Modelling of Distorted Electrical Power and Its Practical Compensation in Industrial Plant

MODELLING OF DISTORTED ELECTRICAL POWER AND ITS PRACTICAL COMPENSATION IN INDUSTRIAL PLANT

JAN HARM CHRISTIAAN PRETORIUS

THESIS presented in partial fulfilment of the requirements for the degree

DOCTOR 1NGENERIAE ' (D. Ing)

in the

FACULTY OF ENGINEERING

of the

RAND AFRIKAANS UNIVERSITY

SUPERVISOR: PROF. J.D. van WYK CO-SUPERVISOR: DR. P.H. SWART

DECEMBER 1997 11

PREFACE

Alternating current systems employing single-frequency sinusoidal waveforms render optimal service when the currents in that system are also sinusoidal and have a fixed phase relationship to the voltages that drive them. Under unity- power factor conditions, the currents are in phase with the voltages and optimal net-energy transfer takes place under minimum loading conditions, i.e. with the lowest effective values of current and voltage in the system.

The above conditions were realised in the earlier years, because supply authorities generated 50 Hz sinusoidal voltages and consumers drew 50 Hz sinusoidal currents with fixed phase relationships to these voltages. Static and rotating electrical equipment like transformers, motors, heating and lighting equipment were equally compatible with this requirement and well-behaved AC networks were more the rule than the exception. The fact that three-phase systems conveyed the bulk of the power from one topographical location to the next did not constrain the utilisation of that concept at all, even though poly-phase transmission systems were necessary to increase the economy of transmission and to furnish non-pulsating power transfer. Also, additional theory had to be developed to handle unbalanced conditions in these multi-phase systems and to take care of complex network analysis and fault conditions.

Difficulties begin to manifest themselves when equipment not meeting these requirements is connected to the network and when the currents it draws are not sinusoidal. An increasing number of applications demand DC-voltage supplies from which DC-currents are to be drawn. Because power transmission is carried out by means of AC networks, the DC is furnished by converting or rectifying the AC-supply. Power-electronic circuits, of which the R 2P2 power supplies the AEC employs are no exception, employ line-commutated AC/DC converters in their front-ends, and fall into that category.

Although these line-commutated, phase-controlled AC/DC converters are capable of handling giga-watt power levels, line-frequency commutation causes the currents they draw on the AC-side to be distorted, even though still to be periodic. These non-sinusoidal currents, drawn from the source, along the transmission lines and through other distribution system immittances, also give rise to non-sinusoidal voltage drops between the source and the load, which results in distorted voltage waveforms at other nodes and at the load.

Harmonic penetration studies are essential to evaluate the performance of transmission systems in the presence of current distortion sources. These sources do not only bring about voltage distortion within the confines of their own borders, but extend their influence outside into those of other consumers as well. Supplyutilities are wary of the distortion introduced into their networks by consumers and initial recommendations have now given way to rigid standards for curbing harmonic pollution by consumers. 111

Because conventional steady-state alternating current circuit theory fails in the presence of distortion there are only two ways in which harmonic penetration studies can be carried out. Numerical integration methods are mandatory in the study of transient performance of electrical networks during switching and similar occurrences, but become cumbersome when the networks contain more than just a few nodes and are impossible to use when several tens or hundreds of nodes are encountered. Fortunately, harmonic penetration studies can be confined to steady-state operating conditions in a network in which voltages and currents are distorted but remain periodic and are therefore Fourier transformable.

When viewed in the frequency-domain, non-sinusoidal but periodic current and voltage waveforms can be represented by discrete frequency spectra. Frequency-domain analysis offers a number of advantages. From the frequency-domain point of view, distortion can be quantified in terms of complex phasor values of voltages and currents at discrete harmonic frequencies that individually lend themselves to conventional circuit theory, permitting calculations to be carried out in extensive networks. Solutions that apply to these individual harmonic frequencies can then be summated across the spectrum to furnish aggregate or joint parameters of currents, voltages and powers and can also be transformed back into the time-domain for the reconstruction of the relevant time-dependent waveforms.

Both the frequency and time-domain waveforms, of voltage and current, constructed in the above manner are concise and convey the same numerical information. When attempting, however, to quantify the circuit behaviour in terms of the classical definitions of active, reactive and apparent power, it is soon discovered that different definitions are possible. The different definitions, unfortunately, lead to divergent results and it is impossible to assess the utility of each different theory on a general basis. Only by applying the different theories in dedicated measurements, can their relative worth be established in terms of specific circumstances. That is the main theme of this dissertation. iv

SUMMARY

An increasing number of applications, ranging from the very small power supplies in computers to multi-megawatt traction systems and DC-arc-furnaces, require DC-voltage supplies in their front-ends. The resonant regulating pulse power (R2P2) units, employed by the Atomic Energy Corporation of SA (AEC) to drive their pulse lasers, are no exception. To compensate the harmonic distortion in the plant to acceptable levels and to facilitate the search for the most economical combination of power supplies, filters and conditioning equipment, different compensation schemes are studied. In order to do this, a steady-state harmonic penetration model, based on harmonic superposition through nodal bus-admittance matrix formulation, has been developed on a Mathcad package, to furnish the flexibility by means of which the different configurations can be modelled. In addition to the availability of all the necessary parameters such as THD-values, rms-voltage and current values and power levels, the model also furnishes time-domain waveforms of the voltages and currents against which actual measurements are experimentally contrasted. The accuracy of this model is verified after which it is used to study larger networks.

The main alternative schemes investigated are those of compensation by means of:

Individual L.V. passive filters, distributed through the plant and of local design;

PWM-controlled, L.V.-connected power-electronic dynamic compensators.

To evaluate the findings a set of power definitions is necessary. Three sets of power definitions have been chosen in this study, namely: the theories of Budeanu, the IEEE Working Group and the Czarnecki power definitions. These definitions will be analysed numerically in the admittance matrix model and on measurements to evaluate their utility and drawbacks in a practical environment. V

OPSOMMING

`n Toenemende aantal toepassings, vanaf kragbronne vir rekenaars tot megawatt aandryfstelsels benodig gelykspanning as inset. Die resonante laaikragbron wat deur die Atoomenergie- korporasie as laser kragbronne gebruik word, is ook 'n voorbeeld hiervan. Om die harmoniese distorsie na aanvaarbare vlakke te kompenseer en om die ondersoek te vergemaklik vir die mees ekonomiese kombinasie van kragbronne, passiewe filters en dinamiese kompenseerders, word verskillende kompensasietopologiee bestudeer. Om dit sinvol te kan doen is 'n bestendige toestand harmoniese penetrasiemodel gebaseer op harmoniese superposisie d.m.v. die nodebus- admittansiematriksformulering op Mathcad ontwikkel. Met die model kan verskillende parameters byvoorbeeld THD, wgk-waardes en drywings bereken word. Die verskillende tydgolfvorms kan ook gegenereer word. Die akkuraatheid van die model word bewys d.m.v. eksperimetele metings.

Kompensering d.m.v. laagspanning passiewe filters en PWM ingevoerde dinamiese kompenseerders word ondersoek. Die drywingsteoriee van Budeanu, die IEEE Werksgroep en die Czarnecki definiesies word bestudeer. Die praktiese toepassings van hierdie teoried word ondersoek. vi

BEDANKINGS

Die volgende persone en instansies het elk 'n unieke hydrae gelewer:

Piet Swart vir sy tegniese leiding, aanmoediging, volgehoue ondersteuning asook hulp — dit het my geinspireer en gemotiveer.

Prof. Daan van Wyk vir sy rigtinggewing; dit was 'n voorreg om die studie onder sy leiding to kon doen.

Personeel van die AEK wat my ondersteun het, in besonder Neill Truter vir sy hulp.

Anlia vir jou aanmoediging, ondersteuning en liefde, ook ons kinders Rachelle en Christiaan wat verstaan het.

The Atomic Energy Corporation of South Africa and COGEMA of France for making this work possible. vii

CONTENTS

THE MLIS PLANT ENVIRONMENT 1

1.1 INTRODUCTION 1 1.2 R2P2 PULSE POWER SUPPLY CONFIGURATION 3 1.3 POWER SUPPLY FRONT-END TOPOLOGY 6 1.4 PLANT LAYOUT 8 1.5 SUMMARY 10

FUNDAMENTAL DEFINITIONS 11

2.1 INTRODUCTION 11 2.2 FUNDAMENTAL DEFINITIONS UNDER SINUSOIDAL CONDITIONS 12 2.2.1 Fundamental considerations 12 2.2.2 Complex voltage and current as a function of time 12 2.2.3 RAIS value 14 2.2.4 Steady-state impedance 15 2.3 FUNDAMENTAL DEFINITIONS UNDER NON-SINUSOIDAL CONDITIONS 15 2.3.1 Complex voltage and current 15 2.3.2 Harmonic phasors 17 2.3.3 Space-vector manipulations 17 2.3.4 Effective value 18 2.3.5 Individual harmonic distortion and total harmonic distortion 19 2.4 SUMMARY 19

POWER DEFINITIONS IN THE PRESENCE OF DISTORTION 21

3.1 INTRODUCTION 21 3.2 POWER IN CIRCUITS WITH SINUSOIDAL VOLTAGE AND CURRENT WAVEFORMS 22 3.2.1 Power in single frequency circuits 22 3.2.2 Separate power components in a simple circuit 22 3.2.3 Real power as function of time 23 3.2.4 Average active power 25 3.2.5 Imaginary power as function of time 26 3.2.6 Average imaginary power 27 3.2.7 Complex power as function of time. 27 3.2.8 Conclusions 28 3.3 FUNDAMENTAL CONSIDERATIONS FOR THE DEFINITION OF POWER IN CIRCUITS WITH DISTORTED VOLTAGES AND CURRENTS 28 3.3.1 Fundamental considerations in the definition of power in multi frequency circuits 28 3.3.2 Real power as function of time 29 viii

3.3.3 Average power 30 3.3.4 Imaginary power as function of time 31 3.3.5 Average imaginary power 32 3.3.6 Joint complex power 33 3.4 THE APPROACH ACCORDING TO BUDEANU 34 3.5 THE IEEE WORKING GROUP DEFINITIONS 37 3.5.1 Single-phase Relations 37 3.5.2 Three-phase Relations 39 3.6 THE CZARNECKI DEFINITIONS 39 3.6.1 Single-phase Relations 40 3.6.2 Three-phase Relations 44 3.6.3 Decomposition of load current with source impedance not zero 45 3.7 SUMMARY 46

DISTORTION COMPENSATION 47

4.1 FUNDAMENTAL CONCEPTS 47 4.1.1 Introduction 47 4.1.2 Frequency-domain view 47 4.2 COMPENSATION PRINCIPLES 49 4.2.1 Local and ambient distortion 49 4.2.2 Parallel distortion compensation 51 4.2.3 Series distortion compensation 52 4.2.4 Combined parallel and series compensation 53 4.3 GENERIC COMPENSATOR TYPES 54 4.4 PASSIVE FILTERS 55 4.5 DYNAMIC COMPENSATION 56 4.5.1 Basic considerations 56 4.5.2 Voltage-fed and current-fed topologies 56 4.5.3 Dynamic compensator control 58 4.6 HYBRID COMPENSATOR TOPOLOGIES 66 4.6.1 Series hybrid compensator topology 66 4.6.2 Parallel hybrid compensator topology 67 4.6.3 Series-dynamic hybrid filter topology 68 4.6.4 Series-parallel hybrid topologies 69 4.6.5 Resonance suppression 69 4.7 SUMMARY 70

HARMONIC SUPERPOSITION MODELLING 71

5.1 INTRODUCTION 71 ix

5.2 THREE-PHASE MODELLING TECHNIQUES 71 5.2.1 Compound admittances 72 5.2.2 Nodal matrix 74 5.2.3 Primitive network 74 5.2.4 Connection matrix 75 5.3 SUPPLY AUTHORITY SOURCE 76 5.4 TRANSMISSION SYSTEM COMPONENTS 77 5.5 6-PULSE AC-DC CONVERTER LOADS 79 5.6 PASSIVE FILTER MODEL 80 5.7 DYNAMIC COMPENSATOR 82 5.8 SUMMARY 82

PRACTICAL VERIFICATION OF THE SUPERPOSITION MODEL 83

6.1 INTRODUCTION 83 6.2 MEASUREMENT 1 AT THE PCC WITHOUT COMPENSATION 84 6.3 MEASUREMENT 2 - AT THE PCC WITH PASSIVE FILTER AT THE LOAD 86 6.4 MEASUREMENT 3 - SERIES INDUCTOR ON SUPPLY-SIDE OF FILTER 88 6.5 MEASUREMENT 4 - DYNAMIC COMPENSATOR AT THE LOAD 90 6.6 SUMMARY 92

MODELLING FOR ALTERNATIVE COMPENSATION SCHEMES AND POWER THEORIES 93

7.1 INTRODUCTION 93 7.2 THE MLISX4 PLANT NETWORK TOPOLOGY 94 7.3 MODELLING RUNS 96 7.4 CIRCUIT CONFIGURATION 1 96 7.4.1 Configuration 1 - Modelling Run 1 - Measuring at load, node 14. 96 7.4.2 Configuration 1 - Modelling Run 2 - Measuring at node 3 100 7.4.3 Configuration 1 - Modelling Run 3 - Measuring at node 2 102 7.4.4 Configuration 1 - Modelling Run 4 - Measuring at node 16 106 7.4.5 Summary of the measurements of circuit configuration 1 109 7.5 CIRCUIT CONFIGURATION 2 110 7.5.1 Configuration 2 - Modelling Run 5 - Measuring at the primary side of the step-down transformer, node 2 110 7.5.2 Configuration 2 - Modelling Run 6 - Measuring at node 14 113 7.5.3 Summary for circuit configuration 2 measurements 115 7.6 CIRCUIT CONFIGURATION 3 116 7.61 Configuration 3 - Modelling Run 7 - Measuring at node 14. 116 7.62 Configuration 3 - Modelling Run 8 - Measuring at node 7 119 7.6.3 Configuration 3 - Modelling Run 9 - Measuring at node 3 121 7.6.4 Configuration 3 - Modelling Run 10 - Measuring at node 2 124 7.6.5 Summary of circuit configuration 3 126 7.7 CIRCUIT CONFIGURATION 4 127 7.7.1 Configuration 4 - Modelling Run 11 - Measuring at node 14 127 7.7.2 Configuration 4 - Modelling Run 12 - Measuring at node 3 130 7.7.3 Configuration 4 - Modelling Run 13 - Measuring at the primary side of the step-down transformer, node 2 .132 7.7.4 Summary for the measurements in circuit configuration 4 135 7.8 CIRCUIT CONFIGURATION 5 135 7.8.1 Configuration 5 - Modelling Run 14 - Measuring at load, node 14. 135 7.8.2 Configuration 5 - Modelling Run 15 - Measuring at node 3 138 7.8.3 Configuration 5 - Modelling Run 16 - Measuring at node 2 140 7.8.4 Summary for circuit configuration 5 142 7.9 CIRCUIT CONFIGURATION 6 142 7.9.1 Configuration 6 - Modelling Run 17 - Measuring at node 16. 142 7.9.2 Configuration 6 - Modelling Run 18 - Measuring at node 3 145 7.9.3 Summary for circuit configuration 6 147 7.10 CIRCUIT CONFIGURATION 7 147 7.10.1 Configuration 7 - Modelling Run 19 - Measuring at node 2 148 7.10.2 Configuration 7 - Modelling Run 20 - Measuring at load, node 16. 150 7.10.3 Summary for circuit configuration 7 153 7.11 CONCLUSIONS REGARDING MODELLING CONFIGURATIONS 153 7.12 ALTERNATIVE POWER THEORIES 155 7.12.1 Budeanu 156 7.12.2 Czarnecki 157 7.12.3 IEEE power definitions 158 7.13 MEASUREMENTS AT THE PCC WITH LASERS IN OPERATION 158 7.14 SUMMARY 162

8. CONCLUSION AND EVALUATION 163

8.1 INTRODUCTION 163 8.2 FIELD COVERED AND RESULTS OBTAINED 163 8.3 EVALUATION 165 8.4 RECOMMENDATIONS 166

ANNEXURE A HARMONIC SUPERPOSITION MODELLING OF THE AC/DC CONVERTER Al

A.1 INTRODUCTION Al xi

A.2 DEFINITIONS Al A.3 MODELLING THE PULSES IN THE TIME DOMAIN A2 A.3.1 Positive-going pulses A2 A.3.2 Negative-going pulses A3 A.4 FOURIER TRANSFORMATION A4 A.4.1 Positive- going pulses A4 A.4.2 Negative-going pulses A6 A.4.3 Synthesis of the complete line-line current waveform: A7 A.4.4 Three-phase representation A8

ANNEXURE B - HARMONIC SUPERPOSITION MODEL B1

INTRODUCTION B1 SPECIFICATION OF UNITS B2 PER-UNIT SPECIFICATIONS B2 DISPLAY CONTROL B4 SUPPLY AUTHORITY EQUIVALENT SOURCE B5 TRANSFORMER MODELS B8 CABLE MODELS B11 PASSIVE FILTER MODELS B15 VOLTAGE VECTOR B19 SINGLE LASER MODEL BUS ADMITTANCE MATRIX B20 ELEVEN LASER BUS ADMITTANCE MATRIX B21 CURENT VECTOR FOR SINGLE LASER SYSTEM B25 CURRENT VECTOR FOR ELEVEN LASER SYSTEM B25 CALCULATIONS FOR TIME WAVEFORMS B27 LASER CURRENT INPUT PARAMETERS B27 GENERAL DISPLAY SECTION B31 MEASUREMENT RESULTS ON EXPERIMENTAL SETUP B31 NODE-NODE CURRENT CALCULATIONS B32 POWER CALCULATIONS B34 Budeanu distortion power B34 Czarnecki power definitions B35 IEEE power calculations B41 CALCULATIONS FOR TIME WAVEFORMS B45

REFERENCES R1 xii

LIST OF SYMBOLS AND DEFINITIONS a Voltage phase angle /3 Current phase angle 'an) The nth order lock admittance matrix Susceptance Susceptance for the different harmonic components Bne Czarnecki equivalent three-phase susceptance at harmonic order-n Db Budeanu distortion power Dg Czarnecki generated power Ds Czarnecki scattered power f(t) Time-dependent rms complex space-vector Fe Effective value offn(t) Fn Harmonic phasor of order-n fn(t) Time-dependent rms complex space-vector of the nth harmonic component Conductance Ge Fryze equivalent conductance Gn Conductance for the different harmonic components Gne Czarnecki equivalent three-phase conductance at harmonic order-n h(n) nth order symmetric component operator I Current phasor I rms current i (t) * Conjugate time-dependent rms scaled complex current 1(n) Current vector of order -n and dimension k i(t) Time-dependent rms scaled complex current i(t) Time-dependent scalar current II Fundamental current phasor II rms fundamental current rms value of the positive-sequence fundamental current IQ rms active current IA Primary generated three phase current ia(t) Time-dependent active current 113 Load generated three phase current Ig Czarnecki rms generated current is(t) Time-dependent Czarnecki generated current Ih Czarnecki rms unbalanced current 111 rms harmonic current ih(t) Time-dependent Czarnecki unbalanced current IHD Individual harmonic distortion Inc Czarnecki equivalent three-phase current I„ nth order harmonic current phasor in(t) Time-dependent rms scaled complex current at order-n in(t) nth order simple-harmonic, time-dependent complex current /„* Conjugate current phasor of order-n inR nth harmonic phase R current Ins Source harmonic current phasor of order-n. Ins nth harmonic phase S current InT nth harmonic phase T current Ire Czarnecki rms reactive current Time-dependent Czarnecki reactive current Is Czarnecki rms scattered current Time-dependent Czarnecki scattered current k Upper range of the order of harmonic summation n Harmonic order Total non-active power NA Set of orders n of harmonics for which P 0 NB Set of orders n of harmonics for which Pn<0 NH Total harmonic non-active power Average active power p(t) Real power as function of time p(t) Time-dependent real power in the nth harmonic component pg(t) Time-dependent real power in conductance PH Total harmonic active power ph(t) Time-dependent real power in susceptance P, Scalar-joint value of the active power in the harmonic components P, Average active power at harmonic order-n P„ Harmonic active power of order-n Q Average imaginary (reactive) power q(t) Imaginary power as function of time (time-dependent imaginary power) qb(t) Imaginary time-dependent power supplied to the susceptance qg(t) Imaginary time-dependent power supplied to the conductance Scalar joint value of the reactive power in the harmonic components Qn Reactive power at harmonic order-n Qn Harmonic reactive power of order-n q„(0 Time-dependent imaginary power in the nth harmonic component Qrc Czarnecki reactive power Apparent power s(t) Time-dependent complex or apparent power Fundamental apparent power Si+ Positive-sequence fundamental apparent power Sel Three-phase fundamental apparent power SF Forced apparent power SH Total harmonic apparent power Vector joint complex power

Sjh Joint complex harmonic power SN Non-fundamental apparent power S, Harmonic apparent power of order-n Sm„ Harmonic phasor apparent power of unequal harmonic orders Unbalanced fundamental apparent power Time T Period of fundamental component THD Total harmonic distortion Voltage phasor rms voltage u(t) Time-dependent rms scaled complex voltage u(t) Scalar time-dependent voltage Ul Fundamental voltage phasor rms fundamental voltage rms value of the positive-sequence fundamental voltage UH rms harmonic voltage Czarnecki equivalent three-phase voltage U(n) Voltage vector of order-n and dimension k Un Harmonic voltage phasor of order-n Un(t) Tithe-dependent rms scaled complex voltage at order-n un(t) nth order simple-harmonic, time-dependent complex voltage x v

Un* Conjugate voltage phasor of order-n UnR nth harmonic phase R voltage Uns nth harmonic phase S voltage U;„, Source harmonic voltage phasor of order-n. UnT nth harmonic phase T voltage V11 Passive filter 11th harmonic admittance Y5 Passive filter 5th harmonic admittance Y7 Passive filter 7th harmonic admittance Complex admittance of order-n Yp Passive filter parallel admittance

Ypnm Primitive admittance matrix Vs Passive filter series admittance Zim(n) Complex impedance between nodes / and m at harmonic order-n On Phase-angle of the nth harmonic phasor Fundamental harmonic frequency XV

LIST OF FIGURES Figure 1.2.1 - Power diagram of R2P2 unit 4 Figure 1.2.2 - Block diagram of laser power supply 5 Figure 1.3.1 - Line-current waveform of 6-pulse converter at full load 6 Figure 1.3.2 - Normalised harmonic line current spectrum for the 6-pulse converter at full load 6 Figure 1.3.3 and Figure 1.3.4 - Respective relationships of THD and output DC-voltage against inductor Lr value 7 Figure 1.4.1 - Admittance diagram for the plant 8 Figure 2.2.1 - Complex s-plane representation of voltage u(t) and current i(t) 14 Figure 2.3.1 - Graphical instantaneous representation of the voltage u(t) 16 Figure 3.2.1 - Parallel circuit with conductance and susceptance 22 Figure 3.4.1 - Representation of active, reactive and distortion power in cartesian space 36 Figure 3.6.1 — Single-phase load current decomposition according to Czarnecki 40 Figure 4.1.1 - Simplified transmission system model 48 Figure 4.1.2 - Circuit for fundamental and order-n harmonics 49 Figure 4.2.1 - Principle of parallel distortion compensation 51 Figure 4.2.2 - Fundamental and harmonic parallel compensation 52 Figure 4.2.3 - Principle of series compensation 53 Figure 4.2.4 - Combined parallel and series compensation 54 Figure 4.4.1 - Single-phase example of a passive filter network 55 Figure 4.5.1 - Essential power electronic structure of a voltage-fed dynamic compensator 57 Figure 4.5.2 - Basic current-fed compensator 58 Figure 4.5.3 - Simple single-phase steady-state compensator 60 Figure 4.5.4 - Current and voltage signals for dynamic compensator 61 Figure 4.5.5 - Dynamic compensator reference signal generator 61 Figure 4.5.6 - Steady-state per-phase control strategy for a dynamic compensator 64 Figure 4.6.1 - Series hybrid compensator topology 67 Figure 4.6.2 - Parallel hybrid compensator topology 67 Figure 4.6.3 - Series-dynamic compensator topology 68 Figure 4.6.4 - Series-parallel hybrid topology 69 Figure 4.6.5 - Resonance suppression example 70 Figure 5.2.1 - Admittance of a three-phase series element 73 Figure 5.2.2 - Single compound admittance representation for three series elements 73 Figure 5.2.3 - Actual connected network 74 Figure 5.2.4 - Primitive three-phase network with compound admittances 75 Figure 5.3.1 - Equivalent circuit for supply authority model 76 Figure 5.4.1 - Primitive network for three-phase transformer 78 Figure 5.5.1 - 6-pulse AC-DC converter line current waveform 79 xvi

Figure 6.1.1 - Per-phase admittance-diagram for dummy-load measurement setup 84 Figure 6.2.1- Measurement 1 - PCC without compensation 84 Figure 6.2.2 - Measurement 1 - Comparison of modelled and measured results 85 Figure 6.3.1- Measurement 2 - PCC with passive filter at load 86 Figure 6.3.2 - Measurement 2 - Comparison of modelled and measured results 86 Figure 6.4.1- Measurement 3 - At PCC with series inductor installed on supply-side of Passive Filter 89 Figure 64.2 Measurement 3 - Comparison of modelled and measured results 89 Figure 6.5.1 - Measurement 4 - At PCC with dynamic filter connected as shown in admittance-diagram 91 Figure 65.2 - Measurement 4 - Comparison of modelled and measured results 91 Figure 7.2.1 - MLISX4 power supply network to lasers 95 Figure 7.4.1 - Measuring at load, node 14 97 Figure 7.4.2 - Line-line voltage and current at the R 2P2 terminals 97 Figure 7.4.3 - Normalised harmonic voltage spectrum for node 14 voltage 98 Figure 7.4.4 - Normalised harmonic current spectrum for node 7-14 current 98 Figure 7.4.5 — Measuring at the secondary side of the transformer 100 Figure 7.4.6 - Line-line voltage and current at the secondary side of the transformer 100 Figure 7.4.7 - Normalised harmonic voltage spectrum 101 Figure 7.4.8 - Normalised harmonic current spectrum 101 Figure 7.4.9 — Measuring on the primary side of step-down transformer 103 Figure 7.4.10 - Line-line voltage and current at the transformer primary 103 Figure 7.4.11 - Normalised harmonic voltage spectrum 104 Figure 7.4.12 - Normalised harmonic current spectrum 104 Figure 7.4.13 - Measuring on the input side of an R3P2 power supply unit 107 Figure 7.4.14 - Line-line voltage and current at the R 3P2 terminals 107 Figure 7.4.15 - Normalised harmonic voltage spectrum 108 Figure 7.4.16 - Normalised harmonic current spectrum at the R 3P2 terminals 108 Figure 7.5.1 - Measuring at the primary side of the transformers 111 Figure 7.5.2 - Line-line voltage and current primary of the transformer 111 Figure 7.5.3 - Normalised harmonic voltage spectrum 112 Figure 7.5.4 - Normalised harmonic current spectrum 112 Figure 7.5:5 - Measuring at the laser load 113 Figure 7.5.6 - Line-line voltage and current at the R2P2 terminals 114 Figure 7.5.7 - Normalised harmonic voltage spectrum 114 Figure 7.5.8 - Normalised harmonic current spectrum at R2P2 terminals 114 Figure 7.6.1 - Measuring at the laser load, node 14 117 Figure 7.6.2 - Line-line voltage and current at the R2P2 terminals 117 Figure 7.6.3 - Normalised harmonic voltage spectrum 118 xvii

Figure 7.6.4 - Normalised harmonic current spectrum at R2P2 terminals 118 Figure 7.6.5 - Measuring at the input side of the series inductor to the laser load 119 Figure 7.6.6 - Line-line voltage and current at the R2P2 terminals 120 Figure 7.6.7 - Normalised harmonic voltage spectrum 120 Figure 7.6.8 - Normalised harmonic current spectrum 120 Figure 7.6.9 - Measuring at the secondary side of step-down transformer 1 122 Figure 7.6.10 - Line-line voltage and current at the PCC (node 5) 122 Figure 7.6.11 - Normalised harmonic voltage spectrum 123 Figure 7.6.12 - Normalised harmonic current spectrum at R2P2 terminals 123 Figure 7.6.13 - Measuring at the primary side of step-down transformers 124 Figure 7.6.14 - Line-line voltage and current at the R2P2 terminals 124 Figure 7.6.15 - Normalised harmonic voltage spectrum 125 Figure 7.6.16 - Normalised harmonic current spectrum 125 Figure 7.7.1 - Measuring on the laser load 128 Figure 7.7.2 - Line-line voltage and current at the R2P2 terminals 128 Figure 7.7.3 - Normalised harmonic voltage spectrum 129 Figure 7.7.4 - Normalised harmonic current spectrum at R2P2 terminals 129 Figure 7.7.5 - Measuring at the secondary side of transformer 1 130 Figure 7.7.6 - Line-line voltage and current at the transformer secondary node 3 131 Figure 7.7.7 - Normalised harmonic voltage spectrum 131 Figure 7.7.8 - Normalised harmonic current spectrum at R2P2 terminals 131 Figure 7.7.9 - Measuring at the primary side of the step-down transformers 133 Figure 7.7.10 - Line-line voltage and current at the R2P2 terminals 133 Figure 7.7.11 - Normalised harmonic voltage spectrum 134 Figure 7.7.12 - Normalised harmonic current spectrum 134 Figure 7.8.1 - Measuring at the load 136 Figure 7.8.2 - Line-line voltage and current at the R2P2 terminals 136 Figure 7.8.3 - Normalised harmonic voltage spectrum 137 Figure 7.8.4 - Normalised harmonic current spectrum at R2P2 terminals 137 Figure 7.8.5 - Measuring at the secondary of step-down transformer I 138 Figure 7.8.6 - Line-line voltage and current at the R2P2 terminals 138 Figure 7.8.7 - Normalised harmonic voltage spectrum 139 Figure 7.8.8 - Normalised harmonic current spectrum 139 Figure 7.8.9 - Measuring at the primary side of the step-down transformers 140 Figure 7.8.10 - Line-line voltage and current at the R2P2 terminals 141 Figure 7.8.11 - Normalised harmonic voltage spectrum 141 Figure 7.8.12 - Normalised harmonic current spectrum 141 Figure 7.9.1 - Measuring at a double-laser load 143 xviii

Figure 7.9.2 - Line-line voltage and current at the R2P2 terminals 143 Figure 7.9.3 - Normalised harmonic voltage spectrum 144 Figure 7.9.4 - Normalised harmonic current spectrum at R2P2 terminals 144 Figure 7.9.5 - Measuring at the secondary side of step-down transformer 1 145 Figure 7.9.6 - Line-line voltage and current at the transformer secondary for modelling run 18 145 Figure 7.9.7 - Normalised harmonic voltage spectrum 146 Figure 7.9.8 - Normalised harmonic current spectrum 146 Figure 7.10.1 - Measuring at the primary side of the step-down transformers 148 Figure 7.10.2 - Line-line voltage and current at the transformer primary option 19 148 Figure 7.10.3 - Normalised harmonic voltage spectrum 149 Figure 7.10.4 - Normalised harmonic current spectrum 149 Figure 7.10.5 - Measuring at load 151 Figure 7.10.6 - Line-line voltage and current at the R2P2 terminals 151 Figure 7.10.7 - Normalised harmonic voltage spectrum 152 Figure 7.10.8 - Normalised harmonic current spectrum at R2P2 terminals 152 Figure 7.13.1- - Line-Line voltage and current at node 5 159 Figure 7.13.2 - Comparison of modelled and measured results 159 Figure 7.13.3 - Normalised harmonic voltage spectrum - modelled 160 Figure 7.13.4 - Normalised harmonic voltage spectrum - measured 160 Figure 7.13.5 - Normalised harmonic current spectrum - measured 160 Figure 7.13.6 - Normalised harmonic current spectrum - measured 160 Figure A 1 - Time domain representation of line-line A3 Figure A 2 - Time domain portion of negative portion A4 Figure A 3 —Positive portion of pulse plotted from frequency domain A6 Figure A 4 — Negative-going pulse portion synthesised from Fourier components A7 Figure A 5 — Effect of changing a on the line-line converter current A8 Figure A 6 — Typical line current profile together with line-neutral voltage for a 6-pulse conveter A9 Figure B 1— Twelve laser layout B3 xix

LIST OF TABLES Table 1.2.1 - Salient Specifications for one R 2P2 Module 5 Table 1.3.1 — Salient parameters of R 2P2 input 7 Table 5.6.1 - Characteristics offilter 81 Table 6.2.1 - Measurement 1 - PCC without compensation 85 Table 6.3.1 - Measurement 2 - PCC with passive filter at load 87 Table 6.4.1 - Measurement 3 - PCC with series inductor on supply-side ofpassive filter 90 Table 6.5.1 - Measurement 4 - Dynamic compensator at node 4 92 Table 7.1.1 - List of circuit configurations modelled 93 Table 7.4.1 - Modelling run 1 - Measurements at the R 2P2 terminals for Configuration 1 99 Table 7.4.1 - Modelling run 2 - Circuit Configuration 1 - Measuring at the secondary side of the step- down transformer 102 Table 7.4.1 - Modelling run 3 - Measurements at the primary side of the step-down transformer 104 Table 7.4.1 - Modelling run 4 - Measurements at the R 3P2 terminals 108 Table 7.5.1 - Modelling run 5 - Measurements at the primary side of the step-down transformer 112 Table 7.5.1 - Modelling run 6 - Measurements at the R 2P2 terminals for Configuration 2 115 Table 7.6.1 - Modelling run 7 - Measurements at the R 2P2 terminals for Configuration 3 118 Table 7.6.1 - Modelling run 8 - Measurements at the at the R2P2 terminals configuration 3 120 Table 7.6.1 - Modelling run 9 - Measurements at the FCC (node 5) for Configuration 3 123 Table 7.6.1 - Modelling run 10 - Measurements at the transformer primary for configuration 3 125 Table 7.7.1 - Modelling run 11 - Measurements at the R 2P2 terminals for Configuration 4 129 Table 7.7.1 - Modelling run 12 - Measurements at the transformer secondary for Configuration 4 132 Table 7.7.1 - Modelling run 13 - Measurements at primary of transformer for Configuration 4 134 Table 7.8.1 - Modelling run 14 - Measurements at the R 2P2 terminals for Configuration 5 137 Table 7.8.1 - Modelling run 15 - Measurements at transformer secondary for Configuration 5 139 Table 7.8.1 - Modelling run 16 - Measurements at the primary of the transformer for Configuration 5 141 Table 7.9.1 - Modelling run 17 - Measurements at the R 3P2 terminals for Configuration 6 144 Table 7.9.1 - Modelling run 18 - Measurements at the secondary of the transformer for Configuration 6 146 Table 7.10.1 - Modelling run 19 - Measurements at the transformer primary for configuration 7 149 Table 7.10.1 - Modelling run 20 - Measurements at the R 2P2 terminals for configuration 7 152 Table 7.11.1 — Economic comparison of the different compensation topologies 155 Table 7.12.1 - Reactive powers for the different modelling runs 156 Table 7.12.1 - Harmonic active power components 157 Table 7.13.1 - Modelling and Measurements at PCC 161 Table 7.13.2 - Modelling and measurements results for the three power theories 161 1

1. THE MLIS PLANT ENVIRONMENT

1.1 INTRODUCTION

The Atomic Energy Corporation of South Africa (AEC) decided in the late seventies that it would invest in the Molecular Laser Isotope Separation (MLIS) process for uranium enrichment 1 .2 Research and Development has progressed, at the time of writing, to the point at which laser excitation systems, on the electrical side, have individually clocked multiple hours during endurance testing and are commercially and technically ready to be exploited in a plant environment.

The next stage in the enrichment program will be the composite testing of all the equipment (excitation systems, control schemes, lasers and isotope processing components) in a pilot plant. This pilot plant is currently in its final stages of preparation and will be used to demonstrate the engineering feasibility of the project and to highlight critical parameters that govern the capital and operating cost structures of such a plant. The final stage, planned upon the successful testing and evaluation of equipment in the pilot plant, will be the construction of a full-scale MLIS plant.

A key element in the development of the MLIS process is the industrial CO 2 transversely excited atmospheric (TEA) pulsed lasers that furnish the excitation in the separation process. These lasers have average optical powers in the kilowatt range and peak pulsed optical power outputs in the megawatt range3. Research in the development of these lasers and their associated electrical pulse excitation systems began in 1985 and has led to the establishment of a new technology of lasers and laser excitation systems 4.

The electrical pulse excitation systems, also referred to as pulsers, consist of a number of different elements. The resonant regulating pulse power (R 2P2) supply units, takes power from the mains network and carries out primary pulse conversion. The output pulses from the R 2P2 units are then further processed by thyratron pulsers or multi-stage pulse compressors 6, before they are compatible with the ultra-high peak current and ultra-short pulse durations required by the pulsed lasers.

These R2P2 units used for primary pulse modulation, employ intermediate DC-stages, in like manner to that of other power-electronic equipment. In this case, the DC-voltage levels are required to be very accurately and continuously adjustable to furnish highly repeatable and adjustable output pulse voltage levels.

Several options exist for the conversion of the supply network three-phase AC-power to DC. Conventional line-commutated phase control is presently employed, but the so-called "line- 2 friendly" or force-commutated PWM-modulated converter is also under investigation as an alternative'.

The use of phase-controlled AC-DC converters in the R2P2 front-ends leads to distortion in the currents drawn by these units from the AC network. On a smallscale, this distortion will not be a problem and can even be tolerated in a number of instances. At the scale envisaged for the main plant, which will extend to several hundred megawatt, and even in the case of the pilot plant, the distortion is expected to introduce problems that will extend from malfunctioning of equipment to infringements of supply-authority standards' for consumer harmonics 9.

Aside from the alternative supply-friendly front-end options, that may come in a number of forms and employ alternative topologies and control schemes, a number of external compensation options also exist for the mitigation of the distortion by the phase-controlled converter. The conventional line-commutated 6-pulse converter features very prominently in the present investigation, because of its relatively low cost, simplicity of operation, and its proven track record.

Distortion compensation technology is still in its infancy. Although passive filter networks have been employed for some time now to reduce distortion in supply currents to non-linear plants, the problems they introduce are many. They are most suited for installation on the H.V. side of consumer transformers and, because they tend to react with each other, multiple installations in close electrical proximity are usually impractical. Because of the spread-out nature of a large number of identical, relatively small distortion loads in the envisaged pilot and main plants, compensation of one form or another at the pulse power supply locations themselves may have to be considered to maintain acceptable limits of distortion in the feeders that will be spread throughout the plant.

Dynamic compensation offers a technical solution, but at the large power levels considered for the main plant, an expensive one. The total compensation solution, if that option is exercised, obviously lies in a hybrid compensation topology in which passive and dynamic compensation is integrated to yield both the most cost-effective and best technical results. This solution, and a search for the optimal combination of alternatives, form the theme of this study, and to which the answers are sought in the following chapters.

A comprehensive study of the problem outlined above necessitates extensive modelling. In this modelling, distorted voltage and current waveforms will have to be treated throughout. Established and concise definitions that are used in well-behaved sinusoidal systems are not applicable and other power definitions, that cater for the conditions of distortion, must be resorted to. Because this study concerns a practical plant and practical measurements and assessments, it also affords the ideal opportunity for the parallel testing and evaluation of the most prominent 3 power theories presently under consideration, namely that of Budeanu 15, the IEEE Working Group definitions3' and the Czarnecki power definitions 17.

Because other literature, 10" 1 already gives a basic background to the envisaged plant layout and operation, further explanations of the process will be avoided here. Schematics of the plant electrical layout furnishing details of the number and characteristics of the topographically distributed pulse power supply loads on the electrical distribution system will suffice here for the purpose of modelling and measurement.

The chapter will begin with a description of the R 2P2 units and their front-ends, that will constitute the main power consumers in the plant. Next, the simplified distribution layout for the envisaged plant will be sketched and typical load characteristics, modelled and measured, will be given.

1.2 R2P2 PULSE POWER SUPPLY CONFIGURATION

The pulsers driving the CO 2 pulse lasers in the programme, are driven by R2P2 modules that take three-phase AC power from the supply network at a nominal voltage of 400 V. Because the aggregate output pulse power requirement per laser exceeds the capacity of the individual modules, more than one module is used in a power supply configuration to time-multiplex the output pulses into the input of each laser pulser. In a typical case in which the laser requires to operate at 1 kHz, two R 2P2 modules can be employed at 500 Hz each or four modules can be employed at 250 Hz each. In practice, more than the minimum necessary number of modules will be used for high reliability and faulty modules will be automatically switched out of the pulse power supply upon failure without interrupting the operation of the power supply as a whole.

The R2P2 units at present employ conventional 6-pulse line-commutated phase-controlled converters in their front-ends to furnish an intermediate, continuously controllable DC-source that is in turn used for the generation of power pulses through primary pulse conversion by successive resonant energy transfer stages. The basic, simplified power diagram of an R 2P2 unit is shown in simplified form in Figure 1.2.1.

As shown, the 3-phase mains supply of 400 V is fed to a 6-pulse phase-controlled bridge in the input end of the R2P2 unit. The rectified output of the bridge is smoothed by the DC-choke inductor Lr and reservoir capacitor C I and is delivered to the power circuit at a nominal DC- voltage that can be controlled by adjustment of the firing angle of the phase-control thyristors from very low values to the maximum value of about 530 V (no-load value). 4

R 3

Rb Dm1 1: 2t>11-1 CI b D m 2 ->F C o Dm3 -[>[ L a MMI Tb 1 Dm4 Lr Lm TX • Cr

I I • D De

Figure 1.2.1 - Power diagram of R2P2 unit

Operation of the R2P2 unit is best explained by designating of three successive but separate phases' 2 to its operation.

The three phases are respectively the charging phase, the transfer phase and the regulating (also called the de - queuing) phase. The charging phase is initiated by firing thyristor Tb and is employed to charge the primary capacitor C o from the rectified supply by resonant energy transfer in the loop formed by Cr, La, Tb Co and the primary winding of transformer TX.

During the transfer phase, energy is resonantly transferred (through the step-up pulse transformer TX) from the primary capacitor C o, through the leakage inductance of the pulse transformer TX and the multiplexing diode, to the secondary (high voltage) capacitor C, in Figure 1.2.1.

The regulating phase is responsible for interrupting the charging phase under controlled conditions by the triggering of thyristor Ta, to terminate the delivery of charge to the primary capacitor Co, thereby accurately controlling the pulse voltage (and the energy) that is delivered to the load with each pulse.

Triggering of thyristor Ta during the second half of the charging transfer cycle, when the current gradient through La is negative, will turn Tb off, thereby terminating the transfer of charge to C o. In the same way, triggering of Tb when Co is discharged, after the next transfer cycle, will initiate the next charging cycle and terminate the current flow in the loop formed by L a and Ta. 5 The relevant specifications for the present configuration of the R 2P2 modules, used in the present study, is given in table 1.2.1.

Table 1.2.1 - Salient Specifications for one R 2P2 Module

DESCRIPTION OF PARAMETER UNIT VALUE Peak output pulse voltage kV 24 Repeatability of output pulse voltage % 0.01 Peak output pulse energy J 20 Peak output pulse frequency Hz 500 Peak input power kW 15 Input three-phase voltage V 380 Input power factor 0.84 Input total harmonic distortion % 47 Overall efficiency % 71

The input to the pulse compressor utilises multiplexing diode D m' to Dm4 to connect to the four power supply modules. In practice three modules are used for an oscillator (first laser in a laser train, and delivering only in the region of 17 J of energy) and four for an amplifier. A high-power laser (80 kW) is also under development and this laser may employ as many as eight power modules each delivering 40 J at 250 Hz. The multiplexing configuration is shown in block- diagrammatic form in Figure 1.2.2.

Power supplies 1 3 4-8

Pulse circuit

Laser load

Figure 1.2.2 - Block diagram of laser power supply

The pilot plant laser train will initially employ 11 lasers, operating at 1 kHz. This train will make use of 41 R2P2 modules, of which 22 will be operating at full load, with each drawing 15 kW of active power from the supply, or drawing a total of 330 kW in total. When fully operational, the chain will employ all 41 modules at 2 kHz, requiring a total input active power of 615 kW. The layout may change in future to accommodate the best achievable process parameters. These

6 parameters will be refined during on-going development of the process and the establishment of the most optimal laser configurations, powers and other parameters. The electrical power requirements and plant layout will ultimately depend on the latest findings.

1.3 POWER SUPPLY FRONT - END TOPOLOGY

There is a space constraint in the maximum size of choke inductor L r that can be installed in the R2P2 units. The maximum size of choke that is now capable of accommodating the continuous load with air cooling is 3.2 mH. This inductance is only just capable of furnishing continuous output current under full-load conditions at 1 kHz. At full power, the line current amplitudes yield continuous output current as shown in Figure 1.3.1.

400 320 240 160 Line 80 current 0 -80 -160 -240 -320

-400 -0.03 -0.02 -0.01 0 0.01 0.02 Time in seconds

Figure 1.3.1 - Line-current waveform of 6-pulse converter at full load

The normalised harmonic current spectrum for the above case, assuming steady-state conditions, is shown in Figure 1.3.2.

40 36 32 Harmonic 28 current as a 24 percentage of 20 the fundamental 16 12 8 4 0 U 5 10 15 20 Harmonic order

Figure 1.3.2 - Normalised harmonic line current spectrum for the 6-pulse converter at full load 7 The most important per-phase parameters, assuming phase symmetry for the above conditions are given in table 1.3.1.

Table 1.3.1 — Salient parameters of R2P2 input'

Equivalent total node voltage 239.03 V Equivalent fundamental node voltage: 239.013 V RMS load current: 84.171 A RMS fundamental component of line current: 78.657 A RMS harmonic component of line current: 29.964 A Apparent power: 60.358 kVA Fundamental apparent power: 52.313+21.08 j kVA Current THD: 38.095 % Voltage THD: • 1.196 %

A PSpice simulation of the 6-pulse converter against different values of the DC-choke value furnished the line current THD and output DC voltages, shown respectively in Figure 1.3.3 and Figure 1.3.4.

