . . . . ring angles of a three-phase six-pulse thyristor-bridge recti. er. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.44 Dynamic control of the ring angles of a three-phase six-pulse thyristor-bridge rectier. . 4.45 Dynamic voltage control signals at the output of the proportional-integral (PI) and the limiter control blocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.46 Circuit for the dynamic control of three-phase PWM voltage source inverter (VSI). . . . 4.47 Phase \A" modulation and triangular carrier waveforms for generation of gating signals through sinusoidal pulse width modulation (PWM). . . . . . . . . . . . . . . . . . . 4.48 Node voltage \vSA " generated by a three-phase PWM voltage source inverter (VSI). . . . 4.49 Voltage across the load \vSA NEUTR " and current supplied to the load by a three-phase PWM voltage source inverter (VSI). . . . . . . . . . . . . . . . . . . . . . . . . . . 4.50 Load currents supplied by a three-phase PWM voltage source inverter (VSI). . . . . . . 4.51 Line-to-line voltage generated by a three-phase PWM voltage source inverter (VSI). . . . xi 135 136 138 139 140 141 144 145 146 147 148 149 and to all my relatives for their unconditional love and support. Dommel. my Ph. I most specially appreciate the help. Jose R. and Richard for their careful and kindness personal assistance. sta members. I have also learned with Dr. past and present colleagues and friends at the Department of Electrical and Computer Engineering of the University of British Columbia (UBC) have been the source of inspiration and support to pursue scienti. Luciana. Fernando. Alexa and Aline.D. and my daughters. Mart. Professor Sandoval Carneiro Jr. Martha. Dommel becoming a Professor Emeritus at UBC. Alexandre. Dommel for the honor and opportunities of have being his teaching assistant. Fulvia. program in Canada. right from the start of this Ph. Brazil. I dedicate a very special note of thanks to our special friends Wany.) I also thank Dr. acceptance and advice of many individuals without whom this opportunity would never have become fruitful. Dorival and Isolina. To my wife. I owe a tremendous debt of gratitude to Dr. thesis cosupervisor. (The responsibility for any remaining errors is solely mine. share of wisdom and support for the development of this thesis. Dunford for kindly accepting to be my Ph. thesis supervisor. with Dr. My most sincere thanks to my parents. Hermann W. for all his personal and professional encouragement. my love and my heartfelt thanks for their strong participation in this life project altogether. from the Federal University of Rio de Janeiro (UFRJ). has gently been very supportive. Professors.D. William G.D. I also thank Dr. who has excellent teaching skills.Acknowledgements I would like to thank God for the gift of learning. Braslia . for building and enhancing the foundation of my knowledge in science and engineering.Brazil.c and personal growth. and at the Federal School of Engineering of Itajuba (EFEI). I also thank my former Brazilian professors and colleagues at the State University of Campinas (UNICAMP). I would like to sincerely thank the Fundaca~o Coordenaca~o de Aperfeicoamento de Pessoal de Nvel Superior (CAPES). for the . Mertens Jr. not mentioned. I also thank the Brazilian utility company ELEKTRO . Brazil.Centro Estadual de Educac~ao Tecnologica Paula Souza. Vancouver.C. for all the teaching experiences I was able to conduct.Eletricidade e Servicos S. for providing opportunities for a practical interaction in power quality analysis. not forgotten.. 2001. I thank and acknowledge the contributions of many people. B. since August 24. which enriched my communication and leadership skills.C. S~ao Paulo. with a special reference to Francisco Alfredo Fernandes. thesis project. Canada Benedito Donizeti Bonatto October 09. Without it.. who certainly have had an impact and in uence on my living and studying at UBC. through a professional cooperation with the engineer Ernesto A.nancial support to this Ph. B. Armando Bayeux da Silva. Canada. Vancouver. 1997. professors and sta at the ETE Prof. my dream would never come true..A. Last but not least. I acknowledge and thank students. xii . a technical high school of the CEETEPS ..D. \Engineering: engineering is the application of mathematical and scienti. and operation of economical and eÆcient structures. It's art and communication. equipment. and systems. politics and .c principles to practical ends. as in the design construction. nance. modeling and simulation. measurement and estimation.eg3.com xiii ." http:nnwww. approximation. invention. It's a way to think about problems. and more. Such time step delays can cause inaccuracies or numerical instabilities. at possibly reduced prices. thus eliminating any time step delay in the digital time domain simulation. 1. followed by a description of the motivations for this Ph. and equipment manufacturers to develop new structures for the electricity market. or in similar programs. utility companies. thesis research and its relevant contributions.D. This chapter presents an introduction to power quality problems and their relation with power electronics.1 Introduction: Better Electricity Quality at "Possibly" Lower Prices? The demand of electricity customers for increased quality of power. Deregulation of the electricity 1 . A \circuit approach" is used for the simultaneous solution of the control and electric power system equations. The advantages of the circuit approach is its \generality and exibility" for modelling multi-terminal linear and nonlinear control devices.Chapter 1 Electric Power Quality and Power Electronic Devices: An Overview T HE 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 Electromagnetic Transients Program (EMTP) [1]. is forcing governments. which are needed in the analysis of electromagnetic phenomena aecting the quality of power. regulatory agencies. [2]. which are used more and more by industry. All these problems point out that more attention must be paid to power quality problems. Power electronic devices. The costs tend to grow as the sensitivity and use of microprocessor-based devices tend to increase. However. with such modernization new issues have emerged regarding the quality of electricity. industry in lost time and revenue due to power related problems were estimated in 1993 as US$26. A. S. As an example. Nevertheless. of damage to equipment. traditional economic analysis. of power electronic devices. which may cause problems for adjustable speed drives (ASD's). this may require some investments in the infrastructure of the power system. industries have been able to produce goods faster and with increasing quality. and of modern manufacturing techniques.1. With the growing utilization of automation and control based on the use of microprocessors. as in any business. the annual approximate value would be in the order of hundreds of million of dollars in damage. would be controlled by power electronics. A.000 [3]. Another problem is capacitor switching in the utility system. The Electric Power Research Institute (EPRI) stated that in the year 2000. and such devices exist today. however. Sensitive loads tend to shut down if there are small variations in the network voltage.000. might show that these investments are only feasible with concurrent increases in electricity taris.1. are also able to \guarantee" a certain expected level of electricity quality to a sensitive or special load. In this scenario. Introduction: Better Electricity Quality at "Possibly" Lower Prices? 2 industry has been proposed as a solution to make the present utility companies more competitive in oering better services and better quality at lower prices to their customers. 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. compared to 30% in 1995. harmonic distortions caused by nonlinear loads may result in wrong operation or may increase the losses in power system components. S. Also. 60% to 70% of total utility power generated within the U. However. The paradox of more quality for less money still remains a topic for discussion in forums such as government regulatory agencies. such as pay back return or rate of interest. the cost per year to U. to cope with the new demands of the modern types of loads (power electronics and microcomputer based).000. Economic losses expressed in terms of interrupted production. Flexible AC . Estimating the cost of poor power quality is a diÆcult task. Introduction: Better Electricity Quality at "Possibly" Lower Prices? 3 Transmission Systems (FACTS) technology.1.1. Custom Power Controllers. active . thus allowing the accurate evaluation of the impact of high power electronics on the quality of power. . The aim of this research project was the development of new EMTP-based models for control and power electronic devices. Therefore. New models for innovative equipment. The propagation of electromagnetic phenomena into the industrial or utility network must be evaluated as well [4]. It is not enough to evaluate the electric quality conditions only at the interface of power electronic systems with the electric power system. will then be required.lters. but also under transient conditions. The introduction of more and more power electronic devices into the network will create issues of compatibility of operation not only in steady state. software packages such as the ElectroMagnetic Transients Program (EMTP). as well as new philosophies for their control and operation. have become important and necessary tools to analyze the impact of large power electronic devices on the quality of electric power. oer promising solutions for improving the quality of power in transmission and distribution systems. As part of the project. among other power electronics applications. eld tests were conducted in cooperation with a Brazilian utility company. to provide realistic power quality data measurements. 1.and multiphase networks is well established [1].1.1 Computer Analysis and Simulation of Electric Power Quality Phenomena Digital computer simulation of electromagnetic transients in single. Since the publication of [1] in 1969. signi. g.cant improvements have been made in models for electrical power system components such as transmission lines [5]. e. [11]. transformers. [10]. [8]. [7]. turbine-generators and cables [6]. the complexity of periodic switching in power electronic devices can only be studied thoroughly through time-domain simulations. [12]. Although steady-state solutions at fundamental and harmonic frequencies have been proposed to analyze power quality problems. Of particular interest for this research project are the necessary advances in the simulation of power electronic devices within power systems. with EMTP-based programs [4]. [9]. . [13].. The performance obtained from some prototypes. Introduction: Better Electricity Quality at "Possibly" Lower Prices? 4 EMTP-type simulation is particularly useful for the analysis of the dynamic interaction of distributed Custom Power Controllers in a power system. how these new devices will interact within each particular power system is still an open question. or from simulations with simpli.1.1. Despite international demonstration projects for some Custom Power Controllers already with some years of operating experience. ed models. despite the impressive accuracy obtained with the models available today. [16]. In Canada. Nevertheless. This could avoid more expensive corrective actions after installation.1. may not be suÆcient for real applications. the Canadian Electricity Association (CEA) developed a guideline for the power quality that utility customers experience. [15]. In many of the real-world problems. The Electric Power Research Institute (EPRI) commissioned an extensive survey of distribution system power quality in the U. 1. does still represent the major challenge. A quantitative measure of the deterioration of the ideal sinusoidal waveform from the growing utilization of power electronic devices resulted from these surveys. The . A. Measurements and simulations also become necessary for performance evaluations under dierent network and load conditions.2 Electric Power Quality Monitoring In recent years. S. These concerns make more realistic electromagnetic transient program based simulations important when detailed models of Custom Power Controllers become available. where dangerous resonance and other unforeseen problems may occur. methods and guidelines for the appropriate use of EMTP-based programs as potential tools for power quality studies. because of the use of microprocessors and digital signal processing techniques [14]. momentary voltage variations have been the main cause for shutting down microprocessor controlled industrial processes. good engineering judgment in setting up the power quality problem and representing the physical phenomena with appropriate models. in a three-year project [17]. tremendous improvements have also been made in digital measuring instruments. All those facts have encouraged the development of new practical models. This made it possible to conduct power quality surveys in many countries around the world. and their complex interaction. [34]. [36]. [22]. as discussed in [38]. [23]. as reported by the related literature [15]. [31]. [30]. According to the author's experience as a \power quality engineer" in a Brazilian utility company. The need for more detailed information on power disturbances. [20]. [27]. Introduction: Better Electricity Quality at "Possibly" Lower Prices? 5 diversity of problems to cope with. [3]. with the present deregulation process in the electricity industry. Moreover. [41]. [29]. The experience in monitoring power quality phenomena has increased in the latest years. [37]. and the search for new techniques to process the amount of available measured data. [25].1. [24]. has created a need for more research on power quality issues [18]. [35]. there are many dierent reasons why electricity customers become dissatis. power quality has become a key factor for utilities and customers in a competitive market. [42]. [33].1. [32]. [40]. [19]. [39]. have motivated research into applications of modern theories in the power quality area. [21]. Well known power quality problems have been summarized and at the same time new problems have been discussed in [28]. [26]. ed with the quality of the electric power delivered by utilities. On the technical side. [46]. which cannot be easily avoided or minimized. [44]. as everybody tries to survive in a competitive and aggressive business environment. Partly. It is also common to . Most of these voltage sags are due to faults in transmission and distribution systems. the motivation comes from the need to reduce electricity costs in the industrial production process. This happen partly because there may indeed be technical problems. and why they complain. looking for better taris and better contracts. the most common power quality problems are caused by a fatal combination of sensitive electronic-based loads and a high incidence of voltage sag phenomena [43]. caused by lightning phenomena. [45]. causing frequent process malfunction or interruption with . [46]. This aggravates the impact of voltage sags.nd poor voltage regulation within the industry electric system [45]. which are rarely presented explicitly by the industry personnel. unless any kind of .nancial losses. nancial compensation is legally required. 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 . There is a wide range of alternatives for possible solutions to technical problems in the quality of the electric power supply. 1. in some cases compensation through the use of power electronic devices might be a promising alternative. but the customer process is much more sensitive and some kind of electronic compensation would be necessary. Typically. However. voltage sag problems can be minimized by proper adjustments in the sensitivities of the load or load control.1. The potential con ict is sometimes created when a possible technical solution requires high . whenever this is technically feasible. Usually. the utilities comply with the standards of supply set by the regulatory agency. Introduction: Better Electricity Quality at "Possibly" Lower Prices? 6 phenomena involved. either for an individual sensitive load or for an entire industrial process. nancial investments. A cost versus bene. impulse and oscillatory transients. which are brie y discussed here. voltage unbalance among the phases in a three-phase system. The Institute of Electrical and Electronics Engineers. (IEEE) has therefore developed standards to address power quality problems.3 Power Quality Standards Power quality has become an important issue because of the increasing use of power electronic devices. However. The problems related to the quality of electricity are not new. in the absence of clear regulations one needs \to live with the problem!" 1. 1159-95 [47] de. with frequency and voltage exactly at their rated values. Inc. short. voltage uctuations (causing visual icker).and long-duration voltage variations. with the changes in the type and sensitivity of the loads in recent years. and other electromagnetic phenomena are increasingly causing power quality problems. IEEE Std.t analysis usually leads to a cheap compromise solution.1. harmonic current and voltage distortions. power frequency deviations. since there was never an ideal sinusoidal waveshape. This standard presents recommended practices and requirements for harmonic control in electric power systems. [50].nes and characterizes electromagnetic phenomena which may cause power quality problems. Most utility regulations dealing with harmonics are based on IEEE Std 519-1992. [48]. [49]. It addresses most of the issues of harmonic generation . It also provides recommended practices for monitoring electric power quality. 2. The IEEE Std. Introduction: Better Electricity Quality at "Possibly" Lower Prices? 7 by power electronic converters.  eects of harmonics. static VAR compensators and power electronic controlled drives.  harmonic analysis methods.  recommended practices and harmonic limits for individual customers. IEEE Std.  reactive power compensation and control. which is presented in Fig. 519-1992 [48] is currently been revised to account for interharmonics in power systems and the possible application of probabilistic approaches in harmonics evaluation. shown in Fig.1. It also discusses:  system response characteristics. It thus can help engineers to provide some answers in detecting harmonic related or other power quality problems. 446-1987 [51] covers the recommended practice for emergency and standby power systems for industrial and commercial applications. arc furnaces. a voltage sag of 30 percent for 0:5s. It shows that computer.1.  methodology for evaluating new harmonic sources. properly designed by the manufacturers. It also de. 1. In the later task. should be able to withstand a complete interruption of voltage supply for up to 20ms.  harmonic measurements. The Information Technology Industry Council (ITIC) revised the CBEMA curve. time-domain simulation can be particularly useful to predict equipment and power system behaviour.1.and power electronics-based loads. 1. A computer \voltage tolerance envelope".  recommended practices and harmonic limits for utilities. 20 percent for 10s or 10 percent in steady state. also known as the CBEMA curve (Computer Business Equipment Manufacturing Association curve) is presented in this standard. nes the upper limits in the input voltage that should be tolerated. The CBEMA (ITIC) curve has been widely used as an important \reference" . 0 10.0 2s Figure 1.1 30% 0.01 0. and their response to transient variations in the supply voltage. Introduction: Better Electricity Quality at "Possibly" Lower Prices? 8 400% PERCENT VOLTAGE 300% 200% VOLTAGE BREAKDOWN CONCERN 115% 106% COMPUTER VOLTAGE TOLERANCE ENVELOPE 100% 87% LACK OF STORED ENERGY IN SOME MANUFACTURERS' EQUIPMENT 0% 0.and power electronics-based loads. However. 446-1987. It addresses the multidisciplinary area of power quality. there are cases where the load sensitivity is much more strict than the CBEMA (ITIC) curve.5 1. (Source: IEEE Std. which is considering the adoption of existing standards and practices .0 1000. Voltage uctuations causing visual icker are being studied by the Task Force IEEE P1453 on Light Flicker.0 TIME IN CYCLES [60Hz] 100." ) for the susceptibility level of computer.1. due to the great variety of products and processes. \IEEE Recommended Practice for Emergency and Standby Power Systems for Industrial and Commercial Applications.1. IEEE Std 1100-1992 [52] presents the recommended practice for powering and grounding sensitive electronic equipment. which has to be determined then case-by-case for an adequate power quality assessment and proposal of solutions. giving practical guidelines on load and source compatibility concerns.1: Typical Design Goals of Power-Conscious Computer Manufacturers.001 0. 1.1. Introduction: Better Electricity Quality at "Possibly" Lower Prices? 9 Percent of Nominal Voltage (RMS or Peak Equivalent) 600 500 400 300 Applicable to Single−Phase 120−V Equipment 200 140 120 100 80 70 40 0 Voltage Tolerance Envelope −4 10 1us −3 10 110 90 −2 10 −1 0 1 2 10 10 10 10 1ms 3ms 20ms 0.5s Duration in Cycles (c) and in seconds (s) 3 10 10s 4 10 Steady State 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. 1.1. Introduction: Better Electricity Quality at "Possibly" Lower Prices? 10 High voltage direct current (HVDC) and exible AC transmission systems (FACTS technology) 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 transmission systems [53], [54], [55]. HVDC, static Var compensator (SVC), thyristor controlled series compensations (TCSC), static synchronous compensator (STATCOM), static synchronous series compensation (SSSC) and unied power ow controller (UPFC) are examples 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 electronics applications designed to mitigate power quality problems in industrial and distribution 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 dynamic compensation schemes, with dierent acronyms, as solutions to specic power quality problems. the D-STATCOM can be designed for reactive power control. or for voltage control of the fundamental frequency. has a wider range of applications. but it may also include higher frequencies as in shunt active power . In fact. \The D-STATCOM. although based on the STATCOM. " The integration of series.and shunt active .lters. referred to as uni.lters. is promising to be the de. [61].ed power quality conditioner (UPQC) [60]. nite solution for the majority of power quality problems. its high cost may make it useful only in some special cases. \However. the shunt or series devices such as the D-STATCOM or the voltage sag compensator will probably play a signi. On the other hand. RJ. Maurcio Aredes. The ongoing deregulation process in many countries is also fostering competition in the 1 From personal communication with Dr.cant role in future distribution systems" 1 . Brazil.-Ing. COPPE/UFRJ. . Rio de Janeiro. the MOS controlled thyristor (MCT). thyristors. in 1997.  http://sac.for SoftSwitching Technologies products.1. state of the art of these devices can be found in reference [62].com/prods/FPQD/cp. and many others. static Var compensator (SVC). Case studies with practical applications of Custom Power Controllers can also be downloaded directly from the web sites of some manufacturers.com . respectively. Transfer Switch and Premium Power Park.com/prods/FPQD/dvr.  http://www.asp . For example. The so considered. static synchronous series compensation (SSSC) and uni. thyristor controlled series compensation (TCSC).html .for S & C UPS products.siemenstd. which accelerates the application of new technologies in the transmission and distribution system. used in an industrial power electronic converter and in a HVDC system.softswitching.1. as for example:  http://www. the static induction thyristor (SITh). along with a description of the main characteristics of HVDC.general power quality information on DVR. and the insulated gate bipolar transistor (IGBT).  http://www. 1.html .siemenstd. which will provide voltage support for a few seconds to sensitive processing equipment during times of voltage sags. static synchronous compensator (STATCOM). Other types of semiconductors being used are the gate turn-o thyristor (GTO). Solid State Breaker.case studies for the voltage sag compensator DVR (dynamic voltage restorer of Siemens).3 (a) and (b) present semiconductor power devices.com/products/purewave/ups pubs. D-STATCOM. Introduction: Better Electricity Quality at "Possibly" Lower Prices? 11 electric power industry.sandc. Figs. there are applications being developed for superconducting magnetic energy storage devices (SMES) for low voltage distribution systems. Thyristors are considered the \backbone" of the high power electronics revolution. ed power ow controller (UPFC). The bene. ts of the application of FACTS technology in a power system depend on the reliability of the speci. c FACTS device. which in turn depends on the reliability of the semiconductor . 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. Moreover. devices used. with a worldwide deregulation process in the electricity industry. power quality analysis . 1. Although the semiconductor power devices act as switches. (b) Thyristors in a high voltage direct current (HVDC) System.2. Motivation for Thesis Research (a) 12 (b) Figure 1. once the appropriate models and methods are developed or improved. This research project evaluated the available EMTP models for power quality studies. they are not ideal switches and many physical limitations do apply.3: (a) Thyristor in an industrial power converter.1. 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. and developed new models where the existing ones needed improvements. Contributions of this Research Project 13 will rise in importance and urgency. as part of this project .3.1. Therefore.  The diversity of power quality phenomena requires an interdisciplinary approach and specialized engineering skills. especially in the power electronics area.eld tests were developed in a Brazilian electric utility company. appropriate courses were attended. where realistic power quality cases were analyzed and simulations were performed.  The opportunity to conduct practical . With opportunities available for interaction with other researchers at The University of British Columbia (UBC). which was helpful for the understanding and development of new models for implementation in MicroTran. the UBC version of the EMTP. 3 Contributions of this Research Project This Ph. as an alternative to the approach of A. and necessary for the validation of digital computer models. thesis oers new models for the digital computer simulation of control and power electronic devices. E. Araujo [63] developed in 1993. An innovative \circuit approach" was developed for the simultaneous solution of control and power systems equations.D.eld tests in cooperation with an electric utility company was a valuable experience. These models were developed for implementation in EMTP-based programs or in similar programs. The main dierences and important advantages are summarized as follows:  With the addition of ideal operational ampli. 1. A. ers. where the circuit elements R. Operational ampli. If integration methods are changed in the EMTP. C are solved by the main code of the EMTP. no extra coding is needed. for example from trapezoidal rule to backward Euler as done in some versions at instants of discontinuities with the CDA technique. L. transfer functions can be implemented with a \circuit approach". Moreover.ers are not aected by integration rule changes. if ideal operational ampli. the frequency response of linear .ers are implemented in steady-state solution. A.  A. this function and similar functions are pre-de. Araujo [63] uses FORTRAN-like statements for control functions such as Y = COS (X ) in the input.1. E. Contributions of this Research Project 14 control systems could be easily calculated in EMTP-based programs by just using the frequency scan option.3. In this thesis. which are then handled with a FORTRAN interpreter. Finally. and also points out the author's recommendations for future work. thus providing an easy EMTP-based modelling of any linear or nonlinear control device. the UBC version of the EMTP. such as distributed FACTS and Custom Power Controllers in transmission and distribution power systems. . Simulation guidelines for the evaluation of the impact of power electronic devices on the quality of power are summarized in Chapter 4. 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. This Ph.D. Chapter 3 discusses the developments made for power electronics models in EMTP-based simulations. This is very useful for the dynamic analysis of novel power electronic controllers.ned control block types. thesis.D. Chapter 4 presents simulation cases of power quality assessment with the use of the existing features of MicroTran. Chapter 5 presents the main conclusions and contributions made in this Ph. thesis is organized as follows: Chapter 2 presents the simultaneous solution method for control and electric power system equations (SSCPS) in EMTP-based programs.  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". 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"). which makes the solution method non-simultaneous.1. Moreover. which then returned information to the main program for use one time step later. with a time delay of one
