of Information Engineering at the University of Padova, with whom I had the ...... subscripts R, S, T the phase currents can be arranged in a vector as follows: ...... Miller: âReactive Power Control in Electric Systemsâ 1st Ed, 1982, John Wiley.
UNIVERSITA' DEGLI STUDI DI PADOVA
Sede Amministrativa: Università degli Studi di Padova Dipartimento di Tecnica e Gestione dei Sistemi Industriali
SCUOLA DI DOTTORATO DI RICERCA IN INGEGNERIA INDUSTRIALE INDIRIZZO: MECCATRONICA E SISTEMI INDUSTRIALI CICLO XXI
COOPERATIVE CONTROL OF DISTRIBUTED COMPENSATION SYSTEMS IN ELECTRIC NETWORKS UNDER NONSINUSOIDAL OPERATIONS
Direttore della Scuola : Ch.mo Prof. Paolo F. Bariani Supervisore : Ch.mo Prof. Paolo Mattavelli Correlatore : Ch.mo Prof. Paolo Tenti
Dottoranda : Elisabetta Tedeschi
ABSTRACT
In the last decades an increasing number of non linear and time variant loads has been connected to the distribution network, thus affecting voltage and power quality in transmission and distribution networks. On the other hand, there is an increasing demand for premium electric power, in terms of quality and reliability, calling for regulatory provisions to limit the harmonic and unbalance impact of loads and technical provisions to reduce voltage distortion and asymmetry. Moreover, following international policies toward environmentsustainable development, power sources making use of renewable energy are developing fast and become more and more diffused in electrical networks. In the coming years, we therefore expect a capillary distribution of electronic interfaces controlling the energy flow from distributed “green” power sources and the distribution network. A completely new scenario for energy generation and management will therefore appear, in a situation where the distribution backbone remains substantially unchanged. The work of this dissertation takes origin from the above context, which shows the need for coordinated operation of the various power sources and compensation equipment distributed in the grid, in a situation where the supply is affected by distortion and asymmetry and the loads pollute harmonics, reactive currents and phase unbalance. The goal of this work is to establish a theoretical background and to develop criteria and control algorithms to manage cooperatively a system of distributed electronic interfaces and compensation equipment, so as to optimize the network operation in terms of power quality and efficient use of energy. This means to face the issue of reactive, harmonic and unbalance compensation, not only at local level, as commonly done at the time of writing, but at a system level. The primary concern is to exploit every equipment at its best, while avoiding detrimental interactions among them. An innovative and comprehensive approach is presented here, which is capable to deal with actual networks (characterized by unbalance, distortion and asymmetry) and is developed in the distributed and cooperative perspective. In particular, supervision (global) and local control algorithms are developed, which apply to every
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type of compensation equipment, from Static VAR Compensators to Active Power Filters and Electronic Interfaces. The control approach lays on a set of conservative quantities, which keep their meaning irrespective of voltage level and phase shift and are therefore capable to provide a general descriptive and communication background of the electrical system. Several control techniques are analyzed, underlining their advantages and drawbacks, and eventually a simplified control method is developed which can easily be implemented and applies both to singlephase and threephase systems. Such approach is analyzed in detail and extended to face the problems of unbalanced and asymmetrical systems. Several simulation results are provided to illustrate properties and operation of the solutions analyzed and developed in the various steps of the work.
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SOMMARIO
Negli ultimi decenni il crescente numero di carichi non lineari e tempovarianti collegati alla rete di distribuzione elettrica ha condizionato l’andamento della tensione e la qualità dell’energia nelle reti di trasmissione e distribuzione. Per contro si è parallelamente registrata un’importante domanda di potenza elettrica di elevata qualità, sia per quanto riguarda l’affidabilità della fornitura, sia per quanto riguarda il soddisfacimento di specifici requisiti sulle forme d’onda, il che ha richiesto interventi normativi per limitare l’inquinamento armonico e lo sbilanciamento dovuto ai carichi e provvedimenti tecnici per ridurre la distorsione di tensione e l’asimmetria. Inoltre, a seguito delle politiche internazionali per lo sviluppo sostenibile, i dispositivi che sfruttano l’energia delle fonti rinnovabili si stanno rapidamente sviluppando e stanno divenendo sempre più diffusi nelle reti elettriche. Viene così a delinearsi uno scenario completamente nuovo nella generazione e gestione dell’energia, che si inserisce però in una struttura della rete di distribuzione che rimane sostanzialmente inalterata rispetto al passato. Il presente lavoro di tesi nasce proprio in questo contesto e sottolinea la necessità di una gestione coordinata dei vari generatori e compensatori distribuiti in una rete elettrica, di per sé affetta da distorsione e asimmetria e ricca di carichi reattivi, distorcenti e sbilanciati. L’obiettivo della ricerca è quello di fornire una base teorica e sviluppare criteri e algoritmi di controllo per la gestione cooperativa di un sistema distribuito di interfacce elettroniche e sistemi di compensazione, così da ottimizzare il funzionamento della rete in termini di power quality ed efficienza nell’impiego dell’energia. Ciò significa affrontare il problema della compensazione armonica, reattiva e dello sbilanciamento, non più a livello locale, come a tutt’oggi viene fatto, ma piuttosto a livello di sistema. Il primo obbiettivo è quello di sfruttare ciascun compensatore al massimo delle proprie possibilità, evitando inoltre l’insorgere di risonanze o interazioni indesiderate tra le varie unità. Viene dunque presentato un approccio alla compensazione innovativo e completo, che risulta applicabile alle reti reali (caratterizzate da sbilanciamento, distorsione e
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asimmetria) ed è sviluppato nell’ottica distribuita e cooperativa. In particolar modo sono presentati diversi algoritmi di controllo, sia locali che centralizzati (globali), applicabili ad ogni tipo di compensatore, dagli Static Var Compensator ai filtri attivi di potenza e alle interfacce elettroniche. La strategia di controllo è basata su grandezze conservative, che mantengono uno specifico significato fisico indipendentemente dal livello di tensione e dallo sfasamento, risultando così un efficace mezzo per la descrizione e il trasferimento delle informazioni attraverso la rete elettrica. Vengono dunque analizzate diverse tecniche di controllo, sottolineandone vantaggi e svantaggi, fino ad arrivare ad una strategia semplificata, di implementazione particolarmente elementare e applicabile sia a sistemi monofase che trifase. Tale approccio è analizzato nel dettaglio e infine esteso alla considerazione di sistemi sbilanciati o asimmetrici. I risultati di numerose indagini simulative illustrano man mano proprietà e funzionamento delle varie soluzioni proposte.
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ACKNOWLEDGEMENTS
I would like to deeply thank my advisors Prof. Paolo Tenti and Prof. Paolo Mattavelli: a solid guide during my research activities, they helped me and understood me in every moment of these three years. Working with them was a pleasure and their stature, scientific, but above all human, will always represent a precious example. My acknowledgements go to Prof. Mauro Zigliotto and to the past and present colleagues of the Mechatronics Laboratory in Vicenza: Alessandro Costabeber, Filippo Dal Sasso, Roberto Losco e Luca Peretti for the positive atmosphere they helped to establish. I also warmly thank the whole Power Electronics Group (PEL) of the Department of Information Engineering at the University of Padova, with whom I had the chance to collaborate and share my experiences, both professional and personal. A special thank goes to Benoit Bidoggia, Luisa Coppola, Luca Corradini, Jonas Gazoli, Enrico Orietti, Vladimir Scarpa and Rosa Paola Venturini. Prof. Stefano Saggini, Walter Stefanutti and Daniele Trevisan from the University of Udine made pleasant and fruitful the whole time of our collaborations. I cannot forget the colleagues of the LC Magazine for helping me in broadening my horizons and for excusing my absences whenever deadlines of my research work would approach. Many friends shared with me the important path of these three years, both in happy and in difficult moments. Last, but not least, a big acknowledgement goes to my parents who supported my choices and respected my decisions, showing always their trust and love to me. I also thank the “Fondazione Studi Universitari di Vicenza” which supported this work with a PhD fellowship. Elisabetta Tedeschi
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COOPERATIVE CONTROL OF DISTRIBUTED COMPENSATION SYSTEMS IN ELECTRIC NETWORKS UNDER NONSINUSOIDAL OPERATIONS
CHAPTER I FRAMEWORK OF THE RESEARCH WORK 1.1. Introduction..................................................................................................1 1.2. The origin of distortion in electric networks.............................................2 1.3. Historical Power Theories...........................................................................3 1.3.1. 1.3.2. 1.3.3. 1.3.4. 1.3.5.
Budeanu Theory......................................................................................3 FryzeBuchholtzDepenbrock Theory (FBD Theory).............................4 Kusters and Moore Theory.....................................................................7 Czarnecki Theory (CPC Theory)............................................................9 Akagi and Nabae Theory (pq Theory)..................................................12
1.4. Compensation technologies presently applied in electric networks.......15 1.4.1. Passive filters.........................................................................................15 1.4.1.1. Detrimental effects of passive filters insertion..................................18 1.4.1.2. Effects of distorted voltage on passive filters...................................19 1.4.2. Static Var Compensators.......................................................................20 1.4.2.1. Thyristor Switched Capacitors..........................................................20 1.4.2.2. Thyristor Controlled Reactors...........................................................21 1.4.2.3. Hybrid solutions................................................................................22 1.4.2.4. Effects of distorted voltage on Static Var Compensators.................23 1.4.2.4.i. Effects of distorted voltage on Thyristor Switched Capacitors....23 1.4.2.4.ii. Effects of distorted voltage on Thyristor Controlled Reactors.....23 1.4.3. Active Power Filters..............................................................................24 1.4.3.1. APF Structure....................................................................................24
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1.4.3.2. 1.4.3.3. 1.4.3.4.
APF Topology...................................................................................25 APF Supply System..........................................................................26 APF Control......................................................................................27
1.5. Development trends in electric networks..................................................29 1.5.1. 1.5.2. 1.5.3.
The impact of distributed generation on electric grids.........................29 Electronic interfaces general topology..................................................32 Trends in the development of electronic interfaces for DE applications...........................................................................................33
1.6. The work of this dissertation.....................................................................34
CHAPTER II POWER QUANTITIES AND CURRENT COMPONENTS UNDER NON SINUSOIDAL OPERATION 2.1. Introduction.................................................................................................37 2.2. Approach to the study of networks working under distorted conditions.......................................................................................................38 2.3. Definition and properties of homovariables...........................................38 2.3.1. Scalar operators....................................................................................38 2.3.1.1. Demonstration of homovariables properties....................................40 2.3.2. Vector operators....................................................................................41
2.4. Homovariables in electric networks.........................................................42 2.4.1.
Conservation of homopowers...............................................................42
2.5. Instantaneous power definitions................................................................44 2.5.1. 2.5.2. 2.5.3. 2.5.4.
Instantaneous real power terms.............................................................45 Instantaneous imaginary power terms..................................................45 Instantaneous complex power................................................................46 Extension to multiphase systems..........................................................46
2.6. Average power definitions.........................................................................47 2.6.1. 2.6.2.
Active power.........................................................................................48 Reactive power.......................................................................................48
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2.7. Power terms associated to elementary bipoles.........................................49 2.8. Power absorption in a linear passive network.........................................50 2.9. Current terms definition............................................................................51 2.10. Complex, apparent and distortion power..............................................56 2.11. Extension to multiphase systems...........................................................58 2.12. Other properties of the proposed power definitions.............................61 2.13. Comparison between the proposed theory and other ones...................63 2.13.1. 2.13.2. 2.13.3.