Figure 1.3.3 and Figure 1.3.4 - Respective relationships of THD and output DC-voltage against inductor L,. value

As shown in Figure 1.3.3 and Figure 1.3.4 the distortion on the line current drops from 120% to 40% and the line voltage from 500 to 445 when L4 increases from 300 1.tH to 3.2 mH. The module operates in the discontinuous mode with Lr= 300 11H and changes to the continuous mode when L1 approaches 3 mH. Both THD and output DC voltage tend to their asymptotic values when increasing I., beyond 3 mH. The choice of inductance for L r of 3.2 mH is therefore a

i The fundamental and harmonic components are separated in accordance with the theory discussed in chapter 3.

8 practical one. The line-current THD and output DC voltage behaviour is also borne out in the literature 13.

1.4 PLANT LAYOUT

The simplified, per-unit, line diagram of the proposed electrical installation that will typically serve the plant is shown in figure 1.4.1.

r-NLTi t Load Single laser PASSIVE ,- at A-EBP0002 FILTER.

15 Load Double laser at PASSIVE‘ A-EBP00:13 FILTER

(-19Toir 16 Load PASSIVE t Double laser at A-EBP0004 FILTER

I-1 10 Load 000) _L 17 Single laser at A-EBP0005 PASSIVE FILTER

(-Nroon} 18 Load v I Single laser at PASSIVE A-EBP009 FILTER

Load (--,12 ino-o. Double laser at PASSIVE 1 A-EBP008 FILTER

Load r-, (130-u 20 Doublelaser at PASSIVE t A-EBP0007 FILTER

r121_(7)()0) 22 Load _I_ Single laser at PASSIVE A-EBP0006 FILTER

Figure 1.4.1 - Admittance diagram for the plant

This diagram has been derived from the electrical reactance diagrams for the installation and only equivalent compound admittances and current sources are employed. Admittances and nodecurrent values are obtained by source transformation from their respective source, line, cable, 9 transformer and other transmission component values. Admittance values are taken from their fundamental frequency values and are scaled for other harmonic frequencies as explained later.

As the diagram shows, power is taken from a single source at 11 kV as in the single-load study above, which in fact consists of several substations in a ring network. The Norton equivalent single source represents this network of substations. A single radial feeder, consisting of a 150mm2 11 kV PEX cable of length 391 m, brings this power to the substation, where it is transformed down to 380 V by two separate 2 MVA 5,69% impedance 11 kV to 380 V AY transformers, each feeding a separate busbar. In the event of a transformer failure, a bus-coupler can link the busbars and the plant is then operated on one transformer only. This configuration will also have to be investigated separately, because of the higher levels of voltage distortion that it will bring about.

Each transformer secondary is coupled to a dynamic compensator as well as to a passive filter, at nodes 3 and 4 for transformers 1 and 2 respectively. This provision again enables these components to be enabled or inhibited as the study demands, merely by redefining the coupling admittances in the case of the passive filters, or by toggling the current that the dynamic compensators inject into the network.

Each substation busbar, located at nodes 5 and 6, supplies a set of lasers. The particular allocation of lasers is representative of the present cable installations in the building and corresponds to the information presently known with respect to the physical location of the lasers. Alterations will only introduce marginal differences in cable and can be accommodated easily when final layouts have been finalised. A number of load models cater for double lasers and other for single lasers as shown.

The R2P2 loads are supplied from the two busbar sections by twin 95 mm 2 PVCAPVC cables that are generally about 60 m long. Provision is also made in the model for accommodating a smaller passive filter at each laser, and to equip each of the filters with a series inductor. The passive filters can be optionally inserted at nodes 14 to 20 and 22 in the model to substitute the parallel admittances as shown. The series admittances between the previous nodes and the laser nodes, such as Yp7_14, permit the installation of series decoupling inductors between the passive filters and the supply cables at each laser load.

The series inductors have important functions and certain configurations may not be possible or feasible without them. Firstly, their use is mandatory when multiple filters are installed to reduce parallel resonance between the multiple filters. They may also assist in reducing harmonic current distortion in the cables, provided that the ancillaries at each laser can be powered from a point before them and provided that voltage distortion will be permitted on the inputs of the AC/DC converters themselves. 1 0

1.5 SUMMARY

The basic plant layout and electrical requirements for the laser plant was discussed in this chapter. The layout and the modelling make provision for the installation of passive and dynamic filters in different locations in the circuit. The essential loads consist of 6-pulse line-commutated phase-controlled converters operating under steady load conditions. The ensuing study, commencing in the next chapters, will comprise finding the optimal configuration and ratings of compensating equipment for this plant. This study will also ideally lend itself to an analysis of different power theories, by means of which the modelling and measurement results can be analysed. 11 2. FUNDAMENTAL DEFINITIONS

2.1 INTRODUCTION

Steady-state linear AC circuit theory is derived from the forced response of circuits when they are excited by sinusoidal forcing functions'''. The definitions of complex phasor quantities and impedances, that form the basis of this theory, however, are confined to the requirement that the forcing functions be sinusoidal and that the circuits be linear.

When the forcing functions in a linear circuit are no longer sinusoidal, or if the circuit is not linear or is time-variant, non-sinusoidal responses take place and either the currents or the voltages or both may be non-sinusoidal. This deviation from the sinusoidal form is referred to as distortion and the voltages and currents are said to be distorted. When these distorted, but periodic responses are analysed in the frequency domain, they are found to possess more than one discrete frequency, and such responses may also be referred to as multi frequency responses.

The distorted or multi-frequency behaviour of AC circuits prohibits the use of complex analysis and the defined concepts of phasors, impedances and power are no longer applicable. When the distorted responses of voltage and current are periodic and when the circuit has a steady-state response, Fourier analysis yields discrete responses in the frequency domain. Each of these responses, unique in phase and magnitude are, however, sinusoidal and analysis becomes amenable to all the first-named definitions for conventional AC circuits. This is where the similarity to conventional treatment ends, however, for each discrete frequency component. New difficulties arise when the collective or joint power components have to be calculated. To do that, new definitions are required.

Studies in the field of circuit-analysis in the presence of distortion is becoming more important every day and has developed into a major field of research, yielding a proliferation of publications in which new analytical theories are advanced. Notable work in this regard, but not necessarily in any order of importance, is that of Budeanu 15, Fryze16, Czarnecki",

Depenbrock' 8 Custers and Moore 19 , Enslin2° and Akagi and Nabae21 . Before beginning with a discussion of some of these theories, a convenient set of definitions will first be made to cover the underlying theory and to gradually lead the reader from the conventional definitions for the sinusoidal case through to multi-frequency topologies in a number of steps. 12

2.2 FUNDAMENTAL DEFINITIONS UNDER SINUSOIDAL CONDITIONS

2.2.1 Fundamental considerations

Steady-state conditions in an AC circuit here refers to periodic behaviour, in the time domain, of the time-dependent values of voltage, current and power. Transient conditions in an AC circuit refer to deviations from periodic behaviour that are relatively short-lived and disappear within a fraction of a period to a few periods. Steady-state theory, on which Fourier analysis is based, does not cater for transient behaviour and both the duration and magnitude of the transient occurrences are assumed to be negligible over the period of operation to which this study applies.

2.2.2 Complex voltage and current as a function of time

A circuit is fully characterised by the topology of its active and passive components. Active components are either voltage or current sources and passive components are those of resistance, capacitance and inductance which, collectively or individually, constitute the immittance of the circuit'''.

Under sinusoidal, steady-state conditions, the voltage and current as time functions, may be represented in a circuit respectively by:

u(t) = 42 U cos(cot + a) (2.2.1) and

i(t) = /coot + p) (2.2.2) in which U and I respectively represent the scalar rms-values of voltage and current, and a and p the phase angles.

The above, measurable, time-dependent quantities represent the values of the respective current and voltage that can be expressed in accordance with the given equations for u(t) and i(t) in the circuit at any instant t. These values of voltage and current will be referred to as the time- dependent voltage and the time-dependent current in the circuit, in preference to the designations of instantaneous voltage and instantaneous current.

Equations (2.2.1) and (2.2.2) may also be expressed in a more general form through Euler's relation as:

u(t) = Re{ U e j(6)t + a)} (2.2.3) and 13

i(t) = yL Re{ I e i(cot + 13)} (2.2.4)

The generalisation may be taken further, to represent the time-dependent quantities in equations (2.2.3) and (2.2.4) directly in their complex form as:

u(t) = U e J(cot + a)) (2.2.5) and

i(t) = / e .i(cot 13)) (2.2.6)

The quantities are now complex and do not represent the normal physically measurable values of voltage and current. They do, however, still represent time-dependent values of voltage and current and lend themselves to a number of operations that will be very convenient later in subsequent analysis. It is worthwhile to note that these complex quantities represent the more general form from which the scalar values of u(t) and i(t) can be constructed. They cannot be designated as instantaneous values. In this regard, u(t) and i(t) are only time-dependent values. Note also that u(t) and i(t) are scaled as rms-quantities. This formulation is expedient to avoid the inevitable .42 and its derivatives in power calculations. It now follows that the complex time-dependent values of equations (2.2.5) and (2.2.6) are related to the scalar values by:

u(t) = 42 Re { u(t)} (2.2.7) and

i(t) = 42 Re { i(t)} (2.2.8)

It also follows from the previous two equations (2.2.7) and (2.2.8) that

u(t) = u e ja e icot (2.2.9) and

i(t)=IeJP a Jwt (2.2.10) in which:

u(t) = U e jc" (2.2.11) and:

i(t) = / e Jwt (2.2.12) 14

The complex quantities U and I are immediately recognised to represent the respective complex rms -phasor values of voltage and current.

The sketches in Figure 2.2.1 show that u(t) and i(t) represent rms-scaled vectors of constant magnitude that rotate in an anti-clockwise direction in the complex s-plane with angular frequency co. Rotation will be centred around the origin in the absence of a DC-component, or be shifted by the appropriate value on the real axis in the presence of one. Note that the respective phase angles a and p of the voltage and the current represent the angles that these vectors make with the real positive axis at time t = 0. u t) imaginary axis imaginary axis i ( t)

t t

a real axis real axis

Figure 2.2.1 - Complex s-plane representation of voltage u(t) and current i(t)

Reflection on the above discussion will show that the measurable physical values of u(t) and i(t), as expressed by equations (2.2.7) and (2.2.8) are simply numerically equal to the projections of the rotating vectors on the real axes, multiplied by the rms-scaling factor 42 to convert the vector magnitudes to peak values.

The term complexor has been suggested for the quantities u(t) and i(t) in the literature22. The corresponding phasor voltage U and phasor current I are complex constants that do not change under steady-state conditions, but which may vary 'gradually' to take the circuit operation from one steady-state condition to the next. In accordance with the foregoing discussion, measurement is not defined during these transitions.

2.2.3 RMS value

It can be seen through inspection that the complex (RMS) value may be obtained by expressing u(t) or i(t) at t = 0:

u(0) = U or i(0) = / (2.2.13), (2.2.14)

Alternatively, it can easily be shown that the scalar (RMS) value of u(t) or i(t) may be directly calculated from the time-dependent complex value as: 15

U= qu(t)u(t) * or I =.\, (2.2.15), (2.2.16)

Because of their identical mathematical nature, both complexors and phasors conform to the definition of space vectors and identical mathematical operations are equally valid in both cases.

2.2.4 Steady-state impedance

The relationship of phasor voltage U to phasor current I in a circuit under sinusoidal conditions is a function of the steady-state or AC-impedance 14 :

Z = 11 (2.2.17)

Instead of expressing impedance through the relationship between U and I, it is analytically valid, for example, to define the AC - impedance directly in terms of the complex time- dependent voltage u(t) and current i(t). That this is valid can easily be shown by respectively substituting equations (2.2.9) and (2.2.10) for u(t) and i(t) in eq. (2.2.17) and cancelling out the e Jwt

u(t) (2.2.18) Z = i(t)

2.3 FUNDAMENTAL DEFINITIONS UNDER NON-SINUSOIDAL CONDITIONS

2.3.1 Complex voltage and current

Time-dependent, single frequency, simple harmonic basic theory was reviewed in the previous section for circuits in which the voltage and current are sinusoidal. Circuits operating under steady-state, but periodic conditions, also yield simple harmonic component waveforms under Fourier analysis. This approach is equally applicable to each separate harmonic frequency and furnishes a convenient vehicle by means of which existing power concepts and definitions can be extended to all frequencies to provide a composite distortion power theory in the frequency domain.

The analysis in the case of single-frequency circuits employ complex values. In steady-state distorted periodic circuit analysis, summations are introduced to represent all the relevant harmonic components in the equations. Consider an instantaneous complex function of a periodic steady-state voltage or current represented in general by f(t):

It can be shown by simple manipulation of the exponential Fourier series 23 that:

k fit) = fFn ekca (2.3.1) n=0 16 in which fit) and the Fn are again rms-scaled complex values, just as in the single-frequency case. Note that n ranges from 0 to k in eq. (2.3.1) in order to include a DC-value when it is present and for which F0 will be real. Because f(t) is scaled to its rms-value, it is necessary to multiply it by -42 to obtain the peak value. The value of k in the summation is usually chosen lower than oo for practical reasons, but at the sacrifice of accuracy of representation. The nth order harmonic phasor Fn is defined for the nth order harmonic, identically to that of the single-frequency case.

It follows from eq. (2.3.1) that the nth order complexor is given by:

= Fn ejncot (2.3.2)

In eq. (2.3.2), fn(t) represents a vector of constant magnitude that rotates in an anti-clockwise direction in the complex plane with angular velocity nco. The corresponding "measurable" part offn(t) is then expressed as:

fn( t) = Re{ fn(t)} (2.3.3)

Because fn(t) is sinusoidal and periodic, it follows that the complex AC-circuit theory developed in the previous section is applicable at each separate harmonic order-n.

The complex quantity fn(t) can, for example, be represented on the complex plane as a time- dependent vector that will be rotating about a point F0 on the real axis. F0 represents the average or DC-value of the waveform. F0 will always be real and will lie on the real axis. The complexor fit) is represented graphically in Figure 2.3.1 for k= 3. It is shown to be formed by the vector sum of the instantaneous values of the fundamental and two harmonic components.

Figure 2.3.1 - Graphical instantaneous representation of the voltage u(t) 17

Although each of the fn(t) components have constant magnitude and a constant angular velocity of rotation about the origin of nco, the complexor fn(t) has neither constant magnitude nor does it rotate about the origin with a constant angular velocity. The average angular velocity of _At) about its point of rotation is, however, still equal to co because it makes one revolution in one fundamental period T. When the real part off(t) is plotted against time, it will yield the periodic but distorted waveform that its Fourier series represents.

2.3.2 Harmonic phasors

The nth harmonic phasor Fn can be calculated directly from the complex Fourier series 23 and adapted for unilateral application as:

r Fn = —TEO) ejnwtdt (2.3.4)

It follows from eq. (2.3.4) that:

Fn = Fn ei(1)11 (2.3.5)

Equation (2.3.5) can conveniently be written in the conventional polar form as:

Fn = Fn EL° (2.3.6)

2.3.3 Space-vector manipulations

Because the harmonic quantities conform to sinusoidal requirements at each harmonic order, it is simple to show that the following operations are equivalent:

The magnitude of the complex quantity fn(t) can be calculated as:

ln(t)fn(0 * =1In(t)12 (2.3.7)

Iffn(t) is substituted in eq. (2.3.7) from eq. (2.3.2), the exponents cancel, reducing it to:

FnFn* = iFni2 = Fn2 (2.3.8)

in which fn(t) and Fn represent the rms-scaled magnitude of the two quantities, therefore:

4.1,2(tYn(0* = Fn (2.3.9) 18 2.3.4 Effective value

The effective value of a waveform is defined to be the root mean square value of the measurable time-dependent value over a representative interval of time. For the periodic but non-sinusoidal function fit) of period T, the scalar effective value can be obtained from:

= 11 —T1 SNERe{ f (t)}2 dt (2.3.10)

By substituting forfit) from equations (2.3.3) and (2.3.3) the following equation is obtained:

~/ = iJ Re{f„ dt (2.3.11)

Because fn(t) has constant magnitude and makes a full revolution on the complex plane in one period, it can be shown that eq. (2.3.11) is also equivalent to the following:

T fn (t) fn (t) * dt (2.3.12) n=0 T

The order of summation and integration may be exchanged in eq. (2.3.12) to give:

(2.3.13)

If eq. (2.3.2) is used to substitute for fn(t) in the last equation, the time-dependent quantities cancel out and integration only yields T which cancels out with the 1/T before the summation sign. This yields the following expression:

k Fn Fn * (2.3.14) n=0

By substituting for Fn from eq. (2.3.5), the above equation can be further reduced to:

(2.3.15) 19 The above result is significant. It shows that the effective value of the distorted waveform may be calculated directly from the individual harmonic phasors, with a consequent potential saving in computation time. The fact that it can be calculated in this way is also by virtue of the fact that the different harmonic phasors are mutually orthogonal - a result that follows from the orthogonality requirement for the Fourier expansion.

2.3.5 Individual harmonic distortion and total harmonic distortion

The above results may be used to calculate the individual harmonic distortion and the total harmonic distortion of a distorted periodic waveform. The individual harmonic distortion (IHD) of a waveform is defined to be the relative magnitude of the rms value of a given harmonic order, relative to that of the fundamental component: JF,71 IHD = (2.3.16) n iFli

The total harmonic distortion (THD) of f(t) is defined to be the ratio of the rms value of the harmonic components to that of the fundamental component.:

Ik 11,1212

n=2 THDF — ,\1 (2.3.17) IFI

For the purpose of modelling in the frequency domain, fit) is synthesised in its complex form directly as shown in eq. (2.3.1). Both i(t) and u(t) can be modelled to conform to the conditions of eq. (2.2.5) and (2.2.6).

2.4 SUMMARY

In this chapter the rms-scaled current i(t) or voltage u(t) complex value in a circuit under sinusoidal conditions were discussed. It was shown that the real part of this respective quantity, when multiplied by -V2 , represents the conventional, measurable, time-dependent value of this quantity at the given instant t. It was shown that this time-dependent value represents a vector of constant magnitude revolving in an anti-clockwise direction about the zero on the Cartesian complex plane. It was shown next that the conventional rms-values of voltage and current can be directly derived from the complex time-dependent values.

The definitions of time-dependent values were extended to circuits under non-sinusoidal excitation. It was shown that for periodic, steady-state non-sinusoidal voltages and currents, the same concepts could be applied to the harmonic components by complex Fourier analysis, and that a periodic but distorted voltage or current merely consisted of the discrete harmonic components of these quantities, which when summed again yielded complex time-dependent 20 values of voltage or current. The concept of harmonic phasors was then introduced and relations were derived by means of which the joint time-dependent and phase values of voltage and current can be expressed in the circuit. 21 3. POWER DEFINITIONS IN THE PRESENCE OF DISTORTION

3.1 INTRODUCTION

The definitions for time-dependent real, imaginary and complex power in single-frequency circuits have classical definitions and are universally accepted. When it comes to multi-frequency circuits, these definitions have been shown to apply to individual harmonic orders, but new definitions are needed when the joint or total power requires to be calculated across the whole spectrum. The same is true, regardless of whether the analysis is carried out in the frequency or in the time domain.

As pointed out previously, a number of researchers have proposed different theories for the definition of power in circuits with distortion. The different definitions do not necessarily complement each other, but often lead to confusion and unnecessary complexity, because basic definitions are repeated in different ways and arguments still exist as to which theory should really be adopted in future. This is often bewildering, especially to the man in industry who needs to understand it in order to design, specify and service equipment. Only rarely can industry afford to sacrifice the time and effort demanded by understanding of these theories. If possible, a unified theory is needed that is merely an extension of already-existing definitions comprehensive enough to satisfy all the practical needs at once.

There is no single generalised power theory, for example, that can at the same time be used in the measurement of power for commercial purposes, characterise distortion, evaluate power quality, pinpoint the sources of distortion in a network and be a utility in the design of compensating equipment. This study pertains to the design of distribution systems in a practical plant with non- linear electrical loads. All the above facets required of a comprehensive power theory is touched in this investigation from the modelling of network behaviour to final measurement and correlation with design criteria. The wide spectrum of application that this presents, also presents an ideal opportunity for the evaluation of the present contending power theories on a practical basis. The theories that will be evaluated through use in this way will principally be those of Czarnecki" and the IEEE Working Group. The Freeze-Bucholz-Depenbrock (FBD) power theory24 and that of Akagi and Nabae21"25 have been proved to be of great utility in the real-time control of dynamic compensating equipment, but is generally limited to that facet and not for general use and will not be dealt with here in any depth. Although proved to be misleading 29 with regard to the magnitude of his so-called "distortion power", Budeanu's approach 15 to distortion power analysis is still the oldest and the most widely used theory in industry, and will be used in a number of calculations to examine its general validity from a practical point of view.

The Czarnecki theory furnishes comparable and technically acceptable magnitudes under all conditions. It furnishes reliable information that can be used in practical design. It does, however,

22 suffer from a number of disadvantages among which its complexity of calculation rate is the worst. The Czarnecki theory makes minimum use of classical power definitions, but introduces a bewildering number of new concepts and definitions that make it less attractive to the user that only wants to measure and design. This theory is also difficult to follow because it was not formulated in its entirety in the initial publications and the theory underwent new development and additions in subsequent publications. It employs established symbols in new context which is most confusing" 26.

The definitions proposed by the IEEE Working Group under the chairmanship of Prof. A. Emmanue131, on the other hand, attempt to overcome both drawbacks at the same time. Instead of proposing new concepts and definitions, this theory extends the well-established concepts and definitions that have been used in single-frequency power systems, with a minimum new number of additional definitions, to systems with distortion. Although relatively simple for the newcomer to distortion power theory to grasp, its calculation results are not misleading, even under the most exceptional conditions, as are those of the Budeanu theory.

3.2 POWER IN CIRCUITS WITH SINUSOIDAL VOLTAGE AND CURRENT WAVEFORMS

3.2.1 Power in single-frequency circuits

Active, reactive and apparent power are terms that have been defined in circuits with scalar sinusoidal voltage and current waveforms. These definitions of power can be expressed even more readily from the more general complex definitions of u(t) and i(t):

3.2.2 Separate power components in a simple circuit

i ( t )

(t )

Figure 3.2.1 - Parallel circuit with conductance and susceptance

The total complex current i(t) supplied to the circuit in Figure 3.2.1 can be expressed as:

i(t) = ig(t) + ib(t) (3.2.1)

The term "primary current" is usually associated in electrical circuits with transformers. In later publications this term is used to designate the positive sequence component of the fundamental harmonic. 23 from which it is shown to be made up of the two components of current that flow through the conductance and the susceptance respectively. By pre-multiplying the complex conjugate of all the terms in this equation with the applied voltage, the complex power in the circuit is shown to be:

u(t)i(t) * = u(t)ig(t) * + u(t)ib(t) * (3.2.2)

If i (t) and ib(t) be substituted for in terms of the applied complex voltage u(t), the conductance G and the susceptance B, then the following equation results:

u(t)i(t) * = u(t)u(t) * G + ju(t)u(t)* B (3.2.3)

By defining the complex power as:

s(t) = u(t)i(t) * (3.2.4) and multiplication of the complex u(t) with its own complex conjugate yields:

s(t) = lu(t)12G +j lu(012B (3.2.5)

When u(t) is replaced in eq. (3.2.5) in terms of eq. (2.3.9), the following simplified expression results:

S = U2G + j U2B (3.2.6)

from which s(t) is seen to lose its time dependence. It is then possible to substitute S for s and it is seen that S is constituted of a real power component P and a reactive or imaginary component Q.

Because the conductance is purely dissipative, P represents the average active power supplied to it, and because B is completely non-dissipative, but only an energy-storage element, Q is recognised to represent the average reactive power supplied to it. The average active power represents that component of the complex power which is solely responsible for transferring net energy to the circuit. Similarly, the average reactive power component represents that component of the complex power solely responsible for reciprocating the stored energy in the susceptance between it and the load, without performing any net energy transfer. The active power component is therefore accounted for by the first term in eq. (3.2.6) and the reactive power component by the second term.

3.2.3 Real power as function of time

Again consider the circuit in Figure 3.2.1 that is formed by a conductance G and an inductor with susceptance B in parallel. Denote the complex voltage applied across the circuit as u(t) and the total current flowing into it as i(t). The current i(t) will be constituted of the two currents ig(t) 24 and ib(t) which are the currents that flow through the conductance and the susceptance respectively. The diagram on the right shows a vector diagram of the voltages and currents at time t = 0.

Let the respective values of voltage and current be given by:

u(t) = U e Jwt (3.2.7) and

i(t) = / e *at - (3.2.8)

If the value of the real power delivered to the whole circuit is defined to be the product of the measurable values of voltage and current, then by standard definition n:

p(t)=Re{42u(t)}Re-42{i(t)} (3.2.9)

Substitute for the real parts of u(t) and i(t) and expand by means of the Euler identity. This yields:

p(t)= 2 U/ cos cot cos (cot - 4) (3.2.10)

By expanding eq. (3.2.10) and replacing the expanded terms with trigonometrical identities, this equation reduces to:

P(t) = U/ (1 + cos 2wt) cos 0 + U/ sin 2cot sin 0 (3.2.11)

Eq. (3.2.11) shows that the real power p(t) in a single-phase circuit pulsates sinusoidally with angular frequency 2cot. The co-sinusoidal relation in the first term confines the pulsation in it between zero and a maximum. The cos0 term scales this maximum between zero and the value 2UL The sin2cot factor in the second term also represents a pulsation in the magnitude of the power at twice the system frequency, but symmetrically about the zero-axis. It is scaled between - U/ and +U/ in this instance. Together the two terms set the conditions for the total power directed in the circuit as time function.

Consider the first term of eq. (3.2.11) first. The component of current i(t) in phase with u(t) is Icosq5, and the real component of the current i(t) is therefore 42./cos0 coscot. Therefore, by the same definition as above, the power in the conductance is:

pg(t) = N12 Ucoscot \12/coscot cos0 (3.2.12) which, through manipulation and substitution of a trigonometrical identity, is reduced to: 25

pg(t) = U/ (1 + cos 2o.)t)cos0 (3.2.13) and which corresponds to the first term of eq. (3.2.11).

Similarly, the current through the susceptance has a magnitude I sinq5 and the component of the current i(t) in phase with the current through the susceptance is therefore 42/coscot sin4 and the power in the susceptance is therefore:

ph(t)= Nriucosc. \a/coscot sink (3.2.14) which after manipulation and trigonometric substitution yields:

ph(t)= U I sin2cot sin0 (3.2.15)

This expression corresponds to the second term of eq. (3.2.11).

The above results show that the two terms of eq. (3.2.11) represent the real power supplied to the conductance and the real power supplied into the susceptance respectively.

Equation (3.2.11) can be rewritten as:

p(t) = pg(t)+ ph(t) (3.2.16)

emphasising that the total real power delivered to the circuit is the sum of the two components that supply the conductance and the susceptance respectively. The first is pulsating in nature and the second one oscillating.

3.2.4 Average active power

If the real power flow persists for a representative period of time, say for the duration of a period, then it follows that the average power over that period must be obtained by integration over that period: The integration ofp(t) in eq. (3.2.11):

P = fUI (1 + cos2cot) cos0 + Ulsin 2cot sin0 dt (3.2.17) T yields:

P = Ulcosq$ (3.2.18)

in which P is recognised to be the (average) active power in to the circuit. This active power is consumed by the conductance alone. 26

3.2.5 Imaginary power as function of time

The time-dependent power in a circuit was defined to be the product of the in-phase components of complex time-dependent voltage and complex time-dependent current in a circuit. It was shown that this power is made up of two components, namely the time-dependent power pg(t) supplied to the conductance and the time-dependent power ph(t) supplied to the susceptance. A complementary definition may now be made for a quantity q(t), which will be referred to here as the time -dependent imaginary power.

The time-dependent imaginary power, supplied to the circuit in Figure 3.2.1, is defined to be the product of the quadrature components of the complex time-dependent voltage u(t) and the complex time-dependent current i(t):

q(t) = 2 Re{u(t)}Im{i(t) * } (3.2.19) in which the asterisk above i(t) denotes its complex conjugate form.

In a similar manner to that done earlier, substitute now for the real part of u(t) and for the imaginary part of i(t) from equations (2.2.9) and (2.2.10) in eq. (3.2.19) by means of the Euler identity to yield:

q(t) = 2 U/ coscot sin(cot + (3.2.20)

By expanding eq. (3.2.20) and replacing the expanded terms with trigonometric identities, this equation reduces to:

q(t) = UI sin 2cot cos 0 + UI (1 + cos 2cot) sin 0 (3.2.21)

The above equation also contains two terms, as in the case of the real power, and may be written in the following form:

q(t) = qg(t) + qh(t) (3.2.22) in which qg(t) represents the imaginary time-dependent power supplied to the conductance and qh(t) the imaginary time-dependent power supplied to the susceptance. Proof of this can be carried out in an analogous manner to that used before in the case of the real power. Note here in this regard that, in accordance with eq. (3.2.21), the first term in this equation represents the time-dependent imaginary power supplied to the conductance and that the second term represents the time-dependent imaginary power supplied to the susceptance. Note that the time-dependent imaginary power to the conductance oscillates about the zero axis and encloses no net positive or negative area in time, whereas the time-dependent imaginary power to the susceptance pulsates and will enclose a net area. 27

3.2.6 Average imaginary power

Substitute for q(t) in eq. (3.2.21) and integrate.:

Q= fUl sin aot cos0 + U/(1 + cos 2cot) sinqi dt (3.2.23) T

This yields:

Q=UI sin0 (3.2.24)

in which Q is recognised to be the average reactive power supplied to the circuit. In the above context, Q represents the average value of the power that reciprocates between the load and the source without carrying out any net energy transfer. This component of power is numerically equal to the conventional reactive power in the circuit.

Note that the second term in eq. (3.2.21) has an analogous form to that of the first term in eq. (3.2.11). The concepts of real time-dependent power p(t) and imaginary time-dependent power q(t) are therefore identical, with the exception that one leads the other in phase by 90°, which makes them orthogonal and complementary.

3.2.7 Complex power as function of time.

It has been shown that p(t) and q(t) represent the real and imaginary components of the time- dependent complex power respectively. On this basis then, it is possible to define the complex power as:

s(t) = p(t) + jq(t) (3.2.25)

which may be written in terms of equations (3.2.11) and (3.2.21) as:

s(t) = (UI (l+cos2cot) cos 4+U/sin 2cotsin4)+AU/sin2cotcos4)+U/(1+cos2cot) sin 4))(3.2.26)

Equation (3.2.26) reduces to the following form after manipulation and substitution of the Euler form:

s(t)= UI (1 + cos2cot ) ei°+ U/ sin2cot ei(0 90) (3.2.27)

When eq. (3.2.27) is integrated over a period, the conventional complex (apparent) power is obtained as:

s = u i ei sis (3.2.28)

or: 28

S = Ulcos0 + j Ulsinq5 (3.2.29)

It is possible to calculate the complex power directly from the voltage and the current as:

s(t) = u(t) i(t) * (3.2.30)

By substituting for u(t) and i(t) from equations (2.2.9) and (2.2.10) into equation (3.2.30), the ' following relationship is obtained:

s(t) = U eia eJot I e JR e -jot (3.2.31) or simply:

S= U/ei(a-A = U/eIfb (3.2.32)

This is similar to the result obtained in eq. (3.2.28). Note that this latter manipulation also does not furnish the complex power, but a constant complex value S which represents the conventional apparent power in the circuit.

3.2.8 Conclusions

The reasoning in this section has shown that the real (or active) power in a circuit represents that component of the total power in the circuit responsible for transferring the real power and that the imaginary (or reactive) power represents that component of the total power in the circuit solely responsible for reciprocating power transmission between two parts of the circuit.

These definitions apply to single-frequency circuits only, but now furnish a convenient platform from which definitions may be launched that will also be applicable to circuits with distorted voltage and current waveforms.

3.3 FUNDAMENTAL CONSIDERATIONS FOR THE DEFINITION OF POWER IN CIRCUITS WITH DISTORTED VOLTAGES AND CURRENTS

3.3.1 Fundamental considerations in the definition of power in multi-frequency circuits

The relationships for power, real power and imaginary power was examined above for simple harmonic conditions. These relationships can be applied to each individual harmonic frequency which, it will be remembered, are orthogonal to all the others. These relationships can now be expanded through the complex Fourier relationships to cater also for steady-state periodic non- sinusoidal voltages and currents in a circuit. 29

3.3.2 Real power as function of time.

The relationships derived in the single-frequency analysis in the previous section may now be applied on a per-harmonic order basis. The voltage and current at order-n may then be respectively expressed as:

un(t) =Un ej(nca + an) (3.3.1)

and

in(t) =Inei(wt Pn) (3.3.2)

In accordance with eq. (2.3.1), the total voltage and current in the circuit are respectively:

k u(t) = Eu n(t) (3.3.3) n=0

and k i(t) = /in(t) (3.3.4) n=0

The total real power can therefore be directly calculated in the time domain as:

p(t) = Re {q2u(t)} Re-42 {i(t)} (3.3.5)

In addition, it can be shown that all the relationships for the sinusoidal case can be expanded and still hold at each harmonic frequency. The real power in the nth harmonic component is given in its simplest form by:

pn(t) = Re{q2un(t)}R42{in(t)} (3.3.6)

The total real power' in the circuit must now make provision for the distortion as well. Eq. (3.3.6) can be summed to furnish the real power in the multi-frequency circuit as:

k p(t) = Epn(t) (3.3.7) n=0

or:

in Mathematically (3.3.6) and (3.3.7) are not the same. Because u„ and in terms are each orthogonal at different n, the cross products in the summation of (3.3.6) will be zero and (3.3.6) = (3.3.7). 30 k p(t) = E[ Un In (1 + cos 2not) cos 4n + Un In sin 2not sin 4)n] (3.3.8) n=0

In the last equation (3.3.8), it again follows by analogous reasoning to that of the single- frequency case that the real power in the conductance is given by:

k pg(t) = Un In (1 + cos 2not) cos 4n (3.3.9) n=0

and that in the inductor by:

k ph(t) = Un In sin 2not sin 4n (3.3.10) n=0

Note here also, as in the single-frequency case, that pg(t) in eq. (3.3.9) will trace the time- dependent value of the power into the conductance in time. The profile of this trace will also enclose a net area with respect to the zero axis over a representative period, just as in the single- frequency case. Clearly, this trace will not be as simple as in the single-frequency one, but may be quite involved because of the presence of other harmonic frequencies.

3.3.3 Average power

The average power in the circuit can now be calculated from the previous result by integrating eq. (3.3.7) over one fundamental period:

1 P = 7,j p(t) dt (3.3.11)

Substitute for p(t) from eq. (3.3.8)

1 P = 7, Un In (1 + cos 2not) cos (I)n + Un In sin 2not sin 4)n] dt (3.3.12) n=0

Exchange the order of integration and summation:

1 P = — Un In (1 + cos 2not) cos n + Un In sin 2not sin 4)n] dt (3.3.13) T f [ 4

n=0

Integration and cancellation of the resulting T yields: 31 k P = EUnln cos4)n (3.3.14) n=o which again makes it possible to calculate the total active power in the distorted waveform directly from the phasor values of voltage and current. Note that (1) 12 represents the angle of lag of the current vector behind the voltage vector for harmonic order-n or: a n - Pn = (I)n•

3.3.4 Imaginary power as function of time

The definition given for imaginary power in eq.(3.2.19) is a general one and is also applicable to the non-sinusoidal case. It is given by:

q(t) = Re{4.2-u(t)}In42{i(t)} (3.3.15)

As in the case of the real power, all the relationships for the simple harmonic case will also hold at each harmonic frequency. In accordance with eq. (3.2.21), the imaginary power in the nth harmonic component can be defined in its simplest form by:

qn(t) = Re{un(0}1111{in(0) (3.3.16)

The total imaginary power' in the circuit must now make provision for the distortion as well. Eq.(3.3.16) can be adapted to furnish the imaginary power in the multi-frequency circuit as:

k q(t) = Eqn(t) (3.3.17) n=0

furnishing, after substitution and manipulation:

k q(t) = E[un In sin 2ncut cos cl)n + Un In (1 + cos 2ncot) sin fin] (3.3.18) n=0

In the last equation (3.3.18), it again follows by analogous reasoning to that of the real power, that the instantaneous imaginary power delivered to the conductance is given by:

q g(t) = Eun In sin 2n of cos il)n (3.3.19) n=0

and in the susceptance by:

i" Mathematically (3.3.16) and (3.3.17) are not the same. Because un and i„ terms are each orthogonal at different n, the cross products in the summation of (3.3.16) will be zero and (3.3.16) = (3.3.17). 32 k

qh(t) = EUn In (1 + cos 2ncot) sin (I)n (3.3.20) n=0

3.3.5 Average imaginary power

The average imaginary power in the circuit is calculated by integrating eq. (3.3.17) over one fundamental period:

Q = r q(t) dt (3.3.21)

Substitute for q(t) from eq. (3.3.18)

1 In sin 2ncot cos Un In (1 + cos 2ncot) sin (I)n] dt (3.3.22) Q=7, L[un (I)n n=0

Exchange the order of integration and summation:

1 n In + Un In (1 + cos 2ncot) sin ] dt (3.3.23) Q = —T f[U sin 2ncot cos (I)n (1)n

n=0

Integration and cancellation of the resulting T yields:

k Q= lunin sin (3.3.24) n=0

Defining (1)„, as the principle angle then: [-n<4,1

which again makes it possible to calculate the average imaginary power in the distorted waveform directly from the phasor values of voltage and current. The phase angle do still represents the angle of lag of the current vector behind the voltage vector for harmonic order-n as in the convention used in the classical case.

The even cosine function in (3.3.14) ensures that P is the sum of the real powers seated in the individual harmonic orders, but the uneven sine function in (3.3.24) can yield arithmetic results that make Q zero28, even in the presence of imaginary powers in the constituent harmonic components. These properties furnish utility to (3.3.14) but cast doubt on the usefulness of (3.3.24). This means that (3.3.14) can be used to calculate the energy that has to be supplied to the system in a given time, for example, but that (3.3.24) cannot be used, for example, to 33 calculate the capacity of a dynamic compensator by means of which the reactive power in the circuit can be compensated. Although this is true, the physical significance implied in (3.3.24) is that the sum of all the reactive power components can be zero, even though the individual reactive power components in the different harmonic orders are not.

If fundamental power factor correction can be neglected,' the capacity of a dynamic compensator that is necessary for harmonic suppression only, is a function of the time-dependent values depicted in (3.3.20). The time-dependent values of imaginary power yielded by the latter equation would have been quite suitable for rating a dynamic compensator, if it could be rated on that basis. In practice, however, compensators, like all other electrical equipment, are rated on an rms- basis and it is necessary therefore to calculate the required capacity from the rms-values of current and voltage of the harmonic components. What is important to observe in (3.3.20) and (3.3.24), is that "harmonic cancellation" does not only take place at identical harmonic orders between different loads, but also takes place between the different harmonic orders in a single load, as is evidenced in these equations. This topic will be dealt with in greater detail later in this document. This present discussion, however, is merely aimed at bringing the fundamental concepts relating to (3.3.24) into the correct perspective.

The erroneous results that calculation of the imaginary power across the harmonic orders can bring about, inter alia, have prompted researchers to seek alternative definitions by means of which the composite power in the different components can be calculated in circuits with periodic distortion.

3.3.6 Joint complex power

Equations (3.3.14) and (3.3.24) respectively furnish a means of summing the real and imaginary powers across all the harmonic orders. The summation may be selectively performed, for example, to calculate the powers in the fundamental component and in all the harmonic components separately, or to select single harmonic orders or groups of harmonics for which the calculations are required. A number of practical applications require such calculations; in filter design, for example, or when attempting to localise distortion sources in networks 37 .

The classical definition for apparent power (or plainly complex power) valid for sinusoidal conditions, can be applied to any individual harmonic frequency:

Sn = Pn j Qn (3.3.25) or:

Sn = Un In * (3.3.26)

The joint complex power Si can then be calculated directly in its complex form by:

Si =ESn (3.3.27)

The joint complex harmonic power may be defined as that component which is seated in the harmonic components alone, excluding the fundamental, and it is calculated by summing from n=2 upwards:

k Sjh= Eun In * (3.3.28) n=2

The fundamental complex power or the fundamental apparent power simply refers to the complex power of the fundamental components of voltage and current as:

S1 = UI /1" (3.3.29)

3.4 THE APPROACH ACCORDING TO BUDEANU

Initial attempts at defining power in multi-frequency circuits began with the work of Budeanu n. Although mathematically correct, this theory was shown to have practical shortcomings, in that the definitions of his "distortion power", for example, was misleading under certain conditions 29. The work of Budeanu is, nevertheless, not only noteworthy, but have been used extensively in the past in distortion power analysis. It is also true that these definitions do not always give misleading results, as numerical calculations in the analysis later will show. It is therefore worthwhile now to look into the basics of this theory.

The rms-voltage and current of the distorted voltage and current waveforms can be respectively calculated in accordance with eq. (2.3.15) by:

U = Un 2 and / = lin/n 1 2 (3.4.1),(3.4.2) n=0 n=0

The magnitude of the apparent power in a circuit is given by definition as:

S = U- I (3.4.3)

By substituting for U and I in the above equation from equations (3.4.1) and (3.4.2) and combining them under the same root sign yields: S = El unI2 (3.4.4) n n 35 This multiplication may be carried out and the sums of equal and unequal harmonic orders may be separated to yield:

S2 =Elu,,1214/12 /0#01unI2vm12 (3.4.5) n nm

If:

Snm — Unlm * (3.4.6) then, by substitution:

* S2 =ESnSn * + Dn#01Snmnrn (3.4.7) n nm

In eq. (3.4.7), S2 represents the apparent power in the circuit, regardless of whether the waveforms of voltage and current are distorted or not.

The first term on the right-hand side represents a summation of the products of voltage and current harmonic phasors of equal harmonic orders. The second term on the RH side in eq. (3.4.7) represents the summation of the products of harmonic phasors of unequal harmonic orders.

In accordance with eq. (2.3.8), the joint harmonic power may also be written as:

(3.4.8)

Substituting for Si in terms of equations (3.3.26) and (3.3.27)

= zunin * EUn*In (3.4.9) and multiplying out yields:

Sj2 = ZUnIn*Un*In + E(n#m)UnIn*Um*Im (3.4.10) n nm which, after substitution and manipulation, may now be written as:

s2=Esnsn* + E(n.in)SnSin* (3.4.11) n nm

The first term in this last equation (3.4.11) is recognised to correspond with the first term in eq. (3.4.7), and represents the power of the equal harmonic terms. The joint harmonic power is therefore also constituted of two terms; one containing only like harmonic products and the other 36 unlike ones. (It is important to note the difference, however, between Snm and Sn and Sm in equations (3.4.7) and (3.4.11).)

Equation (3.4.11) may now be transposed to the following form:

DA?* = S2 - E(n#m)S nSm * (3.4.12) nnm

By replacing for ESnSn * from eq. (3.4.12) into eq. (3.4.7), an equation is obtained expressing n the apparent power in the circuit in terms of the joint complex power and another power component.

S2 = S72 + ( Dn#172)1SnmSnm* Drirn)SnSyn *) (3.4.13) m nm

This latter component numerically corresponds to the "distortion power" D, as originally defined by Budeanu 15 :

Db2 = /(17#m)Isnmsnm* - E(nin)SnSin* (3.4.14) nm nm

Equation (3.4.13) may now be written in its complete form as:

S2 = P.1 2 + Q12 + D b2 (3.4.15)

Figure 3.4.1 - Representation of active, reactive and distortion power in cartesian space

The distortion power D will only be present when the waveforms of voltage and current become non-sinusoidal. When the distortion reduces and the waveforms reduce to pure sinusoidal form, the height of the box in Figure 3.4.1 vanishes to form a plane in which only P and Q remain, to represent the apparent power in a single-frequency circuit. As is shown in the literature, the Schwartz inequality30 is valid in this case for distorted voltage and/or current waveforms:

37 s2 p2 + Q2 (3.4.16)

Failure of the Budeanu theory is illustrated simply in Figure 3.4.1, in that cases may occur in which there should be distortion power, but in which D actually calculates to zero 29

3.5 THE IEEE WORKING GROUP DEFINITIONS

A number of researchers followed up on the work of Budeanu and have subsequently proposed more appropriate definitions by means of which distortion powers can be calculated"'". These definitions all concur with respect to active power, the essence of which is defined in (3.3.5) to (3.3.14) and with apparent or loading power, as defined in (3.4.3). That, unfortunately, is where agreement ends and aside from the academic insight that they furnish, these theories present the average engineer and technician with a bewildering number of new concepts and divergent definitions. It is true that the different definitions suit different applications better than others and that the diversity is sometimes advantageous. It would, however, be a great advantage if a single theory could be found that would be equally suitable for all types of analyses under all circumstances.

The recent work of the IEEE Working Group on non-sinusoidal situations, under the chairmanship of A. Emanue1 31 presents a set of practical definitions for distortion power in terms of the total, fundamental and harmonic constituents.

It has been decided that these definitions will be analysed numerically in the measurements later, to evaluate their practical utility. These basic definitions will now be reviewed as an extension of the basic definitions made above in the previous section.

3.5.1 Single-phase Relations

The rms value of a general complex time-dependent periodic waveform is derived from (2.3.10), in (2.3.15) and can be expressed for voltage as:

U = (Liu: = lunr (3.5.1)

Similarly, the effective value of the current is:

I = in in* = (3.5.2)

The harmonic rms-components in (3.5.1) and (3.5.2) can be separated into their fundamental and harmonic components:

u12 + UH2 and 12 =112 +'H2 (3.5.3) (3.5.4)

with:

UH lt/n 12 and /H = \III/n12 (3.5.5) (3.5.6) =gr71 n#1

The total apparent power S is defined as:

S = unI211n12 (3.5.7) n

and the total nonactive power N as:

N =4S 2 - P 2 (3.5.8)

Alternatively, S can be defined as:

S2 = (U111)2 + (U1/H)2 + (UH/02 + (UHIH)2 (3.5.9)

and grouped as:

S2 = 512 ± SN2 (3.5.10)

In which SI is the fundamental apparent power and SN the nonfundamental apparent power:

In turn:

si2 = (U1/1)2 = (Ul/1cos01)2+ (Wisin01)2= P12 + Q12 (3.5.11)

d SN2 = (UI/H)2 + (UH/1)2 + (UH/H)2 (3.5.12)

The term (UIIH) is named the current distortion power, (UHI1) the voltage distortion power and (UHIH) the harmonic apparent power. The latter term can, in turn, be broken down into the total harmonic active power PH and the total harmonic non-active poiver NH.

sH2 pH2 ± NH2 (3.5.13)

in which NH is derived analogously to N in (3.5.8).

