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EMTP modelling of control and power electronic devices Bonatto, Benedito Donizeti 2001

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E M T P Modelling of Control and Power Electronic Devices by B E N E D I T O DONIZETI B O N A T T O M.A.Sc. in Electrical Engineering, State University of Campinas, Brazil, 1995. .A.Sc. in Electrical Engineering, Federal School of Engineering of Itajuba, Brazil, 1991. A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Electrical and Computer Engineering) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A October 2001 © Benedito Donizeti Bonatto, 2001 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Electrical and Computer Engineering The University of British Columbia Vancouver, B.C., Canada DE-6 (2/88) 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 increas-ing use of power electronics to enhance process efficiency 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 (fixed or controlled, for line reactance compensation) will increase the potential risk of transient disturbance amplifications 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 offer the means for adequate "power conditioning", to meet the special requirements of electric power quality in a system. To evaluate the promising solutions offered 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; • experimental implementation in MicroTran® (the UBC version of the EMTP), based on SSCPS, of a "simultaneous solution" for: linear and, nonlinear current and voltage dependent sources; independent current and voltage sources, which can also be con-nected between two ungrounded nodes; hard and soft limiters; transfer functions; math-ematical and transcendental FORTRAN functions; special control devices and some digital logic gates; transformation of variables (such as the abc to a/30 transformation i i A B S T R A C T i i i 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-off 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 rectifier, and a three-phase PWM voltage source inverter (VSI)) and evaluation of current and voltage waveforms; • interaction with a Brazilian utility company and industries for the realization and anal-ysis of field measurements of electromagnetic phenomena affecting the quality of power, such as voltage sags and voltage swells; harmonic current and voltage distortions; tran-sients, etc., with determination of causes, consequences and investigation of possible solutions for power quality problems, as for example, the application of Custom. Power Controllers; • synthesis of simulation guidelines for the evaluation of the impact of power electronic devices on the quality of power, based on realistic field measurements and EMTP time and frequency domain simulations. The assessment of electric power quality, with the use of EMTP-based programs, and the evaluation of the technical impact of power electronic devices on the quality of power, can hopefully be performed with the models developed in this Ph.D. thesis project. C o n t e n t s Abstract ii List of Tables vi List of Figures vii Acknowledgements xii Quote xiv 1 Electric Power Quality and Power Electronic Devices: An Overview 1 1.1 Introduction: Better Electricity Qual i ty at "Possibly" Lower Prices? 1 1.1.1 Computer Analysis and Simulation of Electric Power Qual i ty Phenomena 3 1.1.2 Electric Power Quali ty Monitor ing 4 1.1.3 Power Quali ty Standards 6 1.1.4 Custom Power Related Publications 9 1.2 Mot ivat ion for Thesis Research 12 1.3 Contributions of this Research Project 13 2 Simultaneous Solution of Control and Electric Power System Equations (SSCPS) in EMTP-based Programs 15 2.1 Previous Developments on Transient Analysis of Control Systems (TACS) . . 15 2.2 Current and Voltage Dependent Sources in E M T P - b a s e d Programs 18 2.2.1 Compensation Method 18 2.2.2 Dependent Sources 21 2.2.3 Ideal Transformers 32 2.2.4 Independent Sources 33 2.2.5 Newton-Raphson Algor i thm 36 iv C O N T E N T S v 2.2.6 Possible Applications 40 2.3 Development of Control Transfer Functions in EMTP-based Programs . . . . 42 2.4 Development of Limiters for Control Systems in EMTP-based Programs . . . 48 2.5 Development of Intrinsic F O R T R A N Functions in EMTP-based Programs . 54 2.6 Development of Control Devices in EMTP-based Programs 58 3 Power Electronics Modelling in EMTP-based Simulations 64 3.1 Modelling Power Electronics in Electric Power Engineering Applications . . . 65 3.2 Simultaneous Solution for Voltage-Controlled Switches in EMTP-based Pro-grams 74 3.3 Implementation of Nonlinear Diode Model in EMTP-based Programs . . . . 78 3.4 Control Modelling Aspects of Power Electronic Devices 88 4 Evaluation of the Impact of Power Electronic Devices on the Quality of Power 90 4.1 Dynamic Interaction between Power Electronic Devices and Power Systems . 91 4.2 Power Quality Assessment through EMTP-based Programs 104 4.2.1 Induction Furnace Harmonic Study 104 4.2.2 Voltage Sag Analysis with EMTP-based Simulation 120 4.2.3 Welding Industry Voltage Fluctuation Study - A Visual Flicker Case 121 4.3 EMTP-based Simulation Cases with SSCPS 125 4.3.1 Basic Control and Control Devices Simulation Cases 125 4.3.2 Power Electronics Simulation Cases 133 4.4 Synthesis of Simulation Guidelines for Studies with EMTP-based Programs . 150 5 Conclusions and Recommendations for Future Work 153 5.1 Conclusions and Main Contributions 153 5.2 Recommendations for Future Work 156 Bibliography 160 L i s t o f T a b l e s 3.1 Comparison between voltage and current in a diode as a function of its parametric values. 80 4.1 Global harmonic distortion limits for the system voltages recommended in Brazil 110 4.2 Comparison between field measurements and E M T P simulation results for the operating condition with the harmonic passive filters turned O F F 119 4.3 Comparison between field measurements and E M T P simulation results for the operating condition with the harmonic passive filters turned O N 120 v i L i s t o f F i g u r e s 1.1 Typical Design Goals of Power-Conscious Computer Manufacturers. (Source: I E E E Std. 446-1987, "IEEE Recommended Practice for Emergency and Standby Power Systems for Industrial and Commercial Applications.") 8 1.2 C B E M A curve revised by the Information Technology Industry Council (ITIC) 9 1.3 (a) Thyristor in an industrial power converter, (b) Thyristors in a high voltage direct current, (HVDC) System 12 2.1 E M T P and T A C S interface with 1 time step delay 16 2.2 M-phase Thevenin equivalent circuit 19 2.3 Representation of branch equation A; as a voltage source in series with a resistance. . . . 20 2.4 Representation of branch equation k as a current source in parallel with a resistance. . . 21 2.5 Current-controlled voltage source (CCVS) 23 2.6 Current-controlled current source (CCCS) 24 2.7 Voltage-controlled voltage source (VCVS) 25 2.8 Symbol for operational amplifier 27 2.9 Inverting amplifier circuit 29 2.10 Non-inverting amplifier circuit 29 2.11 Adder circuit with operational amplifier 29 2.12 Ideal integrator circuit with operational amplifier 30 2.13 Generalization of inverter amplifier circuit 30 2.14 First-order lag circuit using ideal operational amplifier 31 2.15 Voltage-controlled current source (VCCS) 31 2.16 Ideal transformer 33 2.17 Independent current source 34 2.18 Independent voltage source 35 2.19 Newton-Raphson algorithm experimentally implemented in MicroTran 39 2.20 Circuit with ideal operational amplifier 41 vii LIST OF FIGURES viii 2.21 Simulation results of circuit with ideal operational amplifier (noninverting amplifier circuit). 41 2.22 Transfer function 42 2.23 Observer form block-diagram of transfer function in equation 2.80 44 2.24 Possible computer implementation of the transfer function block-diagram in Fig. 2.23. . . 45 2.25 Block-diagram representation of a first-order transfer function 45 2.26 Observer form block-diagram of first-order transfer function of Fig. 2.25 46 2.27 Possible computer implementation of first-order transfer function of Fig. 2.25 46 2.28 Realistic first-order lag circuit 47 2.29 Time domain simulation of first-order transfer function 47 2.30 First-order transfer function with windup (static) limiter 49 2.31 First-order transfer function with non-windup (dynamic) limiter 49 2.32 Transient response of a first-order transfer function with windup and non-windup limiter. 50 2.33 Soft limits 52 2.34 Zero-order transfer function with soft limits 53 2.35 Time domain response for a sinusoidal excitation input u(t) illustrating the effects of hard and soft limits on the output x(t) 53 2.36 Open loop control system with "supplemental devices SI,52 and 53" 54 2.37 Nonlinear control block-diagram with a sinusoidal intrinsic F O R T R A N function 55 2.38 Circuit implementation for the simultaneous solution of a sinusoidal F O R T R A N function. 55 2.39 Transport delay control device 58 2.40 Circuit implementation for the simultaneous solution of a transport delay control device. . 59 2.41 Transient simulation of a transport delay control device 60 2.42 Transient simulation of a pulse delay control device 61 2.43 Pulse delay control device with arbitrary input signal 62 2.44 Logic gate " N O T " 62 2.45 Circuit implementation of a logic gate " N O T " for simultaneous solution 63 3.1 Power semiconductor devices 66 3.2 Voltage-controlled switch in EMTP-based programs 70 3.3 Test cases for transient simulation of voltage-controlled, bipolar in voltage and bidirectional current flowing switch, thyristor and G T O 71 3.4 Simulation of a voltage-controlled bidirectional current flowing switch 72 3.5 Simulation of a simplified model for thyristors 72 3.6 Simulation of a simplified model for G T O ' s 73 LIST OF FIGURES ix 3.7 Circuit with "simultaneous solution" of a voltage-controlled switch 76 3.8 Simulation with simultaneous solution of a voltage-controlled switch 76 3.9 One time step delay in EMTP-based switches 77 3.10 Diode symbol 78 3.11 V - I diode characteristic and network Thevenin equivalent circuit equation 81 3.12 Circuit implementation for the simultaneous solution of a nonlinear diode model 82 3.13 V - I diode characteristic and different network Thevenin equivalents 82 3.14 Dc-dc converter 85 3.15 Half-wave rectifier with freewheeling diode 85 3.16 Electric circuit with a nonlinear diode model 86 3.17 Transient simulation of a nonlinear diode model in an EMTP-based program. . . . . . . 86 3.18 Detail of the transient simulation of a nonlinear diode model in an EMTP-based program. 87 3.19 V- I nonlinear characteristic of the diode resulting from the E M T P simulation 87 4.1 Circuit with a single-phase diode-bridge rectifier 93 4.2 Current drawn from the source by a single-phase diode-bridge rectifier 94 4.3 Harmonic amplitude spectrum of the current drawn from the source by a single-phase diode-bridge rectifier 95 4.4 Current through and voltage across the total inductance, and voltage waveform distortion at the point of common coupling (PCC) 96 4.5 Harmonic amplitude spectrum of the voltage waveform distortion at the point of common coupling (PCC) 97 4.6 Four-wire, three-phase system with "balanced" single-phase diode-bridge rectifiers 97 4.7 Current flowing through the neutral conductor 98 4.8 Harmonic amplitude spectrum of the current flowing through the neutral conductor. . . . 99 4.9 Voltage waveshape measured at the outlet of the Power Electronics Laboratory of the Department of Electrical and Computer Engineering at U B C , Vancouver, B . C . , Canada. . 100 4.10 Measured voltage waveshape, its fundamental component and its harmonic distortion. . . 101 4.11 Harmonic amplitude spectrum of the outlet waveshape voltage 102 4.12 Phase-angle of the harmonic components of the outlet waveshape voltage 103 4.13 (a) Metal melting by an induction furnace, (b) Induction furnace operation 106 4.14 Current measurements in a distribution feeder supplying induction furnaces at the time of maximum voltage distortion 107 4.15 (a) Phase " A " current measured with harmonic passive filters turned off. (b) Phase-to-phase " A - B " voltage measured with harmonic passive filters turned off 108 LIST OF FIGURES x 4.16 (a) Phase "A" current measured with harmonic passive filters turned on. (b) Phase-to-phase "A-B" voltage measured with harmonic passive filters turned on 109 4.17 THD harmonic trend, with harmonic passive filters turned off from 12:00 midnight to 06:00am. 109 4.18 THD harmonic trend, with harmonic passive filters turned on all the time 110 4.19 Distribution substation 113 4.20 Current-source, parallel-resonant inverter for induction heating 114 4.21 Amplitude of the positive sequence system impedance at the PCC with harmonic filters. . 114 4.22 Phase angle of the positive sequence system impedance at the PCC with harmonic filters. 115 4.23 (a) Phase "A" current simulated with harmonic passive filters turned off. (b) Phase-to-phase "A-B" voltage simulated with harmonic passive filters turned off 116 4.24 (a) Phase "A" current measured with harmonic passive filters turned off. (b) Phase-to-phase "A-B" voltage measured with harmonic passive filters turned off 116 4.25 (a) Phase "A" current simulated with harmonic passive filters turned on. (b) Phase-to-phase "A-B" voltage simulated with harmonic passive filters turned on 117 4.26 (a) Phase "A" current measured with harmonic passive filters turned on. (b) Phase-to-phase "A-B" voltage measured with harmonic passive filters turned on 117 4.27 Instantaneous ideal compensation current to be "injected" by a shunt active filter 118 4.28 Voltage sag measurements (%RMS versus time duration) with an overlay of the CBEMA curve. For time durations less than 1 cycle the equipment seems to measure peak values. . 122 4.29 (a) Phase-to-phase "A-B" measured voltage sag. (b) Phase-to-phase "A-B" simulated volt-age sag 123 4.30 Instantaneous voltage fluctuations causing light flickering effect 124 4.31 Modulated voltage and respective amplitude frequency spectrum 124 4.32 Control block diagram of a second order differential equation with poles on the imaginary axis of the complex plane 126 4.33 Solution of system with bounded resonance oscillations 126 4.34 Introduction of a one time step delay in the control block diagram 127 4.35 Solution of system with unstable resonance oscillations caused by the introduction of one time step delay 127 4.36 Classical linearized "swing equation", used in power system small-signal stability studies of a single machine connected to an infinite bus 130 4.37 Simulation results of the synchronous machine rotor angle deviation, in the presence of a positive damping torque coefficient 130 4.38 Simulation results of the synchronous machine rotor angle deviation, in the presence of negative damping torque coefficient 131 4.39 Canonical second order transfer function representation of the single-machine infinite bus system 131 LIST OF FIGURES ' xi 4.40 Circuit for the dynamic control of the firing angle ("a") of a thyristor 135 4.41 Voltages and currents in a circuit with dynamic control of the firing angle of a thyristor. . 136 4.42 Circuit for the dynamic control of the firing angles of a three-phase six-pulse thyristor-bridge rectifier 138 4.43 Voltages and currents with dynamic control of the firing angles of a three-phase six-pulse thyristor-bridge rectifier 139 4.44 Dynamic control of the firing angles of a three-phase six-pulse thyristor-bridge rectifier. . 140 4.45 Dynamic voltage control signals at the output of the proportional-integral (PI) and the limiter control blocks 141 4.46 Circuit for the dynamic control of three-phase P W M voltage source inverter (VSI). . . . 144 4.47 Phase " A " modulation and triangular carrier waveforms for generation of gating signals through sinusoidal pulse width modulation ( P W M ) 145 4.48 Node voltage "VSA" generated by a three-phase P W M voltage source inverter (VSI). . . . 146 4.49 Voltage across the load "VSA-NEUTR" and current supplied to the load by a three-phase P W M voltage source inverter (VSI) 147 4.50 Load currents supplied by a three-phase P W M voltage source inverter (VSI) 148 4.51 Line-to-line voltage generated by a three-phase P W M voltage source inverter (VSI). . . . 149 Acknowledgements I would like to thank God for the gift of learning. My most sincere thanks to my parents, Dorival and Isolina, and to all my relatives for their unconditional love and support. To my wife, Luciana, and my daughters, Alexa and Aline, my love and my heartfelt thanks for their strong participation in this life project altogether. I dedicate a very special note of thanks to our special friends Wany, Fernando, Fulvia, Alexandre, Martha, and Richard for their careful and kindness personal assistance. I owe a tremendous debt of gratitude to Dr. Hermann W. Dommel, my Ph.D. thesis supervisor, for all his personal and professional encouragement, share of wisdom and support for the development of this thesis. (The responsibility for any remaining errors is solely mine.) I also thank Dr. Dommel for the honor and opportunities of have being his teaching assistant. I also thank Dr. William G. Dunford for kindly accepting to be my Ph.D. thesis co-supervisor, with Dr. Dommel becoming a Professor Emeritus at U B C . I have also learned with Dr. Jose R. Marti, who has excellent teaching skills. Professor Sandoval Carneiro Jr. from the Federal University of Rio de Janeiro (UFRJ) Brazil, has gently been very supportive, right from the start of this Ph.D. program in Canada. I most specially appreciate the help, acceptance and advice of many individuals without whom this opportunity would never have become fruitful. Professors, staff members, 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 scientific and personal growth. I also thank my former Brazilian professors and colleagues at the State University of Campinas (UNICAMP) , and at the Federal School of Engineering of Itajuba (EFEI), for building and enhancing the foundation of my knowledge in science and engineering. I would like to sincerely thank the Fundagao Coordenagao de Aperfeigoamento de Pessoal de Nivel Superior (CAPES) , Brasilia - Brazil, for the financial support to this Ph.D. thesis project. Without it, my dream would never come true. I also thank the Brazilian utility company E L E K T R O - Eletricidade e Servigos S.A., with a special reference to Francisco Alfredo Fernandes, for providing opportunities for a practical interaction in power quality analysis, through a professional cooperation with the engineer Ernesto Alberto Mertens Jr.. I acknowledge and thank students, professors and staff at the E T E Prof. Armando Bayeux da Silva, a technical high school of the C E E T E P S -Centro Estadual de Educagao Tecnologica Paula Souza, Sao Paulo, Brazil, for all the teaching experiences I was able to conduct, which enriched my communication and leadership skills. Last but not least, I thank and acknowledge the contributions of many people, not mentioned, not forgotten, who certainly have had an impact and influence on my living and studying at U B C , Vancouver, B.C. , Canada, since August 24, 1997. Vancouver, B .C. , Canada Benedito Donizeti Bonatto October 09, 2001. xii C h a p t e r 1 E l e c t r i c P o w e r Q u a l i t y a n d P o w e r E l e c t r o n i c D e v i c e s : A n O v e r v i e w H E P U R P O S E of this research project was to develop reasonably accurate models for 1 control systems and power electronic devices to evaluate their impact on the quality of power. These models and methods were developed for implementation in the Electromag-netic Transients Program ( E M T P ) [1], [2], or in similar programs. A "circuit approach" is used for the simultaneous solution of the control and electric power system equations, thus eliminating any time step delay in the digital time domain simulation. Such time step delays can cause inaccuracies or numerical instabilities. The advantages of the circuit approach is its "generality and flexibility" for modelling multi-terminal linear and nonlinear control de-vices, which are needed in the analysis of electromagnetic phenomena affecting the quality of power. This chapter presents an introduction to power quality problems and their relation with power electronics, followed by a description of the motivations for this P h . D . thesis research and its relevant contributions. 1.1 Introduction: Better Electricity Quality at "Possi-bly" Lower Prices? The demand of electricity customers for increased quality of power, at possibly reduced prices, is forcing governments, regulatory agencies, ut i l i ty companies, and equipment manu-facturers to develop new structures for the electricity market. Deregulation of the electricity 1 1.1. Introduction: Better Electricity Quali ty at "Possibly" Lower Prices? 2 industry has been proposed as a solution to make the present ut i l i ty companies more com-petitive in offering better services and better quality at lower prices to their customers. However, as in any business, this may require some investments in the infrastructure of the power system, to cope wi th the new demands of the modern types of loads (power electron-ics and microcomputer based). In this scenario, tradit ional economic analysis, such as pay back return or rate of interest, might show that these investments are only feasible with concurrent increases in electricity tariffs. The paradox of more quality for less money sti l l remains a topic for discussion in forums such as government regulatory agencies. W i t h the growing uti l ization of automation and control based on the use of microproces-sors, of power electronic devices, and of modern manufacturing techniques, industries have been able to produce goods faster and with increasing quality. However, with such modern-ization new issues have emerged regarding the quality of electricity. Sensitive loads tend to shut down if there are small variations in the network voltage. Also, harmonic distortions caused by nonlinear loads may result in wrong operation or may increase the losses in power system components. Another problem is capacitor switching in the ut i l i ty system, which may cause problems for adjustable speed drives (ASD's ) , which are used more and more by industry. A l l these problems point out that more attention must be paid to power qual-ity problems. Economic losses expressed in terms of interrupted production, of damage to equipment, and of time delays in the processing of goods and the consequent negative impact on customers have caused a rising number of complaints about power quality problems in many electric ut i l i ty companies. Est imat ing the cost of poor power quality is a difficult task. Nevertheless, the annual approximate value would be in the order of hundreds of mil l ion of dollars in damage. As an example, the cost per year to U . S. A . industry in lost time and revenue due to power related problems were estimated in 1993 as US$26,000,000,000 [3]. The costs tend to grow as the sensitivity and use of microprocessor-based devices tend to increase. The Electric Power Research Institute (EPRI ) stated that in the year 2000, 60% to 70% of total ut i l i ty power generated within the U . S. A . would be controlled by power electronics, compared to 30% in 1995. Power electronic devices, however, are also able to "guarantee" a certain expected level of electricity quality to a sensitive or special load, and such devices exist today. Flexible A C 1.1. Introduction: Better Electricity Quality at "Possibly" Lower Prices? 3 Transmission Systems (FACTS) technology, Custom Power Controllers, active filters, among other power electronics applications, offer promising solutions for improving the quality of power in transmission and distribution systems. The introduction of more and more power electronic devices into the network will create issues of compatibility of operation not only in steady state, but also under transient con-ditions. New models for innovative equipment, as well as new philosophies for their control and operation, will then be required. It is not enough to evaluate the electric quality condi-tions only at the interface of power electronic systems with the electric power system. The propagation of electromagnetic phenomena into the industrial or utility network must be evaluated as well [4]. Therefore, software packages such as the ElectroMagnetic Transients Program (EMTP) , have become important and necessary tools to analyze the impact of large power electronic devices on the quality of electric power. The aim of this research project was the development of new EMTP-based models for control and power electronic devices, thus allowing the accurate evaluation of the impact of high power electronics on the quality of power. As part of the project, field tests were conducted in cooperation with a Brazilian utility company, to provide realistic power quality data measurements. 1.1.1 Computer Analysis and Simulation of Electric Power Qual-ity Phenomena Digital computer simulation of electromagnetic transients in single- and multiphase net-works is well established [1]. Since the publication of [1] in 1969, significant improvements have been made in models for electrical power system components such as transmission lines [5], transformers, turbine-generators and cables [6]. Of particular interest for this research project are the necessary advances in the simulation of power electronic devices within power systems. Although steady-state solutions at fundamental and harmonic frequencies have been pro-posed to analyze power quality problems, the complexity of periodic switching in power elec-tronic devices can only be studied thoroughly through time-domain simulations, e.g., with EMTP-based programs [4], [7], [8], [9], [10], [11], [12], [13]. 1.1. Introduction: Better Electricity Quali ty at "Possibly" Lower Prices? 4 E M T P - t y p e simulation is particularly useful for the analysis of the dynamic interaction of distributed Custom Power Controllers in a power system. Despite international demon-stration projects for some Custom Power Controllers already with some years of operating experience, how these new devices wi l l interact wi thin each particular power system is still an open question. The performance obtained from some prototypes, or from simulations with simplified models, may not, be sufficient for real applications, where dangerous resonance and other unforeseen problems may occur. These concerns make more realistic electromag-netic transient program based simulations important when detailed models of Custom Power Controllers become available. This could avoid more expensive corrective actions after in-stallation. Measurements and simulations also become necessary for performance evaluations under different network and load conditions. A l l those facts have encouraged the development of new practical models, methods and guidelines for the appropriate use of E M T P - b a s e d programs as potential tools for power quality studies. Nevertheless, good engineering judgment in setting up the power quality problem and representing the physical phenomena with appropriate models, does st i l l represent the major challenge, despite the impressive accuracy obtained wi th the models available today. 