Comparison with Kusters and Moore Theory...................................63 Comparison with Czarnecki Theory.................................................69 Comparison with Akagi and Nabae Theory.....................................73
2.14. Conclusions...............................................................................................80
CHAPTER III COMPENSATION CONTROL PRINCIPLES UNDER STATIONARY AND DYNAMIC CONDITIONS 3.1. Introduction................................................................................................83 3.2. Fundamentals about compensation .........................................................84 3.3. Stationary and dynamic control................................................................86 3.4. Goal of the compensation...........................................................................87 3.5. Distributed, cooperative and delocalized compensation.........................88 3.6. Advantages of cooperative and distributed compensation.....................90 3.7. State of the art of distributed and cooperative compensation................91 3.8. Proposed approach to distributed and cooperative compensation........92 3.9. Differences in the approach to quasistationary and dynamic compensation.................................................................................................93 3.10. Strategy of compensation of the different current components...........94 3.11. Typical control scheme for distributed compensation..........................97 3.11.1. 3.11.2.
Central Control Unit........................................................................98 Local Control Unit..........................................................................100
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3.12. Conclusions.............................................................................................101
CHAPTER IV IMPLEMENTATION OF LOCAL CONTROL UNITS IN SINGLE PHASE SYSTEMS 4.1. Introduction...............................................................................................103 4.2. Main goal of compensation: first approaches .......................................104 4.2.1.
Compensation based on q zeroing: application example....................105
4.3. Introduction of complex power...............................................................107 4.3.1.
Minimization problem in terms of power...........................................110
4.4. Basic principle of a possible implementation.........................................111 Comparison with the optimization technique in the frequency domain......................................................................................................113 4.4.2. Stability analysis..................................................................................116 4.4.3. Dynamic algorithm: application example...........................................117 4.4.4. Limits of the proposed solution and alternative strategies.................121 4.4.1.
4.5. Conclusions................................................................................................122
CHAPTER V APPLICATION OF INSTANTANEOUS POWER CONTROL TECHNIQUE TO THREEPHASE SYSTEMS 5.1. Introduction...............................................................................................125 5.2. Local Control Units implementations in three phase systems .............126 5.2.1. Derivation of power transformation matrices.....................................126 5.2.1.1. Full rank transformation matrix....................................................128 5.2.1.2. Non full rank transformation matrix.............................................130
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5.3. Calculation of power matrix coefficients from measures at transformers terminals.......................................................................................................133 5.3.1.
Calculation of matrix parameters: application examples...................136
5.4. LCU implementations in three phase systems........................................138 5.4.1.
Application example............................................................................138
5.5. Conclusions................................................................................................142
CHAPTER VI SIMPLIFIED POWER COMPENSATION STRATEGY FOR SINGLE PHASE AND THREEPHASE SYSTEMS 6.1. Introduction...............................................................................................145 6.2. Simplified definition of instantaneous complex power.........................146 6.3. Compensators control by complex power command.............................146 6.3.1. Threephase threewire units..............................................................147 6.3.2. Singlephase units................................................................................147 6.3.2.1. Choice of the weights for the cost function..................................148 6.3.3. Threephase fourwire units................................................................151
6.4. Basic principle of implementation...........................................................153 6.4.1. 6.4.2.
Threephase threewire systems: application example........................155 Singlephase systems: application example.........................................157
6.5. Conclusions................................................................................................158
CHAPTER VII POWER COMPENSATION STRATEGIES FOR THREEPHASE UNBALANCED AND ASYMMETRICAL SYSTEMS 7.1. Introduction...............................................................................................161 7.2. Compensation of load unbalance using APFs: application example...162
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7.3. Use of SVCs for unbalance compensation..............................................163 7.3.1. Extension of the Steinmetz method to nonsinusoidal systems............164 7.3.1.1. Symmetrical voltage source..........................................................166 7.3.1.2. Asymmetrical voltage source........................................................167 7.3.2. Generation of active and reactive power references...........................169 7.3.3. The distributed compensation approach applied to load balancing...170 7.3.4. Application example: cooperative compensation of load unbalance..172
7.4. Generalization of symmetrical components for nonsinusoidal systems..........................................................................................................175 7.4.1. 7.4.2. 7.4.3. 7.4.4. 7.4.5.
Definition of symmetrical components under distorted operation......175 The relation to the Fourier series expansion.......................................178 Properties of the generalized symmetrical components......................182 Generalized symmetrical components: application example..............185 Comparison with Depenbrock decomposition.....................................186
7.5. Conclusions................................................................................................188
CHAPTER VIII CONCLUSIONS AND FUTURE WORK...........................................................191
APPENDIX A: BASICS OF VECTOR ALGEBRA ............................................193 APPENDIX B: BASICS OF MATHEMATICAL OPTIMIZATION .................197
BIBLIOGRAPHY..................................................................................................201
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LIST OF FIGURES
Fig. 1.1  Bode diagram of the impedance of an ideal LC series passive filter with resonant frequency fr = 250 Hz............................................................................. Fig. 1.2  Bode diagram of the impedance of an LC series passive filter with resonant frequency fr= 250 Hz and merit factor Mf= 40.................................................... Fig. 1.3  Scheme of principle of the functioning of a passive filter for harmonic compensation, with the corresponding impedance Bode diagram......................... Fig. 1.4  Thyristor Switched Capacitors (TSCs).................................................................. Fig. 1.5  Thyristor Controlled Reactor (TCR)...................................................................... Fig. 1.6  Hybrid filter made up of TSCs, TCR, a low pass filter and a capacitor................ Fig. 1.7  Currentfed shunt Active Power Filter................................................................... Fig. 1.8  Voltagefed shunt Active Power Filter.................................................................. Fig. 1.9  Series Active Power Filter..................................................................................... Fig. 1.10  Hybrid Filter made up of a passive LC series and a series APF............................ Fig. 1.11  Costs of power electronics compared to total costs for DE systems...................... Fig. 1.12  General block diagram of typical DE power electronic system............................. Fig. 1.13  The context of the present research work.............................................................. Fig. 2.1  Source sinusoidal voltage and load current waveforms....................................... Fig. 2.2  Load current decomposition into active and reactive current according to K&M theory.................................................................................................................... Fig. 2.3a  Load current decomposition into active, inductive reactive and residual inductive reactive term according to K&M Theory............................................... Fig. 2.3b  Load current decomposition into active, capacitive reactive and residual capacitive reactive term according to K&M theory............................................... Fig. 2.4  Load current decomposition into active, reactive and void terms according to the proposed theory............................................................................................... Fig. 2.5  Load void current decomposition into scattering active/reactive and generated terms according to the proposed theory................................................................. Fig. 2.6  Source distorted voltage and load current waveforms........................................... Fig. 2.7  Load current decomposition into active and reactive current according to K&M theory.................................................................................................................... Fig. 2.8a  Load current decomposition into active, inductive reactive and residual inductive reactive term according to K&M theory................................................ Fig. 2.8b  Load current decomposition into active, capacitive reactive and residual capacitive reactive term according to K&M theory...................................................... Fig. 2.9  Load current decomposition into active, reactive and void terms according to the proposed theory............................................................................................... Fig. 2.10  Load void current decomposition into scattering active/reactive and generated terms according to the proposed theory................................................................. Fig. 2.11  Active and reactive powers (sinusoidal operations)............................................... Fig. 2.12  Active and reactive powers (distorted operations).................................................
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16 16 17 21 21 22 26 26 26 26 31 32 35 65 65 65 65 66 66 68 68 68 68 68 68 69 69
Fig. 2.13  Load current decomposition according to CPC theory......................................... 70 Fig.2.14a Power decomposition according to Czarnecki theory.......................................... 71 Fig.2.14b Power decomposition according to the proposed theory....................................... 71 Fig.2.15  Phase to neutral source voltage of the considered threephase four wire system... 74 Fig.2.16  Phase currents of the considered threephase four wire system............................... 74 Fig.2.17  Instantaneous and average active and reactive powers according to the AkagiNabae theory........................................................................................................ 74 Fig.2.18  Instantaneous and average active and reactive powers according to the proposed theory.................................................................................................................... 74 Fig.2.19  Instantaneous active (phase) current components according to the AkagiNabae theory.................................................................................................................... 75 Fig.2.20  Instantaneous active balanced current components according to the proposed theory.................................................................................................................... 75 Fig.2.21  Instantaneous reactive current components according to the AkagiNabae theory 75 Fig.2.22  Instantaneous reactive balanced current components according to the proposed theory.................................................................................................................... 75 Fig.2.23  Instantaneous zero sequence current component according to the AkagiNabae theory.................................................................................................................... 76 Fig.2.24  Instantaneous void current components according to the proposed theory............ 76 Fig.2.25  Instantaneous unbalanced active currents according to the proposed theory......... 76 Fig.2.26  Instantaneous unbalanced reactive currents according to the proposed theory...... 76 Fig.2.27  Phase to neutral source voltages of the considered threephase fourwire system 77 Fig.2.28  Phase currents of the considered threephase fourwire system............................. 77 Fig.2.29  Instantaneous and average active and reactive powers according to the AkagiNabae theory........................................................................................................ 78 Fig.2.30  Instantaneous and average active and reactive powers according to the proposed theory.................................................................................................................... 78 Fig.2.31  Instantaneous active (phase) current components according to the AkagiNabae theory.................................................................................................................... 78 Fig.2.32  Instantaneous active balanced current components according to the proposed theory.................................................................................................................... 78 Fig.2.33  Instantaneous unbalanced active currents according to the proposed theory......... 79 Fig.2.34  Instantaneous reactive (phase) currents according to the AkagiNabae theory.... 79 Fig.2.35  Instantaneous balanced reactive current according to the proposed theory........... 79 Fig.2.36  Instantaneous unbalanced reactive current according to the proposed theory...... 79 Fig.2.37  Instantaneous zero sequence current component according to the AkagiNabae 79 theory.................................................................................................................... Fig.2.38  Instantaneous void current according to the proposed theory............................... 79 Fig.3.1  Single phase sinusoidal system: non ideality due to phase displacement between voltage and current fundamentals.......................................................................... 84 Fig.3.2  Single phase nonsinusoidal system: non ideality due to current harmonics, even if no fundamental displacement is present............................................................. 84 Fig.3.3  Single phase nonsinusoidal system: non ideality due to both phase displacement between voltage and current fundamental and to current harmonics............................................................................................................. 85 Fig.3.4  Threephase sinusoidal system: non ideality due to load current unbalance.......... 85 Fig.3.5  Distributed compensation system.......................................................................... 89 Fig.3.6  Generic structure of a distributed control system................................................... 98 Fig.4.1  Load voltage u and current i without compensation (top) and with compensation (bottom)................................................................................................................ 105 Fig.4.2  Compensation instantaneous imaginary power vs. the opposite of load instantaneous imaginary power.............................................................................. 105
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Fig.4.3  Voltage u and current i at load terminals before compensation in presence of a load current step in t = 0.1 s................................................................................... Fig.4.4  Voltage u and current i at load terminals after compensation in presence of a load current step in t = 0.1 s................................................................................... Fig.4.5a  Triangular current waveform in case of fixed switching period and variable current slopes......................................................................................................... Fig.4.5b  Triangular current waveform in case of variable switching period and fixed current slopes......................................................................................................... Fig.4.6  Application example of distributed compensation strategy................................... Fig.4.7  Distorted load current (iL) and sinusoidal applied voltage (uL).............................. Fig.4.8  Voltage at the load section (uL) and at the APF section (uF).................................. Fig.4.9  Real power reference (p*) and actual real power obtained at the APF section (p) Fig.4.10  Imaginary power reference (q*) and actual real power obtained at the APF section (q).............................................................................................................. Fig.4.11  Real power obtained by our “instantaneous” approach (p) and by the Matlab minimization (pstat)................................................................................................. Fig.4.12  Imaginary power obtained by our “instantaneous” approach (q) and by the Matlab minimization (qstat)..................................................................................... Fig.4.13  Load current (iL) and APF current (iF) with sinusoidal voltage source.................. Fig.4.14  Value of the Fourier coefficients as a function of the harmonic order for the APF current, with sinusoidal voltage source.......................................................... Fig.4.15  Load current (iL) and APF current (iF) with distorted voltage source.................... Fig.4.16  Value of the Fourier coefficients as a function of the harmonic order for the APF current, with distorted voltage source............................................................ Fig.4.17  Distorted load current (iL) and sinusoidal applied voltage (uL).............................. Fig.4.18  Voltage at the load section (uL) and at the APF section (uF).................................. Fig.4.19  APF current reference elaborated in a transient simulation using the dynamical choice of stability coefficients, a and b.................................................................. Fig.4.20  APF current reference elaborated in a transient simulation using fixed unity stability coefficients, a and b................................................................................. Fig.4.21  Dynamically selected weights (a in blue and b in green) and stability function trend (in red).......................................................................................................... Fig.4.22  Load current (iL) and APF current (iF)................................................................... Fig.4.23  Real power reference (p*) and actual real power (p) obtained at the APF section with the dynamically adjusted algorithm............................................................... Fig.4.24  Imaginary power reference (q*) and actual imaginary power (q) obtained at the APF section with the dynamically adjusted algorithm........................................... Fig.4.25  Real power reference (p*PCC) and actual real power (pPCC) obtained at the PCC with the dynamically adjusted algorithm............................................................... Fig.4.26  Imaginary power reference (q*PCC) and actual imaginary power (qPCC) obtained at the PCC with the dynamically adjusted algorithm............................................. Fig.5.1  Three phase three wire ideal transformer............................................................... Fig.5.2  Vector representation of primary and secondary voltage fundamentals................ Fig.5.3  Vector representation to show that each secondary voltage can be express as a linear combination of two primary voltages.......................................................... Fig.5.4  Transformer input voltages for test 1..................................................................... Fig.5.5  Transformer output voltages for test 1................................................................... Fig.5.6  Transformer input voltages for test 2..................................................................... Fig.5.7  Transformer output voltages for test 2................................................................... Fig.5.8  Application example: three phase network............................................................ Fig.5.9  Control scheme for the generation of the APF current reference........................... Fig.5.10  Voltage and current of thyristor rectifier...............................................................