Another useful relationship is obtained when (3.5.12) is divided by (3.5.11):

L)2 (L)2 (vH)2 rsiv)2 (3.5.14) LSI) V) ±LU1) %Si)

to yield the normalised non fundamental distortion power:

39 (SJ )2 2 2 — (ITHD) + (UT HD) + (UTHD•ITHD) 2 (3.5.15) s1

The three RH-terms of (3.5.15) represent the current THD, the voltage THD and the product of the previous two respectively. The significance, applicability and utility of the different components defined above are expounded in 31 , but will also come out in the modelling and measuring evaluation later on.

3.5.2 Three-phase Relations

The single-phase relationships of (3.5.1) and (3.5.2) are extended to unbalanced three-phase networks by the following definitions that lay down the equivalent three-phase voltages and currents:

Ub2 + \I ua2 + u,2 Ue = 3 (3.5.16) and

\14,2 + 42 ± Ic2 (3.5.17) 4— 3

The identical procedures used in (3.5.1) to (3.5.15), for single-phase values of voltage, current and power, can be obtained for their equivalent three-phase values by replacing U and I respectively with Ue and /e, furnishing a means by which unbalanced three-phase network calculations can be carried out in systems with periodic distortion. In the case of unbalanced polyphase systems, the definition of another apparent power component becomes inevitable. The degree of unbalance in the fundamental apparent power Se/ can be divided into two terms:

sL Ss+i 2+ s2Sul1 (3.5.18)

Where Si+ = 3 • Ui+ • Ii+ is the positive-sequence fundamental apparent power, and Ui+ , II are the rms values of the positive-sequence fundamental voltage and current. The component Sal is the unbalanced fundamental apparent power.

3.6 THE CZARNECKI DEFINITIONS

The definitions of distortion power by Emanuel in the previous section only represents one possible set. Other definitions have been advanced, of which the Czarnecki contribution is also noteworthy, and deserves consideration here. 40

3.6.1 Single-phase Relations

The load current is decomposed by Czarnecki 32.33, into a number of components as shown in Figure 3.6.1.

10)

u(t)

Figure 3.6.1 — Single-phase load current decomposition according to Czarnecki

In the Czarnecki decomposition, the currents are defined to be orthogonal. These effective current values are set out in Figure 3.6.1 and make up the apparent power, delivered by the circuit as:

S 2 = (U/,, ) 2 + (U/r ) 2 + (U/s ) 2 + (U/g ) 2 (3.6.1)

From which:

S 2 = P2 + Q,2 +D32 + Dg2 (3.6.2)

Assume that the voltage is composed of harmonics of order-n from a number set Nu. A non- linear load can further generate harmonic frequencies, in the source current, not present in the source voltage. Let this larger set of harmonics be equal to Ni . Define a new set Ng as:

Ng = Ni 0 Nu (3.6.3)

Remember that Nu is a subset of N,.

The source current is then decomposed into two components, io (set Nu ) and ig (number set Ng). This current separation is important in the calculation of the generated current by means of formula 3.6.3.

i(t) = io (t)+ig (t) (3.6.4)

Because the components io and ig are composed of harmonics of different frequencies, they are mutually orthogonal. Thus 41 12 = 102 + 1g 2 (3.6.5)

If the source has an internal impedance, the decomposition of (3.6.4) is not applicable, because the harmonics generated in the load has the same frequency as in the source. A new composition is proposed as discussed at the end of the paragraph.

The different Czarnecki components will now be discussed in turn, starting with the active current ia(t), from which the active power P is calculated. Notice that the voltage u(t), applied to the circuit, can be of an arbitrary shape.

ACTIVE POWER

The equivalent conductance is specified for the load in terms of the active power that has been calculated in the conventional manner, as:

Ge = u2 (3.6.6) where U is the effective value of the voltage.

The active current in the single-phase load can now be calculated as:

is (t) = Geu(t) (3.6.7)

It is important to note that this active current is (t) differs from the time-dependent value of the current i(t), which is the vector sum of the different Czarnecki current components. It must be noted at this point in time, however, that the time-dependent value of the (total) current i(t) also conveys the reactive power (and with it, the apparent power), whereas the active current transmits only the active power. The active current waveform is always the same as that of the voltage, as is to be expected. It is also possible to calculate the equivalent conductance from the harmonic coefficients, based on the relation:

EG,jun r (3.6.8) Ge = -^c 2 /lunl neN„

REACTIVE POWER

All power theories agree about the definition of active power. The definition of reactive power, however, differs. When voltage u(t) and i(t) are sinusoidal, the classic theory applies and all calculations agree that the apparent power is the vector sum of the active and reactive components. This is not true, unfortunately, for non-sinusoidal currents and voltages. The 42 difficulty has been "solved" by Czarnecki who defines reactive power in a different way. In an analogous way to Fryze's16 definition of equivalent conductance, he defines harmonic susceptance for each of the different harmonic components. This is a very practical definition, because it characterises the "source" of the reactive current of that harmonic. In accordance with this definition, each reactive harmonic component can be compensated for individually by a separate parallel conjugate susceptance. Czarnecki describes this reactive power as the passively compensatable power (compensated by means of tuned parallel passive LCR filter networks) in a circuit34.

The susceptance B„ for the different harmonic components is calculated by using:

„ Y„ = n (3.6.9) (I

Gn = Re(Y,, ) (3.6.10)

B„= Im(Y,2 ) (3.6.11)

The time - dependent Czarnecki reactive current ir(t) can now be calculated as:

i„(t) = -5Re( jB„U„-e ) (3.6.11) neN„ The rms-value of this reactive current can then be calculated as:

T I „ 11 2 - (it 1ysiT " (0 (3.6.13) o The rms. value of the Czarnecki reactive current can also be calculated from the harmonic coefficients as:

I „ = ,2 3 • I n (3.6.14) neN,, U 12 The reactive power in the circuit is then:

Q„=U1„ (3.6.15)

SCATTERED POWER

Czarnecki's definition assigns a physical significance to scattered power. Mathematically, the scattered power component derives from the scattering of the individual harmonic conductances G„ around the equivalent conductance Ge. Physically scattered power implies that the 43 conductances of the circuit are different for different frequency components, in order for scattered power to be present.

From basic circuit theory, it follows that loads in which the real components of the circuit solely consist of conductances and in which the imaginary components consisted solely of susceptances would not have scattered power. In real circuits, however, these real and imaginary immittances can be in series, and for a series circuit of impedance:

Z=R+JX (3.6.16) in which the reactance Xis frequency-dependent, the admittance is:

Y=G–j•B (3.6.17)

X With G – R2 and B – (3.6.18) +

And the conductance G is shown to be frequency-dependent because of the presence of X in its denominator, showing that scattered power will be present in linear systems.

The scattered current component can be described in the frequency domain as:

is (t) = Iii Re E(G„—Ge )•U„- en't (3.6.19) nEN.

The rms-value of the scattered current is calculated in the normal way as:

Is = 1—1 1 ii (02 • dt (3.6.20) T 0 s

Alternatively, it can be calculated from the harmonic components as:

I = – G e )2 „I)2 (3.6.21) neN„

The scattered power component can now be calculated as:

= U•Is (3.6.22)

NON - HARMONIC POWER / GENERATED POWER

The nonharmonic current can be calculated by subtracting the different current components.

ig (t) = (t)– is (t)–i „(t) (3.6.23) 44 Alternatively, it can be calculated from the harmonic components as:

(3.6.24) 1g =

The harmonic power component can now be calculated as:

Dg = U g (3.6.25)

3.6.2 Three-phase Relations

In the following zero sequence components are assumed to be absent, because from a practical point of view zero sequence components represent fault conditions that must be absent during normal measurements. In the single phase case there are three causes of useless current flowing in a circuit, in the three-phase case there are four causes of useless current flow in asymmetrical circuits with a non-linear and/or periodically variable load supplied from a symmetrical source of nonsinusiodal voltage. These causes are: 1) Reciprocating energy transmission between source and load such as due to capacitors or inductors in the load. This is the reactive current component similar to the single-phase case. 2) Load conductance dependency on frequency, thus the equivalent conductance Gne of the load changes with harmonic order-n. This is due to the skin effect on transmission lines. This is the scattered current component similar to the single-phase case. 3) Due to the load asymmetry Czarnecki defined the unbalanced current component. This is the only component added for the three-phase case. 4) The last current component is due to the load non-linearity or periodical variance of load parameters such as arc furnace parameters. This is the generated current component and is also similar to the single-frequency case.

At a particular frequency no , an equivalent conductance Gne and susceptance Bne can be defined as:

Pn and (3.6.26) , (3.6.27) Gne = 772 Bne = —Qn nc U nc where

(Inc = Vun2 + un2s + un2T (3.6.28) and

nR2 + In2 s I n2T Inc = (3.6.28)

The source current can be decomposed into five orthogonal components, and for the three-phase case an additional current component is introduced namely the unbalanced current i u .

i(t) = is (t)+ it (t) + is (0 + (t) + ig (t) (3.6.29) 45

IZ = 1 „2 +1,2, +1,2 +1,24 + Ig (3.6.30)

The different current components are:

ACTIVE CURRENT

This is the generalisation of the Fryze active current. This current is responsible for the active power transmission.

I a= GeU (3.6.31) where Ge and U are the equivalent 3-phase values of conductance and voltage respectively.

REACTIVE CURRENT

I „ = B -IL I na 2 (3.6.32) nEN. SCATTERED CURRENT Is = I (G„ — G ) 2 (It „a l) 2 (3.6.33) neN.

UNBALANCED CURRENT /h = —(Gn2 +/3h2 )•1Unc 12 (3.6.34) neN„

GENERATED CURRENT

)2 /8, Onc (3.6.35) nEN, 3.6.3 Decomposition of load current with source impedance not zero

The case where the source impedance is zero has purely academic value and in practical systems of interest the source impedance is never zero. Czamecki 35 addressed this in 1990. In accordance with this reference the number sets Ari, and Ng are determined by the sign of P„ where

P„ = UnInCos0„ (3.6.36)

(see equation 3.3.14). Positive values of P„ show that the distortion source is at the source side of the measuring cross section. Negative values of P„, show that the distortion source is at the load side of the measuring cross section. With the number sets known, the decomposition is followed as in the cases for the single phase and three phase. This decomposition forms the basis of work carried out on the localisation of distortion sources in networks with harmonics 37. Since the 46 presence of the current components iA and iB in the current due to the two number sets NA and NB, forces additional apparent power, called the forced apparent power (SF). The apparent power S can then be expresed as:

= s 0A ± ± 2 S2 s 0B s F (3.6.37) with

so2A = IA2 . uA2 and so2, = 1132 - uB2 (3.6.38) where, IA and /B are the primary and load generated three phase current and UA and UB are the primary and reflected three-phase voltages. The practical value of this decomposition is evaluated in the modelling and measurements described later.

3.7 SUMMARY

Because of the presence of distortion, the definition of a single phasor to represent the voltage or current is no longer possible and the classical definition of the components of power also fails. To overcome the problem, researchers in the field have advanced different power theories that attempt to reconcile the different components of power in the circuit. Some of these power theories are based on the time domain, whilst others operate in the frequency domain.

Because of the diversity of the approaches used in the different power theories, different aspects of the power measurements are emphasised. Not all the definitions have equal utility in a given application, and different theories are employed under different circumstances for different applications.

The relationships for total power, real power and imaginary power were examined for simple harmonic conditions. These relationships can be applied to each individual harmonic frequency. The relationships have been expanded through the complex Fourier relationships to cater also for steady-state periodic non-sinusoidal voltages and currents in a circuit.

The power theories of Budeanu, the IEEE Working Group under chairmanship of A Emanuel and the Czarnecki power definitions have been dealt with. These definitions will be analysed numerically in the admittance matrix model and on measurements later to evaluate their practical utility. 47 4. DISTORTION COMPENSATION

4.1 FUNDAMENTAL CONCEPTS

4.1.1 Introduction

The advent of faster and more powerful semi-conductor devices has led to a proliferation of loads to distribution networks that had not been foreseen. These loads are usually detrimental to the power quality in one way or another and contribute towards injecting harmonics, causing flicker and unbalance in the networks. In the case at hand, the situation is somewhat simplified by the envisaged steady-state operation of the non-linear loads that will be used in the plant. As a consequence, the control strategy that will be necessary for the applied dynamic compensators, can be relatively unsophisticated.

As pointed out earlier, the pulse lasers used in the enrichment process will be operating under steady-state conditions and will be drawing a steady load. The line-commutated, phase-controlled AC-DC front-end converters of the laser pulse power supply units will normally be operating at steady firing angles, except when changes in conditions of operation, such as slight variations in supply voltage, take place. Under these conditions, steady-state operation can be assumed and where needed, it is possible to design a control strategy for dynamic compensation to cater specifically for these simplified conditions. The work, following in this chapter, will therefore only cater for steady-state operation and will not need to take transient conditions into account''.

4.1.2 Frequency - domain view

It can be assumed that voltage distortion in power networks is caused solely by the voltage drops in supply network impedances, brought about by the non-sinusoidal currents that loads draw at various points in the network. Consider, for example, a simple single-phase linear- and a non- linear load in parallel on a point of common coupling (PCC), as shown in Figure 4.1.1. The two loads are supplied from a sinusoidal voltage source Us through a source impedance Zs between the source and the PCC. Let the impedances separating the linear load and the non-linear load from the PCC be given by Zai and ZCB respectively. In this example, the non-linear load is modelled as a current source'.

This does not imply that transient stability can be ignored completely, however, because lasers and plant will still have to be turned on and off, but such transient behaviour will be negligible when evaluated against the periods of continuous operation. Modelling applications do occur where this assumption is not completely valid. In the modelling done in this document, however, this simplified approach has given excellent agreement with practical measurements and it will therefore be used in this argument. 48 The source current, flowing through Z., will be comprised of the two currents of the linear and the non-linear loads respectively, and will be distorted in proportion to the magnitude of the loads and their respective degrees of distortion. The voltage at the PCC will, of course, also be distorted and the degree of distortion will be a function of the voltage drop in Zs by this distorted current.

ZCA

Figure 4.1.1 - Simplified transmission system model

In fact, it will deviate from the sinusoidal form of Us by the magnitude of distortion present in the voltage drop in Zs so that:

UnpCC = [Ira InS ZnS (4.1.1) and in which LisZns represents the voltage drop in Zns. Because the voltage at the PCC is now distorted, it follows from (4.1.1) that the current drawn by the linear load will now also be distorted, and will itself contribute to further distortion in the PCC voltage.

The analysis of steady-state distortion becomes much simpler when viewed in the discrete frequency domain as shown in the literature 36, when the analysis is based on linear circuit theory on a per-harmonic basis. Consider, for example, the above circuit in the presence of a single harmonic-n, reducing to the circuits in Figure 4.1.2(a) and Figure 4.1.2(b) that represent the fundamental frequency and a harmonic frequency of order-n respectively.

In (a), in which n = 1, the sinusoidal voltage source is seen to supply power both to the linear load and the non-linear load, that is again represented as a sinusoidal current, albeit, with a possible phase difference. The real power supplied to the non-linear source at the fundamental frequency serves to drive the load, but is in part also converted to real power at the harmonic frequencies to drive the losses that occur in the circuit at those frequencies'''. Obviously, the PCC voltage at the harmonic frequency will be subject to the source-impedance voltage drop. 49 In (b) the source can be omitted, since it is a short circuit to the harmonic frequencies. In this case, however, the distortion current source can either supply energy to, or draw energy from the rest of the circuit, depending upon the phase of the sinusoidal nth harmonic current. It has been shown in the literature also, that the real power supplied by this source at the harmonic frequency n is a function of the magnitudes of the real network immittances presented to that frequency. In practice the dissipation of real power at harmonic frequencies is small, but still present, when compared to the fundamental component37.

ZCA ZCA

(a) (b)

Figure 4.1.2 - Circuit for fundamental and order - n harmonics

What Figure 4.1.2(b) really shows is that the nth harmonic currents icAn , generated by the non- linear load will distribute through the circuit in accordance with the current-division rule and will therefore be present in the lines towards the source, as well as in the lines to consumer A. The portions of the currents flowing through Zs and ZCA will introduce voltage drops across the respective two immittances and that will lead to the deviation from pure sinusoidal behaviour, that will be observed over the terminals at the PCC and at the other sets of terminals A. 1-A.2 and B.1-B.2.

4.2 COMPENSATION PRINCIPLES

4.2.1 Local and ambient distortion

It is seen above that harmonic compensation, as its name implies, is basically a technique by means of which the harmonic currents are "deviated" to prevent them from flowing in selected circuit parts. It must be noted that current distortion itself, introduced by one consumer, will not have a detrimental effect on the operation of the equipment of other consumers, but the presence of current distortion in the immittances in other parts of the network, however, introduces voltage drops at the harmonic frequencies and the resulting voltage distortion on the terminals of the other 50 consumers is that adverse quantity leading to problems on their equipment. This does not mean that current distortion itself is permissible, because it does lead to problems of its own, such as low overall power factor and compulsory derating of rotating machinery, but it does not introduce voltage distortion, except for the presence of immittances in the network.

The principle of compensation can now be illustrated best by examining the circuit behaviour in the frequency domain and assuming"" that the circuit behaviour is linear with respect to the frequency spectrum that steady-state behaviour imposes on it. That makes harmonic superposition practical, as discussed in the initial chapters, to examine circuit behaviour on a per- harmonic basis.

As noted before, it is expedient to view the circuit behaviour separately in terms of the fundamental component (n = 1) and in terms of the collective harmonic behaviour 31 .

Consider first the case of the fundamental component, as illustrated in Figure 4.1.2(a). In this case the source supplies a (sinusoidal) current at line frequency, that is the sum of the currents supplied to the linear load at terminals A. 1-A.2 and at terminals B.1-B.2. Obviously the phase of the current supplied by Us will depend on the angle of the impedance at A. 1-A.2 and on the phase angle of the fundamental component of the current at B.1-B.2. Compensation of this fundamental component, at the PCC, will therefore require the injection of a quadrature current at the fundamental frequency, which, when summed with the total current, will produce the Fryze Curren-06, which in this simplified case is just the in-phase current that will yield unity power factor. This process of compensation is classically known as power factor correction and may now be referred to as fundamental (frequency) compensation in a collective sense, when discussing harmonic compensation.

When considering the compensation of the n # 1 harmonics, Figure 4.1.2(b) illustrates that compensation when applied at the PCC will consist merely of injecting the identical harmonic current components to IcA at the PCC so that ICA will be short-circuited there and prevented from flowing through Zs and ZcA and polluting the rest of the circuit.

Because it is only theoretically possible to generate ideal circuit conditions, complete compensation as discussed above is rarely possible in practice and there are additional considerations that must be borne in mind when dealing with the problem (refer to next sub- section).

vu Circuit non-linearities such as the effects of transformer core saturation and the non-linear behaviour of magnetic circuits in rotating equipment, are usually dominated over by the relative magnitudes of linear behaviour. 51 There are two possible sources of distortion that have to be considered in practice. They are respectively:

The distortion produced by the consumer viewed from the PCC. The distortion originating outside the PCC.

For the sake of clarity and economy of discussion, the distortions referred to in (a) and (b) above will be referred to as local distortion and ambient distortion respectively. In (a), the consumer wishes (or is forced) to keep the distortion current at the PCC terminals at a low level. In (b) the high level of ambient distortion at the PCC may force the consumer to effect correction at the PCC to prevent this distortion from interfering with the operation of apparatus. There are two basic approaches to the compensation of local distortion and ambient distortion referred to as parallel and series distortion compensation respectively.

4.2.2 Parallel distortion compensation

To keep the local distortion current out of the supply in the circuit in Figure 4.2.1(a), a current source /comp is installed across the PCC. If

Icomp„ = - # 1) . (4.2.1)

then all the harmonic components generated by the non-linear load will be prevented from flowing through the source and the source impedance. It is possible, also, that the fundamental current component drawn by the non-linear load may have a large phase angle relative to the PCC voltage, and if required this fundamental compensation would have to be carried out as well, either by including the required component in /comp or by injecting it separately at the PCC.

Zs PCC Zs PCC

Us IL

0 (a) (b)

Figure 4.2.1 - Principle of parallel distortion compensation

52 The equivalent compensated circuit can again be viewed separately for the fundamental component and for the collective harmonic components as shown in Figure 4.2.2(a) and (b) respectively.

Z., PCC Zs PCC

Icompn S IL

0 (a) (b) Figure 4.2.2 - Fundamental and harmonic parallel compensation

It is not only interesting, but also of significant practical importance, to note that both the fundamental compensation and the harmonic compensation can be carried out by /comp without injecting any significant magnitude of real power into the circuit. In the case of the fundamental, /comp/ can be a pure quadrature current. Considered individually, the harmonic components will invariably require the injection of real power components in a practical installation to compensate them because of their individual phase angles. Considered collectively, however, the net cumulative real power component in the harmonic components generated by the non-linear load will be a function of the magnitude of the real immittance components in the circuit; the real component of Zs if no compensation is effected. Without the compensator the harmonic currents will flow through Zs and will be subject to losses in the real part of Zs. When the compensator injects all of these currents at their appropriate individual magnitudes and phases, none of these harmonic currents will flow through Zs and because the load between I compn and IL contains no resistance, the compensator will only supply reactive power. In a practical situation this model may be simplistic and it may mean that real losses in that circuit will require small real power input into the circuit by the compensator.

4.2.3 Series distortion compensation

If a consumer wishes to keep ambient voltage distortion out of his equipment, a series voltage source can be installed as shown in Figure 4.2.3. This type of compensation can be extended to harmonic components. The use of series compensation on long transmission lines at the fundamental frequency38 and its principles and application, lies beyond the scope of this work and it will not be discussed here. 53

Now consider the harmonic components, from which it follows that if there are harmonic voltages present at the PCC, namely Upcc, as simulated by a Thevenin equivalent voltage distortion source Us„, then that distortion can be eliminated at PCC' by injecting the measured distortion components into the circuit between PCC and PCC' in Ideally, this series voltage injection should take place without the expenditure of real power. Consider first the case in Figure 4.2.3 without the compensator, i.e. with U.=0. For a given value of n, a distortion current Us / — will flow through the load from the ambient source. By now installing Uc. ” Z + ZL between PCC and PCC' as shown, and controlling Ucon..= , the loop distortion potential is negated, so that In =0. That means that under the above conditions, even though the compensator will inject a voltage U. as shown, it will supply no power at that frequency to the network. When viewed collectively over all the harmonic frequencies the principle remains the same. In practice the net power that must be supplied by the compensator to the network is a function of the voltage injected and the total current through it. Normally this net product is small, although the instantaneous power levels over a fundamental period may assume large values. As in the case of parallel compensator, this compensator will also have to supply the internal losses from its own power source.

Zr. USn

Figure 4.2.3 - Principle of series compensation

4.2.4 Combined parallel and series compensation

Where both ambient and local distortion are present, parallel and series compensation may be applied simultaneously as shown in the simplified single-phase circuit in Figure 4.2.4. In this figure, the equivalent source is shown to consist of a fundamental component Us, in series with the distortion source Us,. 54 Because the superposition principle can be applied here as well, it follows that the two compensators will each perform their respective functions as before.

It is more interesting than useful to note, lastly, that the roles of the two compensators may be reversed; i.e. for the series compensator to compensate for the locally produced distortion and for the parallel compensator to compensate for the ambient compensation. It has been shown, however, that the advantages achieved by reversing the roles invariably results in other adverse circuit behaviour36.

Usn

Us]

Figure 4.2.4 - Combined parallel and series compensation

4.3 GENERIC COMPENSATOR TYPES

In the discussion above, the compensator merely consisted of a controlled voltage or current source. The mechanism or principle of operation of this source was not discussed, and is not important with respect to the theoretical discussion up to this point. The type of compensator is important, however, from a practical point of view; not only with respect to its functioning in the circuit, but also from a capital and operating cost point of view.

Now, although there are other generic types available, harmonic compensators can be grouped under the two almost exclusive headings, namely that of passive compensators and active compensators. Because of the potential ambiguities that the term active compensator can elicit within the context of this subject, it would be more expedient to rename the latter to a dynamic compensator with less possibility of misunderstanding and confusion. In the case of the passive compensator, also, it would be more prudent to refer to it as a passive filter, which it is in reality, and the term passive compensator would then only be used in a generic context. Harmonic compensators are used in many diverse applications today, for power ratings that range from 50 kVA to 50 MVA39. 55

4.4 PASSIVE FILTERS

Although the word filter summons images of ladder 4° network topologies in the electrical engineer's mind, the topology of the passive filters used for passive harmonic compensation are constrained by practical requirements to follow series RLC construction. A typical passive filter, used in parallel compensation schemes, is depicted in Figure 4.2.1(b), Figure 4.2.2(b) and Figure 4.2.4. In these examples, the passive filter will furnish a path for /comp to flow and could replace the controlled current source depicted there in each case.

The diagram in Figure 4.4.1 depicts a typical filter topology for a filter with 5th and 7th harmonic filter arms and with a high-pass filter arm.

Figure 4.4.1 - Single - phase example of a passive filter network

In the passive filter network shown, the two left series RLC arms are tuned to 250 Hz and 350 Hz, representing the 5th and the 7th harmonics respectively. The right-hand arm has a high- pass topology, presenting a low impedance to frequencies above those values. The choice of the filter arm frequencies, the number of arms and the presence and configuration of the high-pass arm depends upon the application and on the parameters of the network in which it is installed.

Passive filter design for harmonic suppression is a specialist task and calls for a thorough knowledge of the network topology and for analytical tools, by means of which the presence of the filter can be modelled in the network. Because passive filters make use of inductive and capacitive energy storage elements, their introduction into a network in which inductance and capacitance are inherent, can be prone to all kinds of transient and steady-state misbehaviour if it is not well catered for in the design. Every passive filter installation must therefore be tailor- designed for the application and it is not possible to purchase these units off the shelf. For the 56 same reasons, these filters are prone to instability when associated network topologies change. The origin of these instabilities are sometimes extremely difficult to establish in practice.

Because passive filter design is not new and the principles and techniques used is well established, the related intricacies will not be discussed here at any length and the reader is referred to the literature in which every aspect is dealt with in dep th41 ,58. The exposure given here is only intended to bring passive filters into perspective with respect to the hybrid compensating structures that will be discussed later.

4.5 DYNAMIC COMPENSATION

4.5.1 Basic considerations

Although passive filters can fulfil a number of requirements with respect to harmonic compensation and can fulfil the functions demarcated for the current source in most of the cases depicted in Figure 4.2.1 to Figure 4.2.4, the behaviour of these filters is not controllable in all the desired respects, often causing them to lead to difficulties of a diverse nature in operation. These difficulties are becoming more pronounced with time as networks become more complex and the magnitude of the collective non-linear loading on the network increases.

Dynamic compensators, on the other hand, can be designed to operate with great stability, and because their control structures can be versatile, the scope of usage of these units is much more extensive from a technical point of view. They do, however, suffer from a number of constraints of which the majority concern cost and complexity.

4.5.2 Voltage-fed and current-fed topologies

Dynamic compensators are power electronics based devices that can be categorised into two generic types, namely voltage-fed and current-fed. In essence, the only difference between the two is the nature of the energy store. In the former, capacitors are usually employed, whilst in the latter inductive energy storage is used. The basic three-phase power electronic structure of a voltage-fed compensator that is capable of compensating only for positive and negative phase sequence components are shown in Figure 4.5.1. 57

Figure 4.5.1 - Essential power electronic structure of a voltage-fed dynamic compensator

The general symbols for the switching elements represent force-commutated devices with built-in anti-parallel diodes for reverse conduction. These devices may consist of insulated gate bipolar transistors (IGBT), field-effect transistors (FET) or gate turnoff thyristors (GTO). IGBT designs prove to be the most popular at the moment for smaller systems (below a megawatt), but GTO- based systems are beginning to be increasingly used for multi-megawatt systems, in which special arrangements have to be used to overcome the relatively slow recovery capability of the high- voltage and high-current devices used and the relatively low modulation frequencies that can be achieved with these high power devices. The structure shown can handle balanced and unbalanced line conditions. It is also possible to expand the circuit to cater for zero-sequence unbalance at the expense of more complicated control.

The function of the added line inductance is to reduce the switching di/dt and to reduce high- frequency current ripple in the supply lines. Unfortunately, the low pass filter network formed by the inserted inductance reduces the bandwidth of control. Design trade-offs have to be made in practice in this regard.

The size of the energy storage capacitor is also dependent upon the application. In the absence of negative and zero-sequence components in the compensating current, this capacitance can be theoretically reduced to zero. When the compensator is used merely for reactive control (no average real power component over a representative cycle), the capacitance is chosen just large enough to limit the voltage excursion over the cycle to chosen and practical minimum values. If the DC link, formed by this capacitor-output end of the compensator is to be connected, then larger values of capacitance is needed to limit the DC voltage ripple. The latter arrangement is, however, not normally used in dynamic compensator applications but only where the compensator has to fulfil other special functions as well, such as when back-to-back operation is necessary, for example.

Figure 4.5.2 shows the basic power electronic circuit for a current-fed dynamic compensator. In this circuit the energy storage element is an inductor and the level of energy stored in it is a

58 function of the level of the circulating current. The discussion relating to the requirements for the switching elements for the voltage-fed topology also applies here, except that the anti-parallel diode requirement mentioned there may be dispensed with.

inductance capacitance

1 I TTT

Figure 4.5.2 - Basic current - fed compensator

In this case, also, filter capacitors are required in addition to the line inductance to reduce switching current ripple in the lines.

The relative merits between the two compensators essentially reside in the maintenance of minimum losses with respect to the level of energy storage and with respect to the bandwidth of the response that can be achieved. At this point in time, voltage-fed units are receiving preference, especially in larger installations, because of the lower inherent losses in the capacitive energy store. New developments in superconducting materials and in cryogenic energy storage may shift this emphasis in the future, but it is still dependent upon further development.

The deployment of the different switching techniques, power-electronic structures and different control strategies are well covered in the literature and the established techniques will not be discussed here in detail42'43' 4'45. Because the requirements for steady-state operation of the case at hand present a special case, however, the specific control technique employed will be elaborated on here in some detail.

4.5.3 Dynamic compensator control

In addition to illuminating a new very fascinating theoretical field, the proliferation of power theories applicable to distorted voltage and current waveforms, has also led to a number of difficulties. Because it is possible to ascribe different interpretations and definitions to the same basic concepts, divergent opinions now reign as to which theories are the most suitable for dynamic compensator control. Although these theories make excellent academic sense, their application in dynamic compensator control calls for different requirements from case to case and different theories become more applicable to individual cases. One aspect that most of these 59 theories have in common though, is the demand for fast computation necessitated by the sophisticated control algorithms they demand. In a number of cases these demands are mandatory because of special requirements. These special requirements mostly have to do with transient response and transient compensation. In most other cases, however, where the call is predominantly for steady-state compensation under conditions of periodic distortion, complex and high-speed computation can be dispensed with almost entirely by adopting a simple principle as the basis of control.

The principle referred to here is that the current or voltage waveform will be periodic over the fundamental period, regardless of the degree of harmonic distortion. Periodic variation, when it is present, will take place relatively slowly over a number of periods. Under such conditions, all waveform characterisation and computation can be carried out over the preceding period or even over a number of preceding periods, extending the available computation time from the microsecond into the millisecond regime. To be precise, in a 50 Hz system, a minimum of 20 ms now becomes available between the moment of waveform measurement and the call for control action. What is even more important is that the adoption of this principle now brings this control requirement within reach of simple analogue control on a per-phase basis. For the sake of simplicity in the ensuing discussion, consider three independent, star-connected equivalent phases in the control circuit. Other combinations are possible and sometimes mandatory, but their consideration for the time being at least, may be ignored.

A suitable control strategy for voltage-fed, force-commutated compensators have a number of requirements. The AC-DC converter, forming the back-bone of the dynamic compensator, employs two inter-related control strategies. The first is the PWM-control strategy generating commands for the switches and the second is the overall control strategy that co-ordinates the different steady-state variables. In the strategy proposed for the latter, the variables are:

The AC-voltage fundamental phase angle, The AC-voltage rms-amplitude, The AC-current rms-amplitude The required periodic profile of the AC-current waveform The DC-link voltage level, where applicable, and The AC-DC average active power level, where applicable.

A simple single-phase illustration will elucidate the proposed control strategy. Refer to Figure 4.5.3. 60

Zs PCC

Rfal Icon, [1.0 I T Figure 4.5.3 - Simple single-phase steady-state compensator

Assume the load current //, to be a periodic waveform consisting of a fundamental component on which only a third harmonic component is superimposed. In this case the compensator may be called upon to compensate both for the harmonic component and for the fundamental out-of- phase component or both. For harmonic compensation alone, it will be necessary to separate the harmonic component and the fundamental component of current. The compensator can then be controlled to draw a current equal to the harmonic component alone but in antiphase with it. That would leave only the fundamental component of the current in the lines from the source, but with its inherent phase angle. In the case of line-commutated converter loads, this phase angle will normally lag and the power factor of the harmonic compensated load will therefore still not be optimum, even in the presence of a low current THD. To improve the fundamental power factor as well, therefore, a further fundamental frequency component of current will also have to be drawn by the compensator.

A dynamic compensator will draw whatever temporal current profile it is programmed to draw, within the bounds of its specifications. This current profile must therefore be furnished to it on a continuous basis and it must conform to the requirements to remove the harmonic current content from the lines or to furnish fundamental power factor correction or both. Although many different control schemes and theories exist for this purpose, a control scheme implemented by means of simple analogue circuitry will be described here. This is the scheme essentially in use in the dynamic compensators.

Figure 4.5.4 shows the necessary measurement signals required for the control scheme used. 61

Figure 4.5.4 - Current and voltage signals for dynamic compensator

Only instantaneous values of current and voltage are measured. The source current is(t) and source voltage us(t) are measured on the load-side of the compensator and the load current iL(t) is measured on the load-side. These are instantaneous per-phase analogue signals. The processing of these signals is shown in block-diagrammatic form in Figure 4.5.5 to furnish the spatial control profile of the current that must be generated by the dynamic compensator.

As shown, is(t), us(t) and IL(t) are input as time-dependent analogue values to the controller.

LPF phase Is' comparator integrator /At) lJ

Us1-90°

us(t) < output signal to dynamic compensator

4L1 1L(t) -Kh

Figure 4.5.5 - Dynamic compensator reference signal generator 62 Both source-side signals first pass through zero-phase shift low-pass filter networks that reject all but the 50 Hz frequency. The fundamental frequency source voltage signal Usi is then integrated and inverted to furnish a fundamental frequency sinusoidal waveform leading the voltage by 90°. This waveform represents the required quadrature phase of a fundamental power factor correction current, necessary to improve the fundamental power factor. A phase comparator compares the phase of the actual fundamental source current component /s, with Us1-90°, generating an analogue value for the phase difference U s/-90° - Z/s/ . Positive values of this analogue output signify a lagging fundamental component and vice versa. The integrator output is multiplied by the quadrature sinusoidal signal Us1-90° to yield the fundamental component compensator driving signal. It will become immediately clear that the fundamental compensating current control operates in a closed loop mode. Whenever the phase of /si lags that of Us, then the integrator output integrates upward, increasing the magnitude of the leading quadrature signal U51-90°. Integration will continue until the fundamental phase difference is zero. Although the integral control mode gives satisfactory operation when adjusted for a slow response, a proportional facility can easily be added when required. The function of the summer between the integrator and the multiplier in the fundamental phase control loop is to provide an adjustable offset for the required fundamental phase angle, by adjusting the magnitude of the offset 0 as shown. By virtue of this closed loop control, the fundamental power factor can be controlled independently of the presence or absence of harmonics in the load current.

Harmonics present in the load current are monitored by input ii,(t) to the controller, after the fundamental current component has been removed by a high-pass filter network. This network may consist either of a passive or active filter or a more sophisticated phase-locked loop arrangement, depending on the requirements. The time-dependent current value of iLh(t) is inverted and multiplied by a factor Kh, permitting the degree of compensation of these components.

It is important to note that the required phase angles of the fundamental current and the third harmonic current generated by the compensator, . must be independently measured and independently controlled with respect to each other.

When employing this mode of independent control of the fundamental and the harmonic components, the DC link capacitor voltage becomes of great importance, because the net periodic energy in this capacitor furnishes the source of the power that resides in the respective frequency components. Because the harmonic components are generated by the distortion load, it was shown37 that the sum of the real power components in the combined harmonic frequencies are brought about by dissipation in the real immittance components in the network. It is therefore possible that the compensator will also have to supply real power at the harmonic frequencies. That being so, it follows that the net energy in the DC link capacitor will reduce over a complete 63 period, even though it will oscillate from beginning to end. This energy will have to be replenished from another source. The converse is also true, i.e. that there may be a net energy supplied to the compensator in the harmonic components over a period, in which case the voltage across the DC-link capacitor will increase and will have to be "bled" through another means.

The same is true for the fundamental component. If the compensator supplies a true quadrature current, the net voltage excursion over a full period will be zero. In practice, however, the compensator will have internal losses (switching and conduction) and it will therefore be necessary to adjust the phase angle of the fundamental component relative to the supply voltage phase angle to maintain the net real power flow at the level necessary to maintain the desired average DC-link capacitor voltage. In a single-phase system such as that described here, the DC- link voltage will vary both on account of the fundamental and harmonic frequency energy- exchange requirements. In three-phase systems, however, the DC link voltage will not have ripple if there is only positive sequence currents flow into the compensator, even in the presence of harmonics. This statement confirms the findings of Venter'''.

By virtue of the above principle, control of the DC-link voltage now becomes a simple matter and it is merely a question of adjusting the phase of the fundamental current component drawn by the compensator for that purpose. If this phase angle is made lower than 90° with respect to the voltage on the PCC, then the link voltage will increase and vice versa. It therefore merely becomes necessary to control this phase angle with a relatively large time constant to maintain the desired DC-link voltage. By monitoring the average DC-link voltage therefore, all other losses, including the real power supplied by the compensator in the harmonic frequencies, can be compensated.

Figure 4.5.6 shows, in block-diagrammatic form, the essential components of the proposed simple phase control strategy for steady-state dynamic compensator control. The current waveform generator in Figure 4.5.6 represents the circuit of Figure 4.5.5, setting the spatial profile of the required compensating waveform that the compensator must generate. 64

0 LP + fil v ter DC

4X i ac zero- ossing det ctor v MIS dcact current DC voltage generator Ui setpoint v dcreq PWM dcerr /aceff carrier current generator DC wave-form LP WA amp filter generator PWM modulator Vcos(0) t)

T v phase dccon shifter

Figure 4.5.6 - Steady - state per - phase control strategy for a dynamic compensator

The diagram in Figure 4.5.6 shows this strategy in block-diagrammatic form for a single-phase converter. Three-phase converters will employ three identical controllers on a per-phase basis. To eliminate zero-sequence current generation by the compensator, however, measurement will take place in two phases only and the third-phase control signal will be the negative sum of the others.

The control scheme shown is intended for operation under steady-state periodic conditions that are Fourier analysable. When controlled in the above way, the compensator is required to be connected to an AC-source, such as the power network. The AC-voltage waveform will therefore be required to be periodic and so will the compensator current. The PWM switching frequency is required to be synchronised with the fundamental frequency in order to limit the switching harmonics to fixed multiples of the fundamental frequency.

It is therefore a good starting point to measure the instantaneous value of the AC-voltage waveform. The PCC is represented here by a voltage source V AC, feeding the compensator. A phase-locked loop is employed to generate the average zero-crossing points of this waveform, with the assumption that it may be distorted and to prevent the crossing-points from dithering in time. The phase-locked loop employs a low-pass filter with a relatively large time-constant. The output of the zero-crossing detector is a symmetric square wave. It is furnished to both the PWM carrier generator and the current waveform generator. 65 Another phase-locked loop is employed in the PWM carrier generator, with a divide-by circuit in its frequency feedback circuit. This circuit is responsible for generating a triangular waveform havingas a frequency which is an exact multiple of the fundamental frequency, and that has a fixed phase with respect to it.

A broad-band current transformer is used to measure the instantaneous value of the converter current after filtering out the PWM switching current ripple. The instantaneous current signal is fed to the rms current generator that generates an analogue signal equivalent to the rms-current value. A low-pass filter with a relatively large time constant is used to remove any residual ripple out of the analogue signal representing the rms current laceff.

The required compensating current waveform is generated by the current waveform generator as outlined above. Both the waveform shape and the amplitude are required.

The DC-link voltage must be monitored in order to be controlled. This is achieved with a galvanically isolated voltage transformer as shown. The scaled-down DC-link voltage analogue signal is supplied to a summer circuit that compares the actual link voltage with the required DC- link voltage, outputting an error signal that is proportional to the difference. The difference is amplified by an amplifier with an adjustable gain.

The PWM-modulator is the recipient of the steady-state control signal that will drive the inverter to furnish the required averaged current signal, as discussed earlier. The PWM-modulator modulates the synchronised triangular carrier wave. If the peak amplitude of the PWM carrier- signal is 10 volt, for example, the input analogue control signal to it will range from +9 V to - 9 V, with a maximum modulation index of 90%.

The phase shifter employs a time-delayed analogue output. Time delay is defaulted to 20 ms with a zero analogue input voltage, corresponding to a full 50 Hz period and with Vcicc„ at zero, the time-dependent output of the phase shifter will therefore be delayed by 20 ms. An increase or decrease in vdeccon will respectively increase 0 (and decrease the time delay) The waveform output profile will therefore remain unchanged but will be delayed or advanced in time. Phase shifter implementation is most easily implemented by means of a digital shift register with the necessary clocking and latching facilities.

If the amplitude of the DC-link voltage is below the required voltage vdcreq, then it will generate a positive error signal v dcerr This signal will be amplified by the DC amplifier and supplied to the phase shifter. The phase shifter has a default setting that will entice the inverter to draw a fundamental DC-component of current that will be 90° behind the fundamental component of the voltage across the inverter. If the DC-link voltage is therefore below the required value, the phase shifter will advance the phase of the required current waveform to generate an in-phase current 66 that will (as discussed before) increase the average active power supply to the DC-link capacitor, causing the voltage across it to increase. This will have the desired result and the DC-link voltage will increase until the setpoint is reached when the phase shifter will begin to retard the requested current waveform fundamental component sufficiently to stabilise the voltage level on the DC- link.

The above method of operation is only possible if the magnitude of the desired current is sufficient to achieve the desired power throughput that will raise the DC-link voltage to the required level..

The function of the DC-amplifier is to increase the closed-loop gain of the system, in a similar manner to that employed in other typical proportional control systems. The amplifier is not shown, but the error signal supplied into the PWM-modulator is also amplified in a similar manner. High closed-loop gain reduces steady-state error, but at too high values instability will result. In addition to the simple proportional control functions described, both integral and differential control functions can also be inserted.

4.6 HYBRID COMPENSATOR TOPOLOGIES

The advantages of the individual generic types of compensators can be realised more fully in practice in hybrid structures that cater for specific requirements. These requirements are varied and specific. The following discussion will deal with some of them, but only those that appertain to the requirements of the question at hand will be discussed.

4.6.1 Series hybrid compensator topology

Passive filters have the practical limitation that their quality factor Q must be artificially reduced in most high-voltage applications to cater for frequency drift and for drift in capacitance with age. Under these conditions, the residual impedance at or near resonance of RLC-arms of the filter is still so high that significant proportions of the distortion current flow through the source terminals and the filter has a limited effectiveness in reducing the source current MD.

The circuit in Figure 4.6.1 shows a series hybrid compensator, employing 5th, 7th and high-pass passive filter arms with a voltage-fed dynamic compensator. Operation is best understood by examining the operation on a per-harmonic basis.

load

Figure 4.6.1 - Series hybrid compensator topology

Under those conditions it is seen by inspection, that to prevent the load harmonic components from flowing through the source, the series voltage of passive and dynamic compensators must be maintained at zero volt. The dynamic compensator is therefore required to inject harmonic voltages in series with the passive filter of the exact opposite polarity and phase to the voltage drops across the passive filter, at the given values of load currents. Because the harmonic current voltage drops across passiVe filters will be low, the kVAr rating of the dynamic compensator will therefore also be low and the passive filter will usually do the lion's share of the work, permitting the use of a much lower rated dynamic filter. This topology allows retrofit applications with existing L-C filters which is an advantage to utilities where off-tuned passive filters are a problem.

The above series hybrid compensator topology has the disadvantage that it will only be able to supplement passive filter operation at the harmonic frequencies to which the arms are tuned and the otherwise wide-band capabilities of the dynamic filter will therefore be wasted. The implementation of this hybrid compensator is given in48,49,50,51.

4.6.2 Parallel hybrid compensator topology

S load

Figure 4.6.2 - Parallel hybrid compensator topology

68 The parallel hybrid topology depicted in Figure 4.6.2 is not practical with normal dynamic compensator control, because the dynamic compensator will be doing all the work if the frequencies permitted in its control coincide with those of the passive filter 52. It will, therefore, not be capable of supplementing the operation of the passive filter as was the case with the series hybrid topology. In the parallel topology depicted, passive and dynamic compensators can, however, be employed for the compensation of separate harmonic components 53'56. It is possible, for example, to install only a 5th harmonic arm in the passive filter and to filter the 5th harmonic out of the control signal to the dynamic filter. In that way, the passive filter will perform its work normally on the 5th harmonic while the dynamic compensator will take care of all the other frequencies within its range. An arrangement like that could, for example, be used with a DC arc furnace, in which there is a large 5th harmonic present, but in which the spectrum of the other harmonics are not discrete but random.

4.6.3 Series - dynamic hybrid filter topology

The series-dynamic hybrid topology depicted in Figure 4.6.3 is used to enhance the performance of the passive filter in the network 54.55'56.

S O load

Figure 4.6.3 - Series-dynamic compensator topology

This topology consists of a series-dynamic filter placed between the source and load, and a tuned passive filter network connected across the load. At the fundamental frequency, the dynamic filter allows the fundamental current for the passive filter and the load to flow through it. The dynamic filter acts as a harmonic isolator forcing the load harmonic current to flow through the passive network. This topology has the advantage that harmonics introduced at the supply cannot overload the passive filter.