t between them. 15 . This non-simultaneous approach was probably used because it was easier to write a code separated from the main program. also require a one-time-step delay between the solution of control and power systems equations. as illustrated in Fig. such as PSIM [65]. By separating the solution into two parts. The main program passed information to the TACS program. with a simple interface.Chapter 2 Simultaneous Solution of Control and Electric Power System Equations (SSCPS) in EMTP-based Programs 2. whereas the network elements in the EMTP result in symmetric matrices. was an expedient way to implement control system equations. 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 . The solution in two parts. control system equation matrices in TACS are usually unsymmetric. the code for symmetric matrices in the EMTP could be maintained. but it proved to be the cause of critical numerical 1 Many other software programs. [63]. In 1993 A. as stated in [69]. Araujo proposed a simultaneous solution of both sets of equations. such as better ordering of its variables to minimize the number of delays inside TACS [70]. [67]. development of a new TACS program \MODELS" [72] and its possible applications for simultaneous solution of power electronics systems equations [73]. using the compensation method to eliminate the one-time-step delay in the EMTP-TACS interface [71]. electric network equations and control systems equations. even more time-step delays were introduced by the internal solution algorithm of TACS. in order to deal with nonlinearities in feedback control loops. and also sequential for its implemented devices.1: EMTP and TACS interface with 1 time step delay. Besides the time delay between TACS and EMTP.1.E. as a way to eliminate the one-time-step . instabilities and inaccuracies in some cases in the time domain simulation of electric and power electronic system transients [66]. The TACS internal solution is therefore non-simultaneous for some control cases. the eect of the interface delay cannot always be eliminated by using a small step size
t. Improvements have been made through the years in the TACS subroutine of some versions of the EMTP.A. In cases where the EMTP and TACS elements form a closed loop (or feedback system according to control theory). [68]. Previous Developments on Transient Analysis of Control Systems (TACS) 16 Electric Network Solution ( EMTP ) Time Delay 1 ∆t Control System Solution ( TACS ) Figure 2.2. E. The control equations. The augmented matrix with the control equations becomes unsymmetric due to the structure of the control equations. Previous Developments on Transient Analysis of Control Systems (TACS) 17 delay problem at the interface. 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]. as well as the internal control delays [67]. were developed inside the subroutine \CONNEC". A proper partition of the system of equations would allow the solution to be separated into two subsystems. Most of the equations of both the electric network and the control systems are usually linear. one linear and another nonlinear. A.2. [68]. Araujo chose to solve the system of linear equations inside the EMTP. which is a user-de.A. while some are nonlinear. both linear and nonlinear.1. [63]. the trapezoidal rule of integration was used to numerically integrate the .ned subroutine in the MicroTran version of the EMTP of the University of British Columbia. \Similarly to TACS. With this novel approach for EMTP-based programs. while elements missing inside 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 control system. in the implementation of transfer functions. but as far as the author knows. for example. The code was written to prove the ideas. such as resistances and capacitances. was not implemented in a production version of the EMTP. elements of the control circuit which already exist in the EMTP.rst-order dierential equations inside CONNEC. are solved by the EMTP proper. such as ideal operational ampli. ers 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 de2 The author acknowledges the help of Mr. Jesus Calvi~no-Fraga for indicating in 1998 in his technical report 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 EMTPbased Programs Since the publication of [1] describing the for example. it becomes easy to simulate operational ampli.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. With a voltage-controlled voltage source. Dependent sources expand the capabilities of EMTP-based programs considerably for modelling many electric and electronic circuits and devices. dependent sources of all possible types have not been implemented in any EMTP-based program. ers. These can then be used to set up control circuits with analog-computer blockdiagrams. they could be added directly to the network equations used in EMTP-based programs (with the modi. As long as the equations of the dependent sources are linear. Nonlinear eects arise with the inclusion of saturation or limits in the dependent sources. which can also handle nonlinear eects with a Newton-Raphson algorithm. but the matrix would then become unsymmetric and a linear equation solver for unsymmetric matrices would have to be used. this approach con.ed nodal analysis (MNA) presented in [76] and [77]. as well as of independent sources. which can also be connected between two ungrounded nodes. 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. If the nonlinear elements are not too numerous.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]. This section provides then the fundamental equations for the implementation of dependent sources in EMTP-based programs. Another alternative discussed here in more detail is based on the compensation method. 2.2. nes the iterations to a . allows the simultaneous solution of the nonlinear equations with the rest of the linear network [2]..2: M-phase Thevenin equivalent circuit.2: [ vOPEN ] [ rTHEV ] [i] [ v] i1 i2 i3 i4 . vOPEN _ 1 iM vOPEN _ M ZM . v4 vs(M) Z4 v3 vs(4) Z3 v2 vs(3) Z2 v1 vs(2) Z1 vs(1) Figure 2. vM . the equations are linear. but with an unsymmetric matrix.1) 3 7 7 7 5 (2. as illustrated in Fig. Without limiters. When there are M nonlinear elements in a circuit. This approach is used here for solving the equations of dependent sources as a special case of nonlinear elements....2) .1 to 2.6. the following system of equations 2.. vOP ENM 1 2 (2.2.. ..2. which is then represented by its M-phase Thevenin equivalent circuit. 2. [vOP EN ] + [rT HEV ]
[i] + [v ] = 0 where: 2 6 [vOP EN ] = 6 6 4 vOP EN vOP EN . Current and Voltage Dependent Sources in EMTP-based Programs 19 relatively small system of equations. compared to the nodal equations for the entire system.[78]. . as illustrated in Fig.6 are the branch equations of the nonlinear elements: vk = fk ([v ] . For other types of impedances. . 2. they can be represented in the form of a voltage source behind an impedance. . .2. It is assumed here that the branch impedances are not coupled. . 2. .6 are linear. as in the case of dependent sources.. etc::::) k = 1. Current and Voltage Dependent Sources in EMTP-based Programs 2 [rT HEV ] = 6 6 6 4 r11 r12


r1M r21 r22


r2M . t.... the equations would have to be modi..5) Equations 2. [i] . . as shown in Fig. or in the form of a current source in parallel with an impedance. :::M (2. . and that they are resistive (Rk ).4. rM 1 rM 2


rMM 2 6 [i] = 6 6 4 2 [v ] = 6 6 6 4 i1 i2 . .3) 3 7 7 7 5 (2. vM 20 3 7 7 7 5 (2.2. . iM v1 v2 .3.6) If the branch equations in 2.4) 3 7 7 7 5 (2. .3: Representation of branch equation k as a voltage source in series with a resistance. ck ik Rk vk vsource(k) dk Figure 2.ed. [e] = [eOP EN ] [zT ]
[i] (2. which adds the eect of the M nonlinear branches to the previously calculated open-circuit solution for all nodes with unknown voltages.2.4: Representation of branch equation k as a current source in parallel with a resistance. the currents [i] of 2. After the two systems of equations 2.6 have been solved in subroutine CONNEC.4 are returned to the main program.7) where: [e] is a column vector with the .1 and 2. Current and Voltage Dependent Sources in EMTP-based Programs 21 ik ck Rk vk isource(k) dk isource(k) = vsource(k) / Rk Figure 2.2. . [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 .2. 2.2 Dependent Sources This section presents the necessary equations for implementing current and voltage dependent sources in EMTP-based programs by using the compensation method. [eOP EN ] is a column vector with the previously calculated open circuit solution for all the N nodes with unknown voltages. and the calculation of matrix [zT ]. see reference [74]. please.nal solution for the N node voltages. [i] = column vector with the M compensating branch currents. The following important assumptions are made: 3 For further details about the compensation method. is connected between nodes c and d with branch voltage vk = vc vd (2. the equations from the Thevenin equivalent circuit are the same.8) and for the dependent source branch k vOP ENk + rk1 i1 + ::: ::: + rkj ij + rkk ik + ::: + rkM iM + vk = 0 (2. such that vj is its branch voltage and ij is its branch current. The following models are derived: Current-Controlled Voltage Source (CCVS). In cases where this calculation fails.11) and that the dependent source.12) . CCVS. CurrentControlled Current Source (CCCS). In all cases.e.  Proper precautions are taken to handle extremely large numbers and zero values.9) where: vOP ENk = voltage vk for [i] = 0 (open circuit).2. vj = va vb ij = iab (2. Current and Voltage Dependent Sources in EMTP-based Programs 22  A Thevenin equivalent circuit can be calculated where the dependent source is to be connected. i. rkk = Thevenin resistance (self resistance of branch k). Voltage-Controlled Voltage Source (VCVS) and VoltageControlled Current Source (VCCS).. 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. namely. and also where the controlling current or controlling voltage is to be measured. the connection of large resistors in parallel may make a Thevenin equivalent circuit possible.2.10) (2. for the controlling branch j vOP ENj + rj 1 i1 + ::: ::: + rjj ij + rjk ik + ::: + rjM iM + vj = 0 (2. Rout = Output resistance of the dependent source in branch k.9. Current and Voltage Dependent Sources in EMTP-based Programs 23 and branch current ik = icd (2.14 into 2.9.5: Current-controlled voltage source (CCVS). results in: vOP ENj + rj 1 i1 + ::: ::: + (rjj + Rin ) ij + rjk ik + ::: + rjM iM = 0 (2.16) vOP ENk + rk1 i1 + ::: ::: + (rkj + ) ij + (rkk + Rout ) ik + ::: + rkM iM = 0 (2.14) vk = ij + Rout ik (2. vj = Rin ij (2.15) where: Rin = Input resistance of branch j .2. = Gain over the controlling or measured current.8 and equation 2. applied as voltage dependent source at branch k.2.15 into 2.5 are 2. 2. as well as: ij vOPEN j [ rTHEV j ] ik Rin Rout vj vk [ rTHEV k ] Ω ij vOPEN k Figure 2.13) The necessary equations for the implementation of a current-controlled voltage source as illustrated in Fig.8 and 2.17) . Inserting equation 2. 6 are 2. Current-Controlled Current Source (CCCS) The necessary equations for the implementation of a current-controlled current source as illustrated in Fig. 2.8 and 2. vj = Rin ij (2. as well as: ij vOPEN j [ rTHEV j ] ik Rin vj vk Β ij Rout [ rTHEV k ] vOPEN k Figure 2.16 and 2.2.14 and 2.2.8. the voltages should be eliminated in this reduction from 4 to 2 equations. from which results: vOP ENj + rj 1 i1 + ::: ::: + rjj ij + rjk ik + ::: + rjM iM = 0 (2.9.6: Current-controlled current source (CCCS).20) vk = Rout Bij + Rout ik (2. Whenever possible. because the solution will then produce the currents. 2. Current and Voltage Dependent Sources in EMTP-based Programs 24 Using the two equations 2. since matrix element j k is no longer equal to matrix element k j .9. Rin = 0 and Rout = 0. one can see that the matrix becomes unsymmetric. For an ideal current-controlled voltage source.19) If expressed in matrix form. which are the variables that have to be passed back to the main program.18) vOP ENk + rk1 i1 + ::: ::: + (rkj + ) ij + rkk ik + ::: + rkM iM = 0 (2.15. 2. because it reduces the number of equations which have to be solved in subroutine CONNEC from 4 to 2.17 is preferable to using the four equations 2.21) . 2. Rin = 0 and Rout ! 1.7 are 2.25) Voltage-Controlled Voltage Source (VCVS) The necessary equations for the implementation of a voltage-controlled voltage source as illustrated in Fig. the following equations: vOP ENj + rj 1 i1 + ::: ::: + (rjj + Rin ) ij + rjk ik + ::: + rjM iM = 0 vOPENk k1 i + ::: + Rrout Rout 1   rkj kk ::: + Rout + B ij + Rrout (2. For an ideal current-controlled current source. vOPEN k . applied as dependent current source at branch k.9.24) Bij + ik = 0 (2.8 and equation 2.21 into 2. 2. as well as: ij vOPEN j [ rTHEV j ] ik Rin Rout vj vk [ rTHEV k ] Α vj Figure 2. By inserting equation 2.23. Current and Voltage Dependent Sources in EMTP-based Programs 25 where: B = Gain over the controlling or measured current. resulting in: vOP ENj + rj 1 i1 + ::: ::: + rjj ij + rjk ik + ::: + rjM iM = 0 (2.20 into 2.2.7: Voltage-controlled voltage source (VCVS).9 one can also obtain. respectively.8 and 2.22)  + 1 ik + ::: + RrkM out iM = 0 (2.23) Observe that the division by Rout as done in equation 2. allows the use of very large numbers for Rout without numerical diÆculties. 2. By inserting equation 2.26 inserted into 2.28.9) and divided by the gain A].2.28 and 2. This procedure eliminates Rin in the resulting equation 2. applied as dependent voltage source at branch k. and also to allow the use of very large numbers for the gain A. In order to eliminate the voltages and keep only the currents as variables.28) (2.30 and 2. the following calculations are done: (equation 2. if  A ! 1. results in equation 2.8 and dividing the resulting equation by Rin to avoid numerical diÆculties.27 inserted into 2.  Rin ! 1. Current and Voltage Dependent Sources in EMTP-based Programs 26 vj = Rin ij (2. and  Rout ! 0.27) where: A = Gain over the controlling or measured voltage.31 are obtained.29 for a voltage-controlled voltage source. which can be used to model \ideal operational ampli.26 into 2.26) vk = Avj + Rout ik = ARin ij + Rout ik (2. then equations 2.8) minus the result of [(equation 2.29) Based on the equations 2.29:   vOPENj rj 1 + Rin i1 + ::: R in  rjj +Rin ::: + Rin ij + Rrjkin ik + ::: + rRjMin iM k + r vOP ENj + vOPEN j1 rkj
A ::: + rjj A i
j + rjk ::: + rjM rkM iM = 0 A =0 rk1
i + ::: 1 A rkk +Rout
i + ::: k A (2. ers". which create the \rjk " coupling resistance. Note that the use of equation 2.31 only makes sense if there are feedback paths modelled in the network part.31 . Then equation 2. 2.31) Ideal Operational Ampli. (Please.8. Note also that the equation 2. Current and Voltage Dependent Sources in EMTP-based Programs 27 will produce the correct current ik .2.31 is exactly stating that vj = 0.) ij = 0 (2. see equation 2. which is returned to the main program for the calculation of voltages by compensation.30) vOP ENj + rj 1 i1 + ::: ::: + rjj ij + rjk ik + ::: + rjM iM = 0 (2. ers The commercially available operational ampli. constructed essentially with many transistors and resistors in an integrated package. Operational ampli.er is in reality an integrated-circuit chip. often called OP AMPS. in active . are frequently used in sensor circuits to amplify signals.ers. 2.8 presents the symbol used for representation of an operational ampli. Fig. [82].ltering and control circuits for compensation purposes and endless applications in analog electronics [79]. [81]. [80]. 8: Symbol for operational ampli.er. The voltage placed across the two input termi- Figure 2. er. is to be ampli. nals (the non-inverting terminal (+) and the inverting terminal( )). but this grounding is usually omitted on the symbol). Since the gain of the operational ampli.ed and to appear at the output terminals (one of which is grounded. er is very high. \In practice the input resistance . it is necessary to have an external feedback circuit to make it stable. the voltage gain A is at least 105 . In the ideal operational ampli. and the output resistance Rout is a few tens of ohms " [79]. Current and Voltage Dependent Sources in EMTP-based Programs 28 Rin of an OP AMP is usually well in excess of 1M .2. and then it can usually be modelled as a voltage-controlled voltage source (VCVS). have to be considered though in realistic applications. Many other electrical properties. which are temperature and frequency dependent.2. er. and the gain would be in. 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). Therefore. the fundamental concepts for the analysis of circuits with ideal operational ampli.nite (A = 1 so that the voltage at the non-inverting input terminal would be equal to the voltage at the inverting input terminal). ers are to assume that the two input terminals of the ideal operational ampli. the assumption is made that all operational ampli. AND  \a virtual short-circuit" (equation 2.er constitute \at the same time" [77]:  \an open circuit" (equation 2.31).30). if not otherwise clearly indicated. \In this thesis. There are many variations and combinations of OP AMP circuits. The two basic ones are the inverting ampli.ers are ideal"!. 9) and the non-inverting ampli.er (Fig. 2. 33 respectively.11 illustrates an adder.10). 2.32 and 2. 2. with the transfer functions are given by equations 2. a special case of the inverting ampli. Fig.er circuit (Fig. with the transfer function as of equation 2.35.12 shows an ideal integrator.34) . with the transfer function given by equation 2.er.33) Ei3 (s) R3  (2. Fig. where the output is a linear sum of the input voltages.34.32)
Ei2 (s) R2 + (2. 2. E o ( s) Ei (s) Eo (s) Ei (s) Eo (s) R4 =  = R2 R1 = 1 + RR21 Ei1 (s) R1 + (2. 2. Current and Voltage Dependent Sources in EMTP-based Programs 29 R2 R1 ei ( t ) eo ( t ) Figure 2.9: Inverting ampli.2. er circuit.10: Non-inverting ampli. eo ( t ) R2 ei ( t ) R1 Figure 2. er circuit.11: Adder circuit with operational ampli. R1 R4 R2 ei1 ( t ) R3 eo ( t ) ei2 ( t ) ei3 ( t ) Figure 2. er. 2.35) Fig. Eo (s) Ei (s) = 1 RCs (2.13 presents a generalization of the inverting ampli. which is very useful to obtain Laplace transfer functions by using the impedance approach [81]. With the ideal operational ampli.er circuit. .er. a \virtual ground" potential appears at the inverting input terminal. 12: Ideal integrator circuit with operational ampli. Current and Voltage Dependent Sources in EMTP-based Programs 30 C R ei ( t ) eo ( t ) Figure 2.2.2. 13: Generalization of inverter ampli.er. Therefore. no current ows into the input terminals of the ideal OP AMP. The transfer function for this generalized inverter circuit is given by equation 2.36. I2 ( s ) I1 ( s ) Z2(s) Z1(s) Eo ( s ) Ei ( s ) Figure 2. since the non-inverting input terminal is grounded. Moreover. resulting in Ei (s) = Z1 (s)I (s) and Eo (s) = Z2 (s)I (s). the same current owing through the complex impedance Z1 (s) has to ow through the complex impedance Z2 (s). the transfer function is derived with ideal operational ampli.14.er circuit. 2. in the circuit shown in Fig. Eo (s) Ei (s) = Z2 (s) Z1 (s) : (2.36) For example. er using the impedance approach. The complex impedances Z1 (s) and Z2 (s) for this circuit are: Z1 (s) = R1 Z2 (s) = 1 Cs+ R12 = (2.37) R2 R2 Cs+1 (2.38) . 2.2. Current and Voltage Dependent Sources in EMTP-based Programs 31 R2 C R1 eo ( t ) ei ( t ) Figure 2.14: First-order lag circuit using ideal operational ampli. 2. The transfer function is therefore obtained as Eo (s) Ei (s) = R2 1 R1 R2 Cs+1 : (2.39) Voltage-Controlled Current Source (VCCS) The necessary equations for the implementation of a voltage-controlled current source as illustrated in Fig.9.41) .8 and 2.15: Voltage-controlled current source (VCCS). vj = Rin ij (2.40) vk = Rout vj + Rout ik (2.er.15 are 2. as well as: ij vOPEN j [ rTHEV j ] ik Rin vj vk Γ vj Rout [ rTHEV k ] vOPEN k Figure 2. 40 into 2.8) minus the result of [(equation 2.2.45) Ideal Transformers Even though an ideal transformer model has already been implemented in most EMTPbased programs.41 inserted into 2. with the equations described in [2] . the following calculations are done: times (equation 2.9.42.8 and 2. an ideal transformer can also be implemented as a special dependent source. results in equation 2. or with similar equations. By inserting equation 2.2.16 are 2. This procedure eliminates Rin in the resulting equation 2. and also to allow the use of very large numbers for Rout .44) vOP ENj + rj 1 i1 + ::: ::: + rjj ij + ( rjk 1) ik + ::: + rjM iM = 0 (2. The necessary equations for the implementation of the ideal transformer as illustrated in Fig.43) For an ideal voltage-controlled current source.40 inserted into 2. Current and Voltage Dependent Sources in EMTP-based Programs 32 where: = Gain over the controlling or measured voltage. Rin ! 1 and Rout ! 1. as well as: .2.42) (2.3 ij = 0 (2. In order to eliminate the voltages and keep only the currents as variables.43:   vOPENj rj 1 + R Rin i1 + ::: in  ::: + rjjR+inRin ij + Rrjkin ik + ::: + rRjMin iM  k + rj 1 vOP ENj + vOPEN Rout    kj i + ::: + rjj Rrout rjk j   ::: + rjM RrkM out iM = 0 =0  rk1 i + ::: Rout 1  rkk +Rout i + ::: k Rout (2.8 and dividing the resulting equation by Rin to avoid numerical diÆculties. resulting in: 2.9) and divided by Rout ]. 2. applied as dependent current source at branch k. which may include saturation eects. From the equations above and from 2. one can easily obtain: ij + nik = 0 k + r vOP ENj + vOPEN j1 rkj
n ::: + rjj n i
j + rjk ::: + rjM rkM n iM = 0 (2. It is important to mention that.46) ij = n ik (2.48) rk1
i + ::: 1 rkkn
i + ::: k n (2.2.47) where: nj 1 n = nk = turns ratio of the ideal transformer. This can be accomplished .48 and 2. are used in EMTP-based simulations. please see.2. 2.4 Independent Sources It may be useful in a circuit or device model to have an independent current or independent voltage source connected between two ungrounded nodes. for example.49 can be used to model an ideal transformer.16: Ideal transformer.8 and 2. Current and Voltage Dependent Sources in EMTP-based Programs 33 1 :n ij ik vj vk Figure 2.49) Equations 2. better models for electric transformers. normally. For further details. references [2] and [6]. vj 1 = vk n (2.2.9. 17: Independent current source.2. Independent Current Source Assuming that the independent current source is connected between nodes c and d with branch voltage vk = vc vd (2.51) then the necessary equations for the implementation of an independent current source as illustrated in Fig.52) vk = Rout isource + Rout ik (2.17 are: ik vk isource [ rTHEV k ] Rout vOPEN k Figure 2.50) and branch current ik = icd (2. where: vOP ENk + rk1 i1 + ::: ::: + rkk ik + ::: + rkM iM + vk = 0 (2. 2.2. but using only one equation in this case. Current and Voltage Dependent Sources in EMTP-based Programs 34 with the same technique used for dependent sources.53) . 18 are: ik Rout vk [ rTHEV k ] vOPEN k vsource Figure 2. 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]. From the equations above. one can also obtain the following equation: vOPENk k1 i + ::: + Rrout Rout   1 r +Rout kk ::: + Rout ik + ::: + RrkM out iM (2. 2. vOP ENk + rk1 i1 + ::: ::: + rkk ik + ::: + rkM iM + vk = 0 (2. Current and Voltage Dependent Sources in EMTP-based Programs 35 isource = independent current source of branch k.2.57) . Independent Voltage Source The necessary equations for the implementation of an independent voltage source as illustrated in Fig.55) Of course.56) vk = vsource + Rout ik (2. resulting in: ik + isource = 0 (2. Rout ! 1.2.18: Independent voltage source. which can be constant or a function of time.54) + isource = 0 For the ideal current source. 2. 2. (2.60) is given by xk+1 = xk +
xk = xk f (xk )=f 0(xk ). Current and Voltage Dependent Sources in EMTP-based Programs 36 where: vsource = independent voltage source of branch k. which can be constant or a function of time.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. Rout = 0.2. one can also obtain the following equation: vOP ENk + rk1 i1 + ::: ::: + (rkk + Rout ) ik + ::: + rkM iM + vsource = 0 (2.2. 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.5 Newton-Raphson Algorithm The equations for current and voltage dependent sources have been presented in the previous sections.61) . with a voltage source to ground on the other side. if more detailed information is needed. The solution of these equations is based on the compensation method.58) For an ideal voltage source. In the scalar case the N-R iteration to solve f (x) = 0 (2. resulting in: vOP ENk + rk1i1 + ::: ::: + rkk ik + ::: + rkM iM + vsource = 0 (2. For completeness. as well as the equations of independent sources which may be connected between two ungrounded nodes. widely used and has quadratic convergence if the initial estimate is close to the solution. 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. From the equations above. .. the vector of variables by [x] and the vector of functions by [f (x)]. @fM (x fM (x ) = fM (x) + @f@xM (x1 x1 ) + @f@xM (x2 x2 ) +


+ @x xM ) +


M M 1 1 1 1 2 2 2 2 1 2 1 2 Assuming that x is close to x . Using superscripts to indicate the iteration count results in: 
  f xk + [J ] xk+1   k 
x = 0 (2.65 is set to zero and solved.66) jx is the Jacobian matrix of the function [f (x)]. fM (x1 .. xM ) = 0 f2 (x1 . denote it by [x ] and expand each function in a Taylor series about [x]: @f (x x ) + @f (x x ) +


+ @f (x f1 (x ) = f1 (x) + @x xM ) +


1 2 1 2 @x @xM M @f @f @f     f2 (x ) = f2 (x) + @x (x1 x1 ) + @x (x2 x2 ) +


+ @xM (xM xM ) +


(2.62 has a compact form: [f (x)] = 0 (2. . higher order terms may be neglected and the system may be written in linearized form: [f (x )]  [f (x)] + [J ] ([x ] where [J ] jx = 2 @f1 @x1 6 @f2 6 @x1 6 . @fM @x1 [x]) @f1 @x2 @f2 @x2





@f1 @xM @f2 @xM @fM @x2