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106 106 109 109 112 113 113 114 114 114 114 115 115 116 116 118 118 118 118 119 119 119 119 120 120 126 134 134 136 136 137 137 139 140 141
Fig.5.11  Voltage and current of ohmicinductive load......................................................... Fig.5.12  Voltage and currents of the SVC........................................................................... Fig.5.13  Effect of the SVC at the PCC: voltage and current occurring before (i’PCC, dotted line) and after (i’’PCC, continuous line) insertion......................................... Fig.5.14  Voltage and current of the APF............................................................................. Fig.5.15  Effect of the APF at the PCC: voltage and current occurring before (i’’PCC, dotted line) and after (i’’’PCC, continuous line) insertion....................................... Fig.6.1  Trend of the function Ψ as a function of the weight b in the case of u 2 > u) 2 ...... Fig.6.2  Trend of the weight a as a function of the weight b in the case of u 2 > u) 2 .......... Fig.6.3  Trend of the weights as a function of u) 2 / u 2 in the case u 2 > u) 2 ......................... Fig.6.4  Trend of the weights as a function of u 2 / u) 2 in the case of u) 2 > u 2 ...................
141 141
Fig.6.5  Ideal representation of a threephase fourwire compensator................................ Fig.6.6  Generic implementation of a distributed compensation system based on instantaneous complex power................................................................................ Fig.6.7  Application example: three phase network............................................................ Fig.6.8  Three phase threewire application: phase 1 input current (plotted with a 0.8 scale factor) and voltage........................................................................................ Fig.6.9  Threephase application: instantaneous real and imaginary power in steady state conditions............................................................................................................. Fig.6.10  Application example: single phase network........................................................... Fig.6.11  Singlephase application: input voltage and APF voltage before and after compensation........................................................................................................ Fig.6.12  Singlephase application: input voltage and current before and after compensation........................................................................................................ Fig.7.1  Application example: threephase network, with a single phase load.................... Fig.7.2  Currents of the three phases at the input section.................................................... Fig.7.3  Currents of the three phases of the Active Power Filter......................................... Fig.7.4  Steinmetz compensator.......................................................................................... Fig.7.5  Application example.............................................................................................. Fig.7.6  Currents of the three phases at the PCC section.................................................... Fig.7.7  Phase currents (ic1, ic2, ic3) provided by the TCR + APF compensators................. Fig.7.8  Line (to neutral) voltages of the considered example............................................ Fig.7.9  Homopolar sequence component........................................................................... Fig.7.10  Generalized heteropolar sequence components..................................................... Fig.7.11  Generalized positive sequence components........................................................... Fig.7.12  Generalized negative sequence components.......................................................... Fig.7.13  Generalized residual components.......................................................................... Fig.7.14  Line to line voltages corresponding to the line to neutral voltages of the proposed example.................................................................................................. Fig.7.15  Homopolar sequence component........................................................................... Fig.7.16  Generalized direct sequence components according to Depenbrock decomposition...................................................................................................... Fig.7.17  Generalized inverse sequence components according to Depenbrock decomposition......................................................................................................
152
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141 141 141 149 149 150 150
153 155 156 156 158 158 157 162 163 163 164 173 174 174 185 185 186 186 186 186 187 187 187 187
CHAPTER I
FRAMEWORK OF THE RESEARCH WORK
1.1 Introduction The goal of this chapter is to introduce the problem of waveform distortion and other nonidealities of electric networks, to create a suitable background for the following research work, which is mainly oriented to compensation. Here, at first, the origin of distortion is briefly explained, then it is pointed out how the presence of reactive power, harmonic pollution and unbalance in electric systems lets them operate in a very different condition with respect to what expected in traditional sinusoidal analysis. The effect of distortion and the peculiarity of energy transmission phenomena under nonsinusoidal conditions have been deeply investigated for several decades, and a lot of different studies, also known as “Power Theories”, were proposed. It is worth to note that such analysis can be extremely different from one another, especially because of the different goals they devote to. When dealing with nonlinear and time invariant networks, the main objective of the study can be: 
The analysis of network nonidealities in association to energy related phenomena. The compensation of undesired current and power terms, which correspond to wasted energy. The introduction of measuring procedures, aimed at identifying the sources of unwanted current and power components and their relevance on network operation, so that a correct distribution of the costs associated to compensation can be applied.
1
The debate on these issues continues to be very participate and productive and some of the proposed contributions have become “historical” in this field. This is the reason why the first part of this chapter is dedicated to some of the most important Power Theories which have been up to now elaborated, underlining their major contributions to the discipline, but also their weak points. After creating this necessary historical background, the attention will be focused on a specific and real problem, i.e. the compensation in electric networks. In particular, the goal of this introductive part will be to present the main technologies that are currently applied to solve harmonic and reactive power problems. A brief review of passive filters, Static Var Compensators and Active Power Filters is presented, so that the state of the art in compensation technologies is hinted. The last part of this chapter gives a perspective on future developments, in the sense that the major trends in energy generation and distribution, which are related to the exploitation of renewable energy resources, are outlined. From this starting point also the future scenario for harmonic, reactive and unbalance compensation is presented, together with the very promising opportunities offered in the distributed compensation perspective. 1.2 The origin of distortion in electric networks The number of nonlinear and timevariant electric loads is continuously increasing both in singlephase (domestic lighting and heating, TV, computers and telecom, air conditioners, cooking sets, etc.) and in threephase applications (variable speed motors, industrial loads like arc furnaces, etc.). As a consequence, many nonidealities are introduced in the distribution systems, in a sort of dominoeffect. In fact if a non linear and timevariant load, which absorbs many current harmonics, is connected to the electric grid, a detrimental effect also for all the other loads which are connected to the same supply line is detected. This depends on the non ideality of the distribution system, which is characterized by a non zero line impedance. The non sinusoidal current absorption causes a corresponding voltage drop on the line impedance, so that the voltage results distorted, too. As a consequence, also the linear and time invariant loads which are connected to the same port are supplied by a nonideal voltage source and many negative consequences may occur. Harmonic pollution determines malfunctioning in electric equipment, excessive heating, undesired vibrations up to fatal failures. This explains the more and more
2
urgent need for harmonic, unbalance and reactive compensation and the many efforts made in this direction. 1.3 Historical Power Theories In the following paragraphs the basics about the most important Power Theories [134] which were developed in the last decades are presented, while further details related to the comparison with the main theory proposed in this research work will be provided in the next chapter. 1.3.1 Budeanu Theory Budeanu Theory [1], which dates back to 1927, represents the first approach to the analysis of electric networks working under nonsinusoidal conditions. It is considered one of the classic Power Theories even if it has been widely and reasonably questioned, and its historical importance is the reason why it is here briefly recalled. It is developed in the frequency domain, thus it exploits the Fourier series to analyze the electric quantities. Neglecting the continuous component, if voltage and current of a specified network port are considered, it can be written: (1.1a) u (t ) = ∑ 2 ⋅ U n sen(nωt + α n ) n
i (t ) = ∑ 2 ⋅ I n sen(nωt + β n )
(1.1b)
n
The instantaneous power is defined as follows: p (t ) = u (t ) ⋅ i (t )
(1.2)
and its average value, which represents the active power, obviously results: T 1 P = ∫ p(t )dt = ∑ U n ⋅ I n cos(ϕ n ) (1.3) T 0 n Budeanu’s idea was to define reactive power by analogy with active one, i.e.: Q B = ∑ U n ⋅ I n sen(ϕ n ) = ∑ Q n (1.4) n
n
To complete the power balance, Budeanu introduced the distorting power: D B2 = S 2 − P 2 − Q B2 (1.5) The main criticisms against this theory, which were moved by Czarnecki in [2] are that the defined power terms are not related to specific physical phenomena and, at the same time, they do not provide any useful information for compensation.
3
Moreover, the so called “distorting power” is not an index of the actual voltage and current distortion. As regards reactive power, each of its harmonic components can be associated to the energetic meaning of energy storage while their sum loses this property since opposite sign terms can cancel each other. As demonstrated by Czarnecki, it is possible that at a network port QB is null, but at the same time the power factor is not equal to unity. Moreover, the distortion power does not quantify waveform distortion, in the sense that it is possible to obtain DB = 0 (when complex impedances are equal at each of the harmonic frequency) but voltage and current have different waveforms. On the other hand it is possible to have DB ≠ 0 when voltage and current maintain the same waveform and are only shifted one another.