4.6.4 Series-parallel hybrid topologies

load

Figure 4.6.4 - Series-parallel hybrid topology

In the topology shown in Figure 4.6.4, dynamic compensator 1 supplements the passive filter in reducing THD on the 5th and 7th harmonic frequencies to very low values, whilst 2 takes care of the other harmonics present. Compensator 1 will have control signals on the 5th and 7th harmonics alone, whilst 2 will have those frequencies blocked in its control input. In that way the filter can be designed to furnish extremely low THDs of current in the source circuit at all the harmonic frequencies, but with the bulk of the kVAr contribution by means of the cheaper passive filter network.

4.6.5 Resonance suppression

There are sometimes occasions where equipment topologies demand that parallel sets of passive filters be installed in parallel. Because of their identical tuning, these filters are prone to resonance with each other. On other occasions, passive filters are used in circuits of which the circuit constants can change by switching, for example.

Series hybrid topologies can be used to suppress the above types of resonances, by generating the control signals differentially, and feeding additional control signal components to the appropriate compensators that are composed of those differential signals. Consider, for example, the circuit in Figure 4.6.5.

70 zs

load

Figure 4.6.5 - Resonance suppression example

In this example, the two compensator currents are measured and subtracted. This differential result is fed as additional components to the two supplementary dynamic compensators and it can be seen that circulating currents between the two filters will generate a signal. These signals can then be configured in the control circuits of the two dynamic compensators in such a way that the circulating currents will be suppressed, while common-mode signals will be ignored, permitting normal control.

4.7 SUMMARY

The basic principles of compensation were discussed by examining the circuit behaviour in the frequency domain. Passive and dynamic compensation principles were reviewed. Despite the known disadvantages of passive filters they are still widely utilised in industry because of their superiority in cost when compared to that of dynamic filters, and the established simplicity of their design. By combining passive and dynamic filters in their so-called hybrid configurations, the best can be had out of both technologies. Simple but practical control strategy commonly used for dynamic filters operating in steady state was also discussed. 71 5. HARMONIC SUPERPOSITION MODELLING

5.1 INTRODUCTION

Harmonic power measurements in electrical systems call for complicated and tedious measuring procedures. The cost of sophisticated measuring equipment normally limits the diversity of measurement while the work requires specialist knowledge. Practical measurements normally necessitate interruption in normal plant operation and the procedures are cumbersome and time- consuming. Modelling presents an easy way out and provided that reasonably accurate data- bases are available for the networks and their components, modelling can be carried out accurately enough to predict the network behaviour for establishing the required ratings of compensating equipment and the severity of harmonic injection back into the network in terms of supply authority standards, for example.

There are basically two types of investigations that can be conducted by using circuit-models, namely transient behaviour and steady-state behaviour. When circuits contain more than a few nodes, however, analytical transient analysis becomes cumbersome and the high orders of the resulting differential equations usually make the task impractical. In this type of analysis, numerical integration techniques are then normally resorted to, yielding accurate and quick results only for systems that contain a limited number of nodes. Numerical integration techniques for steady-state AC-behaviour, also require those initial calculations first be made to exclude transient behaviour. This part of the calculation is itself also time-consuming, with the result that numerical integration is basically always only used in transient modelling.

Nodal network analysis techniques are well documented in the literature 5758.59. Their implementation in multi-frequency superposition modelling, in the form used in this investigation, warrants a brief review.

5.2 THREE - PHASE MODELLING TECHNIQUES

Consider a general multi-frequency network of n orders that contains k nodes and in which any two respective nodes 1 and m are connected by a complex impedance Z, (n) in series with a voltage source 111„,(n). In reality there are n networks with the same configuration, but every harmonic order has unique impedance and series voltage values for each value of n. Normally, it is customary only to number major nodes but the numbering may even be extended to minor nodes. The reference node is usually identified as node-O. By means of source transformations, each series impedance may be replaced by its equivalent parallel admittance as:

1 (5.2.1) Yfrn(n)— Zi„,(n )

Similarly, each series voltage source can be replaced by its equivalent current source as: 72 Ufr„(n) I(n) zi,„(n) (5.2.2)

It is customary to drop second subscripts for a connection between a numbered node and the reference node. The equivalent current sources depicting voltage sources always connect nodes to the reference node and has single subscripts. Voltages of given nodes with respect to the reference node, similarly, employ a single subscript. By employing this formulation, a circuit equation can then be given for a multi-node network in terms of the bus-admittance matrix ['l b.] as:

[1(n)] = [Abus(n)] [U(n)] (5.2.3) in which [1(n)] and [U(n)] are vectors of order-n and of dimension k, and [Abus(n)] is the nth order k x k admittance matrix:

y2 (n) y 3(n) Y1 (n) Y, 4 (n) " Yik (n) -

Y21 (n) Y22(n) Y23 (n) Y24 (n) Y2 k (n) Y32 (n) Y33 (n) Y34 Y31(n) (n)( " Y3 k ( n [2,b„s(Pl)] Y42 (n) Y43 (5.2.4) Y4 (n) (n) Y44 (n) " Y4 k (n)

II II II

Yki (n) Yk2 (n) Yk3 (n) Yk 4 (n) Y kk (n)

The inverse of the bus-admittance matrix [21,,,Xn)] may be calculated analytically or by means of recursive methods as in the Gauss-Seidel method 59 and the node harmonic phase values are obtained from:

[U(n)] = [2b,,,(n)]-1 [1(n)] (5.2.5) as in the conventional single-frequency case.

The network analysis in this section will be carried out for linear passive networks and references to harmonic orders will be ignored for simplicity in representation. The relations and concepts developed here are normally used only in fundamental frequency analysis, but can be extended equally well to the behaviour of transmission circuits at any individual harmonic order.

5.2.1 Compound admittances

The diagram in Figure 5.2.1 represents three elements in a three-phase system with self-

admittances -Y aa, - bb and Y. as well as mutual coupling admittances Icb, Ybc and Ka. 73 Y I as a

Figure 5.2.1 - Admittance of a three-phase series element

The currents and voltages shown are related by the following matrix equation:

la Yaa Yab Ybc Ua

Ib Yba Ybb Ybc Ub (5.2.6)

c_ Yca Ycb Ycc Uc

in which the diagonal matrix elements Yaa, Ybb and Ycc represent the self-admittances of the three series elements and the off-diagonal elements the transfer admittances brought about by mutual coupling. Note that in linear passive networks it can be shown that Yab = Yba, Ybc = Ycb and

Yea = Yac. Also, in circuits with no coupling between the elements, the mutual coupling admittance will be zero and (5.2.6) will have zero entries except on the diagonal.

+u I

Figure 5.2.2 - Single compound admittance representation for three series elements

It is possible to represent the three separate elements of Figure 5.2.1 by a single compound element, as shown in Figure 5.2.1 for which (5.2.6) may be written in compact form as: [1] = [ nu] (5.2.7) 74

5.2.2 Nodal matrix

The actual connection diagram of a three-phase circuit may be represented as shown in Figure 5.2.3 in which the admittances V11 to 1755 represent compound series admittances and the nodes a, b and c are common to all three phases.

Ic Ia lb

Ua V33

Reference node

Figure 5.2.3 - Actual connected network

The nodal voltages Ua, Ub and //c represent the voltages of the major nodes numbered respectively with respect to the reference node, and the nodal currents la, lb and Ic obtained through the appropriate source transformations represent the currents injected into the respective nodes (from the reference node).

The nodal matrix equation for the network in Figure 5.2.3 can be written by inspection as:

Ia Y11 + Y33 0 Ua Ub (5.2.8) Ib Yll Yll + Y22 ± Y44 —Y22

I c 0 — Y22 Y22 ± Y55 _ _ Uc

In (5.2.8), the diagonal elements of [vabc ] represent the sum of the admittances connected to the respective node and the off-diagonal elements the negative of the interconnecting admittance between the respective nodes. Although this approach is straightforward in this instance, more involved circuit approaches generally warrant a more detailed approach that will be discussed briefly.

5.2.3 Primitive network

A primitive or unconnected network can be formed of the three-phase network in Figure 5.2.3 by considering the branches separately in an unconnected form. Remembering that the elements all represent three-phase compound admittances and that voltages and currents in Figure 5.2.3 represent three-phase values, let the currents h to IS through each element and the voltages U1 to

U5 across each element in this primitive or unconnected network be the same as in the actual 75 connected network in Figure 5.2.3 The primitive network for the three-phase circuit can now be drawn as in Figure 5.2.4

I, 1 12 I3 '4 15

U3+ Y44 V11 14+ 1/22 U2+ Y33 U4+ Y55 U5+

Figure 5.2.4 - Primitive three-phase network with compound admittances

The matrix relationship for the primitive admittances in Figure 5.2.4 is:

11 171 1 0 0 0 0 (11

12 0 Y22 0 0 0 U2

0 0 133 0 0 U3 (5.2.9) 14 0 0 0 1'44 0 U4 5 0 0 0 0 Yss Us in which the primitive current vector is obtained by post-multiplying the primitive admittance matrix by the primitive voltage vector.

5.2.4 Connection matrix

The primitive voltages of the actual network in Figure 5.2.3 can be obtained by post-multiplying the connection matrix [C] with the nodal voltages: - -

U , 1 —1 0

U2 0 1 — 1 Ua U3 1 0 0 Ub (5.2.10) U4 0 1 0 U5 0 0 1

The nodal admittance matrix of (5.2.8) can then be obtained from the primitive admittance matrix and the connection matrix: T [yabd= [c] [naml[c] (5.2.11) 76 When (5.2.10) is applied as above, the nodal admittance matrix as in (5.2.8) is obtained in which it must be remembered that the admittance elements each consist of a 3x3 matrix and the voltage and current vectors of 3x1 vectors.

Equation (5.2.9) represents the simple case in which the admittances in Figure 5.2.4 have no mutual coupling. If coupling exists between these elements, the primitive admittance matrix in (5.2.9) would, in addition, contain off-diagonal elements. That being the case, (5.2.8) would not be as easy to write directly from Figure 5.2.3 but that the more elaborate way above would simplify the procedure. In that case the primitive admittance matrix in (5.2.9) would also contain off-diagonal entries.

5.3 SUPPLY AUTHORITY SOURCE

The supply authority source can be represented as a compound Norton equivalent source, consisting of a current source in parallel with an impedance as shown in Figure 5.3.1

Figure 5.3.1 - Equivalent circuit for supply authority model

Normally, the equivalent source impedance and open-circuit voltage at fundamental frequency can be furnished directly by the supply authority for given supply points. In other cases, as an approximation, the equivalent fundamental frequency source admittance can be calculated from the symmetric fault level and the off-load (or nominal) supply voltage:

Ys = 2 (5.3.1) IQ

More accurate information, pertaining to the fundamental frequency, can be obtained by also knowing the negative-sequence impedance to source. In the type of modelling to be undertaken here, negative-sequence effects can be safely ignored and knowledge of the negative-sequence impedances are therefore not essential. Further, by confining all modelling to include only positive and negative sequence conditions, compound admittance matrices can be retained at 3x3 order.

Obtaining the harmonic immittances of networks, such as for the equivalent source, is no simple matter and such data are not normally directly available from supply authorities. Both the real 77 and imaginary immittance components are frequency-sensitive. The former is a function of electromagnetic effects such as skin-depth of conduction and proximity effects 60.61, resulting in an increase in series resistance with increases in frequency. In the latter case, series inductance in the current-carrying components increases with frequency and capacitive reactance between conductors (and ground) reduces with frequency.

Although more extensive models could have been derived for the purpose of this study, excellent correlation with measured results were obtained with the following simple models that assumed resistance to be constant over the applicable frequency range.

The balanced, driving-point equivalent source impedance, which is predominantly inductive, can be modelled as:

Z(n)= R+ jrkaL (5.3.2) leading to the following form for the series admittance:

ncoL (5.3.3) Y (n) = R

Mutual line-line inductance and other parasitic effects can normally be ignored at the harmonic frequencies in this instance, yielding the following compound equation for [i 5 } and [lc in Figure 5.3.1.

(5.3.4) [vs 1-- EY.1 1 [Is]

5.4 TRANSMISSION SYSTEM COMPONENTS

Transformer models are obtained by transforming the primitive models of the series- frequency- dependent leakage impedance and parallel magnetisation and core-loss characteristics into compound nodal form. Inter-winding capacitance, inter-phase coupling and other parasitic effects are neglected. Cable models presently only include series resistance and frequency-dependent series impedance and parallel insulation capacitance. Transformer and cable models are respectively based on T and L-network formulation in accordance with this simplistic approach.

The cable series admittance and parallel capacitance can modelled respectively as:

1 E c(n) — and Ycc(n) = jconG (5.4.1),(5.4.2) R+ jamLc

Both YLc and Ycc are 3x3 matrices at a specific harmonic order. 78 The degree of completeness of the nodal admittance sub-matrices of a transformer depends upon the accuracy of the modelling required and upon the availability of frequency-domain model information with respect to driving-point and transfer admittances. In the transformer model interphase coupling can be ignored, therefore the coupling between the primary and secondary coil is modelled as for the single-phase unit, giving rise to the primitive network of Figure 5.4.1

11111141112111511i31116 M14 M36 Ys6 Ypl Ys4 Yp2 Ys5 Yp3

Figure 5.4.1 - Primitive network for three-phase transformer

The matrix equation is:

Yp1 0 0 M14 0 0 /2 0 Yp2 0 0 M25 0 U2 /3 0 0 Yp3 0 0 M36 U3 • (5.4.3) /4 M41 0 0 Ys4 0 0 U4 /5 0 M52 0 0 Ys5 0 U5 0 0 M63 0 0 Ys6 U6

The design procedure followed for the transformer is as follows: The series admittances are calculated from the leakage impedance magnitude and current phase angle at fundamental frequency. Consider, as an example, the derivation of the admittance matrix for a star-star configuration. In practice, power transformers mostly employ delta-star winding configurations. The connecting matrix is given by:

1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 (5.4.4) 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1_

To determine the series admittance the specified transformer leakage impedance magnitude at fundamental frequency and specified transformer fundamental component leakage impedance angle are used with values of 5% and 88 ° respectively. This relates to the complex fundamental leakage impedance of the transformer: 79

=Z mt Z 3,11 . ej.°1-' (5.4.5)

Real and imaginary components of transformer leakage admittance are calculated from the real and imaginary components of the fundamental leakage impedance.

1 (5.4.6) Y2,n(n) Re(Z m1 ) + j n-Im(Z mi)

To determine the parallel admittance the specified magnitude of fundamental component of the transformer magnetising admittance and specified fundamental component magnetising admittance angle are used with values of 1.5% and 81° respectively. The parallel conductance and susceptance can be calculated as:

YTps (n) = G Tps— j • nB Tps (5.4.7)

With G Tps = Re(Ymae"m.) and B Tps = Im(Ymae-Hsitl.) (5.4.8)

The primitive admittance matrix can be calculated as shown in 5.4.9 for all the harmonic orders: - _ 0.5 . Yrp (n) 0 0 —YTS (n) 0 0 0 0.5- Yrp (n) 0 0 —YTS (n) 0

0 0 0.5. Yi.p (n) 0 0 — VT., (n) Y, (n) = — Yrs (n) 0 0 0.5 - YTp (n) 0 0 0 —YTS (n) 0 0 0.5 - YTp (n) 0 0 0 —YTS (n) 0 0 0.5 . YTp (n) .(5.4.9)

5.5 6-PULSE AC-DC CONVERTER LOADS

500

500 0.01 0.02 0.03 0.04

Figure 5.5.1 - 6-pulse AC-DC converter line current waveform 80 Modelling of the 6-pulse current waveforms in the frequency domain requires special consideration. Although excellent work has been done in this regard62' 63, 64, for modelling purposes here, the 6-pulse converter models must be continuously adaptable to cater for the different possible operating current waveforms of converters used in the R 2P2 units. A convenient approach in designing the 6-pulse converter models is to match the time waveforms against the observed waveforms of the R2P2 units. By providing continuous adjustment in the ripple-ratio, commutation angle, peak current value and firing angle in these models, time waveform matches are simple to achieve, by furnishing general variables that control each of these parameters in the waveform time-domain construction. Figure 5.5.1 depicts the general form of the line current in a 6-pulse converter with continuous DC output current.

In order to ensure the absence of zero-sequence currents in the supply to the converters, the modelling is initially carried out on a line-line basis and then transformed through simple A-Y transformation in the frequency-domain as detailed in Annexure A. This procedure automatically prevents the introduction of zero-sequence currents, but permits unbalanced phase currents by the permitted introduction of negative-phase sequence currents.

Trigonometrical Fourier series expansion is employed. It is implemented in the form of the real parts of exponential expressions, which are more compact and convenient to work with. Complex Fourier coefficients are employed. These coefficients are convenient to use and are expressed as rms-values with the additional advantage that they represent the harmonic phasors of the particular waveform at the given harmonic numbers. Separate coefficients are used for each of the three line-to-line currents and the 120 degrees fundamental phase angles are incorporated in them.

5.6 PASSIVE FILTER MODEL

Tuned series passive LCR-filters are designed in the literature in accordance with standard procedures58. The design procedure followed, for passive filters, employs compound admittances as in the rest of this work'''. To prevent parallel resonance with the source, it becomes necessary to insert additional series impedance in the lines between the source and these filters. These reactors are also essential to eliminate series resonance between equivalent-order harmonic arms in the different filters. The latter parallel resonance is suppressed by virtue of the increase in the frequency difference between series and parallel resonance of these adjacent filters as will be described in chapter 6. The filter used in the experiments was designed for 100 kVA and the parameters below are used. The total parallel capacitance is chosen to correct the fundamental power factor to unity. Note that in the model different filter parameters are used, depending on the location in the network and the load in question. 81

Table 5.6.1 - Characteristics of filter

5th 7th 11th

R 104 mf2 106 mc2 70 mfg

L 2.03 rnH 1.03 mH 419µH

C 200 1..LF 200 i..LF 200 1.1.F

Lseries 200 !JR

Rseries 60 mc2

Q 30.61 21.4 21

The 5th 7th and 11th filter admittance elements:

Y5,7,11 — 1 R5,7,11 j(conL5711 — ) (5.6.1) and for the 3-phase system:

Y5,7,11 0 0

Y5,7,11 = 0 Y5,7,11 0 (5.6.2)

0 0 Y5,7,11_

Filter parallel admittance:

Yp = 175 +Y7 + Y1 1 (5.6.3)

Filter series admittance:

1 (5.6.4) — RS + jconLs 82

5.7 DYNAMIC COMPENSATOR

A voltage-fed 150 kVA, PWM-controlled, IGBT dynamic compensator has been purchased from EEI52 in Italy. The performance of this compensator, against the different harmonic frequencies was measured, and a model of the compensator, with its measured practical constraints and identical control scheme, has been constructed. The compensator is installed between the source and the load and measures the line currents between it and the load. It then simply separates the fundamental and harmonic components and injects the harmonic components into the line at its installed node. It also furnishes the option of injecting the required fundamental component reactive kVAr required for displacement power factor correction. Where possible, the latter correction would rather be gleaned through static capacitor installation, because of its lower cost and would normally be turned off in the dynamic compensator.

5.8 SUMMARY

The useful tools that are based on the Admittance Matrix concepts used in single-frequency systems may be extended to multi-frequency systems as well by adding another dimension to the definitions, namely that of harmonic order. In accordance with this formulation, existing data bases can be extended to include impedances at other frequencies by adding an additional dimension for the harmonic order to equations. This, in effect, changes the single matrix equation to n equations and permits the complete set of calculations, including matrix inversion, to be carried out independently in terms of each harmonic order. The equations obtained in this way enable the analysis and modelling of non-linear networks and networks in which the waveforms of current and/or voltage are distorted through harmonic superposition and all defined values of voltages and currents can be calculated and displayed. The time-domain waveforms of these parameters may be plotted by summation. 83 6. PRACTICAL VERIFICATION OF THE SUPERPOSITION MODEL

6.1 INTRODUCTION

The harmonic superposition model will be required later to investigate the optimal compensation for a pilot plant with a large number of lasers and must be verified. For this purpose, an R 2132 power supply, feeding a pulse dummy-load, was installed at one of the positions reserved for lasers in the plant. The future plant site that was used, was already equipped with power transformers and cables and was eminently suited for the purpose. The admittance diagram for this measurement setup is shown in Figure 6.1.1, in which compound admittances are shown and in which the harmonic order subscripts are omitted for clarity.

The source is represented at node-1 in Norton-equivalent form. Y12 - Y20 represents the feeder cable to the MLIS plant and Y34 - Y40 the feeder cable from the main substation to the laser. The relevant, 11 kV/400 V, 2 MVA 6% AY plant transformer is modelled by 1723-1730. Node-3 represents the low-voltage substation busbar. Admittance V50 represents the tuned parallel series 5th, 7th, and 1 lth harmonic legs of the passive filter at the load. This filter can be turned on and off in the model. Admittance 1'45 makes provision for the series inductor by means of which the load and passive filter can be "uncoupled" from the supply when required.

Unfortunately, system voltages and other operating characteristics of the load could not be held constant between consecutive measurements and this will be encountered in the measuring results below. All the measurements record node-3 line-neutral voltages and nodes 3 to 4 line currents. This immediately becomes evident upon examining the relative fundamental phase angles. The line-line voltage lead the line-neutral voltages, but is not shown in the oscillograms for the sake of clarity. The admittance between nodes 3, 4, 5 and 0 are adjusted to suit the requirements of the measurement at hand, both in the model, as well as in the measuring setup. Modelling is implemented by an iterative procedure, details of which are given in 66.

The twin waveforms shown in Figure 6.2.1 to Figure 6.5.1 show time-reconstructed modelled results on the left and direct-measured results on the right. The graphs in these figures are imported from the main Mathcad worksheet model. The measured results were stored in situ, on disk by means of a Tektronix TDS-620 digital storage oscilloscope and imported to the worksheet in which frequency-domain data of identical format to the modelled data are obtained by fast Fourier transform (FFT). The voltage probe used is a Tektronix P6009 and the current probe is a Tektronix CT-4. The error of direct measurements is 5% Note that the way the measurements results are given sometimes reflect a better error than 5% which must be seen in context. 84

Dynamic

1_71 compensator

140

I'127 Y23 IP34 Y4S 1 -- 2 5 rig

50 Hz High Trans- Main Low Passive 6-pulse source voltage fomer board voltage filter converter cable cable

Figure 6.1.1 - Per-phase admittance-diagram for dummy-load measurement setup

6.2 MEASUREMENT 1 AT THE PCC WITHOUT COMPENSATION

Measurement 1 compares modelling and measured results at the point of common coupling (PCC) in Figure 6.1.1 and is shown in the oscillogram in Figure 6.2.1. and Figure 6.2.2. The modelled and measured results in Figure 6.2.1 is used to construct Figure 6.2.2 by overlaying the two graphs in Paintbrush.

400 400 320 320 240 240 160 4\11111111111111111A11111111111M 160 80 ' 11111111111,111 80 0 1111111/111111MUMINI 0 -80 ■ 80 160 11110111111111 111M111 160 240 11111111111111111KIIT/M111 -240

320 -320 400 400 0.01 0.02 0.03 0.04 0.05 - 0.03 -0.02 -0.01 0 0.01 0.02

a) - Modelled result b) - Measured result

Figure 6.2.1- Measurement 1 - PCC without compensation

A number of remarks will be in order here, as they will also illustrate the utility of the modelling theory used. To optimise the system, with respect to overall cost, the load current waveforms in Figure 6.2.1 can still be improved with respect to THD, which is 38% here as reflected in Table 6.2.1. By further increasing the DC-choke inductance L, beyond 3.2 mH, in the phase controlled AC-DC converter (refer Figure 1.2.1), the inherent THD in the current waveform will also be 85 further improved. The additional cost associated with this improvement, will be lower than that incurred in additional compensation, without upgrading this inductance.

Figure 6.2.2 - Measurement 1 - Comparison of modelled and measured results

There are, unfortunately, space constraints in the present R 2P2 cabinets making this change impossible, and this restriction will have to be tolerated for the time being. This relatively high TIM is also reflected in the current distortion power defined in (31). The relatively low fundamental power factor of 0.927 gives rise to the high fundamental reactive power of 21.07 kVAr as shown in Table 6.2.1. Although not excessive here, at 1.196%, the voltage THD will lead to problems when a large number of lasers are operated together on the same feeder system, and compensation of one type or another will then become mandatory.

Table 6.2.1 - Measurement 1 - PCC without compensation

Description Model Measured % Unit Difference Total node voltage 241.63 239.03 1.08 V Total node current 83.2 84.17 1.17 A Total active power 52.50 52.30 0.38 kW Total non-active power 29.68 30.12 1.48 kVA Total apparent power 60.31 60.35 0.07 kVA Fundamental apparent power 56.21 56.40 0.34 kVA Fundamental active power 52.53 ' 52.31 0.41 kW Fundamental reactive power 20.00 21.07 5.35 kVAr Harmonic apparent power 214.98 257.06 19.57 VA Current distortion power 21.80 21.48 1.46 kVA Voltage distortion power 554.41 674.81 21.75 _ VA 86

Current THD 38.77 38.09 1.75 % Voltage THD 0.986 1.196 21.3 % Normalised Har. App. power 0.383 0.455 18.8 % Fundamental power factor 0.934 0.927 0.75 - Total power factor 0.870 0.866 0.45 -

6.3 MEASUREMENT 2 - AT THE PCC WITH PASSIVE FILTER AT THE LOAD

400 400 320 320 240 1111/1/111/01111111/NNA 240 160 160 80 MIAN 80 0 0

-80 -80 -160 41111F/11111111EILTall 160 -240 11//1111•=1111/111111 240

-320 320 400 400 U 0 01 0 02 0.03 0 04 0.05 -0.03 -0.02 -0.01 0 001 0.02

a) - Modelled result b) - Measured result Figure 6.3.1- Measurement 2 - PCC with passive filter at load

Figure 6.3.2 - Measurement 2 - Comparison of modelled and measured results

In this measurement, the star-connected passive filter of which the layout is given in (67), with 5th, 7th and 11th harmonic filter arms is installed at node 5 and is represented by compound admittance V50. 87 Table 6.3.1 - Measurement 2 - PCC with passive filter at load

Description Model Measured % Unit Difference Total node voltage 242.10 239.15 1.2 V Total node current 87.73 86.39 1.5 A Total active power 59.49 59.27 0.4 kW Total non-active power 22.81 18.116 20.57 kVA Total apparent power 63.71 61.98 2.7 kVA Fundamental apparent power 59.51 59.64 0.4 kVA Fundamental active power 59.52 59.27 0.4 kW Fundamental reactive power -11.51 -6.58 42.8 kVAr Harmonic apparent power 167.85 167.21 0.4 VA Current distortion power 19.57 16.89 13.7 kVA Voltage distortion power 520.07 590.24 13.5 VA Current THD 32.27 28.33 12.2 % Voltage THD 0.857 0.989 15.4 % Normalised Har. App. power 0.276 0.2804 1.6 % Fundamental power factor 0.982 0.994 1.22 - Total power factor 0.934 0.956 2.3 -

The results are shown in Figure 6.3.1 and Table 6.3.1. A comparison of Figure 6.3.1 with Figure 6.2.1 is not very encouraging and the current waveform at the PCC with the filter appears to be worse than without it. It appears that a certain degree of resonance is taking place, presumably with other imaginary immittance components in the rest of the circuit. However, an examination of the measurement results shows improvement. The total node voltage and total node current are basically the same as in measurement 1, but both the power factors have increased, increasing the total active power from 52 to 59 kW. Note also that the passive filter is only succeeding in reducing the current THD by 10% from 38.09% to 28.33% as reflected in Table 6.2.1 and Table 6.3.1 respectively.

This simple example illustrates another of the rather serious disadvantages of passive filters, especially when employed in low voltage applications. As mentioned in chapter 4 paragraph 4.4, the effectiveness of a parallel-installed tuned filter arm with the load of sinking the load-generated harmonk is governed by current division. The filter will only be capable of sinking the harmonic effectively if its impedance for that harmonic is low, compared to the impedance of the source- circuit. The quality factor Q of passive filters is deliberately kept low to ensure that it does not drift from the frequency for which it is designed through various reasons. This mandatory low Q 88 then works against the low impedance that the filter should really exhibit at resonance and makes it less than ideal for sinking the harmonic, as is clearly illustrated in this example.

The filter furnishes an added advantage in another respect though, namely that it furnishes a means of raising the fundamental power factor from 0.927 to 0.994 as reflected in the above- named tables. This is an advantage offered by the passive filter in topologies where line- commutated switching elements like thyristors are used. In contrast to forced commutated devices like GTOs or IGBTs of which the turn-off can be controlled, when thyristors are used in phase- control applications, large firing angles lead to low fundamental power factors.

6.4 MEASUREMENT 3 - SERIES INDUCTOR ON SUPPLY - SIDE OF FILTER

A unique possibility in a plant of the type planned here, and in which a large number of relatively small distortion loads are distributed over an extended reticulation topology, is to permit voltage distortion to the actual 6-pulse converters themselves, but to tap off "clean" voltage for all the auxiliaries that serve the power supplies at points just before the filters. This can be accomplished by installing series inductors in each phase before the filters. The disadvantage of the passive filter tuned circuits of exhibiting relatively large impedance at tuned resonance can then be overcome to a major extent. The additional reactance of the series inductor serves to increase the impedance to source relative to that of the filter for the given harmonics and forces current division through the filter.

In the setup under discussion, the series inductor is installed between nodes 4 and 5 in Figure 6.1.1. This inductance reduces the tendency of the circuit to exhibit parallel resonance between passive filters installed in adjacent loads in the network and at the same time forces a larger part of the load-generated harmonic current to flow through the filter, rather than through the source circuit, reducing the magnitude of the harmonic currents in the latter-mentioned part of the circuit.

It is significant to note that an adjustment had to be made to the firing angle and peak current magnitude of the converter in the model, to acquire a proper match with the measured resuls (when compared to the previous run). This was necessary to compensate for a change in the fundamental voltage magnitude and the relative phase angle brought about by the removal of the series inductor. In real life, this firing angle adjustment is automatically accomplished through the feedback circuit in the R2P2, to maintain a constant active power throughput.

If not taken care of, the presence of the additional series inductive reactance can lead to other problems. These inductors will introduce additional voltage regulation at the load, but if the fundamental load power factor can be maintained close to unity by optimal passive filter design, in a circuit like the one described here, this regulation can be maintained at a minimum. 89

400 400 320 320 240 wrsisrAimussir 240 IMINIONA111111■111/ 160 ■ 160 80 111111111117 11MIIIIIINT 80 4111A - 0 0 ) ■ -80 1111111111111 -80 17/11I411111 WW11 160 1 160 11111M1 240 67.1111:111 -240 111MA

320 -320 400 400 u 001 0 02 0 03 0 04 0.05 0.03 -0.02 -0.01 0 001 0.02

a) - Modelled result b) - Measured result Figure 6.4.1- Measurement 3 - At PCC with series inductor installed on supply-side of Passive Filter

Figure 6.4.2 Measurement 3 - Comparison of modelled and measured results

The simulated and measured line-neutral voltages and line currents at node 4 in Figure 6.1.1, shown in Figure 6.4.1, show a marked improvement on those shown in Figure 6.3.1. This improvement is also reflected in Table 6.4.1 in which current THD, now reduced to 9.63, as compared to that of 28.33% without the series inductor. Note also the improvement of the voltage THD.

Clearly the voltage 11-1D at the load itself will be too high to connect to any other apparatus, but with proper design the 6-pulse converter can be persuaded to work satisfactorily in spite of it. Normally the power electronic circuits of these converters will be content with distorted voltage, but the control circuits require pure waveforms to generate accurate timing for the firing circuits. These requirements should easily be met by the utilisation of phase-lock loops or properly designed filter circuits in the control circuitry. 90 Table 6.4.1 - Measurement 3 - PCC with series inductor on supply-side of passive filter

Description Model Measured % Unit Difference Total node voltage 242.1 239.75 0.97 V Total node current 73.89 74.27 0.5 A Total active power 53.05 53.11 0.1 kW Total non-active power 8.1 5.72 29 kVA Total apparent power 53.67 53.42 0.47 kVA Fundamental apparent power 53.26 53.17 0.17 kVA Fundamental active power 53.06 53.11 0.1 kW Fundamental reactive power -4.7 -2.62 38 kVAr Harmonic apparent power 19.88 36.21 45.1 VA Current distortion power 6.28 5.12 19.53 kVA Voltage distortion power 168.76 376.01 55 VA Current THD 11.7 9.63 17.7 % Voltage THD 0.32 0.707 . 121 % Normalised Har. App. power 0.0373 0.0681 54.8 % Fundamental power factor 0.996 0.998 0.2 - Total power factor 0.989 0.994 0.5 _ -

6.5 MEASUREMENT 4 - DYNAMIC COMPENSATOR AT THE LOAD

The dynamic compensator model is used in this simulation and compared with the measurement results on the actual dynamic compensator. The results are depicted in Figure 6.5.1 and Table 6.5.1. In this measurement, both the passive filter and the series inductor used in measurement number 3 has been removed.

The prominence of the switching distortion in the right-hand oscillogram of Figure 6.5.1 is a result of the relatively low absolute level of the current. When operating at full current capacity, the switching envelope almost disappears. Table 6.5.1 reflects the relevant modelled and measured parameters for measurement 4. Although the measured waveforms are shown to be sinusoidal, the presence of high-frequency switching distortion is immediately apparent. In spite of its presence, Table 6.5.1 shows how effective the dynamic compensator is in reducing every component other than the active power components to mere fractions of their former values. The individual components in Table 6.5.1 hardly need further discussion. 91

400 400 320 320 AL mg 240 240 ■ 160 A11111111 11/1 160 80 121111111111111111A 80 0 INIMINIA11111111111111,4111 0

1111111111111111111111RAMM -80 .1 timmatowiim 1 wit=mavatia 160 2 wassmoommi 240 32 -320 • 400 001 0 02 0 03 0 04 0.05 -0.03 -0.02 -0.01 0 0.01 0.02

a) - Modelled result b) - Measured result Figure 6.5.1 - Measurement 4 - At PCC with dynamic filter connected as shown in admittance-diagram

Figure 6.5.2 - Measurement 4 - Comparison of modelled and measured results

Incidentally, the apparent ripple in the simulated current waveform of Figure 6.5.1 is not representative of switching ripple, but is caused by the Gibbs phenomenon 68. In the case of the measured current and voltage waveforms depicted in Figure 6.5.1, however, the distortion is caused by the 5 kHz modulation frequency through switching in the dynamic compensator. This also explains the difference between the model and measured voltage THD of 0.93% and 1.24% respectively. The voltage ripple was actually bad enough, during the test, to cause mal-operation of the phase controller in the R2P2 phase controller and will have to be overcome when the compensators are installed in the future pilot plant. 92

Table 6.5.1 - Measurement 4 - Dynamic compensator at node 4

Description Model Measured % Unit Difference Total node voltage 241.98 239.73 0.9 V Total node current 72.79 73.48 1 A Total active power 52.77 52.8 0.1 kW Total non-active power 2.833 2.241 21 kVA Total apparent power 52.84 52.84 0 kVA Fundamental apparent power 52.77 52.81 0.1 kVA Fundamental active power 52.77 52.80 0.1 kW Fundamental reactive power -.0045 1.220 kVAr Harmonic apparent power 10.03 21.60 115 VA Current distortion power 2.533 1.892 25 kVA Voltage distortion power 209.14 602.61 188 VA Current THD 4.800 3.583 25 % Voltage THD 0.396 1.141 188 % Normalised Har. App. power 0.019 0.0409 115 % Fundamental power factor 1.000 0.9997 0 - Total power factor 0.9986 0.9991 0.1 -

6.6 SUMMARY

This chapter dealt with the verification of the harmonic superposition model against measurement resulting in a circuit consisting of an R 2P2 load, a passive harmonic filter, a series inductor and a dynamic compensator. The excellent agreement reached between modelled and measured results places little doubt on the suitability of the method and on the accuracy of the model. The latter can now be used with confidence to model the pilot plant circuit, consisting of 12 lasers, as will be reported in the following chapter. 93 7. MODELLING FOR ALTERNATIVE COMPENSATION SCHEMES AND POWER THEORIES

7.1 INTRODUCTION

The harmonic superposition model was verified in chapter 6 by comparing modelled results with those measured on a single-laser installation. All the sub-systems encountered in the plant are also present in the model. They include: passive filters, dynamic compensators, AC-DC converters, series inductors and other power reticulation components encountered in practice. In this chapter, the model is extended to incorporate all the nodes in the complete plant, as set out in Figure 1.4.1. The modelling programs used in this chapter is listed in Annexure B.

As pointed out in chapter 1, the plant has a complement of 12 lasers, of which 10 employ R 2P2 pulse power units and the other two lasers are equipped with R 3P2 units. The two types of pulse power supplies utilise identical power electronic topologies but the latter are capable of accommodating higher pulse energies. When the investigation was initiated, the R 2P2 units were all equipped with under-designed 300 1.tH chokes on the DC side of their AC/DC converters. The R3P2 units were newer, on the other hand, and employed 3.1 mH DC chokes. Inter alia, one of the investigations carried out in the modelling was to asses the relative cost of replacing the smaller inductors in the R2P2 units against that of the additional compensation that would be required.

Modelling results are obtained by sampling circuit data at different nodes in each of 7 different circuit configurations. A circuit configuration represents a given circuit topology with a specific set of parameters. One or more modelling runs are made with each configuration. A modelling run refers to a set of data that is generated for a given circuit configuration at a given set of nodes. Separate modelling runs for a given circuit configuration are made to extract node voltages and node-node currents for the nodes specified. A total of 22 modelling runs are made to extract the modelling results in this chapter for 7 circuit configurations. The different circuit topologies are listed in Table 7.1.1.

Table 7.1.1 - List of circuit configurations modelled

Number Circuit configuration 1 Ten R2P2 pulse power supply units are equipped with 300 [tH inductors and two R3P2 units with 3.2 mH. No compensating gear or passive filters are connected in the circuit. Modelling run 1-4. 2 Ten R2P2 pulse power supply units are equipped with 300 1111- and two R3P2 units with 3.2 mH inductors. Two dynamic compensators of 753 kVAr are installed; one each on the secondary sides of the two supply transformers. No compensators or passive filters at the loads. Modelling 94

runs and 6. 3 Ten R2P2 pulse power supply units are equipped with 300 i_tH inductors and two R3P2 units with 3.2 mH. Each laser load, as depicted in Figure 1.4.1, is separated from the PCC by a series inductor and is equipped with a passive filter on the load-side of the inductor. No compensators or filters at the transformers. Modelling run 7-10. 4 Ten R2P2 pulse power supply units are equipped with 300 pH inductors and two R3P2 units with 3.2 mH. Each laser load has a passive filter connected, but without a series inductor between the load and the PCC. No compensators or filters at the transformers. Modelling run 11-13. 5 The ten R2P2 pulse power supply units are equipped with 3001.tH inductors and the two R3P2 units with 3.2 mH. Each transformer secondary is connected to a suitably rated passive filter. No filters or compensators at the loads. Modelling run 14-16. 6 Both the R3P2 units and the R2P2 units are equipped with 3.2 mH chokes on their DC converter outputs. No dynamic compensators or passive filters at the loads or on the transformer secondaries. Modelling run 17 and 18. 7 The R3P2 units and the R2P2 units are all equipped with 3.2 mH chokes on their DC converter outputs. No compensating gear installed at the loads, but each transformer is equipped with a 150 kVAr dynamic compensator on its secondary. Modelling run 19 and 20.

7.2 THE MLISX4 PLANT NETWORK TOPOLOGY

Figure 7.2.1 depicts a simplified per-phase compound admittance power diagram for the network that supplies the lasers in the MLISX4 plant. This diagram is derived from the one-line distribution diagram shown in Figure 1.4.1 and also conforms to the associated circuit description given in chapter 1.

In accordance with the modelling function descriptions in Chapter 5, the source is modelled as a Norton-equivalent circuit, consisting of compound admittance Y, 01 and compound sinusoidal current source Ism in parallel. The main 11 kV feeder to the two plant transformers is represented by the L-section compound admittances Yc12 and Arm. Node 2 represents the 11 kV plant busbar from which the two transformers are modelled in L-equivalent form as series admittance YT23 with YT3o in parallel on its load-side. Similarly YT24 and Y40 represent the other transformer. The two dynamic filters are modelled in this diagram on the transformer secondary windings as IDO3 and 604. 95 In order to cater for tests using additional series inductors between the transformer secondary windings and the two busbar sections, two admittances Yp50 and Yp60 are inserted. The option of inserting the inductors here permit additional modelling exercises that will be described later.

Provision is also made at the busbars, at node 5 and node 6, for passive filters. Admittances Y50 and Y60 represent the filters that are in the form of the configuration depicted in Figure 4.4.1.

Y C57 7 YP714 CI

Load Single laser at A-EBP0002 I D03 V140 , 1:4 15 PASSIVE PG Load FILTER 1 111 V Double laser at C80 P15_0 v I 0 15_0 A-EBP0003 YT23 P35 Y Y T C59 0 P9_16 CO Y V T30 P50 Load "C90 P16_0 I 0 Double laser at T T 16_0 A-EBP0004

Load Single laser at VC10_0 7 YP17_0 VI A-EBP0005 Zr- PASSIVE

FILTER 2 Y C6 11 P11180 T24 P46 —I I Load ' C11 _0 YP18_0 180 Single laser at T40 P60 I I 0 A-EBP009 C6_12 YP12_196 I DO4 Load 1(C12_0 YP19_0 Vail I 0 V19_0 Double laser at T T A-EBP008 V C6_13 • sca 0 r+e) V C12 Load Doublelaser at V20_0 A-EBP0007 CIO C621 VP21 Load Single laser at SOURCE V 22_0 A-EBP0006

Figure 7.2.1 - MLISX4 power supply network to lasers

Each of the admittances Y57 to Y5_10 and Y6_11 to Y6.21 represents separate feeder cables from the plant substation to the laser load locations.

The configurations at each of the lasers are identical: That of the single laser load at position A- EBP0002 is identical to that of the other single laser loads. The AC/DC converter input is 96 represented by the current-source IV140. Compound admittance Yp140 represents the star- connected passive filter, also of the configuration depicted in Figure 4.4.1. Series connected admittance YP7-14 makes provision for a series inductance that serves basically the same purpose as the series inductances provided for in admittance Yp35. The harmonic current division between the passive filter at the load and the source can be adjusted experimentally by altering the value of this inductor as discussed above.

Supply points A-EBP000-2, -5, -9 and -6 model single lasers. The other supply points each has two lasers connected to it and handles twice the level of power that the single lasers do. Provision is made for this difference in the models of the loads, the series inductors and the filters.

7.3 MODELLING RUNS

In the following, each modelling run will be separately described. Several modelling runs will be undertaken for each circuit configuration to illustrate and substantiate the different conclusions. This information will then be used to select the options that are technically feasible. The costs involved in each of these options will lastly be estimated and compared to select the optimal one.

7.4 CIRCUIT CONFIGURATION 1

In the configuration modelled in this run, depicts one of the circuit configurations for the plant at the design stage when the R2P2 power supply units were still equipped with the smaller 300 1.tfl DC choke inductors. The laser amplifiers are equipped with these R 2P2 supplies and oscillator with R3P2 units constructed with 3.2 mH choke inductors from the outset. Double oscillators are located at nodes 16 and 20 and the amplifiers at the other nodes. Note that nodes 15, 16, 19 and 20 feed double laser power supplies and 14, 17, 18 and 22 feed single power supplies. No dynamic compensators or passive filters are installed in this configuration and the relevant admittances series inductors are inhibited.

7.4.1 Configuration 1 - Modelling Run 1 - Measuring at load, node 14.

Ten R 2P2 pulse power supply units are equipped with 300 ,uH inductors and two R3P2 units with 3.2 mH. No compensating gear or passive filters are connected in the circuit. Measurement is carried out at an R2P2 laser load.

YCS7 14 z --t P. 2 CABLE ' C70

YT23 -4 Load 15 [300 u1-1] Y Load 16 [3.2 n1}1] sO T30 CABLE F- Load 17 [MO uH] zcol 1=1 Y C12 C10 Load 18 [300 u1-1) YT24 Load 19 POO —1 1 Load 20 [3.2 mil] T40 Load 22 [300 di]

Figure 7.4.1 - Measuring at load, node 14

The line-line voltage at node 14 and the line current between nodes 7 and 14 are shown in the waveforms in Figure 7.4.2.

400 320 240 100 110111111111111111 Current 80 0 111111111.111•11111111 Voltage -80 — -180 -240 -320 1111K17,11111111111111111#11111 -400 0 0.01 0.02 0.03 0.04 0.05 t

Figure 7.4.2 - Line-line voltage and current at the R2P2 terminals

The current waveform in Figure 7.4.2 depicts the phase-controlled converter current and the waveform shown is typical of a converter with an undersized DC choke inductance in the converter. The double-peaked currents represent discontinuous DC-output current. As mentioned above, the older R2P2 pulse power supplies, of which one is modelled at note 14, was equipped

98 with 300 .tH chokes for historical reasons. One of the purposes of this modelling run is to show how undesirable this inadequate choke size is on the current THD.

The high voltage distortion of 6% THD, shown in Table 7.4.2 at the power supply terminals will lead to difficulties with the other apparatus there. The phase-controllers on the power supplies, for example, are not equipped with smoothing filters for their zero-crossing detectors and will not operate correctly. The effect that this discontinued current will have on the system as a whole is further augmented by the fact that all the phase-controlled converters have identical firing angles, because all the units have identical output needs and operate under steady-state conditions with identical load requirements. This is an undesirable condition, because there will be no statistical cancellation of harmonics (especially of the higher orders) as there would be in general networks where the firing angles are dependent on varying load conditions.

The normalised harmonic spectrum for the current and voltage at the relevant nodes are shown in Figure 7.4.3. and Figure 7.4.4. 4 100 3.6 00 32 so 2.8 70 2.4 60 2 50 1.6 40 12 30 0.8 20 0.4 10 n 0 n 0 0 5 10 15 20 0 5 10 15 20

Figure 7.4.3 - Normalised harmonic Figure 7.4.4 - Normalised harmonic current voltage spectrum for node 14 voltage spectrum for node 7-14 current

The high current harmonic content of 80% 5th and 65% 7th is to be expected and is highly undesirable. The low triplin current harmonics are indicative of negative sequence currents, but the measured levels are so low that they will be tolerable in this instance. An examination of the voltage waveform in Figure 7.4.2 explains the presence of the node 14 harmonics that lead to the observed THD of 6%.