1.1.2 Electric Power Quality Monitoring In recent years, tremendous improvements have also been made in digital measuring in-struments, because of the use of microprocessors and digital signal processing techniques [14], [15]. This made it possible to conduct power quality surveys in many countries around the world. The Electric Power Research Institute (EPRI ) commissioned an extensive survey of distribution system power quality in the U . S. A . [16]. In Canada, the Canadian Elec-tricity Association ( C E A ) developed a guideline for the power quality that ut i l i ty customers experience, in a three-year project [17]. A quantitative measure of the deterioration of the ideal sinusoidal waveform from the growing uti l ization of power electronic devices resulted from these surveys. In many of the real-world problems, momentary voltage variations have been the main cause for shutting down microprocessor controlled industrial processes. The 1.1. Introduction: Better Electricity Quali ty at "Possibly" Lower Prices? 5 diversity of problems to cope with, and their complex interaction, has created a need for more research on power quality issues [18]. Moreover, with the present deregulation process in the electricity industry, power quality has become a key factor for utilities and customers in a competitive market. The experience in monitoring power quality phenomena has increased in the latest years, as reported by the related literature [15], [19], [20], [21], [22], [23], [24], [25], [26], [27], Wel l known power quality problems have been summarized and at the same time new problems have been discussed in [28], [29], [30], [31], [32], [33], [34], [35], [3], [36], [37]. The need for more detailed information on power disturbances, and the search for new techniques to process the amount of available measured data, have motivated research into applications of modern theories in the power quality discussed in [38], [39], [40], [41], [42]. According to the author's experience as a "power quality engineer" in a Brazi l ian uti l i ty company, there are many different reasons why electricity customers become dissatisfied with the quality of the electric power delivered by utilities, and why they complain. This happen partly because there may indeed be technical problems. Partly, the motivation comes from the need to reduce electricity costs in the industrial production process, looking for better tariffs and better contracts, as everybody tries to survive in a competitive and aggressive business environment. O n the technical side, the most common power quality problems are caused by a fatal combination of sensitive electronic-based loads and a high incidence of voltage sag phenom-ena [43], [44], [45], [46]. Most of these voltage sags are due to faults in transmission and distribution systems, caused by lightning phenomena, which cannot be easily avoided or minimized. It is also common to find poor voltage regulation wi thin the industry electric system [45], [46]. Th i s aggravates the impact of voltage sags, causing frequent process mal-function or interruption with financial losses, which are rarely presented explicitly by the industry personnel, unless any kind of financial compensation is legally required. There is a wide range of alternatives for possible solutions to technical problems in the quality of the electric power supply. Usually, the immediate most cost-effective measure is to minimize the cause or effects of the problem close to its origin, depending on the type of electromagnetic 1.1. Introduction: Better Electricity Quali ty at "Possibly" Lower Prices? 6 phenomena involved. Typically, voltage sag problems can be minimized by proper adjust-ments in the sensitivities of the load or load control, whenever this is technically feasible. However, in some cases compensation through the use of power electronic devices might be a promising alternative, either for an individual sensitive load or for an entire industrial pro-cess. Usually, the utilities comply with the standards of supply set by the regulatory agency, but the customer process is much more sensitive and some kind of electronic compensation would be necessary. The potential conflict is sometimes created when a possible technical solution requires high financial investments. A cost versus benefit analysis usually leads to a cheap compromise solution; in the absence of clear regulations one needs "to live with the problem!" 1.1.3 Power Quality Standards Power quality has become an important issue because of the increasing use of power electronic devices. The Institute of Electrical and Electronics Engineers, Inc. ( IEEE) has therefore developed standards to address power quality problems, which are briefly discussed here. The problems related to the quality of electricity are not new, since there was never an ideal sinusoidal waveshape, with frequency and voltage exactly at their rated values. How-ever, with the changes in the type and sensitivity of the loads in recent years, harmonic current and voltage distortions, short- and long-duration voltage variations, impulse and os-cillatory transients, voltage fluctuations (causing visual flicker), power frequency deviations, voltage unbalance among the phases in a three-phase system, and other electromagnetic phenomena are increasingly causing power quality problems. I E E E Std. 1159-95 [47] defines and characterizes electromagnetic phenomena which may cause power quality problems. It also provides recommended practices for monitoring electric power quality. Most ut i l i ty regulations dealing wi th harmonics are based on I E E E Std 519-1992, [48], [49], [50]. This standard presents recommended practices and requirements for harmonic control in electric power systems. It addresses most of the issues of harmonic generation 1.1. Introduction: Better Electricity Quality at "Possibly" Lower Prices? 7 by power electronic converters, arc furnaces, static V A R compensators and power electronic controlled drives. It also discusses: • system response characteristics; • effects of harmonics; • reactive power compensation and control; • harmonic analysis methods; • harmonic measurements; • recommended practices and harmonic limits for individual customers; • recommended practices and harmonic limits for utilities; • methodology for evaluating new harmonic sources. In the later task, time-domain simulation can be particularly useful to predict equipment and power system behaviour. It thus can help engineers to provide some answers in detecting harmonic related or other power quality problems. The IEEE Std. 519-1992 [48] is currently been revised to account for interharmonics in power systems and the possible application of probabilistic approaches in harmonics evaluation. IEEE Std. 446-1987 [51] covers the recommended practice for emergency and standby power systems for industrial and commercial applications. A computer "voltage tolerance en-velope", shown in Fig. 1.1, also known as the C B E M A curve (Computer Business Equipment Manufacturing Association curve) is presented in this standard. The Information Technology Industry Council (ITIC) revised the C B E M A curve, which is presented in Fig. 1.2. It shows that computer- and power electronics-based loads, properly designed by the manufacturers, should be able to withstand a complete interruption of voltage supply for up to 20ms, a voltage sag of 30 percent for 0.5s, 20 percent for 10s or 10 percent in steady state. It also defines the upper limits in the input voltage that should be tolerated. The C B E M A (ITIC) curve has been widely used as an important "reference" 1.1. Introduction: Better Electricity Quality at "Possibly" Lower Prices? 8 Figure 1.1: Typical Design Goals of Power-Conscious Computer Manufacturers. (Source: IEEE Std. 446-1987, "IEEE Recommended Practice for Emergency and Standby Power Systems for Industrial and Commercial Applications.") for the susceptibility level of computer- and power electronics-based loads. However, due to the great variety of products and processes, and their response to transient variations in the supply voltage, there are cases where the load sensitivity is much more strict than the C B E M A (ITIC) curve, which has to be determined then case-by-case for an adequate power quality assessment and proposal of solutions. IEEE Std 1100-1992 [52] presents the recommended practice for powering and grounding sensitive electronic equipment. It addresses the multidisciplinary area of power quality, giving practical guidelines on load and source compatibility concerns. Voltage fluctuations causing visual flicker are being studied by the Task Force IEEE P1453 on Light Flicker, which is considering the adoption of existing standards and practices 1.1. Introduction: Better Electricity Quality at "Possibly" Lower Prices? 9 Figure 1.2: C B E M A 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 flexible A C transmission systems (FACTS tech-nology) have been used for some time to extend power transfer capability, to improve power system stability, and for other reasons. Dr. Narain G. Hingorani introduced the acronym FACTS (Flexible A C Transmission System) for high power electronics applications in trans-mission systems [53], [54], [55]. H V D C , static Var compensator (SVC), thyristor controlled series compensations (TCSC), static synchronous compensator (STATCOM), static syn-chronous series compensation (SSSC) and unified power flow controller (UPFC) are exam-ples of the so called FACTS devices. Reference [56] provides an annotated bibliography of H V D C and FACTS devices. It also includes a list of FACTS installations, with data on manufacturers, utility companies, countries, etc. It shows that, despite the high costs of these high power electronic devices, they are gaining in acceptability around the world. The term "Custom Power" was also introduced by Hingorani, to represent power elec-tronics applications designed to mitigate power quality problems in industrial and distribu-tion systems [57], [58], [59]. The distribution static condenser (D-STATCOM), the voltage sag compensator (also known as D V R - dynamic voltage restorer), the solid-state breaker (SSB), the solid-state transfer switch (SSTS), among others, are examples of such Custom Power Controllers. Various manufacturers have proposed shunt, series, or shunt/series dy-namic compensation schemes, with different acronyms, as solutions to specific power quality problems. "The D - S T A T C O M , although based on the S T A T C O M , has a wider range of applications. In fact, the D - S T A T C O M can be designed for reactive power control, or for voltage control of the fundamental frequency, but it may also include higher frequencies as in shunt active power filters." The integration of series- and shunt active filters, referred to as unified power quality conditioner (UPQC) [60], [61], is promising to be the definite solution for the majority of power quality problems. "However, its high cost may make it useful only in some special cases. On the other hand, the shunt or series devices such as the D - S T A T C O M or the voltage sag compensator will probably play a significant role in future distribution systems" 1 . The ongoing deregulation process in many countries is also fostering competition in the 'From personal communication with Dr.-Ing. Mauricio Aredes, C O P P E / U F R J , Rio de Janeiro. R J , Brazil . 1.1. Introduction: Better Electricity Quality at "Possibly" Lower Prices? 11 electric power industry, which accelerates the application of new technologies in the trans-mission and distribution system. For example, there are applications being developed for superconducting magnetic energy storage devices (SMES) for low voltage distribution sys-tems, which will provide voltage support for a few seconds to sensitive processing equipment during times of voltage sags. Case studies with practical applications of Custom Power Controllers can also be down-loaded directly from the web sites of some manufacturers, as for example: • http://www.siemenstd.com/prods/FPQD/dvr.html - case studies for the voltage sag compensator D V R (dynamic voltage restorer of Siemens); • http://www.siemenstd.com/prods/FPQD/cp.html - general power quality information on D V R , D - S T A T C O M , Solid State Breaker, Transfer Switch and Premium Power Park; • http://sac.sandc.com/products/purewave/ups_pubs.asp - for S & C UPS products; • http://www.softswitching.com - for SoftSwitching Technologies products; and many others. Figs. 1.3 (a) and (b) present semiconductor power devices, thyristors, used in an industrial power electronic converter and in a H V D C system, respectively. Thyristors are considered the "backbone" of the high power electronics revolution. Other types of semiconductors being used are the gate turn-off thyristor (GTO), the MOS controlled thyristor (MCT), the static induction thyristor (SITh), and the insulated gate bipolar transistor (IGBT). The so considered, in 1997, state of the art of these devices can be found in reference [62], along with a description of the main characteristics of H V D C , static Var compensator (SVC), thyristor controlled series compensation (TCSC), static synchronous compensator (STATCOM), static synchronous series compensation (SSSC) and unified power flow controller (UPFC). The benefits of the application of FACTS technology in a power system depend on the reliability of the specific FACTS device, which in turn depends on the reliability of the semiconductor 1.3. Contributions of this Research Project 13 will rise in importance and urgency. Therefore, as part of this project field tests were developed in a Brazilian electric utility company, where realistic power quality cases were analyzed and simulations were performed; • The diversity of power quality phenomena requires an interdisciplinary approach and specialized engineering skills. With opportunities available for interaction with other researchers at The University of British Columbia (UBC), appropriate courses were attended, especially in the power electronics area, which was helpful for the under-standing and development of new models for implementation in MicroTran, the U B C version of the E M T P ; • The opportunity to conduct practical field tests in cooperation with an electric utility company was a valuable experience, and necessary for the validation of digital computer models. 1.3 Contributions of this Research Project This Ph.D. thesis offers new models for the digital computer simulation of control and power electronic devices. These models were developed for implementation in EMTP-based programs or in similar programs. An innovative "circuit approach" was developed for the simultaneous solution of control and power systems equations, as an alternative to the ap-proach of A . E. A . Araujo [63] developed in 1993. The main differences and important advantages are summarized as follows: • With the addition of ideal operational amplifiers, transfer functions can be imple-mented with a "circuit approach", where the circuit elements R, L, C are solved by the main code of the E M T P . If integration methods are changed in the E M T P , for example from trapezoidal rule to backward Euler as done in some versions at instants of discontinuities with the C D A technique, no extra coding is needed. Operational amplifiers are not affected by integration rule changes. Moreover, if ideal operational amplifiers are implemented in steady-state solution, the frequency response of linear 1.3. Contributions of this Research Project 14 control systems could be easily calculated in E M T P - b a s e d programs by just using the frequency scan option. • A . E . A . Arai i jo [63] uses F O R T R A N - l i k e statements for control functions such as Y = COS(X) in the input, which are then handled with a F O R T R A N interpreter. In this thesis, this function and similar functions are pre-defined control block types. • The "multi-terminal voltage-controlled voltage source concept" implemented in this thesis wi th the compensation method and the Newton-Raphson iterative algorithm is "general and flexible", thus providing an easy E M T P - b a s e d modelling of any linear or nonlinear control device. This is very useful for the dynamic analysis of novel power electronic controllers, such as distributed F A C T S and Custom Power Controllers in transmission and distribution power systems. This P h . D . thesis is organized as follows: Chapter 2 presents the simultaneous solution method for control and electric power system equations (SSCPS) in E M T P - b a s e d programs. Chapter 3 discusses the developments made for power electronics models in E M T P - b a s e d simulations. Chapter 4 presents simulation cases of power quality assessment with the use of the existing features of MicroTran, the U B C version of the E M T P . S S C P S simulation cases with the new models of Chapter 2 and the developments for the dynamic control of power semiconductors presented in Chapter 3 are illustrated in practical power electronics controllers. Simulation guidelines for the evaluation of the impact of power electronic devices on the quality of power are summarized in Chapter 4. Finally, Chapter 5 presents the main conclusions and contributions made in this P h . D . thesis, and also points out the author's recommendations for future work. C h a p t e r 2 S i m u l t a n e o u s S o l u t i o n o f C o n t r o l a n d E l e c t r i c P o w e r S y s t e m E q u a t i o n s ( S S C P S ) i n E M T P - b a s e d P r o g r a m s 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 E M T P (acronym for "Electromagnetic Transients Program"). The general philosophy of the solution method adopted at that time required a one-time-step delay at the interface between TACS and the electric network solution l . This non-simultaneous approach was probably used because it was easier to write a code separated from the main program, with a simple interface. The main program passed information to the TACS program, which then returned information to the main program for use one time step later, as illustrated in Fig. 2.1. Moreover, control system equation matrices in TACS are usually unsymmetric, whereas the network elements in the E M T P result in symmetric matrices. By separating the solution into two parts, the code for symmetric matrices in the E M T P could be maintained. The solution in two parts, with a time delay of one At between them, was an expedient way to implement control system equations, but it proved to be the cause of critical numerical 1 Many other software programs, such as P S I M [65], also require a one-time-step delay between the solution of control and power systems equations, which makes the solution method non-simultaneous. 15 2.1. Previous Developments on Transient Analysis of Control Systems (TACS) 16 Electric Network Solution (EMTP ) Time Delay lAt 1 Control System Solution g (TACS) Figure 2.1: EMTP and TACS interface with 1 time step delay. instabilities and inaccuracies in some cases in the time domain simulation of electric and power electronic system transients [66], [67], [68], [63]. In cases where the E M T P and TACS elements form a closed loop (or feedback system according to control theory), the effect of the interface delay cannot always be eliminated by using a small step size At , as stated in [69]. Besides the time delay between TACS and E M T P , even more time-step delays were introduced by the internal solution algorithm of TACS, in order to deal with nonlinearities in feedback control loops. The TACS internal solution is therefore non-simultaneous for some control cases, and also sequential for its implemented devices. Improvements have been made through the years in the TACS subroutine of some versions of the E M T P , such as better ordering of its variables to minimize the number of delays inside TACS [70], using the compensation method to eliminate the one-time-step delay in the E M T P - T A C S interface [71], development of a new TACS program " M O D E L S " [72] and its possible applications for simultaneous solution of power electronics systems equations [73]. In 1993 A . E . A . Araiijo proposed a simultaneous solution of both sets of equations, electric network equations and control systems equations, as a way to eliminate the one-time-step 2.1. Previous Developments on Transient Analysis of Control Systems (TACS) 17 delay problem at the interface, as well as the internal control delays [67], [68], [63]. The aug-mented matrix with the control equations becomes unsymmetric due to the structure of the control equations. Most of the equations of both the electric network and the control systems are usually linear, while some are nonlinear. A proper partition of the system of equations would allow the solution to be separated into two subsystems, one linear and another non-linear. A . E . A . Araiijo chose to solve the system of linear equations inside the E M T P , and the system of nonlinear equations (including nonlinearities from the electric network and from the control system) with the compensation method in an iterative Newton-Raphson algorithm as in [74]. The control equations, both linear and nonlinear, were developed inside the subroutine " C O N N E C " , which is a user-defined subroutine in the MicroTran version of the E M T P of the University of British Columbia. "Similarly to TACS, the trapezoidal rule of integration was used to numerically integrate the first-order differential equations inside C O N N E C , for example, in the implementation of transfer functions. The code was written to prove the ideas, but as far as the author knows, was not implemented in a production version of the E M T P . " In this research project, the simultaneous solution of the electric network and control equations in EMTP-based programs is achieved with a "circuit implementation" of the con-trol system. With this novel approach for EMTP-based programs, elements of the control circuit which already exist in the E M T P , such as resistances and capacitances, are solved by the E M T P proper, while elements missing inside the E M T P , such as ideal operational am-plifiers 2 and current and voltage dependent sources, are solved in the subroutine C O N N E C with the compensation method. This circuit approach is an alternative to the mathematical representation adopted by Araiijo, and gives some important advantages, such as "generality and flexibility" 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, different types of limiters, as well as intrinsic F O R T R A N functions and some special control devices, as explained in the following sections. "Among the added elements, the de-2The author acknowledges the help of Mr. Jesus Calvifio-Fraga for indicating in 1998 in his technical report for a graduate course, the need for modeling operational amplifiers in MicroTran [75]. 2.2. Current, and Voltage Dependent Sources in E M T P - b a s e d Programs 18 pendent, sources are the most, important, ones for control system modelling." The F O R T R A N code for the added elements in subroutine C O N N E C has approximately 5,000 lines of code, compared to 15,000 lines of code in the main part of the MicroTran version of the E M T P . 2.2 Current and Voltage Dependent Sources in E M T P -based Programs Since the publication of [1] describing the first version of the E M T P , many others have contributed to the development, of models as documented in [2] and elsewhere. As far as the author knows, dependent sources of all possible types have not been implemented in any E M T P - b a s e d program. Dependent, sources expand the capabilities of E M T P - b a s e d programs considerably for modelling many electric and electronic circuits and devices. W i t h a voltage-controlled voltage source, for example, it becomes easy to simulate operational amplifiers. These can then be used to set up control circuits wi th analog-computer block-diagrams. A s long as the equations of the dependent sources are linear, they could be added directly to the network equations used in E M T P - b a s e d programs (with the modified nodal analysis ( M N A ) presented in [76] and [77], but the matrix would then become unsymmetric and a linear equation solver for unsymmetric matrices would have to be used. Another alternative discussed here in more detail is based on the compensation method, which can also handle nonlinear effects wi th a Newton-Raphson algorithm. Nonlinear effects arise wi th the inclusion of saturation or l imits in the dependent sources. The main motivation for the use of the compensation method is its "generality and flexibility" in modelling linear and nonlinear devices in E M T P - b a s e d programs. This section provides then the fundamental equations for the implementation of dependent sources in E M T P - b a s e d programs, as well as of independent sources, which can also be connected between two ungrounded nodes. 2.2.1 Compensation Method The compensation method has long been used in E M T P - b a s e d programs for solving the equations of nonlinear elements with the Newton-Raphson iterative method [74], If the nonlinear elements are not too numerous, this approach confines the iterations to a 2.2. Current and Voltage Dependent Sources in E M T P - b a s e d Programs 19 relatively small system of equations, compared to the nodal equations for the entire system. This approach is used here for solving the equations of dependent sources as a special case of nonlinear elements. Without limiters, the equations are linear, but wi th an unsymmetric matrix. When there are M nonlinear elements in a circuit, the following system of equations 2.1 to 2.6, allows the simultaneous solution of the nonlinear equations with the rest of the linear network [2],[78], which is then represented by its M-phase Thevenin equivalent circuit, as illustrated in F i g . 2.2: [ VOPEN 1 6666 I rTHEV 1 [ ' ] [V] 0 - A / V V - W v -A A A r - W V M S(M) c x c X vs(2)\ 6 K s(I) Figure 2.2: M-phase Thevenin equivalent, circuit. vopen} + [tthev] • [i] + [v] = 0 (2.1) where: [vopen] vOPENI VOPEN2 VQPENM (2.2) 2.2. Current and Voltage Dependent Sources in EMTP-based Programs 20 Vthev] — rMl ?M2 T2M TMM (2.3) t2 (2.4) V2 vM Equations 2.6 are the branch equations of the nonlinear elements: vk = fk ( M , [?;], t, etc..) k = l,...M (2.5) (2-6) If the branch equations in 2.6 are linear, as in the case of dependent sources, they can be represented in the form of a voltage source behind an impedance, as illustrated in Fig. 2.3, or in the form of a current source in parallel with an impedance, as shown in Fig. 2.4. It is assumed here that the branch impedances are not coupled, and that they are resistive (Rk)-For other types of impedances, the equations would have to be modified. Figure 2.3: Representation of branch equation k as a voltage source in series with a resistance. 