1.3.2 FryzeBuchholzDepenbrock Theory (FBD Theory) The FBD Theory collects the contribution of three authors [3,4,6,7], in the sense that Depenbrock extends the Fryze concept of active and non active power and current, from single to multiphase systems and, at the same time, exploits some of the definitions of apparent power which were originally elaborated by Buchholz. The first theory in the timedomain for singlephase systems under nonsinusoidal conditions [3] was developed by Fryze in 1931. His contribution is, now, the unavoidable basis for each following timedomain study. The starting point was the concept of apparent power, which is defined as follows: S =U ⋅I (1.6) where capital letters denote the RMS value, that, for the generic quantity X results: T
1 2 x (t )dt . T ∫0
X =
(1.7)
Apparent power can be decomposed into active and non active power according to: P =
1 T
T
∫
T
p(t )dt =
0
1 u (t )i (t )dt T ∫0
Q F2 = S 2 − P 2 .
(1.8) (1.9)
This decomposition can also be expressed in current terms. In fact, let Ge be the equivalent conductance, defined as:
4
Ge =
Pa
(1.10)
U2
the active current is defined as: i a (t ) = G e u (t )
(1.11)
and the non active current is: i f (t ) = i (t ) − i a (t ) .
(1.12)
Fryze had the merit of introducing for the very first time the concept of active current, which reproduces the voltage waveform and is responsible for the whole power transmission between the source and the load. It corresponds to the current which passes through a resistive load, once the voltage of the original load is applied to its terminals. On the other hand, non active current is not involved in power transmission at all. Moreover, active and non active currents are orthogonal in the L2 space (see Appendix A): i
2
= ia
2
+ if
2
(1.13)
Buchholz [4] contributed in extending these concepts to multiphase (Nphase) systems. In doing this, none of the N phases is treated as a special conductor. This means that irrespective of the presence of the neutral wire, all the phase voltages are referred to the virtual star point. In other words, the neutral wire, if present, is considered exactly as the other phases and similarly included in the vector analysis. Introducing instantaneous collective values of voltage and current, and corresponding RMS values: N
i∑ =
∑ iυ2
u∑ =
uυ ∑ υ
i∑
2
υ =1
N
= ∑ iυ
2
(1.14a)
υ =1
N
2
u∑
N
= ∑ uυ
2
(1.14b)
υ =1
=1
Depenbrock
2
[67]
defined
collective
instantaneous
power
p ∑ , collective
instantaneous conductance Gp, instantaneous collective power current i ∑ P and instantaneous phase power current iυp , respectively as: M
M
υ =1
υ =1
p Σ (t ) = ∑ uυ iυ = ∑ pυ (t )
(1.15)
5
G p (t ) =
pΣ
(1.16)
u Σ2 = G p (t )u Σ
(1.17)
iυp = G p (t )uυ
(1.18)
i Σp
The introduction of instantaneous phase power current, which is responsible for the instantaneous power and includes harmonics and unbalances, immediately leads to the definition of the corresponding instantaneous phase powerless current, i.e. iυz = iυ − iυp . (1.19) Such current does not contribute to the energy transfer and can be compensated without need of energy storage. When the average level is considered, instead of the instantaneous one, the phase active current is defined as in Fryze case, i.e.: i aυ = Guυ
(1.20)
where G is the equivalent active conductance, defined as the ratio between collective average power and the square of collective RMS voltage, i.e: P G = Σ2 . UΣ
(1.21)
The active current is, obviously, responsible for the whole average energy transfer from the source to the load. The difference between the phase power current and the phase active current represents the phase variation component: iυv = iυp − iυa ,
(1.22)
which is caused by the oscillation of the instantaneous equivalent conductance around its average value (or, correspondingly, by the variation of p Σ around PΣ ). Finally the non active current is obtained by subtracting the active current from the total phase current iυn = iυ − iυa
(1.23)
and it does not contribute to the average energy transfer. The various current terms defined by the FBD theory are mutually orthogonal, i.e. iυ
2
6
2
= iυ p + iυ z
2
2
2
2
= iυ a + iυ v + iυ z .
(1.24)
1.3.3 Kusters and Moore Theory In 1980 Kusters and Moore extended Fryze decomposition [8], based on the possibility to compensate for nonactive current components through passive devices. Assumed the conventional definition of active power P, their theory, (which was later generalized by Page [9]) introduces the same definitions as Fryze one, regarding instantaneous active current, that they denote with the subscript “p”: P ip = 2 u , u with the corresponding RMS value: P Ip = . u
(1.25)
(1.26)
Consequently the instantaneous reactive power is defined by difference as: iq = i − i p (1.27) with the corresponding RMS value:
I q = I 2 − I p2 .
(1.28)
The reactive power is thus defined as:
Q = S 2 − P2 .
(1.29)
Kusters and Moore Theory proposes a further decomposition of the current, into either an inductive or a capacitive component and a corresponding residual (inductive or capacitive) term. The instantaneous reactive term is: T 1 ) u idt T ∫0 ) i ql = ) 2 u u
(1.30)
)
(where the symbol x here denotes the instantaneous value of the alternating component of the integral of the quantity x). The corresponding residual component is: i qlr = i − i p − i ql . On the other hand, the instantaneous capacitive component results: T 1 ( u idt T ∫0 ( i qc = ( 2 u u
(1.31)
(1.32)
7
(
(where the symbol x here denotes the instantaneous value of the derivative of the quantity x). Thus the capacitive residual term becomes: i qcr = i − i p − i qc
(1.33)
Corresponding RMS values can be easily obtained for both reactive and capacitive components. As regards power terms, inductive reactive power can be defined as: T 1 ) u idt T ∫0 Ql = u I ql = u , ) u while capacitive reactive power is: T 1 ( uidt T ∫0 Qc = u I qc = u ( u
(1.34)
(1.35)
Residual reactive terms are defined by difference, resulting in:
Qlr = S 2 − P 2 − Ql2
or
(1.36)
Qcr = S 2 − P 2 − Qc2 . The fundamental meaning of Kusters and Moore decomposition is that the load current is split into a component which is proportional to the load voltage (i.e. a resistive current), a component which is proportional to the voltage integral (i.e. an inductive current), a component which is proportional to the voltage derivative (i.e. a capacitive current) and finally a residual current. Kusters and Moore Theory is aimed at passive compensation, in the sense that whenever a negative (inductive or capacitive) reactive current component is detected, it can be immediately compensated by inserting a suitablesized reactive element (inductive or capacitive, respectively). The major drawback of such theory, besides the unusual possible presence of negative RMS values, is that the so defined current components are not mutually orthogonal, since capacitive and inductive terms do not exclude each other, thus losing the physical meaning of the decomposition.
8
1.3.4 Czarnecki Theory (CPC Theory) Czarnecki studies about power transfer phenomena started more than two decades ago [1011], thus resulting in several different contributions, which have recently converged into the so called Currents’ Physical Components (CPC) Theory [1517]. Czarnecki considerations started from the most simple case of singlephase networks and are mainly developed in the frequency domain, thus representing port voltage and current through the Fourier series: u = U 0 + 2 Re ∑ U n e jnω1t
(1.37a)
i = I 0 + 2 Re ∑ I n e jnω1t
(1.37b)
n∈N u
n∈N i
In the most simple case of linear loads, the active current can be defined, substantially reproducing Fryze concept: ia (t ) = Ge u (t ) = GeU 0 + 2 Re ∑ GeU n e jnω1t ,
(1.38)
n∈N
where the equivalent conductance Ge is defined exactly as in eq. (1.10). Moreover, Czarnecki defines the reactive current as the sum of the current harmonics that are shifted by 90° with respect to the corresponding voltage harmonics, i.e.: (1.39) ir (t ) = 2 Re ∑ jBnU n e jnω1t . n∈N
Finally the scattered current is introduced, which depends on the fact that harmonic conductances Gn are generally scattered around the value of the equivalent conductance Ge: i s (t ) = (G0 − Ge )U 0 + 2 Re ∑ (Gn − Ge )U n e jnω1t .
(1.40)
All these three components result orthogonal: 2 2 2 2 i = ia + i r + i s .
(1.41)
n∈N
The case of presence of harmonic generating loads is more complex and was faced by Czarnecki in different ways. In the most historical formulations Czarnecki splits the current into two components, the homologous one, io, which contains current harmonic components that are present in the voltage, too: io = ∑ i h , (where Nυ collects the voltage harmonics)
(1.42)
h∈N v
and the (independently) generated one, ig, which collects the harmonics that are present only in the current:
9
ig =
∑i
h h∈( N i − N v )
.
(1.43)
Thus the ig component adds to the other orthogonal terms in the current decomposition as follows: i
2
= ia
2
+ ir
2
+ is
2
+ ig
2
.
(1.44)
On the other side, when an unbalanced threephase system is analyzed, a further current component is introduced, which takes account of the load unbalance. If a delta load configuration is considered, where the three phases are denoted with subscripts R, S, T the phase currents can be arranged in a vector as follows: i R I R i = i S = 2 Re I S e jωt = 2 Re I e jωt (1.45) iT I T In case of symmetrical voltage source, once defined: U R U R # U = U S and U = U T U T U S
(1.46)
the current can be written as: # i = 2 Re (Ge + jBe )U + AU e jωt
(1.47)
where the equivalent admittance is: Ye = Ge + jBe = YRS + YST + YTR
(1.48)
and the unbalanced admittance is defined as: A = Ae jΨ = −(YST + αYTR + α *YRS ) .
(1.49)
{[
] }
In this case the current can be decomposed as follows: i a = 2 Re ∑ Ge U n e jnω1 t active current n∈N i r = 2 Re ∑ jBen U n e jnω1 t reactive current n∈N # i u = 2 Re ∑ AU e jnω1 t unbalanced current n∈N i s = 2 Re ∑ (Gen − Gn )U n e jnω1 t scattered current n∈N
10
(1.50) (1.51) (1.52) (1.53)
The four current components are mutually orthogonal and, if the independently generated current (which is orthogonal to them all) is also taken into account, it results: i
2
= ia
2
+ ir
2
+ iu
2
+ is
2
+ ig
2
(1.54)
From the eq. (1.54), through the multiplication for the corresponding voltage, power terms can be immediately derived: 2 2 S 2 = P 2 + Q 2 + Du + D s + D g2 .
(1.55)
The previous decomposition is used by Czarnecki as a starting point for some considerations about the possibility to compensate the various components especially by passive means. He observes that reactive and unbalanced terms can be compensated by means of shunt reactors. On the other hand, scattered current can be also theoretically eliminated by passive means, but series branch is required in addition to the shunts [14]. Finally the generated current needs the use of active devices to be compensated. When dealing with the compensation by switching devices, however, in the most recent contributions [1718] the author abandons the previous current decomposition, while splitting the current only into the working and the detrimental components. Starting from the assumption that the useful current components, especially with reference to synchronous or induction motor, are associated to the fundamental positive sequence, Czarnecki defines as working current the one which is proportional to the positive sequence (p) component of the supply voltage fundamental, i.e.: i w = Gw u w P where G w = w 2 , with Pw = P1 p uw The detrimental term is then defined by difference: id = i −iw .
(1.56) (1.57)
(1.58)
As it will be better clarified through a comparative analysis in the following chapter, CPC theory has the main disadvantage of being essentially based on a mathematical decomposition (i.e Fourier series), which causes computational complexity and loss of a precise physical meaning and the electrical quantities it derives are strongly dependent on the considered network terminals.