Table 7.4.1 furnishes a summary of power calculations for the node 14 voltages and nodes 7-14 currents that represent power flow at that location. The table is configured in two parts. The first part depicts general values and the second part relates to power, calculated in accordance with the definitions of Budeanu, Czarnecki and the IEEE working group. The calculations are made 99 directly in the Mathcad model (Annexure B, pp B34-B44) and transferred to the word processor by copying.

Table 7.4.1 - Modelling run 1 - Measurements at the R2P2 terminals for Configuration 1

Voltage node: 14 Current THD: 106.5 % Start-end current node 7 - 14 Apparent 97.884 kVA RMS Node Voltage: 240.6 V Joint harmonic -11-j4060 VA Node-node RMS cur. 135.6 A 5th harmonic active +90.7 watt Voltage THD: 6.0 % 7th harmonic active -83.1 watt Fund. power factor: 0.793 11th harmonic active -16.3 watt Power factor: 0.542 13th harmonic active -4.6 watt

Budeanu Czarnecki IEEE Units Active 53.013 Active 53.048 Fundamental active 53.019 kW Reactive 36.694 Reactive 67.408 Fundamental reactive 40.755 kVAr Distortion 73.655 Scattered 0.967 Fundamental apparent 66.872 kVA Unbalanced 0.431 Non-fund. apparent 71.480 kVA Generated 2.268 Current distortion 71.235 kVA Forced app. 47.009 Voltage distortion 4.04 kVA Harmonic apparent 4305.8 VA Unbal. fund. app. 61.4 VA Normalised har. app. 6.44 %

As expected, the harmonic active power levels are extremely low 37. Only the 5th, 7th, 11th and 13th are above a few tens of watt and are tabulated. Because the superposition model carries out calculations accurate to a high number of significant decimal digits, these values are meaningful. In practice, measurement accuracy will be much lower and it is doubtful if any significant interpretation can be derived from them with regard to the origin of the distortion sources. Because the firing angles on all the converters are identical, it follows that the levels of these active harmonic powers at the measuring points will correspond. It has been shown by Swart and van Wyk, however, that that will no longer be the case when the converters operate at different firing angles. It is felt that the incorporation of the "Generated Power" components in the Czarnecki power definitions is superfluous and only serves to further complicate it, because these power components are not capable of indicating the direction of the distortion sources 37.

Note that all three theories agree on the level of the active power, almost to second decimal accuracy. The level of harmonic compensation required just for node 14 is 71 kVA in terms of the IEEE theory. The high current and voltage distortion brought about by the inadequate DC chokes installed in the R2P2 power supplies, will increase the level of the required compensation and will require to trade off the cost of increasing the DC choke size against the additional higher required levels of compensation.

100

7.4.2 Configuration 1 - Modelling Run 2 - Measuring at node 3.

Ten R2P2 pulse power supply units are equipped with 300 pH inductors and two R3P2 units with 3.2 mH. No compensating gear or passive filters are connected in the circuit. Measurement takes place at the secondary side of Stepdown Transformer 1.

gg C57 z P.2 T23 -I 1 Load 15 [300 ull]

so "T30 Load 16 [3.2 rali] CABLE cv a4 Load 17 [300

`Q. 1M Y C12 VCl$ E; 2 Load.18 [300

"T24 Load 19 POO uH) —I Load 20 [3.2 inH] T40 Load 22 [300 Oil

Figure 7.4.5 - Measuring at the secondary side of the transformer

The line-line voltage and the line current, at node 3 are shown in the waveforms in Figure 7.4.6.

2000 1600 1200 in•I•wAsi. 800 Current 400 111111111111111 0 cimmiliss. Nur

Voltage-4.5 —400 —800 1200 1600 —2000 0 0.01 0.02 0.03 0.04 0.05 t

Figure 7.4.6 - Line-line voltage and current at the secondary side of the transformer 101 In this instance the total current, as measured at node 3, is made up of the current both from R 2P2 and R3P2 units. The summation accounts for the new current waveform. The voltage waveform distortion shown here at the transformer is slightly improved with respect to that shown for node 14 in Figure 7.4.2. The voltage distortion has improved from 6% at node 14 to 4.87% as shown in Table 7.4.2. The relatively large voltage distortion here is to be expected on account of the transformer leakage impedance. This voltage distortion will now, unfortunately, also be propagated to other loads that are connected to this transformer secondary. A diverse range of problems can be expected as a result of it, ranging from the overheating of transformers and power factor capacitors, to odd phenomena such as the overheating of lighting circuit breaker trip coils by resonance in the fluorescent circuit ballast circuitsv iii.

The normalised harmonic spectrum at node 3 is shown in Figure 7.4.7 and Figure 7.4.8. 3 60 2.7 54 2.4 48 2.1 42 1.8 36 1.5 30 12 24 0.g 18 0.6 12 0.3 6 0 n fl 0 0 5 10 15 20 0 5 10 15 20

Figure 7.4.7 - Normalised harmonic Figure 7.4.8 - Normalised harmonic current voltage spectrum spectrum

A comparison of Figure 7.4.7 and Figure 7.4.8 with Figure 7.4.3 and Figure 7.4.4 shows a reduction both in the voltage and in the current distortion for the measurement at this point, compared to that at the laser. This reduction would be expected because the source impedance is lower and because the current THD is lower as a result of the contribution of the lower distortion R3P2 power supply units. The results for modelling run 2 are summarised in Table 7.4.2.

"in The author was involved in the solving of a problem in another department of the AEC where several large AC/DC phase-controlled converters are employed. Lighting distribution board circuit breakers were overheating and it was observed that the overheating originated in the magnetic trip coils and associated iron magnetic circuits as the result of high frequency losses. 102

Table 7.4.1 - Modelling run 2 - Circuit Configuration 1 - Measuring at the secondary side of the step-down transformer

Voltage node: 3 Current THD: 76.89 % Start-end current node 3 - 5 Apparent 481.49 kVA RMS node voltage: 241.46 V Joint harmonic -155- j 13563 VA Node-node RMS cur. 664.69 A 5th harmonic active -88.3 watt Voltage THD: 4.87 % 7th harmonic active -49.6 watt Fund. power factor: 0.845 11th harmonic active -11.9 watt Power factor: 0.669 13th harmonic active -1.8 watt Budeanu Czarnecki IEEE Units Active 321.87 Active 322.03 Fundamental active 322.03 kW Reactive 190.51 Reactive 204.073 Fundamental reactive 204.073 kVAr Distortion 303.22 Scattered 0 Fundamental apparent 381.246 kVA Unbalanced 2.613 Non-fund. apparent 294.09 kVA Generated 14.283 Current distortion 293.147 kVA Forced app. 293.735 Voltage distortion 18.576 kVA Harmonic apparent 14283 VA Unbalanced fund. app 390 VA Normalised har. app 3.75 %

Although the topology modelled here will not be used because of the large voltage distortion that it brings about on the LV circuits, even so, it will not produce unacceptable levels of voltage distortion on the supply authority terminals as seen in the following modelling run.

7.4.3 Configuration 1 - Modelling Run 3 - Measuring at node 2.

Ten R 2P2 pulse power supply units are equipped with 300 pH inductors and two R3P2 units with 3.2 mH. No compensating gear or passive filters are connected in the circuit. Measurement is carried out on the primary side of step-down transformer I.

103

Y C57 4' CABLE 2 YC70 7 YT23 V CI

—1 Load 15 [300 ail] 'V sO T30 — Load16 [3.2 inli] H- CAB Load 17 [3oo di]

C12 vC10 E4. Load 18 [300 uli] T24 Load 19 [300 41] H 1 Load 20 [3.2 rall] T40 Load 22 [300 u1-1]

Figure 7.4.9 — Measuring on the primary side of step-down transformer

The line-line voltage and the line current at the transformer primary, at node 2, are shown in the waveforms in Figure 7.4.10. For expediency, as in the other current and voltage graphs, a common voltage and current scale is used. Notice that the measured current value is multiplied by a factor 80, only to obtain a comparable numerical magnitude with that of the voltage.

4 1'10 8000 ,1 i A 6000 4000 Current -80 2000 0

Voltage —2000 —4000 —5000 —8000 —1.104 0 0.01 0.02 0.03 0.04 0.05

Figure 7.4.10 - Line-line voltage and current at the transformer primary

The same current waveform is measured on the primary of the step-down transformer as in its secondary as expected. An inspection of the voltage waveform shows, however, that voltage distortion is now much reduced. This is expected on account of the much lower source impedance and illustrates a very important principle, namely that transformer impedances always dominate

104 the source impedance in any distribution system. There is a practical reason for it, of course, namely that of confining the fault level to manageable and economic values. This same transformer impedance, unfortunately has a compounded effect of the voltage drop at frequencies above the fundamental'.

The normalised harmonic spectrum at primary of transformer at node 2 is shown in Figure 7.4.11 and Figure 7.4.12. In contrasting them to those measured at the load and on the transformer secondary, very little extra needs to be pointed out. 60 0.15 0.135 54 0.12 48 0.105 42 0.09 36 0.075 30 0.08 24 0.045 18 0.03 12 0.015 0 0 n n 0 15 20 5 10 0 5 10 15 20

Figure 7.4.11 - Normalised harmonic Figure 7.4.12 - Normalised harmonic current voltage spectrum spectrum

The results for modelling run 3 are summarised in Table 7.4.3.

Table 7.4.1 - Modelling run 3 - Measurements at the primary side of the step-down transformer

Voltage node: 2 Current THD: 71.838 % Start-end current node 1 - 2 Apparent 993.08 kVA RMS node voltage: 6445.1 V Joint harmonic -157-j 911 VA Node-node RMS cur. 51.36 A 5th harmonic active -90.7 watt Voltage THD: 0.165 % 7th harmonic active -50 watt Fund. power factor: 0.8108 11th harmonic active - 11.3 watt Power factor: 0.6583 13th harmonic active -1.7 watt

As remarked elsewhere in an earlier chapter, the leakage impedance of multi-winding power transformers is related to the frequency. The leakage impedance of the transformers for the harmonic frequencies is therefore higher at the higher the harmonic order and the voltage drops across the transformer are accentuated for the

higher frequencies. 105

Budeanu Czarnecki IEEE Units Active 653.78 Active 653.93 Fundamental active 653.93 kW Reactive 471.16 Reactive 472.07 Fundamental reactive 472.07 kVAr Distortion 580.35 Scattered 0 Fundamental apparent 806.52 kVA Unbalanced 5.226 Non-fund. apparent 579.42 kVA Generated 0.959 Current distortion 579.40 kVA Forced app. 579.40 Voltage distortion 1.334 kVA Harmonic apparent 958.56 VA Unbalanced fund. app 333.5 VA Normalised har. app 0.1188 %

The apparent power level observed, of almost 1MVA, on the transformer begins to approach its maximum rating. The low overall power factor of 0.65, compared to the fundamental power factor of 0.81, is brought about by the relatively high level of harmonic distortion. In practice, transformer capacities must be derated in the presence of harmonics, or overheating will result. If this configuration is used, it will not be able to be operated without effective compensation, or the transformer capacity will have to be increased.

The relatively low source impedance permits the high current distortion of 71.8% without causing unacceptable voltage distortion (0.17%) on the transformer primary. This voltage distortion falls well within the prescribed maximum by the utility of 6% THDv

It is informative to note that the real part of the joint harmonic power is only 157 watt and it is so low because of the relatively large, real equivalent source admittance component. In comparing the measurement results of the three theories in this case, it is seen that there is almost a complete correspondence in the numerical values of the active and reactive power components of the three power theories.

Table 7.4.3 contains both useful and useless information. Aside from the very conventional per- phase RMS current and voltage, the voltage and current THD values are very useful and immediately convey the status of the information at the measuring point. The voltage TI-1D of 0.165% tells one that the other consumers connected to the transformer primary will not be troubled. The current 11-ID of 71.838% is high, but as a result of the high equivalent source admittance, it can be easily tolerated. A low ambient voltage 11-ID confirms that other consumers draw sinusoidal currents and by the time that the distorted current from this plant is blended with the undistorted current of the other consumers, no one will notice. The discrepancy between the fundamental and the overall power factor is again a confirmation that there is considerable distortion in the local plant load. 106 The Budeanu distortion power of 580 kVA means nothing and may as well not be calculated. When compared to the level of the apparent power of 993 kVA, it merely means that there is distortion, but quantifying it in this way conveys very little, if any, useful information. As with the Budeanu figures, the Czarnecki quantities are also not very meaningful. Aside from the active and reactive power levels quoted, the others are not very significant. The level of generated power is of no significance, even though it shows that the plant is pumping active harmonic power in the direction of the source. If the firing angles of the converters change, the direction of this power can easily change as well as pointed out in the literature 37. The unbalance power gives an indication of the level of negative-sequence power and the "forced apparent power" is not useful at all because its author states that it is a term that has no physical significance 69.

The measurements by the IEEE Working Group convey very meaningful information: The fundamental apparent power (806 kVA) and the fundamental reactive power (472 kVAr) shows how large the fundamental power factor correction capacitors will have to be to overcome the effects of the relatively large firing angles used in the natural commutated phase-control converters. This reactive power can be corrected easily by means of static capacitors used in a passive filter configuration. The non-fundamental apparent power (579 kVA) and the (not shown) non-fundamental reactive power furnishes indications of the required dynamic compensator capacity when used for non-fundamental distortion correction alone. The current distortion power of 579 kVA is also of significance, telling what the level of current distortion is and the voltage distortion power conveys the same thing with respect to the voltage distortion. As set out in the respective theory, the current and voltage THD values can be directly calculated therefrom. These observations illustrate the merits of the latter theory, when compared with the former two.

7.4.4 Configuration 1 - Modelling Run 4 - Measuring at node 16.

Ten R2P2 pulse power supply units are equipped with 300 ,uH inductors and two R3P2 units with 3.2 mH. No compensating gear or passive filters are connected in the circuit. Measurement takes place on the AC input side of an R 3P2 power supply unit.

107

z Load 14 [3oo u}i] 0.2 Load 15 [300 uH] l v-T23 --1 —4 CABLE 'Y 501 T30 CABLE cq Load 17 [300 • 1."' Y C12 P.2 vC10 Load 18 [300 uH] Y T24 Load 19 [300 uI-1] —4 1 Load 20 [3.2 rnH]

T40 Load 22 [300 uH]

Figure 7.4.13 - Measuring on the input side of an R 3P2 power supply unit

Modelling run 4 also measures at the input to a laser power supply, just as modelling run 1 did, with the exception that node 16 supplies an R 3P2 unit, whereas modelling run 1 had to do with the input of a R2P2 unit. The line-line voltage and the line current at the power supply terminals, at node 16, are shown in the waveforms in Figure 7.4.14.

400 320 240 kimisswkemsom 160 Current 80 fill111111111111/A 0 mourrinumaram Voltage -80 M111111111111111111111111 -160 -240 -320 -400 0 0.01 0.02 0.03 0.04 0.05 t

Figure 7.4.14 - Line-line voltage and current at the R 3P2 terminals

The improvement in the current waveform is immediately apparent. This improvement is also evident in the reduced level of current TM) reflected in Table 7.4.4, in which the current THD is shown to have reduced to 38.94%, compared to that of the R 2P2 unit of 106.5%. Although there has been this improvement in the current THD, the voltage waveform is still shown to be severely distorted as shown in Figure 7.4.14. In spite of the better current THD, the voltage THD is still

108 5.3% as shown in Table 7.4.4, compared to 6% for that of the R2P2 case. This high voltage distortion is conducted to this point through the step-down transformer secondary and is the result of the distortion by the R2P2 units with their larger current distortion.

The normalised harmonic voltage and current spectrum at the R 3P2 terminals at node 16 are shown in Figure 7.4.15 and Figure 7.4.16.

4 40 3.6 38 32 32 2.8 28 2.4 24 2 20 1.6 16 1.2 12 0.8 8 0.4 4 0 0 n 5 10 15 20 0 5 10 15 20

Figure 7.4.15 - Normalised harmonic Figure 7.4.16 - Normalised harmonic current voltage spectrum spectrum at the R3P2 terminals

The information depicted in these graphs confirms the data in Table 7.4.4. What is significant in the current spectrum, is the relative high value of the 5th harmonic when compared to the others. The results for modelling run 4 are summarised in Table 7.4.4.

Table 7.4.1 - Modelling run 4 - Measurements at the R3P2 terminals

Voltage node: 16 Current THD: 38.94 % Start-end current node 9 — 16 Apparent 123.849 kVA RMS node voltage: 239.64 V Joint harmonic -1516-j 750 VA Node-node RMS cur. 172.27 A 5th harmonic active -1279 watt Voltage THD: 5.327 % 7th harmonic active -163 watt Fund. power factor: 0.937 11th harmonic active -32.1 watt Power factor: 0.859 13th harmonic active -0.8 watt 109

Budeanu Czarnecki IEEE Units Active 106.43 Active 106.60 Fundamental active 107.96 kW Reactive 39.55 Reactive 40.32 Fundamental reactive 40.298 kVAr Distortion 49.46 Scattered 1.33 Fundamental apparent 115.227 kVA Unbalanced 2.036 Non-fund. apparent 45.4 kVA Generated 2.30 Current distortion 44.87 kVA Forced app. 45.24 Voltage distortion 6.139 kVA Harmonic apparent 2390.5 VA Unbalanced fund. app 310 VA Normalised har. App 2.07 %

As in the previous cases, the Budeanu distortion power, and the Czarnecki scattered, unbalance, generated and forced apparent power figures are not of much use. The traditional THID values of current and voltage and the fundamental and overall power factor levels are of significance and convey a great deal of information. Note the improved fundamental power factor for the R 3P2, brought about by smaller firing angles in the R3P2 converters than in the R 2P2 ones, in which small firing angles lead to instability'.

Inspection of the measurement results at this node 16, supplying the R 3P2 unit with the larger choke, immediately shows that it produces improved results, compared to that of the R 2P2 units on the other nodes.

7.4.5 Summary of the measurements of circuit configuration 1.

The voltage distortion at both the R 2P2 terminals (6.2% THD) and the R3P2 terminals (5.4% TI-1D) is unacceptably high in terms of industry standards. The flattening of the crests of the voltage waveforms can clearly be seen on the graphs and illustrates Why the available DC output voltage of the converters reduces when the load current is increased. With this type of voltage

It is informative to calculate the average power input to the converter from the AC-side for the R 2P2 current and voltage waveforms as shown, for example, in Figure 7.4.2. Because of the relative shortness of the double current-spike, the average power is seen to increase as the firing angle is initially reduced from its maximum value. When the first current spike's peak passes into the first half of the positive voltage sinusoid, the average power reduces at first, until the second one also passes that latter point. When the automatic phase control transits through this region, advancement of the firing angle does not have the desired effect and the control becomes unstable. The same is also true for the R 3P2 waveforms with their improved current waveforms, but the effect is much less marked and does not lead to instability in these units. 110

distortion, other problems usually also begin to emerge and this type of operation is to be avoided in industrial installations.

The lower displacement power factor of AC/DC converters with discontinuous output current, compared to those with continuous output current, becomes evident when the graphs for the R 2P2 units are compared with the R3P2 ones. An examination of the operation of an AC/DC line- commutated converter shows that it becomes unstable at certain firing angles, and it is not possible to advance the firing angles sufficiently to obtain high fundamental power factor. The lack of this feature alone, is against the use of the smaller DC-choke in the power supplies.

ESKOM specifies the maximum voltage distortion at 6% in this instance'. They do not specify a limit for the current distortion. The voltage distortion at the primary side of the step-down transformers is therefore acceptable (0.17%), but the current distortion at 71.8% is not considered to be a factor here, because of the relatively low load level. The low power factor of 0.66 is indicative of an unnecessarily high maximum demand and consequently inflated power cost, that will have to be offset against the cost of displacement power factor correction.

Although the voltage distortion at the PCC will be acceptable to ESKOM, the excessive voltage distortion on equipment terminals in the plant will have to be dealt with. It is the cost of the necessary corrective measures to overcome the latter defect, that will have to be traded off against the cost of modification of the R 2P2 units and minimal harmonic compensation. The next modelling exercises will investigate those alternative configurations.

7.5 CIRCUIT CONFIGURATION 2

In circuit configuration 2 the conditions are still as in circuit configuration 1 but dynamic power filters are installed at nodes 3 and 4 which form the secondary of each of the two step-down transformers. The power supply configurations are identical to those in circuit configuration 1: The oscillator laser loads are still equipped with the newer R 3P2 units with 3.2 mH inductors and the amplifier lasers are equipped with the original R 2P2 supplies with 300 pH inductors. Double oscillators are located at nodes 16 and 20 and the amplifiers at the other nodes as before.

7.5.1 Configuration 2 - Modelling Run 5 - Measuring at the primary side of the step-down transformer, node 2.

Ten R 2P2 pulse power supply units are equipped with 300 pH inductors and two R3P2 units with 3.2 mH inductors. Dynamic power filters are installed at nodes 3 and 4, on the secondary side of each of the two step-down transformers. No passive or other filters are connected in the circuit. This measurement takes place on the primary side of transformer I. 111

CABLE 1 D0

—I I Load 15 [300 111-1]

— Load 16 C3.2 rain Ys01 CAB —I I- Load 17 [3oo uH]

ca YC1 2 1 501 , SOURCE VC10 z Load 18 [300 Load 19 POO uli] 1 Load 20 [.2 rnI-1] I Do4 Load 22 [300

Figure 7.5.1 - Measuring at the primary side of the transformers

The line-line voltage and the line current at the primary of the transformer at node 2, are shown in the waveforms in Figure 7.5.2.

4 1•10 8000 6000 4000 Current 100 2000 0

Voltage —2000 —4000 8000 8000 4 —1'10 0 0.01 0.02 0.03 0.04 0.05

Figure 7.5.2 - Line-line voltage and current primary of the transformer

Although there is still evidence of distortion, the current waveform is almost sinusoidal and the improvement brought about by the dynamic compensator becomes immediately apparent when comparing Figure 7.5.2 with Figure 7.4.2. The voltage, in this case is almost purely sinusoidal, as would be expected with such a relatively "clean" current waveform.

112 The normalised harmonic voltage and current spectrum at the primary of the step-dwon transformer at node 2 is shown in Figure 7.5.3 and Figure 7.5.4. 0.02 0.018 5.4 0.018 4.8 0.014 4.2 0.012 3.8 - 0.01 3 0.008 2.4 0.008 1.8 0.004 1.2 0.002 0.8 0 0 0 5 10 15 20 0 5 10 15 20

Figure 7.5.3 - Normalised harmonic Figure 7.5.4 - Normalised harmonic current voltage spectrum spectrum

The results for modelling run 5 are summarised in Table 7.5.1. Table 7.5.1 - Modelling run 5 - Measurements at the primary side of the step-down transformer

Voltage node: 2 Current THD: 8.49 % Start-end current node 1 - 2 Apparent 662.31 kVA RMS node voltage: 6451.6 V Joint harmonic -1.466-j 17.6 VA Node-node RMS cur. 34.22 A 5th harmonic active -0.1 watt Voltage THD: 0.035 % 7th harmonic active -0.1 watt Fund. power factor: 0.996 11th harmonic active -0.7 watt Power factor: 0.992 13th harmonic active -0.1 watt Budeanu Czarnecki IEEE Units Active 657.03 Active 657.91 Fundamental active 657.91 kW Reactive 61.759 Reactive 61.9 Fundamental reactive 61.77 kVAr Distortion 56.184 Scattered 0 Fundamental apparent 6599 kVA Unbalanced 3.72 Non-fund. apparent 56.147 kVA Generated .02 Current distortion 56.02 kVA Forced app. 56.03 Voltage distortion 0.228 kVA Harmonic apparent 19.4 VA Unbalanced fund. app 213.8 VA Normalised har. app 0.0029 %

The accompanying results show that the dynamic compensators improve the voltage THD to 0.04%, the current THD to 8.49% and the power factor to 0.99, which are all well within the

113 bounds of the supply-authority specifications. Note that the dynamic compensator is more effective at lower harmonic orders than at the higher ones. Note also that it is not capable of entirely compensating the current distortion, still at 7.8% THD.

7.5.2 Configuration 2 - Modelling Run 6 - Measuring at node 14.

Ten R2P2 pulse power supply units are equipped with 300 pH inductors and two R3P2 units with 3.2 mH ones. Dynamic power filters are installed at nodes 3 and 4, the secondary side of each of the two step-down transformers. No passive filters are connected in the circuit. Measuring is Carried out at the laser load on node 14.

IDO

—I I Load 15 [300 uH] — Load 16 [3.2 }50 1 --I 1— CABLE Load 17 [3oo uH)

C12 1 501 SOURCE V C10 Load 18 [300 uH] Load 19 [300 uH] H I Load 20 [3.2 mH] I DO4 Load 22 0300 uHl

Figure 7.5.5 - Measuring at the laser load

The line-line voltage and the line current at the power supply terminals, at node 14, are shown in the waveforms in Figure 7.5.6.

114

400

320 240 160 80 1111111M 0

-80 11111111111111111111111111111 -180 ■ -240 11111111111111

-320 muctiommis.

-400 0 0.01 0 02 0 03 0.04 0.05

Figure 7.5.6 - Line-line voltage and current at the R 2P2 terminals

Note that the current waveform is still that drawn by an R 2P2 unit with an inadequate DC choke, but that the voltage waveform is now greatly improved when compared with a similar measurement without compensation as depicted in Figure 1.3.1. This comparison shows that the previous distortion at the power supply input terminals are primarily caused by the harmonic voltage drop in the step-down transformer leakage impedance and not in the cable to the individual laser site.

The normalised harmonic voltage and current spectrum for R 2P2 load terminals at node 14 are shown in Figure 7.5.7 and Figure 7.5.8. The difference in the voltage spectrum is again immediately apparent with the 7th harmonic magnitude reducing from 3.6% of the fundamental to only 0.9%. As expected, the spectrum of the R 2P2 load itself will only change marginally, as a result of the change in the supply voltage to it. 1 100 0.9 90 0.8 80 0.7 70 0.6 60 0.5 50 0.4 40 0.3 30 0.2 20 0.1 10 0 0 H n n 0 5 10 15 20 0 5 10 15 20

Figure 7.5.7 - Normalised harmonic Figure 7.5.8 - Normalised harmonic current voltage spectrum spectrum at R2P2 terminals

The results for modelling run 6 are summarised in Table 7.5.2. 115 Table 7.5.1 - Modelling run 6 - Measurements at the R 2P2 terminals for Configuration 2

Voltage node: 14 Current THD: 106.52 % Start-end current node 7 - 14 Apparent 98.319 kVA RMS node voltage: 241.64 V Joint harmonic -304-j 1063. VA Node-node RMS cur. 135.63 A 5th harmonic active -170 watt Voltage THD: 2.1 % 7th harmonic active -109 watt Fund. power factor: 0.793 11th harmonic active -19 watt Power factor: 0.54 13th harmonic active -4 watt Budeanu Czarnecki IEEE Units Active 53.057 Active 53.288 Fundamental active 53.361 kW Reactive 39.918 Reactive 41.237 Fundamental reactive 40.981 kVAr Distortion 72.518 Scattered 0.375 Fundamental apparent 67.277 kVA Unbalanced 0.409 Non-fund. apparent 71.697 kVA Generated 1.334 Current distortion 71.666 kVA Forced app. 71.532 Voltage distortion 1.413 kVA Harmonic apparent 1506.01 VA Unbalanced fund. app 44.8 VA Normalised har. app 2.239 %

Although the voltage at the PCC has almost no distortion, the harmonics in the load current still cause voltage distortion (2.1%) at the R 2P2 terminals. The flattening of the voltage crests are still visible on the graph as a result of this and still renders this type of operation less desirable.

7.5.3 Summary for circuit configuration 2 measurements

The total non-active power can be calculated from the tabled information under measurement 3 above and is shown to be 748 kVA. The fundamental reactive power to the plant is shown to be 472 kVAr. There are basically two compensation configurations possible here. In the first, only dynamic compensators are installed at the two transformer secondaries to compensate for harmonic distortion as well as for displacement reactive power. In the other, dynamic compensators are used only for the harmonic compensation and static capacitors (in a simple series tuned circuit configuration) is used at the same nodes for displacement power factor correction.

In the first scheme, if dynamic compensators alone are used, the total cost will be in the region of $134 000,00 for the total of 743 kVA. Basis for calculation of dynamic compensation is: $0.18/VA. 116 If dynamic compensators are used only to compensate for the harmonic distortion of 579 kVA, as in the second scheme, the cost of these compensators will be in the order of $100 000,00 and an additional figure of about $20 000,00 will be required for passive (static) power factor correction, again bringing the total to $120 000,00. Basis for calculation of fundamental power factor correction: $0.042NA.

The second scheme is technically less satisfactory than the first, because it introduces all kinds of possibilities for parallel resonance between filters and requires more complex control on the part of the dynamic filters so that it should rather be discarded when compared to the first. The high level of harmonic distortion is the main reason for the high cost. As shown later, this high value of harmonic distortion power can be solved more economically by improving the R2P2 input current characteristics by also increasing their DC choke values to 3.2 mH.

Failing other solutions, the installation of expensive dynamic compensators at the secondary sides of the two step-down transformers will solve the problem and will yield acceptable levels of voltage distortion at the power supply terminals where other sensitive equipment is also connected. The relatively high cost of the compensators makes this solution less attractive, however.

7.6 CIRCUIT CONFIGURATION 3

The same pulse power supply configurations are used as in options 1 and 2. The oscillator loads are equipped with the newer R3P2 units, already equipped with 3.2 mH inductors. Amplifiers are equipped with the original R2P2 supplies with 300 RH inductors. Double oscillators are located at nodes 16 and 20 and the amplifiers at the other nodes. A passive harmonic filter is installed at the terminals of each pulse power supply. An inductor is placed in the supply line on the supply side of each passive filter to de-couple the different filters and the source for parallel resonance and to improve the "clean" terminal voltage THD on the basis explained earlier. The passive filters are modelled in accordance with those verified earlier. The examination is begun with the measurements at the load terminals and the current into the load, which is in parallel with the passive filter on node 14 where the previous measurements were also made.

7.6.1 Configuration 3 - Modelling Run 7 - Measuring at node 14.

Ten R2P2 pulse power supply units are equipped with 300 ,uH inductors and two R3P2 units with 3.2 mH. The dynamic compensators are not connected and a passive harmonic filter with a series inductor is installed at the terminals of each R3P2 and R2P2 pulse power supply. Measurement is taken at the same R 2P2 load as before for comparison purposes. 117

Load 15 [300 uli]

V sO — Load 16 [3.2 CABLE Load 17 [300 uH] I—1 — 2 I sOl 1- SOURCE VC1i Load 18 [300 ull] Load 19 [300 uHJ Load 20 [3.2 ira] Load 22 [300 uki]

Figure 7.6.1 - Measuring at the laser load, node 14

The line-line voltage and the line current at the power supply terminals, at node 14, are shown in the waveforms in Figure 7.6.2.

400

320 240 160 Current •2 80 0 MI=IMUM1111111101111 Voltage -80 MR1111111111111111111EMI -160 -240 1111111111M111111111117AM -320

-400 0 0.01 0.02 0.03 0.04 0.05

Figure 7.6.2 - Line-line voltage and current at the R 2P2 terminals

The deterioration of the current and voltage waveforms is immediately apparent when compared with the much improved previous conditions in modelling run 6 in which dynamic compensators were installed on the transformer secondaries. There is, however, an improvement when compared with the uncompensated waveforms as depicted in Figure 7.4.2. 118

The normalised harmonic voltage and current spectrum for R 2P2 load terminals at node 14 are shown in Figure 7.6.3 and Figure 7.6.4. 4 30 3.0 27 32 24 2.8 21 2.4 18 2 15 1.6 12 12 0 0.8 0.4 • 3 0 rt 0 5 10 15 20 0 n n 0 5 10 15 20

Figure 7.6.3 - Normalised harmonic Figure 7.6.4 - Normalised harmonic current voltage spectrum spectrum at R2P2 terminals

Although there is a major improvement in both the current- and voltage harmonic magnitudes, the residual level of 3.2% 5th and 2.4% 7th voltage harmonics could be improved. Note that the 19th voltage harmonic has risen to almost 2% when compared to 0.6% in modelling run 6 and about 1.2% in modelling run 1 (Figure 7.4.3). This shows that there is a degree of resonance at the frequency represented by this harmonic order.

The results for modelling run 7 are summarised in Table 7.6.1.

Table 7.6.1 - Modelling run 7 - Measurements at the R 2P2 terminals for Configuration 3

Voltage node: 14 Current THD: 28.23 % Start-end current node 7 - 14 Apparent 55.814 kVA RMS node voltage: 236.9 V Joint harmonic -83-j 705 VA Node-node RMS cur. 78.5 A 5th harmonic active -55 watt Voltage THD: 5.3 % 7th harmonic active -23 watt Fund. power factor: 0.992 11th harmonic active -0.7 watt Power factor: 0.952 13th harmonic active -2 watt Budeanu Czarnecki IEEE Units Active 53.13 Active 53.075 Fundamental active 53.216 kW Reactive 6.009 Reactive 6.911 Fundamental reactive 6.714 kVAr Distortion 16 Scattered 0.881 Fundamental apparent 53.638 kVA Unbalanced 0.414 Non-fund. apparent 15.433 kVA Generated 0.758 Current distortion 15.141 kVA Forced app. 15.295 Voltage distortion 2.847 kVA Harmonic apparent 803.8 VA 119 Unbalanced fund. app 103.2 VA Normalised har. app 1.5

The current distortion is reduced to 28% TI-ID, but remaining current harmonics in the series inductor still induce a voltage distortion of 5.3% at the converter input terminals. If these terminals can only be used to supply the converters themselves and "clean" power can be taken at nodes such as node-7 before the inductors, this may bring about a workable proposition. The increase in certain harmonic voltage levels cautions one, however, that the filters are resonating and that this configuration may prove to be unstable in practice.

7.6.2 Configuration 3 - Modelling Run 8 - Measuring at node 7.

Ten R2P2 pulse power supply units are equipped with 300 pH inductors and two R3P2 units with 3.2 mH ones. A passive harmonic filter with a series inductor is installed at the terminals of each pulse power supply. Measuring is carried out at the input side of the series inductor to the laser at node 14.

Load 15 [300 uH]

Load 16 [3.2 mil]

Load 17 p00

Load 18 [300 uH] Load 19 0300 uH] Load 20 [3.2 mH] Load 22 [300

Figure 7.6.5 - Measuring at the input side of the series inductor to the laser load

The line-line voltage and the line current at at the R 2P2 terminals are shown in the waveforms in Figure 7.6.6. 120

320 240 hvisussrAMENNwi 160 li111111111111111111111MINIM Current .2 80 11E111111111111MAIMMISh

0 111111111111111/11111111•MM Voltage —80 111111111ffillEUMFM 160 1111111MMIIIIMPRIN

—240 —320

—400 0 0.01 0.02 0.03 0.04 0.05,-,

Figure 7.6.6 - Line-line voltage and current at the R2P2 terminals

The normalised harmonic voltage and current spectrum for the PCC at node 7 are shown in Figure 7.6.7 and Figure 7.6.8. 30 0.8 27 0.8 24 0.7 21 0.6 18 0.5 15 0.4 12 0.3

0.2 6 0.1 3 r1 f-1 0 ri n n 0 n 0 5 10 15 20 0 5 10 15 20

Figure 7.6.7 - Normalised harmonic Figure 7.6.8 - Normalised harmonic current voltage spectrum spectrum

The results for modelling run 8 are summarised in Table 7.6.2.

Table 7.6.1 - Modelling run 8 - Measurements at the at the R2P2 terminals configuration 3

Voltage node: 7 Current THD: 28.228 % Start-end current node 5 - 7 Apparent 56.947 kVA RMS node voltage: 241.75 V Joint harmonic -1.31-j 166.9 VA Node-node RMS cur. 78.52 A 5th harmonic active +2 watt Voltage THD: 1.214 % 7th harmonic active -3.7 watt Fund. power factor: 0.9899 11th harmonic active -0.1 watt Power factor: 0.953 13th harmonic active -0.4 watt 121

Budeanu Czarnecki IEEE Units Active 54.242 Active 54.211 Fundamental active 54.244 kW Reactive 7.621 Reactive 15.232 Fundamental reactive 7.788 kVAr Distortion 15.578 Scattered 0.274 Fundamental apparent 54.799 kVA Unbalanced 0.416 Non-fund. apparent 15.490 kVA Generated 0.07 Current distortion 15.469 kVA Forced app. 8.244 Voltage distortion 0.665 kVA Harmonic apparent 187.8 VA Unbalanced fund. app 56.7 VA Normalised har. app 0.343 %

Note that there is a definite improvement in the voltage THD in Table 7.6.2, when compared to the uncompensated case, depicted in Table 7.4.1, from 6% down to 1.214%. It is significant also to note that this value of THD is lower than in the case when dynamic compensators are installed on the transformer secondary sides as depicted in Table 7.5.1. When viewed in that light, the desired result can be achieved, namely that an acceptable level of harmonic distortion is achieved in the plant, at the lasers.

7.6.3 Configuration 3 - Modelling Run 9 - Measuring at node 3.

Ten R 2P2 pulse power supply units are equipped with 300 pH inductors and two R3P2 units with 3.2 mH. A passive harmonic filter with a series inductor is installed at the terminals of each pulse power supply. Measurement is carried out at the secondary side of step-down transformer I. 122

series L

CABLE Pas fil

V Load 15 [300 u1-1]

YsO — Load 16 [3.2 CABLE 1 1— Load 17 [300 uH] I-1— cs, 1501C C12 SOURCE ViC . _ Load 18 [300 uFl] 2 — Load 19 [300 u1-1] H — Load 20 [3.2 m1-1] Load 22 [300

Figure 7.6.9 - Measuring at the secondary side of step-down transformer 1

The line-line voltage and the line current at the PCC, at node 5, are shown in the waveforms in Figure 7.6.10.

1000 800 800 400 A Current 200 011111111IIMMINIINITA

0 IIIIIIIMIL11111111111 Voltage _200 11111MWM1113111111111 11111M5=111111M5M11 —400

—BOO

—800

—1000 0 0.01 0.02 0.03 0.04 0.05 t

Figure 7.6.10 - Line-line voltage and current at the PCC (node 5)

The normalised harmonic voltage and current spectrum for the R2P2 load terminals at the PCC are shown in Figure 7.6.11 and Figure 7.6.12. 123

0.6 16 0.54 13.5 0.48 12 0.42 10.5 0.36 0.3 7.6 0.24 8 0.18 4.5 012 3 0.06 1.5 n 0 n 0 0 5 10 15 20 0 5 10 15 20

Figure 7.6.11 - Normalised harmonic Figure 7.6.12 - Normalised harmonic current voltage spectrum spectrum at R2P2 terminals

The results for modelling run 9 are summarised in Table 7.6.3.

Table 7.6.1 - Modelling run 9 - Measurements at the PCC (node 5) for Configuration 3

Voltage node: 3 Current THD: 16.88 % Start-end current node 3 - 5 Apparent 334.310 kVA RMS node voltage: 242.60 V Joint harmonic -5.5-j 476.86 VA Node-node RMS cur. 459.35 A 5th harmonic active -3.8 watt Voltage THD: 0.95 % 7th harmonic active -1.2 watt Fund. power factor: 0.999 11th harmonic active 0 watt Power factor: 0.9858 13th harmonic active 0 watt Budeanu Czarnecki IEEE Units Active 329.56 Active 329.572 Fundamental active 329.572 kW Reactive 5.313 Reactive 5.789 Fundamental reactive 5.789 kVAr Distortion 55.864 Scattered 0 Fundamental apparent 329.62 kVA Unbalanced 2.666 Non-fund. apparent 55.784 kVA Generated 0.529 Current distortion 55.629 kVA Forced app. 55.718 Voltage distortion 3.131 kVA Harmonic apparent 528.484 VA Unbalanced fund. app 370 VA Normalised har. app 0.1603 %

The voltage THD at the transformer secondary is entirely acceptable at 0.95%, although the current THD stands at 16.8%. 124 7.6.4 Configuration 3 - Modelling Run 10 - Measuring at node 2.

Ten R 2P2 pulse power supply units are equipped with 300 itH inductors and two R3P2 units with 3.2 mH. A passive harmonic filter with a series inductor is installed at the terminals of each pulse power supply. Measurement is carried out on the primary side of step-down transformer 1.

Load 15 [300 0.1]

vs0 — Load 16 [3.2 m1-1] Load 17 [300 al] CABLE I sOl SOURCE Load 18 [300 u1-1] Load 19 [300 u1-1] H Load 20 [3.2 rali] Load 22 [300 uK

Figure 7.6.13 - Measuring at the primary side of step-down transformers

The line-line voltage and the line current at the transformer primary, at node 2, are shown in the waveforms in Figure 7.6.14.

4 1'10

8000 8000 4000

Current - 120 2000 0 Voltage -2000 -4000 6000 -8000 4 -1'10 0 0.01 0.02 0.03 0.04 0.05

Figure 7.6.14 - Line-line voltage and current at the R 2P2 terminals

125 The normalised harmonic voltage and current spectrum at node 2 are shown in Figure 7.6.15 and Figure 7.6.16. 0.03 15 0.027 13.5 0.024 12 1.11, 0.021 10.5 0.018 0.015 7.5 0.012 0.009 4.5 0.008 3 0.003 1.5 n 1-1 0 n n 11 n n 5 10 15 20 0 5 10 16 20

Figure 7.6.15 - Normalised harmonic Figure 7.6.16 - Normalised harmonic current voltage spectrum spectrum

The results for modelling run 10 are summarised in Table 7.6.4.

Table 7.6.1 - Modelling run 10 - Measurements at the transformer primary for configuration 3

Voltage node: 2 Current THD: 16.23 % Start-end current node 1 - 2 Apparent 681.886 kVA RMS node voltage: 6451 V Joint harmonic -5.57-j 31.9 VA Node-node RMS cur. 35.23 A 5th harmonic active -4 watt Voltage THD: 0.0321% 7th harmonic active -1.2 watt Fund. power factor: 0.994 11th harmonic active -0.1 watt Power factor: 0.981 13th harmonic active -0.1 watt 126

Budeanu Czarnecki IEEE Units Active 669.00 Active 669.009 Fundamental active 669.009 kW Reactive 73.723 Reactive 73.755 Fundamental reactive 73.755 kVAr Distortion 109.40 Scattered 0 Fundamental apparent 673.06 kVA Unbalanced 5.331 Non-fund. apparent 109.347 kVA Generated 0.035 Current distortion 109.217 kVA Forced app. 109.218 Voltage distortion 0.216 kVA Harmonic apparent 35 VA Unbalanced fund. app 310.8 VA Normalised har. app 0.0052 %

The voltage THD of 0.0321% is better than the requirements of the supply authority. This configuration also achieves a superior voltage TIM to that of 0.035% obtained by installing dynamic compensators at the transformer's secondaries as shown in Table 7.5.1. These differences are marginal, however, and are only quoted to show that passive filters at the load can yield superior results.

7.6.5 Summary of circuit configuration 3

The configuration at the load-ends with passive-tuned filters in parallel with the loads and series "de-coupling" inductors between the supply cable-ends and the loads introduce a new concept in harmonic compensation. The high voltage distortion on the converter sides of the series inductors can be tolerated, provided that all ancillary equipment is powered from the source-side of these inductors. It is true that these inductors will introduce additional voltage regulation, but that can also be negated by designing the tuned filter capacitance to bring the fundamental power factor of the loads, as seen by the inductors, to near unity, as is the case with this configuration.

The filters will not operate as effectively without the series inductors and it is noticed that the incoming current distortion can only be reduced at the expense of a worsening of the voltage distortion after the series inductors. The series inductors increase the impedance to source, and because the effectiveness of the filter, in reducing the current distortion from the source, depends on the relative admittances of the filters compared to that of the source, it succeeds in reducing the current distortion in the supply.

This concept is new and relatively untried. If the AC/DC converters will operate satisfactorily under high supply voltage distortion conditions this method can be implemented. It does mean, however, that the R2P2 units and the R3P2 units will have to be rewired and a considerable amount of alterations will be necessary for them to procure "clean" supply voltage for all their ancillaries 127 from the connection on the incoming side of the series inductors. Another drawback of this configuration is that the different filters may still tend to resonate with each other and with the source. Before this concept can be implemented, it will be necessary to first carry out frequency- sweep analysis for parallel resonance.

An estimated price per filter would be about $7 000,00, bringing the total to about $84 000,00. Basis for calculation of passive filter and series inductor: $0.07/VA.

7.7 CIRCUIT CONFIGURATION 4

Because of the relatively high cost of altering the power supply wiring, to connect only the AC/DC converters on the load-sides of the series inductors, the use of passive filters at the loads is investigated, but without the presence of the additional series inductors. In this modelling option, the series inductors are removed from the supply line, leaving only the filters in-circuit at the load-ends.

7.7.1 Configuration 4 - Modelling Run 11 - Measuring at node 14.

Ten R2P2 pulse power supply units are equip. ped with 300 ,i1H inductors and two R3P2 units with 3.2 mH. A passive harmonic filter is installed at the terminals of each pulse power supply, but the series inductors between the AC/DC converters and the PCCs are not installed. Measurement is carried out on the laser load itself at node 14 as before for ease of comparison. 128

CABLE

H I Load 15 [300 al]

V.1 — Load lb [3.2 s0 1 z —1 I- CABLE — Load 17 [300 uH] 1—i- YC12 1 ‘1 501 VS W SOURCE C z — Load 18 [300 u1-1] — Load 19 0300 uH]

— Load 20 [3.2 m1-1] — Load 22 [300

Figure 7.7.1 - Measuring on the laser load

The line-line voltage and the line current at the power supply terminals, at node 14, are shown in the waveforms in Figure 7.7.2.

400 320 240 100

Current 80 lisur1kulr4 0 Voltage —80 =MUMMER= — 180 —240 11111•WMIUM1111111 —320

—400 0 0.01 0.02 0 03 0.04 0.05

Figure 7.7.2 - Line - line voltage and current at the R2P2 terminals

A glance at the current waveform in Figure 7.7.2 immediately shows that the current waveform has deteriorated when the series inductors are removed.