2.2. Current and Voltage Dependent Sources in E M T P - b a s e d Programs 21 ^source(k) ^sourcefk) F i g u r e 2.4: Representation of branch equation A; as a current source in parallel with a resistance. After the two systems of equations 2.1 and 2.6 have been solved in subroutine C O N N E C , the currents [i] of 2.4 are returned to the main program, which adds the effect of the M non-linear branches to the previously calculated open-circuit solution for all nodes with unknown voltages, [e] = [e0pEN] ~ [zt] (2.7) where: [e] is a column vector with the final solution for the N node voltages: [zopen] is a column vector with the previously calculated open circuit solution for all the N nodes with unknown voltages; [zr] is a rectangular matrix with N rows and M columns (N = number of nodes with unknown voltages and M = number of branches solved wi th the compensation method) 3 ; [i] — column vector wi th the M compensating branch currents. 2.2.2 Dependent Sources This section presents the necessary equations for implementing current and voltage de-pendent sources in E M T P - b a s e d programs by using the compensation method. The following important assumptions are made: 3 For further details about the compensation method, and the calculation of matrix [zt], please, see reference [74]. 2.2. 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, and also where the controlling current or controlling voltage is to be measured. In cases where this calculation fails, the connection of large resistors in parallel may make a Thevenin equivalent circuit possible. • Proper precautions are taken to handle extremely large numbers and zero values. The following models are derived: Current-Controlled Voltage Source (CCVS), Current-Controlled Current Source (CCCS), Voltage-Controlled Voltage Source (VCVS) and Voltage-Controlled Current Source (VCCS). In all cases, the equations from the Thevenin equivalent circuit are the same, namely, for the controlling branch j -VoPENj + Tjilx + ... ,2 g^ ••• + rjjij + rjkik + ... + r j M i M + Vj = 0 and for the dependent source branch k -VoPENk + rk\i\ + ... ^ g\ ... + rkjij + r k k i k + ... + r k M i M + vk = 0 where: voPENk = voltage vk for [i] = 0 (open circuit). rkk = Thevenin resistance (self resistance of branch k). rkj = Thevenin resistance (coupling or mutual resistance between branches k and j). Current-Control led Voltage Source ( C C V S ) Assume that the controlling current is measured through a branch between nodes a and b in a circuit, such that Vj is its branch voltage and ij is its branch current, i.e., Vj = v a - v b (2.10) ij = iab (2.11) and that the dependent source, C C V S , is connected between nodes c and d with branch voltage (2.12) 2.2. Current and Voltage Dependent Sources in E M T P - b a s e d Programs 23 and branch current 2fc ted (2.13) The necessary equations for the implementation of a current-controlled voltage source as illustrated in F i g . 2.5 are 2.8 and 2.9, as well as: ' 6 - A A / V [ rTHEVyl -AA/V-[rTHEV k ] 0 Figure 2.5: Current-controlled voltage source (CCVS) . Vj = Rinij (2.14) vk = Qij + Routh (2.15) where: Rin = Input resistance of branch j. Rout = Output resistance of the dependent source in branch k. £1 = G a i n over the controlling or measured current, applied as voltage dependent source at branch k. Inserting equation 2.14 into 2.8 and equation 2.15 into 2.9, results in : — VoPENj + Vjiii + . . . , x ... + (tjj + Rin) ij + rjkik + ... + rjMiM = 0 -VoPENk + rki%i + ... ... + (rkj + ft) {• + (rkk + Rout) ik + ... + rkMiM = 0 (2.17) 2.2. Current and Voltage Dependent Sources in EMTP-based Programs 24 Using the two equations 2.16 and 2.17 is preferable to using the four equations 2.8, 2.9, 2.14 and 2.15, because it reduces the number of equations which have to be solved in subroutine C O N N E C from 4 to 2. Whenever possible, the voltages should be eliminated in this reduction from 4 to 2 equations, because the solution will then produce the currents, which are the variables that have to be passed back to the main program. For an ideal current-controlled voltage source, Rin = 0 and Rout = 0, from which results: -VoPENj + Tjiii + ... •• + rjfl:i + rjkik + ••• + r j M i M = 0 (2.18) -VoPENk + rkih + ... ... + (rkj + CT) ij + r k k i k + ... + r k M i M = 0 (2.19) If expressed in matrix form, one can see that the matrix becomes unsymmetric, since matrix element j — k is no longer equal to matrix element k — j. Current-Controlled Current Source (CCCS) The necessary equations for the implementation of a current-controlled current source as illustrated in Fig. 2.6 are 2.8 and 2.9, as well as: OPEN j 6 - A / V V \-rTHEV y] Figure 2.6: Current-controlled current source (CCCS) . Vj R^nij vk — R0UtBii + Routik (2.20) (2.21) 2.2. Current and Voltage Dependent Sources in E M T P - b a s e d Programs 25 where: B = Ga in over the controlling or measured current, applied as dependent current source at branch k. B y inserting equation 2.20 into 2.8 and equation 2.21 into 2.9 one can also obtain, re-spectively, the following equations: -vopeNj + rjxiA + ... • + (rjj + Rin) ij + rjkik + ••• + r j M i M = 0 (2.22) VOPENk o + j ^ M i + ... • • • + t e + B H + t e + i ) * * + . . . + £ > = 0 (2.23) Observe that the division by Rout as done in equation 2.23, allows the use of very large numbers for R,out without numerical difficulties. For an ideal current-controlled current source, Rin = 0 and Rout —• oo, resulting in: -VOPENj + Tjiii + ... • + rnh + rjkik + ... + r j M i M = 0 (2.24) Bij + ik = 0 (2.25) Voltage-Controlled Voltage Source ( V C V S ) The necessary equations for the implementation of a voltage-controlled voltage source as illustrated in F i g . 2.7 are 2.8 and 2.9, as well as: vOPEN j - A / W 0 R.. A V ; - A A A -[ rTHEV k\ 0 F i g u r e 2.7: Voltage-controlled voltage source ( V C V S ) . 2.2. Current and Voltage Dependent Sources in E M T P - b a s e d Programs 26 (2.26) vk = Avj + Routik = ARinij + Routik (2.27) where: A = Ga in over the controlling or measured voltage, applied as dependent voltage source at branch k. B y inserting equation 2.26 into 2.8 and dividing the resulting equation by Rin to avoid numerical difficulties, results in equation 2.28. In order to eliminate the voltages and keep only the currents as variables, and also to allow the use of very large numbers for the gain A , the following calculations are done: (equation 2.26 inserted into 2.8) minus the result of [(equation 2.27 inserted into 2.9) and divided by the gain A] . This procedure eliminates Rin in the resulting equation 2.29: (2.28) I vOPENh . ( -VOPENj H ^ + [Tji - + (rjj ~ it) h + {rjk -rkk + Rout A + -1.1.4- R„... \ • (2.29) Based on the equations 2.28 and 2.29 for a voltage-controlled voltage source, if • A ->• oo, • Rin —> oo, and • Rout -> 0, then equations 2.30 and 2.31 are obtained, which can be used to model "ideal operational amplifiers". Note that the use of equation 2.31 only makes sense if there are feedback paths modelled in the network part, which create the "r,-fc" coupling resistance. Then equation 2.31 2.2. Current and Voltage Dependent Sources in EMTP-based Programs 27 will produce the correct current ik, which is returned to the main program for the calculation of voltages by compensation. Note also that the equation 2.31 is exactly stating that Vj = 0. (Please, see equation 2.8.) ij = 0 (2.30) —vopeNj + fjih + ••• /2 3-n ••• + '[if1./ + rjkik + ... + r j M i M = 0 Ideal Operational Amplifiers The commercially available operational amplifier is in reality an integrated-circuit chip, constructed essentially with many transistors and resistors in an integrated package. Oper-ational amplifiers, often called OP A M P S , are frequently used in sensor circuits to amplify signals, in active filtering and control circuits for compensation purposes and endless ap-plications in analog electronics [79], [80], [81], [82]. Fig. 2.8 presents the symbol used for representation of an operational amplifier. The voltage placed across the two input termi-Figure 2.8: Symbol for operational amplifier. nals (the non-inverting terminal (+) and the inverting terminal( —)), is to be amplified and to appear at the output terminals (one of which is grounded, but this grounding is usually omitted on the symbol). Since the gain of the operational amplifier is very high, it is neces-sary to have an external feedback circuit to make it stable. "In practice the input resistance 2.2. Current and Voltage Dependent Sources in E M T P - b a s e d Programs 28 R,in of an O P A M P is usually well in excess of lMfl, the voltage gain A is at least 10 5 , and the output resistance Rout is a few tens of ohms " [79], and then it can usually be modelled as a voltage-controlled voltage source ( V C V S ) . Many other electrical properties, which are temperature and frequency dependent, have to be considered though in realistic applications. In the ideal operational amplifier, no current would flow into the input terminals (Rin — oo as in an open circuit) , the output voltage would not be affected by the load connected to the output terminal (Rout — 0), and the gain would be infinite (A = oo so that the voltage at the non-inverting input terminal would be equal to the voltage at the inverting input terminal). Therefore, the fundamental concepts for the analysis of circuits with ideal operational amplifiers are to assume that the two input terminals of the ideal operational amplifier constitute "at the same time" [77]: • "an open circuit" (equation 2.30), A N D • "a vir tual short-circuit" (equation 2.31). "In this thesis, i f not otherwise clearly indicated, the assumption is made that all opera-tional amplifiers are ideal"!. There are many variations and combinations of O P A M P circuits. The two basic ones are the inverting amplifier (Fig. 2.9) and the non-inverting amplifier circuit (Fig. 2.10), with the transfer functions are given by equations 2.32 and 2.33 respectively. F i g . 2.11 illustrates an adder, a special case of the inverting amplifier, where the output is a linear sum of the input voltages, wi th the transfer function given by equation 2.34. F i g . 2.12 shows an ideal integrator, wi th the transfer function as of equation 2.35. Eo(s) _ _R2 Ei(s) m (2.32) (2.33) (2.34) 2.2. Current and Voltage Dependent Sources in E M T P - b a s e d Programs 29 R2 R l e . ( t ) •—vw-Figure 2.9: Inverting amplifier circuit. R2 Rl *o(0 T F i gure 2.10: Non-inverting amplifier circuit. ^(0 Rl R4 - A A A / — i r - A A A ^ - i R2 A A A -R3 • A A A / — 1 ^(0 Figure 2.11: Adder circuit with operational amplifier. E0{s) _ l _ Ei(s) RCs (2.35) F ig . 2.13 presents a generalization of the inverting amplifier circuit, which is very useful to obtain Laplace transfer functions by using the impedance approach [81]. W i t h the ideal operational amplifier, a "virtual ground" potential appears at the inverting input terminal, 2.2. Current and Voltage Dependent Sources in EMTP-based Programs 30 R -A/VV Figure 2.12: Ideal integrator circuit with operational amplifier. since the non-inverting input terminal is grounded. Moreover, no current flows into the input terminals of the ideal OP A M P . Therefore, the same current flowing through the complex impedance Zx(s) has to flow through the complex impedance Z2(s), resulting in Ei{s) = Zl{s)I(s) and E0{s) = —Z2(s)I(s). The transfer function for this generalized inverter circuit is given by equation 2.36. Z,(s) ZJs) Et{s) E0(s) Figure 2.13: Generalization of inverter amplifier circuit. For example, in the circuit shown in Fig. 2.14, the transfer function is derived with ideal operational amplifier using the impedance approach. The complex impedances Z\(s) and Z2(s) for this circuit are: Zl{s) = Rl (2.37) Z*(*) = crf^r = T^TT (2.38) 2.2. Current and Voltage Dependent Sources in EMTP-based Programs 31 R2 C R l A A / V e.(t) F i gure 2.14: First-order lag circuit using ideal operational amplifier. The transfer function is therefore obtained as Eq(s) ft.2 I Ei(s) ~ i?., R2Cs+-[ (2.39) Voltage-Controlled Current Source ( V C C S ) The necessary equations for the implementation of a voltage-controlled current source as illustrated in Fig. 2.15 are 2.8 and 2.9, as well as: V, OPEN j •AA/V-0 [rTHEVj\ Figure 2.15: Voltage-controlled current source (VCCS). Vk — RoutTVj + Routik (2.40) (2.41) 2.2. Current and Voltage Dependent Sources in E M T P - b a s e d Programs 32 where: T = G a i n over the controlling or measured voltage, applied as dependent current source at branch k. B y inserting equation 2.40 into 2.8 and dividing the resulting equation by Rin to avoid numerical difficulties, results in equation 2.42. In order to eliminate the voltages and keep only the currents as variables, and also to allow the use of very large numbers for Rout, the following calculations are done: V times (equation 2.40 inserted into 2.8) minus the result of [(equation 2.41 inserted into 2.9) and divided by Rout\- This procedure eliminates R.in in the resulting equation 2.43: ^ i n Y ^ t n J fey A c\\ •••+ +£*+•••+3_'»=°. - + ( r r « - £ : ) v + ( r r * - ™Sr) * + - <2 -43> - + (Tr>« ~ S ) = 0 For an ideal voltage-controlled current source, Rin —¥ oo and Rout —> oo, resulting in: ij = 0 (2.44) -Fvopenj + rr-j-i.i + ... ... + Trjjij + (IYj-jfc - 1) ik + ... + rrjMiM = 0 2.2.3 Ideal Transformers Even though an ideal transformer model has already been implemented in most E M T P -based programs, with the equations described in [2] , or with similar equations, an ideal transformer can also be implemented as a special dependent source. The necessary equations for the implementation of the ideal transformer as illustrated in F i g . 2.16 are 2.8 and 2.9, as well as: 2.2. Current and Voltage Dependent Sources in EMTP-based Programs 33 1 :n VJ l. J j ( V V,. Figure 2.16: Ideal transformer. Hi = I vk n (2.46) - = -n (2.47) where: - = ^ = turns ratio of the ideal transformer. From the equations above and from 2.8 and 2.9, one can easily obtain: ij+nik=0 (2.48) . VOPENh . I r n \ • , -VQPENj + + {Tjl - - £ ) « ! + ... + b M - ^ ) i M = 0 Equations 2.48 and 2.49 can be used to model an ideal transformer. It is important to mention that, normally, better models for electric transformers, which may include saturation effects, are used in EMTP-based simulations. For further details, please see, for example, references [2] and [6]. 2.2.4 Independent Sources It may be useful in a circuit or device model to have an independent current or indepen-dent voltage source connected between two ungrounded nodes. This can be accomplished 2.2. Current and Voltage Dependent Sources in EMTP-based Programs 34 with the same technique used for dependent sources, but using only one equation in this case. Independent Current Source Assuming that the independent current source is connected between nodes c and d with branch voltage vk = vc - vd (2.50) and branch current 1-k = led (2.51) then the necessary equations for the implementation of an independent current source as illustrated in Fig. 2.17 are: C^ ) hource Rout [ rTHEV k\ \ 0 "OPEN k F i gure 2.17: Independent current source. -voPENk + r k \ i \ + ... • + r k k i k + . . . + r k M i M + vk = 0 Vk R-outisource "b Rout^k (2.52) (2.53) where: 2.2. Current and Voltage Dependent Sources in EMTP-based Programs 35 isource = independent current source of branch k, which can be constant or a function of time. From the equations above, one can also obtain the following equation: l F t + _ i u - i i + ... = 0 For the ideal current source, Rout -> oo, resulting in: (2.54) 1'k ~f~ ^ source 0 (2.55) Of course, there is a much easier way to represent an independent current source between nodes c and d directly in the nodal equations of the E M T P : inject the current source into node c and with a negative sign into node d [2]. Independent Voltage Source The necessary equations for the implementation of an independent voltage source as illustrated in Fig. 2.18 are: *• J Rout A / W [rTHEV k~\ 6 "OPEN k Figure 2.18: Independent voltage source. •VoPENk + rkih + ... • + r k k i k + ... + r k M i M + vk = 0 Vk Vsource -\- Routik (2.56) (2.57) 2.2. 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. From the equations above, one can also obtain the following equation: ~VOPENk + rkiU + ... ••• + (rkk + Rout) ik + •••+ TkM^M + vsource = 0 For an ideal voltage source, R o u t = 0, resulting in: -v0pENk + r k i i i + ... ••• + fkkik + ... + rkMi]W + Vsource = 0 (2.58) (2.59) Another approach for voltage sources between ungrounded nodes frequently used in EMTP-based programs is the insertion of an ideal transformer between the two ungrounded nodes, with a voltage source to ground on the other side. 2.2.5 N e w t o n - R a p h s o n A l g o r i t h m The equations for current and voltage dependent sources have been presented in the previous sections, as well as the equations of independent sources which may be connected between two ungrounded nodes. The solution of these equations is based on the compensation method, which is already being used to solve nonlinear equations associated with nonlinear elements in electric or electronic circuits with Newton-Raphson (N-R) iteration schemes. The Newton-Raphson algorithm is well known, widely used and has quadratic convergence if the initial estimate is close to the solution. For completeness, the Newton-Raphson algorithm is presented in this section as it is in [77] and the reader is referred to mathematical books or numerical analysis books or network solutions books, if more detailed information is needed. In the scalar case the N-R iteration to solve f{x) = 0 (2.60) is given by xk+i = Xk + Axk = xk_ f(xk)/f'(xk), (2.61) 2.2. Current and Voltage Dependent Sources in E M T P - b a s e d Programs 37 where the iteration count is denoted by the superscripts. Consider now the system of M nonlinear equations ft in M variables x{ fi(xi,x2, • • -,xM) = 0 f2(xl,x2, • • -,xM) = 0 Im(xi,X2, . . .,xM) = 0 (2.62) Denote, for easy notation, the vector of variables by [x] and the vector of functions by [f(x)]. Then 2.62 has a compact form: [/ (*)] = o (2.63) Assume that the system has a solution; denote it by [x*] and expand each function in a Taylor series about [x\: dxl* (x*m Xm) + fAxn=f2(x) + ^(xl-Xl) + l^(x*2-X2) + . . . + ^(x*M-XM) + . . . fM(Tn = fM(x) + ^ ( x t - X 1 ) + ^(x*2-X2) + . . . + ^-(x*M-XM) + (2.64) Assuming that x is close to x*, higher order terms may be neglected and the system may be written in linearized form: [/ (X')]*[f(x)] + [J]([X*]-[X]) (2.65) where [J] Ix = r dh_ dxi dh dx\ Ml .. 0X2 dh dX2 . 9h -j dm dXM dfM - dxi dfM dx2 dfM 0xM -1 (2.66) is the Jacobian matrix of the function [f(x)\, which has to be calculated at each iteration step. If equation 2.65 is set to zero and solved, the result w i l l not be the vector [x*] (because the higher-order terms have been neglected) but some new value for [x]. Using superscripts to indicate the iteration count results in: [f(xk)]+[J] ([xk+l] - [xk]) = 0 (2.67) 2.2. Current and Voltage Dependent Sources in EMTP-based Programs 38 Formally, the solution of 2.67 is obtained by writing [xk^] = [xk] - [J]"1 [/ (xk)] (2.68) In practice, the Jacobian matrix is not inverted. Instead define Axk] = [xk+l] — [xk (2.69) Then [J] [Axk] [f (**)] (2.70) is solved by L U factorization and the new [xk+1] is obtained from (2.71) Formulae 2.70 and 2.71 represent the Newton-Raphson algorithm for systems of equa-tions, which reduce the error norm iteratively so that This iterations scheme is repeated until the errors are lower than a specified tolerance. For the case of a system of linear equations, as in the case of linear dependent sources, convergence to the solution is achieved with just one iteration step. More iteration steps are required for the solution of a system of nonlinear equations depending on how close the initial guess is to the final solution. For highly nonlinear functions, the standard application of the iteration scheme of the Newton-Raphson algorithm may cause numerical problems involving computer overflows. The solution algorithm experimentally implemented in the MicroTran version of the E M T P at The University of British Columbia is presented in Fig. 2.19. This method presents "generality and flexibility properties", thus looking promising for future work in detailed modelling of circuits and devices, as will be explained later in this thesis. [f (*k+1)}\\ < \\[f (*")] (2.72) 2.2. Current and Voltage Dependent Sources in EMTP-based Programs 39 INPUT DATA INITIAL G U E S S F O R C U R R E N T S C A L C U L A T I O N O F B R A N C H V O L T A G E S [from Eq . (2.6) or (2.1)] S E T V A R I A B L E T O ITS LIMIT C A L C U L A T I O N O F R IGHT H A N D S IDE [negative values o f E q . (2.1)] Yes C H E C K F O R LIMITS BUILD J A C O B I A N MATR IX [PARTIAL derivatives o f E q . (2.1)] S O L V E F O R C U R R E N T S R E T U R N C U R R E N T S T O MAIN P R O G R A M U P D A T E C U R R E N T S Figure 2.19: Newton-Raphson algorithm experimentally implemented in MicroTran. 2.2. Current and Voltage Dependent, Sources in EMTP-based Programs 40 2.2.6 Possible App l i ca t ions The methodology presented in the previous sections for the implementation of dependent sources in EMTP-based programs permits the computational development of many practical applications, such as: 1. Current, and voltage sensors; 2. Operational amplifiers; 3. User-defined coupled branches in a circuit; 4. Modelling of electronic components, where the physical behavior would need to be represented by nonlinear equations; 5. User-defined linear and nonlinear functions; 6. User-defined modelling of linear and nonlinear devices, limited only by the creativity and ingenuity of the user. Figure 2.20 and Fig. 2.21 illustrate the solution method with an example of a noninverting amplifier circuit, as commonly used in practical analog electronics. It consists of a sinusoidal voltage source, an ideal operational amplifier and 2 resistors (Rf and Rg). The ideal opera-tional amplifier was modelled using equations 2.30 and 2.31, whereas the sinusoidal voltage source and the resistors are part of the network, represented through a Thevenin equivalent circuit. If Rf = 2Rg, then voutput = 3 ^ n p „ t , as shown in Fig. 2.21. Alternatively, to get an amplification of 3 in circuit simulation, one could just use a voltage-controlled voltage source, with equations 2.28 and 2.29 with the gain A set to a value of 3. In this case a load resistor should be connected in the output of the dependent source, to avoid numerical floating subnetwork problems. Indeed, in theory this noninverting amplifier circuit should result in: (2.73) 2.2. Current and Voltage Dependent Sources in EMTP-based Programs 41 v. input input =1.QUE [V] /=60 [Hz] v. output Figure 2.20: Circuit with ideal operational amplifier. Figure 2.21: Simulation results of circuit with ideal operational amplifier (noninverting amplifier circuit). The next sections will present a technique for the simulation of transfer functions in EMTP-based programs, some improvements already made for the implementation of satura-tion or limits for the elements or sources presented in this work, as well as the implementation of some other special control devices. 2.3. Development of Control Transfer Functions in EMTP-based Programs 42 2.3 Development of Control Transfer Functions in E M T P -based Programs A transfer function as in Fig. 2.22, is defined in the frequency domain (Laplace transfor-mation of a continuous time system) by the equation 2.74, which represents the output signal X(s) as a function of the input signal U(s) for a particular linear time-invariant system. H(s) Figure 2.22: Transfer function. H(s) = X(s) U(s) with n>m, and an f 0. bmsm + 6 m _ 1 s m - 1 + ... + M 1 + b0' ansn + an-isn-1 + ... + a-iS1 + a0 . = k B(s) A(s) (2.74) It is possible to reorganize the terms of equation 2.74 as follows: (ansn + a n _ i s n _ 1 + ... + axs1 + a0) X(s) = k {bmsm + b m _ l S m - 1 + ... + M 1 + & o ) U(s) (2.75) (b-^sr „m—1 -•" + ...+ t j - n ) U(s) (2.76) X(s) = k (£. -n + -1-71 + (2.77) 2.3. Development of Control Transfer Functions in E M T P - b a s e d Programs 43 For m = n, 4 this results in: X(s) = k(b-^ + ^ i s - i + ... v ' \o,n an + iLgl-n + bLs-n\ rffg) _ ( + ... + oj. l-n + aos~n\ W s ) (2.78) X(8) = k±U(s) + s-* kb-^U{s)-^X{s) + +s l-n k^U(s) - ^X(s) k^U(s) - ^X(s) (2.79) .. + s X(s) = kbtU{s) + s-1 { (kb-^U(s) - *^X{8)) + ... (2.80) A transfer function block-diagram realization of equation 2.80 is presented in F ig . 2.23, which, according to [79], is called the observer form. The solution technique proposed in this thesis is based on the fact that a practical realization of a transfer function block-diagram can be accomplished wi th the use of circuit components, such as operational amplifiers, resistors and capacitors 5 . In practice, an analog signal processing scheme is usually designed as the first step for a digital signal processing derivation. Moreover, in this work, the derivation of an analog circuit model for the transfer function implementation takes advantage of al l the options already implemented in E M T P -based programs, and redundant computational work is avoided. This way, the equations for the digital model of a transfer function are automatically constructed inside the E M T P , which 4 If m < 77, then bn = = ... = 6m+i = 0. 5 Analog computers were commonly used in the past to solve power system control and stability differential equations [83]. 2.3. Development of Control Transfer Functions in E M T P - b a s e d Programs 44 U(s) X(s) F i gure 2.23: Observer form block-diagram of transfer function in equation 2.80. uses the trapezoidal integration rule, or the backward Euler rule whenever C D A technique is applied [84], [85]. W i t h analog circuit modelling based on operational amplifiers, the method presented here is general and allows an arbitrary design of transfer functions (or many special control devices) by the users of E M T P - b a s e d programs. A simplification can be made in a computer transfer function implementation, in contrast to a physical circuit implementation, in order to reduce the number of operational amplifiers needed: resistances and capacitances can assume negative values 6 . As long as the system eigenvalues (or poles of the transfer function) remain on the left hand side of the complex plane, the system is stable. Negative values then may be used here for capacitances and resistances connected on the feedback path of the "ideal operational amplifiers", which re-sults in a stable solution for transfer functions of linear systems. Such a possible computer implementation of the transfer function block-diagram of F i g . 2.23 is presented in F ig . 2.24. Assume for example, that the first-order transfer function illustrated in the block-diagram of F i g . 