@fM @xM . the result will not be the vector [x ] (because the higher-order terms have been neglected) but some new value for [x]. : : : .67) . . . x2 .2. If equation 2. which has to be calculated at each iteration step..63) Assume that the system has a solution. .62) Denote. Consider now the system of M nonlinear equations fi in M variables xi : f1 (x1 .65) 3 7 7 7 5 (2... Current and Voltage Dependent Sources in EMTP-based Programs 37 where the iteration count is denoted by the superscripts. .2. x2 . : : : . for easy notation. : : : . xM ) = 0 . .64) . x2 . (2.. xM ) = 0 (2. 4 . Then 2. the solution of 2. Current and Voltage Dependent Sources in EMTP-based Programs 38 Formally. Instead de.2.2. the Jacobian matrix is not inverted.68) In practice.67 is obtained by writing  k+1   k  x = x [J ] 1  f xk
 (2. ne      
xk = xk+1  k x : (2. which reduce the error norm iteratively so that  f xk+1
   f xk
 (2.71) Formulae 2.72) This iterations scheme is repeated until the errors are lower than a speci.70 and 2.69) Then  [J ]
xk = f xk
 (2.71 represent the Newton-Raphson algorithm for systems of equations.70) is solved by LU factorization and the new [xk+1 ] is obtained from  k+1   k   k  x = x +
x : (2. ed tolerance. More iteration steps are required for the solution of a system of nonlinear equations depending on how close the initial guess is to the . as in the case of linear dependent sources. For the case of a system of linear equations. convergence to the solution is achieved with just one iteration step. The solution algorithm experimentally implemented in the MicroTran version of the EMTP at The University of British Columbia is presented in Fig. . This method presents \generality and exibility properties". the standard application of the iteration scheme of the Newton-Raphson algorithm may cause numerical problems involving computer over ows.nal solution.19. For highly nonlinear functions. as will be explained later in this thesis. 2. thus looking promising for future work in detailed modelling of circuits and devices. NO.1)] IS IT ACCURATE ENOUGH? Yes Yes CHECK FOR LIMITS Yes STOP IS THERE ANY LIMIT VIOLATION? No MAX.6) or (2. OF ITERATIONS WAS EXCEEDED? No No BUILD JACOBIAN MATRIX [partial derivatives of Eq. 39 .19: Newton-Raphson algorithm experimentally implemented in MicroTran. Current and Voltage Dependent Sources in EMTP-based Programs INPUT DATA INITIAL GUESS FOR CURRENTS CALCULATION OF BRANCH VOLTAGES [from Eq.1)] SET VARIABLE TO ITS LIMIT CALCULATION OF RIGHT HAND SIDE [negative values of Eq.2. (2.2. (2. (2.1)] RETURN CURRENTS TO MAIN PROGRAM SOLVE FOR CURRENTS UPDATE CURRENTS Figure 2. Current and voltage sensors.2. Current and Voltage Dependent Sources in EMTP-based Programs 2.2.2. 2. Operational ampli. such as: 1.6 40 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. User-de.ers. 3. 5. where the physical behavior would need to be represented by nonlinear equations. User-de. 4. Modelling of electronic components.ned coupled branches in a circuit. ned linear and nonlinear functions. User-de. 6. limited only by the creativity and ingenuity of the user. Figure 2.21 illustrate the solution method with an example of a noninverting ampli.ned modelling of linear and nonlinear devices. 2.20 and Fig. It consists of a sinusoidal voltage source.er circuit. as commonly used in practical analog electronics. an ideal operational ampli. The ideal operational ampli.er and 2 resistors (Rf and Rg ). as shown in Fig. to get an ampli. then voutput = 3vinput . whereas the sinusoidal voltage source and the resistors are part of the network. represented through a Thevenin equivalent circuit. Alternatively.21.30 and 2.31. 2.er was modelled using equations 2. If Rf = 2Rg . with equations 2.29 with the gain A set to a value of 3. to avoid numerical oating subnetwork problems.cation of 3 in circuit simulation.28 and 2. In this case a load resistor should be connected in the output of the dependent source. Indeed. in theory this noninverting ampli. one could just use a voltage-controlled voltage source. 73) .er circuit should result in:  voutput R = 1+ f vinput Rg  (2. 20: Circuit with ideal operational ampli.2.2.0 0o [V] f=60 [Hz] Rg Figure 2. Current and Voltage Dependent Sources in EMTP-based Programs 41 Rf voutput vinput vinput=1. er. 4 voutput 3 2 Voltage ( V ) 1 vinput 0 −1 −2 −3 −4 0 5 10 15 20 25 30 35 40 45 50 Time ( ms ) Figure 2.21: Simulation results of circuit with ideal operational ampli. er (noninverting ampli. some improvements already made for the implementation of saturation or limits for the elements or sources presented in this work. . The next sections will present a technique for the simulation of transfer functions in EMTP-based programs. as well as the implementation of some other special control devices.er circuit). 3 Development of Control Transfer Functions in EMTPbased Programs A transfer function as in Fig. 2.22. Development of Control Transfer Functions in EMTP-based Programs 42 2. is de.3.2. 74.74) with n  m and an 6= 0. It is possible to reorganize the terms of equation 2.74 as follows: (an sn + an 1 sn 1 + ::: + a1 s1 + a0 ) X (s) = k (bm  k  sm + b m 1 sm 1 + ::: + b1 s + b0 ) U (s) 1 + anan s 1 + ::: + aan s1 1 bm sm n + an X (s) = k + ab1n s1 n + ab0n s n  1 bm 1 m an s  U (s) 1 n + a0 s n an n + ::: + b1 s1 n an bm sm n + bm 1 sm an an  (2.22: Transfer function. which represents the output signal X (s) as a function of the input signal U (s) for a particular linear time-invariant system.75) 1 an 1 an s 1 + ::: + 1  X (s) = + abn s 0 n n  (2.77) .ned in the frequency domain (Laplace transformation of a continuous time system) by the equation 2.76) U (s) + ::: a1 1 n an s + a0 n an s  X (s) (2.   X (s) b sm + b sm 1 + ::: + b1 s1 + b0 B (s) H (s) = =k m n m 1n 1 =k 1 U (s) an s + an 1 s + ::: + a1 s + a0 A(s) (2. U(s) H(s) X(s) Figure 2. 23.78) (2. 2. which. 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.80) A transfer function block-diagram realization of equation 2. 4 this results in: X (s) = k + ab1n s1 n + ab0n s n X (s) = +s1   U (s) k abnn U (s) + s h n k b1 U (s) an ::: + s 1 k abn U (s) 1 + bnan s 1 + ::: bn an 1 an 1 an s 1 h 1 1 + ::: + n +s a1 1 n an s h n k b0 U (s) an a1 an X (s) +s 1  X (s) + ::: i a0 an X (s)  an 1 X ( s ) + ::: an 1  + an 1 an X (s) k bnan U (s)  a0 n an s i k bnan 1 U (s) i a1 X (s) an X (s) = k abnn U (s) + s h  k abn U (s) 0 a0 an X (s) io (2.3.80 is presented in Fig. Development of Control Transfer Functions in EMTP-based Programs 43 For m = n. is called the observer form.2. such as operational ampli.79) (2. according to [79]. an analog signal processing scheme is usually designed as the . In practice. resistors and capacitors 5 .ers. This way. . in this work. which 4 If m < n then bn = bn 1 = ::: = bm+1 = 0: 5 Analog computers were commonly used in the past to solve power system control and stability dierential equations [83]. Moreover. the derivation of an analog circuit model for the transfer function implementation takes advantage of all the options already implemented in EMTPbased programs. and redundant computational work is avoided.rst step for a digital signal processing derivation. the equations for the digital model of a transfer function are automatically constructed inside the EMTP. or the backward Euler rule whenever CDA technique is applied [84].3. Figure 2.. Development of Control Transfer Functions in EMTP-based Programs 44 U(s) . [85]. uses the trapezoidal integration rule. kb0 ___ an + kb1 ___ an 1 ___ s a0 ___ an + + - kbn-1 ___ an 1 ___ s + .2..23: Observer form block-diagram of transfer function in equation 2.80. With analog circuit modelling based on operational ampli... + - a1 ___ an kbn ___ an 1 ___ s + + X(s) an-1 ___ an ... A simpli. 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.ers. cation can be made in a computer transfer function implementation. in order to reduce the number of operational ampli. in contrast to a physical circuit implementation. the system is stable. Negative values then may be used here for capacitances and resistances connected on the feedback path of the \ideal operational ampli. As long as the system eigenvalues (or poles of the transfer function) remain on the left hand side of the complex plane.ers needed: resistances and capacitances can assume negative values 6 . 2. Such a possible computer implementation of the transfer function block-diagram of Fig. that the .24.23 is presented in Fig. which results in a stable solution for transfer functions of linear systems. 2.ers". Assume for example. 2.80 becomes in 6 Proper precautions should be taken.25 is to be implemented with the proposed technique. though.rst-order transfer function illustrated in the block-diagram of Fig. . The connection of large resistances to the node can easily solve this problem. whenever the equivalent digital self admittance of a node becomes equal to zero. Equation 2. Development of Control Transfer Functions in EMTP-based Programs 45 u(t) .2.. 10 __________ U(s) X(s) 0..3.24: Possible computer implementation of the transfer function block-diagram in Fig. Figure 2.25: Block-diagram representation of a .23.. -1M Ω 1M Ω an _ ___M Ω a1 x(t) 1M Ω an _ ___ MΩ an-1 . 2..01 s + 1 Figure 2... an ___ M Ω kb0 -1µ F an ___ M Ω kb1 -1µ F 1M Ω an _ ___ MΩ a0 an ___ MΩ kbn-1 an ___ MΩ kbn -1µ F 1M Ω . rst-order transfer function.81 is presented in Fig. and its \possible computer implementation" is illustrated in Fig. The observer form block-diagram for equation 2.81) where.27. 2.26. for illustration purposes. kb0 = 10. a0 = 1 and a1 = 0:01 seconds. with n = 1: X (s) = s  1 b k 0 U (s) a1  a0 X (s) a1 (2. 2. this case. The realization of this . rstorder transfer function can also be done with a physically-based realistic . rst-order lag circuit as shown in Fig. 2. which requires two inverting ampli.28. er circuits 7 . Jesus Calvi~no-Fraga in a practical laboratory experiment for . instead of just one 7 The author acknowledges the help of Mr. 29 presents the time domain transient response x(t) of the . 2.2. which the proposed method can handle as well without any restrictions.3. Development of Control Transfer Functions in EMTP-based Programs 46 required in the \more economic" computer implementation. Fig. There may be cases where the realistic implementation is needed. for a pulse u(t) of 1V with a duration of 25ms.27 and as in Fig. 2.28.26: Observer form block-diagram of . 2. U(s) kb0 ___ a1 + - X(s) 1 ___ s a0 ___ a1 Figure 2.rst order transfer function implemented as in Fig. 1µ F . u(t) a1 ___ M Ω kb0 .1µ F x(t) = u(t) 1 kΩ x(t) a1 _ ___ MΩ a0 Figure 2. 2.25.27: Possible computer implementation of .10 kΩ .rst-order transfer function of Fig. 25. which validated the simulation results presented in Fig. the circuit of Fig. 2.rst-order transfer function of Fig.28.29. 2. . 2. 3. Development of Control Transfer Functions in EMTP-based Programs 47 10 k Ω 1µ F u(t) 10 k Ω .28: Realistic .2.x( t ) 1 kΩ x(t) 10 k Ω Figure 2. 29: Time domain simulation of . 10 9 8 x(t) Voltage ( V ) 7 6 5 4 3 2 u(t) 1 0 0 5 10 15 20 25 30 35 40 45 50 Time ( ms ) Figure 2.rst-order lag circuit. rst-order transfer function. This technique can easily be used with the solution for the \ideal operational ampli. ers" implemented in subroutine CONNEC. The modi. and the solution of both systems of equations is simultaneous. There is no time delay between the electric network and control equations as in TACS. ed nodal analysis method (MNA) [76] could also be used for the solution of operational ampli. ers and other \linear branch equations" as presented in [77] . . However. Development of Limiters for Control Systems in EMTP-based Programs 48 with the modi.4.2. The implementation of limits associated with . Also. which may eventually decrease the computational eÆciency. as in the CONNEC subroutine of MicroTran. and the remaining nonlinear equations could be solved with the compensation method. the network and control system equation matrix becomes unsymmetric. if the simulation time becomes a very important issue. a combined solution could be investigated. 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.ed nodal analysis (MNA). the dimension of the matrix may become very large. and zero diagonal elements may appear. Possibily. if proper techniques are not used. requiring extra columns and extra rows. which requires pivoting techniques. as the number of added branch equations increases. such that all linear control and system equations could be solved using the MNA with appropriate techniques. as in the case of digital real-time simulators. particularly because the branch Thevenin equivalent circuit is readily available. [67]. With the Newton-Raphson iterative algorithm in the compensation method. 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. a simultaneous solution for limiters can be found. There are two types of limiters associated with . This can result in inaccuracies and instabilities [66]. The technique proposed in this thesis for the solution of limiters overcomes this diÆculty. as well as special devices and intrinsic FORTRAN functions. are presented in the following sections.rst order transfer functions. rst-order transfer functions : windup (also referred to as static limiter) and non-windup (dynamic limiter) [66]. "Non-windup limiters should only be used with . [2]. For second and higher-order transfer functions it is no longer clear which variables should be limited..rst-order transfer functions. . Even for the .. rstorder transfer function. the meaning of the limiting function is confused if it has any zeros" . In a lead-lag control function block.2.4. for example. the way in which a non-windup limiter can be realized is not unique. the . 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. 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 interpretation of the limiting action should therefore be based on the electronic implementation of the physical device [86]. Development of Limiters for Control Systems in EMTP-based Programs 49 [2]. 2.31. but x(t) backs o the limit .30: First-order transfer function with windup (static) limiter. The time domain simulation of the output variable x(t) for both cases. is presented in Fig.25 is assumed to have a windup limiter as in Fig. Note that the output variable x(t) reaches its limit at the same time for both cases.rst-order transfer function presented earlier in Fig. 2.30. 2.01 s + 1 Figure 2. and a non-windup limiter as in Fig. 2.31: First-order transfer function with non-windup (dynamic) limiter.01 s + 1 Figure 2.32. for a pulse input excitation u(t) of 1V. + 5 slope=1 U(s) 10 __________ X(s) 0. + 5 slope=1 10 __________ U(s) X(s) 0. rst for the non-windup (dynamic) limiter. whereas in the non-windup limiter the dierential equation is actually modi. The reason is that for the windup limiter the output variable is just clipped at the limit. [2]. [87].ed [66]. . Development of Limiters for Control Systems in EMTP-based Programs 50 10 x(t) without limiter 9 8 Voltage ( V ) 7 6 5 x(t) with windup limiter (static) 4 x(t) with non−windup limiter (dynamic) 3 2 1 u(t) 0 0 5 10 15 20 25 30 35 40 45 50 Time ( ms ) Figure 2.32: Transient response of a .4.2. a simultaneous system solution is . The implemented solution for limiters uses the methodology proposed in Section 2.1.2.5. As indicated in the algorithm illustrated in Fig.rst-order transfer function with windup and non-windup limiter. 2.19 of Section 2.2. Then.rst found without considering any of the limits. each limit violation is veri. The maximum (xmax ) and minimum (xmin ) limiting values are part of the input data. Also. the solution for limiters is simultaneous without time delays. and has given the correct solution for all limiters. If a particular limit has been reached. . because the compensation method is properly applied 8 . For 8 TACS seems to use a pseudo-compensation method to solve limiters [2]. and a solution is found for this particular limiter and all of its consequences on the other limiters.ed in the sequence of the input data given by the user. \independent of the ordering of the input data given by the user". This cause-consequence iterative process has been found to be a very \robust method" in all cases tested. all previous indications of limit violations are cleared. 2. in the case of the .4. Development of Limiters for Control Systems in EMTP-based Programs 51 example. rst order transfer function with a non-windup limiter illustrated in Fig. with a computer model as in Fig. 2. 2.31. it is possible to represent the non-windup hard limiting action with a simple change in the equations for the ideal operational ampli.27. in a realistic . such that equations 2.82 and 2. In practice.31 are replaced by equations 2.83.er.30 and 2.82) =0 (2. or vk limit limit (2. respectively: vOP ENj + rj 1 i1 + ::: ::: + rjj ij + rjk ik + ::: + rjM iM = 0 vOP ENk + rk1 i1 + ::: ::: + rkj ij + rkk ik + ::: + rkM iM + vk where vk limit = xmax . By using these equations it becomes easy to observe the limits accurately.83) = xmin . 2.28.rst order lag circuit. as in Fig. the clamping action is done with the use of Zener diodes connected in parallel with the capacitor in the feedback loop of the . one could use the equations for an \ideal voltage-controlled voltage source" including the limiting values in the output voltage.84) limit =0 (2. or vk limit (2. In this case. as follows: ij = 0 vOP ENk + rk1 i1 + ::: ::: + rkj ij + rkk ik + ::: + rkM iM + vk where vk limit = xmax .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 . Implementation of soft limits The limiters presented in the previous section assume . Another example is the simple limiter control block.85) = xmin . 2. . the slopes in the limited region are nonzero. as recommended in [2]. 9 The author acknowledges the help of Mr. It may be useful to allow soft limits as well. With soft limits.32. which validated the simulation results presented in Fig.xed values (hard limits) for the maximum and minimum of the output variable. Jesus Calvi~no-Fraga in a practical laboratory experiment. if Ku(t)  xmin.33. 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. Development of Limiters for Control Systems in EMTP-based Programs 52 Hard limits are then just a special case of soft limits when the slopes are set to zero. with the notation from Fig. x(t) = xmin + Kmin[u(t) umin]. the zero-order transfer function (constant gain) in Fig. x(t) slope=Kmax x max SLOPE = GAIN "K" u min u max slope=Kmin u( t) x min Figure 2.35. for example. The time domain response for a sinusoidal excitation input u(t) of 1V is presented in Fig.4. illustrating the eects of hard and soft limits on the output x(t).2. In this thesis project. Consider.86) : xmax + Kmax[u(t) umax].2. . 2. The equations for soft limits. 2. 2.33: Soft limits. if Ku(t)  xmax . are: 8 if xmin < Ku(t) < xmax . < Ku(t).34. Limits can also be easily implemented for all the FORTRAN functions and special devices which will be discussed in the following sections. .1 ( 10% ) + 2 K=4 U(s) 53 X(s) . 4 x(t) with soft limit 3 2 Voltage ( V ) 1 u(t) x(t) with hard limit 0 −1 −2 −3 −4 0 5 10 15 20 25 30 35 40 45 50 Time ( ms ) Figure 2.2.2 K min= + 0.35: Time domain response for a sinusoidal excitation input u(t) illustrating the eects of hard and soft limits on the output x(t). Development of Limiters for Control Systems in EMTP-based Programs K max= + 0.4.1 ( 10% ) Figure 2.34: Zero-order transfer function with soft limits. 5 Development of Intrinsic FORTRAN Functions in EMTP-based Programs \Supplemental variables and devices".2.5. as de. Development of Intrinsic FORTRAN Functions in EMTP-based Programs 54 2. they are not solved with the matrix of the set of linear equations in TACS. \In Fig. dier in TACS from the transfer function blocks. 2. The solution would still be simultaneous in this case. are solved in TACS in a sequential way. The sequential solution requires a de. Supplemental variables. the reader should consult the EMTP Rule Book" [2]. the sequence of calculations is more complicated. and S 2 would be solved after G3 has been solved. For details. In general. the special device S 1 would be solved after G2 has been solved.ned by the EMTP Rule Book. as follows: 1. with non-simultaneous solutions through time delays. instead of simultaneously (so that the data cards must be ordered accordingly). such as intrinsic FORTRAN functions and special devices. they are calculated sequentially.36. 2. S 2 in Fig. and no special ordering is necessary. the input of data by the user is arbitrary. i. S 3 in Fig. Assume. and use input and output devices instead whenever possible" [2]. 2.36". In this research project a \truly simultaneous solution" is achieved for intrinsic FORTRAN functions.S 2 and S 3". for . S 1 in Fig.g.36: Open loop control system with "supplemental devices S 1. 2.e.g.36)" and \inside (e.2 for the implementation of current and voltage dependent sources. \output (e. \To make the solution as much simultaneous as possible.nition by the user of \input (e.. The solution technique applies the compensation method in a similar way as done in Section 2.g.36)" groups of devices. 2. A special ordering of these device blocks is necessary to minimize the time delays introduced by this sequential solution method. S1 G1 ( s ) G2 ( s ) S2 G3 ( s ) S3 Figure 2. the user should keep the number of internal devices as low as possible. 88 and the branch equations 2. For the controlling branch.5. vj = Rin ij (2. the control block-diagram of Fig. 2.38: Circuit implementation for the simultaneous solution of a sinusoidal FORTRAN function.90.89 and 2. 2. These equations resemble those of a voltage-controlled voltage source presented in Section 2.87) ::: + rjj ij + rjk ik + ::: + rjM iM + vj = 0 and for the dependent source branch it is vOP ENk + rk1 i1 + ::: (2. the equation is vOP ENj + rj 1 i1 + ::: (2.87 and 2.89) . rkk = Thevenin resistance (self resistance of branch k).37. 2.37 in the form of an electric circuit as shown in Fig.2. rkj = Thevenin resistance (coupling or mutual resistance between branches k and j ).38. with a nonlinear relationship between the output voltage vk (t) and the input voltage vj (t).37: Nonlinear control block-diagram with a sinusoidal intrinsic FORTRAN function. ij vOPEN j [ rTHEV j ] ik Rin Rout vj vk [ rTHEV k ] vOPEN k K2 sin ( K1 vj ) Figure 2. A simultaneous solution can be obtained for this nonlinear function by representing the block-diagram of Fig.88) ::: + rkj ij + rkk ik + ::: + rkM iM + vk = 0 where: vOP ENk = voltage vk for [i] = 0 (open circuit). The necessary equations are 2. Development of Intrinsic FORTRAN Functions in EMTP-based Programs 55 example.2. vj vk K2 SIN ( K1 vj ) Figure 2. the following nonlinear intrinsic FORTRAN functions were implemented. Development of Intrinsic FORTRAN Functions in EMTP-based Programs vk = K2 (sin (K1 vj )) + Rout ik = K2 (sin (K1 Rin ij )) + Rout ik 56 (2.19. From the equations above. and of course.  SIN 10 In theory Rin ! 1. Rout is obviously assumed to be equal zero. which would result in a trivial solution for equation 2. of Section 2. .90) where: K1 = Gain over the controlling or measured voltage. 2. Note that for the proper application of the compensation method there must always be possible a Thevenin equivalent circuit. These equations can then be solved with the implemented Newton-Raphson algorithm illustrated in Fig. then the insertion of a big resistance between this node and ground easily overcomes this problem. making ij = 0. This allows the convergence to a non trivial solution for the current ik . K2 = Gain applied to the dependent source in branch k. one can also obtain the following equations:   vOPENj rj 1 + R Rin i1 + ::: in  ::: + rjjR+inRin ij + Rrjkin ik + ::: + rRjMin iM =0 vOP ENk + rk1 i1 + ::: ::: + [rkj ij + K2 (sin (K1 Rin ij ))] + (rkk + Rout ) ik + ::: + rkM iM = 0 (2.2..91) (2. Applying this technique. i.e.5. then the solution for the current ij will be very small but not exactly equal to zero. Therefore.2. for the input and output voltages of this nonlinear control block of a sine function.92) If a large number is used for Rin 10 . in cases where there is a oating subnetwork.92. as is usually done in some versions of the EMTP for the solution of nonlinear elements [2]. a node without connection to ground (as for example if the output of an intrinsic FORTRAN function has no circuit elements connected to ground). which is then converted to radians internally. exponentiation of negative number with non-integer exponent. The inverse of trigonometric functions gives the answer already converted to degrees. Development of Intrinsic FORTRAN Functions in EMTP-based Programs  COS  TAN  COTAN  SINH  COSH  TANH  ASIN  ACOS  ATAN  EXP  LOG  LOG10  SQRT. etc.2. square root of negative number. The evaluation of trigonometric functions accepts the argument in degrees.5. 11 Addition (+) and subtraction ( ) can be implemented with the use of just one ideal operational ampli. logarithm of zero or of negative values.. Proper precautions were taken to handle mathematical and computational problems such as division by zero. 57 as well as the mathematical operations 11 :  multiplication ()  division (=)  exponentiation (). .er. by using the same approach presented in the previous section.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.2. Their input can be in arbitrary order de. or conversely.93) and for the dependent source branch vOP ENk + rk1 i1 + ::: (2. the equations are derived including Rin and Rout .93 and 2. for example. To illustrate the potential of this technique. . Transport Delay Assume. all the current and voltage variables are the instantaneous values at the present time t.95 and 2.ned by the user.e. If not otherwise indicated.. 2. The transport delay of Fig 2.40. with circuit equations 2. the output voltage vk (t) only depends on the past history value of the input voltage.94.39. the detailed development of some useful special devices will be presented. and branch equations 2.96. For the controlling branch. (also called "transport delay" in TACS). 2.39: Transport delay control device. vj (t  ). the time delay control block-diagram of Fig.39 can be represented in the form of an electric circuit 12 as in Fig. the equation is vOP ENj + rj 1 i1 + ::: ::: + rjj ij + rjk ik + ::: + rjM iM + vj = 0 (2. where the input voltage vj (t) only aects the output after the elapsed time t +  . however the ideal equations are used in the computer implementation.94) ::: + rkj ij + rkk ik + ::: + rkM iM + vk = 0 12 For generality reasons. i. vj DELAY vk Figure 2. to create From the equations above. where: vOP ENk = voltage vk for [i] = 0 (open circuit).97) (2. then ij = 0 (2.98) If Rin ! 1 and Rout ! 0.96) where: K2 = Gain applied to the controlling or measured past history voltage vj (t an independent source at time t.2.99) vOP ENk + rk1i1 + ::: ::: + rkj ij + rkk ik + ::: + rkM iM + K2 vj (t  ) = 0 (2. in branch k. vj = Rin ij (2. rkk = Thevenin resistance (self resistance of branch k).τ ) Figure 2.6. rkj = Thevenin resistance (coupling or mutual resistance between branches k and j ). Development of Control Devices in EMTP-based Programs ij vOPEN j ik Rin [ rTHEV j ] 59 Rout vj vk [ rTHEV k ] vOPEN k K2 vj ( t .100) .95) vk = K2 vj (t  ) + Rout ik (2. one can also obtain the following equations:   vOPENJ rj 1 + Rin Rin i1 + :::  ::: + rjjR+inRin ij + Rrjkin ik + ::: + rRjMin iM =0 vOP ENk + rk1 i1 + ::: ::: + rkj ij + (rkk + Rout ) ik + ::: + rkM iM + K2 vj (t  ) = 0 (2.  ).40: Circuit implementation for the simultaneous solution of a transport delay control device. it is possible to develop a model for a pulse delay control device. K 2 = 1.41 that the output voltage signal vk (t) is actually equal to the input voltage signal vj (t). 2. the input voltage signal vj (t) is a sinusoidal source of 1V.19.5 1 Voltage ( V ) 0.2. some type of interpolation must be used. Development of Control Devices in EMTP-based Programs 60 which can be solved with the implemented Newton-Raphson algorithm illustrated in Fig. [2].1667ms.6.2. Pulse Delay Applying the same technique used for the implementation for a transport delay.5 −1 −1.