11
Moreover, despite several attempts it fails in defining the compensation provisions in a unified and straightforward way, while taking into account the different available compensation technologies.
1.3.5 Akagi and Nabae Theory (pq Theory) In 1983 Akagi and Nabae gave a fundamental contribution in the field of Power Theories thanks to the so called “Instantaneous pq Theory” [2122]. It is focused on instantaneous quantities and works on threephase applications, considering them as a whole. The basic idea derives from the application of Clark transformation to three phase systems, with or without neutral wire. Thus, port voltages (here denoted with subscripts a, b, c) are transformed into the orthogonal αβ0 frame through the following:
u 0 u α = u β
2 3
1 2 1 0
1 2 1 − 2 3 2
1 2 u a 1 − u b 2 3 u c − 2
(1.59)
and the same transformation can be similarly applied to the port currents ia, ib, ic. Then the fundamental concepts of instantaneous real power (p) and instantaneous imaginary power (q) are introduced. If a four wire system is considered, the instantaneous zero sequence power p0 is also taken into account.
Instantaneous power terms can be obtained as follows: u 0 0 0 i0 p0 p = 20 u u β iα . α 3 0 u β − u α i β q
(1.60)
It can be noted that the sum of instantaneous real and zero sequence power results in the traditional instantaneous power of threephase systems: p 3φ = p + p 0 = u α iα + u β i β + u 0 i 0
(1.61)
where the homopolar component obviously disappears for threephase threewire systems. As regards the reactive power concept, it results to be a mathematical definition with no physical meaning, and this represents one of the main weak points of this approach, as will be better clarified later.
12
Instantaneous power quantities are then used to introduce a correspondent current decomposition. Instantaneous active components on the α and β axis are respectively: uβ u and iαp = 2 α 2 p i βp = 2 p uα + u β uα + u β2
while instantaneous reactive components result: − uβ u and iαq = 2 q i βq = 2 α 2 q . 2 uα + u β uα + u β
(1.62)
(1.63)
If the zero sequence term is also taken into account, all the current components can be reported into the abc axes frame with an inverse transformation: 1 1 0 i a 0 2 i0 i0 i = 2 1 − 1 − 3 0 =∆ M 0 , b0 3 2 2 2 0 ic 0 0 1 − 1 − 3 2 2 2 i ap 0 i aq 0 and ibp = M iαp ibq = M iαq . icp i βp icq i βq As a consequence, instantaneous phase currents can be decomposed as: i a i a 0 i ap i aq i = i + i + i . b b 0 bp bq i c i c 0 i cp i cq
(1.64)
(1.65)
(1.66)
Furthermore, it is also possible to decompose real and imaginary powers into an average and an oscillating term, i.e.: p= p+ ~ p ~ q =q +q
(1.67)
Correspondingly, also into the current components of (1.66) the two contributions can be separated, so that the final decomposition of the phase currents results: i a i a 0 i ap i a~p i aq i aq~ i = i + i + i ~ + i + i ~ . (1.68) b b 0 bp bp bq bq i c i c 0 i cp i c~p i cq i cq~
13
The pq theory has been widely and fruitfully applied to the control of switching compensators, however it does not succeed in explaining the origin of the different physical phenomena. In particular, as it has been already shown in [23], also in the simple case of sinusoidal systems, it leads to misleading results when unbalanced loads are considered. In this case reactive components may appear also in the case of purely resistive load, together with unexpected current harmonic components. Moreover, when dealing with the power decomposition into average and oscillating values, it is not clear how these terms are related to the current components, in the sense that, when an hypothetical compensation strategy is based on the cancellation of some of these power terms, it is not immediate to understand what will result in terms of current.
14
1.4 Compensation technologies presently applied in electric networks The following paragraphs are dedicated to describe the main technical devices that are nowadays adopted in electric networks, when current compensation provisions are needed. At the beginning the most traditional solution, i.e. the insertion of passive filters is considered, which is characterized by the presence of fixed energy storage elements. This is a cheap and standardized provision, which however has some important drawbacks. Then the analysis of devices whose energy storage capability can be modified and that are generally adopted for reactive power compensation is introduced. Since their main limitation is due to their slow regulation, more flexible and innovative solutions are also considered. In fact switching compensators are presented, which don’t require large energy storage elements and permit more complex control schemes, so that faster intervention and better performances can be obtained.
1.4.1 Passive filters Whenever in an electric network a problem related to an excess of reactive power or current/voltage harmonics is detected, the first provision that is usually adopted is the insertion of a passive filter, due to its high degree of standardization and reasonable cost. For sure this strategy derives from the traditional frequency domain approach to networks working under nonsinusoidal condition, which results useful to explain the passive filter functioning principle in a straightforward way. Let’s assume that a Harmonic Generating Load is connected to the Point of Common Coupling of an electric network. According to what previously explained, the PCC voltage results distorted due to the effect of line impedance and possibly of the transformers reactance. The simplest way to avoid that a harmonic current influences the network, is to insert an LC series filter in parallel with the distorting load and to tune it on the harmonic of interest. It is worth to recall that an ideal LC filter is formed by the series connection of an inductor L and a capacitor C and it is characterized by a resonant frequency, which is defined as: 1 fr = 2π LC
(1.69)
15
Resonant frequency represents the frequency where inductive reactance and capacitive reactance are equal. This means that the voltage drops across the inductance and the capacitance have the same amplitude and inverse polarity, so that the LC filter results a short circuit at that particular frequency. For this reason, the LC filter results a very low impedance path for the selected harmonic. Thus the harmonic current generated by the load, which would flow back towards the source, passes instead through the LC filter which is ideally a short circuit. The Bode diagram of the impedance of an LC filter which is tuned on the 5th harmonic is depicted in fig 1.1. It can be easily seen that below the resonant frequency the filter behaves as a capacitive load, thus helping also for the compensation of load reactive power associated to the fundamental frequency, while it appears as an inductance above such frequency. In practical implementation LC filters have also a series resistive component R, which is mainly due to the reactor windings and which influences the filter selectivity. Such resistive component is generally quantified through the merit factor, which represents the ratio between the inductive component (at the resonant frequency) and the resistive one, i.e: 2π f r L 1 Mf = = R 2π f r C R
(1.70)
As a consequence, at the resonant frequency, the passive filter does not appear as a pure short circuit but as a resistance, whose value corresponds to R.
Bode Diagram
Bode Diagram
30
40
ZLC
ZLC
250 Hz
20
20
0
40
sL
1/sC
Magnitude (dB)
Magnitude (dB)
20
1/sC
sL
R
10
60 80
0
10
20
100
30 120
40
140 160 1 10
2
3
10
10
4
10
Frequency (Hz)
Fig. 1.1 Bode diagram of the impedance of an ideal LC series passive filter with resonant frequency fr = 250 Hz
16
50 1 10
250 Hz 10
2
10
3
4
10
Frequency (Hz)
Fig. 1.2 Bode diagram of the impedance of an LC series passive filter with resonant frequency fr= 250 Hz and merit factor Mf= 40
In practical applications merit factor is usually comprised between 20 and 200. As much the merit filter is high, as much the filter will be selective for the considered harmonic and less this harmonic will influence the rest of the network. In fig. 1.2 an LC filter tuned on the 5th harmonic, having a merit factor Mf = 40 is depicted. Usually a passive filter is composed by many LC series filters, put in parallel one another. Each of the LC cell is tuned on a different harmonic, chosen among the load dominant ones. Since the number of tuned LC branches is limited, a highpass filter is added in parallel too, for the compensation of higher order harmonics. If a sufficient selectivity is ensured for each LC branch, mutual interferences are prevented and above presented considerations can be extended to the whole structure (fig. 1.3). A few further observations about passive filters connection and localization can be made. When three phase applications are considered, both star and delta connection can be used. Compensators usually come in delta connection even if, for passive filters, this choice will be only partially effective, since the third harmonics, that are prevalent in most office environments due to single phase equipment, flow between phase and neutral and result not to be compensated in this way. Another aspect that needs to be taken into account when locating a passive filter in proximity of a distorting load, is that, even if this prevents harmonic currents to influence the network, there is an increment of the current circulating between the distorting load and the filter due to the diminished impedance. This explains why HV
MV
Zcc
LV
PCC Electric network Bode Diagram
ZLC
Antiresonance freqs.
Magnitude (dB)
0
Distorting load
50
High pass effect
100
Current harmonics
150
200
250 350 550 Hz Hz Hz 3
10
10
4
Frequency (Hz)
Fig. 1.3 Scheme of principle of the functioning of a passive filter for harmonic compensation, with the corresponding impedance Bode diagram
17
the connection of passive filters in close proximity to the distorting loads is preferred whenever possible.
1.4.1.1 Detrimental effects of passive filters insertion Even if passive filters insertion is the most diffused choice whenever a compensation is required, being cheap and standardized, several negative aspects have to be taken into account. The first one is that they origin a high level of interaction with the electric grid, whose effects can be severely detrimental. The first risk is the parallel resonance with the network, which is not ideal, thus being represented by a non zero line series inductance. At the parallel resonance frequency between distorting load and electric network, the total impedance which is seen from the load is extremely high. This means that, if the load generates even a minimal current component at such frequency, the passive filters suffers large voltage stresses. Another important aspect is that when a passive filter is connected to the electric grid it offers a low impedance path for the corresponding harmonic currents of all the loads that are connected to the same point. This needs to be carefully taken into account in the design process and generally requires an over sizing of the passive filters. The importance of carefully studying the insertion of passive filters in an electric grid is confirmed also by another important possibility of interaction with other devices connected to the network. For instance, if two LC filters for harmonic compensation result to be in parallel into the grid, the following detrimental effects can occur. Due to the fact that they cannot be truly identical, if one of them has a resonant frequency which is slightly higher than foreseen, while the other ones is slightly lower, the two filters together form an excellent passive resonant filter, which has the opposite of the desired effect. On the other side, if one of the LC filters is precisely tuned and the other is not, the first one may be overloaded, while the second is idling. Thus, direct parallel must be avoided and also lowering the filters merit factors helps in avoiding such problems.
18
1.4.1.2 Effect of distorted voltage on passive filters It is of capital importance to consider the effects that a distorted voltage source has on the passive filters behaviour, since, in practical applications, a perfectly sinusoidal voltage source is almost never available. The first negative effect of voltage distortion relates to possible series resonances between the passive filter (LC) and the line impedance (Ls). As a consequence the real resonant frequency of the passive filters is lower than expected, i.e.: 1 (1.71) fr = 2π ( L + L s )C For this reason if the supply voltage is distorted and contains a voltage harmonic which is close to fr, a very high current can pass through the LC filter and consequent damages can occur. Thus, two main provisions are taken to avoid harmonic currents amplification and series resonance problems: the first one is the limitation of the merit factor of the passive filter, which consequently, results to be a tradeoff choice among filter selectivity and resonance avoiding issues. On the other hand, at the design stage, the filter resonance is lowered through an increment of the inductive component. This process is known as filter detuning, where the detuning factor is defined as: f −f δ= r f
(1.72)
where fr indicates the theoretical resonant frequency and f the considered one. It is also worth to take into consideration that beside the desired filter detuning, a similar effect is provoked by components aging, so that filter performances are also affected by the passing of time. The need of accurately consider the effect of line inductances on compensation provisions is not limited only to passive filters for current harmonic compensation, but regards also the power factor correction capacitors that are used to control load reactive power. Also in this case the need of avoiding resonance requires that a reactor is connected in series with the capacitive elements so that, even if the desired capacitive effect is maintained at the fundamental frequency, a controlled behaviour at harmonic frequencies is also ensured. The effect of voltage distortion on passive filters, however, is not limited to the possible risk of series resonance. In fact whenever the voltage supply contains an harmonic which is present also in the load current spectrum, harmonics cause
19
additional power losses, which need to be taken into account as another undesired effect. As a conclusion, when the employment of purely passive compensators is considered all the present aspects must be carefully considered: 
They use passive components, thus resulting in cheap and standardized solutions for the compensation of reactive power and harmonic currents.