129 The normalised harmonic spectrum for R2P2 load terminals at node 14 is shown in Figure 7.7.3 and Figure 7.7.4.

80 2 1.8 72 1.6 64 1.4 56 1.2 48 1 40 0.8 32 0.6 24 0.4 16 0.2 8

0 n 0 n 11 nn 0 5 10 15 20 0 5 10 15 20

Figure 7.7.3 - Normalised harmonic Figure 7.7.4 - Normalised harmonic current voltage spectrum spectrum at R2P2 terminals

The results for modelling run 11 are summarised in Table 7.7.1.

Table 7.7.1 - Modelling run 11 - Measurements at the R 2P2 terminals for Configuration 4

Voltage node: 14 Current THD: 85.16 % Start-end current node 7 - 14 Apparent 71.158 kVA RMS node voltage: 241.88 V Joint harmonic -321-j 1241 VA Node-node RMS cur. 98.06 A 5th harmonic active -155 watt Voltage TIAD: 2.98 % 7th harmonic active -158 watt Fund. power factor: 0.992 11th harmonic active -11 watt Power factor: 0.750 13th harmonic active -2.3 watt Budeanu Czarnecki IEEE Units Active 53.359 Active 53.632 Fundamental active 53.687 kW Reactive 5.823 Reactive 7.895 Fundamental reactive 7.064 kVAr Distortion 46.716 Scattered 0.508 Fundamental apparent 54.150 kVA Unbalanced 0.41 Non-fund. apparent 46.166 kVA Generated 1.322 Current distortion 46.116 kVA Forced app. 46.002 Voltage distortion 1.614 kVA Harmonic apparent 1374 VA Unbalanced fund. app 55.7 VA Normalised har. app 2.539 % 130 The observed current and voltage deterioration after removing the series inductor is confirmed in the voltage and current spectra in Figure 7.7.3 and Figure 7.7.4. The two respective values increase from 0.95% and 16.88% to 2.98% and 85.16%.

This exercise shows that there is a definite merit in installing the series inductors at the filters. The reason for the deterioration is that the filter impedances, especially when designed at 380 V level, are high. The reason for that is the relatively low Q that is inherent in high-current low- voltage inductors. These inductors have to employ cores and this results in higher losses. The losses can be reduced, but only by using more refined core materials and thinner laminations.

7.7.2 Configuration 4 - Modelling Run 12 - Measuring at node 3.

Ten R2P2 pulse power supply units are equipped with 300 ,uH inductors and two R3P2 units with 3.2 mH. A passive harmonic filter is installed at the terminals of each pulse power supply. Measuring is performed at the secondary side of transformer 1.

Load 15 [300 u.li] — Load 16 [3.2 ml-1] Load 17 poo

Load 12 [300 uli] Load 19 [300 uH] Load 20 [3.2 mH] Load 22 [300 uli]

Figure 7.7.5 - Measuring at the secondary side of transformer 1

The line-line voltage and the line current at the transformer secondary, at node 3, are shown in the waveforms in Figure 7.7.6.

131

1000

800 600 400 Current 200 0

Voltage -200 400 BOO 800

- 1000 0 0.01 0.02 0.03 0.04 0.05

Figure 7.7.6 - Line-line voltage and current at the transformer secondary node 3

The current waveform in Figure 7.7.6 is clearly not acceptable and is just a replica of that at each laser load.

The normalised harmonic voltage and current spectrum at node 3 is shown in Figure 7.7.7 and Figure 7.7.8. 1.5 40 1.35 38 12 32 1.05 28 0.9 24 0.75 20 0.6 16 0.45 12 0.3

0.15 4 1••■••■ 0 0 1-1 nn n fl 0 5 10 15 20 0 5 10 15 20

Figure 7.7.7 - Normalised harmonic Figure 7.7.8 - Normalised harmonic current voltage spectrum spectrum at R2P2 terminals

Note the relatively high 11th, 13th 17th and 21' harmonics in the voltage spectrum. These are most definitely the result of resonance in the system and will be confirmed if a frequency sweep is carried out. 132 The results for modelling run 12 are summarised in Table 7.7.2

Table 7.7.1 - Modelling run 12 - Measurements at the transformer secondary for Configuration 4

Voltage node: 3 Current 11-1D: 43.71 % Start-end current node 3 - 5 Apparent 355.546 kVA RMS node voltage: 242.67 V Joint harmonic -36-j 2974.7 VA Node-node RMS cur. 488.37 A 5th harmonic active -25.1 watt Voltage THD: 2.224 % 7th harmonic active -8.9 watt Fund. power factor: 1 11th harmonic active -0.5 watt Power factor: 0.916 13th harmonic active -0.5 watt Budeanu Czarnecki IEEE Units Active 325.66 Active 325.695 Fundamental active 325.695 kW Reactive -1.104 Reactive 1.870 Fundamental reactive 1.871 kVAr Distortion 142.69 Scattered 0 Fundamental apparent 325.698 kVA Unbalanced 2.634 Non-fund. apparent 142.59 kVA Generated 3.166 Current distortion 142.352 kVA Forced app. 142.538 Voltage distortion 7.243 kVA Harmonic apparent 3165 VA Unbalanced fund. app 363 VA Normalised har. app 0.972 %

There is a general improvement here with respect to that of the system without any filters as depicted in modelling run 2, but the presence of the higher harmonics and the potential instability that these filters can elicit are negative factors that mitigate against the adoption of this circuit configuration.

7.7.3 Configuration 4 - Modelling Run 13 - Measuring at the primary side of the step-down transformer, node 2.

Ten R 2P2 pulse power supply units are equipped with 300 pH inductors and two R3P2 units with 3.2 mH. A passive harmonic filter is installed at the terminals of each pulse power supply, but without the series inductors between the converters and the source. Measuring takes place at the primary side of step-down transformer 1.

133

CABLE

I I Load 15 [3OO 0-1] c4 Y sO — Load.16 [3.2 mi-1] I E- Load 17 [3OO 1211] --I Hy- 1CABLE I $01 SOURCE Load 18 [300 u1-1] Load 19 [300 W-1] H I Load 20 [3.2 mH] Laad 22 [3001111]

Figure 7.7.9 - Measuring at the primary side of the step-down transformers

The line-line voltage and the line current at node 2, are shown in the waveforms in Figure 7.7.10.

4 1'10

8000 IL

8000 6111111111111MIIIIIIA

4000 Current 120 2000 0 Voltage -2000 t. -4000 5000 aniummiviuml 8000 vir 4 lr -1'10 0 0.01 0.02 0.03 0.04 0.05

Figure 7.7.10 - Line-line voltage and current at the R 2P2 terminals

134 The distortion in the primary current is expected, as it basically carries through from the transformer secondary side.

The normalised harmonic voltage and current spectrum at the transformer primary at node 2 are shown in Figure 7.7.11 and Figure 7.7.12. 0.08 40 0.054 38 0.048 32 0.042 28 0.036 24 - 0.03 20 0.024 16 0.018 12 0.012 8 0.006 4 0 nn n II 0 0 6 10 15 20 0 5 10 15 20

Figure 7.7.11 - Normalised harmonic Figure 7.7.12 - Normalised harmonic current voltage spectrum spectrum

The results for modelling run 13 are summarised in Table 7.7.3. Table 7.7.1 - Modelling run 13 - Measurements at primary of transformer for Configuration 4

Voltage node: 2 Current THD: 42.2 % Start-end current node 1 - 2 Apparent 721.276 kVA RMS node voltage: 6451.6 V Joint harmonic -36.7-j 203 VA Node-node RMS cur. 37.27 A 5th harmonic active -25.8 watt Voltage THD: 0.0769% 7th harmonic active -8.9 watt Fund. power factor: 0.995 11th harmonic active -0.4 watt Power factor: 0.9167 13th harmonic active -0.4 watt Budeanu Czarnecki IEEE Units Active 661.21 Active 661.25 Fundamental active 661.25 kW Reactive 65.59 Reactive 65.791 Fundamental reactive 65.794 kVAr Distortion 280.60 Scattered 0 Fundamental apparent 664.513 kVA Unbalanced 5.267 Non-fund. apparent 280.466 kVA Generated 0.216 Current distortion 280.416 kVA

As mentioned previously, the step-down transformers are modelled as Y-Y, whereas configurations are used in

the practical installation. If the model also employed a 0-Y configuration, the shape of the current harmonic would change, because of the selective nature of harmonic cancellation in the latter winding configuration. 135

Forced app. 280.42 Voltage distortion 0.511 kVA Harmonic apparent 216 VA Unbalanced fund. app 0.305 VA Normalised har. app 0.0324 %

The low source impedance brings about a low distortion in the supply-authority terminal voltage here, of only 0.077%, in spite of a still relatively high current distortion of 42.2%.

7.7.4 Summary for the measurements in circuit configuration 4

This arrangement is generally ineffective and cannot even be considered as a practical alternative. The probleni is not only that the filters are generally incapable of reducing the voltage distortion at the load-side satisfactorily, but parallel resonance is shown to occur with the many filters that operate in close electrical proximity to each other. It is possible to carry out frequency scans and to ascertain at which frequencies parallel resonance would be most prone, but the variable nature future configurations, additions of lasers or the connecting of other loads will make this circuit configuration a risky one and it is not recommended for consideration as a solution to the problem at all.

7.8 CIRCUIT CONFIGURATION 5

7.8.1 Configuration 5 - Modelling Run 14 - Measuring at load, node 14.

Ten R2P2 pulse power supply units are equipped with 300 ,uH inductors and two R3P2 units with 3.2 mH ones. Passive filters are installed at nodes 3 and 4 which is the secondary of each of the two step-down transformers. No series inductors installed with the passive filters. No dynamic filters are connected in the circuit. Measurement is carried out at the R2P2 laser at node 14 as before.

136

Load 15 [300 0-1]

Vs01 Load 16 [3.2 mil] --i 1— Load 17 [300 uH] 1 CABLE 1I sOl SOURCE Load 18 [300111-1] Load 19 [300 u1-1] Load 20 [3.2 rnFl] Load 22 [300 uli] [

Figure 7.8.1 - Measuring at the load

The line-line voltage and the line current at the power supply terminals, at node 14, are shown in the waveforms in Figure 7.8.2.

400 320 ■ 240 itM1111 11111,111111111111M 160 Current 80 11111111111111111 0 1111111111111111111111111111

Voltage -80 111111111111111111111111111111 180 -240 11111111111111111 320 -400 0 0.01 0.02 0.03 0.04 0.05

Figure 7.8.2 - Line-line voltage and current at the R 2P2 terminals

Notice the very prominent dips in the voltage waveform, co - inciding with the current peaks.

The normalised harmonic voltage and current spectrum for the R2P2 load terminals at node 14 are shown in Figure 7.8.3 and Figure 7.8.4. 137

4 100 3.6 80 32 80 2.8 70 2.4 60 • 2 50 1.6 40 12 30 02 20 0.4 10 n n n 0 0 0 6 10 15 20 0 6 10 15 20

Figure 7.8.3 - Normalised harmonic Figure 7.8.4 - Normalised harmonic current voltage spectrum spectrum at R2P2 terminals

The results for modelling run 14 are summarised in Table 7.8.1.

Table 7.8.1 - Modelling run 14 - Measurements at the R 2P2 terminals for Configuration 5

Voltage node: 14 Current THD: 106.5 % Start-end current node 7 - 14 Apparent 98.46 kVA RMS node voltage: 241.98 V Joint harmonic -1019-j 3930 VA Node-node RMS cur. 135.6 A 5th harmonic active -327.8 watt Voltage THD: 5.82 % 7th harmonic active -497.5 watt Fund. power factor: 0.793 11th harmonic active -162.3 watt Power factor: 0.532 13th harmonic active -15.5 watt Budeanu Czarnecki IEEE Units Active 52.349 Active 53.306 Fundamental active 53.368 kW Reactive 37.037 Reactive 41.223 Fundamental reactive 40.968 kVAr Distortion 74.719 Scattered 0.366 Fundamental apparent 67.279 kVA Unbalanced 0.409 Non-fund. apparent 71.892 kVA Generated 4.116 Current distortion 71.663 kVA Forced app. 71.623 Voltage distortion 3.917 kVA Harmonic apparent 4173 VA Unbalanced fund. app 62 VA Normalised har. app 6.203 %

As far as the loads are concerned, the current THD is unchanged and the voltage THD on the load terminals is only marginally reduced from 6.05% to 5.82%. From this point of view, the configuration is completely ineffective for solving the problem. 138 7.8.2 Configuration 5 - Modelling Run 15 - Measuring at node 3.

Ten R 2P2 pulse power supply units are equipped with 300 pH inductors and two R3P2 units with 3.2 mH ones. Passive filters are installed at nodes 3 and 4, namely the secondary of each of the two step-down transformers. No dynamic filters are connected in the circuit and no series inductors are installed in conjunction with the passive filters. Measurements are carried out on the secondary of step-down transformer 1.

CABLE

Load 15 [300 u.1-1]

lfsO 1 — Load 16 [3.2 n}1]

—1 1--- Load 17 [300 uH]

CABLE 1 501 SOURCE Load 18 [300 AO] Load 19 [300 41] Load 20 [3.2 mH] Load 22 [300 u1.1]

Figure 7.8.5 - Measuring at the secondary of step-down transformer 1

The line-line voltage and the line current at the transformer secondary, at node 3, are shown in the waveforms in Figure 7.8.6.

2000 1800 1200 1411111111111111.111111111111VA 800 11.==111=1=11 Current 400 itamminsimmo 0 111111111111111.111111111111 Voltage .4.5 -100 -800 11111111111111111111111111111111 -1200 1111111MIUMMITIM -1800 -2000 0 0.01 0.02 0.03 0.04 0.05 t

Figure 7.8.6 - Line-line voltage and current at the R2P2 terminals

139 The current and voltage waveforms immediately show that this circuit configuration does not yield the desired results, not even at the points of installation of the filters. Clearly the filters are not performing as desired.

The normalised harmonic voltage and current spectrum at node 3 are shown in Figure 7.8.7 and Figure 7.8.8. 4 100 3.6 GO 3.2 80 2.8 70 2.4 BO • 2 50 1.6 40 1.2 30 0.8 20 0.4 10

0 n n 0 n 0 5 10 15 20 0 5 10 16 20

Figure 7.8.7 - Normalised harmonic Figure 7.8.8 - Normalised harmonic current voltage spectrum spectrum

The results for modelling run 15 are summarised in Table 7.8.2.

Table 7.8.1 - Modelling run 15 - Measurements at transformer secondary for Configuration 5

Voltage node: 3 Current THD: 91.27 % Start-end current node 3 - 5 Apparent 449.248 kVA RMS node voltage: 242.9 V Joint harmonic -163-j 13409 VA Node-node RMS cur. 616.6 A 5th harmonic active -105.5 watt Voltage THD: 4.60 % 7th harmonic active -47.6 watt Fund. power factor: 1 11th harmonic active -6.3 watt Power factor: 0.737 13th harmonic active -0.7 watt Budeanu Czarnecki IEEE Units Active 331.29 Active 331.457 Fundamental active 331.457 kW Reactive -12.469 Reactive 0.94 Fundamental reactive 0.94 kVAr Distortion 303.17 Scattered 0 Fundamental apparent 331.459 kVA Unbalanced 2.63 Non-fund. apparent 303.248 kVA Generated 13.91 Current distortion 302.53 kVA Forced app. 302.917 Voltage distortion 15.243 kVA Harmonic apparent 13912 VA Unbalanced fund. app 366 VA Normalised har. app 4.197 % 140 The 4.6% voltage 11-1D here is brought about by harmonic voltage drop in the transformer leakage impedance by the high Current THD. This current THD of 91%, compared to that with no filters in modelling results 2 of only 77%, is indicative of parallel resonance between the two filters. So, instead of reducing current distortion, this configuration will actually increase it.

7.8.3 Configuration 5 - Modelling Run 16 - Measuring at node 2.

Ten R2P2 pulse power supply units are equipped with 300 inductors and two R3P2 units with 3.2 mH ones. Passive filters are installed at nodes 3 and 4, namely the secondary of each of the two step-down transformers. No dynamic filters are connected in the circuit and the series inductors associated with the passive filters are also not installed. Measurements are carried out on the primary side of the step-down transformer 2.

-1 Load 15 [300 WI]

so 1 — Load 16 [3.2241] —I 1— — Load 17 [3oo u1-1]

1,01 CABLE ,, W SOURCE Load 18 [300 ul-r]

M." Load 19 [300 u1-1] Load 20 [3.2 111-1] Load 22 (300 u14] [

Figure 7.8.9 - Measuring at the primary side of the step-down transformers

The line-line voltage and the line current at the transformer primary, at node 2, are shown in the waveforms in Figure 7.8.10.

141

4 2'10 4 1.6'10 4 1.2 -10 8000 111111111•1111111•11

Current -120 4000 IVIMMTAIIMMIf

0 111111111111111111111111W

Voltage —4000 1111111/1111111111011/ —8000 1111111.1511111111111114111 4 -1.2'10 —1.6 . 104 4 —210 0 0.01 0 02 0.03 0.04 0.05

Figure 7.8.10 - Line-line voltage and current at the R2P2 terminals

The normalised harmonic voltage and current spectrum at node 2 are shown in Figure 7.8.11 and Figure 7.8.12.

0.15 80 0.135 72 0.12 84 0.105 58 0.09 48

- 0.075 40

0.06 32 0.045 24 0.03 18 0.015 a r-t 0 0 n 0 5 10 15 20 0 5 10 15 20

Figure 7.8.11 - Normalised harmonic Figure 7.8.12 - Normalised harmonic current voltage spectrum spectrum

The results for modelling run 16 are summarised in Table 7.8.3.

Table 7.8.1 - Modelling run 16 - Measurements at the primary of the transformer for Configuration 5

Voltage node: 2 Current THD: 88.27 % Start-end current node 1 - 2 Apparent 901.473 kVA RMS node voltage: 6451.6 V Joint harmonic -166.3-j 918.0 VA Node-node RMS cur. 46.58 A 5th harmonic active -108.5 watt Voltage THD: 0.160 % 7th harmonic active -47.5 watt Fund. power factor: 0.996 llth harmonic active -6 watt Power factor: 0.746 13th harmonic active -0.7 watt 142

Budeanu Czarnecki IEEE Units Active 672.62 Active 672.783 Fundamental active 672.784 .kW Reactive 63.198 Reactive 64.117 Fundamental reactive 64.117 kVAr Distortion 596.86 Scattered 0 Fundamental apparent 675.832 kVA Unbalanced 5.257 Non-fund. apparent 596.577 kVA Generated 0.956 Current distortion 596.552 kVA Forced app. 596.553 Voltage distortion 1.082 kVA Harmonic apparent 956 VA Unbalanced fund. app 307 VA Normalised har. app 0.142 %

The voltage THD has improved to 0.16% on the primary, compared to 4.60% on the secondary, as in modelling run 15. This improvement is only the result of the transformer leakage impedance, present in the former measurement, when compared to this latest one. Note that although the displacement power factor is 0.99 through the action of the passive filter capacitances, the overall power factor is very low, at 0.75 and is brought about by the high harmonic content in the current.

7.8.4 Summary for circuit configuration 5

Parallel resonance between passive filters is evident here and renders this type of installation impractical. Although the passive filter installations on the transformer secondary sides accomplish fundamental power factor correction, their usefulness for harmonic compensation is negligible. In fact, they increase the harmonic content in the individual transformer secondaries because of parallel resonance. This option does not provide a technically feasible solution and need not be investigated further.

7.9 CIRCUIT CONFIGURATION 6

Both the R3P2 units and the R2P2 units are now equipped with 3.2 mH chokes on their DC converter outputs (no dynamic compensators or passive filters are connected at the loads or on the transformer secondaries).

7.9.1 Configuration 6 - Modelling Run 17 - Measuring at node 16.

All the pulse power supply units are equipped with 3.2 mH inductors. No dynamic compensators or passive filters are connected in the circuit. Measurement at a double-laser load. (Both oscillator and amplifier laser power supplies now draw identical currents.) 143

Load 14 C3.2 mll]

Load 15 [3.2 mI-1]

Load 17 [3.2 m1-1] CABLE I sin SOURCE Load 18 [3.2 nai] Load 19 [3.2 mEl] Load 20 [3.2 in.H]

r— Load 22 [3.2 m.H]

Figure 7.9.1 - Measuring at a double-laser load

The line-line voltage and the line current at the power supply AC-input terminals, at node 16, are shown in the waveforms in. Figure 7.9.2.

400 320 240 160 Current 80 1111117111111/11 0 milmorinnoimi Voltage -go 11111111111111M11111111111 -160 111111E1/11111 -240 -320 111111111p11111111111W11111 -400 0 0.01 0.02 0.03 0.04 0.05

Figure 7.9.2 - Line-line voltage and current at the R 2P2 terminals

The normalised harmonic voltage and current spectrum for the R2P2 load terminals at node 16 are shown in Figure 7.9.3and Figure 7.9.4. 144

3 40 2.7 36 2.4 32 2.1 28 1.8 24 1.5 20 1.2 16 0.0 12 0.8 8 0.3 4 n 0 n n 0 n 0 5 10 15 20 0 5 10 15 20

Figure 7.9.3 - Normalised harmonic Figure 7.9.4 - Normalised harmonic current voltage spectrum spectrum at R2P2 terminals

The results for modelling run 17 are summarised in Table 7.9.1.

Table 7.9.1 - Modelling run 17 - Measurements at the R3P2 terminals for Configuration 6

Voltage node: 16 Current THD: 38.72 % Start-end current node 9 - 16 Apparent 124.065 kVA RMS node voltage: 239.99 V Joint harmonic -137-j 1109 VA Node-node RMS cur. 172.32 A 5th harmonic active -125.2 watt Voltage THD: 2.7 % 7th harmonic active -2.5 watt Fund. power factor: 0.937 11th harmonic active -0.6 watt Power factor: 0.872 13th harmonic active -0.1 watt Budeanu Czarnecki IEEE Units Active 108.20 Active 108.34 Fundamental active 108.34 kW Reactive 39.313 Reactive 40.422 Fundamental reactive 40.422 kVAr Distortion 46.25 Scattered 0 Fundamental apparent 115.635 kVA Unbalanced 2.031 Non-fund. apparent 44.953 kVA Generated 1.205 Current distortion 44.783 kVA Forced app. 44.89 Voltage distortion 3.112 kVA Harmonic apparent 1205 VA Unbalanced fund. app 345 VA Normalised har. app 1.042 %

The reduction in the harmonic content in the converter line currents brings the THD down from 106%, as shown in modelling run 1 to 38.72% here. This reduction also brings about a reduction in the voltage THD from 6.1% to 2.7%, that can almost be tolerated as it stands without further need for compensation. Note also the improvement in both the displacement and in the overall power factor of the loads, for the reasons stated earlier. 145 7.9.2 Configuration 6 - Modelling Run 18 - Measuring at node 3.

All the pulse power supply units are equipped with 3.2 mH. No compensating gear or passive filters are connected in the circuit. Measurement is carried out on the secondary side of step-down transformer 1.

-- Load 14 [3.2 min

Load 15 [3.2 rrit-i]

I CABLE

. s0 1

--I 1-- IT — Load 17 [3.2 mH] CABLE 1 501

SOURCE z `.4 — Load 18 [3.2 rail — Load 19 [3.2 mH] H — Load 20 [3.2 mH] — Load 22 [3.2 mH]

Figure 7.9.5 - Measuring at the secondary side of step-down transformer 1

The line - line voltage and the line current at the transformer secondary, at node 3, are shown in the waveforms in Figure 7.9.6.

1000 800 BOO ill Att 400 113111M1111111111A11111=1/; Current 200 MINIII11111111Nri 0 1111111111111111111111111111111114

Voltage -2 —200 11111111111111M1111111 400 IIIIIRIM1111111111111114 600

—800

—1000 0 0.01 0.02 0.03 0.04 0.05 t

Figure 7.9.6 - Line-line voltage and current at the transformer secondary for modelling run 18 146 The normalised harmonic voltage and current spectrum at node 3 are shown in Figure 7.9.7 and Figure 7.9.8. 2 40 1.8 36 1.6 32 1.4 28 12 24 20 0.8 16 0.6 12 0.4 8 0.2 4 F." n n 0 0 it n 0 a 10 15 20 0 5 10 16 20

Figure 7.9.7 - Normalised harmonic Figure 7.9.8 - Normalised harmonic current voltage spectrum spectrum

The results for modelling run 18 are summarised in Table 7.9.2. Table 7.9.1 - Modelling run 18 - Measurements at the secondary of the transformer for Configuration 6

Voltage node: 3 Current TI-ID: 38.7 % Start-end current node 3 - 5 Apparent 368.715 kVA RMS node voltage: 241.82 V Joint harmonic -32-j 2456 VA Node-node RMS cur. 508.24 A 5th harmonic active -29.1 watt Voltage THD: 2 % 7th harmonic active -0.6 watt Fund. power factor: 0.937 11th harmonic active -1.4 watt Power factor: 0.873 13th harmonic active 0 watt Budeanu Czarnecki IEEE Units Active 321.92 Active 322.024 Fundamental active 322.024 kW Reactive 117.72 Reactive 120.179 Fundamental reactive 120.179 kVAr Distortion 135.7 Scattered 0 Fundamental apparent 343.718 kVA Unbalanced 5.945 Non-fund. apparent 133.447 kVA Generated 2.632 Current distortion 133.116 kVA Forced app. 133.289 Voltage distortion 6.796 kVA Harmonic apparent 2632 VA Unbalanced fund. app 843 VA Normalised har. app 0.766 %

The merits of this configuration hardly require pointing out. It is interesting to note that the THD of the collective current from several loads is almost identical to that of the individual loads. In this modelling exercise, all the firing angles in the different loads are identical. In practice they 147 will differ individually and cancellation of harmonics can be expected, which will further improve the THD of the secondary current here. In the present case, however, operation will be completely steady state and because all the power supply units are identical, the converter firing angles are expected to be identical. The further fact that the cable impedances to the different power supply units will be almost identical will not help to bring about differences that will call for different firing angles in the converters.

It would be possible to artificially induce different converter firing angles by, for example, changing the pulse transformer winding ratios in different units, but the complications that that will bring about in the selecting of spare parts, for one, will not justify the potential gain that such an arrangement may have.

7.9.3 Summary for circuit configuration 6

The reduction in the current THD by increasing the DC-choke inductances in the R 2P2 units is observed to be large, reducing it from a former 106% to only 39% without any compensation.

The displacement power factor is observed to be 0.94, which is also significantly better than the 0.79 observed for the R2P2 units when equipped with the small chokes. As discussed previously, it is possible to operate AC/DC converters at such high fundamental power factors only when equipped with chokes furnishing them with continuous DC output current. This improvement will require a lower capacity for fundamental power factor correction. The required harmonic compensation power level is now only 134 kVA (IEE non-fundamental apparent power) at each transformer secondary, when compared to that for modelling run 2 of 294 kVA. This reduction is significant and presents an option worth considering.

Considering the cost of dynamic compensation to be $0.18/VA, at 134x2 = 268 kVA, the capital cost of dynamic compensators will come to $50 000,00. The additional cost of converting 10 pulse power supply units to 3.2 mH chokes from 300 pH chokes will be $26 000,00. The total cost is therefore $76 000,00. When contrasted with the cost of dynamic compensator installation in accordance with circuit configuration 2, in which the total cost is estimated at $134 000,00, the first option takes precedence in the line-up for a solution.

7.10 CIRCUIT CONFIGURATION 7

The same pulse power supply configurations are used as in circuit configuration 6. The oscillator loads are driven by R2P2 units, equipped with 3.2 mH inductors. Amplifiers are driven by the original R2P2 supplies and their 300 [tH inductors are replaced with 3.2 mH units as well. Double oscillators are located at nodes 16 and 20 and the amplifiers at the other nodes. In this option, only dynamic compensators are installed in parallel with the secondary terminals of the two step- down transformers and no passive filters are installed anywhere in the system. 148 7.10.1 Configuration 7 - Modelling Run 19 - Measuring at node 2.

All the pulse power supply units are equipped with 3.2 mH DC choke inductors. Dynamic power filters are installed at the secondary of each of the two building transformers at nodes 3 and 4. No passive filters are connected in the circuit. This measurement is made at the primary side of step-down transformer I.

Load 14 [3.2 mi-1]

Load 15 [3.2 mH] I DO

Vso Hto 14 Load 17 [3.2 rril-1] CABLE 1 501 , / SOURCE W Load 18 [3.2 /TIN Load 19 [3.2 m1-1] H I Load 20 [3.2 mH] I DO4 Load 22 [3.2 mH] [

Figure 7.10.1 - Measuring at the primary side of the step-down transformers

The line-line voltage and the line current at the transformer primary, at node 2, are shown in the waveforms in Figure 7.10.2.

4 2'10 1.6.104 4k 4 1.2 -10 8000 Current -240 4000 0 IM=EfinaillEM Voltage -2 -4000 1111111111111111111111 -8000

4 -1.810 -2'10 WW1 0 0.01 0.02 0.03 0.04 0.05 t

Figure 7.10.2 - Line-line voltage and current at the transformer primary option 19

149 The normalised harmonic voltage and current spectrum at node 2 are shown in Figure 7.10.3 and Figure 7.10.4. 0.008 2 0.0072 1.8 0.0084 1.8 0.0056 1.4 0.0048 1.2 0.004 1 0.0032 0.0024 0.8 0.0015 0.8 0.0008 0.4 0 n 0.2 0 6 10 15 20 0 rgn Ii 0 5 10 15 20

Figure 7.10.3 - Normalised harmonic Figure 7.10.4 - Normalised harmonic current voltage spectrum spectrum

The results for modelling run 19 are summarised in Table 7.10.1. Table 7.10.1 - Modelling run 19 - Measurements at the transformer primary for configuration 7

Voltage node: 2 Current THD: 3.961 % Start-end current node 1 - 2 Apparent 659.53 kVA RMS node voltage: 6451.7 V Joint harmonic -0.32-j 5.1 VA Node-node RMS cur. 34.08 A 5th harmonic active 0 watt Voltage THD: 0.021 % 7th harmonic active 0 watt Fund. power factor: 0.996 11th harmonic active 0 watt Power factor: 0.995 13th harmonic active 0 watt Budeanu Czarnecki IEEE Units Active 656.06 Active 656.058 Fundamental active 656.058 kW Reactive 61.83 Reactive 61.833 Fundamental reactive 61.833 kVAr Distortion 27.45 Scattered 0 Fundamental apparent 61.897 kVA Unbalanced 8.439 Non-fund. apparent 27.431 kVA Generated 0.005 Current distortion 26.101 kVA Forced app. 26.101 Voltage distortion 0.138 kVA Harmonic apparent 6 VA Unbalanced fund. app 487 VA Normalised har. app 0.0008 % 150 As shown in Table 7.9.2, the IEEE current distortion is only 133 kVA, compared to the Czarnecki Reactive power of 120 kVA. In terms of the definition of the former parameter, it would be safe to say that the order of the harmonic compensating capacity needed is 150 kVA. On the strength of this, two 150 kVAr units were ordered and commissioned and tested under minimal load conditions.

This condition compares very favourably with the operation of the system in which the amplifier power supplies, driven by R2P2 units with 300 11F1 DC chokes as depicted in Table 7.4.3. In this earlier case the IEEE current distortion power and non-fundamental apparent power were both close to 579 kVA per transformer. This is a large difference when compared to the equivalent figure of 133 kVA per transformer in Table 7.9.2 in which all the power supplies are equipped with 3.2 mH DC choke inductors.

The dynamic compensators are capable of virtually removing 5th and 7th current harmonics and reducing higher order harmonics. This enables the current THD in the secondaries of the transformers to be reduced to the values shown, which are more than asked for. In practice, the initially low values of distortion may even be tolerated with reduced dynamic compensator capacity, or the compensators can be adjusted to compensate only the harmonics without correcting the fundamental power factor as they are seen to do in Table 1.2.1.

7.10.2 Configuration 7 - Modelling Run 20 - Measuring at load, node 16.

All the pulse power supply units are equipped with 3.2 mH DC choke inductors. Dynamic power filters are installed at the secondary of each of the two step-down transformers at nodes 3 and 4. No passive filters are connected in the circuit. Measurement is carried out at one of the double lasers directly on the converter- input terminals. 151

— Load 14 [3.2 m11]

— Load 15 [3.2 mEl]i, 1 D0 —1 I CABLE of W 50 1 z

—I I-- — Load 17 [3.2 mI-1] CABLE 1 501 , SOURCE `z1 Load 18 [3.2 rnH]

- 2 Load 19 [3.2 inFl]

Load 20 [3.2 m1-1]

I DO4 Load 22 [3.2 rt&I] [

Figure 7.10.5 - Measuring at load

The line-line voltage and the line current at the power supply terminals, at node 16, are shown in the waveforms in Figure 7.10.6.

1000

800 600 400 tiM111111FAIRINI/Ndi Current .2 200 MILIIIIIIIIMETA1111111111111 0 W1111111/11111111111M11 Voltage •2 _200 MIIMMEMLUINIM IIIILWAVM=W1i111 -400 600 11111111p1111111111FAII 800

- 1000 0 0.01 0.02 0.03 0.04 0.05

Figure 7.10.6 - Line-line voltage and current at the R2P2 terminals

The normalised harmonic voltage and current spectrum for R2P2 load terminals at node 16 are shown in Figure 7.10.7 and Figure 7.10.8. 152

0.8 40 0.72 38 0.84 32 0.58 28 0.48 24 • 0.4 20 0.32 18 0.24 12 0.16 8 0.08 4

0 ri n [1 0 0 5 10 15 20 0 5 10 15 20

Figure 7.10.7 - Normalised harmonic Figure 7.10.8 - Normalised harmonic current voltage spectrum spectrum at R2P2 terminals

The results for modelling run 20 are summarised in Table 7.10.2.

Table 7.10.1 - Modelling run 20 - Measurements at the R 2P2 terminals for configuration 7

Voltage node: 16 Current THD: 38.72 % Start-end current node 9 - 16 Apparent 124.472 kVA RMS node voltage: 240.8 V Joint harmonic -127-j 351 VA Node-node RMS cur. 172.3 A 5th harmonic active -115.5 watt Voltage THD: 1.3 % 7th harmonic active -2.3 watt Fund. power factor: 0.937 11th harmonic active -0.1 watt Power factor: 0.873 13th harmonic active -5.6 watt Budeanu Czarnecki IEEE Units

Active 108.61 Active 108.734 Fundamental active 108.734 kW

Reactive 40.191 Reactive 40.543 Fundamental reactive 40.543 kVAr

Distortion 45.634 Scattered 0 Fundamental apparent 116.047 kVA

Unbalanced 2.035 Non-fund. apparent 45.017 kVA

Generated 0.572 Current distortion 44.943 kVA

Forced app. 44.967 Voltage distortion 1.476 kVA Harmonic apparent 571 VA Unbalanced fund. app 295 VA

Normalised har. app 0.493

The voltage THD is reduced to 1.3% and is hardly visible in the voltage trace of Figure 7.10.6. The vast improvement in the power quality at the load is also depicted by the IEEE current and voltage distortion values. 153

7.10.3 Summary for circuit configuration 7

The excellent performance of the dynamic compensators cannot be doubted from the results of Table 7.10.2. In all the modelling runs without compensation, the 5th and 7th harmonic components bring about the distortion and are mainly the harmonics that must be catered for in compensation. Note that the fundamental reactive power is 120 kVAr. If that were to be corrected by passive filter, only about 260 kVAr of dynamic compensation would be required to bring the current waveforms to almost pure sinusoidal form. Under those conditions if one of the EEI 52 150 kVAr dynamic compensators should fail, only one of them would be capable of achieving an acceptable voltage THD at the load ends, until the situation could be remedied'".

7.11 CONCLUSIONS REGARDING MODELLING CONFIGURATIONS

The series of 20 different measurements, carried out on 7 different circuit configurations, were conducted on the harmonic superposition program developed in this thesis. All the results shown in this chapter are simulated. Unfortunately it will not be possible to carry out practical measurements at the full operating capacity at which the modelling was carried out. That work was scheduled for at least two years in the future and lies beyond the available time frame of completion of this thesisxiii.

That this final experimental evaluation and verification will not be possible was realised from the outset and that is why detailed verification of the program was carried out in chapter 6. Because no additional circuit components or topologies has been introduced in the circuit topologies in the larger model, little doubt can exist that a further extension of the program, only in the number of nodes, can or will introduce incompatibilities or inaccurate modelling now. A thorough examination of the results in the above 20 modelling exercises should be sufficient to dispel any scepticism in that regard.

Although the lasers and other loads at all 22 nodes, as shown in Figure 7.2.1, were in place at the time of writing, they have only been operated together at the relatively low pulse repetition rate of 200 Hz. Unfortunately the power system loading and harmonic generation levels are too low under those conditions to give really meaningful measurements. The power supplies and lasers

This condition has not been modelled, as it would further extend the scope of an already bulky document. It is true, however, that even during failure of both dynamic compensators, the voltage THD of 2.7% at a load as typically depicted in Table 7.9.1 would be acceptable. Even that possibility has now been eliminated as the MLIS X4 project on which this study is based has been closed down on account of other factors having nothing to do with the operation of the power systems and equipment dealt with here. 154 are, however, capable of operating at their full 2 kHz capacity and a limited number of lasers were operated at the maximum frequency the rest of the equipment could handle and the results shown later in this chapter derive from that measurement.

It became clear, early during the simulation in this chapter, that replacing the undersized DC- choke inductors in the R2P2 converters were mandatory. Aside from the greater expense of higher levels of compensating power that will be required to accommodate the higher levels of harmonics, the levels of voltage distortion that they introduce at the load-ends are unacceptable and expensive to remove. As mentioned too, the double current peaks per half-cycle on the AC- sides of the converters lead to instability in the voltage-control circuitry of the power supplies. It is for these reasons, more than the extra cost of compensation, that configurations 1 to 5 can be excluded.

The modelling has not been repeated to compare the operation of passive filters with- and without decoupling series inductors in conjunction with the changed power supplies. Although the levels of the individual harmonics would reduce with the installation of larger DC inductors, the required basic filter configurations would still be the same and would still be prone to parallel resonance and instability. If series inductors were to be installed, the same alterations would still be required to rewire the circuits so that "clean" power can be taken off on the source-sides of these inductors, introducing the same unknowns and other operational difficulties.

As a result of the studies carried out in this chapter, it was decided about two years before the time of writing here, to increase the size of the DC choke inductors on all the R 2P2 units to 3.2 mH. It was even realised at the time that 3.2 mH was still inadequate and another optimum value could be derived, but other constraints, such as available space in the cabinets precluded any larger inductor implementations.

Other alternatives were investigated, such as the replacement of the pulse power supply AC-DC converters with supply-friendly PWM units. Another alternative that was considered was the installation of dynamic compensators at each power supply unit. The former possibility was abandoned because of the delay the program would suffer if commercial supply-friendly converters were sought, or if the one unit developed and tested had to be commercialised. Without furnishing the details, knowledgeable readers will agree that the second alternative would be prone to technical difficulties and that the equivalent cost of twelve 25 kVAr dynamic compensators would far overshadow that of two 150 kVAr units.

At that point in time, all the available energy in R&D was directed towards other difficulties in the program and could not be afforded to be spent on the development of dynamic compensators. It was therefore decided to purchase two standard 150 kVAr units of a known make and from a proven manufacturer52. 155 The following table compares the cost of the alternative options:

Table 7.11.1 — Economic comparison of the different compensation topologies.

OPTION DESCRIPTION REMARKS

1 R2P2 fitted with 30011H inductors, R3P2 Technically unacceptable. with 3.2mH. No compensation.

2 As for 1 but with dynamic filters of Estimated cost $13400 000. 753kVA required at transformers.

3 As for 1 but with passive filters and series Estimated cost: $84 000. inductors at all the loads Not an ideal configuration.

4 As for 3 but without the series inductors Possibility of inter-filter resonance. Ineffective method of compensation.

5 As for 1 but with passive filters without Evidence of inter-filter resonance. series inductors on transformer Distortion at load still too high. secondaries.

6 R3P2 units with 3.2mH inductors and R 2P2 Estimated cost: $26 000. units now similarly equipped. No Distortion on the limit of compensation. acceptability.

7 As for 6 but with two 150 kVA dynamic Estimated cost: $26 000 plus compensators. $50 000 = $76 000.

The implementation of option 7 satisfies the requirements of the plant and this option is recommended.

7.12 ALTERNATIVE POWER THEORIES

Under conditions of voltage and current distortion, the well-known, unambiguous power calculations failed and fell short of completely describing the power phenomena. It was realised that the definitions must be extended. The new theories had different goals in mind. Budeanu wanted to understand the different power phenomena and at the time did not even anticipate compensator control, as do some of the other more recent theories. Czarnecki developed a new theory that he claims are more physically meaningful than the theory of Budeanu. Unfortunately the Czarnecki theory is quite involved and introduces new concepts that are difficult to be taken up by the ordinary power engineer and technician. The IEEE definitions was developed to give guidance to the quantities that should be measured for revenue purposes, specifically for economical despatching and for voltage profile determination in systems with harmonics.

In this section the different theories will be compared with their practical utility in mind. The comparisons will be done on the basis of the foregoing study of 7 different circuit configurations 156 and the associated 20 sets of measurements. It must be kept in mind that this study is about a practical plant and the detrimental effects that harmonics have on the system and its associated equipment. Note that all this study requires from the point of view of the supply authority is for the distortion levels to fall within the permissible bounds. A study of tariffs and the determination of the origins of distortion therefore do not enter into it. Once the conditions have been established that will keep harmonic levels within the required regimes at the supply authority terminals, the only remaining objective of the study is to ensure that internal plant harmonic levels will be compatible with the requirements of the equipment used. Only internally generated harmonics are considered and experience has shown that ambient harmonic levels on the supply authority side have never been a problem. That aspect will, therefore, also not enter into the study.

7.12.1 Budeanu

As mentioned earlier, the Budeanu theory is still the most widely adopted in industry generally as well as in the design of equipment and it will be used in the comparison even though it was shown by Czarnecki29 to be misleading. Table 7.12.1 compares the reactive power (Budeanu), reactive power (Czarnecki) and the fundamental reactive power of the IEEE. It is informative to note that the Budeanu reactive power is smaller than the fundamental reactive power of the IEEE. In all the modelling runs the Budeanu reactive power is smaller then the fundamental reactive power, it is therefore not clear what the reactive power means.

Table 7.12.1 - Reactive powers for the different modelling runs

Run Budeanu Czarnecki IEEE Run Budeanu Czarnecki IEEE Run Budeanu Czarnecki IEEE Units

1 36.694 67.408 40.755 8 7.621 15.232 7.788 15 -12.469 0.94 0.94 kVAr

2 190.51 204.07 204.07 9 5.313 5.789 5.789 16 63.198 64.116 64.116 KVAr

3 471.16 472.07 472.07 10 73.723 73.755 73.755 17 39.3 40.422 40.422 KVAr

4 39.55 40.32 40.298 11 5.823 7.895 7.064 18 117.72 120.179 120.179 KVAr

5 61.759 61.9 61.77 12 -1.104 1.872 1.871 19 61.83 61.833 61.833 KVAr

6 39.918 41.237 40.981 13 65.59 65.79 65.794 20 40.191 40.543 40.543 KVAr

7 6.009 6.911 6.714 14 37.037 41.223 40.968 _ kVAr

It is also possible in modelling run 15 to change the firing angle of the thyristors to get the reactive power to zero, while the fundamental reactive power is not. This casts doubt over the use of this theory because due to harmonic cancellation this cannot be used and the values cannot be 157 designed with confidence. To illustrate this even further the total apparent power in modelling run 12 is increased to 4.2 MVA. This leads to a fundamental reactive power of 357 kVAr while Budeanu's reactive power is -0.253 kVAr. Commercial instruments designed to analyse the harmonics in the power system still make use of the collective reactive power concept of Budeanu and it will therefore not be accurate in its readings. This theory will therefore not be used to design or to specify ratings of equipment in any situation and its use in industry must be abandoned in total.

7.12.2 Czarnecki

The basis of this theory rests on the presence of two number sets, which must be known before any calculation can take place. As pointed out in chapter 3 it is not possible to establish if a given harmonic is brought about on the source or load-side of the measuring cross-section and therefore Czarnecki introduces a new method of determining the two number sets. The sign of the harmonic active power P. is used to determine if that specific harmonic belongs to the source or the load. In the modelling runs this procedure is used to specify the different number sets and the results of the harmonic powers are tabulated for the different modelling runs. The individual and collective harmonic active power values are very small. The measured harmonic components on the dummy load as described in 'section 6.2 are tabled in Table 7.12.2.

Table 7.12.1 - Harmonic active power components

n Harmonic active power Unit n Harmonic active power Unit

1 52.313 kWatt 5 -14.1 Watt

2 -13.6 Watt 7 7 Watt

3 10.2 Watt 9 0.5 Watt

4 -0.2 Watt 11 4.6 Watt

The harmonic powers are very small and it is not easy to measure these small quantities in a practical situation. This is in line with the findings of De Beer 7° who stated that it is difficult if not impossible to measure the components of the different number sets accurately in a noisy environment. If there is a measuring error in this quantity the results will be erroneous. Observe that in Table 7.12.2 the 7th harmonic order is positive; not a single modelling run described gives positive values of the 7th harmonic order. The only modelling run that indicates positive harmonic active power is that of modelling run 1 and 8, for the 5th harmonic. When this 158 measurement of Pn is not accurate the power calculations as proposed by Czarnecki will not be accurate and the subdivision of the different current components is not valid.

Czarnecki defines the reactive current as the summation of the imaginary harmonic current JB,,Un associated with each of the harmonic admittances. The physical interpretation of this reactive current is that if a one-port shunt network of susceptance B„ for every nw frequency is connected at the load terminals, then the reactive current will be cancelled out completely at the source. The modelling shows that the reactive power of Czarnecki is the same in most cases as the fundamental reactive power, and no new information can be gained from this except of course the design specifications for fundamental power-factor correction capacitors. Czarnecki criticises the Budeanu theory in that the Budeanu total reactive power cannot be used because of reasons already mentioned. On the other hand, the Czarnecki reactive power does have a clear physical meaning but is totally impractical in the collective sense and cannot be used as planned and is therefore also useless in this respect. There are definitely 5th and 7th order harmonic power components not shown in the Czarnecki reactive power analysis. Other means must therefore be sought to determine the different harmonic sets.