t = 166:6667s.5 −2 0 5 10 15 20 25 30 35 40 45 50 Time ( ms ) Figure 2.5 0 delay 4. but delayed in time in 4. 2. Considering that the delay time  is not usually an integer multiple of the simulation time step
t. 2 delayed signal 1. the negative to .41 illustrates the time response of a transport delay control device where  = 4:1667ms. Linear interpolation has been chosen for that purpose. In a pulse delay. Fig. of Section 2. 2. in a way similar to the transient time domain simulation of a transmission line model [1]. Note in Fig.41: Transient simulation of a transport delay control device.1667ms −0. 2.6. Development of Control Devices in EMTP-based Programs 61 positive and positive to negative zero crossings of the input signal are detected and a pulse is created with the speci. which is a \more computational economic" delay if the output signal will always have to be a pulse.42: Transient simulation of a pulse delay control device. and the input voltage signal vj (t) is a 1V pulse source. This way there is no need to store all the past history values of the input signal. K 2 = 1.43 shows the time response of a pulse delay control device where  = 20ms. 2 delayed pulse 1.5 Voltage ( V ) 1 0.5 −1 −1. and the input voltage signal vj (t) is an arbitrary signal source. K 2 = 1. irrespective of the shape of the input signal.5 0 20ms delay −0. Fig.5 −2 0 5 10 15 20 25 30 35 40 45 Time ( ms ) Figure 2.ed delay and with the width of respective time between the two zero crossings of the input signal. 2. 50 . 2.42 illustrates the time response of a pulse delay control device where  = 20ms. Fig. just the respective times of zero crossing. 2. .5 −2 0 5 10 15 20 25 30 35 40 45 50 Time ( ms ) Figure 2.44 and Fig.5 −1 −1.95 and vk = K2 (1 vj ) + Rout ik (2. vj vk Figure 2.93.6.43: Pulse delay control device with arbitrary input signal. as illustrated in Fig.94. 2.2. Logic Gate "NOT" The necessary equations for the simultaneous solution of a logic gate NOT.45.5 1 Voltage ( V ) 0.44: Logic gate "NOT". 2. 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. Development of Control Devices in EMTP-based Programs 62 2 delayed pulse 1.5 0 20ms delay −0. are 2. From equations 2.103) If Rin ! 1. such as "AND". .6.102) (2. "OR".2. Development of Control Devices in EMTP-based Programs ij vOPEN j [ rTHEV j ] 63 ik Rin Rout vj [ rTHEV k ] vk vOPEN k K2 ( 1 . Rout ! 0 and K2 = 1. then the following equations are obtained: ij = 0 (2.105) Other logic gates. 2.45: Circuit implementation of a logic gate "NOT" for simultaneous solution.104) vOP ENj vOP ENK + 1 + (rj 1 + rk1 ) i1 + ::: ::: + (rjj + rkj ) ij + (rjk + rkk ) ik + ::: ::: + (rjM + rkM ) iM = 0 (2. "NOR".101. "NAND". etc.vj ) Figure 2. one can also obtain the following equations:   vOPENJ rj 1 Rin +  Rin i1 + ::: ::: + rjjR+inRin ij + Rrjkin ik + ::: + rRjMin iM =0   vOPENK r k 1 vOP ENj + 1 + rj 1 + K2 i1 + ::: K  2   ::: + rjj + rKkj2 ij + rjk + rkk +KR2 out ik + :::   ::: + rjM + rKkM2 iM = 0 (2. 2.93. can be implemented in a similar way.95 and equation 2.94. Chapter 3 Power Electronics Modelling in EMTP-based Simulations \The application of semiconductor devices in the electric power . simulation of power electronics systems is made challenging by the following factors: 1) extreme nonlinearity presented by the switches. This is important in order to minimize the overall simulation time. and 3) a lack of models. design and education of Power Electronics. The power semiconductor devices. which distinguishes the subject of power electronics from many other engineering studies" [88]. and a study of power electronics (as it is commonly called) is now a feature of most electrical and electronics engineering courses. such as the diode. However. 2) time constants within the system may dier by several orders of magnitude. The simulation package should also be able to represent the controller portion of the converter system by its functional features 64 . The development of theory and application relies heavily on waveforms and transient responses. \In system level investigation. it is often adequate to represent semiconductor switches within converters by ideal switches. thyristor. and power transistor. Therefore. \Computer simulation can greatly aid in the analysis. it is important that the objective of the computer analysis be evaluated carefully and appropriate simulation packages be chosen" [89]. But. it is very desirable if the same simulation package has the detailed device models to design snubbers and gate drives. are used in power applications as switching devices.eld has been steadily increasing. triac. Modelling Power Electronics in Electric Power Engineering Applications 65 in as simpli.1.3. it should be able to model the controller on a component level if needed" [89].ed a manner as possible. The preceding quotations show the need for the development and implementation of simpli. Yet. 1 presents the major power semiconductor devices: Diode. 3. the implementation of a nonlinear diode model. Some of these semiconductor devices are already well known. Models for the existing and new power electronics devices are therefore necessary for analyzing existing or future applications. The following sections discuss some of the IEEE recommendations for power electronics modelling. Gate Turn-O (GTO) Thyristor and Gate-Controlled Thyristor (GCT). MOS Controlled Thyristor (MCT). 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. High-power semiconductor devices with increasing switching and power capabilities appear on the market every year (such as the Integrated Gate-Commutated Thyristors (IGCTs)).ed as well as detailed nonlinear models of semiconductor devices in EMTP-based programs. Thyristor. For power quality studies. MOS Turn-O Thyristor (MTO). and some aspects of control of power electronic devices.1 Modelling Power Electronics in Electric Power Engineering Applications Fig. They are especially useful for the detailed evaluation of the impact of power electronic devices on electric power quality. Emitter Turn-O Thyristor (ETO). and the impact of transients coming from the supply power system on the electronic devices have to be analyzed as well. Insulated Gate Bipolar Transistor (IGBT). and Metal Oxide Semiconductor Field Eect Transistor (MOSFET) [90]. 3. Transistor. the simultaneous solution of voltage-controlled switches in EMTP-based programs. Such models can be used for the transient simulation of electromagnetic phenomena in low and high power circuit networks. which is the main emphasis of this thesis. which extend the potential applications of modern power electronics techniques to basically all voltage levels in utility companies and industrial sites. . Modelling Power Electronics in Electric Power Engineering Applications DIODE THYRISTOR Cathode GTO and GCT Cathode Cathode Gate ( turn-on & turn-off ) Gate ( turn-on ) Anode 66 Anode Anode MTO ETO MCT Cathode Cathode Cathode Turn-off Gate Turn-on Gate Turn-off Gate Gate ( turn-on & turn-off ) Turn-on Gate Anode Anode Anode TRANSISTOR IGBT MOSFET Emitter Emitter Source Base Collector Gate Collector Figure 3.1: Power semiconductor devices.1. Gate Drain .3. especially for use in EMTP-based programs. Mohan [93].3.1. R-C snubbers are connected across them. To avoid this problem of extreme nonlinearity such as that associated with diodes. The approximately 60 test cases in the computer exercise collection of Dr. Ned Mohan also presents in [89] a power electronics library with special sub-circuits. [92]. Modelling Power Electronics in Electric Power Engineering Applications 67 Guidelines for modeling power electronics in electric power engineering applications. In particular. According to [89] \in case of extreme nonlinearity. PSpice uses extremely small time steps and is also prone to problems of voltage convergence. Dr. [94]. N. the dierences between existing EMTP and PSpice models may give hints for the improvement of EMTP models. can be found in [91]. These guidelines can also be useful when using other digital simulation tools. are also useful for power electronics digital simulations. The values of R and C in these snubbers are not optimized. which are used to represent switching electronic devices and some control devices. rather these are based on speeding up the simulation without distorting the system voltage and current waveforms signi. [89] which are available for both R EMTP and PSpice simulations. where the power electronics load is connected. in many applications such as adjustable speed drives. FACTS and Custom Power Controllers. [92]. Variations of the system parameters. frequency. etc. active power conditioning. need to be used by the power electronics control. Depending on the type of study. These currents are assumed to be independent of voltage variations at the point of common coupling (PCC).. power electronics modelling depends on the objectives of the study. In steady-state or harmonic analysis.cantly. etc. The power electronic subsystem is then modelled as \known" harmonic current sources. such as voltage and current amplitude.." As stated in [91]. the main concern is the injection and propagation of harmonic currents into the transmission and distribution system. to properly adjust the . phase-angle displacement among the phases in a three-phase system. However. dierent software tools and solution techniques can be applied. instantaneous or average power. the operation of a power electronics device closely depends on and can aect the dynamics and the electric transient behavior of the connected system. which may cause unacceptable voltage distortions and dangerous resonances. Therefore. for transient analysis. which in turn might have a feedback eect on the system. a more complex and detailed representation of the power electronics devices as well as of the supplying power .ring time of the semiconductors. 3. Modelling Power Electronics in Electric Power Engineering Applications 68 system is required.1. Some model simpli. and that it . provided that the equivalent model is validated against practical measurements. This can be acceptable.cation and system reduction [92] might be necessary for practical reasons. The representation of semiconductor switching devices is commonly simpli.ts the study investigation needs. ed in power level application studies. Therefore. the nonlinear characteristic of a diode is usually represented in a simpli. series resistances must be included to allow multiple switch connections on the same circuit node. In some EMTP-based programs. Series on-state and parallel ostate resistances can be added to represent the semiconductor losses. such as MicroTran. Parallel resistances can provide a resistive connection between the DC sides of recti.ed form as a two-terminal uncontrollable unidirectional current owing switch. or in some programs as a voltage-controlled switch. ers and inverters and the AC local ground. thus avoiding oating sub-network problems [95]. The use of simpli. ed switch models for power electronics devices may be justi. especially related to semiconductor commutation phenomena. Also.ed to speed up the simulation time for system level studies. the EMTP solution at . but it may also give wrong and misleading results. 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. it is much more important to use such techniques than to reduce the time step size. [99]. causing unrealistic high frequency transients in the simulation of power electronic devices. [85]) is eective in the elimination of numerical oscillations. The use of techniques such as CDA (\Critical Damping Adjustment" [84]. . Therefore. or even the Clock Synchronized Structure Changing Concept (CSSC) [97] can be used to minimize the problem. as in the software PSCAD/EMTDC [98]. MicroTran has CDA implemented. for EMTP-based simulations of power electronics.xed discrete time intervals
t may result in inaccurate turn-on or turno switching times. Backtracking techniques [67] and/or resynchronization techniques ([96] pages 185. 207). but other EMTP versions may not. or may use dierent approaches. 204. Numerical oscillations caused by the trapezoidal rule of integration in solving the system of equations may also be a problem for EMTP-based simulations. controllable. Modelling Power Electronics in Electric Power Engineering Applications 69 If the gating circuit is not considered in the study.1. unidirectional current owing semiconductor devices can be represented by simpli.3. three-terminal. Dierent .ed switches with gate turn-on and turn-o controls. only allows pre-de. The implementation of a \basic power electronic unit" in digital programs requires special care with respect to \instantaneous commutation phenomena" [95]. The UBC version MicroTran of the EMTP. [96]. [88]. etc. [97]. a reversal diode (free wheeling diode) is used in parallel with a controllable switching device to form the basic power electronic unit" [91]. \in many actual power electronics applications. in order to provide a continuous current ow path for an inductive load. IGBT's.ring controls can be applied to represent thyristors. GTO's. However. up to now. This is done through the de.ned timing for the closing and opening of switches representing semiconductor devices. nition of a modi. ed . 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 (\") de. and because it ignores the dynamics of the control circuits. MicroTran users could write their own .ned from a zero crossing detection related to the voltages at the semiconductor terminals. and users must be aware of that.ring control subroutine based on an available ALPHA subroutine. sensing voltages and currents from the main program would introduce a one time step delay with this approach. but only a few users have used this option. the auxiliary program \PWM" can be useful to calculate and de. Also. For PWM control techniques. ne the closing and opening times. the subroutine GATE allows a simpli. As part of this thesis project. a subroutine \GATE" was developed to simulate power electronics dynamic control schemes with more accuracy and exibility. which limits the application of \PWM" to steady-state behavior. As its name indicates. The dynamics of the control circuit would be ignored. which would then be read in as a switching table. ed gate . and. for some devices. The control signal is assumed to be a gate voltage signal. it can control its turn-on time.e. de. also its turn-o time.ring control of a semiconductor. i. This subroutine was derived from the subroutine ALPHA (which can be user . between the gate node and ground.ned for simplicity. 3. Modelling Power Electronics in Electric Power Engineering Applications 70 de.1. may conduct irrespective if it is forward or reverse biased) and bidirectional current owing switch.3 presents some test cases for the transient simulation of:  voltage-controlled. dynamically controlled semiconductor devices as voltage-controlled switches.  Thyristor (simpli. With the subroutine GATE. Fig.ned). most of the three-terminal controllable power semiconductor devices can be represented. 3.. to implement new four-terminal. 3.2.e.2: Voltage-controlled switch in EMTP-based programs. \bipolar in voltage" (i. i vgate VoltageControlled Bidirectional Switch Figure 3. as illustrated in Fig. may only conduct when forward biased).  GTO (simpli.ed model. and unidirectional current owing switch). as a voltage-controlled.e. \unipolar in voltage" (i.. 6 illustrate the controlling properties of the bidirectional switch. 3. unidirectional current owing switch. with turn-o capabilities). the change of switch position (\status" on or o) only happens one time step after the enabling gate signal [100]. an alternative implementation for a \simultaneous" solution for voltage-controlled switches is presented in the next section of this chapter. Since the solution for switches follows the algorithm already implemented in most EMTP-based programs [1].4. 3. as a voltage-controlled. including MicroTran. .5 and 3. If it becomes necessary to avoid this delay problem for certain types of simulations. Figs. \unipolar in voltage".ed model. thyristor and GTO. respectively. thyristor and GTO. bipolar in voltage and bidirectional current owing switch. Modelling Power Electronics in Electric Power Engineering Applications 71 1 Ω i VoltageControlled Bidirectional Switch vsource vgate 1 Ω i THYRISTOR vsource vgate 1 Ω i GTO vsource vgate vsource = 10 sin (ω t ) [ V ] vgate = 1 [ V ] f = 60[ Hz ] Figure 3.3.1.3: Test cases for transient simulation of voltage-controlled. . Source Voltage Thyristor Current Current ( A ) 10 5 Gate Pulse Voltage ( V ) 0 −5 −10 0 2 4 6 8 10 12 14 Time ( ms ) Figure 3.4: Simulation of a voltage-controlled bidirectional current owing switch.3.1.5: Simulation of a simpli. Modelling Power Electronics in Electric Power Engineering Applications Switch Current 10 Current ( A ) Source Voltage 5 Gate Voltage Voltage ( V ) 0 −5 −10 0 2 4 6 8 10 12 14 16 Time ( ms ) Figure 3. 16 72 .ed model for thyristors. Modelling Power Electronics in Electric Power Engineering Applications Source Voltage GTO Current Current ( A ) 10 5 Gate Pulse Voltage ( V ) 0 −5 −10 0 2 4 6 8 10 12 14 Time ( ms ) Figure 3.3.1.6: Simulation of a simpli. ed model for GTO's. 16 73 . then vk = Ronik . can be done with the compensation method presented in Section 2. vOP ENj + rj 1 i1 + ::: ::: + rjj ij + rjk ik + ::: + rjM iM + vj = 0 (3.2) where: vOP ENk = voltage vk for [i] = 0 (open circuit). From equation 3. in such a way that a simultaneous solution is found at each time step for the controlling gate voltage and the voltage-controlled switch.3. for the voltage-controlled switch connected at branch k. (3.4) . Simultaneous Solution for Voltage-Controlled Switches in EMTP-based Programs 74 3. for the controlling branch with the gate voltage. It is possible to assume as branch equations: vj = Rin ij (3.1) and for the voltage-controlled switch branch vOP ENk + rk1 i1 + ::: ::: + rkj ij + rkk ik + ::: + rkM iM + vk = 0 (3. and Roff = O-state resistance.2 Simultaneous Solution for Voltage-Controlled Switches in EMTP-based Programs The implementation of voltage-controlled bidirectional current owing switches in EMTPbased programs. The necessary equations are. rkj = Thevenin resistance (coupling or mutual resistance between branches k and j ).3) and vk = [vj (Ron Roff ) + Roff )] ik where: Ron = On-state resistance. rkk = Thevenin resistance (self resistance of branch k).2.4 it is easy to verify that:  if vj = 1.  if Ron = 0. then vk = Roff ik . as illustrated in Fig. the switch turns on at the same time the enabling signal is received.8. 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.5) vOP ENk + rk1i1 + ::: + rkj ij + rkk ik + ::: + rkM iM + + [vj (Ron Roff ) + Roff )] ik = 0 (3.  else if Roff ! 1. [85]. 3. When the gating signal becomes zero (0V). In the case of a conventional switch. for example. would have to be triggered to avoid numerical oscillation problems in the simulation of power electronic devices.7.9. the switch turns o at the corresponding same time.1 (i. Every time a switch changes its status. special computer techniques. In order to sense the gate control voltage. the Critical Damping Adjustment (CDA) [84]. Observe. 3. Assume. such as.. Applying an enabling control voltage signal (1V) at the node gate. which results in ij = 0 (3.3.e. which is always correct and less prone to numerical problems. as it is implemented in most EMTP-based programs. vj = vOP ENj rj 1 i1 ::: rjj ij rjk ik ::: rjM iM ) in the solution algorithm. the turn-on and turn-o would occur 1 time step later. that the voltage signal vj is calculated using equation 3. . for example.6) which then can be solved with the implemented Newton-Raphson algorithm illustrated in Fig.2. then vk = 0. then ik = 0. Simultaneous Solution for Voltage-Controlled Switches in EMTP-based Programs  75 else if vj = 0. 2. as illustrated in Fig. Moreover. the simple circuit with a simultaneous solution for a voltagecontrolled switch presented in Fig. 3. however. assume Rin ! 1.19. 76 . Simultaneous Solution for Voltage-Controlled Switches in EMTP-based Programs 1 Ω i "Simultaneous" VoltageControlled Bidirectional Switch vsource vgate vsource = 10 sin(ω t ) [ V ] vgate = 1 [ V ] f = 60[ Hz ] Figure 3.2.7: Circuit with \simultaneous solution" of a voltage-controlled switch.8: Simulation with simultaneous solution of a voltage-controlled switch. Source Voltage "Simultaneous" Switch Current Current ( A ) 10 5 Gate Pulse Voltage ( V ) 0 −5 −10 0 2 4 6 8 10 12 14 16 Time ( ms ) Figure 3.3. 8 77 .2. Simultaneous Solution for Voltage-Controlled Switches in EMTP-based Programs 12 "Simultaneous" Switch Current 10 EMTP Switch Current Voltage ( V ) Current ( A ) Source Voltage 8 6 4 Gate Pulse 2 0 −2 0 1 2 3 4 5 6 7 Time ( ms ) Figure 3.9: One time step delay in EMTP-based switches.3. 3 Implementation of Nonlinear Diode Model in EMTPbased Programs The use of simpli. Implementation of Nonlinear Diode Model in EMTP-based Programs 78 3.3.3. [101]. However.10: Diode symbol. A detailed nonlinear model for a diode is also needed in the synthesis of equivalent networks to represent. . from anode to cathode. voltage-controlled switches and piecewise linear representation. with its symbol shown in Fig.. is described by h qv t i(t) = Is e( kT ) ( ) i 1 (3. with respect to current and voltage as shown in 3. v (t) is the voltage across the diode. as for example. the most common nonlinear element in power electronics.10. i. The terminal behavior of a diode [77].7) where: i(t) is the current through the diode.10. ANODE i (t) v (t) CATHODE Figure 3. a bipolar transistor with the Ebers-Moll model [77].ed models for diodes. for example. according to [89]. the potential dierence between the anode and cathode terminals. usually gives simulation results with acceptable accuracy for most of the power system studies.e. 3. The semiconductor diode. detailed nonlinear modelling of semiconductors is needed to design snubbers and gate drive circuits. is therefore. 3. The value Is is usually referred to as the saturation current. i(t)  Is . Table 3. I0 ) is called the operating point of the diode. If the diode is forward biased with v (t) > 4VT (over 100mV ). Implementation of Nonlinear Diode Model in EMTP-based Programs 79 Is is a constant which depends on the physical properties of the diode.3. and T is the temperature in degrees Kelvin (273:16Æ K = 0Æ C ). and is usually in the range of 10 6 [A] to 10 9[A] 1 .1 expresses the relationship between v (t)=VT and i(t)=Is . a constant current I0 ows through it. q = 1:6022  10 19 C is the charge of an electron. For v (t) < 3VT ( 75mV ).3. equation 3.7 may be approximated by i(t) = Is e(qv(t)=kT ) . the constant VT  kT=q  25mV . the diode is in the conducting region. k = 1:3806  10 23 J=Æ K is the Boltzmann's constant.8) When a constant voltage V0 is applied to the diode. The pair of values (V0 . For each operating point along the characteristic curve of the diode. When the polarity of v (t) is as shown in Fig. At 17Æ C  290Æ K . which are derived from i(t) =    v(t) Is e VT  1 (3. one could de.10. with a NewtonRhapson solution algorithm. 