2.25 is to be implemented with the proposed technique. Equation 2.80 becomes in 6 Proper precautions should be taken, though, whenever the equivalent digital self admittance of a node becomes equal to zero. The connection of large resistances to the node can easily solve this problem. 2.3. Development of Control Transfer Functions in EMTP-based Programs 45 Figure 2.24: Possible computer implementation of the transfer function block-diagram in Fig. 2.23. 10 0.01s + 1 Figure 2.25: Block-diagram representation of a first-order transfer function. this case, with n = 1: X{S) = s-' (k^U(s) - ^X(s)) (2.81) where, for illustration purposes, kb0 = 10, a n = 1 and a-[ = 0.01 seconds. The observer form block-diagram for equation 2.81 is presented in Fig. 2.26, and its "possible computer implementation" is illustrated in Fig. 2.27. The realization of this first-order transfer function can also be done with a physically-based realistic first-order lag circuit as shown in Fig. 2.28, which requires two inverting amplifier circuits 7 , instead of just one 7The author acknowledges the help of Mr. Jesus Calvifio-Fraga in a practical laboratory experiment for 2.3. Development of Control Transfer Functions in EMTP-based Programs 46 required in the "more economic" computer implementation. There may be cases where the realistic implementation is needed, which the proposed method can handle as well without any restrictions. Fig. 2.29 presents the time domain transient response x(t) of the first order transfer function implemented as in Fig. 2.27 and as in Fig. 2.28, for a pulse u(t) of IV with a duration of 25ms. U(s) kbn X(s) Figure 2.26: Observer form block-diagram of first-order transfer function of Fig. 2.25. u(t) kbn - IJIF MQ x(t) u(t) 1 kCl - A A A r -- io kn -AAAr - l|iF — 1 6 -r 1 x(t) Figure 2.27: Possible computer implementation of first-order transfer function of Fig. 2.25. the circuit of Fig. 2.28, which validated the simulation results presented in Fig. 2.29. 2.3. Development of Control Transfer Functions in EMTP-based Programs 47 u(t) 1 kfi - A A / V 10 kCl luF io kn x(t) io kn - A A A r -x(t) 1 1 1 F i gure 2.28: Realistic first-order lag circuit. 20 25 30 Time (ms ) 35 40 45 50 Figure 2.29: Time domain simulation of first-order transfer functi> This technique can easily be used with the solution for the "ideal operational amplifiers" implemented in subroutine C O N N E C . There is no time delay between the electric network and control equations as in TACS, and the solution of both systems of equations is simulta-neous. The modified nodal analysis method (MNA) [76] could also be used for the solution of operational amplifiers and other "linear branch equations" as presented in [77] . However, 2.4. Development of Limiters for Control Systems in EMTP-based Programs 48 with the modified nodal analysis (MNA), the network and control system equation matrix becomes unsymmetric, and zero diagonal elements may appear, which requires pivoting tech-niques. Also, as the number of added branch equations increases, requiring extra columns and extra rows, the dimension of the matrix may become very large, which may eventually decrease the computational efficiency, if proper techniques are not used. Possibily, if the sim-ulation time becomes a very important issue, as in the case of digital real-time simulators, a combined solution could be investigated, such that all linear control and system equations could be solved using the M N A with appropriate techniques, and the remaining nonlinear equations could be solved with the compensation method. The main motivation here in this thesis for using the compensation method with a Newton-Raphson iterative algorithm is its "generality and flexibility" to model nonlinear (and linear as a special case) control devices in EMTP-based programs, particularly because the branch Thevenin equivalent circuit is readily available, as in the C O N N E C subroutine of MicroTran. The implementation of limits associated with first order transfer functions, as well as special devices and intrinsic F O R T R A N functions, are presented in the following sections. 2.4 Development of Limiters for Control Systems in EMTP-based Programs The use of limiters in control loops may introduce extra time delays in the solution method implemented in TACS. This can result in inaccuracies and instabilities [66], [67]. The technique proposed in this thesis for the solution of limiters overcomes this difficulty. With the Newton-Raphson iterative algorithm in the compensation method, a simultaneous solution for limiters can be found. There are two types of limiters associated with first-order transfer functions : windup (also referred to as static limiter) and non-windup (dynamic limiter) [66], [2]. "Non-windup limiters should only be used with first-order transfer functions. For second and higher-order transfer functions it is no longer clear which variables should be limited. ... Even for the first-order transfer function, the meaning of the limiting function is confused if it has any zeros" 2.4. Development of Limiters for Control Systems in EMTP-based Programs 49 [2]. Reference [66] presents an appropriate model for a proportional-integral (PI) controller (which can be represented as a transfer function with one zero) with a non-windup limiter. In a lead-lag control function block, for example, the way in which a non-windup limiter can be realized is not unique; the interpretation of the limiting action should therefore be based on the electronic implementation of the physical device [86]. The main difference between windup and non-windup limiters is the way in which the limited variable comes off its limit. To illustrate that, the first-order transfer function pre-sented earlier in Fig. 2.25 is assumed to have a windup limiter as in Fig. 2.30, and a non-windup limiter as in Fig. 2.31. The time domain simulation of the output variable x(t) for both cases, for a pulse input excitation u(t) of IV, is presented in Fig. 2.32. U(s) 10 / 0.01 s + 1 / Figure 2.30: First-order transfer function with windup (static) limiter. U(s) + 5 / slope=l 10 0.01s + 1 X(s) F i gure 2.31: First-order transfer function with non-windup (dynamic) limiter. Note that the output variable x(t) reaches its limit at the same time for both cases, but x(t) backs off the limit first for the non-windup (dynamic) limiter. The reason is that for the windup limiter the output variable is just clipped at the limit, whereas in the non-windup limiter the differential equation is actually modified [66], [87], [2]. 2.4. Development of Limiters for Control Systems in EMTP-based Programs 50 10 & 5 x(t) with non-windup limiter (dynamic) u(t) / x(t) without limiter y x(t) with . A windup limiter (static) 10 15 20 25 30 35 40 45 Time (ms ) 50 Figure 2.32: Transient response of a first-order transfer function with windup and non-windup limiter. The implemented solution for limiters uses the methodology proposed in Section 2.2.1. As indicated in the algorithm illustrated in Fig. 2.19 of Section 2.2.5, a simultaneous system solution is first, found without, considering any of the limits. Then, each limit violation is verified in the sequence of the input, data given by the user. If a particular limit has been reached, all previous indications of limit violations are cleared, and a solution is found for this particular limiter and all of its consequences on the other limiters. This cause-consequence iterative process has been found to be a very "robust method" in all cases tested, and has given the correct solution for all limiters, "independent of the ordering of the input data given by the user". Also, because the compensation method is properly applied 8 , the solution for limiters is simultaneous without time delays. The maximum (xmax) and minimum (xmin) limiting values are part of the input data. For TACS seems to use a pseudo-compensation method to solve limiters [2]. 2.4. Development of Limiters for Control Systems in EMTP-based Programs 51 example, in the case of the first order transfer function with a non-windup limiter illustrated in Fig. 2.31, with a computer model as in Fig. 2.27, it is possible to represent the non-windup hard limiting action with a simple change in the equations for the ideal operational amplifier, such that equations 2.30 and 2.31 are replaced by equations 2.82 and 2.83, respectively: —vopen3•+ Tjii\ + ... ('2 82) ••• + rijij + rj>dk + ... + rjMiM = 0 -v<DPENk + rkiii + ... (2 83) ... + rkjij + r k k i k + ... + r k M i M + vk-.lirnit = 0 where vk—nmn x m a x > or vk—nmit %min-By using these equations it becomes easy to observe the limits accurately. In practice, in a realistic first order lag circuit, as in Fig. 2.28, the clamping action is done with the use of Zener diodes connected in parallel with the capacitor in the feedback loop of the first OP A M P for a non-windup (dynamic) limiter, or with Zener diodes connected in parallel with the resistor in the feedback loop of the "second" OP A M P for a windup (static) limiter 9 . Another example is the simple limiter control block. In this case, one could use the equations for an "ideal voltage-controlled voltage source" including the limiting values in the output voltage, as follows: ij = 0 (2.84) -v0PENk +rklH + ... « 2 8 5 v ••• + rkjij + r k k i k + ... + r k M i M + vk_[imit = 0 where vk—nmn — ^-man or vk—nmn = xmin. Implementation of soft limits The limiters presented in the previous section assume fixed values (hard limits) for the maximum and minimum of the output variable. It may be useful to allow soft limits as well, as recommended in [2]. With soft limits, the slopes in the limited region are nonzero. 9The author acknowledges the help of Mr. Jesus Calvino-Fraga in a practical laboratory experiment, which validated the simulation results presented in Fig. 2.32. 2.4. 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. The equations for soft limits, with the notation from Fig. 2.33, are: Ku(t), if xmin < Ku(t) < x m a x , x(t) = \ xmin + Kmin[u(t) - um-m], if Ku(t) < xm-m, (2.86) £max + Kmax[u(t) - Umax], if Ku(t) > Xm!iX. Consider, for example, the zero-order transfer function (constant gain) in Fig. 2.34. The time domain response for a sinusoidal excitation input u(t) of I V is presented in Fig. 2.35, illustrating the effects of hard and soft limits on the output, x(t). In this thesis project, soft limits (and hard limits as a special case) have been implemented for all the current and voltage dependent sources presented in Section 2.2. Limits can also be easily implemented for all the F O R T R A N functions and special devices which will be discussed in the following sections. 2.4. Development of Limiters for Control Systems in EMTP-based Programs 53 Figure 2.35: Time domain response for a sinusoidal excitation input u(t) illustrating the effects of hard and soft limits on the output x{t). 2.5. Development of Intrinsic F O R T R A N Functions in EMTP-based Programs 54 2.5 Development of Intrinsic FORTRAN Functions in EMTP-based Programs "Supplemental variables and devices", as defined by the E M T P Rule Book, differ in TACS from the transfer function blocks, as follows: 1. they are not, solved with the matrix of the set of linear equations in TACS. 2. they are calculated sequentially, instead of simultaneously (so that the data cards must be ordered accordingly). Supplemental variables, such as intrinsic F O R T R A N functions and special devices, are solved in TACS in a sequential way. "In Fig. 2.36, the special device 51 would be solved after G2 has been solved, and 52 would be solved after 673 has been solved. The solution would still be simultaneous in this case. In general, the sequence of calculations is more complicated, with non-simultaneous solutions through time delays. For details, the reader should consult the E M T P Rule Book" [2]. The sequential solution requires a definition by the user of "input (e.g. 51 in Fig. 2.36", "output (e.g. 53 in Fig. 2.36)" and "inside (e.g. 52 in Fig. 2.36)" groups of devices. A special ordering of these device blocks is necessary to minimize the time delays introduced by this sequential solution method. "To make the solution as much simultaneous as possible, the user should keep the number of internal devices as low as possible, and use input and output devices instead whenever possible" [2]. G , ( s ) G2(s) A. G3(s) 17 F i gure 2.36: Open loop control system with "supplemental devices S1,S2 and 53". In this research project a "truly simultaneous solution" is achieved for intrinsic FOR-T R A N functions, and no special ordering is necessary, i.e., the input of data by the user is arbitrary. The solution technique applies the compensation method in a similar way as done in Section 2.2 for the implementation of current and voltage dependent sources. Assume, for 2.5. Development of Intrinsic F O R T R A N Functions in EMTP-based Programs 55 example, the control block-diagram of Fig. 2.37, with a nonlinear relationship between the output voltage vk(t) and the input voltage Vj(t). v. j K2 SIN(K,v.) v. F i gure 2.37: Nonlinear control block-diagram with a sinusoidal intrinsic FORTRAN function. A simultaneous solution can be obtained for this nonlinear function by representing the block-diagram of Fig. 2.37 in the form of an electric circuit as shown in Fig. 2.38. The nec-essary equations are 2.87 and 2.88 and the branch equations 2.89 and 2.90. These equations resemble those of a voltage-controlled voltage source presented in Section 2.2. 'OPEN j - A A / V [ rTHEVy] R,, AA/V-[ rTHEV /J K2 sin (KjVj) 0 Figure 2.38: Circuit implementation for the simultaneous solution of a sinusoidal FORTRAN functi ion. For the controlling branch, the equation is -vopen; + r n i i + ... (2 87") ••• + rjjij + rjkk + ••• + r j M i M + Vj = 0 and for the dependent source branch it is -voPENk+rkih + - , 2 8 8 N ••• + rkjij + r k k i k + ... + r k M i M + vk = 0 where: voPENk = voltage vk for [i] = 0 (open circuit). rkk = Thevenin resistance (self resistance of branch k). rkj = Thevenin resistance (coupling or mutual resistance between branches k and j). Vj = RtJj (2.89) 2.5. Development of Intrinsic F O R T R A N Functions in EMTP-based Programs 56 vk = K2 (sin {KlVj)) + Routik = K2 (sin {KxRini3)) + Routik (2.90) where: R~i = Gain over the controlling or measured voltage. K2 = Gain applied to the dependent source in branch k. From the equations above, one can also obtain the following equations: J L in \ -•'•in / -voPENk + rk-[ix + ... ... + [rkiij + K2 (sin (iTii?.^^))] + (rkk + Rout) ik + ... + r k M i M = 0 (2.91) (2.92) If a large number is used for Rin 1 0 , then the solution for the current ij will be very small but not exactly equal to zero. This allows the convergence to a non trivial solution for the current ik, and of course, for the input and output voltages of this nonlinear control block of a sine function. These equations can then be solved with the implemented Newton-Raphson algorithm illustrated in Fig. 2.19, of Section 2.2. Note that for the proper application of the compen-sation method there must always be possible a Thevenin equivalent circuit. Therefore, in cases where there is a floating subnetwork, i.e., a node without connection to ground (as for example if the output of an intrinsic F O R T R A N function has no circuit elements connected to ground), then the insertion of a big resistance between this node and ground easily over-comes this problem, as is usually done in some versions of the E M T P for the solution of nonlinear elements [2]. Applying this technique, the following nonlinear intrinsic F O R T R A N functions were im-plemented, • SIN 1 0In theory Rin -> oo, making ij = 0, which would result in a trivial solution for equation 2.92. Rout is obviously assumed to be equal zero. 2.5. Development of Intrinsic F O R T R A N Functions in E M T P - b a s e d Programs 57 • C O S • T A N • C O T A N • S I N H • C O S H • T A N H • A S I N • A C O S • A T A N • E X P • L O G . L O G 1 0 • S Q R T , as well as the mathematical operations 1 1 : • mult ipl icat ion (*) • division (/) • exponentiation (**). Proper precautions were taken to handle mathematical and computational problems such as division by zero, square root of negative number, exponentiation of negative number wi th non-integer exponent, logarithm of zero or of negative values, etc.. The evaluation of trigonometric functions accepts the argument in degrees, which is then converted to radians internally. The inverse of trigonometric functions gives the answer already converted to degrees. 1 1 Addition (+) and subtraction (—) can be implemented with the use of just one ideal operational amplifier. 2.6. Development of Control Devices in E M T P - b a s e d Programs 58 2.6 Development of Control Devices in EMTP-based Programs A simultaneous solution is also obtained for special control devices, by using the same approach presented in the previous section. Their input can be in arbitrary order defined by the user. To illustrate the potential of this technique, the detailed development of some useful special devices wi l l be presented. Transport Delay Assume, for example, the time delay control block-diagram of F i g . 2.39, (also called "transport delay" in T A C S ) , where the input voltage Vj(i) only affects the output after the elapsed time t + r, or conversely, the output voltage Vk[t) only depends on the past history value of the input voltage, i.e., v3(t — r ) . DELAY ) • vk Figure 2.39: Transport delay control device. The transport delay of F i g 2.39 can be represented in the form of an electric circuit 1 2 as in F i g . 2.40, wi th circuit equations 2.93 and 2.94, and branch equations 2.95 and 2.96. If not otherwise indicated, al l the current and voltage variables are the instantaneous values at, the present time t. For the controlling branch, the equation is —vopeNj + r + ... (2 93) and for the dependent source branch -VoPENk + rk\i\ + ... ^ 2 94) ... + rkjij + rkkik + ... + r k M i M + vk = 0 1 2 For generality reasons, the equations are derived including Rin and R,out, however the ideal equations are used in the computer implementation. 2.6. Development of Control Devices in E M T P - b a s e d Programs 59 yOPEN j -AA/V 0 [VTHEV j] R. -A/VV R„. [rTHEV k\ 0 K2Vj(t-x) Figure 2.40: Circuit, implementation for the simultaneous solution of a transport delay control device, where: voPENk = voltage vk for [i] = 0 (open circuit). rkk — Thevenin resistance (self resistance of branch k). rkj = Thevenin resistance (coupling or mutual resistance between branches k and j). Vj RinZj (2.95) vk = K2Vj (t-r)+ Routik (2.96) where: K2 = Gain applied to the controlling or measured past history voltage Vj(t - r ) , to create an independent source at time t, in branch k. From the equations above, one can also obtain the following equations: -v<DPENk + rkiii + ... ... + r^ij + (rkk + Rout) ik + ... + r k M i M + i C ^ - (t - r ) = 0 -voPENk + rk]ii + ... • + rfcj? ;j + rfcfcU + ••• + r k M i M + i C ^ y ( i - r ) = 0 (2.97) (2.98) If Rin —>• oo and i ? o u t —> 0, then ij = 0 (2.99) (2.100) 2.6. Development of Control Devices in E M T P - b a s e d Programs 60 which can be solved wi th the implemented Newton-Raphson algorithm illustrated in F ig . 2.19, of Section 2.2. Considering that the delay time r is not usually an integer multiple of the simulation time step At, some type of interpolation must be used. Linear interpolation has been chosen for that purpose, in a way similar to the transient time domain simulation of a transmission line model [1], [2]. F i g . 2.41 illustrates the time response of a transport delay control device where r = 4.1667ms, A t = 166.6667/its, K2 = 1; the input voltage signal Vj(t) is a sinusoidal source of I V . Note in F i g . 2.41 that the output voltage signal Vk(t) is actually equal to the input voltage signal Vj(t), but delayed in time in 4.1667ms. 1.5 0.5 -0.5 -1.5 delay 4.1667ms delayed signal -j 1 i i i i i i_ 0 5 10 15 20 25 30 35 40 45 50 Time (ms) Figure 2.41: Transient simulation of a transport delay control device. Pulse Delay Apply ing the same technique used for the implementation for a transport delay, it is possible to develop a model for a pulse delay control device. In a pulse delay, the negative to 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 specified delay and with the width of respective time between the two zero crossings of the input signal. This way there is no need to store all the past history values of the input signal, just the respective times of zero crossing, which is a "more computational economic" delay if the output signal will always have to be a pulse, irrespective of the shape of the input signal. Fig. 2.42 illustrates the time response of a pulse delay control device where r = 20ms, K2 = 1, and the input voltage signal Vj(t) is a I V pulse source. Fig. 2.43 shows the time response of a pulse delay control device where r = 20ms, K2 = 1, and the input voltage signal Vj(t) is an arbitrary signal source. 1.5 0.5 <3 -0.5 -1 -1.5 delayed pulse 20ms delay 10 15 20 25 30 Time (ms) 35 40 45 50 F i gure 2.42: Transient simulation of a pulse delay control device. 2.6. Development of Control Devices in EMTP-based Programs 62 1.5 0.5 §J, 0 -0.5 -1h -1.5p ~i r 1 r delayed pulse 20ms delay -i i_ _i i_ 10 15 20 25 30 Time (ms ) 35 40 45 50 F i gure 2.43: Pulse delay control device with arbitrary input signal. Logic Gate " N O T " The necessary equations for the simultaneous solution of a logic gate NOT, as illustrated in Fig. 2.44 and Fig. 2.45, are 2.93, 2.94, 2.95 and vk = K2(l-Vj) + Routik (2.101) where: K2 = 1 is the gain over the controlling or measured voltage (with either Vj = 0 or Vj = 1), being applied as a dependent source in branch k. Figure 2.44: Logic gate "NOT". 2.6. Development of Control Devices in EMTP-based Programs 63 OPEN j 6 A A A r [ rTHEV j\ R. - A / V V R.. [ rTHEV k\ 0 K2( 1 - v} ) F i gure 2.45: Circuit, implementation of a logic gate "NOT" for simultaneous solution. From equations 2.93, 2.94, 2.95 and equation 2.101, one can also obtain the following equations: + (2.102) -VOPENJ ~ ^ + 1 + ( r „ + FT) i , + K2 J "J JjM + T-ff) IM = 0 (2.103) If Rin —» oo, Rout -» 0 and K2 = 1, then the following equations are obtained: (2.104) -VOPENJ - VOPENK + 1 + {rjx + rkl) ii + ... • + (rjj + rkj) ij + (rjk + rkk) ik + ... • + {tjm + rkM) %m = 0 (2.105) Other logic gates, such as " A N D " , " N A N D " , "OR", "NOR" , etc. can be implemented in a similar way. Chapter 3 Power Electronics Model l ing in E M T P - b a s e d Simulations "The application of semiconductor devices in the electric power field has been steadily increasing, and a study of power electronics (as it is commonly called) is now a feature of most electrical and electronics engineering courses. The power semiconductor devices, such as the diode, thyristor, triac, and power transistor, are used in power applications as switching devices. The development of theory and application relies heavily on waveforms and transient responses, which distinguishes the subject of power electronics from many other engineering studies" [88]. "Computer simulation can greatly aid in the analysis, design and education of Power Electronics. However, simulation of power electronics systems is made challenging by the following factors: 1) extreme nonlinearity presented by the switches, 2) time constants within the system may differ by several orders of magnitude, and 3) a lack of models. Therefore, it is important that the objective of the computer analysis be evaluated carefully and appropriate simulation packages be chosen" [89]. "In system level investigation, it is often adequate to represent semiconductor switches within converters by ideal switches. This is important in order to minimize the overall simulation time. But, it is very desirable if the same simulation package has the detailed device models to design snubbers and gate drives. The simulation package should also be able to represent the controller portion of the converter system by its functional features 64 3.1. Modelling Power Electronics in Electric Power Engineering Applications 65 in as simplified a manner as possible. Yet, it should be able to model the controller on a component level if needed" [89]. The preceding quotations show the need for the development and implementation of simplified as well as detailed nonlinear models of semiconductor devices in EMTP-based programs. Such models can be used for the transient simulation of electromagnetic phe-nomena in low and high power circuit networks. They are especially useful for the detailed evaluation of the impact of power electronic devices on electric power quality, which is the main emphasis of this thesis. For power quality studies, transient phenomena have to be evaluated not only at the interface between power electronic devices and the power system, but, the propagation of transients through the supply network and neighboring systems, and the impact of transients coming from the supply power system on the electronic devices have to be analyzed as well. The following sections discuss some of the I E E E recommendations for power electronics modelling, the simultaneous solution of voltage-controlled switches in EMTP-based programs, the implementation of a nonlinear diode model, and some aspects of control of power electronic devices. 3.1 Modelling Power Electronics in Electric Power En-gineering Applications Fig. 3.1 presents the major power semiconductor devices: Diode, Thyristor, Gate Turn-Off (GTO) Thyristor and Gate-Controlled Thyristor (GCT), MOS Turn-Off Thyristor (MTO), Emitter Turn-Off Thyristor (ETO), MOS Controlled Thyristor (MCT), Transistor, Insulated Gate Bipolar Transistor (IGBT), and Metal Oxide Semiconductor Field Effect Transistor (MOSFET) [90]. Some of these semiconductor devices are already well known. High-power semiconductor devices with increasing switching and power capabilities appear on the market every year (such as the Integrated Gate-Commutated Thyristors (IGCTs)), which extend the potential applications of modern power electronics techniques to basically all voltage levels in utility companies and industrial sites. Models for the existing and new power electronics devices are therefore necessary for analyzing existing or future applications. 3.1. Modelling Power Electronics in Electric Power Engineering Applications 66 DIODE Cathode A Anode THYRISTOR Cathode £ Gate zx (turn-on) Anode GTO and GCT Cathode A Anode Gate (turn-on & turn-off) MTO Cathode Turn-on Gate * \ Anode Turn-off Gate ETO Cathode £ Anode Turn-off Gate -• Turn-on Gate MCT Cathode Gate (turn-on & turn-off) Anode TRANSISTOR Emitter Collector Base IGBT Emitter Collector Gate MOSFET Source - x Drain -« Gate Figure 3.1: Power semiconductor devices. 