7.12.3 IEEE power definitions

The supply authority generates pure sinusoidal voltage waveforms at 50 Hz frequency. The object of transmission is to deliver as much of the power as possible through the 50 Hz positive sequence component to the consumer. It therefore makes sense to separate the fundamental and the harmonic components from each other in analysis. It is from this basis that the IEEE theory was developed. The concepts of fundamental active and fundamental reactive power are well known to engineers. They can easily grasp this extension. Practical measurements are more accurate than for determining the Czarnecki components because only collective voltages and currents are measured for the fundamental and harmonic components. The accuracy of measuring the total active power (P) and the fundamental positive sequence power (P+1) has been proven". Measurement of the index P//),./ is accurate and thus suitable for revenue purposes. The fundamental reactive power Q1 is of utmost importance in an electrical system since it governs the bus voltage magnitude, is an indication of the energy stored in magnetic and electrostatic fields and also effects network stability and power loss. The remaining non-active power terms, as tabulated in the different modelling runs, monitor the effectiveness of dynamic compensators and passive filters.

7.13 MEASUREMENTS AT THE PCC WITH LASERS IN OPERATION

All the pulse power supply units are equipped with 3.2 mH inductors. No compensating gear or passive filters are connected in the circuit. 159

The first part of this chapter was done before any measurement could take place in the plant. It is due to the simulation that the change on the power supplies could take place. Measurements can only be performed after the design of the plant. This measurement compares modelling and measured results at the point of common coupling (node 5) in Figure 7.2.1 and are shown in the oscillogram in Figure 7.13.1.

640 500 480 WEI it A 1 480 320 360 MO Illr 160 LIIMMIL1111111111/11 240 11M11 Mr 120 wino iri 0 WM. V IM1111/4111111111VIIINI 0 160 120 awstiimmEz imalrill1111111111111M11 240 lawn LWA -320 111041 MOM -360 -480 11 If I MUM WM -480 640 00 EYE a) - Modelled result b) - Measured result

Figure 7.13.1 - - Line - Line voltage and current at node 5

Figure 7.13.2 - Comparison of modelled and measured results

In this measurement and modelling the motors are included in the model to show the practical case. Motor loads were not included in the configurations as described. Excellent agreement between the two graphs can be seen. This illustrates that the modelling done can be relied on. Take note that the lasers could not be operated at full design specifications because the plant is still in a commissioning stage.

The normalised harmonic voltage spectrum for the modelling and measurement at node 5 are shown in Figure 7.13.3 and Figure 7.13.4.

160

1 1 0.9 0.9 0.8 0.8 T 0.7 0.7 0.8 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 n 11 0.1 0 n11 r 0 5 10 15 20 0 n n nfln Fin 0 5 10 15 20

Figure 7.13.3 - Normalised harmonic Figure 7.13.4 - Normalised harmonic voltage voltage spectrum - modelled spectrum - measured

Note the presence of the third harmonic at the measurement, which is to be expected of loads generating third order harmonics. Excellent correlation between the 5th and 7th order harmonic can be seen. With a total voltage distortion of 1.2% this distortion is not serious and can be tolerated.

The normalised harmonic current spectrum for the modelling and measurement at node 5, are shown in Figure 7.13.5 and Figure 7.13.6. 40 40 30 36

32 32 28 28 24 24 20 20 18 16 12 12

4 n n rt 4 0 n n n _ 0 5 10 15 20 0 0 5 10 15 20

Figure 7.13.5 - Normalised harmonic Figure 7.13.6 - Normalised harmonic current current spectrum - measured spectrum - measured

Again when comparing the current graphs there is little doubt that there is a good correlation between the results. With a current TIM of 35% this is not serious at this particular power consumption of the lasers.

The results for both modelling and practical measurements are summarised in Table 7.3.20. 161 Table 7.13.1 - Modelling and Measurements at PCC

Description Model Measured % Unit Difference Total node voltage 226.68 226.44 0.1 V Total node current 302.02 302.63 0.2 A Total active power 177.35 178.33 0.6 kW Total apparent power 205.38 205.58 0.1 kVA Current THD 35.09 35.88 2.2 % Voltage THD 0.9 1.2 25 % Fundamental pnwer factor 0.915 0.922 0.8 - I Total power factor 0.864 0.868 0.5 _ -

The reason for the difference in the voltage THD is due to the third harmonic that is measured.

The difference between modelled and practical results for the theories under discussion is tabulated in Table 7.13.2.

Table 7.13.2 - Modelling and measurements results for the three power theories

Budeanu Model Measured % Difference Unit Active 177.346 178.333 0.6 kW Reactive 77.398 74.216 -4.3 kVAr Distortion 68.853 70.382 2.2 kVA Czarnecki Model Measured % Difference Unit Active 177.354 178.464 0.6 kW Reactive 78.039 80.281 2.3 kVAr Scattered 0 7.156 kVA Unbalanced 3.030 0 kVA Generated 0.658 0.564 -16.7 kVA Forced app. 68.0335 62.597 -8.7 kVA 5th harmonic active -6.5 -109.3 94 watt 7th harmonic active -1.9 33.4 10 watt 11th harmonic active -0.1 -7.7 98 watt 13th harmonic active -0.01 -7.3 99 watt 162

IEEE Model Measured % Difference Unit Fundamental active 177.354 178.400 0.6 kW Fundamental reactive 78.039 74.905 -4 kVAr Fundamental apparent 193.765 193.488 -0.1 kVA Non-fund. apparent 68.104 69.475 2 kVA Current distortion 68.008 69.431 2.1 kVA Voltage distortion 1.875 2.323 19 kVA Harmonic apparent 658 834 21 VA Unbalanced fund. app 322 0 VA I Normalised har. app 0.340 0.431 21 I% I

A few remarks are necessary to explain some of the differences between the measurement and modelling results. In the measurement results the unbalanced component is zero, because only one phase is measured. The harmonic active powers difference is 100% and can therefore not be used, this again confirms the findings that this component must be kept out of power measurements. There is a good correlation between the other components and can therefore use when designing a compensator strategy.

7.14 SUMMARY

In this chapter the pilot plant was modelled with different circuit configurations tested. It was found that it is an economical viable option to upgrade the R 2P2 power supplies, also if it is kept in mind that the plant does not consist only of 12 lasers but the full scale plant will be more. A dynamic filter compensated the existing plant, which is adequate for now. It was shown that the power theory formulated by the IEEE Working Group is of great use in an industrial plant. The theory of Budeanu shows results that are not correct and the power theory of Czarnecki is difficult to measure and errors can easily make the results erroneous. The measurements done at the lasers confirm the statement that the model developed is of practical use in an industrial plant. 163 8. CONCLUSION AND EVALUATION

8.1 INTRODUCTION

When a relatively large number of power supplies are operated synchronously, as is envisaged for the single chain of the first MLIS pilot plant now being commissioned, difficulties can be anticipated, not only with the power supply units themselves, but also with respect to the magnitude of distortion generated on the internal supply network. Solution strategies must be formulated at an early stage of the proposed final MLIS plant, which is planned to contain eight laser chains with an anticipated total maximum power demand of well into the multi- megawatt range. The overcoming of harmonic distortion in the current single-chain installation was one of the goals of this study. The results of this study will not necessarily suffice for the final plant, and planning and design should encompass both scenarios, even at this early stage.

Because the nature of this study lent itself perfectly to it and presented the ideal opportunity in the form of frequency-domain network performance results, it was combined with an analysis in terms of three power theories that represent the most feasible approach to this type of analysis in future industry. The power theories referred to are those of Budeanu, Czarnecki and the IEEE Working Group. Interpretation of the modelled and measured results is done on the basis of these three theories. The utility and drawbacks of the different theories are determined and compared in a practical environment in order to assess their worth from a practical point of view.

8.2 FIELD COVERED AND RESULTS OBTAINED

The foregoing study in this document begins, in Chapter 1, with a description of the topologies and equipment that will be used in a Molecular Laser Isotope Separation (MLIS) plant for uranium enrichment. It outlines the basics of the pulse power supplies that will be used, with an emphasis on the front-end designs of the AC/DC phase controlled 6-pulse converters that concern the interface between the plant and the power supply network.

The fundamental concepts of power measurements in electrical circuits are introduced in Chapter 2, with an initial discussion of the classical definitions that relate to single-phase sinusoidal behaviour. The study then progresses to circuit behaviour in which the voltages and currents are distorted, and illustrates the shortcomings of the classical definitions through frequency-domain analysis.

Chapter 3 describes the relationship for total power, real power and imaginary power for simple harmonic and steady-state periodic non-sinusoidal voltages and currents in a circuit. The construction of the harmonic phasor that can be defined for periodically distorted waveforms through discrete spectra is introduced. It is shown that time-dependent quantities of voltage, 164 current and complex power, as well as the equivalent harmonic phasor quantities that can be defined for the former, are space vectors and can be treated similarly. It is shown that fundamental relationships such as orthogonality apply to both and through these active, reactive and complex power can be defined on a per-harmonic basis. At this point the difficulties that are encountered when summation is carried out over the harmonic orders emerge and the Budeanu distortion power definition is introduced, illustrating one of the oldest attempts to accommodate the difficulty of defining power in systems with distortion. Chapter 3 then brings in the theories of the IEEE Working Group and those of Czarnecki for single-phase systems, which are then extended for three-phase systems.

Chapter 4 outlines the principles of harmonic compensation from a frequency-domain point of view. It shows that normal network principles can be applied on a per-harmonic order basis and that distortion sources and harmonic-compensating current and voltage sources can be applied on the same basis for steady-state operation in which both the distortion and the compensating sources exhibit strict periodic behaviour. This approach simplifies the network analysis and furnishes insight into the optimal approaches and topologies needed to counteract the distortion produced in networks from both outside and inside the network boundary of the consumer PCC. Chapter 4 also shows that both passive and dynamic compensation apparatus operates in a similar way from the network point of view. The only difference is that the dynamic compensator behaviour can be controlled in a more extended way than the passive filter, with significant advantages for stability and accuracy. The generic compensator types are outlined and optimal circuit topologies that apply to different needs are discussed.

Chapter 5 introduces concepts of harmonic superposition modelling. Broadly speaking this approach makes it possible to carry out the network analysis on a per-frequency basis after transforming all the network and time-domain data into the discrete frequency domain. This approach is, of course, only applicable to steady-state operation in the network on current and voltage waveforms that are strictly periodic in nature. The method cannot be used in transient analysis or when non-periodic harmonics are present. In this way the fundamental frequency behaviour of the network can be separated from that of the other harmonics and separately analysed. After analysis the individual harmonic order results are synthesised into time-domain data again after inverse Fourier transformation. The principles of harmonic superposition modelling is applied to the three-phase application used in this thesis, through the definitions of compound immittances and an extension of the utility of the nodal-network approach that has been developed for the classical analysis used in three-phase network analysis.

The remaining parts of Chapter 5 discuss modelling topologies used for the different transmission system components, beginning with the supply-authority source and progressing through 165 transformers, passive filters, dynamic compensators, cables and modelling of the 6-pulse phase- controlled converters that are used in the pulse power supplies.

Chapter 6 is devoted to the verification of the superposition model. It employs a circuit with five nodes and is made up of all the transmission components encountered in the plant. These components include the supply-authority source, the high-voltage supply cable, the step-down transformer, the low-voltage distribution cable, a dynamic compensator, a passive filter with associated series inductor, and a six-pulse converter load. The circuit was set up in the pilot-plant building and consists of a pulse dummy load and an R2P2 pulse power supply with an experimental low-voltage passive filter and a commercial dynamic compensator of 150 kVA. Measurements were taken on the above set-up under different conditions and are compared, in Chapter 6, with modelled results obtained from the superposition model. The different conditions catered for measurements at different points in the circuit and represent operation with and without the passive filter, the dynamic compensator and the series inductor between the load and the source. This inductor was used to minimise resonance of the passive filter with the source and other energy storage components in the network.

The excellent agreement between the measured and modelled results in Chapter 6 furnished the necessary confidence on which the further work in the thesis and planning of the project is based. The work in Chapter 7 consists of twenty modelling runs carried out on seven different circuit configurations. The different circuit configurations cater for the installation of passive and dynamic compensators in different combinations at different points in the circuit they also compare the capital cost of different circuit topologies with respect to further work on the pulse power supply input converters, which will make them more circuit-friendly from the point of power supply quality. Chapter 7 conclude with a discussion in which the practical utility of the different power theories stated is compared and assessed on the basis of an analysis of the results. The results in Chapter 7 show that replacing the undersized DC choke inductors in the R 2P2 converters was mandatory. If this is done, the cost of compensation is reduced drastically and only two 150 kVA dynamic compensators are needed to compensate the plant.

8.3 EVALUATION

Because of an untimely termination of the MLIS project by the AEC, final extensive measurement and evaluation of the pilot plant electrical distribution system will not be possible. The measured and modelled results obtained from the five-node network as set out in Chapter 6, and the measurements on the pilot plant, as set out in Chapter 7, are sufficiently accurate and representative to confirm that more measurements on the pilot plant would only have confirmed the modelling results set out in Chapter 7. The final conclusion of this study is that the model can be used for different applications. In this study the model was used to search for the most technical and economical solution. The model can also be used to determine the permissible 166 harmonic levels in the total distribution circuit, i.e. harmonic penetration exercises. The model is set out in a generalised form and is not only suitable for modelling the plant in its present state, but any extension can be dealt with. It is shown to produce excellent correlation with practical results.

The Czarnecki theory is highly dependent on the precise measurement of the active power per harmonic order. It is a finding of this study that the equipment that was used could not measure the different components very accurately.

It was proven that the Budeanu power theory gives erroneous results when distortion is present in a power system. The use of Budeanu should be abandoned and the use of the IEEE Working Group theory should be encouraged instead. This theory is easy to understand and implement under industry conditions and gives technically correct answers. It accurately describes power compensation equipment that ought to be used in practice and is an excellent theory to indicate the severity of distortion in power systems.

8.4 RECOMMENDATIONS

Firstly, even though the superposition model as programmed on a Mathcad package is adequately capable of performing the intended function, it is slow and takes about five minutes to compute one modelling run, as described in Chapter 7. It also requires an extensive amount of computer memory (40 MB) as a result of the structure used. At present the model functions by performing an analytical calculation of the network equations at each and every harmonic order by inversion of the nodal admittance matrix and the associated matrix operations necessary to furnish frequency and time-dependent voltages and currents at required nodes. In networks using a large number of nodes even more memory will be required and the computation time will be slow. Approved methods of overcoming this problem exist and are based on Gaussian elimination with partial pivoting and back-substitution. By implementing these methods in the algorithms in the program, memory requirements can be significantly reduced and the time requirement per simulation run will also be reduced.

Another drawback of the present model is that it cannot perform frequency scans to investigate resonance between passive filters and other components. A frequency or impedance scan must be added to the program whereby a plot of the magnitude and phase of the driving point impedance at a point of interest versus frequency can be generated. The frequency scan can provide some useful insight into the system response at harmonic frequencies. Furthermore, the program could be made more user-friendly. If this is done, it could be used as a general tool in power analysis. 167 In the present study all the alternatives are based on the use of phase-controlled 6-pulse converters in the front-ends of the power supplies. In broad outline, two different front-end designs of the R2P2 can be considered:

Employ the present line-commutated, phase-controlled 6-pulse AC-DC converters, but install distortion-compensating equipment.

Replace the present line-commutated phase-controlled AC-DC converters in the front- ends of the R2P2 power supplies with supply-friendly units that draw sinusoidal currents, a prototype unit has been developed but has not been installed in a power supply unit for evaluation.

Lastly, the evaluation of the modelled results is not as complete and extensive as one would wish. The analysis of the modelling results is based on almost-arbitrary three-phase unbalances in the network in Chapter 7. It is recommended that measurements be done simultaneously on all three phases, for both the voltage and the current. Synchronous measurements are really necessary in practical measuring environments to correctly evaluate the different symmetric components and powers and call for more sophisticated equipment than was available at this point in time. The measurements should be completed, possibly in another project, and additional comparison of the utility of the different power theories would then also become possible. Al ANNEXURE A HARMONIC SUPERPOSITION MODELLING OF THE AC/DC CONVERTER

A.1 INTRODUCTION

This appendix develops a frequency-domain model for the line currents of a 6-pulse AC/DC converter. The model is intended for use as a sub-model in frequency-domain models of electrical distribution networks. The model is not intended to be based on the conventional analysis of AC/DC converters in the form discussed in the literature, in which the current waveforms follow circuit and converter parameters. Such rigid analyses have been carried out in the literature n' y mewing of 74 . In this instance, the need was for a flexible b which the observed waveforms measured on actual systems in the plant can be matched in the frequency domain, simply by setting a number of parameters only. These parameters are simply:

The peak current magnitude The conduction angle The commutation angle, and The half-sinusoidal pulse width

By matching the above parameters for the line-line currents in the 6-pulse converter, and subsequently converting them to line values only by frequency-domain Delta/Star conversion, the line currents are obtained. It is further possible merely by changing these parameters, to simulate the converter throughout its DC output operating range, from discontinuous converter output current to continuous output.

The procedure of beginning with the line-line converter currents and then converting the line- values ensures the absence of zero-sequence components being present in the line current, that will not be permissible in the main circuit models in which this sub-model is to be used.

DEFINITIONS

RMS AC current /max A Fundamental frequency f Hz

Fundamental angular frequency a) = 2 TCf rad/s Highest harmonic order k Conduction angle w deg Commutation angle u deg Periodic time T =1 s f Pulse width factor (ratio of conduction to a sinusoidal pulse width)

A.3 MODELLING THE PULSES IN THE TIME DOMAIN

A.3.1 Positive-going pulses

General periodic pulse waveforms can be modelled in the time domain by the following equations, which are set up in a Mathcad PLUS package:

The angular increment for the purpose of plotting is set up as:

:- w-h 2.0 100 (A.3.1)

The harmonic order range is defined as:

n :=0 -h 10-19 (A.3.2)

The harmonic frequency is defined as:

(o(n) (A.3.3)

The half-sinusoidal pulse width of the current waveform is defined as:

T :=— (A.3.4)

For the purpose of plotting, the discrete time is defined as:

T t :=-T , - T t — T 50 (A.3.5.)

and the time-frequency products as:

2 \ 2 2 (A.3.6)

During commutation, the rising current waveform is modelled as:

1 1 imin(wt) :=- .(2.0 w 2.o)t)•Imaxcos — .a•w (A.3.7) 2-u 2

The centre part of the waveform is modelled as:

imid(co t) := Imaxcos (a q(o t) (A.3.8)

and the declining commutation profile in the waveform is modelled as: A3 - irrliv(w t) :=- .((- 2.0 – w) 2.03 )• Imax cos 1 .a • NV 2.0 2 (A.3.9)

The converter line-line current can then be plotted for checking by providing for the different conditions, depending if the converter output is continuous or discontinuous, and by incorporating the three parts of the waveform in accordance with equations (the three last equations):

w il(w t) := if( w t<- — , in-Lin(w t),imid(wt) 2 (A.3.10)

w . i(wt) = <— ,11(wt),ima:,(wt) 2 (A.3.11)

It is now possible, merely by changing the above parameters, to model the continuous or the discontinuous output current modes of the converter and to adjust the commutation angles and the - - ripple ratio of the line-line current.

The following waveform portion is plotted for /max = lIcA,f = 50 Hz, w = 60°, u = 2° and a = 1.5 in Figure Al..

1000

1 (wt) 500

—0.002 —0.0015 —0 001 —0.0005 0 0.0005 0.001 0.0015 0.002 we 6)( 1)

Figure A.1 - Time domain representation of line-line pulse

A.3.2 Negative-going pulses

The three equations that model the negative edge, the half-sinusoidal centre section and the rising edges of the pulse respectively, are:

1 I iminn( t) := - - .(2.u w -f- 2.6) 0 • [max cos — . • w 2.0 2 (A.3.12) A4

imidn(o) t) := - ( Imax cos (a (I) 0) (A.3.13) and

- 1 1 imam( co := - - ( (- 2.0 — w) 2.6) t) • Imax cos (- • a • w) 2- u 2 (A.3.14) similarly as for the positive-going pulses.

Again, the following conditional statements make for plotting provision for continuous- and discontinuous output current conditions of the converter:

i2n(ut) := if(o) t , iminn( t ), imidn( co 0 2 (A.3.15)

in(wt) = if '( (0 t<-‘X,i2n(co t),imaxr(cat) 2 (A.3.16)

The plot, for the same parameters as in the positive-going pulse conditions, is shown in Figure A.2.

,o,

in(wt) 500

1000, — 1000 —0.002 —0.0015 —0.001 —0 0005 0 0.0005 0.001 0.0015 0.002 ,.-0.001777713,. cot ,0.00177778, (DM

Figure A.2 - Time-domain portion of negative portion

A.4 FOURIER TRANSFORMATION

The positive and negative time-domain equations for the pulses can now be Fourier transformed:

A.4.1 Positive- going pulses

The rising part of the positive pulse: A5 w 2 I -j •ncot Frp(I1)1.-1. 1.(2.11 w 2-w 0.1max cos (- .a w -e dw t 2•u 12 w --- U - 2 (A.4.1)

yielding:

1 1 (j -•j-.j1 -n-w •u•n f exp(--j .n • w) - exp(j •n • u -- - .j • n. w)) 2 2 2 Frp(n) :=Imaxcos ( la•w • 2 [ n•(n2•u )1 (A.4.2)

The centre part of the positive -going pulse:

w 2 Fcp(n)=1-1. Imaxcos(a•w t)•e i .n.°)t t

- 2 (A.4.3)

yielding:

I [sin[-1 .w.(a ni a - sin[-I -w-(a n) • n t sin 1 . w. (a - n) •a. t sin - .w.(a - 2 2 2 2 Fcp(n) := Imax [n. (- n 2 a 2)] (A.4.4)

and the declining part of the positive pulse:

w - 2 I Ffp(n)m- • 2.0 - w) -t- 2.w 0 • Imax cos 1 .a • w • . n.wt clw t 2 I 2 w

- 2 (A.4.5)

yielding:

1 - - 1 exp - j - -.j -n-w) t j •exp(—j •n.w)•u.n - exp(—j •n.w)) 2 2 2 Ffp(n) :=- Imax cos (I. a •w ) [n . (n2. un (A.4.6)

If all three positive parts are now added, the following composite equation is yielded for the positive-going pulse:

A6 1 Fp( n) := - [2 -cos (- . a w) • cos I n • ( 2. u + w) • (n2 - a 2) + 2. n. cos -I • a • w • u. sin -I .w.n • (n2 - a 2) Inv (n ) 2 2 2 2 1 1 1 I + 2. cos . a • w) • cos - . w. n • n2 - 2- cos - . a • w • cos - . w-n -a 2 - n2• u. sin -1 . w. ( a + n) -a 2 2 2 2 2 2 . 1 3 [ 1 , + n3 . u• sin w. ( a + n) + n • u. sm -- w. ( - a + n) • a + n • u. sin --- w.(- a + n) 2 2 2 (A.4.7) and in which:

Imax Inv(n) - [n2-(u.((- a + n)•((a + n).A )))] (A.4.8)

The total positive pulse can now be plotted out of the frequency domain, as a function of the harmonic order as shown in Figure A.3.

1000

800

600

fp( t ) 400

200

200 -0.01 -0.005 0 0.005 0.01 0.015

Figure A.3 - Positive portion of pulse plotted from frequency domain

A.4.2 Negative-going pulses

A similar treatment of the negative-going pulses is now possible, yielding the negative pulse portion by synthesis of the frequency domain components as shown in Figure A.4.

-200

400

f1(1-T) 600

800

1000

1200 -0.01 -0.005 0 0.005 001 0.015

Figure A.4 — Negative-going pulse portion synthesised from Fourier components

A.4.3 Synthesis of the complete line-line current waveform:

The derivation of the complete line-line current waveform followed analytically in a much less complex procedure, by adding the Fourier components of the rising middle and falling portions together first and then summing these three components to finally obtain the composite waveform:

1 sin (--I- •w•n)•u•n — cos (- • w • n) 2 2 1 . ( 1 + (cos (- •w•n)-cos(u•n)— si n - •w•n)•sin(u•n)) 1 2 2 F(n) =2-Imaxcos (---a-w -(- 1+ cos(n -n) + j • sin(n • n)) • 2 [n •(n 2-u)] 1 1 Imax 2•(- 1+ cos(n•n) + j -sin(n•n)). a •sin( —•a-w -cos -- -w-n ... 2 2 (n•((a — n)•(a +n))) 1 +- (n-cos (-1 •a-W)•sni - wn)) 2 1 2 (A.4.9)

F(n), in equation (A.4.9) represents the nth harmonic phase of the line-line current in the 6-pulse converter.

The four waveforms in Figure A.5 represent values of a from 0.5, 1, 2 and 3.

150

1000

1151 0 15

-50

100

150 -001 ,005 0005 001 0 01

1000

100

0n o

5u0

-100r.

-001 -0005 0 005 001 0015 0005 01 0015

Figure A.5 - Effect of changing a on the line-line converter current

A.4.4 Three-phase representation

Equation (A.4.9) is used in the main model in three-phase form to represent the three-phase currents in vector form as:

1 •91 vl 1 n (hn) ° •e • la vi n

i 01 v1 2 n Illa yin = „r2• 010) 1 ' e • la vi n

2 -1 el v1 3 n (h n) • e la vi n (A.4.10) in which:

1 cos( --•0 ...vi .w v1)•(cos(ri-n) - 1 +j sin(A•n)) 2 Ilao n la On := ( 2 Ti • n 'll v) (A.4.11) and in which, in turn:

1 Ila yin z sin ( 1 w v1 . ' n -sin (uv n)) - cos (- .w vt n) • (1 - cos (u v- n)) ... 2 ( 1 - cos(ii•n)) - j •sin•ri) 1 8 vl sini — - 8vl .w vi - cos l ' w vl . n •-• 2)1 \ 2 2 / [ I. ( 6 vi e — n 1 . + - (n• cos (--• 8 0. V1) . sin .---w vi- n 2 - ( 21 ) \ (A.4.12) A9 The breakdown, into component parts, of equation (A.4.12) has been carried out here for the purpose of inserting it into the text of this document. In its original form, Mathcad is capable of handling the equation without the breakdown shown here. The line-line current waveforms can be converted to line currents, by delta-star transformation to yield the typical current profile as shown in Figure A.6.

0 01 0 02 003 004 0.05

Figure A.6 - Typical line current profile together with line-neutral voltage for a 6-pulse converter

In the example shown, the DC output current is just becoming continuous. With larger DC choke values the other typical waveforms are obtained. ANNEXURE B - HARMONIC SUPERPOSITION MODEL INTRODUCTION

This model makes use of harmonic superposition. It makes use of a nodal admittance matrix and matrix inversion for its operation. Both driving-point and transfer nodal admittance sub-matrices are 3x3 matrices and mutual and self-admittances are modelled by filling in the 9 different elements in each matrix. Nodal currents are modelled as 3x1 vectors. Both the single and twelve laser model is include in the annexure. The twelve laser model can be seen in Figure Bl.

Y C57 YP7 14 0

Load It1Ser YC70 YP14_0 V14_0 at A-EBP0002 I DO3 V1 W C58 ® YP8 15 PASSIVE 0 FILTER 1 Load Double laser at P15_0 I 0 Y C80 V 15_0 A-EBP0003 T23 YP35 YC59 0 YP9_16 Y T30 P50 Load C90 P16_0 I 0 V Double laser at 16_0 A-EBP0004

YC6_10 P10 17

Load VI Single laser at 470 A-EBP0005 PASSIVE 2 FILTER 2 Y Y T24 P46 -4 I Load YC11_0 P18_0 I 0 V Single laser at T40 P I Y 18 0 A-EBP009 YC6_12 YP12 19 I DO4 Load. C12_0 I YP19 0 I 0 V 19 Double laser at A-EBPOOS Y C6_13 YP13 2 V C12 Load "C13_0 YP200 Doublelaser at V200 A-M:10007 C10 C6_21 YP21 22

Load v Single laser at SOURCE 22_0 A-EBP0006

Figure B1 - Twelve laser layout. CNI cci

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c>, a)c al a) c _c(T3 ih co a) E 6- (r) E CD 0 0 C1 TD a) (1) _a 2 >- I II u) o 2 win c O a ca o 17) m C CI) CI) ..e" cu co 4- ...___.)- , D 73 a) _c I-- ------. • 7) ..._____.- <-- r-) a) - cs, ir) ai CI) a r) ...----. ------. c.) 2 C \I ,,--,. r's -d- Z- =.. (n 2) 'rT) 0 C \I 1- A--- ) -____, ...___., ....._.., a) u) U) cr , a) 1 >- o o a 0.1 --, ---, ----. _c .--- ,.....__..., c\I l— ••-• ',. I— 8 a U) C it a) >- 1 I I II .,- ai u) ..___, .,- c c c u) a) ci u) E. > 0 .1 t a -----.. I I .vi...... __....>- — 0_o (0= C O a a 5 ,-- C 7) -0 0 W 7) c U) CD 0 '7) c D u) a) In U) U) 7) ...._____, = -0 Trn O >- >- >- c,) 0 01 U) u) _CIa) E N- D ›.... CD ai Ecu a) C \J .c CV -o c7) a.) ca 1 -a ..- a) 2 O a) C1) V' ci a a C = i/i ascn 3 _za) a) (r) -o = II ,7 1) a) o To - > 1:70 7 a) 7) a) 6 ....E :..r... u) U) E >-•, .c ' c 03 2 :1.1 al c .c2 >- = S' o o a) "ai 2 c ° W ... u) - -al O_ - :4- o a) (1) c 1). ci) _cv -2 u) > _Nd ai 2 C -. --E ..- o c U) ‘- Z .5`) = ,... cm C LI • - -0 L u_a) ( -C a) 0.°) ca 2 u a) 9 CL ,- °) — E ,._ a, U CIS 0 a3 .... a) >., . CD 0 .•-•Ea- (13U) 7) ...a c 5 -ta CU 17 0- C = a) a) >- E ' a) 03 0 6 a)> E E c E a) E -13 > C 0) 2 RI CL as 2 u U ci) 114 1C3 E E ... .5- N--a) o= °- c -6 (n . Zi > u) ,....= 8 < o (/) 5 O •

, -cs o 2 `a-.) a) '5 siT) ,_a) o-a., _c a) To 0) ■-- L' _c = . -c° •E'. "-•-•'' 4-0) U CD = C C 0 a) F.- -o .6a' c _c cri o -5 CO 1— 0) E 0_ 0 a3 .:z .—'5 (E) a) CC D. 0 .= V) _o>s >s o o 7.1 rii) 2 o --,-, -o 2 _a U..1 a) c 0 0 a) E _c ILn .c o o 0- -E.

E m u) = l 1 lis ... a) a),_ ,T) a) —• a)

773 de CIII -C■ ,- (..) a) c '0 z .1- C 'TT < ••■• .,-- a) :- as = 380 V N co O U) mo

g § cs) u) deg u) c a) fii. c c c 2 CO D _i .0 .5,. o er I- ii) kV to c c >,v, 0) 6-, o_ t-3 N E II

7) c) cn -o al o = 88• ,- c E (-) 2 I1 N 8 L... form : 10 >- U) co (1) o.a) 0,- _›...0 -(a7) i >- c <1) ..- t 11 II o -0 .S 2 O Cn

CE I- I— ... I. ,,, a) LI) Ts D c r v, Q) Trans N O oc (/) < >:. , a) -o CO al > 15 .= .° 0 a) ci. cr) cn _I 2 Er SO UR >- '5 ir=3 E (2'5; `) CNI crj UJ O_ o E .G -o 0 > o CD =' CU 0 a tn In -CI E -CI CO CO 't 7 al to o 22 -10z _ > .cia) a) c0 >.o Ora) IL 2 0 03 a ,._ 4-0 a) = CD (n ...1C E 70' E cr= 03 2 re ,-. a) a) T. • cn — E lc 2 w c(pu CJ) 0) J CU .... a) CO S. >, -cs E ..)c Cli ru a, 0 0 >, ..-. CO a) CO 0.)0- MI . . IEI 0) a) (r) C3 Z. E _c) >-• CI) C c W .__CO c c ..,,.— >., .- CD 0 C O C CD a) 6

= -0. c o_ a) >_ : co _c cu a:' -c- E ..2 g E °- u a) to 0 (I) a) 4-. -Hi E (-) c ■_,. E c C a) ,_ _ > , u) ing CO t O cr, 0- u 0 as i] COQ L, ,..- CD C/) c..) LI . a_ cu o3 < -Q= cn as i- a) CO cn . . ra -o

ittance c ,_ E 2 a) u) c E 4- O a) _a) .._ moi CU -cr3 C Z 7:u" 8 0) a) ..- C '''''c E a) c dm U c C.) -o E 8 c (-) ...= lc c a) -a C

0 2 cu a a) 03 c a)

'di a former CU 8 Fa) lima) u) 6. c c ) a) (/`; 0. -o o) ies Z. 5 i a) Ce .Ev)u) c g E -o E 7) r a) a. 73 CD a co 0_ _c as ci = a, O

Se .E I- '- I-- c.) as a) -o Trans cncl E c a cf) LL --I - - -

N." F— co 1-- C -----. — ,--- ^ a) ..... ,—.— -- .'11.-' a: ..— a_Q . a_ Zir:L• :' -.:ig 0) 03 1`.. Lt) a) Nt .,— Li) LI '....__....' CO "----..-• F- T CD C■1 0 T Ev: I I ct II ii 03 6 _ N." . .

C T II II TC T ... ■ T T cn cn CU O , .1. 4.• ... I-- I— Y fa. 0- H. . H H CC X 0 0 co cc)

a) a) (.) C C C

o cu ...= Co ca ca E u) 'c-- ce: a) -..= a) 6 E c O aC O cc ittan -o (a a_ c

C E cu dm ...-.= c)E o a) c.) 0 a c `8 '

03 03 ion

E t t' .. o 8 .

a) isa

= t

:.., in e C o cuE(II m n ittance a u_ 03 ..L. Co = = dm E z• iti ‘•••• al W (-) -1= u) mag l a a) c a) .=.' = o cu = a) lex lle U C. (.7a E E a) o) -o o. CU Para (/) C.o) 03 (n E Comp ▪

0 0 0 0 0

0 0 0 0 0 74j

9878j 00

ix: 0 0 CD tr 19. 0 0 0 - 0.

ma 0 0 0 (0 ion 698 + t 0012 0.

0 0 0 0 0.

Connec 0 9878j 0074j 19. - 0.

+ 698 0012 Cli 0. 0 0 0. 0 0 0 0 (/) 0 U

8j -CI 987

0074j C) 9. 1 0 (1) 0 0 0 - 0.

ti) 698 +

O 0012 0. 0 0 0. 0 0 -

0 0 0 0 9878j 0074j O 19. - 0.

98 + 6 0012 0.

0 0 0. 0 0

a 0074j co 0 0 - 0. Li) O 0012

0 0.

0 0 0 0

ix: tr ix: 0074j tr 0. ma - ma ce ce ittan 0012 ittan 0. dm dm l a a da o ive it n im Pr The ▪ •

(0 (.0

•••••

X x 8 8 E E E _o _o .o = = N u) II

ce:

: 5 tan ce: it a) dm ittance ittan o- a a) dm dm C fer a) a t a t E _c a) hun hun a) _c • a) C s 0 a) O co 2 3 s 23 trans E C a5 O E co de de de a) C a) No No No a) O _o a)2) 0 C 0 0 C U) ca a) a .o ----. 0 0 ------. cu ...-----, ..------. CO O a) a) 3 0 1:3 C.") 0 a) _c .4- l) CD a) 2 C .o .4- T123 5 E `-- co de 0 ‘.... co co cw a) _a a) E C) a) u) ,-- .4- mo C a) c r c c >- 60 ui a) > • >- .,,, 4 4 .- C = a) U c o) ..-- , ..- lase a) _c C C ..- .- c C *§" ZT) c c cu C a) ....____-,>- 8 .._-_,› .....____.>- 0. U) .o TRANSFORMER .x 1-1- x x Eleven a) a) ( TRANSFORMER 8 8 (U 8 7.1 _o 0 n3 is E a) a) E 2 ca 1:3 E E _a -Cl -0 a) 0 rn _o .o = = (/) _c a) = = CO Cia (/) I E Fr, (.1) CO I _c a) 0_ Z II II II O 0. II II O . . c c C N 2 (I) C C C 0 0 VI O -o 0 c) co (N 01 N_I 0 a) _c C a) cv VI CNI N- ca _o U 1- _o o- 2 E ›- U) a) co t, ters:

U) C e o.

a) _c U) caw C a) O a) a3

8 2 a) aram -ru • c) ce: ce: ce: a) O f p ittance: ittance: rn UJ _c E a) o CI- 5 O U) dm

dm a) ittance: ittan ittan ittan CD

_a ion a a _o L1.1 11-1 t dm dm dm dm CD 0 fer fer U) < a a a a 77 ina s t t t t O 2 < 0 O hun hun hun hun tran rn term a) 17 s s U) CD CABLE MODELS De 23 trans 3 2 s 3 s 23 CC co de 2 de de de de de de No No No No No No

C E 0 0 CU 0 W CV ›- 0 380 a c 0 CV -1 0 .------.. 0 ,----- 0 OS0 N

CV CO Zbase

0 C•1 sc 0 0 • in 0 a) E 0 2 >- e u) Li

H.m CU iNc -I- ii 0 -0 c 0 (-) CV m NI N .:) Cc

.4 c..) CL U CV ll- • 0 a n a CD CD 0 (...) c

09 cp a 0 ,*z1 . 0003. u)

CY •(o 01 >- j l 2 ci 0 c:i in = 0. 11 0. de 11 II c = c CV • • CV CV 0 CV CV C.) c2

mo 0 C.) 0 0- (11 Q 0 a a a 0 ble u) 0 o o >-a_ Rc --I J -J >- 0 0 0 Ca

0 0 ca ci) >- 7. a 11 a — se 0 (..)

.-- 1 J 0 CU 0 Zba - -.--- • c C-) • .- 3 (/) E 1 m m >- e Q• u)U Li_ 7.) H• 0 c -. ....0■ 92• _a .1_ T F

m Ci)

NE H NJ .4 0 .n •

t CV 8-

0 7, n

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583 0 c) a 58 l 1 0 •co j

0 a O . 0003 >- ci c.) U) tx c.) de 0. 48.

rn 0. J 1 1 109. =

= c 1 c c .- mo .- 7) .-- 0 =

i 0 7) a C.) 7) C. CO e ble CU 0 a a a 0 a 0 0 Cl) a 0 cs) C. Ca (/) >-• 0 (-) 0 >- Lc >-

tre: :

ea : me ce ce: er l Ar p itance:

tan tan c

is iona a ce: t

duc itance tan c tan ce: Res In cap a is le ittance ble b duc ittan Crossec hase

res in cap dm ca ca p dm tor l a ble ble er a

_C hase p lle hase hase duc ca ca p ies p p

c ble er ara er

a) er con c ser p hase hase ca l l p

a) l p p p ble ble ble ble ta ta Ca Ca Per To Ca Per To

0 Tota Ca

C CNI

0 0

C (NI 0 O CU 0

>- "A

C CNI "A 0001 CU 0 0 CU 0.

4774

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C;') =

-o

l) from de ble mo ca ser DS001-0003 to he

C la ch f t le E0-

l o C V-

co de b Sing

( 0 Mo su

0 .%

C.) "A

(U 2081 C.) MVA CU 0. 0915

›- U) =

482. 7.) 01 N 0. = 1. (7) O II

C EBP0003.

S001- II D

: E0- V- ble

ca a_ CU O on

>, co ce U l) hase N cl) ittan

cr t: arameters mode

2 dm d per p d per l f p a ta o loa n ble

O laser e

d a. le d curren ion t ro ca te u ing dam ec S p l loa lida l CU ( ies Fu Exp Fun Va Ser •

..->,o M EL ....,o >, o 2. a_ c .f..._., ,_ ,,..„, , a t-.-) E co a a) c) in el U O a a -o a) C.) a, ,... 0 , c) (.) a) c O _o a c/) an 9 (f)■ E al "' MI -. Nc N •r- ›- >- 0 :0 g a) (.) (..) NI ° cf) ' 11 ) CD O E c‘:s 0 (n c.) a) .C..(1, , 0 U (0 a) (.) Ct. cot N ci) 0 -al ,,- 0 0 cr) CB a) 0 .nr >- )- 0 LL1 10 1.0 (if ..0 , 0 c __ , a_ _ N a 1l co 0 C) { I ul a) I L) ■I I ), (1) > LI. (C) QC C 0 LL1 (g) c CO a) -1:3 _a u.I ca ul 0) < L5 ci) (7, .,,c. 2 0 0 I 0 co > cT3 12 r) a) >- >-),_ > 0 03Q< —„a) lz)o _c,= ui, oal 0 0)_ c...) u) < .o 0 2 >.. T E 0 a- 0 0 C C ,- ,... 0 = l) 07 ce) a) a a) 9 `" 17 9 w C de N Nc.) o O Za ,T; `cr) (..) In

3 >. )"- E c'a 0 ' (..) c.) m U u).

t.) (i) r CO a 1 t

C e Zii h. N CO a) O M > ›- s

6_ C O C....) in 0) ..0 , C C

1 la 0, ::: 0 0 . . C 0 a 01 o 0 U..1 a M I 0 ILI 0 n 0. C coc (DC E U.1 e a. >' L 03 a) v a)> CO -.i > (1) ca /2 1.0 00 B .S7 tiJ cct a) -o _c Li' as 0 0 le a 0 0 0 o m , 0 A < E a >- )-. ( (Q c) 0 ILI 2 in < ..0 C .4- 0 >- 2 >, .2 ›.. a •Cr Er)o ...„. a E cn Ta. --- ,_ L... = ...... 2 8 0- a (n c ... = Mc Mc cp , . N (r) 0 a) C.) fq 0 -0 C.) .13 G IE o 0 .0 -c; a) ,, , ra c.) O. ... — Et) a c-4 a'c..) iv) as'.. 1. IT) a) 0 c-\) >- >.- 'l. ‘..•a) a) 0 co ► 0) -C t a II 0 0 0 ai ,....-* 0 c) c.c)I co LLI a a 0 0 Lu c 0 C 0 — "5woo u.i a a) -,-,) j, 9-c3 a t-- c) (.1.1 a .... CL .0 -.■ > Lc) h- _.1 co a) > co ._. co a) -10 .z) Lu as 0 0 > o , , o cO 0) -Co3 .0 = al, 003 IS-I 2 u) < -0 >- >- < 0 Ui 2 u) < _a

-o- X ri 2 >, c cr) c) E c") "ii. . . c.---. o _. g ,—, 2 c) = . c.,-) ,-- . ... , 0_ a) ce) c.) c.) -a co .. c c a) 9 r_ ) V) 0 co .-- -0 0 ci o -15 7:7, a s U Uo.. 0. E -g 0 Li 0 0 cn a. , 0 v- E ai0 0(.1) 2'3 ›- >. > aCI T.. C•I a) 0 u) ,- a) „..- i • E "9 C c) >- >" a) ,asn 0 ur o (r) col 0) -.E. 1 c) . 3 ‘.... , -.7... C..) CB 0 0 ,--` ac LU c "- co . IS Ly 2 .T. 1 N c al c -1 c9> .__ a) (f) ° if) ' 0 Nc C) -.1 a) a) > co '2 CO 0 I 0 -SI al N > CO >- ct 0 co a) lzo _az sul as0 c_) C.) tri (3 Li cn > 3 < >- t..) >- )" 0 Ill 2 0 < _o >-

a) o Fo ..- >,.. E r) :5_

o a a. ...-.. ,_ 0 c c e ._ cl el c EE a) a) 9 'n t.) ° n 17 0 tO 0 Jo G 'a).- c)(n 0. tta i a) E a"(.) (/) 0 co ' >- ›. LI I i

dm ili C1) 0 I.0 3 ocz .F., , 0

t a ,.... 0 0 c)C a` n a o LU 5) ,- 0. C Lill fo hu u) 1 76 > OS 12 '0 ja Lu (13 s 0 0 C.: = • o le a) u.I 2 0 < _o >- >-. b 1 (f) ( Ca

CONVERTER MODELS

Include:CAJHdata\JAN-H\ding2\Model\SUBCOMP\Sixpu ls. mcd

(Single laser m odel)

6 PULSEC ONVERTER LOADS II o

Eleven laser models 6 PULSE CONVERTER LOAD V14_0 6 P ULSE CONVERTER LOADV1 5_0 6 P ULSE CONVERTER LOAD V17_ 0 6 P ULSECON VERTER LOADV1 6_ 0 v 5 II > C C O

I C

C I > w 0. M -1 (/) V 0 z 3 w 0 O O tD — a. V) 6 PULS E CONVERTER LOAD V22_ 0 J 0C I 6 P ULSECON VERTER LOAD V20__ 0 CO O it > CV c) O C \I c II C\I C

> I c

Include:CAJHdata\JAN-H\ding2\Model\SUBCOMP\Pftrfoff.MC D Cl) Cl) a

: a) a)

e: C.) C ts: ts: C C ai ai ce: Cri RS C-) 0 en tance: C C

:•-• t.- tan tanc tance 0 C.) a7 RI CU is c RS (U d (0 (/) lemen lem duc 0- O. m .r/5 . (7) es duc du

in RS RI e e 0 2 a) r

in in

ce lter ciiI `ei5 `a5 Icr) lter ce: fi fi lter ce:

fi t.= I= 1.= ittan ittance

fi lter Ut ic ic

0 0 0 tan ic ic tan

._ c

on * C E "E C C is dm o o o dm du a a E E E E es

as in r harm harmon lter as 'iii 'Er Iiii lter harmon fi fi ..c -C -C -C ies ies h h harmon t h h C ..0 -C -C h 11t 5t N- 111 h. r (C) I,- a- Ser Ser 5t 7t

0 C 0 0 (\IC C T) CN1 O a TD co >- a) CL r- (/) ›- (II C O 0 ■ a o o 1 1 TD. C a) C >- 0- C N CD 0 r-- a a) >- Lc) 0 0 >- C II 0 0 N

CL 0-

ts: en

: lem e dmittance ittance a l dm lle a ra a lter h fi 11t Filter p Filter .0 co O cr, a) ›- NC Filter se ries admittance : )-• C

PASSIVEFI LTER MODELS (Single laser mode l) -cr tr, 0_ a 0_

Eleven laser models:

PASSIVE FILTER P35 PASSIVE FILTER P46 PASSIVE FILTER P7_14 PASSIVEFILTER P8_1 5PASS IVE FILTER P9_16 >- O- c0 U) >- I to 4- 1- c c >- a. ID CD > n. 4- 1- c . c a. < >- U) CI) W e- 1.7. -I I-• 11.1 >"' re U)

PASSIVEFILT ER P21_22 ▪ 0 CD N.0 1

- I >- CI- N- a ›- 0 H N - C I

co c"

VOLTAGE VECTOR NODAL ADMITTANCE MATRIX IL 7=1 17 E"0 2 "E' a' -5 .x „,,Cu _c -2 § —' _o 45 73 w. C . 0 ,7 co ,- .) 0a as c o a Cu — C • 8 c 2 9C N 2 E ca 0 Cu 4 :0.,,, a cr = a) Cu 2 a) 43 ..5x • c E o a) U 3 , ) - )17 -C a-a) " c_cu)45 ,cr ca ... 13 t--Ca) 4. _c -.E.-aal . o , ‹. -g Ta (-.1 :,.=. c -c= a)3 7 Ti .4_. r _c .7. _ca-c a,Eu)a) cDC = .._.a, r) ..= ca a) a C E a) u) ,T, a)i= „ = = 3 IT " a a) cn cr o (4 0 ■ , 3 , =..-. C . -,-, E

.._•c . _c ._, . .. Cu 8 c` ° Ir.) ° E .. a) a) -,--. E c$2 a) al ‘... (.0 c Ca. C.) a' a) c,,,, ( a) a) a E. 0 C a)E 2 .(,) ii -ca a .0 3 .-Cu - .). )) ,,_ E2.c.) I-C. Eus 2 2 . *i7i 2 a ci, 23 a ).c- a) a) 0 o -c E CI) a) c E a) 6 a) al U) 0 a) E ..c Cli (u 0c u) - 7 , 3

a) ..c Cu _C 46 0 0) - ( o o 0- 8 a a) c) " 6 )

Number of phases : 0 C a) a) (Eleven lasers) U) Number of n odes in system: Nodes : = 22 Nodes : = 50 E Heig ht of matrix per harm onic o rder: mh Nph• Nodes II X 0 Admittan ce matrix order: Xord : = Nph• Nodes II Yord : = k. Nph. Nodes Yord = 2640 Rang e-varia bles for prog ram use: _c a) a O C U) x E C

Input of adm ittance su b-matrices into main matrix I- c.)..G• -c ‘I 7:) 5a) _c C .;_. V t Cu' 1 L : .,,,- fu - a) 9 5 ''''0-=0.> .... a).a..'CO4- C Cn.,.. cm o u) ,_ > - E a) _cc.) OC .. Cfl 0_c- Cu a) ■ .... -0-I.- Cu c S u) a) c.)=,- a) :-0 cL 0 o c o) = .6-,cnt- 03 _C c 8 >9 CU a), 03 (0,--(/) E ■ E 2 ,- = - , -- c.)coS 4-4 -....•

-0 4.... .c 2 • - 4 i ...... a) . ; 1.... o eLco 2E w ,3) a) o . 6- c (,) -, 0 c a) as , U) (1) 03-... T,' caal 0 aS E c( C '"ac u) r.) C 13)2 ow o ._ 51 . ETA-aZ ..... a) ..°8 -o- - .... - 2.2o) -- •• ._ ,- - u a) C -, 3-C F. I.... 1 -i, 0 3 a) , 03 o 10 0 CcT E C cr Cu Cu as Cu 0 ca 6; o c>) ■• l 2 . C ., ) . - )

(y, Cu jul, i-:5ts_c c . .21 4... .2 1'1 (7i 0u).cCu a c c,..t.-.-a, c) 0oa) = Cu a) ca ■ o) -0 a) =5c 0 E = Cu , = n., .0 0 : - a), ) , > ) • ,

Cu-c.• -t= E." 0) _ooa) 03 .'-- = 2 E,.. :L= -0 ._ 8 2 • '5 -C . ..-, -6.- a) 0 x (n 1.— _CIC 0 0)2 0 a) aas C c o 1 5 Cu >, U3 uC c o. a) 0 -0Cu a) o in 0 CD . .... ,. : .