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.ne a dynamic resistance of the diode. which is solved together with the linear network equa1 Actually. including a sub-circuit for a diode model with reverse recovery.2.8 and inclusion of a series resistance Rout results in branch equation 3. The reader is referred to [77]. is used for the solution of the nonlinear model of a diode. Reference [89]. the compensation method presented in Section 2. additional physical eects come into play and the diode may no longer be treated as a simple nonlinear resistor. For higher frequencies. a function of the voltage across the diode. (pp. . In this thesis project. Inversion of the diode characteristic curve from equation 3. in general. discuss the PN junction diode switching characteristics. which relates increments of the voltage to the increments of the current (dv (t)=di(t) for v (t) = V0 and i(t) = I0 ). [101] and other references for further details. 71-1 to 71-7). Is is temperature dependent and may assume default values of 10 14[A] at 27ÆC [101].10. 9 to determine the operating point of the diode for a particular network condition.. 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 .999 . . Implementation of Nonlinear Diode Model in EMTP-based Programs 80 Table 3.633 402.10) where: vOP ENk = voltage vk for [i] = 0 (open circuit). as illustrated in Fig. v(t) VT i(t) Is . voltage across the diode (the potential dierence vanode vcathode ) plus the voltage drop across the series resistance Rout ..086 6.982 -0.e.11: vOP ENk + rk1 i1 + : : : + rkk ik + : : : + rkM iM + vk = 0 vk = VT ln h ik Is +1 i + Rout ik (3.. .9) (3.1: Comparison between voltage and current in a diode as a function of its parametric values. . tion 3. .865 -0. 1095. i.3.. i.632 -0.. the current owing from the anode to the cathode terminal.998 -0. . 3.429 147..950 -0.413 53. Rout = series resistance for the diode model (which may be assumed to be equal zero).898 19.718 0 -0. vk = voltage of branch k. . rkk = Thevenin resistance (self resistance of branch k). ik = current through the diode.3.e.983 -0.389 1. with a change in vOP EN (t).5 1 1. for simplicity.5 2 2. Note that the Thevenin equivalent circuit equation changes from step to step.12.11 is illustrated in Fig.9 results in: vOP ENk + rk1 i1 + : : : h i : : : + [rkk + Rout ] ik + VT ln Iiks + 1 + : : : + rkM iM = 0 (3. Inserting equation 3. 3. The iterative solution with the Newton-Raphson algorithm requires an initial guess.5 Current ( A ) Figure 3. where it is assumed. Implementation of Nonlinear Diode Model in EMTP-based Programs 81 3 Linear Network Thevenin Equivalent Circuit 2 vOPEN (t) Nonlinear Diode Characteristic 1 V Voltage ( V ) 0 0 iSC (t) I 0 i −1 SC (t) = vOPEN (t) / (rkk + Rout ) −2 Resulting nonlinear equation to be solved with the Newton−Raphson method −3 −0.10 into equation 3.11: V-I diode characteristic and network Thevenin equivalent circuit equation. that the Thevenin equivalent resistance does not change along the simulation time.13 illustrates this. Equation 3. Fig.5 0 0. 3.11) A general circuit representation of equation 3.3. and in general with a change in the Thevenin equivalent resistance (slope dv=di).3. Depending on how close the initial guess is to the .11 is solved iteratively at each time step. nal solution. convergence can be very fast or . Implementation of Nonlinear Diode Model in EMTP-based Programs 82 ik [ rTHEV k ] vOPEN k Rout vk vT ln ( ik / Is + 1 ) Figure 3.3.13: V-I diode characteristic and dierent network Thevenin equivalents.3.12: Circuit implementation for the simultaneous solution of a nonlinear diode model. 3 . 3 Network Thevenin equivalent at time "t" v 2 OPEN (t) V−I Diode nonlinear characteristic Voltage ( V ) 1 −I s 0 i −1 SC (t) −2 −3 −3 −2 −1 0 1 2 Current ( A ) Figure 3. The detailed development of such a technique for implementation in EMTP-based programs is presented in the following.  If vOP ENk  0 and vOP ENk  VT ln h ikSC Is i + 1 . Therefore.3.12 is the short circuit current ikSC . then the voltage across the diode is .3. with the particular linear network Thevenin equivalent circuit: ik = ik = ( vOPENk rkk +Rout )  v OPENk  Is e VT (3.12. as calculated with equation 3. A simple inspection of Fig. and speed up the convergence of the solution..e. which becomes the initial guess for the diode current ik in the conduction mode.14) Equation 3. 3. it is important to derive a heuristic computer technique for initial guesses. associated with the network conditions.11 reveals some \physical" candidates for the initial guess of the nonlinear diode current. the initial current is assumed to be equal to ikSC . or numerical problems such as computer over ow may even arise. Then. A robust rule should be general and insensitive to the network or diode parameters.  if vOP ENk  0 and vOP ENk > VT ln h ikSC Is i + 1 .13) (3. to accommodate the highly nonlinear exponential characteristic of a diode. i. Implementation of Nonlinear Diode Model in EMTP-based Programs 83 on the contrary.12)  1 ik = Is (3. convergence can be very slow for some cases.  On the other hand. and a better estimate for the current is calculated with equation 3.13 (Another simple alternative would be just to assume ik = 0). if vOP ENk < 0 and vOP ENk  6:0VT .rst estimated to be equal to vOP ENk . then again the voltage across the diode is . one could use a linear function instead of using equation 3. Alternatively.14.13.14. .rst estimated to be equal to vOP ENk . and the initial current is then calculated with equation 3.  If vOP ENk < 0 and vOP ENk < 6:0VT . then the diode current is assumed to be as calculated with equation 3. piecewise linear approximations could be used . For example.3.3. Implementation of Nonlinear Diode Model in EMTP-based Programs 84 There are other possibilities for the initialization of the variables when a Newton-Raphson method is being used [77]. This is particularly true if there are many diodes to be solved. with up to 6 diodes. the method has so far converged in all tested cases. which will then decide. 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. Even though the equations are linear in that case it is not known a priori on which segment the solution ends up. Theoretically. the solution points for the N diodes. the linear characteristic would be replaced by the detailed nonlinear equation. etc. such as diodes. transistors. \In circuits with gate turn-o thyristors (GTO's). iteratively. commutation of the current into a diode must often be instantaneous. An example for such situation is the buck-boost (step-down/up dc-dc) converter shown in `Fig. but by limiting the new voltages and currents to technically reasonable values. 3. without any current interruption. The production code of MicroTran still uses the Newton-Raphson method for the two-slope piecewise linear representation. there is no guarantee that the iterations will converge in such cases. This approach might be useful for speeding up the solution of networks with multiple nonlinear elements.14 [93]' and the half-wave recti. A two-slope piecewise linear approximation of the diode characteristic is accurate enough in many cases.rst to determine the initial guess (either for the current or for the voltage across the diode) and then. 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. because the current in the GTO would already have dropped to zero.er circuit with a freewheeling diode in `Fig. [78]. EMTP-type simulation of \instantaneous commutation" with simple switch models might give wrong results in the case of switching converters. 3. Therefore. 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.. User knowledge of the commutation process can be used to pre-de. . or controlled in this way through TACS)." [95].. This is one time step too late.15'. but it may not work for all cases. which assures a simultaneous solution of all equations [95]. . A better option is the compensation method used in this work.ne the switches which will have simultaneous commutation [78]. 3.14: Dc-dc converter. Implementation of Nonlinear Diode Model in EMTP-based Programs GTO 85 DIODE i Figure 3.50 [ V ] Figure 3.3. DIODE 1 v = 155 sin(ω t ) [ V ] i DIODE 2 .15: Half-wave recti. 3. 3. 3. Fig.16. Detailed modelling of other semiconductor devices. whereas Fig.er with freewheeling diode. where the basic parameters for the diode are: Is = 10 12 [A] and VT = 0:026[V ]. and also forcing at least 1 iteration in the algorithm.19 presents the V-I nonlinear characteristic of the diode resulting from the EMTP simulation.17. assuming a convergence tolerance error of less than 10 10 for the resulting nonlinear equation. assume. was 1 for this case. such as bipolar transistors. for example. and using the proposed heuristic technique for initial guess. The average number of iterations with the Newton-Raphson algorithm. 3. the simple electric circuit with a nonlinear diode as in Fig. To illustrate the solution method presented here. The time response solution for this circuit is presented in Fig.18 shows the detail of the diode current at the zero crossing 2 . . eld ef2 The 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]. . or more complex models (Gummel-Poon) could be used as well. etc. based on functional terminal conditions. can also be derived with the interconnection of circuit elements. [77]. The Ebers-Moll model of a bipolar transistor. Implementation of Nonlinear Diode Model in EMTP-based Programs 86 1Ω iDIODE Vmax = 2 [V] f = 60 [Hz] vDIODE Figure 3.3. dependent sources. can be accomplished with the \synthesis of equivalent networks" (subcircuits) using nonlinear diodes. especially with the use of the dependent sources presented in Section 2. but one should be aware of the fact that even though the .2. 3 Current ( A ) 2 Diode Current Diode Voltage 1 Volltage ( V ) 0 −1 −2 −3 0 2 4 6 8 10 12 14 16 Time ( ms ) Figure 3. voltage-dependent capacitances and resistances [101]. The use of macromodels. Piecewise linear approximations can also be used to model device characteristics.16: Electric circuit with a nonlinear diode model. but their description is beyond the scope of this thesis project.17: Transient simulation of a nonlinear diode model in an EMTP-based program.3. fect transistors. 5 −2 −0.8 1 1. 0 87 Figure 3.2 9.5 1 0.5 −1 −1.6 Current ( A ) 0.5 0 −0. Implementation of Nonlinear Diode Model in EMTP-based Programs 2 1.2 0.5 1 0.18: Detail of the transient simulation of a nonlinear diode model in an EMTP-based program.2 0.0E−12 [A] 8 Diode Current 8.3.4 0.5 Time ( ms ) 9 3.5 1.5 −1 −1.5 0 −0.19: V-I nonlinear characteristic of the diode resulting from the EMTP simulation. Current ( A ) Voltage ( V ) .−12 x 10 Is=1.5 −2 7.4 Figure 3. Spline .4. Control Modelling Aspects of Power Electronic Devices 88 approximations are continuous from segment to segment.3. the derivatives are discontinuous. 3.tting techniques can then be used to approximate the device characteristic by separate low-order polynomials between adjacent segments. with the cubic spline being the most popular one [77].4 Control Modelling Aspects of Power Electronic Devices The digital simulation of control devices for power electronics applications. such as recti. inverters. The derivation of reliable system references requires a growing number of digital and analog components. in general. etc. Their modelling can. AC-AC converters. motor drives. DC-DC converters. is made challenging because of the more frequent use of mixed analog and (real-time) digital signal processing (DSP) control techniques. not be simpli.ers. For example.ed without compromising the simulation accuracy and stability. signal sensoring and zero crossing prediction or detection usually requires an appropriate use of signal . either through analog design (operational ampli.ltering. require particular attention in digital implementations. which are commonly used in voltage-sourced and current-sourced converters.. due to the discrete nature of computer simulation programs. which are of fundamental importance for the control of high power electronics applications. such as FACTS and Custom Power Controllers [90].ers. thus in uencing the simulation results. abc to . 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). Pulse width modulation (PWM) techniques.g. System frequency tracking requires the use of appropriate phase-locked loop (PLL) controls. However. resistances and capacitances) or its respective digital implementation. where the time step size may aect the results of discrete binary comparators. Special transformation of variables into other reference systems (e. which is used in active . 0 transformation. lter control). limits (windup and non-windup) . Digital simulation of hysteresis eects. requires a simultaneous solution of control and system equations. typically related to voltage sag problems. With graphical user-interfaces for MicroTran. this method has been shown to be very robust for the cases simulated up to now. As part of this thesis project. it would become easy to de. 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. or improper designs may become the cause of many power quality problems. compared to TACS (Transient Analysis of Control System) used by many other EMTP versions.e. with variations in the supply voltage caused by power system disturbances. ideally. in most practical power quality cases. The main advantage of this development. the control. Moreover. power electronicsbased loads (i. However. Therefore.4. \basic control devices" were experimentally implemented in MicroTran.. Control Modelling Aspects of Power Electronic Devices 89 and various non-linearities in the control system also requires appropriate models for a successful computer simulation.3. is that a \true simultaneous solution" is found through the compensation method using a Newton-Raphson iteration scheme. provided that appropriate computer techniques are used to allow convergence in the solution. should be able to actively withstand and support its primary regulation functions without disrupting the electric supply to the controlled load. their control) are either extremely sensitive to momentary voltage variations causing frequent shut-downs in industrial process operation. . [103]). thesis. [102]. since a unique simultaneous solution approach would be used.D. based on the method and algorithm presented in this Ph. similarly as in [65]. in such a way that no dierentiation between power and control circuit will then be necessary.ne libraries of control devices as well as of power components (and sub-circuits. D. The objective of this Ph. thesis research project was to develop reasonably accurate models for EMTP-based programs.Chapter 4 Evaluation of the Impact of Power Electronic Devices on the Quality of Power T HE 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. for a . with which one could evaluate the impact of high power electronic devices on the quality of power. ELEKTRO . A. a federal agency of the Brazilian Government. To make the thesis results more valuable to utilities and industries. 1 .Eletricidade e Servicos S. Financial support for this cooperation was provided by CAPES 2 . which kindly agreed to provide data and real power quality problem cases to validate simulation results. the following practical tasks were carried out:  Cooperation with a utility company in Brazil. 2 CAPES . . Anexo I. F. Rua Ary Antenor de Souza.  Appropriate models for time-domain simulations using EMTP-based programs were applied to simulate some of the real power quality problems experienced by ELEKTRO. P. Caixa Postal 00365.D. Esplanada dos Ministerios. CEP 70047-900. 90 .Jardim Nova America.eld activity at ELEKTRO with the duration of three months. Braslia . 1 ELEKTRO . 321 . Sala 215.Fundac~ao Coordenac~ao de Aperfeicoamento de Pessoal de Nvel Superior. BRAZIL. BRAZIL. A. Campinas-S. . CEP 13053-024. .Eletricidade e Servicos S. the type of \front-end" electronic converter.4. This chapter presents some simulation cases of power quality assessment with the use of the existing features of MicroTran. (such as electronic converter controlled electric motor drives).1. through the use of time and frequency-domain techniques with EMTP-based programs. which converts line-frequency AC into DC (diode-bridge recti. the UBC version of the EMTP. Important simulation guidelines for the evaluation of the impact of power electronic devices on the quality of power are also summarized in this Chapter. as for example: 1. a synthesis of simulation guidelines for power quality evaluation was developed. The simultaneous solution of control and electric power system equations (SSCPS).1 Dynamic Interaction between Power Electronic Devices and Power Systems The interaction between the utility supply system and power electronics-based loads. with emphasis to determine the dynamic interaction between power electronic devices and the power systems. Dynamic Interaction between Power Electronic Devices and Power Systems  91 Based on the power quality monitoring and simulations. 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. 4. are illustrated through practical control and power electronics controllers simulation cases. depends on a variety of factors [104]. and the converter con. the number of phases (single-phase. three-phase) from the supply system used by the converter. power ow can be reversed). thyristors converters (which can be made bidirectional in power ow).ers (which are unidirectional in power ow). switch mode converters (in which. 2. which also aects the waveform current distortion..guration (e. determined by its \short-circuit power". 3. 48.g. etc. 24. the \strength" (or \stiness") of the utility system. pulses converters). 6. 12. . Dynamic Interaction between Power Electronic Devices and Power Systems 92 4.. particularly in cases where low-cost choices are made. the design and control of the electronic converter. the point of their connection in the network.e. the number of power electronics based loads. its control) can easily be disrupted by power system disturbances travelling through the utility network. The most common lower-power electronics based load uses a single-phase diode-bridge recti.1. which will in uence their impact on the electric supply system and vice-versa. as is very common in the case of sensitive loads. 5. On the other hand. and the electric power of electronic converters.4. can cause a deterioration on the quality of power of the utility system. the converter operation (i. The choice of parameters and components. and may cause harmonic associated problems.  voltage distortion along the distribution circuits. This type of recti. such as:  overheating in neutral conductors due to the ow of high third harmonic currents. [104] which draws highly distorted waveshape current from the utility system.er [94]. etc.  risk of resonances with utility or industry power factor correction capacitor banks.  increase in power losses. er only draws current close to the maximum peak of the \assumed sinusoidal" utility voltage source. in order to recharge the capacitor . lter on the DC side of the recti. 4.er.1 presents a typical single-phase diode bridge recti. To illustrate that. Fig. [104] 3 . [95]. where Ls1 represents the system Thevenin equivalent inductance (resistance ignored here) and Ls2 represents any series inductance added in the AC side of the recti.er [93]. [94]. er. such that the DC capacitor . Assume that the diodes are ideal and the circuit has been energized a long time ago. e.e. when diodes 1 and 2 are forward biased at 3 Reference [95] presents useful guidelines on power electronics applications using the EMTP. when vSA is greater than the voltage vdc (i.lter has already been charged (i.2 .. Note. . from Fig. the transient energization has gone and \stead state waveforms" are present). that the current is (and consequently id ) only starts to ow during the positive semicycle of the source voltage.. 4. Conversely.. when vSA is less than vdc .1.4. corresponding then to the electronic current recti.e. the current is starts to ow (i. Dynamic Interaction between Power Electronic Devices and Power Systems 93 time t1 ). with id = is owing in the same direction. during the negative subcycle. when diodes 3 and 4 are forward biased at time t4 ). 3 presents the harmonic amplitude spectrum of the current drawn from the source by a single-phase diode-bridge recti.4.cation process). Fig. 5 mH + D3 vA 0. id vSA Ls1 vPCC D1 Ls2 is 0.7 [V] f = 60 [Hz] D4 1000 µ F 20 Ω D2 - Figure 4.5 mH vdc Vmax = 169.1: Circuit with a single-phase diode-bridge recti.er. To better understand the waveform voltage distortions caused by a single-phase diodebridge recti.er. 1. vSA vL vA = 0 (4.2) . i1 = i2 = is . 4.er.1) vL = vL1 + vL2 = Ls1 didt + Ls2 didt = (Ls1 + Ls2 ) didts = L didts 1 where: vL = voltage across the total inductance L = Ls1 + Ls2 . one can apply the Kirchho's second law to the circuit of Fig. and derive the following equations. 2 (4. reaching a maximum peak value when the voltage across the inductance is zero. many DC-DC electronic converters rely on the inductor physical properties during switching transients. vA . From Fig.200 150 100 50 0 −50 −100 −150 −200 t1 5 t4 10 SA v 15 is 20 25 vdc Time ( ms ) vA = vSA Ls1 L 30 dis dt 35 40 vA 45 50 94 (4. such that is starts to grow exponentially. and in the point of common coupling (PCC).3) (4. to either step voltages up (boost) or step voltages down (buck).4 it can be seen that the inductances in uence the rate of change (and consequently the waveshape) of the current. 4. after which vL changes its polarity (because the derivative of the current changes its signal 4 ) physically trying to keep the current owing until the current .4) input. vP CC . where many other loads may 4 As a matter of fact. 4.4 and Fig.4 one can easily understand the voltage waveforms distortions in the recti. Following the equations 4.nally becomes zero. 4.3. er vP CC = vSA dis dt Therefore. the following voltages can be de. 2: Current drawn from the source by a single-phase diode-bridge recti.ned: Figure 4. 0 4.er.1. Dynamic Interaction between Power Electronic Devices and Power Systems Current ( A ) Voltage ( V ) . 1.3: Harmonic amplitude spectrum of the current drawn from the source by a single-phase diodebridge recti. Dynamic Interaction between Power Electronic Devices and Power Systems 95 30 25 RMS Current ( A ) 20 15 Third Harmonic Component 10 5 0 0 5 10 15 20 25 30 35 40 45 50 Harmonic Order Figure 4.4. Fig. composed by single-phase diode bridge recti.er. be connected and may be aected by the waveform voltage distortion.4. In a four-wire. three-phase system with \well balanced" loads.5 presents the harmonic amplitude spectrum of the voltage at the PCC. for a 60Hz fundamental) as shown in Fig. Observe that the third harmonic component of the neutral current in Fig. 4.4.8.8 is approximately 3 times the third harmonic component of the current of a single-phase diode-bridge recti.7 and Fig. 180Hz. 4. the current owing through the neutral conductor has mainly third harmonic components (i. as illustrated in Fig. 4.e.6.ers. 4. thereby creating a potential hazardous risk.3.er as illustrated in Fig. These high currents may cause overheating in the neutral conductor (usually designed with a smaller cross section than the phase conductors). such as . The widespread use of this type of inexpensive recti.re. computers and compact uorescent lamps (CFL) . usually as the front end of low power appliances.er. such as television sets. B. The waveshape distortion is probably caused by the large number of computers in the building. For example. and voltage waveform distortion at the point of common coupling (PCC). Fig.12 show the results of a Fourier analysis of the outlet voltage curve (the DC component present in the harmonic amplitude spectrum at Fig. 4. according to the present standards [48]. It also aects. Vancouver. Fig. 4. 