3.1. Model l ing Power Electronics in Electric Power Engineering Applications 67 Guidelines for modeling power electronics in electric power engineering applications, es-pecially for use in E M T P - b a s e d programs, can be found in [91], [92]. These guidelines can also be useful when using other digital simulation tools. The approximately 60 test cases in the computer exercise collection of Dr . N . Mohan [93], [94], [89] which are available for both E M T P and PSpice® simulations, are also useful for power electronics digital simulations. In particular, the differences between existing E M T P and PSpice models may give hints for the improvement of E M T P models. Dr . Ned Mohan also presents in [89] a power electronics library wi th special sub-circuits, which are used to represent switching electronic devices and some control devices. According to [89] "in case of extreme nonlinearity, PSpice uses extremely small time steps and is also prone to problems of voltage convergence. To avoid this problem of extreme nonlinearity such as that associated with diodes, R - C snubbers are connected across them. The values of R and C in these snubbers are not optimized, rather these are based on speeding up the simulation without distorting the system voltage and current waveforms significantly." As stated in [91], [92], power electronics modelling depends on the objectives of the study. Depending on the type of study, different software tools and solution techniques can be applied. In steady-state or harmonic analysis, the main concern is the injection and propagation of harmonic currents into the transmission and distribution system, which may cause unacceptable voltage distortions and dangerous resonances. The power electronic sub-system is then modelled as "known" harmonic current sources. These currents are assumed to be independent of voltage variations at the point of common coupling ( P C C ) , where the power electronics load is connected. However, in many applications such as adjustable speed drives, active power conditioning, F A C T S and Custom Power Controllers, etc., the operation of a power electronics device closely depends on and can affect the dynamics and the electric transient behavior of the connected system. Variations of the system parameters, such as voltage and current amplitude, frequency, phase-angle displacement among the phases in a three-phase system, instantaneous or average power, etc., need to be used by the power electronics control, to properly adjust the firing time of the semiconductors, which in turn might have a feedback effect on the system. Therefore, for transient analysis, a more complex and detailed representation of the power electronics devices as well as of the supplying power 3.1. Model l ing Power Electronics in Electric Power Engineering Applicat ions 68 system is required. Some model simplification and system reduction [92] might be necessary for practical reasons. This can be acceptable, provided that the equivalent model is validated against practical measurements, and that it fits the study investigation needs. The representation of semiconductor switching devices is commonly simplified in power level application studies. Therefore, the nonlinear characteristic of a diode is usually repre-sented in a simplified form as a two-terminal uncontrollable unidirectional current flowing switch, or in some programs as a voltage-controlled switch. Series on-state and parallel off-state resistances can be added to represent the semiconductor losses. In some E M T P - b a s e d programs, such as MicroTran , series resistances must be included to allow multiple switch connections on the same circuit node. Parallel resistances can provide a resistive connection between the D C sides of rectifiers and inverters and the A C local ground, thus avoiding floating sub-network problems [95]. The use of simplified switch models for power electronics devices may be justified to speed up the simulation time for system level studies, but it may also give wrong and misleading results, especially related to semiconductor commutation phenomena. Also, the E M T P solution at fixed discrete time intervals A t may result in inaccurate turn-on or turn-off switching times, causing unrealistic high frequency transients in the simulation of power electronic devices. Backtracking techniques [67] and/or resynchronization techniques ([96] pages 185, 204, 207), or even the Clock Synchronized Structure Changing Concept (CSSC) [97] can be used to minimize the problem. Interpolation and/or extrapolation as well as resynchronization techniques seem to be more and more applied even in the E M T P - b a s e d solution of modern control for power electronics systems, as in the software P S C A D / E M T D C [98], [99]. Therefore, for E M T P - b a s e d simulations of power electronics, it is much more important to use such techniques than to reduce the time step size. Numerical oscillations caused by the trapezoidal rule of integration in solving the system of equations may also be a problem for E M T P - b a s e d simulations. The use of techniques such as C D A ("Cri t ica l Damping Adjustment" [84], [85]) is effective in the elimination of numerical oscillations. MicroTran has C D A implemented, but other E M T P versions may not, or may use different approaches. 3.1. Modelling Power Electronics in Electric Power Engineering Applications 69 If the gating circuit is not considered in the study, three-terminal, controllable, unidi-rectional current flowing semiconductor devices can be represented by simplified switches with gate turn-on and turn-off controls. Different firing controls can be applied to represent thyristors, GTO's , IGBT's, etc. However, "in many actual power electronics applications, in order to provide a continuous current flow path for an inductive load, a reversal diode (free wheeling diode) is used in parallel with a controllable switching device to form the basic power electronic unit" [91]. The implementation of a "basic power electronic unit" in digi-tal programs requires special care with respect to "instantaneous commutation phenomena" [95], [96], [88], [97]. The U B C version MicroTran of the E M T P , up to now, only allows pre-defined timing for the closing and opening of switches representing semiconductor devices. This is done through the definition of a modified firing angle (" /3 ") and of a "switching frequency" for each semiconductor. The frequency is needed to calculate the switching period and the time of switching from j3, which assumes a fixed reference at time t = 0. This implementation with fixed opening and closing times has limitations, because it does not use the real firing angle ("a") defined from a zero crossing detection related to the voltages at the semiconductor terminals, and because it ignores the dynamics of the control circuits. MicroTran users could write their own firing control subroutine based on an available A L P H A subroutine, but only a few users have used this option. Also, sensing voltages and currents from the main program would introduce a one time step delay with this approach, and users must be aware of that. For P W M control techniques, the auxiliary program " P W M " can be useful to calculate and define the closing and opening times, which would then be read in as a switching table. The dynamics of the control circuit would be ignored, which limits the application of " P W M " to steady-state behavior. As part of this thesis project, a subroutine " G A T E " was developed to simulate power electronics dynamic control schemes with more accuracy and flexibility. As its name indi-cates, the subroutine G A T E allows a simplified gate firing control of a semiconductor, i.e, it can control its turn-on time, and, for some devices, also its turn-off time. The control signal is assumed to be a gate voltage signal, defined for simplicity, between the gate node and ground. This subroutine was derived from the subroutine A L P H A (which can be user 3.1. Modelling Power Electronics in Electric Power Engineering Applications 70 defined), to implement new four-terminal, dynamically controlled semiconductor devices as voltage-controlled switches, as illustrated in Fig. 3.2. With the subroutine G A T E , most of the three-terminal controllable power semiconductor devices can be represented. Fig. 3.3 presents some test cases for the transient simulation of: • voltage-controlled, "bipolar in voltage" (i.e., may conduct irrespective if it is forward or reverse biased) and bidirectional current flowing switch; • Thyristor (simplified model, as a voltage-controlled, "unipolar in voltage" (i.e., may only conduct when forward biased), and unidirectional current flowing switch); • G T O (simplified model, as a voltage-controlled, "unipolar in voltage", unidirectional current flowing switch, with turn-off capabilities). Figs. 3.4, 3.5 and 3.6 illustrate the controlling properties of the bidirectional switch, thyris-tor and G T O , respectively. Since the solution for switches follows the algorithm already implemented in most EMTP-based programs [1], including MicroTran, the change of switch position ("status" on or off) only happens one time step after the enabling gate signal [100]. If it becomes necessary to avoid this delay problem for certain types of simulations, an alternative implementation for a "simultaneous" solution for voltage-controlled switches is presented in the next section of this chapter. _L Voltage-Controlled Bidirectional Switch Figure 3.2: Voltage-controlled switch in EMTP-based programs. 3.1. Modelling Power Electronics in Electric Power Engineering Applications 71 v„ •0 v„ source ^source\^ 1 Q A A A r gate 6 I a A A / V V g a t e V _ 1 Q A A A r V. 7 Vso«rce=10sinfat)[V] \ate=UV] f=60[Hz] Voltage-Controlled Bidirectional Switch THYRISTOR GTO Figure 3.3: Test cases for transient simulation of voltage-controlled, bipolar in voltage and bidirectional current flowing switch, thyristor and GTO. 3.1. Modelling Power Electronics in Electric Power Engineering Applications 72 10 I 3 _ ! " 5 -10 Switch Current / Source Voltage Gate Voltage 2 4 6 8 10 12 14 Time (ms) 16 Figure 3.4: Simulation of a voltage-controlled bidirectional current flowing switch. 1 1 1 1 — 1 1 1 r Source Voltage -10- '•• •' • 1 1 1 _ 1 I i I i _ 0 2 4 6 8 10 12 14 16 Time (ms) Figure 3.5: Simulation of a simplified model for thyristors. 3.1. Modelling Power Electronics in Electric Power Engineering Applications 73 Source Voltage \ GTO Current Gate Pulse i \ -1 1 I 1_ 6 8 10 12 14 16 Time (ms) Figure 3.6: Simulation of a simplified model for G T O ' s . 3.2. Simultaneous Solution for Voltage-Controlled Switches in EMTP-based Programs 74 3.2 Simultaneous Solution for Voltage-Controlled Switches in EMTP-based Programs The implementation of voltage-controlled bidirectional current flowing switches in E M T P -based programs, in such a way that a simultaneous solution is found at each time step for the controlling gate voltage and the voltage-controlled switch, can be done with the com-pensation method presented in Section 2. The necessary equations are, for the controlling branch with the gate voltage, -VoPENj + + ... ••• + rjjij + rjkik + ... + rjMiM + Vj = 0 and for the voltage-controlled switch branch -VoPENk + rk-ih + ... ... + rkjij + rkkik + ... + rkMiM + vk = 0 where: voPENk = voltage vk for [i] = 0 (open circuit). Tkk — Thevenin resistance (self resistance of branch k). rkj = Thevenin resistance (coupling or mutual resistance between branches k and j). It is possible to assume as branch equations: v 3 ( 3 - 1 ) (3.2) Rinij (3.3) and vk = [VJ (Ron - Roff) + Roff)} ik (3.4) where: Ron — On-state resistance, and R0ff = Off-state resistance, for the voltage-controlled switch connected at branch k. From equation 3.4 it is easy to verify that: • ifvj = l, then vk = Ronik\ 3.2. Simultaneous Solution for Voltage-Controlled Switches in E M T P - b a s e d Programs 75 • else if Vj = 0, then vk = R0ffik. Moreover, • if Ron = 0, then vk = 0; • else i f Rnff —> co, then ik = 0. In order to sense the gate control voltage, assume Rin —>• co, which results in ij = 0 (3.5) -VoPENk + rk\ii + ••• + rkjij + rkkik + ... + rkMiM+ ( r . fix + [VJ (Ron - Roff) + Roff)} ik = 0 [ • ] which then can be solved with the implemented Newton-Raphson algorithm illustrated in F i g . 2.19. Observe, however, that the voltage signal Vj is calculated using equation 3.1 (i.e., Vj = VQPENJ — rjiii — ... — Tjjij — Tjkik — . . . — TJMIM) in the solution algorithm, which is always correct and less prone to numerical problems. Assume, for example, the simple circuit wi th a simultaneous solution for a voltage-controlled switch presented in F i g . 3.7. App ly ing an enabling control voltage signal ( IV) at the node gate, the switch turns on at the same time the enabling signal is received. When the gating signal becomes zero (OV), the switch turns off at the corresponding same time, as illustrated in F i g . 3.8. In the case of a conventional switch, as it is implemented in most E M T P - b a s e d programs, the turn-on and turn-off would occur 1 time step later, as illustrated in F i g . 3.9. A similar approach has been proposed in [100] and [73] for the simultaneous solution (also called "synchronized solution" in [73]) of voltage-controlled switches in E M T P - b a s e d programs. Every time a switch changes its status, special computer techniques, such as, for example, the Cr i t i ca l Damping Adjustment ( C D A ) [84], [85], would have to be triggered to avoid numerical oscillation problems in the simulation of power electronic devices. 3.2. Simultaneous Solution for Voltage-Controlled Switches in EMTP-based Programs 76 1 ft A A / V gate \ -/ "Simultaneous" Voltage-Controlled Bidirectional Switch vso=10sin<fot)[V] \ate-UV] f=60[Hz] Figure 3.7: Circuit with "simultaneous solution" of a voltage-controlled switch. T 1 1 r Time (ms) Figure 3.8: Simulation with simultaneous solution of a voltage-controlled switch. 3.2. Simultaneous Solution for Voltage-Controlled Switches in EMTP-based Programs 77 Figure 3.9: One time step delay in EMTP-based switches. 3.3. Implementation of Nonlinear Diode Model in EMTP-based Programs 78 3 . 3 Implementation of Nonlinear Diode Model in EMTP-based Programs The use of simplified models for diodes, as for example, voltage-controlled switches and piecewise linear representation, usually gives simulation results with acceptable accuracy for most of the power system studies. However, according to [89], detailed nonlinear modelling of semiconductors is needed to design snubbers and gate drive circuits. A detailed nonlinear-model for a diode is also needed in the synthesis of equivalent networks to represent, for example, a bipolar transistor with the Ebers-Moll model [77], [101]. The semiconductor diode, with its symbol shown in Fig. 3.10, is therefore, the most common nonlinear element in power electronics. The terminal behavior of a diode [77], with respect to current and voltage as shown in 3.10, is described by where: i(t) is the current through the diode, from anode to cathode, v(t) is the voltage across the diode, i.e., the potential difference between the anode and cathode terminals, ANODE CATHODE Figure 3.10: Diode symbol. i(i) = Is e^> - 1 (3.7) 3.3. Implementation of Nonlinear Diode Model in EMTP-based Programs 79 IS is a constant which depends on the physical properties of the diode, and is usually in the range of 1(T6[,4] to 1(T9L4] \ q — 1.6022 x 10~ 1 9 C is the charge of an electron, k = 1.3806 x I O - 2 3 J / ° K is the Boltzmann's constant, and T is the temperature in degrees Kelvin (273.16°^ = 0°C). When the polarity of v(t) is as shown in Fig. 3.10, the diode is in the conducting region. At 17°C « 290°K, the constant VT = kT/q « 25mV. For v(t) < -WT ( « -75ml/ ) , i(t) « — / , . The value IS is usually referred to as the saturation current. If the diode is forward biased with v(t) > WT (over 100mV), equation 3.7 may be approximated by i(t) = Ise^qv^/kT\ Table 3.1 expresses the relationship between v(t)/VT and i(t)/IS, which are derived from i(t) = is eyyr) - I (3.8) When a constant voltage Vo is applied to the diode, a constant current I0 flows through it. The pair of values (Vo,VTo) is called the operating point of the diode. For each operating point along the characteristic curve of the diode, one could define a dynamic resistance of the diode, which relates increments of the voltage to the increments of the current (dv(t)/di(t) for v(t) = VQ and i(t) = I0). For higher frequencies, additional physical effects come into play and the diode may no longer be treated as a simple nonlinear resistor. Charges stored in the semiconductor material will also require the inclusion of "dynamic capacitive effects" in the nonlinear model of a diode. The value of the capacitance is, in general, a function of the voltage across the diode. The reader is referred to [77], [101] and other references for further details. Reference [89], (pp. 71-1 to 71-7), discuss the P N junction diode switching characteristics, including a sub-circuit for a diode model with reverse recovery. In this thesis project, the compensation method presented in Section 2.2, with a Newton-Rhapson solution algorithm, is used for the solution of the nonlinear model of a diode. In-version of the diode characteristic curve from equation 3.8 and inclusion of a series resistance R o u t results in branch equation 3.10, which is solved together with the linear network equa-1 Actually, Is is temperature dependent and may assume default values of 10 - 1 4[J4] at 27°C [101]. 3.3. Implementation of Nonlinear Diode Model in E M T P - b a s e d Programs 80 Table 3.1: Comparison between voltage and current in a diode as a function of its parametric values. v(t) VT i(t) h 7 1095.633 6 402.429 5 147.413 4 53.898 3 19.086 2 6.389 1 1.718 0 0 -1 -0.632 -2 -0.865 -3 -0.950 -4 -0.982 -5 -0.983 -6 -0.998 -7 -0.999 tion 3.9 to determine the operating point of the diode for a particular network condition, as illustrated in F i g . 3.11: ~vQpENk + T-fc.il + . . . + rkkik + ... + r k M i M + vk = 0 (3.9) vk = VT ln ( £ + l ) ] + Routik (3.10) where: voPENk = voltage vk for [i] = 0 (open circuit). Tkk = Thevenin resistance (self resistance of branch k). vk = voltage of branch k, i.e., voltage across the diode (the potential difference van0(te — vcathode) plus the voltage drop across the series resistance E,out. ik = current, through the diode, i.e., the current flowing from the anode to the cathode terminal. Rout — series resistance for the diode model (which may be assumed to be equal zero). 3.3. Implementation of Nonlinear Diode Model in EMTP-based Programs 81 §}, o -2 i 1 1 1 — Linear Network Thevenin Equivalent Circuit / VOPEN® ^^-^^ Nonlinear Diode Characteristic / -{S C <*> = VOPEN <*) 1 <r* + Ra*> - \ Resulting nonlinear equation to be solved with the Newton-Raphson method 1 i i i - 3 L -0.5 0.5 1.5 2.5 Current (A) Figure 3.11: V-I diode characteristic and network Thevenin equivalent circuit equation. Inserting equation 3.10 into equation 3.9 results in: -voPENk + rk\i\ + . . . . . + [rkk + Rout] ik + VT ln + • • • + r k M i M = 0 A general circuit representation of equation 3.11 is illustrated in Fig. 3.12. (3.11) Equation 3.11 is solved iteratively at each time step. Note that the Thevenin equivalent circuit equation changes from step to step, with a change in VOPEN^), and in general with a change in the Thevenin equivalent resistance (slope dv/di). Fig. 3.13 illustrates this, where it is assumed, for simplicity, that the Thevenin equivalent resistance does not change along the simulation time. The iterative solution with the Newton-Raphson algorithm requires an initial guess. De-pending on how close the initial guess is to the final solution, convergence can be very fast or 3.3. Implementation of Nonlinear Diode Model in EMTP-based Programs 82 v, OPEN k 6 A / W [ rTHEV k\ R out vTln(ik/IS+1)Q v. F i g u r e 3.12: Circuit implementation for the simultaneous solution of a nonlinear diode model. •1 0 1 Current (A) Figure 3.13 V-I diode characteristic and different network Thevenin equivalents. 3.3. Implementation of Nonlinear Diode Model in E M T P - b a s e d Programs 83 on the contrary, convergence can be very slow for some cases, or numerical problems such as computer overflow may even arise. Therefore, it is important to derive a heuristic computer technique for in i t ia l guesses, to accommodate the highly nonlinear exponential characteristic of a diode, and speed up the convergence of the solution. A robust rule should be general and insensitive to the network or diode parameters. The detailed development of such a technique for implementation in E M T P - b a s e d programs is presented in the following. A simple inspection of F ig . 3.11 reveals some "physical" candidates for the ini t ial guess of the nonlinear diode current, associated with the network conditions, i.e., with the particular linear network Thevenin equivalent circuit: VQPENk (Tkk+Rout) (3.12) ik («OPENk\ ;\ VT ) — I (3.13) -L (3.14) Equation 3.12 is the short circuit current iksc, which becomes the ini t ia l guess for the diode current ik in the conduction mode. Then, if v0pENk > 0 and vOPENk > VTln (^ff- + l ) equal to iksc, as calculated with equation 3.12. , the ini t ia l current is assumed to be • If VopENk > 0 and voPENk < ^ r l n {^f2- + l ) , then the voltage across the diode is first estimated to be equal to v0pENk, and a better estimate for the current is calculated with equation 3.13 (Another simple alternative would be just to assume ik = 0); • O n the other hand, if v0PENk < 0 and v0PENk > -6.0VT, then again the voltage across the diode is first estimated to be equal to vopENk, and the ini t ia l current is then calculated wi th equation 3.13; • If v0pENk < 0 and VopENk < —6.0VV, then the diode current is assumed to be as calculated wi th equation 3.14. Alternatively, one could use a linear function instead of using equation 3.14. 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]. For example, piecewise linear approximations could be used first to determine the initial guess (either for the current or for the voltage across the diode) and then, the linear characteristic would be replaced by the detailed nonlinear equation. This approach might be useful for speeding up the solution of networks with multiple nonlinear elements, such as diodes, transistors, etc. A two-slope piecewise linear approximation of the diode characteristic is accurate enough in many cases. Even though the equations are linear in that case it is not known a priori on which segment the solution ends up. This is particularly true if there are many diodes to be solved. The production code of MicroTran still uses the Newton-Raphson method for the two-slope piecewise linear representation, which will then decide, iteratively, the solution points for the diodes. Theoretically, there is no guarantee that the iterations will converge in such cases, but by limiting the new voltages and currents to technically reasonable values, the method has so far converged in all tested cases, with up to 6 diodes. The use of linear piecewise or detailed nonlinear models of electronics semiconductors might also be a requirement for the correct solution of commutation phenomena in power electronics circuits. "In circuits with gate turn-off thyristors (GTO's), commutation of the current into a diode must often be instantaneous, without any current interruption. An example for such situation is the buck-boost (step-down/up dc-dc) converter shown in 'Fig. 3.14 [93]' and the half-wave rectifier circuit with a freewheeling diode in 'Fig. 3.15'. ... If the diode is represented as a switch which closes when the voltage from anode to cathode becomes positive (either built into the code as in MicroTran, or controlled in this way through TACS), then the positive voltage 'across the diode' will only be seen in the next time step immediately following the time step in which the G T O turned off. This is one time step too late, because the current in the G T O would already have dropped to zero." [95], [78]. Therefore, EMTP-type simulation of "instantaneous commutation" with simple switch models might give wrong results in the case of switching converters. User knowledge of the commutation process can be used to pre-define the switches which will have simultaneous commutation [78], but it may not work for all cases. A better option is the compensation method used in this work, which assures a simultaneous solution of all equations [95]. 3.3. Implementation of Nonlinear Diode Model in EMTP-based Programs 85 GTO DIODE 6 Figure 3.14: Dc-dc converter. DIODE 1 v = 155sin(ot) [VJtQJ) J -50[V] Figure 3.15: Half-wave rectifier with freewheeling diode. To illustrate the solution method presented here, assume, for example, the simple electric circuit with a nonlinear diode as in Fig. 3.16, where the basic parameters for the diode are: Is = 10~1 2[A] and VT = 0.026[V]. The time response solution for this circuit is presented in Fig. 3.17. Fig. 3.18 shows the detail of the diode current at the zero crossing 2 , whereas Fig. 3.19 presents the V-I nonlinear characteristic of the diode resulting from the E M T P simulation. The average number of iterations with the Newton-Raphson algorithm, and using the proposed heuristic technique for initial guess, was 1 for this case, assuming a convergence tolerance error of less than 1 0 - 1 0 for the resulting nonlinear equation, and also forcing at least 1 iteration in the algorithm. Detailed modelling of other semiconductor devices, such as bipolar transistors, field ef-2 T h e 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]. 3.3. Implementation of Nonlinear Diode Model in EMTP-based Programs 86 f=60 [Hz] 1 Q A A / V DIODE xz V, DIODE Figure 3.16: Electric circuit with a nonlinear diode model. Diode Current Diode Voltage 6 8 10 Time ( ms ) 12 14 16 Figure 3.17: Transient simulation of a nonlinear diode model in an EMTP-based program. feet transistors, etc. can be accomplished with the "synthesis of equivalent networks" (sub-circuits) using nonlinear diodes, dependent sources, voltage-dependent capacitances and re-sistances [101], [77]. The Ebers-Moll model of a bipolar transistor, or more complex models (Gummel-Poon) could be used as well, but their description is beyond the scope of this thesis project. The use of macromodels, based on functional terminal conditions, can also be derived with the interconnection of circuit elements, especially with the use of the de-pendent sources presented in Section 2.2. Piecewise linear approximations can also be used to model device characteristics, but one should be aware of the fact that even though the 3.3. Implementation of Nonlinear Diode Model in EMTP-based Programs 87 x 10 1.5 0.5 -0.5 -1 .5 1 i i Diode Current • Is=1.0E-12 [A] 1 1 1 7.5 8.5 Time ( ms ) 9.5 F i gure 3.18: Detail of the transient simulation of a nonlinear diode model in an EMTP-based program. " 1 0.5 bo-0-5 -1 .5 - 2 -0.2 0.2 0.4 0.6 O.f Current (A) 1.2 1.4 Figure 3.19: V-I nonlinear characteristic of the diode resulting from the E M T P simulation. 3.4. Control Modelling Aspects of Power Electronic Devices 88 approximations are continuous from segment to segment, the derivatives are discontinuous. Spline fitting techniques can then be used to approximate the device characteristic by sepa-rate low-order polynomials between adjacent segments, with the cubic spline being the most popular one [77]. 