. 4 Cu

co 8 >-0 ..o -c - .c ( . ..- ::- ..c ,_ . _ .,- a) , cel 5 a) aE Cu 3 = Cu a a 0 0 = C.) ow .c a_ ..c 0) c 5 7 (i) `Eua ca a) .... • u ' = u " ) . . , ., , 0 a).4 •-... _ .=. a) 0 C -= . .7 4- ...., .-, 4-. as 0. C 8 0 0 a) 15. C 0 C.1 0 Cu E Cu E (i) x 82 Cu 2 3 ) - 8 )

-0 cu E a) g

0 o II .o '5: o .0 (ci U) o ------.. yr 03- C .o ------. ni c r) C ..----.. C., =e 0 CNI C.) _CI ›- vt I— ).- Cfl) Mccr) >- + = ...... __.-- 0 _CI )- 1 0 C ° -CI >" s..._../ II CO I .------.. '' II C'' (;';' en '5--. ÷ NI- ';--i i- CL 5:. (7:1 C ).- CO 4- C0 (n (NI = C in"r o = ›- vt _o _CI o 0 _a ni )- ).- ------, ai i------. C 4- C C -54 I- In (-n C .1. (N •cr CV CC). CL CC cn U;3"- 0 ------)-- 1 4- = .....___..-)- c _1:1 0 ›- IN\ I Nr Q 5.: -5:. _ oIn .0 c 0 Tr al C\I iri Of' ..----. 0 (1) = C 0 -0 = _CI CV ›- -CI ›- ,-- a, 4- -----, ›.- 0 I- ).- F- C ....._...... C Cn 1 CNI CV a.- 0 o >- .....___.. _a A -5: II II. ..-----. I- c cv cv r, covt =co . "-,5 "-.5, 0 X .sa (-4 cn ...... _.,)- )- ir rc 0 0 I IX la _ta II .0 >- >- r) W < 0- ..---. --5.-. ---- 0 2 CN C o vr .o 0 Lij ).-0 ....--,as =0 ' 2 o ,_ C _,, . (N ).- tx z 0 ,- .- 0 W < u) ›. cn I— >- < _J 2 LLJ -J < c) w-7( Z C 0 > - •

-0 .0

CO I CO CL CL 0 ›- .o > _o -0 _a .L3 4- CO V; al. 4- -I- ci ...-",.. -,----■ ....--Ne ..---:- — C C C Oc Oc I- N— N— IN VI ,-- or c 1— r ,— 1— CN \ (D I I I- NJ' CO CO I CO I CO CO I ›- a. 0 0 0 0 0 4- ›- ›- >- ›- )-- >- co I— C -I-- 1 I I Uc II a II II 11 1 Mc (N "4' CO 0 I C\ C \ "cl• I — IN (1_ Ci p- 1- C,1 VI .- /-• 1- I- 5..... ›- R.' 5. 5", 5:: .-->: irci -1- F- CO CD CD (3 tn."( C ( ..-.. = co .57( cn-.."( cnx cn-5" N '1 U) Cs/ ; (3 = ›... 5. 5.: "5.- ". 5: >- 0- _C2 _CI .C1 _CI ___--)- >- >- >- >-- (7) C) C) ;7 ; Ci cv rn'A.- C0 =co's"- =cnr ` =cg t =u)"." =(1; 5( = CO

O (N I- ›- 4- 0 ,-- I— .o ›.- c o- -I- ,.. X .-----. c c VJ ce (N CD l—. IN -a rei 0 la 0 a. Ca 1". a .12 .0 LL1 < >- .o al- )-- ci >- .0 co- co- ___, i ..---.7- , --, ----. i-- + ----. Ce E ,--'- . ,.... MC Nrc c I— c U I c., IN CNI (NI Lc) c r-- III 0' .1— .— ( \ lc l—• IN cf) in In 1-- 0 U) III 0 0 1— I- 0- M c Q (i) >- >- 0 >- ›- 0- )- < ° ›- •---...- ---__.....- ›.- •-____.." ).- -...._.... --...____... 1 I ...._____... I 1 I ---_--- I Z 1I II ii . < ,-. ■ ---. — -- o Z l .-- cs, N C7 v. oo WI■c >. >. i. --5. -->-: -5.. ---- W ,- —,- R;. cv-- IN CV in In tr) 0. 0. --.- --;-,- ir E u) cn cn-5,- , cn.3.-,- cn co cn . . = = = = = = = = = = _0 _C) JD .0 -CI Lii

_a .ct co- co ------. ------, c c 0 I's- CNI 1- I .0 V, 1 0 e- 07 1.... a. ------. c CL ›- _a )-• CD 4- -1- ______, 1 c c a) c 0 0 0- 0 1 1 ›- cn ,t 0 1 e- T.- 4- 1- 0 -0 CL 0 .o - -0 c .0 _a ›- ). 0 >. - CI ni 0) .ct 4- ----: 4- -1- _ ------.. c ------.. .o 0 ci -----... c 0 H H H c >- ..-----. i- H - _ r-- H c C■I c a ci c c a .- VI On I et vct -I- CD 0 0 I N- Cf) H N-.. l•••• n-- _ I T. I • I . 0 c I- I I I •I• CD CD N- I,- a) cf) a. c a) 1.0 0- 0 0 ›- CL CL cf) CL 0 ■ )- >- ›- ...... _..., _--- ...... ____, )- 0 ...... --, ---__--- -...____..- i I -----.--- ›- 1 1 1 1 ii i 1 . II II —. ^ c•I — — cn o .n. c■i 0 N. co .- csi r-- .- Cn co In ,- ,- '5: -5-; -5:, '5; '5; 1.1.-;- C-n- - 5: 5.. 5, . -, .-. ,-. --, 5: 5: 5, .-, .-. cv c.,, t..-) v. Q o o o Q) 0) 0) .- .- ic"*. ii ic ic" ic *ic "ii" ic -",i U) U) (/) V) U) (/)'. U)-Z.a U) U) U) U) = = = = =

›—›— ›— )— )-• )—

.cs _ca F C C CO l..... I I ..- C`,1 ..- .- a_ >- >- -4- 4- c 0c 0 1 I CNI V.- (.)w .0 (...) .0 -0 > _a ›- mi. co- ci -1- ,..--,, co 4- ..,---,,. „...-----, a ....----... c 4- I- Co I-- H 0 C c o- c c 1- 1-- 1- 1 CNI CV 1 .- N- ..- o f1 .- I I CD1 0- CO CO Cl- 0 0 >- 0 0 )- > "....--/ -....._____.›- ...... _____.>”' -....--, >- I I II II II II II . --.. ..-. CO Cg CO (0 .- .- CD ,- co 117 5,. 5, "5-, 5; '5', -5; 5; 5: 5: „--, -- — .- .- cv c4 c4 ,-, co U) oa -57,- 're .7,-. --X w x tn . U) 'ire U) U) co U) U) U) = = = -CI _CI -CI )— >•• ).— )— ›— ›- co N c ■ I .....---..., 1- >- -0 a. CO 1 tf) ...... ___...-- ,....-----.... = —... I I- ,.-- to ...... '5":.. co ›- -0 N- r CD 1 c >- a. I V) . .0 • co . .- r- ...-. O .._. 1 >. c I- 7 I ( .o a3 . .....-----,, 1- )- -0 a_ 00 'V I.0 ).- 0- r 1-0 0 — = -.....___.-- U) II -I- . to ,-. - 7( ›- -0 54 c >' a- — 1.. h•-• 0 0. , r- C 0 I I . V) i -I- Ja ti r„ '- r- .- .....-.. ii >. c C I I .0 " co' .....--., I- ›- .0 a_ ,l CD (0 .- _.....----..., = CO I I-- it' to '5" -;-- . ›- -0 )- a_ CO .1 r c .,- I C0'K ... .o . co - oo .-.. 5.. . c I- I .ca " to ,...------..... I- >' -0 a_ 1—• CY) CD ...... ____..--- CL r (1) 0 ------.. = U) -k '3"( to (1) .>- ›- -0 5.. a. — ‘-. CO .r c 0- ..-- co C 0 I I V) w -I- .0 ... CU it co .-- oo >. c C I I -0 co . .-- I- -- , )"- -0 r >- 0- N r 0) ›- = > w 0 (0 N r II 7" = Cl, I 7r ,- a) ...... "5:: N - c74 .. csi c - I " >. = t --- I .0 . . -0 m- co „....-----, I- I- )- -0 >- a_ •,- N 0) r >- CL 0) v•-• 0 ›- = U) ).- 0 CO N r N 0.- 0 0 r >- a_ N r II C N -I- = (/) II -I- -1- \I ic ,- a) ...-. ,- c) __ .- CV >. (; c C I I c (NI I I I .. .0 co .0 - ....___- ...-----... •----• ...------. >-* >- CL r CNI CV I- cn 11 >- .0 el >- CL r N 0 = N N - X.. (71 ---. > cn II c I .0 ir CsI o el -- co ------. 5.: c

I .o 0- to - - I- „..-----, 0-• >- a_ N 0.1 N N F- = cn II —. ).- -0 >- a_ Cr) 0 N >- CD) N 0 - CV 74 IN .--.. -5- = (/) - II rt 4- c I , 13 -5( 01 0 tsi o . CO 5.. c

C I I - .0 oi ,....--, I- ›- a_ ).- a. ›— N N CV r N N 0 = (n II 4- ii (N CV -, IN (V ---. >. c c I I

Bus admittance matricix for the network per harmonic order 4- (0 r X 0) a• 0 - _Ne ..c 0 a) cn O ° E C C C) c E X CT) a) U) a) U a) C a) To a) 0 a) — a) U 0 a) C (/) a) (.) 0 E o_ 0 C7) •

..o oa) E >-• 0 U cs)

DYNAMIC COMPENSATOR CURRENT F- 121 V= V) I— C -c a) z T as '- E O E o_ C a) C cn al 8 CD (/) a) a> ( O E ai u) 0 U) U) CU 0 a) C 5 a) a) O ci) C0) a p 101 - -0 1 ( al O a c > 0 cu a a 3 as 0- a) O cl 0 : i ) 3

1-- 0 c‘j cv ,— a) E CN1 (-) >. C 0 13 Z Z Z Z - 0 11) C < O Lij

ix tr

0 bma O c-4 0 0 0 0 = su

C CN E O

_ ,■■•

- a_

l ) l ) e im c 1 C dy dycr OR RN( O c (N 4-- O E 0

EADPRN( a) E

READP E c -o 0

= R C.) COMPENSAT 0 >-.

: ›c i

Cu ix G II MIC

tr E c 0 .ca o 0 c nre 0 bma O C. Idy u DYNA tN O O O ci O 0 .5-2 15c7) 6'3 = s c) 3 8 4— C 1-) CS) O 0 .0 a) = O O a) O

-o VECTOR

C a) '5 v) a_ a) C C CURRENT

(f) a) _c E O -es a)

a) CA a) O

Ui C a) 0=

a) L.

C O E _c

01 CO (0 ..-----■ .,"--", ---- \ as C a C C a) a 0 .2 I I I "a5 I's CO 0 O CA 1- O Cn _c co _c _c UN a) a) cr) C() 8 0

1:5 C O 't:73 E CE _c

a) co _c ts. OC O C O C CQ O In CO ten Cn C .0

con 61) E1) its E II II U iew '10 1.0) (1) v >ca N CA 5( O cc

Ux 0 X d to a) as Cn > > lle O a) o CD cr a) O s ce a) be LZ 1-1-1 to 0 u_ I— C may a) co a) ce >- _c _c cn tor C C6 . ,„ below O Cn 1-- ce vec C ,---, , , tor . 0 al ,_. c\ic e a) E a).

ec u) . 0 ° c. LLI — 0 0 a) a)

v cl) ltag 03 t LZ (r; ...----, E "----= > < ----.. E E .0 a) c o 0 c o o

vo >- a o Lo o 0 0 .— >, > 1-- ....i , (/) -o ,7_, to he > t Z z curren '58 ',.....--, •------I S■...." '...■.' '...■....■' O CU a) C/) I I II ite > II ILI Li II II II of _c

os .0 o) ion

t a) Cn CC .0

la C 5 cs a) a) CD CD CD CD u O > > > > MJ > > > comp C.) CI) lc E

U1.1.1 he a) 0 72 Ca T I— 0 .c CCI r'l = _coa_ I a a D Ci LLI —J —I 8. Z a)'ci) . w _c ...... ID' fp, 1 D TD 73 = 0- = —. . t r) O oco 'a a) vi fs E 8 0 U C > a) o O o 2 `6 2(1) a) C a)CD cu as c.) c C a) 0 E a) a) ` a) u) =cp aa) (/) a) •co Et a) 7 u) • C _a a) - . 2 ) .v) . a 1-71 ...'8 2 - _c c) - _c • • c 8 a c o a) = c.. a) (af2 C.) o c 0 a) o 8 oc a) o 03 a) o 21 E a) 03 E ) > CD( o) ° 1 , " 5 ) - :41-; _ E `C.) 0 l .g. = w - - - 2 = 1.5 cuo a3 a) = 5 2 o a a) ui 2 a c 0) o 8 a ) ..

Ll.l iT) -;_" V c CD co a) wa). 1

e (Eleven laser) z , ..kr Z Z tr) ii II - N 3 3 .

Node at which converter is located:: _a 0 0 c,-) co ED co Cy, _o as° a) a) as u) mu' Q3 cuu' a) II .N. .,— Voltag e and current base: ------. ------,....____- D E r I -1 /I .._... -- -C .. Z ,_. — z . CL + ' - 1 > III , -

1 1--1 ------, a) a) C a) Voltag e at converter node: ...._., N -C -c Z _ N z C CL O. + > I ...----. r, L-I /-` .__.... = z _ cn z . O. O. 4- > 1 ■

/ 24 0. 1268 - 1. 8733j 1 239.2879 - 2. 1614j .0 a) CQ a) C II 121.7146 - 207.0 182j N 121.6021 - 206. 1 51 1j N O CO CO •••-- ao N I _ 117.685 8 + 208. 3125j 0) C) a) C a) C C) a) C zT Cl) C N N -• Joint apparent power to converter : C

Sjgen i . Sbase = 53.0074 + 36.6942j •kVA Sjgen 2• Sbase = 53.2174 + 1 9. 7736j •kVA U) U) -0 0) C) a) C a) Cn Cl) a) C = 64.469 •kVA = 56.7722 •kVA

C 3 a a

I-

C) CO 0) C N N O cc it it

C 2 l un un 0 O tor tor II I mode duc I duc III .--.. •-• case) ser III ... C 0

in N la

c ...... , N ter

N N H in N -CI H C -0 N le a a Co CU co Co C u C C C C co Co N a) a) a) a) a) 2 m C) a) CT Sing E E Conver ( 3. co 0) 300 D 0 D

I- c CO co N C") a) 0 CT C)

a) it 7:3 it C

) M l un C un N

de co N tor a) tor o case O c II co m du 0 duc i in in

laser

rC ter co

C H H le

_ct ver 111 . III Ca Co Cu Co C C > 2 m Sing E (

a) a) Con 300 u 3. CT a) CD FORMS

: VE-

ters WA e hase: tor: IME d p fac t

aram : e t an p en

t n

u le: e: tm ltag s inp ltag l vo t dju curre ang e

t: tra vo en ion in h a l t rr eu dt ta e- i line u n in cu r to- k l to curren e lse-w ine Pu Pea L CALCULATIONS FOR T Line- Lin Lase Comm

O _0—

(0 r-- ai O C3) N O N

it it un un O a) tor case) tor r 130 c

c fII

( Ul

Iv du = du

-0 O CD CA co lase II 2 N in II v

in N N

14") H H

O > c a I 1 a) as N N U) 2 m

Eleven E Imax ( 300u C,C0 3. ti

a) CJ C0. O a) a) _o . 0 (Sr rn I- CES (0 ---, a) ( I- > .O co O co U) 4 0 ...E + O co o) 0 .4- ,-. a) Cu r--- 73 0 CD O ,-- I- 0

>. it

01 ..---. V) it .1- TD LI) >

un II un 1`.... IT,' O N a) tor case) Cn 1 CD er a ...... :.=.

duc CD HI a) ductor 0 (n

in >

las -cr > T3 3 I o in

U) H c.,:s H a (41 K III III (V L" C III a3 E., _ CD C) 2 m 3

Eleven E a) a) ( 300u cAo 3. 0 — ,0 c:::, -0CL C/) ..-.CD 0 E E ....C cias ..c..-

): ,._ c =0 er. 4■•• t ow CD p II) g ive t ac r fo t s Adju

( : le le. ng a ang n ing tio ta fir u e iv t a) ) a la _0) 8 . ?-• - c a)O = 5 0 Re Comm a_ a_ Q ca —3 Time waveform s of line-line voltage an d line cu rrent 0 0

DYN?

Harmonic compensation only Dynamic compensator 1 Dynamic compensator 1

(Sing le laser m ode l) (Sing le laser mode l)

Starting node for current: End ing node for current: a c

Interme diate impedan ce: (Eleven laser mode l) (Eleven laser model) z Z > 0 - U M In 0 U Starting node for current: c6) II

Ending node for current: 5 >— tr)

Interme diate impedance : = Ubase = Ubase380 Ubase C2 Ubase 380 a) !base C2 : = lbase380

Include: C:UHdata\JAN-H\ding2\Mode l\SUBCOMP\Dyncom. MCD

Storag e of compensator requiredharmon ic spectrum - . 11) a) t. O 8 a) 0 t= Ct 6 C c CO C.) 0. CDc— a) 3 2 C . ▪ • > 0 a3 > 0 c.) a) 0 . _cn a) > U) 6- o a 0 ai

0 U)) u c 0- 6 (1) 0 a) (‘ ..) (2 E a) C a) C ai

For compensation of fundamenta l on ly use 0 _c For compensation of harmon ic compon ents on ly use C C

For compensation of fundamenta l and harm on ic components use 5 co '5 c) U) 8 c _a C O C 0 U) a) a)

(a WRITEPRN( dycre l ) = Re

WRITEPRN( dycim l ) . = Im

on w < Ww W z < I CL Lu cn Cl) 0 — —I (9 1.1.1 W z C/) Ce in fa __I < _1 >-. scc W 0 0 1— Z Lij W 0 . -

- I- EL "CI •= - . = >,-c)c..,. 15 CI ... a) .= CU. _ TD I- a) 0 . E cr' Q) 1— 0 E 2 O-0 .2.-=0 (L 8 o a)- O oFe:: to o) 0 > c =ca>•-•(-)..- 2 0 al oa)2 ›... cn a ; 1 E cn (-) a) (..) - cn - C D cu 0) 0 fm.=..-. 0 > 0 ,T)a)o"c a) ...,.. C ° S a) c c o as cm u) o '-.- 0> 1 ,.: ct. crs 2 a) E,...... a) a) 0 U) ) 5 5 a . a) IT=0 Rea ding measured resu lts from disc: I__ _c .511 2 a)` -o cocn .-c, ... 1 ii,- 2 a) ,L a9 .A c g a9 cnu) ° '— ., ) ) I) 4 *- 0 ' " . u) 1 -FP - '-' _ c - _c a) ... , 2 c)a.)a) 3 7 0 a) 0 co => cn 0 .0 0 al 0 ° a) ( 5 c 0 cu 0 ,... .F c a) = E o r„a 0 i Im c : = READPR N(wf51) ) 2 u)= c0 .--; L.. Cn C.) .-E,a) =— . . '1:7) c '.. 0 1 4 1 - - T. i°40.. 1"/ - 0 a. 0= O a) ca) >" (7J ' - 0 c C> .> Cli o c u 01) a)c o -0a) a o o) . ` = 0;,0 1 c D_ ... ..c -I - .-- ' - - ) , 03 ) - 2c CD CID) 25- -- .2 .= - ' ''''' = a .- o -0 11 u) .a3 E c 2 0 a) 5 z 0 .LI ai C E' (a c.9" EL a a a) ..-. > 0) (ng 5 a) ..0 ) 3 0 , ca , '2

)'-' . c

ci) ._-0 44 eL .0 l _c -0 — 0 2 a c . a a) (0.3 0 o a) E a) , cnc 0, c a) c p ) a > 0 13 0 cn c CD a ) = an w -=;-; , .0 - MT '6 _a ,- f 0> ti s._ ..... _5 - - X 0 O C alE. 8 1 a a) a) =3 o c L a) o c CD CU 70 C 0 o) 2 c 0 5 a)t7, a) ›.. u) o ' a) C.)) "Ei C o T = u) a) ( U 5 cu c .._a) 6 6 o 0_ = 5 2 u) a) = N - Urn c : = READPRN(wf52) u ) E.° o .- Co _C 0 73 .0 .... - = t a -c t' 0 a) >. ' a 6 cal ( a = a 0 0 ca a) a .6 a) > t cn , 0 c ° ::-D ca c * 3L .....__.., .„---_, )

) ' 7 (I) . 7.-_.

.. N -a- O o 1--- .- - t 0- .

` c = U) E CCI xE E II CO (1) C.) 0 4- 0 _ _ _ = I... co `a-, ca co u) 2 = ,_ )

.c- = -0 C a) x I Ind iT) _ - ;7: > a) 0 a) cm a) 03 a) (Eleven laser) ',...__-, „--„, .c> -0 ' = 8 75 .,-- NI- 0 0 N N co = x E I tr) Tt) ZZZ5 CDC c.) 5 0 'cl 0 - _ _ _ - . Cn . 11 a) 0 1 I a 0 - c I C C tr) Z 5= ' C0 > 0 H > 0 0 I o I

< ...._. -C -le za, .... > a) ;2 0) > Cl) C) CD o C.3 Q) 0 8 o cn CI)

a) C.) U) II cr 0 0 (D 0 Lr) ..-- CD 0 8 II C.) CD 0. (a 0 1 Z 1 a.

. c..) ° E Upta c = 1.92 .V 0 C C .- ._ .6 -F 0_ 'a I.11C N- (0 ›.- a_ Cr) a a) o CD E c E 2 a. a) 0 -7 ' E E II E 2 I 11 o - - c I c c

75 sa > 2 2 Jo a) cn a) a) 0 u) 0 a) ca co

= Ubase380 Freq uency domain conversion by means of FFT:

Current base to choose (base G : = lbase380 E ...---. ! NODE VOLTAGECALCULATIONS a) C C E c__. .._...- c z Z -C -I- ; 0. : (.9 5 I a_ ,----, .__, ...... c N -C 4- >" 0. o I Voltag e at vo ltag e n ode: C) a) C C) a) C II C ..----. .o co O aS O a) a 4-- a t--I .._.. ..--... c cn .0 4- > 0. o I

239.2879 - 2. 1614j tf) CO 0 0) 0) •er N-: c0 CO co c) C) a) C U) a) co co 0 C) a) C fU a) II 0 - 121.602 1 - 206. 1 511j

- 11 7.6858 + 208. 3125j w [e a) C a) 0.1 F.

Equiva lent tota l node vo ltag e: a) C a) II * o N1 0 co C

C coa co M a) co co a) C a) cn o I

Ugene CT Ubase 380 = 239. 0301 •V N— CO w C) a) C a) a) C a) I a) C 0 Equiva lent fundam enta l voltag e: 0 L.L.1 V zT > a) as C as rn ai c E a) Ugen e ci - Ubase380 = 239. 0 13 •V Ugene Gi • Ubase380 = 239.2995 •V ; C) a) a) C C) a) C 04 0 rt

Equiva lent Harmonic Voltag e: C C) a) C C C

Ugene GH • Ubase380 = 12.7474 •V Ug en e cH • Ubase 380 = 2. 8597 •V

NODE-NODECURRENT CAL CULATIONS

Vo ltag e drop between curren t-measuring nodes :

co M ...._ ------. C ..---, ..".. Z O 0 .. ._, D 0 0 Averag e Current (to check for offset) a) E E 0 ..__ L.--.I ..._ _..., ..._... z .,C C zz — z — z 0. 0 0 4- 0 .. o. 4- -• 0 0 I 0.1 0_ co '/ ...... _. -----. 14. U ._.. ..• O a O a) C I ,___, L--....1 ...._., ...—, - z ..c 0.1 z c .0 • 0 0 4- 0 0. o. 4- 0 I ....___. ..' ...._.... ------. C ..-----... 14. a) 0 ,___, 1..--1 ■ - ...... . c r) z .0 r z z 0.2 0 . 0 4- o. 4- 0 _.., 1 Freq uency-domain current from measured data: ,.... LS 5= ...•- -0 H C) C CD X E a a) C a) O a) N a) - C) a) C) a) 0 0 0 0 ■ . _ Current between nodes:

-0 CO 0 61 (II CD C C C C C 4-

152.5343 + 3 1.6745j co O 7 (3) C co Cs1 0) O C C) a) C U) a) a) - 44.6074 - 1 50.3998j

- 1 07. 9369 + rs1 C) a) C a) a) a) C Equiva lent tota l node-n ode current: II C C

!g ene CT . lbase380 = 176.7591 •A a) C 73 " ILI c C C Cr 0 Ct1 C CI) 0 CD E a) co - 73 0 a)

Ig ene ci . lbase380 = 165. 1794 •A (g en e G1 . 1base380 = 1 57.7204 •A IN /4. C) C C) a) C C a) a) it 0 a) C O a) Equiva lent ha rm onic current: a) II 0 C M C

(g en e GH . Ibase380 = 1 5.4 112 •A (g en e cH . lbase380 = 62.925 •A

POWER CALCU LATIONS N c;) a) c cn c a) C7) cn a) I • I Tota l joint complex power: c up c c _.— vcs co CO -)C > co co a) .-- 0 U) Co -CI co 03 + a) a) cn CU II O - U) U) a) C a) 0) a) a) rn a) 0 H U co Tota l apparent power: 3• Ug en e GT• !g en e GT CI) 1 CN1 CO LO 0) U) Ch N a) 0) (II a) a) II — II

S Ch [.."1 U) 0) a) C II H C C a) a) a) a) C 0 0 0

5th harmonic ca lcu lation: ▪

I a) CO U, CO 1 U) U) 7 C CO C) a) C as ci) a) 11 — . Sbase = 1.4002 • kVA 4 H

r-- [ C 0 U) 0-1 C) a) C a) C 0) a) C N- 0

7th harmon ic ca lcu lation: C

ti II ----- cv 0, (V csi ----- ,-----.. s.—...-, 0 W U) 0 -C1 - U) a) C a) a) c c) 77 C) a) C Budeanu Distortion Power: 0) a) C a) 0 H I H I"••• H 0 H 0 1 — -e* ■

Dbud GT • Sbase = 49.4597 •kVA Dbud cT• Sbase = 45. 5877 •kVA D ca) a) c a) 0 c) a) cl) C C3 0 H 0 a) a) a) a 0 D )

MCD rac. ki p ec COMP\Czarn \SUB l de \Mo 2

C H\ding O \JAN- ta

27j JHda 89j 34 6861j 4249j c 3. 52 CA N 16. 5. 1 2 de: - 208. - 1 lu -

1 C c 2 + 42 3 + In a) 07 7662

64 C 894 0.

09. Cr) 2 1 120.69 229. 17 0. -

C = 380 =

380 base form lbase U - - z the

in C:)

b 199 3 s V Gc

Gcz O

Fe Al tor ene ene l to: Ig vec ua Ug ETEP he eq t is in

d C e der c

C or ang a) a) C) C a) h n a) C) c 0 C ea

are arr are (1) t C C)

for a O o o N I- ions

ower a) 4- 4- 4-

it d curren p

0 t NB fin a) an

a) (1) d e C t de a) C) C) a) aren C) an tar A ltag wer o s II N Nc app o v ts d - lex c Nc e s m

U r a) hase C ki. a) p C be C comp a) a) Ct) C)

rnec N The Num The Cza ▪ •A 1006 8. 27

380 =

CO lbase

- • -c N

N N lArms

C C C N NJ a) NJ a) 0 Nc C CD

CD a) a) C c 4- 4- N N

N N N C C N N N O C AI O NJ V V C C CD a) a) Al C NJ C a) C.) a) NJ C cr) C.) a) a) C C a) a) 4- C/) N a) N N a) a) CC C N NJ a) Nc •V •V 7 o C 0 a) a) 521 C 4- 4- C 599 C a) 2 a) CD a) C a) O C) a) C

21. C) a) 414. = 0 = 0 c 38 N 38 Nc c se -----r G.1 H Ubase CO Uba

• c . s a)C u) s E a) a) a) C a)

UArm ra) D D UBrm 0 V

H a) •W

AI 5611

N 1 522. =

C Sbase N B• P

a) rn 0 a NC c a) ca CO CC oCL 03 0_ c

AI II Q Al C C73 0 N as V H a_ a) •kW

a) V 9691 CL

CN 07.

Nc 1

0 = Nc

c a N •A Sbase

0 0 0 co) 03 H w PA.

+ + a) 1998 CD C:7) CD 0 o o a) rn C/)

108. 9- I I a) NJ CL o 0 C Nc 80 = 0 3 c <

se CL II

03 lba Nc GNI c s. c a) IBrm •kVA •kVA - 3003 392 CD 2 2.

5. c=5 = cy) = 4 =

base (t) A

se CD S 00 a) •kV B• Sba CC CD 59 F (7 S 8 S s •kVA 0 • 3. N co 59 0

12 0 C■I 8 lBrm I— CD - >- = s O 23. a) 0) _E II 1 II a = .....u)_.- UBrm co Sbase

• c (..9— se 0-) 0 cu CD N (1) N 0 CC Sba

GT Al 0 1:1 e S• N Sg en

as a)

kVA U) • 0 C a)

787 V

2 N 5. U) >- 11 Nc a)

= CE a) C) C C C) N CD c Sbase

o F o U") N a) S SA- SA- ca C) a) ower; 6 and 17 1 p t 'Arms

5, CD en

1 Nc n ar e: io t UArms• d ap ua a) ittanc

ce N Eq a) C) For U) U) >- Adm 0 0 a)

Effective conductan ce a) N co co 0 = 0.6284 -siemens

( UArms )• ( UArms ) ca a)

■ 149. 8726 - 1.71 86j 0

Active current 6, a) la • lbase 380 = I -76.51 96 - 0 cq M IN ca Ct 4-

Active power PaA : = UArms • lArms PaA• Sbase = 107.6038 -kW

Reactive power

The tins value of the Czarnecki reactive current can be calculated from the harmonic co - efficients vi 0 0 CNJ a) CD N I

Irh1 • lbase380 = 97.2759 -A

The reactive power in the circuit is then: a •cS UArms • Irh1 QrA• Sbase = 40. 323 -kVA 0 a) cv

a) cn (No 0 •A

-4- Grl 112 2 as: 3. ts

onen •VA 380 = 3183 •VA 3183 comp lbase 0. ic A• 30 Is 2 2300. rmon = = ha

Sbase he Sbase t A• •kVA as: d SB• Dg te la 3311 d from lcu 1. te ca = la A be lu - lcu C

Sbase N now ca n A. be ca UBrms t

Ds 03 a) onen can A t t a) en Dg rren comp

er or ow compon 0388 red cu red d d p 0.

tte a) a) ttere A =

lance ":*-• Iu sca cn sca

ba o 0 .4C 0 The Un The 0 as Ter 0 cb L Sgen e GT - Sbase = 123. 859 •kVA 0 as a) Nct N rv csi csi i U) a_ a cc 0 ct ct CO R C) 4- L

S• Sbase = 123. 5407 •kVA

ower calcula tions N N N U) U) I:t

To C) a) a) U) ta l non-ac C) a) C a)

tive power: a) Q) a) C H a) C) a) C H

Ngen e GT • Sbase = 63.3261 •kVA Ng en e c-r • Sbase = 63. 2579 •kVA U) U) ( C) a) a) a) C a) C 4 a) C a) 0) a) a) C) C 0 0 0 i Fundam enta l apparent power: ) — —

Sg ene Gi • Sbase = 107. 9626 + 40.2 974j •kVA Sg ene ci • Sbase = 1 09.8564 + 44.2677j •kVA <-4 Cl) Displa cement power factor: CD C a)

Xg en e = 0. 9369 .gene ci = 0. 9275 U) U) c;') a) C) a) C it C) a) C a) to) a) C a) Current distortion Power: 0

Sg en G I• Sbase = 44. 8747 •kVA Sgen ci • Sbase = 45. 11 97 •kVA C U) c4 rn a) C l) C) a) a) C) a) C a) C) C a) co a) C a) C) a) C a) . 0 0 )

Voltag e distortion power:

Sg en Gu • Sbase = 6. 1396 •kVA Sg en c u • Sbase = 1.4171 •kVA

A c

I -V en 5j C

0 OH Ug e

921 rn D a) -VA U) 1. -VA en a) 1 1 TH 5 a) Ig .a) •

• I

CL 'A 56 84 0 0 c " • c - 41

-% a) a) 5

cH 0 0 C

a) a) a) a) ene 96 539.

539. I 5078 C c

Cr) lg 1 ene a) a) 095 =

= D a) C) 8. e 17. I— 1.

s C 3 - a) THD = ba Ug C) = c = =

a) = : Sbase

C • • e : c a) H e

H C H a)

I. C en

base a) en cH

C c a) Cn S e e

C • I DUg en en en

ene 5 cH a) I g

S Ng TH THDlg Sg Sg C

en a) ar)

cD Sg C C a) u) A -V

G 01 9j

a) H C ene

a) 50. G Ug -VA -VA U) ne - 7 e 5 67 THD Ig 593 0 •

• 1 4

1 N I

8. 0 0 •% G

6. I l GH e a) a) -% 9

84 390. 51

e C C 7

a) en 1 a) a) a) a) O 34 1 2 C C C en

a) a) 32 9 - a) a) = D D rn C) C) 5. ti e =

Ug HDlg 38. U) = se II CD e

= C ba 3• = = T

= 0 G Sbas S - : • : G Sbas a) • H I a) e C I C C ene

I GH a) GH GH en

Gn a) GH

e a) Ug C C C Dlg en ene

en a) a) en ene a) C) C) I C) Sg THD Ng

TH I- Sg

Sg u) U) Sg z I-

n S er: ow

t er: er: ren ow a

ow p r: ic p t ive we t app l

o en mon ac ta p ar er: t

en har in on- ow d n app jo e t p ic ic ic dam THD: lis t on on fun en aren rm app Norma Curr Ha Harmon Harm Non-

O CL U) CL a) a) a) C) a) 4 0 0 N• - O a) C a) 0 07 U) cr) a) C;) a) a) 0 Tota l powe r factor:

Xgen GT = 0. 8594 kgen CT = 0. 8666 Cl) 43 44 .,1 4i al ii In k gi C In 0 0 2 1— C•1 ...... =I d c4 ..... t:$ t:I a) C 0 cy) a) C O C 4J I 4J -4) aJ 0 0 O 0 a) U) 0 la a) 0 O • N a) C l7 CD 1-1 0 p os it ive and nega t ive phase -sequence harmon ic curren t pha so rs: O CO (NI rn 0 O CNI 0 O Ir•-• co O CNI U, cvl O O (NI O 0 0 0 0 0 0 II • It 0 0

•kVA 5 238 C,1 5. 11

0 0 0— O = a) a) a) II base •kVA S

N 1 • VA

O 0004 .k

U) 0.

= l) S0121

I 31 01 ta O O 0. base en II S =

O I. dam 22 O base S

fun U) 0604j l - S01

I 0. iee for Su

. • I s.4 0 619 + 1 ly O +.) ._, I co 0 0. a) vos ta =. N o 4..) in ro o 0 (on 0 a) a) a al 0. t power: 01 Cn 0 ts: en na fa I 0 a) S0121

.4.) 4.-) (NI ar

-i 04 p

0 0 0 onen > q U a_ ,-- l ap 0,1

,-I U U 0 ta al —I II -,1 CN1 CNI N_

14 en 74 CV /4 comp 4./ 1) CNI 4.) a) O O

= a) D W he U) CS E

a7) dam (Al li 73 1 14 f t 0.1

a) u) US fun 0 H a) ..1 H W h o a) c 4) a) uence a) - U 4.-) S 0) ..-4 = 0 as U)

k a) c) ..-4 a) seq .i.) II I I

W I 14 ea in

ci) ive- as t 1 W t o CNI a) a as V—. ..-4 s-4 N wer N.) .o cn .-I a) c) 04 . 5 Po N D Neg 0) VA •k

•V 558j 8 •A 01 3. •kVA 591 7 39.03 2 7523 8389 +4 176. 0 = 09. 0 = 126. 38 1 38 = se Ubase

base Sbase = Sbase lba • T S T r • - r C - UREMENT C c c

ene ene en ene Ug (g MEAS Sg Sg

3 a — a) •kVA

C .• C 0 1 66j

CU — a) Ca C C C a) C 0 •V VA 39.54 a) co 0 •A C7) a)C D C •k a) 57 59 6388

C c a) 28 0 3 W 8

-a L.41 .. 4465 + a) >- NC 72. 1 C) 0 L4 239. 123. = 106. = = G G

C 0 .01 0

CU CD Cp Sbase

co • Ca ca Ibase C C 0 Ubase Sbase C C • • a) a) a) CI) C CT C) a) II " II G

G Gr e

C) G L1.1 0 0 0 e

o cn cu en en en 0 > C.) ene Sg Sg Z Z Z (g Ug S ORM F

ERS t: TIME WAVE-

en e rr

: FOR e

ltag e ower: cu o de p de: de t o ltag v

l : o ltag PARAMET o a n o ower:

t n t tr vo de t v aren u p e OF t de de-n No o in

urren e lin To o n c l no l j app to-n e curren l l CULATIONS t-

Curren a) to ta ta

e C ltag d- a) ta ta To To En Star CAL To Line- Lin Line To 0 Vo SUMMARY Sg en CH Sbase = -17. 5078 - 411.921 5j -VA Joint harm on ic power: Sg en GH • Sbase = -1516. 1 05 - 750.8019j -VA

Current THD: THDlg ene G = 38.9349 -% THDlg en e c = 38.095 - ')/0 THDUg ene G = 5.327 "'A

Fundam enta l node vo ltag e : Ugene • Ubase 380 = 239.013 -V Ug ene cH • Ubase380 = 2. 8597 •V Harm on ic n ode vo ltag e: Ig en e c • Ibase 380 = 165. 1 794 -A !g ene Gi Abase G = 160. 5462 -A Fundamenta l node-node current: (g ene cH • lbase 380 = 62. 925 -A Ig ene GH • l base G = 62. 5085 •A Harmonic n ode-n ode current: Agene ci = 0. 9275 Fundamenta l power factor: .g ene Gi = 0.9369 co C) a) Tota l power factor: Ag en GT = 0. 8594

5th harm onic Sg en GT5 • Sbase = - 1288. 5346 - 548. 054 9j -VA (/) Sba 734 = 4200. 5 O a) se -VA 0 U, • a) O

Sg ene • Sbase = 1 07. 9626 +4 0.2974j •kVA Sg en e c • Sbase = 109. 8564 + 44.2677j •kVA Fundam enta l apparent power: S1 U) 11 5. 2381 kVA 0) a) a) • U) C) C1) CI) 0 Sbase = 118.440 1 -kVA U) 0 0 (0 a) 0) C a) ca to a) • 1 09.8564 -kW Fundam enta l active power: P 1

Fundam enta l reactive power: Q1

Ngene GT • Sbase = 63. 3261 -kVA Ng ene CT Sbase = 63.2579 -kVA Total non-active power: N Snfapm• Sbase = 45. 1452 •kVA Non fundamenta l apparent power:Sn Snfap • Sbase = 11.4 077 •kVA

Current distortion power: Vl lh Sg en Gl. Sbase = 10.9904 •kVA Sg en c I • Sbase = 45. 1197 •kVA

Voltag e d istortion power:Vhll Sgen GU • Sbase = 2247. 1 557 •VA Harm onic apparent power:Vhlh Sg ene GH • Sbase = 219. 57 37 •VA " U) 3 2 CL a) C' 1Suiee l • Sbase = 0. 5575 •kVA u) 03- U) a) 3 a) c o, a_ a) o 0 .c c I- _a , a) , 0) CL U) cvi co co O a) c a) a) a) U) u) O II 0 a) `----' ....---... U) _a a) ctS a) in II I- 0) 0) -0 CV r- r- CO E a) aJ a) Rea ctive power: II Dbud u• Sbase = 45.5877 •kVA Distortion power Dbud GT • Sbase = 11.4789 •kVA

Czarnecki p ower definitions U a_ 0 a) Pak Sbase = 1 08. 9807 .kW PaAPr • Sbase = 109. 3886 •kW Reactive power: QrA• Sbase = 28.584 •kVA QrAPr• Sbase = 46.2529 •kVA

Scattered power: DsA. Sbase = 0.4684 •kVA

Unba lanced power: DuA• Sbase = 2.0613 • kVA DgA• Sbase = 0.211 5 .kVA Generated power: LL L7 8 U a) co a_ a_ ;. a. 0 a) L.:

1 SF• S base = 11. 1823 .kVA

CN 0 1 N C.

C s: O C 0 form

( C ve wa e ltag vo r o t O O ren C° ("! ° 0 (D N CO NI' 0 (0 N CO •Cr 0

0 o o oo o o o o Tr el VI CV N ur c 0 Cr t C C 0 rn ver

C In • Ci C) cn -r7

0 o 4

co CZ

0 Csi 0 N

0 0 h-

trum trum ec ec sp e sp ic ltag

0 on

vo O N 03 •11* CD C I CO 'V 0 ic VI N 0 0 o co cm vr 0 CD N 03 0 yr VI N /- harm t 0- 0 harmon C) C) C curren C cn line C)

t4. d line d line- lise e 4 a lis rm No Norma O

C C

. 5 O C

C., 0

0 0 0 0 0 0 0 o 0 0 0 CV V CD CO CO co cNi C'') f‘l C C oCC < I ° 5 > I

) . 1 t)

( G b des no ena ug t u hosen c inp e he t ltag o for v t e en in l - curr line ; 1 ine t) ( C G d l a) C) an ena e ug t ltag

u r.- C 'o vo CD

inp ,

C) 7 1 ..a. co ,-- ti) line co co o lay, 0 0 0 0 0 0 0 0 I,- CD CD CD 0 9924 0 0 c%,1 , 0 Cs1 •Cr CD CO 0 000 0 0 v"- (,) CV .—

0. , O a) 6 disp 0 i , f line- f e ca 1106. 1 - ltag CD o JZ1 l v forms o forms

a •••■ tr ve u 0 ne wa .C%C I ine- L ( Time R1

REFERENCES

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