4.10 present the voltage waveshape measured with a digital oscilloscope at the outlet of the Power Electronics Laboratory of the Department of Electrical and Computer Engineering at UBC. [49]. as well as in the entire university.11 and Fig. Although. also propagates through the BC Hydro electric supply system. 16 96 (which are used more and more because of its higher energy eÆciency). 0 4.C. Canada 5 .. Dynamic Interaction between Power Electronic Devices and Power Systems Current ( A ) Voltage ( V ) . 4. Kenneth Wicks in doing this measurement is gratefully acknowledged. such Figure 4. presently with minor consequences.200 150 100 50 0 −50 −100 −150 −200 vSA 2 vLs1 4 s i L v 6 vA vPCC 8 Time ( ms ) 10 12 14 5 The help of Mr. 4. all the other loads in the building supplied from the same common bus.4: Current through and voltage across the total inductance.11 might be caused by inaccuracies or dc osets in the measuring equipment).9 and Fig. commercial and industrial customers [105] . can have a cumulative negative eect on the quality of power supplied to residential. and eventually.1. 2778 % HRMS = 7.1000 Volts Max = 166.6: Four-wire.1.5136 Volts TIF / IT = 121.9200 Volts Creast Factor = 1.3851 Min = −166. Dynamic Interaction between Power Electronic Devices and Power Systems 97 100 90 Percentage of Fundamental ( % ) 80 70 60 Fund = 119. isa Vmax = 169.5: Harmonic amplitude spectrum of the voltage waveform distortion at the point of common coupling (PCC).7 [V] 10 -6 Ω f = 60 [Hz] isc diode bridge diode bridge iNEUTRAL diode bridge isb vSB vSA vSC Figure 4.8154 50 40 30 20 10 0 0 5 10 15 20 25 30 35 40 45 50 Harmonic Order Figure 4. three-phase system with \balanced" single-phase diode-bridge recti.4.1000 Volts THD = 6.6844 Volts RMS = 119. and better alternatives for power conversion are actually available. harmonic voltage distortions are usually within acceptable limits. . other sensitive equipment may be aected.ers. 7: Current owing through the neutral conductor. new technologies with less impact and/or less susceptibility to power disturbances are gaining market acceptance. Examples are active . With the introduction of more strict standards for power quality.200 150 100 50 0 −50 −100 −150 −200 5 10 vSA 15 20 30 Neutral i 25 Time ( ms ) 35 40 Figure 4. 45 50 98 The choice of a power electronic converter is based on its intended application and on the price. and with growing concerns about the dynamic interaction between power electronic devices and the power system. By using.or three-phase controlled thyristor converters can also adversely aect the quality of power. and the poor power factor. the notching of the input voltage waveform caused by the commutation among the thyristors.lters and other dynamic compensating Single. for example. in connection with diode-bridge recti. due to their distorted current waveforms. step-up (boost) DC-DC converters (consisting of IGBT's switching at high frequency with PWM control techniques. free wheeling diodes and inductors). ers.1. Dynamic Interaction between Power Electronic Devices and Power Systems Current ( A ) Voltage ( V ) . power factor corrected (PFC) interfaces can be designed. Such PFC circuits are able to draw almost sinusoidal currents at close to unity power factor. 0 4. For a proper identi.8: Harmonic amplitude spectrum of the current owing through the neutral conductor.1. Dynamic Interaction between Power Electronic Devices and Power Systems 99 30 Third Harmonic Component 25 RMS Current ( A ) 20 15 10 5 0 0 5 10 15 20 25 30 35 40 45 50 Harmonic Order Figure 4.4. devices. or they can be adversely aected by electromagnetic transient phenomena coming from the power system. which include models not only for the power electronics but also the power system part. transient and steadystate analysis are needed. because of the dynamic interaction between them.cation and solution of power quality problems. Reference [24] presents the fundamental de. Power electronics based loads can either be the cause of problems in the power system.  Harmonics.nition of electromagnetic phenomena aecting the electric power quality. such as:  Voltage sags and interruptions.  Transient overvoltages. with realistic cases and practical monitoring results and examples of power disturbances. . .g. such as electronically controlled motor drives. which may propagate through the distribution system and cause ampli. etc. textile.1. B.  Long duration voltage variation.) rely on accurate speed and torque control through the use of power electronics.4. Vancouver. automotive.  Wiring and grounding practices. pulp and paper. Many industrial processes (e. This is actually at the heart of many power quality problems! It is also common that utility capacitor switching creates high frequency transients. Dynamic Interaction between Power Electronic Devices and Power Systems 100 Phase Voltage 200 150 100 Voltage ( V ) 50 0 −50 −100 −150 −200 0 2 4 6 8 10 12 14 16 time ( ms ) Figure 4. and thus become more or less vulnerable and susceptible to power quality problems depending on the sensitivity of these devices. Canada. Voltage sags are by far the most common cause of disruption of operation of power electronics based loads.9: Voltage waveshape measured at the outlet of the Power Electronics Laboratory of the Department of Electrical and Computer Engineering at UBC.C. ed 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 . \nuisance tripping" of sensitive loads. and the country. consequently. plants may excite natural resonance modes of weak distribution and other industrial systems.10: Measured voltage waveshape. because of many available computer models for power systems and power electronics. mainly to control voltage on transmission or distribution lines. The importance of power quality then depends on its economic impact on the industry. such as capacitors failures. Typically. Dynamic Interaction between Power Electronic Devices and Power Systems 101 Phase Voltage 200 Measured Phase Voltage 150 100 Voltage ( V ) 50 Fundamental Voltage 0 −50 Harmonic Distortion −100 −150 −200 0 2 4 6 8 10 12 14 16 time ( ms ) Figure 4. 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".4. the society. the 5th order harmonic current commonly \injected" into the power system by traditional power electronics converters.1. causing high frequency oscillations in the voltages and. the utility. have become a necessary engineering tool for the evaluation of the . EMTP-based programs. [107]. has the potential to cause problems. its fundamental component and its harmonic distortion. Resonances tend to occur more frequently as more power factor and voltage support capacitor banks are used in the system. [2]. impact of power electronic devices on the quality of power.1. 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.11: Harmonic amplitude spectrum of the outlet waveshape voltage.5 Fundamental = 100% Percent of Fundamental ( % ) 2 1. and elsewhere. Specialized conferences.5 1 0.4. 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]. Dynamic Interaction between Power Electronic Devices and Power Systems 102 Harmonic Amplitude Spectrum 2.5 0 0 5 10 15 20 25 30 35 40 45 50 Harmonic Order Figure 4. Dynamic Interaction between Power Electronic Devices and Power Systems 103 Harmonic Phase Angle 150 100 degrees ( º ) 50 0 −50 −100 −150 0 5 10 15 20 25 30 35 40 45 50 Harmonic Order Figure 4.1.12: Phase-angle of the harmonic components of the outlet waveshape voltage.4. . Section 4. the harmonic current sources are not exactly known. . This section presents an application of the EMTP in a distribution system study. 6 . 321 . Rua Ary Antenor de Souza.2. transient impulses caused by switching utility capacitor banks.2 presents a voltage sag case study.2. such as fault analysis. Section 4.Eletricidade e Servicos S. . P. diagnosing the eects of special loads into the system. harmonics from transformer saturation clearly depend on the voltage magnitude and waveform at the transformer terminal. A.2 Power Quality Assessment through EMTP-based Programs This section presents some real cases of power quality assessment through EMTP-based simulations. an electric utility in the southeast of Brazil. S.2. Field test measurements made by the author and comparisons with time and frequency domain computer analysis are also shown for some of the cases. A. because they depend on the behaviour of the power system as well. Campinas.2. . It is taken from a harmonic problem experienced by ELEKTRO Eletricidade e Servicos S. Only time domain simulations of the EMTP type can address the interaction between the system and the harmonic sources. Power Quality Assessment through EMTP-based Programs 104 4. CEP 13053-024. Section 4.2. troubleshooting the failure of sensitive loads.3 is concerned with visual light icker caused by voltage uctuations. For example. The three-phase power system with a digital model of the induction furnace as a power electronic load was 6 ELEKTRO . evaluating the application of \Custom Power" devices as solutions to power quality problems. which use linear solutions at the harmonic frequencies. etc. The characteristic harmonic spectra of non-linear loads are assumed to be known.4. 4. In reality. which can result in non-characteristic harmonics as well.1 Induction Furnace Harmonic Study The problem of harmonic analysis in power systems is usually studied with steady-state solution techniques.1 presents a harmonics case study. Time domain simulations can also be useful to develop other types of power quality studies. where the harmonics injected into a distribution feeder by induction furnaces were the prime concern for power quality. BRAZIL.Jardim Nova America. and are modeled as current sources at the respective harmonic frequency. and allow more detailed modelling of power electronic loads and devices. than steady-state harmonic programs. Electromagnetic transients programs are more accurate in representing nonlinear eects of the supply system. Induction heating has gained wide acceptance in industry because this type of heating process is considered clean. quick and eÆcient. 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.4. Field measurements at the point of common coupling between the utility and the industry are presented as well. causing voltage distortions in the electric supply network. On the other hand. The dynamic interaction of these harmonics with the electric system in terms of system con. the use of power electronic devices for induction heating introduces harmonic currents.2. Voltage and current waveforms were then analyzed with a Fourier analysis program to obtain the harmonic content of the distorted waveforms. Power Quality Assessment through EMTP-based Programs 105 simulated with the EMTP over a time span which was long enough to reach steady state. with consequent economical losses. or even in undesirable steady-state conditions. Series or parallel-resonant inverters are typical con. where the heat in the electrically conducting workpiece to be melted is produced by circulating currents through electromagnetic induction. failure and life reduction of equipment.guration. may result in linear resonances. loading and other conditions. which all can result in misoperation. Induction furnaces are power electronics-based loads. 4. and has created incompatibility problems between these special loads and other sensitive loads connected on the same distribution feeder. Changes in the con. Figs. The operation of these induction furnaces has produced distortions in the current and voltage waveforms.gurations used to supply energy to the induction coil. which can be in a range varying from the power system frequency to a few hundred kilohertz [94].13 (a) and (b) illustrate an induction furnace in operation. at a selected frequency. guration of the power supply system as well as application of passive . but have not completely eliminated the harmonics power quality problem.lters. have minimized the eects. Field measurements have been made for dierent operating conditions to evaluate the eectiveness of the already installed harmonic passive . .lters. this customer was connected to the same feeder where one of the industrial plants of the same customer had been operating induction furnaces.4.Eletricidade e Servicos S. Coincidentally. Power Quality Assessment through EMTP-based Programs (a) 106 (b) Figure 4. At that time. . in assisting an industrial customer to deal with the harmonic problem created by the operation of induction furnaces in a distribution system.2. 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 . The problem emerged after a customer complained that its sensitive loads (computer-based loads) were suering from the poor quality of the utility-supplied power. (a) History of the Induction Furnace Harmonic Problem This section summarizes the actions taken by ELEKTRO . A. (b) Induction furnace operation.13: (a) Metal melting by an induction furnace. the protection fuses of the customer's capacitor banks started to blow. which led the utility to change the capacitor digital overcurrent protection relays to electromechanical relays. Resonance eects caused overcurrents in this capacitor bank.ltering of the injected harmonic currents from the induction furnaces.0 MVAr capacitor bank in the same substation supplying this customer. the solution was postponed by the customer. Two years later. After that. the utility installed a 9. and because of the economic costs involved. Since there was no clear regulation about this issue at that time. measurements were made after the utility capac- . Then.8kV transformer was later installed in the substation. The utility's capacitor bank was turned o until another 25/30 MVA-138/13. 4. Power Quality Assessment through EMTP-based Programs 107 itor bank was connected to a dierent bus bar from the feeder that supplies the induction furnaces customer. especially on the feeder supplying the induction furnaces [108]. new measurements were made at the substation. In August 1995. Sometimes later. Fig. the customer installed 4th and 5th order harmonic tuned passive .4.2.14 shows the feeder current at the moment of maximum harmonic distortion. 7 TIF/IT 101821 120 100 80 Amps 60 40 20 0 0 5 10 15 20 25 30 Harmonic BMI/Electrotek Figure 4. (b) Harmonic Measurements Harmonic measurements were made at the point of common coupling (PCC) between the utility and the induction furnace customer.7 RMS 235.92 (inductive limit from 06:00am to 12:00 midnight and capacitive limit from 12:00 midnight to 06:00am). LIM_IV August 29. which essentially changed from a minimum monthly average power factor of 0. Measurements were made in 1998 to verify the system overall performance on this feeder and in the respective distribution substation. which acquires 256 samples per cycle (60Hz) in each of the voltage channels. with an economic motivation imposed by a new Brazilian power factor legislation.091 -492. and 128 samples per cycle (60Hz) in each of the current .14: Current measurements in a distribution feeder supplying induction furnaces at the time of maximum voltage distortion.6 Max 487.41 HRMS 135. 1995 at 20:40:49 GMT Phase B Current SS Wave 600 Fund 192.85 to a minimum hourly average of 0. with modern monitoring equipment of the type \Dranetz/BMI-PQNode 8010 and 8020".lters.2 THD 70.6 400 200 CF Amps 0 -200 Min -400 -600 0 20 40 60 80 100 120 140 Time (mSeconds) 2. cold load pick-up. outages. [43]. harmonic distortions and frequency deviations [15]. waveshape faults. [108]. Power Quality Assessment through EMTP-based Programs 108 channels. it is possible to monitor power quality phenomena such as voltage sags. With this equipment.2. The following . transient impulses.4. voltage swells. gures illustrate some of the harmonic measurements made in March/April 1998.15 (a) shows the phase \A" measured current and Fig. 4. with their respective harmonic amplitude spectrum.15 (b) shows the measured voltage between phases A and B. 4. Fig. Both measurements were taken at a particular time when the induction furnace operation was with the 4th and 5th order harmonic passive . 400 300 200 100 0 -100 -200 -300 -400 10 20 30 40 50 Time (mSeconds) 60 70 35 30 25 20 15 10 5 0 20000 10000 0 -10000 -20000 -30000 % Fund Amps 0 Fund 198.531 CF Min -308.489 CF Min -20237 Max 19805 THD 7.17). 1998 at 03:58:49 Local M_V Phase A-B Voltage SS Wave Volts Amps April 01.9 30 BMI/Electrotek (b) Figure 4. 1998 at 03:58:49 Local M_V Phase A Current SS Wave 0 5 10 15 20 Harmonic (a) 25 30 BMI/Electrotek 0 10 0 5 20 30 40 50 Time (mSeconds) 60 70 4 3. 4.5 2 1.172 HRMS 972.5 1 0.5 0 10 15 20 Harmonic 25 Fund 13557 13592 RMS 1.15: (a) Phase \A" current measured with harmonic passive .3 TIF/IT 234.4 Max 18.8 RMS 1.9 204.5 313.58 TIF/IT52660 April 01.5 3 2.89 THD HRMS 37.lters turned o (see Fig. (b) Phase-to-phase \A-B" voltage measured with harmonic passive .lters turned o. but during an operating condition when the 4th and 5th harmonic passive .16 (a) and (b) illustrate the same measurements. Figs. 4.lters turned o. lters were turned on (see Fig.18). 4. 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. 4. Fig.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) . 5 Max 294. Power Quality Assessment through EMTP-based Programs 300 200 100 0 -100 -200 -300 -400 7 6 5 4 3 2 1 0 0 (a) Volts 13:19:03 14:29:04 15:39:04 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 THD (%) 8 7 6 5 4 3 2 1 0 12:09:10 From these charts it is possible to conclude that passive .585 HRMS 11.4 1.515 Min -301.589 HRMS 362.431 Min -20083 Max 19896 THD 2.4 0.2 0 0 0 5 10 (b) 15 20 Harmonic 30 40 50 Time (mSeconds) 10 20 25 60 Fund 14011 RMS 14037 CF 1.8 0. 1998 at 03:30:40 Local M_V Phase A-B Voltage SS Wave 4.6 0.0 CF 1.7 TIF/IT 115.2 THD 5.1 109 BMI/Electrotek 30 70 April 09.2 1 0.0 10 5 20 15 20 Harmonic 30 40 50 Time (mSeconds) 10 60 25 70 Fund 198.7 RMS 199.10 TIF/IT24637 BMI/Electrotek 30 20000 15000 10000 5000 0 -5000 -10000 -15000 -20000 -25000 1.2. but not eliminate them completely.17: THD harmonic trend. New power electronic devices such Figure 4. with harmonic passive .lters can minimize the harmonic voltage distortions. 16:49:05 it is easy to see a substantial increase in the THD values. 4.18 presents the THD historic trend for another operating condition. Fig. with the 4th and 5th harmonic .lters turned o from 12:00 midnight to 06:00am. 16: (a) Phase \A" current measured with harmonic passive .lters turned on during all the time. Figure 4. lters turned on. (b) Phase-to-phase \A-B" voltage measured with harmonic passive . lters turned on. % Fund April 09. 1998 at 03:30:40 Local M_V Phase A Current SS Wave Amps Amps . as well as on standards from CIGRE (Conference Internationale des Grands Reseaux Electriques a Haute Tension . new Brazilian legislation about harmonics and other power quality phenomena are currently under development.1: Global harmonic distortion limits for the system voltages recommended in Brazil. 6 2 9. 13 3 8 1 15 to 25 2  27 1 Voltage  69kV : THD = 3% max Odd Even Order Value % Order Value % 3.5 15 to 25 1  27 0. Some recommendations have been used to guide utility system planners and operators in supplying power to special loads. Reference [50] for example.5 8 0. 5.1). based on the experience of Brazilian power utilities.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 4.International Conference on Large High Voltage Electric Systems). 110 As far as the author knows. 5.5 Table 4. 13 1. 7 5 2. Power Quality Assessment through EMTP-based Programs 14:20:54 15:30:47 THD (%) 8 7 6 5 4 3 2 1 0 13:10:54 06:40:33 07:50:34 09:00:35 10:10:36 11:20:28 Voltage < 69kV : THD = 6% max Odd Even Order Value % Order Value % 3. 4. 11. IEC (International Electrotechnical Commission) and IEEE (The Institute of Electrical and Electronics Engineers).2. 11. presents some criteria and global voltage harmonic limits (Table 4. 7 2 2. 4. 6 1 9. as active . 18: THD harmonic trend.lters. might be able to dynamically compensate the distortions and improve the quality of power at the interface of industrial and power systems. with harmonic passive . Figure 4. 12:00:53 .lters turned on all the time. Power Quality Assessment through EMTP-based Programs 111 (c)Digital Modelling of Induction Furnaces in Distribution Networks A three-phase detailed modelling of the distribution system. This would require a complete and well-organized database of the system and load parameters. unless some speci. and so on.2. such details are usually not required. For distribution system planning and operation. including the linear and nonlinear loads.4. transformer saturation eects. in order to analyze power quality phenomena. power electronic loads. unbalanced conditions. would be the ideal and recommended database for an electromagnetic transient simulation. frequency dependent characteristics of the system and of the loads. which is rarely available in practice. automatic control devices. Typically. neither some important data is available.c power quality problem emerges as urgent and important. One must therefore use simpli. nor appropriate models exist to represent the physical behaviour by digital simulation. Therefore. the development of more accurate models is needed for power quality studies.cations. some realistic data was available. For this case study. which may make the simulations unrealistic. and some simpli. angle of . For the frequencies of interest here. In reality. with positive and zero sequence parameters at 60Hz. A Thevenin equivalent circuit with a series connection of coupled resistances and inductances was used to represent the 138kV transmission system. one obtains: Zposp:u: = 1=Ssc3php:u:. a  -circuit representation is reasonably accurate. The actual system under study is shown in Fig.cations had to be made for other data.77. either with constant parameters or .3 MVA. 4. and zero fault resistance for a single-line-to ground fault.1 MVA. a distributed-parameter line model would have to be used. Zzerop:u: = 3=Ssc1phg p:u: 2  Zposp:u ). based on the given three-phase short-circuit power (2881.7 degrees) from which the positive and zero sequence impedances at the frequency of 60Hz can be calculated (assuming 100MVA as a base power. The distribution line was modelled as a three-phase coupled  -circuit.78 degrees) and single-line-to-ground short-circuit power (1734. 138kV as base voltage. The transformer model used was based on three single-phase coupled impedances (\INVERSE" option in MicroTran). the resistance and inductance derived from these impedances are frequency-dependent. which was ignored in this study.19. For higher frequencies. angle of . and for the capacitor banks for power factor correction were available.2. The resonant inverter is used to create variable frequency at the induction coil. [94] was therefore used: a current-source. 4. Not enough information was available though for the induction furnaces. The sixpulse controlled recti.20. The data for all the distribution feeders with their respective loads. Power Quality Assessment through EMTP-based Programs 112 with frequency-dependent parameters. as shown in Fig.4. A digital model based on [93]. parallel-resonant inverter for induction heating. inductance and capacitance were available for the 4th and 5th order harmonic passive . Saturation eects were not considered in this simulation.3.0MVA .48kV .8/0.5. Realistic parameter values of resistance.9% three-phase unit transformer in delta/wye-grounded connection.er on the AC side of each induction furnace was supplied through a 13. EMTP-based programs can perform time-domain transient analysis or frequency-domain analysis.lters. A frequency-domain analysis was done for this case to . but over the continuous frequency range as well.21 correspond to the eects of the 4th and 5th order harmonic .21 and Fig. 4.nd the system impedance as a function of frequency. 4. Not only the impedance at multiples of the fundamental frequency was evaluated. by using a small step increment (