3.4 Control Modelling Aspects of Power Electronic De-vices The digital simulation of control devices for power electronics applications, such as recti-fiers, inverters, D C - D C converters, A C - A C converters, motor drives, etc. is made challenging because of the more frequent use of mixed analog and (real-time) digital signal processing (DSP) control techniques. The derivation of reliable system references requires a growing number of digital and analog components. Their modelling can, in general, not be simplified without compromis-ing the simulation accuracy and stability. For example, signal sensoring and zero crossing prediction or detection usually requires an appropriate use of signal filtering, either through analog design (operational amplifiers, resistances and capacitances) or its respective digital implementation. However, in the simulation of DSP controls it is important to pay attention to the digital sampling frequency with respect to the selection of the time step size (At). System frequency tracking requires the use of appropriate phase-locked loop (PLL) con-trols, which are of fundamental importance for the control of high power electronics appli-cations, such as F A C T S and Custom Power Controllers [90]. Pulse width modulation (PWM) techniques, which are commonly used in voltage-sourced and current-sourced converters, require particular attention in digital implementations, due to the discrete nature of computer simulation programs, where the time step size may affect the results of discrete binary comparators, thus influencing the simulation results. Special transformation of variables into other reference systems (e.g., abc to a(30 transfor-mation, which is used in active filter control), requires a simultaneous solution of control and system equations. Digital simulation of hysteresis effects, limits (windup and non-windup) 3.4. Control Modelling Aspects of Power Electronic Devices 89 and various non-linearities in the control system also requires appropriate models for a suc-cessful computer simulation. Moreover, with variations in the supply voltage caused by power system disturbances, the control, ideally, should be able to actively withstand and support its primary regulation functions without disrupting the electric supply to the controlled load. However, in most practical power quality cases, typically related to voltage sag problems, power electronics-based loads (i.e., their control) are either extremely sensitive to momentary voltage variations causing frequent shut-downs in industrial process operation, or improper designs may become the cause of many power quality problems. As part of this thesis project, "basic control devices" were experimentally implemented in MicroTran. The main advantage of this development, compared to TACS (Transient Analysis of Control System) used by many other E M T P versions, is that a "true simultane-ous solution" is found through the compensation method using a Newton-Raphson iteration scheme. Therefore, provided that appropriate computer techniques are used to allow conver-gence in the solution, this method has been shown to be very robust for the cases simulated up to now, and does not have the 1 time step delay present in the interface of E M T P and TACS and also does not have any internal delays in the linear and non-linear control solution. With graphical user-interfaces for MicroTran, it would become easy to define libraries of control devices as well as of power components (and sub-circuits, similarly as in [65], [102], [103]), in such a way that no differentiation between power and control circuit will then be necessary, since a unique simultaneous solution approach would be used, based on the method and algorithm presented in this Ph.D. thesis. Chapter 4 Evaluation of the Impact of Power Electronic Devices on the Quality of Power H E analysis of the dynamic interaction between power electronic devices and power _L- systems and the assessment of electric power quality phenomena can be thoroughly done with EMTP-based programs. The objective of this Ph.D. thesis research project was to develop reasonably accurate models for EMTP-based programs, with which one could evaluate the impact of high power electronic devices on the quality of power. To make the thesis results more valuable to utilities and industries, the following practical tasks were carried out: • Cooperation with a utility company in Brazil, E L E K T R O - Eletricidade e Servigos S. A . which kindly agreed to provide data and real power quality problem cases to validate simulation results. Financial support for this cooperation was provided by C A P E S 2 , a federal agency of the Brazilian Government, for a field activity at E L E K T R O with the duration of three months. • Appropriate models for time-domain simulations using EMTP-based programs were applied to simulate some of the real power quality problems experienced by E L E K T R O . ' E L E K T R O - Eletricidade e Servicos S. A. , Rua Ary Anterior de Souza, 321 - Jardim Nova America, CEP 13053-024, Campinas-S. P. , BRAZIL. 2 C A P E S - Fundacao Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior, Esplanada dos Ministerios, Anexo I, Sala 215, Caixa Postal 00365, CEP 70047-900, Brasilia - D. F. , BRAZIL. 90 4.1. Dynamic Interaction between Power Electronic Devices and Power Systems 91 • Based on the power quality monitoring and simulations, a synthesis of simulation guidelines for power quality evaluation was developed, with emphasis to determine the dynamic interaction between power electronic devices and the power systems, through the use of time and frequency-domain techniques with EMTP-based programs. This chapter presents some simulation cases of power quality assessment with the use of the existing features of MicroTran, the U B C version of the E M T P . The simultaneous solu-tion of control and electric power system equations (SSCPS), with the new circuit approach presented in Chapter 2 and with the models developed for the dynamic control of power semiconductor devices presented in Chapter 3, are illustrated through practical control and power electronics controllers simulation cases. Important simulation guidelines for the eval-uation of the impact of power electronic devices on the quality of power are also summarized in this Chapter. 4.1 Dynamic Interaction between Power Electronic De-vices and Power Systems The interaction between the utility supply system and power electronics-based loads, (such as electronic converter controlled electric motor drives), depends on a variety of factors [104], as for example: 1. the type of "front-end" electronic converter, which converts line-frequency A C into DC (diode-bridge rectifiers (which are unidirectional in power flow), switch mode converters (in which, power flow can be reversed), thyristors converters (which can be made bidirectional in power flow); 2. the number of phases (single-phase, three-phase) from the supply system used by the converter, and the converter configuration (e.g., 6, 12, 24, 48, etc. pulses converters), which also affects the waveform current distortion; 3. the "strength" (or "stiffness") of the utility system, determined by its "short-circuit power"; 4.1. Dynamic Interaction between Power Electronic Devices and Power Systems 92 4. the number of power electronics based loads, the point of their connection in the network, and the electric power of electronic converters, which wi l l influence their impact on the electric supply system and vice-versa; 5. the design and control of the electronic converter. The choice of parameters and compo-nents, particularly in cases where low-cost choices are made, can cause a deterioration on the quality of power of the uti l i ty system. O n the other hand, the converter opera-tion (i.e., its control) can easily be disrupted by power system disturbances travelling through the ut i l i ty network, as is very common in the case of sensitive loads. The most common lower-power electronics based load uses a single-phase diode-bridge rectifier [94], [104] which draws highly distorted waveshape current from the uti l i ty system, and may cause harmonic associated problems, such as: • overheating in neutral conductors due to the flow of high third harmonic currents; • voltage distortion along the distribution circuits; • increase i n power losses; • risk of resonances wi th ut i l i ty or industry power factor correction capacitor banks, etc. This type of rectifier only draws current close to the maximum peak of the "assumed sinu-soidal" ut i l i ty voltage source, in order to recharge the capacitor filter on the D C side of the rectifier. To illustrate that, F i g . 4.1 presents a typical single-phase diode bridge rectifier [93], [94], [95], [104] 3 , where Lsi represents the system Thevenin equivalent inductance (resistance ignored here) and Ls2 represents any series inductance added in the A C side of the rectifier. Assume that the diodes are ideal and the circuit has been energized a long time ago, such that the D C capacitor filter has already been charged (i.e., the transient energization has gone and "stead state waveforms" are present). Note, from F i g . 4.2 , that the current is (and consequently id) only starts to flow during the positive semicycle of the source voltage, when VSA is greater than the voltage (i.e., when diodes 1 and 2 are forward biased at 3 Reference [95] presents useful guidelines on power electronics applications using the EMTP. 4.1. Dynamic Interaction between Power Electronic Devices and Power Systems 93 time t\). Conversely, during the negative subcycle, when V$A is less than Vdc-. the current is starts to flow (i.e., when diodes 3 and 4 are forward biased at time £4), with id = —is flowing in the same direction, corresponding then to the electronic current rectification process). Fig.4.3 presents the harmonic amplitude spectrum of the current drawn from the source by a single-phase diode-bridge rectifier. I f=60[Hz] d + •• V dc ;iooop.F >2oa Figure 4.1: Circuit with a single-phase diode-bridge rectifier. To better understand the waveform voltage distortions caused by a single-phase diode-bridge rectifier, one can apply the Kirchhoff's second law to the circuit of F i g . 4.1, and derive the following equations, VSA - vL - vA = 0 (4.1) vL = vLl + vL2 = Lslf + Ls2f ( 4 2 ) where: vL = voltage across the total inductance L = L s 1 + L s 2 , i-i = ?;2 = is-4.1. Dynamic Interaction between Power Electronic Devices and Power Systems 94 200 10 15 20 25 30 Time (ms) 35 40 45 50 Figure 4.2: Current drawn from the source by a single-phase diode-bridge rectifier. Therefore, the following voltages can be defined: dis VA = VSA - L— at (4.3) vpcc = V S A - Lsl dis dt (4.4) From Fig. 4.4 it can be seen that the inductances influence the rate of change (and con-sequently the waveshape) of the current, such that is starts to grow exponentially, reaching a maximum peak value when the voltage across the inductance is zero, after which vL changes its polarity (because the derivative of the current changes its signal 4 ) physically trying to keep the current, flowing until the current finally becomes zero. Following the equations 4.3, 4.4 and Fig. 4.4 one can easily understand the voltage waveforms distortions in the rectifier input, VA, and in the point of common coupling (PCC), vpcc, where many other loads may "As a matter of fact, many D C - D C electronic converters rely on the inductor physical properties during switching transients, to either step voltages up (boost) or step voltages down (buck). 4.1. Dynamic Interaction between Power Electronic Devices and Power Systems 95 30 25 20 K a S 10 ! ! Thin i Harmon ic Compt ment III. . _ _ — -i • i 10 15 20 25 30 Harmonic Order 35 40 45 50 Figure 4.3: Harmonic amplitude spectrum of the current drawn from the source by a single-phase diode-bridge rectifier. be connected and may be affected by the waveform voltage distortion. Fig.4.5 presents the harmonic amplitude spectrum of the voltage at the P C C . In a four-wire, three-phase system with "well balanced" loads, composed by single-phase diode bridge rectifiers, as illustrated in Fig. 4.6, the current flowing through the neutral conductor has mainly third harmonic components (i.e, 180Hz, for a 60Hz fundamental) as shown in Fig. 4.7 and Fig. 4.8. Observe that the third harmonic component of the neutral current in Fig.4.8 is approximately 3 times the third harmonic component of the current of a single-phase diode-bridge rectifier as illustrated in Fig.4.3. These high currents may cause overheating in the neutral conductor (usually designed with a smaller cross section than the phase conductors), thereby creating a potential hazardous risk, such as fire. The widespread use of this type of inexpensive rectifier, usually as the front end of low power appliances, such as television sets, computers and compact fluorescent lamps (CFL) 4.1. Dynamic Interaction between Power Electronic Devices and Power Systems 96 (which are used more and more because of its higher energy efficiency), can have a cumulative negative effect on the quality of power supplied to residential, commercial and industrial customers [105] . For example, Fig. 4.9 and Fig. 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 U B C , Vancouver, B.C. , Canada 5 . The waveshape distortion is probably caused by the large number of computers in the building, as well as in the entire university. It also affects, presently with minor consequences, all the other loads in the building supplied from the same common bus, and eventually, also propagates through the B C Hydro electric supply system. Fig. 4.11 and Fig. 4.12 show the results of a Fourier analysis of the outlet voltage curve (the DC component present in the harmonic amplitude spectrum at Fig. 4.11 might be caused by inaccuracies or dc offsets in the measuring equipment). Although, according to the present standards [48], [49], such 5The help of Mr. Kenneth Wicks in doing this measurement is gratefully acknowledged. 4.1. Dynamic Interaction between Power Electronic Devices and Power Systems 97 100 90 80 70 60 50 ft) S -§ & 40 R ft) 30 ft) a, 20 10 • • Fund = 119.6844 Volts RMS = 119.9200 Volts Creast Factor = 1.3851 Min = -166.1000 Volts Max= 166.1000 Volts THD = 6.2778 % HRMS = 7.5136 Volts TIF / IT = 121.8154 10 15 20 25 30 Harmonic Order 35 40 45 50 Figure 4.5: Harmonic amplitude spectrum of the voltage waveform distortion at the point of common coupling (PCC) . sa Vmax= 169.7 IT] f=60[Hz] 6) lNEUTRAL diode bridge diode bridge 1 diode bridge vsc Figure 4.6: Four-wire, three-phase system with "balanced" single-phase diode-bridge rectifiers. harmonic voltage distortions are usually within acceptable limits, other sensitive equipment may be affected, and better alternatives for power conversion are actually available. 4.1. Dynamic Interaction between Power Electronic Devices and Power Systems 98 Time (ms ) F i gure 4.7: Current flowing through the neutral conductor. By using, for example, step-up (boost) D C - D C converters (consisting of IGBT's switch-ing at high frequency with P W M control techniques, free wheeling diodes and inductors), in connection with diode-bridge rectifiers, power factor corrected (PFC) interfaces can be designed. Such P F C circuits are able to draw almost sinusoidal currents at close to unity power factor. Single- or three-phase controlled thyristor converters can also adversely affect the qual-ity of power, due to their distorted current waveforms, the notching of the input voltage waveform caused by the commutation among the thyristors, and the poor power factor. The choice of a power electronic converter is based on its intended application and on the price. With the introduction of more strict standards for power quality, and with growing concerns about the dynamic interaction between power electronic devices and the power system, new technologies with less impact and/or less susceptibility to power disturbances are gaining market acceptance. Examples are active filters and other dynamic compensating 4.1. Dynamic Interaction between Power Electronic Devices and Power Systems 99 30 r 1 1 Third Harmonic Component 25 20 K ft; 15 10 I 10 15 20 25 30 Harmonic Order 35 40 45 50 Figure 4.8: Harmonic amplitude spectrum of the current flowing through the neutral conductor, devices. For a proper identification and solution of power quality problems, transient and steady-state analysis are needed, which include models not only for the power electronics but also the power system part, because of the dynamic interaction between them. Power electronics based loads can either be the cause of problems in the power system, or they can be adversely affected by electromagnetic transient phenomena coming from the power system. Reference [24] presents the fundamental definition of electromagnetic phenomena affecting the electric power quality, with realistic cases and practical monitoring results and examples of power disturbances, such as: • Voltage sags and interruptions; • Transient overvoltages; • Harmonics; 4.1. Dynamic Interaction between Power Electronic Devices and Power Systems 100 Phase Voltage -2001 1 1 1 1 1 1 1 1 I 0 2 4 6 8 10 12 14 16 (ime f ms ) Figure 4.9: Voltage waveshape measured at the outlet of the Power Electronics Laboratory of the Depart-ment of Electrical and Computer Engineering at UBC, Vancouver, B.C., Canada. • Long duration voltage variation; • Wiring and grounding practices. Voltage sags are by far the most common cause of disruption of operation of power elec-tronics based loads, such as electronically controlled motor drives. Many industrial processes (e.g. pulp and paper, textile, automotive, etc.) rely on accurate speed and torque control through the use of power electronics, and thus become more or less vulnerable and suscepti-ble to power quality problems depending on the sensitivity of these devices. This is actually at the heart of many power quality problems! It is also common that utility capacitor switching creates high frequency transients, which may propagate through the distribution system and cause amplified 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 4.1. Dynamic Interaction between Power Electronic Devices and Power Systems 101 Phase Voltage -200' 1 1 1 1 1 1 1 1 ' 0 2 4 6 8 10 12 14 16 rime ('ms ) Figure 4.10: Measured voltage waveshape, its fundamental component and its harmonic distortion. plants may excite natural resonance modes of weak distribution and other industrial systems, causing high frequency oscillations in the voltages and, consequently, "nuisance tripping" of sensitive loads. [107]. Resonances tend to occur more frequently as more power factor and voltage support, capacitor banks are used in the system, mainly to control voltage on transmission or distri-bution lines. Typical ly, the 5th order harmonic current commonly "injected" into the power system by tradit ional power electronics converters, has the potential to cause problems, such as capacitors failures. The importance of power quality then depends on its economic im-pact on the industry, the utility, the society, and the country. Appropriate means to predict problems in the early design stage or to diagnose and mitigate problems in existing systems becomes a very important task of "power quality engineers". E M T P - b a s e d programs, because of many available computer models for power systems and power electronics, have become a necessary engineering tool for the evaluation of the 4.1. Dynamic Interaction between Power Electronic Devices and Power Systems 102 2.5 Harmonic Amplitude Spectrum 1 1 Fundamental = i 100% ! j 1 • 1 l i d .III. .1.1.1 LL.I I i _ l . a l l . i l K 1.5 S "§ K * 1 cu cu 0.5 10 15 20 25 30 Harmonic Order 35 40 45 50 Figure 4.11: Harmonic amplitude spectrum of the outlet waveshape voltage. impact of power electronic devices on the quality of power. An extensive literature survey about EMTP-based models for time and frequency domain analysis of electric and electronic power systems can be found in [92], [6], [2], and elsewhere. Specialized conferences, such as the International Power System Transients Conference (IPST) held every two years, provide opportunities to exchange information about new techniques and practical experiences in the use of EMTP-based programs. 4.1. Dynamic Interaction between Power Electronic Devices and Power Systems 103 Harmonic Phase Angle 1 1 1 1 1 1 i } 1 1 1 J 1 T\ t I I 1 1 I I I I I I I 0 5 10 15 20 25 30 35 40 45 50 Harmonic Order F i gure 4.12: Phase-angle of the harmonic components of the outlet waveshape voltage. 4.2. Power Quality Assessment through EMTP-based Programs 104 4.2 Power Quality Assessment through EMTP-based Programs This section presents some real cases of power quality assessment through EMTP-based simulations. Field test measurements made by the author and comparisons with time and frequency domain computer analysis are also shown for some of the cases. Section 4.2.1 presents a harmonics case study. Section 4.2.2 presents a voltage sag case study. Section 4.2.3 is concerned with visual light flicker caused by voltage fluctuations. 4.2.1 Induc t ion Furnace H a r m o n i c S tudy The problem of harmonic analysis in power systems is usually studied with steady-state solution techniques, which use linear solutions at the harmonic frequencies. The charac-teristic harmonic spectra of non-linear loads are assumed to be known, and are modeled as current sources at the respective harmonic frequency. In reality, the harmonic current sources are not exactly known, because they depend on the behaviour of the power system as well. For example, harmonics from transformer saturation clearly depend on the voltage magnitude and waveform at the transformer terminal. Only time domain simulations of the E M T P type can address the interaction between the system and the harmonic sources, which can result in non-characteristic harmonics as well. Time domain simulations can also be useful to develop other types of power quality studies, such as fault analysis, transient impulses caused by switching utility capacitor banks, diagnosing the effects of special loads into the system, troubleshooting the failure of sensitive loads, evaluating the application of "Custom Power" devices as solutions to power quality problems, etc. This section presents an application of the E M T P in a distribution system study, where the harmonics injected into a distribution feeder by induction furnaces were the prime con-cern for power quality. It is taken from a harmonic problem experienced by E L E K T R O -Eletricidade e Servigos S. A . 6 , an electric utility in the southeast of Brazil. The three-phase power system with a digital model of the induction furnace as a power electronic load was B E L E K T R O - Eletricidade e Servicos S. A. , Rua Ary Antenor de Souza, 321 - Jardim Nova America, CEP 13053-024, Campinas, S. P. , BRAZIL. 4.2. Power Quality Assessment through EMTP-based Programs 105 simulated with the E M T P over a time span which was long enough to reach steady state. Voltage and current waveforms were then analyzed with a Fourier analysis program to obtain the harmonic content of the distorted waveforms. Field measurements at the point of com-mon coupling between the utility and the industry are presented as well. Electromagnetic transients programs are more accurate in representing nonlinear effects of the supply system, and allow more detailed modelling of power electronic loads and devices, than steady-state harmonic programs. Power electronics and power quality have such a strong correlation that can only be fully described and analyzed with the use of time domain simulation techniques. Induction heating has gained wide acceptance in industry because this type of heat-ing process is considered clean, quick and efficient. On the other hand, the use of power electronic devices for induction heating introduces harmonic currents, causing voltage dis-tortions in the electric supply network. The dynamic interaction of these harmonics with the electric system in terms of system configuration, loading and other conditions, may result in linear resonances, or even in undesirable steady-state conditions, which all can result in misoperation, failure and life reduction of equipment, with consequent economical losses. Induction furnaces are power electronics-based loads, where the heat in the electrically conducting workpiece to be melted is produced by circulating currents through electromag-netic induction. Series or parallel-resonant inverters are typical configurations used to supply energy to the induction coil, at a selected frequency, which can be in a range varying from the power system frequency to a few hundred kilohertz [94]. Figs. 4.13 (a) and (b) illustrate an induction furnace in operation. The operation of these induction furnaces has produced distortions in the current and voltage waveforms, and has created incompatibility problems between these special loads and other sensitive loads connected on the same distribution feeder. Changes in the configuration of the power supply system as well as application of passive filters, have minimized the effects, but have not completely eliminated the harmonics power quality problem. Field measurements have been made for different operating conditions to evaluate the effectiveness of the already installed harmonic passive filters. 4.2. Power Quality Assessment through EMTP-based Programs 110 T H D (%) \j , — — — — — c O ' d - T t r ^ o o c D o c s j n ^ ' t f o o c n a ) O T - c o * * L r ) c o o o o o o o o o o o o o o o o N n ^ i n i b s o i d ^ N n o N r i ^ i f l i i s o i d r : ' < - ' < - - ^ ' « - T - T - T - ( M C M C N J C S J O O O O O O O O T - T -Figure 4.18: T H D harmonic trend, with harmonic passive filters turned on all the time. as active filters, might be able to dynamically compensate the distortions and improve the quality of power at the interface of industrial and power systems. As far as the author knows, new Brazilian legislation about harmonics and other power quality phenomena are currently under development. Some recommendations have been used to guide utility system planners and operators in supplying power to special loads. Refer-ence [50] for example, presents some criteria and global voltage harmonic limits (Table 4.1), based on the experience of Brazilian power utilities, as well as on standards from C I G R E (Conference Internationale des Grands Reseaux Electriques a Haute Tension - International Conference on Large High Voltage Electric Systems), IEC (International Electrotechnical Commission) and I E E E (The Institute of Electrical and Electronics Engineers). Table 4 .1: Global harmonic distortion limits for the system voltages recommended in Brazil . Voltage < Q9kV: THDmax = 6% Voltage > &9kV: THDmax = 3% O d d Even O d d Even Order Value % Order Value % Order Value % Order Value % 3, 5, 7 5 2, 4, 6 2 3, 5, 7 2 2, 4, 6 1 9, 11, 13 3 > 8 1 9, 11, 13 1.5 > 8 0.5 15 to 25 2 15 to 25 1 > 27 1 > 27 0.5 4.2. 