They offer a fixed amount of reactive power, whose value is defined during the design stage.

They need to be specifically designed taking into account their location and the configuration and topology of the network they are connected to.

Risk of resonances must be carefully taken into account and suitable provisions need to be adopted to avoid it.

Possibility of undesired compensation also for “external loads” current harmonics requires an over rating of the passive filters, to avoid that overloading results in premature failures.
1.4.2 Static Var Compensators A higher degree of flexibility in the compensation of reactive power can be obtained using quasistationary compensators (i.e. Static Var Compensators, SVCs). In these systems energy storage elements are not fixed, but can vary within a specified range and in defined conditions. Basic elements of quasistationary compensators are Thyristor Switched Capacitors (TSCs), used as variable capacitances, and Thyristor Controlled Reactors (TCRs), used as variable inductances. The insertion of such compensators, however, depends on the line period, thus they do not allow a compensation faster than the line period.
1.4.2.1 Thyristor Switched Capacitors The elementary configuration of a TSC is made up of two basic components: the capacitor and two thyristors, used as a bidirectional switch. A little inductance, rated at a few percent of the capacitor impedance is added, too. Commonly, TSCs use several basic cells in parallel (fig. 1.4), to guarantee a regulation of reactive power, that, anyway is always done in discrete steps. A binary choice (C, 2C, 4C..) for the
20
capacitors is very effective, but, for economical reasons, the solution with few C capacitors and a C/2 one can be preferred. Insertion of TSC deserves special attention: to avoid voltage steps and also oscillations between inductive and capacitive elements, TSC must be turned on when line voltage has a peak and trying to have it equal to the capacitor voltage. In this way, insertion transients are avoided and current has no harmonics, if a sinusoidal voltage source is assumed. This however explains the reason why TSC insertion depends on line period and compensation is necessarily slow.
1.4.2.2 Thyristor Controlled Reactors The elementary configuration of TCRs is formed by an inductance which is controlled by two antiparallel thyristors (fig 1.5). If the gating signal is given at the line voltage peak, a full conduction is obtained and an inductive current is established, with a low active current component which is needed to compensate inductor losses. Control of TCR reactive power is made by changing the thyristors firing angle. For a firing angle increasing from 90 to 180 degrees, inductive current (and also reactive power) is progressively decreased. Differently from TCRs, TSCs make it possible a continuous change in the provided reactive power, however the insertion of TSC can take place only two times per period. One detrimental effect of the phase control lies in the current distortion. When the gating angle α is higher than 90°, TCR current is not sinusoidal anymore. If the conduction angles are equal for both thyristors, only odd harmonics are detected and
Fig. 1.4 Thyristor Switched Capacitors (TSCs)
Fig. 1.5 Thyristor Controlled Reactor (TCR)
21
the RMS value of both current fundamental and current harmonics can be easily derived: U π (1.73a) I L1 = s ( 2π − 2α + sin 2α ), α ∈ [ , π ] 2 πωL 2U s sin( k + 1)α sin( k − 1)α 2 cos(α ) sin( k α ) I Lk = ( + − ), k +1 k −1 k πωL (1.73b) π α ∈ [ , π ], k = 3, 5, 7... 2 Thus, while helping in reactive power control, TCR insertion generally determines harmonic pollution, which needs to be taken into account. This is the reason why deltaTCRs connection is preferred and they are often used with low pass harmonic filters.
1.4.2.3 Hybrid solutions In the most general case, the Static Var Compensator is realized using one or more of the above described schemes. Hybrid solutions (fig.1.6) are generally composed by a suitable number of TSCs for a discrete step regulation of reactive power. Besides, a TCR is used, too, whose rating is established to allow the system, as a whole, to reach a continuous reactive power compensation. Other advantages of such solutions are the absence of transients and a limited harmonic generation, due to the fact that the inductance is rated for a small fraction of total reactive power. Obviously such compensation units are more expensive than single TCR or TSC.
Fig. 1.6 Hybrid filter made up of TSCs, TCR, a low pass filter and a capacitor
22
1.4.2.4 Effects of distorted voltage on Static Var Compensators It is worth to focus on the consequences that a distorted voltage provokes on SVC functioning. In fact in most cases, SVCs work under non ideal conditions and some differences in the behaviour with respect to what expected for sinusoidal operations can be detected. The following analysis will briefly recall the effects of voltage distortion on both TSC and TCR.
1.4.2.4.i Effects of distorted voltage on Thyristor Switched Capacitors The behaviour of a TSC supplied by a distorted voltage can be easily brought back to what presented in par 1.4.1.1 for passive filters. In fact it must be considered that due to the presence of the inductor, practically an LC series filter is obtained. For this reason it is of capital importance to consider the risk of resonance with the line inductance, which, in the case of an imposed distorted voltage, provokes high harmonic currents. Exactly as happens for the capacitors that are used for power factor correction, even if not switched by thyristors, the increment of the current flowing through the capacitors causes further stress on them. Due to the harmonics the value of the voltage which stresses the capacitor may result higher than expected, so that the dielectric may deteriorate faster. Commonly, the risk of capacitor overload due to harmonics is taken into account by simply oversizing it, as is prescribed also by international standards. Major caution must be paid to the resonance problem, according to the TSC location, since resonance may cause major damages [37].
1.4.2.4.ii Effects of distorted voltage on Thyristor Controlled Reactors The presence of distortion in the voltage that supplies a TCR determines several differences with respect to the ideal situation described in 1.4.2.2 for a sinusoidal voltage source. In general terms, each of the harmonics of the supply voltage gives origin to its own series of current harmonics. The mathematical expression corresponding to (1.73a) and (1.73b) can be derived [38]. Besides the more complex mathematical formulation it must be taken into account that the formula are valid only if the conduction angle is the same for both thyristors. It can be seen that each voltage harmonic contributes to the generation of current harmonics of the same parity, thus, if a voltage source with an even harmonic added
23
to the fundamental is considered, the corresponding current flowing through the TCR will contain both even and odd harmonics, while if the voltage source will be composed, for instance, only by odd harmonics, the TCR will have only odd harmonics, too. Moreover, in the case of inductances in series with ideal switches supplied by distorted voltage, resistive type current harmonics are also generated. It is however important to underline that such considerations are derived considering that the conduction angle of the TCR is not influenced by harmonics. If this hypothesis is not verified, the accuracy of the calculation of the firing angle by the mathematical formula is affected.
1.4.3 Active Power Filters In the last decades Active Power Filters (APFs) gained growing interest as compensation means, due to their flexibility and improved control strategy, associated to better performances and decreasing costs [3954]. Active power filters have been shown to be effective in solving a large number of power quality issues. Part of them are related to the voltage waveform, i.e. voltage harmonics, dips and sags, flickers etc. and are mainly dealt by the use of the series active filter topology. The others are “current related” and include current harmonics, reactive power, load unbalance and excessive neutral current. These last issues are generally solved by the use of the shunt topology. In spite of the common physical origin of some of these power quality problems (for instance voltage and current harmonics due to line impedance), this common distinction between voltage and current related power quality issues, reflects somehow the different sharing of responsibilities for their compensation, belonging to the electric utility in case of voltage disturbances, and to the users in case of current nonidealities. This last perspective is the one which will be mainly adopted in the following analysis.
1.4.3.1 APF Structure An active power filter is basically composed by a DC/AC converter. Both Current Source Inverter (CSI) with inductive energy storage and Voltage Source Inverter (VSI) with capacitive energy storage are used. The first one is composed by a currentfed PWM modulation inverter, which acts as a nonlinear current source to compensate harmonic currents. It is considered quite reliable, but has higher losses
24
and higher parallel AC capacitors requirement then the other. The second one stores the energy into a large DC capacitor and is now the most common structure, since it is cheaper and lighter. It has also the advantage of being expandable to multilevel configuration, to obtain improved performances. The structure of an active power filter is extremely important also if considered in the perspective of a possible integration with Distributed Generation (DG) systems, as will be better clarified in the following paragraphs.
1.4.3.2 APF Topology Active Power Filters can be at first classified according to their topology, which can be shunt, series or a combination of both. The shunt APF topology is shown in figs 1.7 and 1.8. This configuration is mainly used to eliminate harmonic current, reactive power and load unbalance. The principle of operation is that the active filter provides the distorting load all the harmonic and reactive current components, so that source current results proportional to corresponding voltage. Fig. 1.9 shows an Active Series Filter, which is generally connected before the loads through a matching transformer. It is typically used to reduce voltage harmonics, and to balance and regulate the terminal voltage of the load or line. Both series and shunt configuration can be used by electric utilities to compensate for voltage harmonics and to damp out harmonic propagation, caused by resonances between line inductances and passive shunt capacitors. Anyway it is worth to underline that the different destination of the two topologies is not a strict one, but simply an indication about major trends. Finally, unified power quality conditioners consist in a combination of active parallel (shunt) and series compensators, sharing the same energy storage element. Even if, due to their flexibility and performance, they can be considered an ideal compensator, their main disadvantage stands in the cost and control complexity. Another interesting solution is represented by hybrid filters (fig 1.10), which conjugate passive elements with active filters technology. Their principle of operation is that a passive LC shunt filter is used to eliminate lower order current harmonics, while active filter provides the residual compensation.
25
iS
iS
iL
iL
Non linear load
Non linear load
iAPF
iAPF APF
APF
vd
id Fig. 1.7 Currentfed shunt Active Power Filter iS
vAPF
Fig. 1.8 Voltagefed shunt Active Power Filter iS
iL
vAPF
iL Non linear load
Non linear load
iAPF
APF
APF
iP Shunt passive filter
vd
Fig. 1.9 Series Active Power Filter
Fig. 1.10 Hybrid Filter made up of a passive LC series and a series APF
Such an approach allows a consistent decrement of active part size and cost.
1.4.3.3 APF Supply System Another important classification of Active Power Filters is related to their supply system type. In fact, different APF structures and control strategies are adopted depending on if dealing with a singlephase or a threephase system. Another relevant distinction must be done between threephase threewire systems and three phase fourwire systems, taking into account that the last ones present major power quality problems, due to the additional need for neutral current compensation. The active filters performance is based especially on the choice of the control strategy. It is worth to underline that there is often a deep difference between three phase and single phase control strategy. From an historical point of view, three phase configuration has been investigated in a wider way starting from the eighties due to
26
Akagi pq theory. Singlephase investigation is more recent since the extension of the pq theory to such kind of APF is not so immediate.