f) in the frequency variation input parameter. 4. seen from the point of common coupling (PCC) of the induction furnace. It was small enough to allow linear interpolation between calculated points.22. as shown in Fig. The two minimum impedance values at 240Hz and 300Hz shown in Fig. lters. Next. 4. 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. 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). from inductive to capacitive and vice-versa.22 also shows the zero crossings of the phase angle of the system impedance. For this simulation case. thus indicating parallel (maximum impedance) or series (minimum impedance) resonant conditions. using the MicroTran version of the EMTP. The induction furnace operating condition selected for this case corresponds to the time of maximum total harmonic distortion (THD) measured at the point . all the distribution feeders were represented. respectively. Fig. 881.0MVAr 13.0 ACSR .8kV 9.48kV Harmonic Passive Filters Induction Furnaces 4th 5th Figure 4.19: Distribution substation. without the harmonic .8kV 477.8kV Bus II .2.2% (25MVA) 25/30MVA 138/13.3 / -78.72% (25MVA) Bus I .1 / -77.8/0.13.13.4 o MVA 113 138 kV 25/30MVA 138/13.734.4 ACSR 0.48kV 5.8kV 10.0 o MVA Ssc 1phg = 1.0MVA 13.9% 0. of commom coupling (PCC). Power Quality Assessment through EMTP-based Programs Ssc 3ph = 2.5km PCC 3.8kV 9.4.2km 336. 4. for the induction furnaces operation with the 4th and 5th order harmonic passive . at the point of common coupling (PCC) between the utility electric system and the customer facilities.17). 4.lters (see Fig. Figs.23 (a) and (b) show the current and voltage waveforms respectively. with their harmonic contents. 15 is repeated here as Fig. Fig.lters turned o. 4.24 to facilitate . 4. parallel-resonant inverter for induction heating.20: Current-source.2. Power Quality Assessment through EMTP-based Programs POS 1 114 Ld 5 3 Lc VA INVA VB Lr Cr VC Rload INVB 4 2 6 INDUCTION FURNACE NEG Figure 4.21: Amplitude of the positive sequence system impedance at the PCC with harmonic .4. 50 Impedance Magnitude ( Ohms ) 45 40 35 30 25 20 15 10 5 0 0 50 100 150 200 250 300 350 400 450 500 frequency ( Hz ) Figure 4. the comparison between the simulated and the measured results.25 (a) and (b) show the current and voltage waveforms. 4. Figs.lters. but for induction furnace . 2. Power Quality Assessment through EMTP-based Programs 115 100 80 Phase Angle ( Degrees ) 60 40 20 0 −20 −40 −60 −80 −100 0 50 100 150 200 250 300 350 400 450 500 frequency ( Hz ) Figure 4.22: Phase angle of the positive sequence system impedance at the PCC with harmonic .4. operation with the 4th and 5th order harmonic passive .lters. but in this case with the harmonic passive . This operating condition corresponds to the time of minimum total harmonic distortion (THD) measured at the point of common coupling (PCC).lters turned on. lters turned on (see Fig. 4.8kV). Fig. 4. . Both operating conditions have approximately the same value of fundamental current (199A RMS at the phase-to-phase RMS voltage of 13. Again.18).26 to facilitate the comparison between the simulated and the measured results.16 is repeated here as Fig. 4. 23: (a) Phase \A" current simulated with harmonic passive .5 1 0.4.5 0 0 5 10 15 20 25 30 Harmonic Order (a) (b) Figure 4.5 3 2.5 2 1.2. Power Quality Assessment through EMTP-based Programs 3 Voltage ( V ) 0 −200 10 20 30 40 50 60 Peak Amplitude ( A ) Time ( ms ) 40 35 30 25 20 15 10 5 0 0 5 10 15 20 25 30 Harmonic Order x 10 2 1 0 −1 −2 −3 0 70 Peak Amplitude ( % Fund ) Current ( A ) 200 −400 0 Phase−to−phase "A−B" Voltage 4 Phase "A" Currrent 400 116 10 20 30 40 50 60 70 Time ( ms ) 4 3. lters turned o. (b) Phase-to-phase \A-B" voltage simulated with harmonic passive . 5 3 2.5 2 1.8 RMS 1.58 TIF/IT52660 April 01.489 CF Min -20237 Max 19805 THD 7.24: (a) Phase \A" current measured with harmonic passive .lters turned o.5 313.5 1 0.3 TIF/IT 234. 35 30 25 20 15 10 5 0 10 20 30 40 50 Time (mSeconds) 60 70 20000 10000 0 -10000 -20000 -30000 % Fund 0 Fund 198.4 Max 18.531 CF Min -308. 1998 at 03:58:49 Local M_V Phase A-B Voltage SS Wave Volts Amps 400 300 200 100 0 -100 -200 -300 -400 Amps April 01.89 THD HRMS 37.5 0 10 15 20 Harmonic 25 Fund 13557 13592 RMS 1. 1998 at 03:58:49 Local M_V Phase A Current SS Wave 0 5 10 15 20 Harmonic (a) 25 30 BMI/Electrotek 0 10 0 5 20 30 40 50 Time (mSeconds) 60 70 4 3.9 30 BMI/Electrotek (b) Figure 4.172 HRMS 972.9 204. (b) Phase-to-phase \A-B" voltage measured with harmonic passive .lters turned o. lters turned o. . 2.25: (a) Phase \A" current simulated with harmonic passive .4 1.2 1 0.8 0.6 0.2 0 0 5 10 15 20 25 30 Harmonic Order (a) (b) Figure 4.4 0.4. Power Quality Assessment through EMTP-based Programs 3 Voltage ( V ) 100 0 −100 −200 10 20 30 40 50 60 Peak Amplitude ( A ) Time ( ms ) 7 6 5 4 3 2 1 0 0 5 10 15 20 25 30 Harmonic Order x 10 2 1 0 −1 −2 −3 0 70 Peak Amplitude ( % Fund ) Current ( A ) 200 −300 0 Phase−to−phase "A−B" Voltage 4 Phase "A" Currrent 300 117 10 20 30 40 50 60 70 Time ( ms ) 1. lters turned on. (b) Phase-to-phase \A-B" voltage simulated with harmonic passive . lters turned on. Some care must be taken when de. for the transient simulation.515 Min -301.431 Min -20083 Max 19896 THD 2.585 HRMS 11.5 Max 294. 1998 at 03:30:40 Local M_V Phase A-B Voltage SS Wave Volts Amps April 09.6 0.1 30 BMI/Electrotek (b) Figure 4.2 THD 5.4 0. 1998 at 03:30:40 Local M_V Phase A Current SS Wave 0 5 10 15 20 Harmonic 25 30 0 10 0 5 20 30 40 50 Time (mSeconds) 70 1.7 RMS 199.589 HRMS 362.26: (a) Phase \A" current measured with harmonic passive .2 0 10 15 20 Harmonic BMI/Electrotek (a) 60 25 Fund 14011 RMS 14037 CF 1.8 0.2 1 0. Relatively large step sizes introduce errors into the results of a time-domain simulation and consequently into the post- 300 200 100 0 -100 -200 -300 -400 10 20 30 40 50 Time (mSeconds) 60 70 7 6 5 4 3 2 1 0 20000 15000 10000 5000 0 -5000 -10000 -15000 -20000 -25000 % Fund Amps 0 Fund 198.0 CF 1.4 1.10 TIF/IT24637 April 09.7 TIF/IT 115.ning the time step size (which corresponds to the inverse of the sampling frequency). (b) Phase-to-phase \A-B" voltage measured with harmonic passive .lters turned on. lters turned on. . 2. [12]. New solutions to this harmonic problem could be investigated using EMTP-based simulations. Moreover. 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 . as for example the possible use of active . It is also recommended that the time step size be such that the period of the fundamental frequency is an integer multiple of
t. Power Quality Assessment through EMTP-based Programs 118 processing frequency-domain analysis. thus producing errors which are a function of the frequency and the time step size (
t) [2].4. in order to avoid the generation of non-characteristic harmonics in the post processing Fourier analysis. then the step size can be calculated as
t = 1=(10  fmax ) = 16:66666s. The explanation is that the dierential equations of inductances and capacitances in EMTP-based programs are solved with the trapezoidal integration rule. at the point of common coupling.062 0.058 0.05 40 0.064 0.056 0.052 0.066 0.05 Time ( ms ) Figure 4.058 0.27. thus improving the quality of power at the interface of industrial and utility power systems. 4.056 0. Such power electronic device should be able to inject a shunt compensated current.lters [60].054 0. as shown in Fig.062 0.27: Instantaneous ideal compensation current to be \injected" by a shunt active .06 0.054 0.064 0.066 Time ( ms ) Ideal Compensation Current for Phase "A" Current ( A ) 30 20 10 0 −10 −20 −30 −40 0.06 0.052 0. Phase "A" Currrent Current ( A ) 400 200 0 −200 −400 0. [61] to minimize harmonic distortions. .lter. 4. A digital model of the induction furnace as a current-source. A comparison between the actual measurements and the EMTP simulations is presented in Table 4. parallel-resonant inverter load. Field measurements were also presented.2 and Table 4. The dierences in the values of the total harmonic distortion (THD) and the telephone in uence factor (TIF) are possibly due to simpli. 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 common coupling.3. where the voltage and current harmonic distortions were produced by induction furnaces. together with three-phase representations of the supplying distribution system were used in a time-domain simulation. Power Quality Assessment through EMTP-based Programs 119 (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. The history of this harmonic resonance problem was described. After reaching a time considered as steady state.2. Considering unbalanced conditions could also improve the simulation results.cations or lack of realistic data in the digital modelling. Both were unfortunately not available. The knowledge of detailed manufacturer data of the induction furnaces and their operating conditions. would allow the improvement of the EMTP simulation. Table 4. such as the natural resonant frequency.2: Comparison between . eld measurements and EMTP simulation results for the operating condition with the harmonic passive . Phase \A" Current [A] Parameters Measured Simulated Fund 198.4 6.6 1. Once the system is modelled for an EMTP- .3 1.4 EMTP-based simulations can be useful tools for harmonic analysis.592 13.1 121.8 204.4 304.444 3.4 2.557 13.237 -19.11 TIF/IT 52.9 18.1 234.814 1.660 64.489 1.6 1.lters turned OFF.950 0.4 19.3 Phase-to-phase \A-B" Voltage [V] Measured Simulated Error % 13.2 22.3 2.8 RMS 204.805 19.5 -304. based on the fact that very detailed eects can be taken into account.0 -20.491 Min -308.4 17.4 972.9 520.172 7.2 CF 1.6 13.89 15.5 0.4 THD [%] 18.9 201.42 HRMS 37.7 7.416 Error % 1.776 1.531 1.4 Max 313.58 31.487 4.950 1.031. 2.3: Comparison between .4. Power Quality Assessment through EMTP-based Programs 120 Table 4. eld measurements and EMTP simulation results for the operating condition with the harmonic passive . 589 3.011 14.10 9. The new models developed in this thesis project. Moreover.5 Max 294.3 2.910 Error % 4.72 TIF/IT 24.515 1.4 49.3 4.211 1.2 19.037 14. 4.896 20.2 Voltage Sag Analysis with EMTP-based Simulation Voltage sags or voltage dips are short duration variations in the supply voltage.3 115. caused by faults in transmission lines or in parallel distribution feeders. An entire process may be shut down when the voltage sags momentarily.7 458.2 RMS 199.6 8. in most cases the equipment ride-through characteristics are not known.5 -279.lters turned ON.7 190.5 362.2. the CBEMA curve [51] (or the ITIC curve) has been widely used as a .220 1.6 -20.1 based software.0 190.5 12.130 0.431 1. any type of studies can be performed. which are very common in today's electricity industry.3 4. For these reasons.7 THD [%] 5. The ride-through characteristics of modern electronic and computer-controlled loads are very sensitive to short duration variations in the supply voltage.2 280.224 24.7 7.9 91.1 219.2 26. The majority of power quality problems are associated with voltage sags.422 0. Phase \A" Current [A] Parameters Measured Simulated Fund 198.6 2.110 HRMS 11.637 36.4 14. hopefully will contribute to make EMTP-based programs more valuable tools for electric utility companies and industrial customers in evaluating power quality problems.3 1. The equipment tolerance characteristics to voltage sags vary very much among equipment manufacturers.585 5.219 1.8 Phase-to-phase \A-B" Voltage [V] Measured Simulated Error % 14.474 Min -301. or caused by the start-up of large induction motors or other types of sudden load variations.4 CF 1.083 -20. Fig. 4. 7 \We typically employ the curve only from 0.1 cycles and higher due to limitations in power quality monitoring instruments and dierences in opinion over de.rst reference for power quality studies related to short duration voltage variations.28 presents actual measurements of voltage sag phenomena with an overlay of the CBEMA curve 7 . ning the magnitude values in the subcycle time frame. 37-38." From [24]. pp. . 4. there are certain levels of light icker which can be easily detected. EMTPbased programs have the exibility to include aggregated load models [109]. is a promising solution for the mitigation of voltage sag phenomena. 4. 4.29 (b) shows the simulation results using MicroTran. Power system protection and operating practices may also aect the success or failure of loads which are sensitive to voltage disturbances. which is beyond the scope of this thesis project.3 Welding Industry Voltage Fluctuation Study { A Visual Flicker Case Flicker is historically considered a problem of perception. 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). controlled by DC drives. because the human ability to visually sense light changes caused by voltage uctuations does vary.2. seems to be needed for the accurate representation in EMTP-based programs of the dynamic behavior of loads. Nevertheless.4.29 (a) shows a voltage sag phenomenon caused by a single-line-to-ground fault in a distribution feeder. however.2. Custom Power Controllers such as the dynamic voltage restorer (DVR). More research. However. because the dynamic behaviour of industrial loads (typically induction motors) were not included in the simulation model. Fig. The industrial process. which is in parallel to a feeder supplying a \PVC" pipe (and other plastics derived products) manufacturer with sensitive loads. This section presents a recent case of light icker and some simpli. Power Quality Assessment through EMTP-based Programs 121 The power system characteristics at the point where the sensitive load is connected is another important issue for possible mitigation of voltage sag problems. though their impact on possible human brain disorders or any other potential health damage is diÆcult to quantify. The simulation does not match the measurements exactly. Fig. ed EMTP-based simulations. connected to a distribution system was simulated with MicroTran. even when the RMS voltage uctuations are in a range of very small percentage deviations. to produce meshed wires for construction.8 Hz. If the duty cycle of the welding process is near 8. the human eye would perceive the maximum visual ickering eect. . The operation of a welding machine. For time durations less than 1 cycle the equipment seems to measure peak values.Max Depth 16/04/96 06:03:38 PQNode Local 300 Max 106. Power Quality Assessment through EMTP-based Programs PQNode Group 122 March 15.01 0. IEC Standard 1000-4-15 provides the speci.2.0 Min 73.1 1 10 100 1000 10000 Time (Cycles) BMI/Electrotek Figure 4. 1996 at 18:56:21 PQNode Local Triggering Phase.001 0.28: Voltage sag measurements (%RMS versus time duration) with an overlay of the CBEMA curve.50 250 % Volts 200 150 100 50 0 0.4. The operating duty cycle.cations for a ickermeter with \lamp-eye-brain" frequency response to light ickering eects. Fig. 4. controlled by the semiconductor .30 shows the simulated instantaneous voltage for a power electronics controlled welding machine . results in a modulating frequency of approximately 7Hz for this case.ring angle. 4. such as the distribution static synchronous compensator (D-STATCOM) can eectively mitigate this type of power quality problem [110]. as shown in Fig. .31. The application of reactive dynamic compensation through power electronic devices. which causes the visual light icker. when this does not aect the industry productivity or the quality of the manufactured product. Cost-eective solutions for voltage uctuation problems are usually related to changes in the load duty cycle. 29: (a) Phase-to-phase \A-B" measured voltage sag.5 −1 −1.5 Time ( s ) (b) Figure 4.1 0. (b) Phase-to-phase \A-B" simulated voltage sag.5 0 −0.4 0.5 −2 −2.5 Ref Cycle 30664 Time (Seconds) % Volts 100 75 50 25 0 -25 -50 -75 -100 0 25 50 75 100 125 150 175 200 Time (mSeconds) Uncalibrated Data (a) 4 2.5 Voltage ( Volts ) 1 0.4 0.8 0 0.45 0.05 0.50 Ave 89.5 0 0. Power Quality Assessment through EMTP-based Programs TIGRE2 March 25.3 0.25 0.2 0.70 Max 100.233 Sec Min 73.1 0.35 0.5 x 10 Instantaneous Phase−to−phase"A−B" Voltage at the PCC 2 1.4. .2 0. 1996 at 08:25:40 PQNode Local Phase A-B Voltage RMS Variation % Volts 110 105 100 95 90 85 80 75 70 123 Trigger Duration 0.15 0.2.3 0. 9 1 % Volts 1.4.1 0.9 1 Time ( s ) Figure 4.5 0.3 0. Phase "A" Voltage Voltage ( Volts ) 1000 500 0 −500 −1000 0 0.31: Modulated voltage and respective amplitude frequency spectrum .3 0.5 0 40 45 50 55 60 65 70 75 80 Frequency ( Hz ) Figure 4.1 0. Power Quality Assessment through EMTP-based Programs 124 Instantaneous Voltage at the Point of Common Coupling 1000 950 Voltage ( Volts ) 900 850 800 750 700 650 600 0 0.8 0.5 1 0.6 0.2.8 Time ( s ) Harmonic Amplitude Spectrum 2 0.4 0.6 0.7 0.2 0.2 0.5 0.4 0.30: Instantaneous voltage uctuations causing light ickering eect.7 0. 4. These results emphasize the \importance of a simultaneous solution approach". In other cases. The intentional introduction of a one time step delay in the control system. give wrong or inaccurate simulation results.3. 4. Where an interesting and already published benchmark simulation case with all the necessary data was readily available. simple circuits were assembled by the author to test and prove the ideas. it was selected because it made comparisons easy. 4.3 EMTP-based Simulation Cases with SSCPS This section presents a collection of test cases to validate the EMTP-based models developed in this thesis. The simultaneous solution is shown in Fig.34. EMTP-based Simulation Cases with SSCPS 125 4.32 presents a control block diagram of a transfer function with a second order dierential equation. or the internal delays inside TACS. Fig. .4. as shown in Fig. 4.35. (a) The Eect of a One Time Step Delay in the Solution of EMTP-based Simulations Reference [66] presents some interesting cases where the one time step delay in the solution of control and system equations. 4. with some applications to test cases.3.1 Basic Control and Control Devices Simulation Cases This section provides more explanations about the theory presented in the previous chapters.33. the simulation of a transfer function with poles on the imaginary axis of the complex plane is very sensitive to time delays. resulting in instability. as illustrated in Fig. leads to an unstable resonance condition. For example. which can \move" the poles to the right half of the complex plane. which is critical in some cases for the correct simulation of control and power system equations with EMTP-based programs. To illustrate this. 0 [V] 1 _____ s Figure 4. 80 .3.33: Solution of system with bounded resonance oscillations. 8 6 4 vout vin Voltage ( V ) 2 0 −2 −4 −6 −8 0 10 20 30 40 50 60 70 Time ( s ) Figure 4. EMTP-based Simulation Cases with SSCPS v IN 126 1 _____ s + - v OUT 1.32: Control block diagram of a second order dierential equation with poles on the imaginary axis of the complex plane.4. 3.34: Introduction of a one time step delay in the control block diagram. EMTP-based Simulation Cases with SSCPS v IN 127 1 _____ s + - v OUT 1.0 [V] 1 _____ s delay 1 ∆t Figure 4.35: Solution of system with unstable resonance oscillations caused by the introduction of one time step delay. . 8 v out 6 4 vin Voltage ( V ) 2 0 −2 −4 −6 −8 0 10 20 30 40 50 60 70 80 Time ( s ) Figure 4.4. lead-lag's for phase compensation. washout . PD. derivative (D). Generally. (and their combinations PI. PID). Transfer Functions and Filters Laplace transfer functions are essential for control design and simulation. classical control blocks such as proportional (P).3. integral (I). EMTP-based Simulation Cases with SSCPS 128 (b) Basic Control Blocks.4. . are present in any analog or digital control scheme.lters. etc. Many if not most of the . second or higher order dierential equations used to model the physical behavior of electrical. Transfer functions can also represent passive and active . mechanical. and any other systems use such transfer functions for their mathematical representation.rst. chemical. lters (i..e. connections of circuits with operational ampli. resistors and capacitors). Depending on their properties and frequency response.ers. these . elliptic. can also be very useful for engineers and scientists. especially for the power system part. It is important to mention that other general purpose and powerful computer tools.lters are referred to with special names. however. Chebyshev. all-pass (phase). EMTP-based programs. high-pass. Legendre-Papoulis. etc. Therefore. especially for the design of control systems. still seem to have more detailed and proven models. which are simulated with the simultaneous solution method implemented experimentally in MicroTran. band-elimination. notch. [80]). such as MATLAB [103]. the methodology 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. Bessel. Butterworth. band-pass. such as low-pass. This section then presents some test cases with classical control blocks. The . the UBC version of the EMTP. The computer program SPICE seems to be more used for electronics and power electronics simulations. all-pass (magnitude). 36.rst control block diagram case is illustrated in Fig. 4. which is used in power system small-signal stability studies of a single machine connected to an in. which represents the classical linearized \swing equation". . H = 3:5 MWs/MVA = inertia constant in MWs/MVA. where: KS = 0:757pu torque/rad = synchronizing torque coeÆcient in pu torque/rad. as extracted from page 731 of reference [86]. KD = 10 or -10 pu/pu = damping torque coeÆcient in pu torque/pu speed deviation.nite bus.
T a =
T m
T eS
T ed = accelerating torque deviation.1pu disturbance in the per unit mechanical torque deviation (
T m).3.
Æ = rotor angle deviation in electrical rad.4. EMTP-based Simulation Cases with SSCPS 129
!r = speed deviation in pu = (!r !0 )=!0 .
T ed = damping torque component. !0 = rated speed in electrical rad/s = 2f0 = 377 rad/s for a 60Hz system. Fig.37 illustrates the simulation results for a -0. 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. 4.
T es = synchronizing torque component in pu. 4. As shown in Fig.38 with a negative damping torque coeÆcient (KD = 10) the rotor angle deviation (
Æ ) presents ampli.