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, including the linear and non-linear loads, transformer saturation effects, unbalanced conditions, power electronic loads, automatic control devices, frequency dependent characteristics of the system and of the loads, and so on, would be the ideal and recommended database for an electromagnetic transient simulation, in order to analyze power quality phenomena. This would require a complete and well-organized database of the system and load parameters, which is rarely available in prac-tice. For distribution system planning and operation, such details are usually not required, unless some specific power quality problem emerges as urgent and important. Typically, nei-ther some important data is available, nor appropriate models exist to represent the physical behaviour by digital simulation. One must therefore use simplifications, which may make the simulations unrealistic. Therefore, the development of more accurate models is needed for power quality studies. For this case study, some realistic data was available, and some simplifications had to be made for other data. The actual system under study is shown in Fig. 4.19. A Thevenin equivalent circuit with a series connection of coupled resistances and inductances was used to represent the 138kV transmission system, based on the given three-phase short-circuit power (2881.3 M V A , angle of - 78 degrees) and single-line-to-ground short-circuit power (1734.1 M V A , angle of - 77.7 degrees) from which the positive and zero sequence impedances at the frequency of 60Hz can be calculated (assuming 100MVA as a base power, 138kV as base voltage, and zero fault resistance for a single-line-to ground fault, one obtains: Zp0Sp.u. = 1/Ssc3php.u., Zzerop.u. = 3/Ssciphgp.u. — 2 * Zposp.u ). In reality, the resistance and inductance derived from these impedances are frequency-dependent, which was ignored in this study. The transformer model used was based on three single-phase coupled impedances ("IN-V E R S E " option in MicroTran). The distribution line was modelled as a three-phase cou-pled 7r-circuit, with positive and zero sequence parameters at 60Hz. For the frequencies of interest here, a 7r-circuit representation is reasonably accurate. For higher frequencies, a distributed-parameter line model would have to be used, either with constant parameters or 4.2. Power Quality Assessment through EMTP-based Programs 112 with frequency-dependent parameters. The data for all the distribution feeders with their respective loads, and for the capaci-tor banks for power factor correction were available. Not enough information was available though for the induction furnaces. A digital model based on [93], [94] was therefore used: a current-source, parallel-resonant inverter for induction heating, as shown in Fig. 4.20. The resonant inverter is used to create variable frequency at the induction coil. The six-pulse controlled rectifier on the A C side of each induction furnace was supplied through a 13.8/0.48kV - 3.0MVA - 5.9% three-phase unit transformer in delta/wye-grounded connec-tion. Saturation effects were not considered in this simulation. Realistic parameter values of resistance, inductance and capacitance were available for the 4th and 5 t h order harmonic passive filters. EMTP-based programs can perform time-domain transient analysis or frequency-domain analysis. A frequency-domain analysis was done for this case to find the system impedance as a function of frequency, seen from the point of common coupling (PCC) of the induction furnace, as shown in Fig. 4.21 and Fig. 4.22. Not only the impedance at multiples of the fundamental frequency was evaluated, but over the continuous frequency range as well, by using a small step increment (Af) in the frequency variation input parameter. It was small enough to allow linear interpolation between calculated points. The two minimum impedance values at 240Hz and 300Hz shown in Fig. 4.21 correspond to the effects of the 4th and 5th order harmonic filters. Fig. 4.22 also shows the zero crossings of the phase angle of the system impedance, from inductive to capacitive and vice-versa, thus indicating parallel (maximum impedance) or series (minimum impedance) resonant conditions, respectively. Next, the three-phase distribution system with the digital model of the induction furnace as a power electronics load was simulated as a transients case (At = 16.6667^s), using the MicroTran version of the E M T P , until a time when steady state was reached. Voltage and current waveforms were then processed through a Fourier analysis program to obtain the harmonic content of the distorted waveforms. For this simulation case, all the distribution feeders were represented. The induction furnace operating condition selected for this case corresponds to the time of maximum total harmonic distortion (THD) measured at the point 4.2. Power Qual i ty Assessment through E M T P - b a s e d Programs 113 Ssc3 h =2,881.3 /-78.00 MVA Ssc1phg = 1,734.1 /-77.40 MVA 138 kV BusI T T 25/30MVA 138/13.8kV i L U j 9.72% (25MVA) ^ r^y^j 1 3 . 8 k V r T j / / 25/30MVA 138/13.8kV 10.2% (25MVA) BusII r 1 3 . 8 k V 1 A / / / 9.0MVAr 13.8kV 477.0 ACSR - 2km PCC 336.4 ACSR • 0.5km 3.0MVA A J j 13.8/0.48kV . ™ 5.9% i 0.48kV A L J J A L J J ^ F p A Fp A; Induction Furnaces A Harmonic Passive Filters A Figure 4.19: Distribution substation. of commom coupling ( P C C ) , without the harmonic filters (see F i g . 4.17). Figs. 4.23 (a) and (b) show the current and voltage waveforms respectively, with their harmonic contents, at the point of common coupling ( P C C ) between the uti l i ty electric system and the customer facilities, for the induction furnaces operation with the 4th and 5 t h order harmonic passive filters turned off. F i g . 4.15 is repeated here as F i g . 4.24 to facilitate 4.2. Power Quality Assessment through EMTP-based Programs 114 POS X X INVA L c Cr INVB X X L i > >Rload NEG INDUCTION FURNACE Figure 4.20: Current-source, parallel-resonant inverter for induction heating. frequency (Hz) Figure 4.21: Amplitude of the positive sequence system impedance at the PCC with harmonic filters. the comparison between the simulated and the measured results. Figs. 4.25 (a) and (b) show the current and voltage waveforms, but for induction furnace 4.2. Power Quality Assessment through EMTP-based Programs 115 100 80 60 40 co 20 CD 0 bo K ^ "20 a; -40 -60 -80 -100 ! ! ^ \ : ? \ vj \ \1 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 filters. operation with the 44 f t and 5th order harmonic passive filters turned on. This operating condition corresponds to the time of minimum total harmonic distortion (THD) measured at the point of common coupling (PCC), but in this case with the harmonic passive filters turned on (see Fig. 4.18). Both operating conditions have approximately the same value of fundamental current (199A RMS at the phase-to-phase RMS voltage of 13.8kV). Again, Fig. 4.16 is repeated here as Fig. 4.26 to facilitate the comparison between the simulated and the measured results. 4.2. Power Quality Assessment through EMTP-based Programs 118 processing frequency-domain analysis. The explanation is that the differential equations of inductances and capacitances in EMTP-based programs are solved with the trapezoidal integration rule, thus producing errors which are a function of the frequency and the time step size (At) [2], [12]. Moreover, It is also recommended that the time step size be such that the period of the fundamental frequency is an integer multiple of At , in order to avoid the generation of non-characteristic harmonics in the post processing Fourier analysis. For example, if the system nominal frequency is 60.0Hz and the maximum frequency expected in the transient simulation is in the order of fmax = 6kHz, then the step size can be calculated as At = 1/(10 * fmax) = 16.66666MS; New solutions to this harmonic problem could be investigated using EMTP-based sim-ulations, as for example the possible use of active filters [60], [61] to minimize harmonic distortions. Such power electronic device should be able to inject a shunt compensated cur-rent, as shown in Fig. 4.27, at the point of common coupling, thus improving the quality of power at the interface of industrial and utility power systems. Phase "A " Currrent 400 -400 1 1 1 1 — 1 1 i I i 0.05 0.052 0.054 0.056 0.058 0 06 0.062 0.064 0.066 Time (ms) 4_ Ideal^ Compensation Current for Phase "A " ^_ 0.05 0.052 0.054 0.056 0.058 0.06 0.062 0.064 0.066 Time (ms) F i gure 4.27: Instantaneous ideal compensation current to be "injected" by a shunt active filter. 4.2. 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, where the voltage and current harmonic distortions were produced by induction furnaces. The history of this harmonic res-onance problem was described. A digital model of the induction furnace as a current-source, parallel-resonant inverter load, together with three-phase representations of the supplying distribution system were used in a time-domain simulation. After reaching a time considered as steady state, the simulated results were processed through a Fourier analysis program to obtain the harmonic contents of the voltage and current waveforms at the point of com-mon coupling. Field measurements were also presented. A comparison between the actual measurements and the E M T P simulations is presented in Table 4.2 and Table 4.3. The differ-ences in the values of the total harmonic distortion (THD) and the telephone influence factor (TIF) are possibly due to simplifications or lack of realistic data in the digital modelling. The knowledge of detailed manufacturer data of the induction furnaces and their operat-ing conditions, such as the natural resonant frequency, would allow the improvement of the E M T P simulation. Considering unbalanced conditions could also improve the simulation results. Both were unfortunately not available. Table 4.2: Comparison between field measurements and E M T P simulation results for the operating con-dition with the harmonic passive filters turned O F F . Phase " A " Current [A] Phase-to-phase " A - B " Voltage [V] Parameters Measured Simulated Er ro r % Measured Simulated E r r o r % F u n d 198.9 201.8 1.5 13,557 13,776 1.6 R M S 204.8 204.2 0.3 13,592 13,814 1.6 C F 1.531 1.491 2.6 1.489 1.444 3.0 M i n -308.5 -304.4 1.4 -20,237 -19,950 1.4 M a x 313.4 304.4 2.9 19,805 19,950 0.7 T H D [%] 18.89 15.42 18.4 7.172 7.487 4.4 H R M S 37.58 31.11 17.2 972.3 1,031.4 6.1 T I F / I T 52,660 1 64,416 22.3 234.9 520.1 121.4 EMTP-based simulations can be useful tools for harmonic analysis, based on the fact that very detailed effects can be taken into account. Once the system is modelled for an E M T P -4.2. Power Quality Assessment through EMTP-based Programs 120 Table 4.3: Comparison between field measurements and EMTP simulation results for the operating con-dition with the harmonic passive filters turned ON. Phase "A" Current [A] Phase-to-phase "A-B" Voltage [V] Parameters Measured Simulated Error % Measured Simulated Error % Fund 198.7 190.2 4.3 14,011 14,211 1.4 RMS 199.0 190.4 4.3 14,037 14,219 1.3 C F 1.515 1.474 2.7 1.431 1.422 0.6 Min -301.5 -279.5 7.3 -20,083 -20,130 0.2 Max 294.2 280.7 4.6 19,896 20,220 1.6 T H D [%] 5.585 5.110 8.5 2.589 3.224 24.5 HRMS 11.10 9.72 12.4 362.7 458.2 26.3 T I F / I T 24,637 36,910 49.8 115.1 219.9 91.1 based software, any type of studies can be performed. The new models developed in this thesis project, hopefully will contribute to make EMTP-based programs more valuable tools for electric utility companies and industrial customers in evaluating power quality problems. 4.2.2 Voltage Sag Analysis with EMTP-based Simulation Voltage sags or voltage dips are short duration variations in the supply voltage, caused by faults in transmission lines or in parallel distribution feeders, or caused by the start-up of large induction motors or other types of sudden load variations. The majority of power quality problems are associated with voltage sags, which are very common in today's electric-ity industry. The ride-through characteristics of modern electronic and computer-controlled loads are very sensitive to short duration variations in the supply voltage. An entire process may be shut down when the voltage sags momentarily. The equipment tolerance charac-teristics to voltage sags vary very much among equipment manufacturers. Moreover, in most cases the equipment ride-through characteristics are not known. For these reasons, the C B E M A curve [51] (or the ITIC curve) has been widely used as a first reference for power quality studies related to short duration voltage variations. Fig. 4.28 presents actual measurements of voltage sag phenomena with an overlay of the C B E M A curve 1 . 7 "We typically employ the curve only from 0.1 cycles and higher due to limitations in power quality monitoring instruments and differences in opinion over defining the magnitude values in the subcycle time frame." From [24], pp. 37-38. 4.2. 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. Power system protection and operating practices may also affect the success or failure of loads which are sensitive to voltage disturbances. Fig. 4.29 (a) shows a voltage sag phenomenon caused by a single-line-to-ground fault in a distribution feeder, which is in parallel to a feeder supplying a " P V C " pipe (and other plastics derived products) manufacturer with sensitive loads. The industrial process, controlled by DC drives, is stopped if the voltage sags at the point of common coupling to less than 90% of the nominal voltage during a time greater than 18 cycles (300ms). Fig. 4.29 (b) shows the simulation results using MicroTran. The simulation does not match the measurements exactly, because the dynamic behaviour of industrial loads (typ-ically induction motors) were not included in the simulation model. Nevertheless, E M T P -based programs have the flexibility to include aggregated load models [109]. More research, however, seems to be needed for the accurate representation in EMTP-based programs of the dynamic behavior of loads, which is beyond the scope of this thesis project. Custom Power Controllers such as the dynamic voltage restorer (DVR), is a promising solution for the mitigation of voltage sag phenomena. 4.2.3 W e l d i n g Indus t ry Vol tage F l u c t u a t i o n S tudy - A V i s u a l F l i cke r Case Flicker is historically considered a problem of perception, because the human ability to visually sense light changes caused by voltage fluctuations does vary. However, there are certain levels of light flicker which can be easily detected, though their impact on possible human brain disorders or any other potential health damage is difficult to quantify. This section presents a recent case of light flicker and some simplified EMTP-based simulations. The operation of a welding machine, to produce meshed wires for construction, connected to a distribution system was simulated with MicroTran. If the duty cycle of the welding pro-cess is near 8.8 Hz, the human eye would perceive the maximum visual flickering effect, even when the RMS voltage fluctuations are in a range of very small percentage deviations. 4.2. Power Quality Assessment through EMTP-based Programs 122 PQNode Group March 15, 1996 at 18:56:21 PQNode Local T r i g g e r i n g Phase,Max Depth 16/04/96 06:03:38 PQNode Local 3 0 0 2 5 0 2 0 0 3 1 5 0 o > 1 0 0 J _ 50 I h Max 106.0 Min 73.50 0 . 0 0 1 0 . 0 1 0 . 1 1 10 1 0 0 1 0 0 0 1 0 0 0 0 Time ( C y c l e s ) BMI/Electrotek Figure 4.28: Voltage sag measurements (%RMS versus time duration) with an overlay of the C B E M A curve. For time durations less than 1 cycle the equipment seems to measure peak values. IEC Standard 1000-4-15 provides the specifications for a flickermeter with "lamp-eye-brain" frequency response to light flickering effects. Fig. 4.30 shows the simulated instantaneous voltage for a power electronics controlled welding machine . The operating duty cycle, con-trolled by the semiconductor firing angle, results in a modulating frequency of approximately 7Hz for this case, as shown in Fig. 4.31, which causes the visual light flicker. Cost-effective solutions for voltage fluctuation problems are usually related to changes in the load duty cycle, when this does not affect the industry productivity or the quality of the manufac-tured product. The application of reactive dynamic compensation through power electronic devices, such as the distribution static synchronous compensator (D-STATCOM) can effec-tively mitigate this type of power quality problem [110]. 4.2. Power Quality Assessment, through EMTP-based Programs 123 Figure 4.29: (a) Phase-to-phase "A-B" measured voltage sag. (b) Phase-to-phase "A-B" simulated voltage sag. 4.2. Power Quality Assessment through EMTP-based Programs 124 1000 950 900 850 800 IP 700 650 600 Instantaneous Voltage at the Point of Common Coupling 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time ( s ) Figure 4.30: Instantaneous voltage fluctuations causing light flickering effect. Phase "A" Voltage Harmonic Amplitude Spectrum i 1 1 1 • Ii I .•• L i l L - i Frequency (Hz) F i gure 4.31: Modulated voltage and respective amplitude frequency spectrum 4.3. EMTP-based Simulation Cases with SSCPS 125 4 . 3 EMTP-based Simulation Cases with S S C P S This section presents a collection of test cases to validate the EMTP-based models de-veloped in this thesis. Where an interesting and already published benchmark simulation case with all the necessary data was readily available, it was selected because it made com-parisons easy. In other cases, simple circuits were assembled by the author to test and prove the ideas. 4.3.1 Bas ic C o n t r o l and C o n t r o l Devices S imula t ion Cases This section provides more explanations about the theory presented in the previous chap-ters, with some applications to test cases. (a) T h e Effect of a One T i m e Step Delay in the Solution of E M T P - b a s e d Simu-lations Reference [66] presents some interesting cases where the one time step delay in the solution of control and system equations, or the internal delays inside TACS, give wrong or inaccurate simulation results. For example, the simulation of a transfer function with poles on the imaginary axis of the complex plane is very sensitive to time delays, which can "move" the poles to the right half of the complex plane, resulting in instability. To illustrate this, Fig. 4.32 presents a control block diagram of a transfer function with a second order differential equation. The simultaneous solution is shown in Fig. 4.33. The intentional introduction of a one time step delay in the control system, as illustrated in Fig. 4.34, leads to an unstable resonance condition, as shown in Fig. 4.35. These results emphasize the "importance of a simultaneous solution approach", which is critical in some cases for the correct simulation of control and power system equations with EMTP-based programs. 4.3. EMTP-based Simulation Cases with SSCPS 126 IN t o OUT 1.0 [V] \ i Figure 4.32: Control block diagram of a second order differential equation with poles on the imaginary axis of the complex plane. Figure 4.33: Solution of system with bounded resonance oscillations. 4.3. EMTP-based Simulation Cases with SSCPS 127 Figure 4.35: Solution of system with unstable resonance oscillations caused by the introduction of one time step delay. 4.3. EMTP-based Simulation Cases with SSCPS 128 (b) Basic Control Blocks, Transfer Functions and Filters Laplace transfer functions are essential for control design and simulation. Generally, classical control blocks such as proportional (P), integral (I), derivative (D), (and their com-binations PI, PD, PID), lead-lag's for phase compensation, washout filters, etc., are present in any analog or digital control scheme. Many if not most of the first, second or higher order differential equations used to model the physical behavior of electrical, mechanical, chemical, and any other systems use such transfer functions for their mathematical repre-sentation. Transfer functions can also represent passive and active filters (i.e., connections of circuits with operational amplifiers, resistors and capacitors). Depending on their properties and frequency response, these filters are referred to with special names, such as low-pass, Butterworth, Chebyshev, Legendre-Papoulis, Bessel, elliptic, high-pass, band-pass, notch, band-elimination, all-pass (magnitude), all-pass (phase), etc. [80]). Therefore, the method-ology proposed in this thesis for the simultaneous solution of transfer functions (Chapter 2) expands considerably the potential applications of EMTP-based programs in time and frequency domain simulation studies. It is important to mention that other general purpose and powerful computer tools, such as M A T L A B [103], can also be very useful for engineers and scientists, especially for the design of control systems. The computer program SPICE seems to be more used for electronics and power electronics simulations. EMTP-based programs, however, still seem to have more detailed and proven models, especially for the power system part. This section then presents some test cases with classical control blocks, which are simu-lated with the simultaneous solution method implemented experimentally in MicroTran, the U B C version of the E M T P . The first control block diagram case is illustrated in Fig. 4.36, which represents the classical linearized "swing equation", which is used in power system small-signal stability studies of a single machine connected to an infinite bus, as extracted from page 731 of reference [86], where: Kg = 0.757pu torque/rad = synchronizing torque coefficient in pu torque/rad; Kp = 10 or -10 pu/pu = damping torque coefficient in pu torque/pu speed deviation: H = 3.5 M W s / M V A = inertia constant in M W s / M V A ; 4.3. EMTP-based Simulation Cases with SSCPS 129 Aur = speed deviation in pu = (tur — UJQ)/OJQ\ A5 = rotor angle deviation in electrical rad; o>o = rated speed in electrical rad/s = 27r/0 = 377 rad/s for a 60Hz system; A T m = — 0.1 pu = mechanical torque deviation in pu; ATes = synchronizing torque component in pu; ATe<i = damping torque component; ATa = ATm — ATes — ATej, = accelerating torque deviation. Fig. 4.37 illustrates the simulation results for a -O.lpu disturbance in the per unit me-chanical torque deviation (ATm). With a positive damping torque coefficient (KD = 10) the rotor angle deviation (AS) presents damped natural oscillations and reaches a new stable operation point in steady state. As shown in Fig. 4.38 with a negative damping torque co-efficient (KD = —10) the rotor angle deviation (AS) presents amplified natural oscillations, as expected, causing small-signal instability. Alternatively, the control block diagram pre-sented in Fig. 4.36 could also be represented by a canonical second order transfer function, as presented in Fig. 4.39, where: (4.5) (4.6) 2 2Hu, 1 KD 2 y/Ks2Hu0 (4.7) 4.3. EMTP-based Simulation Cases with SSCPS 130 Are -0.1 p.u. ATm ^ "1 ATa ± o — © 2Hs K, D ATe, Aco, A5 —• F i g u r e 4.36: Classical linearized "swing equation", used in power system small-signal stability studies of a single machine connected to an infinite bus. 0.1 0.05 s si. § -0.05 ft) Q. .Sj ^ -0 1 ft) * .a C -0.15 -0.2 -0.25 1 1 YATa - > \ ATed 5*Aco T 1 1 1 i i i I * ' l l It I t \ / ATe / ' ATm I I r / \ / - 1 1 I 1 AS i i i i p i i 1 2 3 4 5 6 Time ( s ) 9 10 F i g u r e 4.37: Simulation results of the synchronous machine rotor angle deviation, in the presence of positive damping torque coefficient. 4.3. EMTP-based Simulation Cases with SSCPS 131 200 4 5 6 Time ( s ) 10 F i g u r e 4.38: Simulation results of the synchronous machine rotor angle deviation, in the presence of negative damping torque coefficient. ATm t . • s2 + 2^d)ns +C0n2 0.1 p.u. \ AS Fi gure 4.39: Canonical second order transfer function representation of the single-machine infinite bus system. 4.3. EMTP-based Simulation Cases with SSCPS 132 (c) Voltage and Current Sensors, Control Devices, Function Blocks and Digital Logic Gates The solution method for voltage and current sensors, control devices, function blocks and digital logic gates follows the procedures for dependent sources presented in Chapter 2. For example, the simultaneous solution through the compensation method for a generic linear or nonlinear voltage-controlled voltage source can be used • to sense a voltage signal, with the compensating current at the controlling branch j set to zero, i.e., ij = 0; • then to define any linear or nonlinear voltage function at the controlled branch k, i.e., Vk = f(vj); • and finally to use the Newton-Raphson algorithm presented in Section 2.2.5, (Fig. 2.19) to calculate a solution for the compensating currents ij and ik-If more than one control voltage signal needs to be sensed, their respective compensating branch currents are "simply set to zero". On the other hand, if currents have to be sensed, their respective branch voltages can be set to zero. Since control signals are usually voltage signals (or current signals converted to voltage signals), a current sensor can be represented by a current-controlled voltage source. The implementation of control devices (such as transport or time delay blocks, com-parators, zero crossing detection and generation of gate firing signals, Clark transformation, Park transformation, etc.), F O R T R A N function blocks (such as SIN, COS, T A N , ASIN, ACOS, L O G , E X P , SQRT, M U L T I P L I C A T I O N , DIVISION, etc.) and digital logic gates (AND, OR, NOT, N A N D , NOR, etc.) uses the same concept of the simultaneous solution for voltage-controlled voltage sources presented before. The implementation of a simultaneous solution for limiters requires special attention, as for example in the case of the widely used PI controllers with non-windup limiters. Reference [66] clearly explains the problem and the solution for a correct EMTP-based simulation. 4.3. EMTP-based Simulation Cases with SSCPS 133 Special functions, as for example for the evaluation of average values (e.g. "rolling average power" presented in [66]), root mean square (RMS) values, instantaneous total harmonic distortion (THD) values, etc. can be easily modelled with control blocks, and with sub-circuit implementation. 