1.4.3.4 APF Control General analysis of Active Power Filter control, will be mainly developed in the following with reference to a VSI shunt filter, indeed oriented to the solution of current related power quality issues. In such case, active filters control problem can be divided into three main themes.  The first one is the need of controlling the DC link voltage, which has to be kept stable despite possible power losses.  The second one is related to the generation of the current reference and is fundamental for the success of the control strategy.  The last one regards the current controller design and deals with the generation of a suitable switching pattern to allow the inverter to follow the reference in the best possible way. In the last decades many and very different solutions have been proposed for each of these issues; in the following only a brief overview of some of the possible different approaches, with their respective limitations, is proposed. Regarding the generation of current reference, the problem is mainly related to the need of extracting harmonic content from source or load current. Such extraction can be theoretically done both in the frequency and in the time domain [39]. The frequency domain approach is based on the FFT, thus the performance strongly depends on the periodic characteristics of the distortion. Beside, with the increasing number of harmonics to be compensated also the number of required calculation grows up, resulting in a longer response time. On the opposite, time domain approaches have a simpler implementation, thus resulting in a fast response to system changes. Among them the most diffused are the Instantaneous Reactive Power (IRP or pq) theory and the Synchronous Reference Frame (SRF) [4147]. As regards the APF current regulators, many different approaches have been used [48]. One main issue to be considered is that the use of digital control may suffer of a limited bandwidth, that has to be as high as possible to obtain good dynamic performance. The fastest current control technique is the hysteresis one [4950],
27
which shows also better robustness. It’s main disadvantage, however, is that it generates a variable switching frequency. To maintain a constant switching frequency, PWM techniques are mainly used and to get good dynamic performances peculiar control strategies have been proposed, i.e. deadbeat control, repetitive control, adaptive control [5153]. To reduce control delay also multisampling technique has been applied to active power filters [54].
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1.5 Development trends in electric networks The last part of this chapter will be devoted to briefly outline the general evolution scenario of electric grids. The growing interest for renewable energy sources is considered, especially taking into account the technical changes that these new technologies will imply in the electric system. Also new ideas regarding network and energy management (as in the case of microgrids or energy storage elements) will be hinted. The main focus, however, will be to show how these already unavoidable changes will impact the electric system, but especially how they will offer brand new perspectives also in the field of compensation.
1.5.1 The impact of distributed generation on electric grids The energetic problem is becoming crucial in modern societies. The awareness that conventional energy sources like oil and other fossil fuels are limited and fast decreasing, imposed the issue to public attention. The deep concern for primary energy availability comes both from economical reasons and from an increasing environmental sensitivity. Sustainability and renewable energy development are a main discussion topic at international level [5559], which lead to more and more strict regulation also at regional or national level. As regards Italy, other two aspects are influencing the evolution of energetic choices: the first one is the large dependence from foreign countries for energetic supply, since about 80% of internal need is fulfilled by foreign production. The second one is the recent advent of market liberalization, which means a large number of players, whose relationships and interests are diverse and potentially conflicting. The most direct answer to all these needs and issues lies in the Distributed Generation (DG) philosophy, which makes it possible to exploit both conventional energy sources and renewable ones, throughout the installation of small/medium size plants, spread around the electric grid. This approach has also other advantages: locating generators close to users defers the need to upgrade the existing transmission system and allows a better burdensharing among the suppliers, so that also blackout events become rare and less critical. On the other side, the lower size of the installation shortens the time for construction and commissioning and diminishes the environmental impact.
29
Practically both selfdispatched systems and systems dispatched by local operators can be enclosed in the distribution grid. This transforms the distribution network from passive to active, in the sense that decision making and control are distributed and power flow becomes bidirectional. This also requires an interaction with the final users through smart metering systems and advanced dynamic control approaches, to guarantee the best management of the whole network regarding efficiency, security and reliability. The realization of active distribution network technologies introduces radically new system concepts in implementation. In this perspective one of the most promising structure is the microgrids paradigm. These systems are interconnected to the medium voltage distribution network, but they are also capable of working in isolated mode, for instance in case of faults to the upstream network. Such a new approach to energy distribution systems provides advantages for all the involved parts. The customer experiences enhanced local reliability, improved power quality and potentially receives both electricity and heat at a lower cost. On the distribution network perspective, the advantage derives from the reduction of distribution and transmission facilities demand, together with a significant drop of power losses and assets investment. The distributed generation scenario is thus the ideal context for the improvement of technology based on renewable energies, as photovoltaic plants, wind, fuel cells and many more that were originally confined to niche applications, (i.e. energy supply of remote locations), but that now are gaining interest for widespread application in the sustainability perspective. It’s also interesting to note that distributed generation networks can also integrate energy storage systems, as batteries and flywheels. Their function is to harness excess electricity produced by the most efficient generators during low load, while releasing it onto the grid when needed, obtaining a reduction of highcost generator demand. In the near future distributed energy systems should include also hybrid electric vehicles, which can be used along with the utility network in the form of plugin hybrid vehicles (PHEV) and vehicle to grid (V2G) systems.
30
120
100
80
60
40
20
0 PV
Wind Power Electronics Cost (%)
Fuel Cell
Microturbine
Other Costs (%)
Fig. 1.11 Costs of power electronics compared to total costs for DE systems
It is quite obvious that the penetration of distributed generation and especially of renewable energy sources is subordinated to economical considerations, since presently the main factor that discourages the distributed energy (DE) installation is the initial capital cost. It is here important to underline that the total cost of the single generation plant is not influenced only by the generation system itself, but a significant amount of money is required by the electronic interfaces that are always needed to properly interconnect the power generator to the supply grid. Those electronic interfaces are integral part of the renewable energy plant and they can impact up to the 40% of the total cost (fig 1.11). In the end it’s worth to note that the advent of distributed generation opens many new issues in the network management since the connection/disconnection and stability related problems have to be managed. Then also the DG system influence on fault conditions has to be taken into account and this, in general, requires a different control/coordination at the distribution level. Power electronic interfaces are widely involved in all of these topics.
31
1.5.2 Electronic interfaces general topology Due to the capital importance of the electronic interface for DE systems, their main features are here described, since, irrespective of the kind of selected energy source, they maintain a basic common structure. The generic form is represented in fig. 1.12 through a block diagram. Each electronic interface is composed by four major parts. The first one is the input converter that is the block that is more deeply influenced by the energy source type. When DC power generators (as photovoltaic panels or fuel cells) are considered, this first part is composed by a DC/DC converter, whose goal is to take the uncontrolled, unregulated input DC voltage and convert it into a controlled DC one, whose level is suitable for the following stage. In the case of AC power generation systems (i.e. wind and microturbines), instead of the DC/DC converter the input stage is composed by a rectifier, which converts the alternate voltage to a continuous one for the DC link. The second block of the electronic interface is the DC/AC converter, which is the most generic block, made up of an inverter whose function is to convert the DC power into a regulated AC one, which is compatible with the grid level. The inverter is necessarily followed by an output filter, which represents the third block in the above scheme. The last block contains monitoring and control functionality to ensure DE proper functioning. It also comprises protective functions for the distributed generator and
Distributed Energy Resources
AC/DC or DC/DC converter
DC/AC Converter
Output interface Module
PV, wind, microturbine, fuel cell, etc
PCC
Batteries, flywheel, energy storage
Local loads
Monitoring and Control Module
Fig. 1.12 General block diagram of typical DE power electronic system
32
Utility
also for the local electric power system to allow safe paralleling and disconnection from the grid. Such block can also comprise human machine interface, communication interface and power management. It is worth to underline that when energy storage devices are considered, the corresponding power electronic interface needs the peculiar feature of total bidirectionality. This means that it has to take the power from the grid during the charging phase, while providing power to the network during discharge. This is what is indicated by the double edged arrows in the block diagram, while single edged arrows represent unidirectional power flow.
1.5.3 Trends in the development of electronic interfaces for DE applications As already pointed out, electronic interfaces represent a key point for the future consolidation of distributed and renewable energy systems. The present focus of power electronic research is concentrated at first in improving their reliability. Many of the power electronics used for the DE applications have a low reliability rate, typically operating less than five years before a failure occurs, so an improvement is needed in order to reach a mean time between failures of at least ten years and have the whole plants to be economically convenient. The second imperative is cost reduction. It can be reached developing an architecture for standardized, highly integrated, modularized power electronics interconnection technologies. In particular the development of modular interfaces could avoid the need of redesign for different DE applications, making it possible high volume production at a lower cost. The third key point faces the economical issue from a complementary perspective. In fact, economical convenience can be obtained not only with cheaper devices, but also offering at the same price systems that are capable of many auxiliary features. Optimization of overall electric system performance is one of the most important aspects for the longterm viability of distributed generation systems. In this perspective power electronic interfaces have to acquire other functionalities besides the one to extract the maximum possible power from the power generator, matching the grid requirements, in the sense that they can exploit their capability of providing the so called “ancillary services”. These definitions comprise among the others: voltage control, frequency regulation, spinning reserve, network stability and harmonic compensation. It is quite obvious that to provide these auxiliary services
33
from the energy system, two different subsystems are needed. The first one is the online detection system which registers the need for an ancillary service and creates a proper signal as a consequence. The second one is the actuation system which receives this signal and implements the required function. Quite often many different ancillary services can be potentially provided, since only an additional code is needed in the control part, while no extra rating is required in the DE system itself or in the electronic interface. The possibility to face different needs, however, requires an hierarchical approach in the control strategy, since, when more than one ancillary service is needed, it is necessary to decide which is the priority according to technical feasibility, system reliability and economic issues. It is clear that according to this perspective tremendous opportunities and challenges of research, development and implementation of new technologies, advanced control and comprehensive optimization of the whole network arise.
1.6 The work of this dissertation The present research analysis gets into the crossover point of the fields of investigation that have been briefly presented in the previous paragraphs (fig. 1.13). In fact its starting point is the elaboration of a theoretical framework for the analysis of electric networks working under nonsinusoidal conditions. Thus, according to the tradition of “Power Theories”, it is at first oriented to analyze electrical phenomena undertaking under distorted conditions. The goal is to provide a theoretical explanation which is based on quantities that maintain a precise physical meaning also under distorted conditions. It can be said that this part somehow deals with the past, in the sense that, obviously, a confrontation with previous most important theories is required, so that differences and similarities are highlighted and the advantages of the proposed approach are also presented. It is however worth to note that the present research work is not aimed to offer a pure theoretical and speculative contribution but, on the opposite, it makes a more practical effort in the field of compensation. For this reason it needs to take into consideration all the different means and technologies that are commonly used for the compensation of current nonidealities in electric networks, so that many of them can be taken into account for the elaboration a general compensation strategy.
34
Special attention is paid at first to Static Var Compensators, that are presently the preferred solution for reactive power compensation and for all the issues involving average power absorption and energy storage. Then, the research work focuses on Active Power Filters, that represent a resource of increasing interest for compensation problems. The problem of SVCs but especially of APFs control, which is one of the most important one in order to ensure good compensation performance, is here dealt with, in a new perspective, which is the one of distributed compensation systems. This scenario is considered to be of capital and growing interest in the future due to the need of an optimization of energy generation, distribution and management. The future trends for electric networks will register the diffusion of distributed generation systems and consequently power electronic interfaces dispersed along the grid. Moreover the increased sensibility to the issue of smart design and management of energy systems encourages to adopt a more general perspective of optimization. In the field of compensation this means reasoning, no more at a local level, but, instead, at centralized one. Passive filters, SVC, active filters. Cost/flexibility issues
Main Power Theories (pro/cons)
Theoretical framework for studying electric grids under distorted operation
Compensation technologies currently adopted
Cooperative and distributed compensation approach
Distributed generation and related electronic interfaces
PE ancillary services
Fig. 1.13 The context of the present research work
35
Practically this implies that, when a new compensation provision is required, in a complex system as a distribution network may be, the first thing to consider is the availability of compensation means that are already connected to the grid and that may have a residual compensation capability. Then the exploitation of such compensating units must be organized in a hierarchical way, preferring cheaper units whenever possible. Such global perspective however implies problems that are absolutely new in the field of compensation: the first one is the need for a theoretical framework which allows delocalized compensating actions. The second one regards the capability to deal with compensators that are different in nature, as SVCs and APFs, so that they can be exploited in the most convenient way. Moreover, another crucial issue for the practical implementation of distributed compensation, whose analysis, however, is beyond the scope of this dissertation, is the one regarding communication techniques along the electric networks, whose performance can severely impact the whole compensation performance.