T m = 0:1pu = mechanical torque deviation in pu. 7) .6) 0  = 12 2KH!Dn = 12 pKKS 2DH! 0 (4.36 could also be represented by a canonical second order transfer function. where: K= !n = (4. 4. causing small-signal instability.ed natural oscillations. as presented in Fig.39.5) 1 KS p KS 2!H (4. 4. Alternatively. as expected. the control block diagram presented in Fig. 36: Classical linearized \swing equation". EMTP-based Simulation Cases with SSCPS 130 ∆Tes KS - ∆Tm ∆Ta + - 1 _____ 2H s ω0 _____ s ∆ωr ∆δ .4. used in power system small-signal stability studies of a single machine connected to an in.u.0.1 p.3. KD ∆Ted Figure 4. 0. ) 0.05 ∆Te d 5*∆ω 0 −0.37: Simulation results of the synchronous machine rotor angle deviation.05 ∆Te ∆Tm s −0. .u.25 0 1 2 3 4 5 Time ( s ) 6 7 8 9 10 Figure 4.1 ∆Ta Variables in per unit ( p. in the presence of a positive damping torque coeÆcient.15 ∆δ −0.1 −0.2 −0.nite bus. Figure 4. EMTP-based Simulation Cases with SSCPS 131 200 Rotor angle deviation ( p.38: Simulation results of the synchronous machine rotor angle deviation.1 p.39: Canonical second order transfer function representation of the single-machine in.3.4.u. ∆Tm K ωn2 _________________ s2 + 2 ξωn s + ωn2 ∆δ .u. ) 150 100 50 0 −50 ∆δ −100 −150 0 1 2 3 4 5 6 7 8 9 10 Time ( s ) Figure 4. in the presence of negative damping torque coeÆcient.0. .nite bus system. Control Devices.e. For example.4. EMTP-based Simulation Cases with SSCPS 132 (c) Voltage and Current Sensors. 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. control devices.3. Function Blocks and Digital Logic Gates The solution method for voltage and current sensors. function blocks and digital logic gates follows the procedures for dependent sources presented in Chapter 2.  then to de. with the compensating current at the controlling branch j set to zero. i.. ij = 0. i..ne any linear or nonlinear voltage function at the controlled branch k.  and .e. vk = f (vj ). their respective compensating branch currents are \simply set to zero". 2. Since control signals are usually voltage signals (or current signals converted to voltage signals).2. If more than one control voltage signal needs to be sensed.nally to use the Newton-Raphson algorithm presented in Section 2.5. (Fig. zero crossing detection and generation of gate . The implementation of control devices (such as transport or time delay blocks.19) to calculate a solution for the compensating currents ij and ik . On the other hand. if currents have to be sensed. a current sensor can be represented by a current-controlled voltage source. their respective branch voltages can be set to zero. comparators. SQRT. Reference [66] clearly explains the problem and the solution for a correct EMTP-based simulation. etc.). FORTRAN function blocks (such as SIN. OR. MULTIPLICATION. 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. TAN. . NAND. Park transformation. EXP.) uses the same concept of the simultaneous solution for voltage-controlled voltage sources presented before. etc.ring signals. etc. ACOS. LOG. NOT. Clark transformation. COS. DIVISION.) and digital logic gates (AND. NOR. ASIN. 3. as for example for the evaluation of average values (e. (a) Dynamic Control of the Firing Angle (\") of a Thyristor Fig. etc. [94]. [95]. [89]. 4. can be easily modelled with control blocks. \rolling average power" presented in [66]).40 illustrates a simple circuit to demonstrate three important contributions of this thesis. root mean square (RMS) values. [93].2 Power Electronics Simulation Cases The cases presented in this section are based on references [65]. EMTP-based Simulation Cases with SSCPS 133 Special functions. and with subcircuit implementation. instantaneous total harmonic distortion (THD) values. for the dynamic simulation of power electronic devices in EMTP-based programs:  simultaneous solution for voltage sensors.  zero crossing detection and generation of .4. [104]. 4.g.3. ring pulses with an instantaneous updating of the controlling . The controller is enabled by a voltage signal of \1 volt" and disabled by a voltage signal of \0 volt". the synchronizing signal for the gate . For this case. 4. The voltage sensor in Fig.40 is modelled with the equations for an \ideal" voltagecontrolled voltage source (VCVS) presented in Section 2.9) where A = 1.2.ring angle . which receives the gate signal directly from the controller.  EMTP-based voltage-controlled unidirectional current owing switch. i.8) rk1
i + ::: 1 rkkA
i + ::: k A (4.: ij = 0 k + r vOP ENj + vOPEN j1 rkj
A ::: + rjj A i
j + rjk iM = 0 ::: + rjM rkM A (4.e. to represent the thyristor. 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. 3.4. EMTP-based Simulation Cases with SSCPS 134 The gate . ring controller is modelled as a \multi-terminal voltage-controlled voltage source". vGAT E (a pulse of amplitude equal to 1 with a speci. where three controlled voltages are sensed and used to determine the voltage source at the gate. the time of zero crossings) . or zero volts). with interpolation.ed width. vALP HA (which is the . The output vGAT E depends on the values of vSY NCHR (which is used to detect. ring angle in degrees. based on the given input frequency) and venable=disable (whose value of 1 or zero is multiplied by vGAT E to enable or disable the control . converted to time. ring). The following equations are used to solve for the gate . the use of internal variables for the control could result in a more economic implementation. The time step size used was
t = 16:6667s. Eventually. Internal control variables would actually not have any physical connection with the power network part of the circuit. which is a truly simultaneous solution of the power and control circuit. In cases where this calculation fails. EPSILON=10 12. 4. as mentioned before in the fundamental assumptions of Section 2.10.13. Fig.2. .2. The solution with the compensation method using an iterative Newton-Raphson algorithm requires the calculation of a Thevenin equivalent for each branch.12) vOP ENm + rm1 i1 + ::: (4.ring controller using the compensation method: ij = 0 (4.11) il = 0 (4.13) ::: + rmm im + ::: + rmM iM + vGAT E = 0 In addition to equations 4.10 to 4.10) ik = 0 (4. the connection of a large resistance of 109 and with the near zero tolerance parameter for checking matrix singularities. which receives the gate signal directly from the controller. conditional IF-statements are used to implement the logic described above equation 4. this problem can easily be solved. The thyristor was modelled as an EMTP-based voltage-controlled unidirectional current owing switch.41 shows the resulting voltages and currents simulated with MicroTran and the method proposed in this thesis. in an EMTP production code. the connection of large resistors in parallel with the branch may make a Thevenin equivalent circuit possible 8 . 8 In MicroTran. 0 [V] tstop = 25.0 [ms] Figure 4.4.5 Ω Vmax = 5.0 [V] f = 60 [Hz] 5 mH vGATE Gate Firing Controller α vSYNCHR Voltage Sensor vALPHA f=60Hz pulse width=10 degreees 45.3.40: Circuit for the dynamic control of the .0 [V] Firing Angle (degrees) venable / disable 1. EMTP-based Simulation Cases with SSCPS 135 iLOAD vSA vLOAD 2. .ring angle (\") of a thyristor. 41: Voltages and currents in a circuit with dynamic control of the .5 4 3 2 1 0 −1 −2 −3 −4 −5 LOAD i 5 LOAD v v 25 Time ( ms ) GATE enable / disable 20 v 15 10 vsynchr 30 35 40 45 50 136 Figure 4. 0 4. EMTP-based Simulation Cases with SSCPS Current ( A ) Voltage ( V ) .ring angle of a thyristor.3. EMTP-based Simulation Cases with SSCPS 137 (b) Dynamic Control of the Firing Angles of a Three-Phase Six-Pulse ThyristorBridge Recti.4.3. er Fig.42 illustrates a phase controlled recti. 4. The six-pulse .er with a feedback control system based on the manual of PSIM [65]. \It should be noted that. There is one time step delay between the power and the control circuit solutions" [65]. in PSIM. the power and the control circuit are solved separately. Similarly to the simple .ring controller implemented with the models developed in this thesis project uses a simultaneous solution for the power and control circuit equations. 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 . the dynamic .ring controller are the synchronizing signal (voltage vAC sensed from the supply system). From the gating signal generated to the thyristor with identi.ring angle  (resulting from the ACOS control block. after comparison with the desired reference voltage for the DC load). which receives the signal from the limited PI controller. For starting purposes. all the other gating signals are derived sequentially by adding a time delay corresponding to 60 degrees at the 60Hz frequency.cation number \1". whenever a . another \isolated" .ring signal is sent to a particular thyristor. 43 presents the EMTP simulated voltages and currents. 4. using a
t = 16:6667s and with a dynamic control of the . Fig.ring signal is sent to the previous thyristor as recommended in [88]. ring angles of the three-phase six-pulse thyristor-bridge recti. er.44 shows the dynamic behavior of the control variables. Fig. 4. emphasizing the . and Fig.ring control signal . . 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. 42: Circuit for the dynamic control of the . EMTP-based Simulation Cases with SSCPS 138 5 mH G1 Vmax = 81.3.4.01 KI= 1.0 vSENS= vLOAD vPI vLIM ACOS -1 1 [V] vENABLE / vALPHA G1 G2 G3 G4 G5 G6 DISABLE Figure 4.65 [V] G3 iLOAD + G5 f = 60 [Hz] vDC vLOAD vA vB 400 µ F 2Ω vC G4 G2 G6 - vAC vSYNCHR 1 vREF 100 [V] + vERR PI KP=0. ring angles of a three-phase six-pulse thyristor-bridge recti. .er. 150 100 50 0 −50 −100 −150 vDC iLOAD 5 vA 10 Time ( ms ) LOAD v 15 vsynchr 20 25 139 Figure 4.43: Voltages and currents with dynamic control of the . ring angles of a three-phase six-pulse thyristor-bridge recti. er. EMTP-based Simulation Cases with SSCPS Current ( A ) Voltage ( V ) . 0 4.3. 44: Dynamic control of the .120 100 80 60 40 20 0 −20 5 10 15 20 ALPHA v 25 Time ( ms ) 30 35 vREF ERR v 45 vSENS=VLOAD 40 50 140 Figure 4. ring angles of a three-phase six-pulse thyristor-bridge recti. 0 4. EMTP-based Simulation Cases with SSCPS Current ( A ) Voltage ( V ) .3.er. 3. 0 4.4 1.6 0.1.2 0 5 PI v 10 15 20 vLIM 25 Time ( ms ) 30 35 40 45 50 141 Figure 4. EMTP-based Simulation Cases with SSCPS Current ( A ) Voltage ( V ) .4 0.8 0.45: Dynamic voltage control signals at the output of the proportional-integral (PI) and the limiter control blocks.2 1 0. the power and the control circuit equations are solved simultaneously with the methods proposed in this thesis.4.3. For this EMTP-type simulation it was used a
t = 16:66667s. EMTP-based Simulation Cases with SSCPS 142 (c) Dynamic Control of Three-Phase PWM Voltage Source Inverter Fig.47. Again. 4. The phase \A" modulation and triangular carrier waveforms for generation of gating signals through sinusoidal pulse width modulation (PWM) are presented in Fig.4. With the use of comparators and NOT logic gates the .46 presents a circuit for the dynamic control of a three-phase PWM voltage source inverter (VSI) [65]. by the EMTP-based voltage-controlled switches.ring signals are dynamically generated. The IGBT's with anti-parallel diodes were represented. in a simultaneous solution with the network equations through the compensation method. Here it is opportune to discuss the issue of simultaneous commutation: Let us assume that the current is owing through the IGBT with identi. for simplicity and without much loss of accuracy in this simulation. which were implemented in this thesis through the GATE subroutine. the voltage vSA NEUT R reverses polarity almost instantaneously (due to the behavior of the inductor. which forces the current to keep owing in the same direction). When IGBT number 1 is turned o. thereby forward biasing the anti-parallel diode of the IGBT with identi.cation number \1" from the DC source to the load. which then starts conducting. i. 4.48 presents the node voltage \vSA " generated by the 3-Phase PWM voltage source inverter (VSI). with an IGBT and diode in anti-parallel (which would have to be modelled with piecewise linear or nonlinear model). where the control signals play the role of the commutation. In digital simulation programs this means that there is simultaneous commutation between IGBT 1 and the anti-parallel diode at IGBT 4. or as a voltagecontrolled bidirectional current owing switch.50. 4. Fig.cation number \4". 4..49 shows the voltage across the load \vSA NEUT R " and the current in phase \A" supplied to the load. The dynamically generated 3-phase load currents are illustrated in Fig. The line- . This could be modelled as it is. whereas Fig.e. 4.51. EMTP-based Simulation Cases with SSCPS 143 to-line voltage generated by the three-phase PWM voltage source inverter (VSI) is shown in Fig.4.3. Most of the advanced Custom Power Controllers [60] (and active . -Ing. the models developed in this thesis. Mauricio Aredes. 9 The Ph. can be downloaded by the reader from the the web site http://www. dynamic hysteresis current-band PWM converters are used [60] 9 .lters [111]) apply this type of converter to synthesize voltages or current waveforms according to the desired \dynamic reference modulating signal". especially those aimed at improving the quality of power in utility and industrial systems. will hopefully be useful for the accurate EMTP-simulation of a variety of existing and new power electronic devices.dee. Therefore. . When a current is to be synthesized.ufrj.D thesis \Active Power Line Conditioners" of Dr.br. 87 Ω 7.4.8 [V] f = 60 [Hz] COMPARATOR vTRI 1 [V] f = 1500 [Hz] phase = -180 [degrees] Figure 4.8 [V] vC 0. EMTP-based Simulation Cases with SSCPS IGBT1 450 [V] 144 IGBT3 IGBT5 iSA-NEUTR vSA vSB vSC vNEUTR iSB-NEUTR iSC-NEUTR 3.46: Circuit for the dynamic control of three-phase PWM voltage source inverter (VSI).3.7 mH 10 9 Ω IGBT4 IGBT6 IGBT2 vA 0.8 [V] vB NOT 0. . 5 A v 5 10 TRI v Time ( ms ) 15 20 25 145 Figure 4. EMTP-based Simulation Cases with SSCPS Current ( A ) Voltage ( V ) .1.5 1 0.47: Phase \A" modulation and triangular carrier waveforms for generation of gating signals through sinusoidal pulse width modulation (PWM).5 −1 −1.3. 0 4.5 0 −0. EMTP-based Simulation Cases with SSCPS Current ( A ) Voltage ( V ) .48: Node voltage \vSA " generated by a three-phase PWM voltage source inverter (VSI).600 400 200 0 −200 −400 −600 5 vSA 10 Time ( ms ) 15 20 25 146 Figure 4.3. 0 4. 600 400 200 0 −200 −400 −600 5 vSA − vNEUTR Time ( ms ) 5 * iSA−NEUTR 10 15 20 25 147 Figure 4. EMTP-based Simulation Cases with SSCPS Current ( A ) Voltage ( V ) . 0 4.49: Voltage across the load \vSA NEUTR " and current supplied to the load by a three-phase PWM voltage source inverter (VSI).3. EMTP-based Simulation Cases with SSCPS Current ( A ) Voltage ( V ) 148 .60 40 20 0 −20 −40 −60 5 SA−NEUTR i 10 15 SB−NEUTR i Time ( ms ) iSC−NEUTR 20 25 Figure 4. 0 4.3.50: Load currents supplied by a three-phase PWM voltage source inverter (VSI). EMTP-based Simulation Cases with SSCPS 149 600 vSA − vSB 400 Voltage ( V ) 200 0 −200 −400 −600 0 5 10 15 20 25 Time ( ms ) Figure 4. .3.51: Line-to-line voltage generated by a three-phase PWM voltage source inverter (VSI).4.  Use of statistical and other advanced data analysis methods to produce meaningful information. Synthesis of Simulation Guidelines for Studies with EMTP-based Programs 150 4. which will result in a 3% error with the trapezoidal integration rule [2]. modelling and simulation in EMTP-based programs of power system components and power electronics devices are pointed out. Usually. For example. the time step size is set to a value at least equal to one tenth of the period of the maximum frequency. with the main objective of analyzing their dynamic interaction and of evaluating their impact on electric power quality. It is also recommended that the step size be such that the period of the fundamental frequency is an integer multiple of
t. In the analysis of power quality phenomena through time and frequency domain EMTPbased simulations.  Comparison against national and international power quality standards. [112].4. Important factors regarding power quality monitoring. through the use of instruments with appropriate voltage and current sensors and adequate digital sampling frequency [15].4. in order to avoid the generation of non-characteristic harmonics in the post processing Fourier analysis. taking into consideration system dierences and similarities. if the system nominal frequency is 60:0Hz and the maximum frequency expected in the transient simulation .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. For power quality monitoring the important factors are:  Evaluation based on accurate measurements of power quality phenomena. 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. Synthesis of Simulation Guidelines for Studies with EMTP-based Programs 151 is in the order of fmax = 6kHz .4. then the time step size can be calculated as
t = 1=(10  fmax ) = 16:66666s. particularly in the case of transformer saturation. especially if frequency dependence has to be taken into consideration in the simulated phenomena [92].  the use of simpli.  nonlinearities.  the selection of appropriate models to represent power system and load components. which are usually disregarded in many simulations. They can aect the accuracy of the simulation results.4. ed switch models for power electronics devices. This may be justi. [68] and/or resynchronization techniques ([96] pages 185.  numerical oscillations caused by the trapezoidal rule of integration in solving the system of equations. Therefore. [85]) is eective in the elimination of numerical oscillations.  the one time step delay at the interface between the control and power systems solution. it is much more important to use such techniques than to reduce the time step size. The method SSCPS (\Simultaneous Solution of Control and electric Power System equations") proposed in . especially related to semiconductor commutation phenomena. MicroTran has CDA implemented. which may exist in TACS (\Transient Analysis of Control Systems") and in other software packages. The use of techniques such as CDA (\Critical Damping Adjustment" [84]. 204. as well as other internal time step delays. for better accuracy in EMTP-based simulations of power electronics.ed to speed up the simulation time for system level studies. but other EMTP versions may not. [99]. 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 [98]. causing unrealistic high frequency transients in the simulation of power electronic devices. the EMTP solution at discrete time intervals
t may result in inaccurate turn-on or turn-o switching times. Backtracking techniques [67]. 207) or even the Clock Synchronized Structure Changing Concept (CSSC) [97] can be used to minimize the problem. or may use dierent approaches. Also. but it may also give wrong and misleading results. Synthesis of Simulation Guidelines for Studies with EMTP-based Programs 152 this thesis overcomes this problem. .4. and the user only introduces time delays if needed to represent system physical behaviour.4. 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.Chapter 5 Conclusions and Recommendations for Future Work T HE MAIN GOAL of this Ph. The growing regulatory. with obvious catastrophic consequences. thesis project was the development of EMTP-based models for control and power electronic devices for electric power quality assessment.D. 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. environmental. and motor equipment. This. 153 . heat.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). and the adoption of programs to \save energy" with the promotion for the use of more electricity eÆcient light. the use of FACTS devices to enhance power system stability and control. in turn. might create a deteriorating impact on the quality of power. which are typically power electronics and computer-based loads. and system loss reduction. It is a challenging environment for engineering. Conclusions and Main Contributions 154 due to the manufacturing of usually inexpensive power electronic converters. electricity has become a basic commodity. economics. politics.5. The quality of the electric power delivered to customers by utilities may not be acceptable for some types of sensitive loads. There are cases where the increasing use of power electronics to enhance process eÆciency and controllability creates power quality problems. as well as the use of series capacitors (.1. The growing application of shunt capacitors for voltage support. power factor correction. since in modern human society. which almost everybody and almost everything depends on. etc. particularly in the control of industrial processes. for line reactance compensation) will increase the potential risk of transient disturbance ampli.xed or controlled. 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. new power electronic devices also oer the means for adequate \power conditioning".cations and potential electrical and mechanical resonances in the presence of more and more power electronic devices. can hopefully be performed with the models developed in this work. The assessment of electric power quality and the technical impact of power electronic devices on the quality of power.D. and of steam and gas turbines in distributed and co-generation power plants. to meet the special requirements of electric power quality in a system. so grows its ability to oscillate more! At the same time. computer programs based on the EMTP (Electromagnetic Transients Program) are becoming more useful. As the natural order of the system grows. 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. through the compen- . The main contributions of this Ph. 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. 1. based on SSCPS. A. Conclusions and Main Contributions 155 sation method and the Newton-Raphson iterative algorithm.  experimental implementation in Microtran.5. E. This solution method eliminates not only the one time step delay problem at the interface between the solution of power and control circuits. Araujo in 1993 [67]. as an innovative alternative to the solution presented by A. A \circuit approach" was proposed in this thesis. for example the modelling of operational ampli. the UBC version of the EMTP. but also all the internal delays. of a simultaneous solution for: { linear and nonlinear current and voltage dependent sources (which allow. which may exist in methods based on the transient analysis of control systems (TACS) since 1977 [64]. ideal current and voltage sensors. etc. { independent current and voltage sources. which can also be connected between two ungrounded nodes.).ers. { 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 . ACOS. ASIN. /. etc. -. COS. { mathematical and transcendental FORTRAN functions (such as +. TAN.lters and classical control blocks). comparators.) and some digital logic gates (NOT). EXP. { transformation of variables (such as the abc to . LOG.). SIN. * . etc. { special control devices (such as time delays. 0 transformation and its inverse). thyristors. IGBT's. { voltage-controlled switches.  development of the subroutine \GATE" in MicroTran. . GTO's. { nonlinear model of a diode semiconductor. allowing the dynamic control of the turn-on and turn-o times of semiconductor devices (e..).g. which are modelled as EMTP-based voltage-controlled switches. etc. Recommendations for Future Work 156  development of power electronics simulation cases in MicroTran.2.5. using the simultaneous solution approach (SSCPS) for the dynamic control of semiconductor switching devices (as in a three-phase six-pulse thyristor controlled bridge recti.  interaction with a Brazilian utility company and industries for the realization and analysis of . and in a three-phase PWM voltage source inverter (VSI)) and evaluation of current and voltage waveforms.er. etc.eld measurements of electromagnetic phenomena aecting the quality of power. based on realistic . as for example. such as: { voltage sags and voltage swells. consequences and investigation of possible solutions for power quality problems.  synthesis of simulation guidelines for the evaluation of the impact of power electronic devices on the quality of power. with determination of causes. the application of \Custom Power Controllers". { harmonic current and voltage distortions. { transients. D. allowing for example.eld measurements and EMTP time and frequency domain simulations.  development and implementation in EMTP-based programs of techniques for the calculation of the \frequency response of integrated control and power system equations".2 Recommendations for Future Work The author's main recommendations for future work related to his Ph. 5. the determination of transfer functions between a generic input and a generic output. thesis project are listed below:  development and implementation in EMTP-based programs of methods for the \automatic calculation of initial conditions" in the simultaneous solution of control and power electronic circuits. . ). active .  development and implementation in EMTP-based programs of \detailed nonlinear models of semiconductor devices" (such as transistors. UPFC. UPQC.2. DVR. etc. but suitable for time and frequency domain simulations in power quality studies.5.  development and implementation in EMTP-based programs of \voltage and frequency dependent aggregated load models" [109].  development and implementation in EMTP-based programs of algorithms for the simultaneous solution of \generic nonlinear dependent sources" [101]. such as STATCOM. Recommendations for Future Work 157  development and implementation in EMTP-based programs of techniques for the accurate solution of \digital control systems".  development and implementation in EMTP-based programs of models for FACTS and Custom Power Controllers. etc. and de.lters. if not yet available in the technical literature. possibly with .nition. of benchmark test cases. Some of the author's publications in areas related to the thesis topic are listed here for easy reference:  B.eld validation of the models. [90]. Complex and very important electromagnetic and electromechanical phenomena could then be investigated thoroughly. E. will require detailed and complex modelling of power system components. S~ao Paulo-SP. S. power electronics controllers. Dias. FACTS Controllers and special loads. E. S. Bonatto. March 18-22. F. D. etc. The evaluation of \power system stability and control. and L. based on long term EMTP-type simulations". Mertens Jr.. Brazil. the damping and control of subsynchronous resonance [86]. submitted to the IEEE/PES Transmission and Distribution Latin America Conference (IEEE/PES T&D 2002). A. \Power Quality Assessment at Sensitive Loads". power system stabilizers. 2002. da Silva. . with all their associated control equipment such as turbine-governor controllers.. exciter controllers. as for example. August 08-12. \Power Quality Diagnosis in Industrial Customers . \The Quality of Electric Power in Coordination with the Industrial Safety". and H. 2001. H. E..  B. L. 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