4.3.2 Power Elect ronics S imula t ion Cases The cases presented in this section are based on references [65], [93], [89], [94], [104], [95]. (a) D y n a m i c C o n t r o l of the F i r i n g Ang le ("a") of a Thyr i s to r Fig. 4.40 illustrates a simple circuit to demonstrate three important contributions of this thesis, for the dynamic simulation of power electronic devices in EMTP-based programs: • simultaneous solution for voltage sensors; • zero crossing detection and generation of firing pulses with an instantaneous updating of the controlling firing angle a. The controller is enabled by a voltage signal of "1 volt" and disabled by a voltage signal of "0 volt"; • EMTP-based voltage-controlled unidirectional current flowing switch, to represent the thyristor, which receives the gate signal directly from the controller. The voltage sensor in Fig. 4.40 is modelled with the equations for an "ideal" voltage-controlled voltage source (VCVS) presented in Section 2.2, i.e.: where A = 1. For this case, the synchronizing signal for the gate firing 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. ij = 0 (4.8) .•• + h - M - I f ) « M = 0 (4.9) 4.3. EMTP-based Simulation Cases with SSCPS 134 The gate firing controller is modelled as a "multi-terminal voltage-controlled voltage source", where three controlled voltages are sensed and used to determine the voltage source at the gate, VQATE (a pulse of amplitude equal to 1 with a specified width, or zero volts). The output VQATE depends on the values of VSYNCHR (which is used to detect, with interpolation, the time of zero crossings) , v ALP HA (which is the firing angle in degrees, converted to time, based on the given input frequency) and venabie/disable (whose value of 1 or zero is multiplied by VGATE to enable or disable the control firing). The following equations are used to solve for the gate firing controller using the compensation method: i3- = 0 (4.10) h = 0 (4.11) ii = 0 (4.12) -vopENm + rmiii + ... (4 13) ••• + r m m i m + ... + r m M i M + VGATE = 0 In addition to equations 4.10 to 4.13, conditional IF-statements are used to implement the logic described above equation 4.10. The thyristor was modelled as an EMTP-based voltage-controlled unidirectional current flowing switch, which receives the gate signal directly from the controller. The solution with the compensation method using an iterative Newton-Raphson algo-rithm requires the calculation of a Thevenin equivalent for each branch, as mentioned before in the fundamental assumptions of Section 2.2.2. In cases where this calculation fails, the connection of large resistors in parallel with the branch may make a Thevenin equivalent circuit possible 8 . Eventually, in an E M T P production code, the use of internal variables for the control could result in a more economic implementation. Internal control variables would actually not have any physical connection with the power network part of the circuit. Fig. 4.41 shows the resulting voltages and currents simulated with MicroTran and the method proposed in this thesis, which is a truly simultaneous solution of the power and control circuit. The time step size used was At = 16.6667yLis. 8In MicroTran, the connection of a large resistance of 109fi and with the near zero tolerance parameter for checking matrix singularities, EPSILON=10~ 1 2, this problem can easily be solved. 4.3. E M T P - b a s e d Simulation Cases wi th S S C P S 135 Vmax=5.0[V] f=60[Hz] 'LOAD 'LOAD 2.5Q. 5mH VGATE Gate Firing Controller SYNCHR ALPHA f=60Hz pulse width=10 degreees Voltage Sensor / ( l ) 45.0 [V] Firing Angle (degrees) enable / disable 1.0 [V] tstop=25.0[ms] Figure 4.40: Circuit for the dynamic control of the firing angle ("a") of a thyristor. 4.3. E M T P - b a s e d Simulation Cases with S S C P S 136 Time (ms) F i g u r e 4.41: Voltages and currents in a circuit with dynamic control of the firing angle of a thyristor. 4.3. EMTP-based Simulation Cases with SSCPS 137 (b) D y n a m i c C o n t r o l o f the F i r i n g A n g l e s o f a T h r e e - P h a s e S i x - P u l s e T h y r i s t o r -B r i d g e R e c t i f i e r Fig. 4.42 illustrates a phase controlled rectifier with a feedback control system based on the manual of PSIM [65]. "It should be noted that, in PSIM, the power and the control circuit are solved separately. There is one time step delay between the power and the control circuit solutions" [65]. The six-pulse firing controller implemented with the models developed in this thesis project uses a simultaneous solution for the power and control circuit equations. Similarly to the simple firing controller "a" of the previous test case, a multi-terminal voltage-controlled voltage source is used to model it. The inputs to the six-pulse firing controller are the synchronizing signal (voltage VAC sensed from the supply system), the dynamic firing angle a (resulting from the ACOS control block, which receives the signal from the limited PI controller, after comparison with the desired reference voltage for the DC load). From the gating signal generated to the thyristor with identification number "1", all the other gating signals are derived sequentially by adding a time delay corresponding to 60 degrees at the 60Hz frequency. For starting purposes, whenever a firing signal is sent to a particular thyristor, another "isolated" firing signal is sent to the previous thyristor as recommended in [88]. Fig. 4.43 presents the E M T P simulated voltages and currents, using a A t = 16.6667/is and with a dynamic control of the firing angles of the three-phase six-pulse thyristor-bridge rectifier. Fig. 4.44 shows the dynamic behavior of the control variables, emphasizing the firing control signal a, and Fig. 4.45 illustrates the dynamics of the voltage control signals at the output of the proportional-integral (PI) control block and the limiter control block. 4.3. EMTP-based Simulation Cases with SSCPS 138 Figure 4.42: Circuit for the dynamic control of the firing angles of a three-phase six-pulse thyristor-bridge rectifier. 4.3. EMTP-based Simulation Cases with SSCPS 139 4.3. EMTP-based Simulation Cases with SSCPS 140 4.3. EMTP-based Simulation Cases with SSCPS 141 1.4 4.3. EMTP-based Simulation Cases with SSCPS 142 (c) Dynamic Control of Three-Phase P W M Voltage Source Inverter Fig.4.46 presents a circuit for the dynamic control of a three-phase P W M voltage source inverter (VSI) [65]. Again, the power and the control circuit equations are solved simultane-ously with the methods proposed in this thesis. For this EMTP-type simulation it was used a A t = 16.66667/Lts. The phase "A" modulation and triangular carrier waveforms for generation of gating signals through sinusoidal pulse width modulation (PWM) are presented in Fig. 4.47. With the use of comparators and NOT logic gates the firing signals are dynamically generated, in a simultaneous solution with the network equations through the compensation method. The IGBT's with anti-parallel diodes were represented, for simplicity and without much loss of accuracy in this simulation, by the EMTP-based voltage-controlled switches, which were implemented in this thesis through the G A T E subroutine. Here it is opportune to discuss the issue of simultaneous commutation: Let us assume that the current is flowing through the I G B T with identification number "1" from the D C source to the load. When IGBT number 1 is turned off, the voltage VSA-NEUTR reverses polarity almost instanta-neously (due to the behavior of the inductor, which forces the current to keep flowing in the same direction), thereby forward biasing the anti-parallel diode of the IGBT with iden-tification number "4", which then starts conducting. In digital simulation programs this means that there is simultaneous commutation between IGBT 1 and the anti-parallel diode at IGBT 4. This could be modelled as it is, i.e., with an I G B T and diode in anti-parallel (which would have to be modelled with piecewise linear or nonlinear model), or as a voltage-controlled bidirectional current flowing switch, where the control signals play the role of the commutation. Fig. 4.48 presents the node voltage UVSA" generated by the 3-Phase P W M voltage source inverter (VSI), whereas Fig. 4.49 shows the voltage across the load "VSA-NEUTR" and the current in phase "A" supplied to the load. The dynamically generated 3-phase load currents are illustrated in Fig. 4.50. The line-4.3. EMTP-based Simulation Cases with SSCPS 143 to-line voltage generated by the three-phase P W M voltage source inverter (VSI) is shown in Fig. 4.51. Most of the advanced Custom Power Controllers [60] (and active filters [111]) apply this type of converter to synthesize voltages or current waveforms according to the desired "dynamic reference modulating signal". When a current is to be synthesized, dynamic hysteresis current-band P W M converters are used [60] 9 . Therefore, the models developed in this thesis, will hopefully be useful for the accurate EMTP-simulation of a variety of existing and new power electronic devices, especially those aimed at improving the quality of power in utility and industrial systems. 9The Ph.D thesis "Active Power Line Conditioners" of Dr.-Ing. Mauricio Aredes, can be downloaded by the reader from the the web site http://www.dee.ufrj.br. 4.3. EMTP-based Simulation Cases with SSCPS 144 IGBTj IGBT4 IGBT3 "SB IGBT6 IGBT, ,5 — i v, sc I SA-NEUTR SB-NEUTR IGBT2 NEUTR lSC-NEUTR 3.87Q 7.7mH 10 9 Q 0.8 [VJ\ 0.8/Yjnj 0.8 [V]Ky f= 60 [Hz] 4>°— NOT 4>-^COMPARATOR TRI 4>-liV] f= 1500 [Hz] 1 /?/jtf.se = -750 [degrees] Figure 4.46: Circuit for the dynamic control of three-phase P W M voltage source inverter (VSI). 4.3. EMTP-based Simulation Cases with SSCPS 145 4.3. EMTP-based Simulation Cases with SSCPS 146 600 400 ^ 200 R ft) C3 bo •S -200 -400 -600 10 15 Time (ms ) 20 25 Figure 4.48: Node voltage "VSA" generated by a three-phase P W M voltage source inverter (VSI) . 4.3. EMTP-based Simulation Cases with SSCPS 147 600 -400 r--600 1 10 15 Time (ms) 20 25 Figure 4.49: Voltage across the load "VSA-NEUTR" and current supplied to the load by a three-phase PWM voltage source inverter (VSI). 4.3. EMTP-based Simulation Cases with SSCPS 148 -60L 10 15 Time (ms) 20 25 Fi gure 4.50: Load currents supplied by a three-phase PWM voltage source inverter (VSI). .3. EMTP-based Simulation Cases with SSCPS Time (ms ) Figure 4.51: Line-to-line voltage generated by a three-phase PWM voltage source inverter (VSI) 4.4. Synthesis of Simulation Guidelines for Studies with EMTP-based Programs 150 4.4 Synthesis of Simulation Guidelines for Studies with EMTP-based Programs This section emphasizes the basic issues which are critical for the successful evaluation of the impact of power electronic devices on the quality of power. Important factors regarding power quality monitoring, modelling and simulation in EMTP-based programs of power system components and power electronics devices are pointed out, with the main objective of analyzing their dynamic interaction and of evaluating their impact on electric power quality. For power quality monitoring the important factors are: • Evaluation based on accurate measurements of power quality phenomena, through the use of instruments with appropriate voltage and current sensors and adequate digital sampling frequency [15]; • Use of statistical and other advanced data analysis methods to produce meaningful information; • Comparison against national and international power quality standards, taking into consideration system differences and similarities; In the analysis of power quality phenomena through time and frequency domain EMTP-based simulations, special attention must be paid to: • the simulation step size At, which has to be chosen as a function of the maximum frequency expected (or of concern) in the simulation. Usually, the time step size is set to a value at least equal to one tenth of the period of the maximum frequency, which will result in a 3% error with the trapezoidal integration rule [2], [112]. It is also recommended that the step size be such that the period of the fundamental frequency is an integer multiple of At, in order to avoid the generation of non-characteristic harmonics in the post processing Fourier analysis. For example, if the system nominal frequency is 60.0Hz and the maximum frequency expected in the transient simulation 4.4. Synthesis of Simulation Guidelines for Studies with EMTP-based Programs 151 is in the order of f m a x = 6kHz, then the time step size can be calculated as A t = 1/(10 * fmax) = 16.66666^; • the selection of appropriate models to represent power system and load components, especially if frequency dependence has to be taken into consideration in the simulated phenomena [92]; • nonlinearities, which are usually disregarded in many simulations. They can affect the accuracy of the simulation results, particularly in the case of transformer saturation; • the use of simplified switch models for power electronics devices. This may be justified to speed up the simulation time for system level studies, but it may also give wrong and misleading results, especially related to semiconductor commutation phenomena. Also, the E M T P solution at discrete time intervals A t may result in inaccurate turn-on or turn-off switching times, causing unrealistic high frequency transients in the simulation of power electronic devices. Backtracking techniques [67], [68] and/or resynchroniza-tion techniques ([96] pages 185, 204, 207) or even the Clock Synchronized Structure Changing Concept (CSSC) [97] can be used to minimize the problem. Interpolation and/or extrapolation as well as resynchronization techniques seem to be more and more applied even in the EMTP-based solution of modern control for power electronics sys-tems [98], [99]. Therefore, for better accuracy in EMTP-based simulations of power electronics, it is much more important to use such techniques than to reduce the time step size; • numerical oscillations caused by the trapezoidal rule of integration in solving the system of equations. The use of techniques such as C D A ("Critical Damping Adjustment" [84], [85]) is effective in the elimination of numerical oscillations. MicroTran has C D A implemented, but other E M T P versions may not, or may use different approaches. • the one time step delay at the interface between the control and power systems solu-tion, as well as other internal time step delays, which may exist in TACS ("Transient Analysis of Control Systems") and in other software packages. The method SSCPS ("Simultaneous Solution of Control and electric Power System equations") proposed in 4.4. Synthesis of Simulation Guidelines for Studies with E M T P - b a s e d Programs 152 this thesis overcomes this problem, and the user only introduces time delays if needed to represent system physical behaviour. Chapter 5 Conclusions and Recommendations for Future Work H E M A I N G O A L of this Ph.D. thesis project was the development of EMTP-based 1 models for control and power electronic devices for electric power quality assessment. E M T P simulations can offer theoretical and practical insights into the evaluation of power quality, both by time-domain simulation techniques and by frequency-domain simulation techniques. 5.1 Conclusions and Main Contributions The increasing demand for electricity and other forms of energy in modern society will create issues of conflict and interest. The simple absence of enough power generation or of available transfer capacity may become the cause of scheduled load shedding or more frequent blackouts, with obvious catastrophic consequences. The growing regulatory, environmental, financial and time constraints in building new power plants (typically hydroelectric) and transmission lines has been forcing emergency solutions in the electricity industry all over the world, such as the increasing use of "small" distributed generation (mainly thermal power plants with steam or gas turbines), the use of FACTS devices to enhance power system stability and control, and the adoption of programs to "save energy" with the promotion for the use of more electricity efficient light, heat, and motor equipment. This, in turn, might create a deteriorating impact on the quality of power, 153 5.1. Conclusions and Main Contributions 154 due to the manufacturing of usually inexpensive power electronic converters. It is a challeng-ing environment for engineering, politics, economics, etc. since in modern human society, electricity has become a basic commodity, which almost everybody and almost everything depends on. The quality of the electric power delivered to customers by utilities may not be acceptable for some types of sensitive loads, which are typically power electronics and computer-based loads, particularly in the control of industrial processes. There are cases where the increas-ing use of power electronics to enhance process efficiency 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 (fixed or controlled, for line reactance compensation) will increase the potential risk of transient dis-turbance amplifications and potential electrical and mechanical resonances in the presence of more and more power electronic devices, and of steam and gas turbines in distributed and co-generation power plants. As the natural order of the system grows, so grows its ability to oscillate more! At the same time, new power electronic devices also offer the means for adequate "power conditioning", to meet the special requirements of electric power quality in a system. To evaluate the promising solutions offered with the introduction of more and more power electronic devices in the transmission and distribution systems, as well as to analyze their interaction and impact on either the load or the network side, computer programs based on the E M T P (Electromagnetic Transients Program) are becoming more useful. The development of new EMTP-based models for more accurate representation of controls and power electronic devices has been the main subject of this thesis project. The assessment of electric power quality and the technical impact of power electronic devices on the quality of power, can hopefully be performed with the models developed in this work. The main contributions of this Ph.D. thesis project are summarized as follows: • development of a "simultaneous solution for linear and nonlinear control and electric power system equations" (SSCPS) in EMTP-based programs, through the compen-5.1. Conclusions and Main Contributions 155 sation method and the Newton-Raphson iterative algorithm. This solution method eliminates not only the one time step delay problem at the interface between the so-lution of power and control circuits, but also all the internal delays, which may exist in methods based on the transient analysis of control systems (TACS) since 1977 [64]. A "circuit approach" was proposed in this thesis, as an innovative alternative to the solution presented by A. E. A. Araiijo in 1993 [67]; • experimental implementation in Microtran, the U B C version of the E M T P , based on SSCPS, of a simultaneous solution for: - linear and nonlinear current and voltage dependent sources (which allow, for example the modelling of operational amplifiers, ideal current, and voltage sensors, etc.); - independent, current and voltage sources, which can also be connected between two ungrounded nodes; - hard and soft limiters (which can be used to represent nonlinear effects such as saturation); - transfer functions (which allows the simulation modelling of all types of analog filters and classical control blocks); - mathematical and transcendental F O R T R A N functions (such as + , - , * , / , SIN, COS, T A N , ASIN, ACOS, L O G , EXP, etc.); - special control devices (such as time delays, comparators, etc.) and some digital logic gates (NOT); - transformation of variables (such as the abc to a(30 transformation and its inverse); - voltage-controlled switches; - nonlinear model of a diode semiconductor; • development of the subroutine " G A T E " in MicroTran, allowing the dynamic control of the turn-on and turn-off times of semiconductor devices (e.g., thyristors, GTO's, IGBT's, etc.), which are modelled as EMTP-based voltage-controlled switches; 5.2. Recommendations for Future Work 156 • development of power electronics simulation cases in MicroTran, using the simultaneous solution approach (SSCPS) for the dynamic control of semiconductor switching devices (as in a three-phase six-pulse thyristor controlled bridge rectifier, and in a three-phase P W M voltage source inverter (VSI)) and evaluation of current and voltage waveforms; • interaction with a Brazilian utility company and industries for the realization and analysis of field measurements of electromagnetic phenomena affecting the quality of power, such as: — voltage sags and voltage swells; — harmonic current and voltage distortions; — transients, etc. with determination of causes, consequences and investigation of possible solutions for power quality problems, as for example, the application of "Custom Power Controllers": • synthesis of simulation guidelines for the evaluation of the impact of power electronic devices on the quality of power, based on realistic field measurements and E M T P time and frequency domain simulations. 5.2 Recommendations for Future Work The author's main recommendations for future work related to his Ph.D. thesis project are listed below: • development and implementation in EMTP-based programs of methods for the "au-tomatic calculation of initial conditions" in the simultaneous solution of control and power electronic circuits; • development and implementation in EMTP-based programs of techniques for the cal-culation of the "frequency response of integrated control and power system equations", allowing for example, the determination of transfer functions between a generic input and a generic output; 5.2. Recommendations for Future Work 157 • development and implementation in EMTP-based programs of techniques for the ac-curate solution of "digital control systems"; • development and implementation in EMTP-based programs of "detailed nonlinear models of semiconductor devices" (such as transistors, etc.); • development and implementation in EMTP-based programs of algorithms for the si-multaneous solution of "generic nonlinear dependent sources" [101]; • development and implementation in EMTP-based programs of "voltage and frequency dependent aggregated load models" [109], but suitable for time and frequency domain simulations in power quality studies; • development and implementation in EMTP-based programs of models for FACTS and Custom Power Controllers, such as S T A T C O M , U P F C , D V R , U P Q C , active filters, etc, and definition, if not yet available in the technical literature, of benchmark test cases, possibly with field validation of the models. The evaluation of "power system stability and control, based on long term EMTP-type simulations", will require detailed and complex modelling of power system components, FACTS Controllers and special loads, with all their associated control equipment such as turbine-governor controllers, exciter controllers, power system stabilizers, power electronics controllers, etc.. Com-plex and very important electromagnetic and electromechanical phenomena could then be investigated thoroughly, as for example, the damping and control of subsynchronous resonance [86], [90]. Some of the author's publications in areas related to the thesis topic are listed here for easy reference: • B. D. Bonatto, E. A . Mertens Jr., E. S. da Silva, and L. F. S. Dias, "Power Quality Assessment at Sensitive Loads", submitted to the IEEE/PES Transmission and Dis-tribution Latin America Conference (IEEE/PES T&D 2002), Sao Paulo-SP, Brazil, March 18-22, 2002. 5.2. Recommendations for Future Work 158 • B. D. Bonatto, E. H. Watanabe, E. A . Mertens Jr., H . W. Dommel, L. F. S. Dias, M . Aredes, S. Carneiro Jr., and S. Nosaki, "Power Electronics and Electric Power Quality: Cooperative Research at E L E K T R O , C O P P E / U F R J , and U B C " , submitted to the / Brazilian Congress on Electric Energy Technological Innovation (I CITENEL), Brasflia-DF, Brazil, November 6-7, 2001 (in Portuguese). • B. D. Bonatto, H. W. Dommel, E. H. Watanabe, M . Aredes, S. Carneiro Jr., E. A. Mertens Jr., S. Nosaki, and L. F. S. Dias, "Custom Power Applications for the Improvement of the Quality of Power - Literature Review", IV Brazilian Seminar about the Quality of Power (SBQEE01), Porto Alegre-RS, Brazil, August 12-17, 2001. • B. D. Bonatto, and H. W. Dommel, "Current and Voltage Dependent Sources in EMTP-based Programs", International Conference on Power Systems Transients -(IPST01), Rio de Janeiro-RJ, Brazil, Volume I, pp. 299-304, June 24-28, 2001. • B. D. Bonatto, E. A . Mertens Jr., F. A . Fernandes, and L. F. S. Dias, "The Quality of Electric Power in Coordination with the Industrial Safety", XIV National Seminar on Electric Energy Distribution (XIV SENDI), Foz do Iguassu-PR, Brazil, November 19-23, 2000 (in Portuguese). • B. D. Bonatto, H. W. Dommel, E. A . Mertens Jr., and F. A . Fernandes, "Power Quality Analysis based on E M T P Simulations Harmonics Case Study", 5th. Brazilian Power Electronics Conference (COBEP99), Foz do IguassuPR, Brazil, Volume 1, pp. 135-140, September 19-23, 1999. • B. D. Bonatto, E. A . Mertens Jr., and F. A . Fernandes, "Power Quality Diagnosis in Industrial Customers - Case Study", III Brazilian Seminar about the Quality of Electric Power (SBQEF/99), Brasilia-DF, Brazil, pp. 108-113, August 08-12, 1999 (in Portuguese). • B. D. Bonatto, T. Niimura and H. W. Dommel, "A Fuzzy Logic Application to Repre-sent Load Sensitivity to Voltage Sags", in 8th International Conference on Harmonics and Quality of Power (8th ICHQP), I E E E / P E S , Ed., Athens, Greece, Vol. I, pp. 60-64, October 14-16 1998. 5.2. Recommendations for Future Work 159 • B. D. Bonatto, E. A. Mertens Jr., and F. A. Fernandes, "Power Quality Diagnosis in Distribution System", III Latin-American Congress in Electric Energy Distribution (III CONLADIS), Sao Paulo-SP, Brazil, pp. 37-41, September 8-10, 1998 (in Portuguese). • L. E. O. Pinheiro, B. D. Bonatto, R. Torrezan, and F. A . Fernandes, "Power Quality Monitoring - Practical Cases, Solutions and the Planning Perspective", XIII National Seminar in Electrical Energy Distribution (XIII SENDI), Sao Paulo-SP, Brazil, May 11-16, 1997 (in Portuguese). • L. E. O. Pinheiro, O. S. I. Komukai, B . D. Bonatto, and E. Yoshida, "Measurements for Power Quality Monitoring in Distribution System", I Brazilian Seminar on Power Quality (SBQEE/96), Uberlandia-M.G., Brazil, pp. 92-97, June 10-13, 1996 (in Por-tuguese). • S. M . Deckmann, and B. D. Bonatto, "Damping Introduced by Control Means Consid-ering Generator Capability Limits", IFAC Control of Power Plants and Power Systems (SIPOWER'95), Cancun, Mexico, 1995. • B. D. Bonatto, and S. M . Deckmann, "Damping Introduced by SVC, CSC, and PSS for Operation on the Synchronous Machine Capability Curve", XIII National Seminar on Production and Transmission of Electric Energy (XIII SNPTE), Florianopolis-SC, Brazil, 1995. • B. D. Bonatto, Damping of Electromechanical Oscillations in Electric Systems through Reactive Dynamic Compensation, Master Thesis, The State University of Campinas -U N I C A M P , Campinas-SP, Brazil, 1995 (in Portuguese). • S. M . Deckmann, and B. D. 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