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CHAPTER II
POWER QUANTITIES AND CURRENT COMPONENTS UNDER NONSINUSOIDAL OPERATION
2.1 Introduction The goal of this chapter is to introduce the theoretical framework which is needed to deal with electric systems working under nonsinusoidal conditions. The timedomain approach adopted in the following is developed starting from [33], but it contains the elaboration of such theory to better cope with the compensation problem. The proposed approach will be also compared with other Power Theories: the main differences will be underlined and its advantages will become apparent. In the first part of the chapter, the mathematical background will be set: under the basic hypothesis of periodic operation, a suitable set of homogeneous reference variables will be defined and their properties will be introduced. Then, these mathematical concepts will be applied to the analysis of electrical networks, and the attention will be especially focused on power terms. At first, instantaneous power quantities will be defined and the corresponding average terms will be taken into account, too. The physical meaning and the properties of such terms will be also pointed out. From average power definitions, a corresponding current decomposition is elaborated, which again maintains a physical meaning due to the relationship with power terms and voltage/current distortion. Following, also the concepts of complex, apparent and distortion power are derived and their properties are clarified.
37
To achieve complete generality, all concepts and definitions are extended to vector analysis so that multiphase systems can be treated, too. In the end of the chapter some meaningful testcases are chosen to compare the proposed theory with other wellestablished approaches, both regarding average values and regarding instantaneous ones.
2.2 Approach to the study of networks working under distorted conditions In the following paragraphs a power theory to analyze distorted electric systems is proposed. It is entirely established in the time domain, since its main goal, from a practical standpoint, is to make it possible the development of suitable compensation techniques. As is wellknown, a time domain approach potentially ensures faster computations than theories based on Fourier analysis, since it does not require analysis along the waveform period. This practically means that it represents the best framework for the control of fast compensators like Active Power Filters. Obviously this requires the introduction of suitable instantaneous quantities, whose definition is not univocal and, thus, must be chosen according to the peculiar properties, which are considered fundamental for the development of the whole theory. Moreover, when trying to analyze a distorted network, one of the main requirements for the new theoretical framework is to represent a natural extension of what is consolidated for sinusoidal operation. In other words sinusoidal operation has always to be analyzed as a particular instance of the distorted case. In the specific case, this means that all quantities that have a specific physical meaning under sinusoidal conditions (i.e. active or reactive power) have to maintain the corresponding meaning also under nonsinusoidal operations.
2.3 Definitions and properties of homovariables To build a theoretical background for the analysis of electrical networks under sinusoidal or distorted condition, we assume as a basic hypothesis the one of periodic operation, i.e. we assume that all the quantities involved in the analysis are periodic.
2.3.1 Scalar operators Thus let’s consider x(t) as a real, continuous and periodic variable, of period T and angular frequency ω = 2π T . We also suppose that it is alternating, i.e. with a zero
38
DC value: T 1 x = ∫ x(t ) dt = 0 . T 0
(2.1)
We can also introduce the timeintegral and timederivative of the basic quantity respectively as: t
x ∫ (t ) = ∫ x(τ ) dτ 0
d x(t ) . dt We can define the homointegral of x(t) as: ) x = ω (x∫ − x∫ )
(2.2)
while the homoderivative is: ( 1 x = x'
(2.3)
x' (t ) =
ω
It is worth to underline that both homointegral and homoderivative are homogeneous to the basic quantity, in the sense that they have the same physical dimension and this explains the origin of the prefix homo . Under sinusoidal operations it’s easy to verify that, if: x = 2 X sin(ω t + α ) , then: ) x = − 2 X cos(ω t + α ) ( x = 2 X cos(ω t + α )
(where X is the RMS value of x(t)). Thus, in each instant of period T we have: ) ( x+x =0 ) ( )( x2 + x 2 = x2 + x 2 = x2 − x x = 2 X 2 Let’s now recall the definition of internal product: T 1 x, y = ∫ x(t ) y (t ) dt. T 0
(2.4)
(2.5)
As a consequence the norm is:
39
T
x =
x, x =
1 x 2 (t ) dt = X ∫ T 0
(2.6)
Some basic properties of homovariables are: () )( x=x=x ( ) x, x = x , x = 0 ( ( x, y = − x , y ) ) x, y = − x , y ) ( ( ) x , y = x , y = − x, y and finally: ) ( x , x = − x, x = − x
2
It follows directly from homovariables definition.
( ) x, x = x, x = 0
1 dx 1 ( = x, x = x, ω dt ωT
T
T
∫
x(t )
0
1 x 2 (t ) dx(t ) dt = =0 ωT 2 0 dt
For the second equivalence we can use the (2.7) and what just shown, thus: T T ) ) () ) 1 ) dx (t ) 1 x 2 (t ) ) x, x = x , x = x (t ) dt = =0 ωT dt ωT 2 0 0
∫
D.2.9
( ( x, y = − x , y
Let:
f =xy
we have: df dx dy ( ( ( ( = y+x = ω x y + x ω y = ω (x y + x y ) dt dt dt then: 1 ( ( x , y + x, y = ωT
D.2.10
T
∫ dt dt = 2π [ f (t )] df
0
) ) x, y = − x , y
Based on (2.7) we can write:
40
(2.9) (2.10) (2.11)
(2.12)
2.3.1.1 Demonstration of homovariables properties () )( D.2.7 x=x=x
D.2.8
(2.7) (2.8)
1
T 0
=0
() ) ) x, y = x , y and according to (2.9) it results: () ) ) () ) x, y = − x, y = − x, y
D.2.11 Let:
) ( ( ) x , y = x , y = − x, y ) f = xy
we have: ) df dx ) dy ) ( )( = = ω x y + x ω y = ω (x y + x y ) y+x dt dt dt then: 1 ) ( x, y + x , y = ωT
T
∫ dt dt = 2π [ f (t )] df
1
=0
0
) ( x , x = − x, x = − x
D.2.12
T 0
2
It immediately follows from 2.11
2.3.2 Vector operators All the definitions given above can be easily extended to vectors of real, continuous and periodic variables. Making reference to the Ndimension vectors x and y, defined as:
x1 x1 (t ) .. .. x = x n = x n (t ) .. .. x N x N (t )
y1 y1 (t ) .. .. y = y n = y n (t ) .. .. y N y N (t )
we can introduce the instantaneous scalar product as: N
x ⋅ y = ∑ xn yn n =1
Thus vector magnitude becomes:
x = x⋅x =
N
∑x n =1
2 n
.
Average operators are:
41
x1 T 1 x = x = ∫ x(t ) dt = .. average value T 0 x N T N N 1 x, y = ∑ x n , y n = ∑ ∫ x n (t ) y n (t ) dt internal product n =1 n =1 T 0 and consequently:
x =
x, x =
N
∑X n =1
2 n
.
norm
It’s also useful to recall the CauchySchwartz inequality for the scalar product: x, y ≤ x y . As a consequence, when applying (2.12), it gives: 2 ) ( ) ( x, x = x ≤ x x
2.4 Homovariables in electric networks All the above mathematical definitions are extremely useful when trying to analyze the behaviour of a generic electric network Π working under nonsinusoidal conditions, which, in the general case, can be completely described starting from branch voltage vector u and current vector i.
2.4.1 Conservation of homopowers Before going through further details it is worth to recall the Tellegen’s theorem, which is fundamental for the following analysis. It states that in generic electrical network Π, built up of F branches, if u is a vector of consistent branch voltages, i.e. voltages satisfying Kirkhhoff’s Law for Voltages, (KLV), and i is a vector of consistent branch currents, i.e. currents satisfying Kirkhhoff’s Law for Currents, (KLC), we have: u ⋅i = 0
(2.13)
provided that all branches are referred as loads. If the considered branch voltages u and branch currents i are simultaneous, Tellegen’s Theorem corresponds to the principle of conservation of the instantaneous power. ) ( If, then, voltage homointegral u and homoderivatives u are considered, it can be easily seen that they are consistent with the considered network Π, too.
42
Consistence of homoderivatives Showing homoderivatives consistence is quite straightforward. If a generic network mesh, µ, of M branches is considered, we can define the mesh voltage vector u µ . From the KLV we get: M
∑ uµ = 0 m =1
m
From the derivation of (2.14), it immediately follows: M M du mµ ( = ω umµ = 0 ∑ ∑ m =1 dt m =1
(2.14)
(2.15)
which shows the consistence of the homoderivatives of mesh voltages. The same applies to the currents in a cut set. Since such reasoning can be extended to all the network meshes, it derives that homoderivatives of voltage u and currents i are consistent with the network Π.
Consistence of homointegrals From the integration of (2.14), it results: t M
∫∑u 0 m =1
µ m
(τ ) dτ = 0 .
(2.16)
Defining now, for the sake of brevity: t
u µm = ∫ u mµ (τ ) dτ 0 ∫
(2.17)
(2.16) becomes: M
∑ u∫µ m =1
m
= 0.
The average value of (2.16) is, obviously, null, too. M 1 T M µ u ( t ) dt = u µm = 0 ∑ ∑ ∫ m 0 T m =1 ∫ m =1 ∫ Recalling the definition of homointegral, we get: ) u mµ µ + u µm u m = ∫ ∫ ω and then: 1 M )µ M µ u m = 0. ∑ um + ∑ ω m =1 m =1 ∫
(2.18)
(2.19)
(2.20)
(2.21)
Thus substituting (2.19) in (2.21), we finally get:
43
M
)
∑u µ = 0 , m =1
m
which shows that also mesh voltage homointegrals are consistent with mesh equation. The same applies to the currents in a cut set. Once again this consideration can be extended to each network mesh, thus obtaining ) that homovoltages u are consistent with network Π. ( Similar demonstrations apply also to current homoderivatives i and current homo) integrals i , which result consistent with the network Π, too. Thus, applying Tellegen Theorem to each pair of consistent sets of homovoltages and homocurrents, we get:
) ( u ⋅i = u ⋅i = u ⋅i = 0 ) ) ) ( ) u ⋅i = u ⋅i = u ⋅i = 0 ( ) ( ( ( u ⋅i = u ⋅i = u ⋅i = 0
(2.22)
When the considered sets of voltage and current homovariables are simultaneous, we have the conservation of instantaneous homopowers. This is a property of all the (real, imaginary, active, reactive and complex) power terms that are going to be defined and it is a fundamental property, especially in the compensation perspective. Instantaneous homopower conservation practically means that, in every generic network with consistent current and voltage homovariables, you can sum up power contributions irrespective of their precise location, in a way that the total instantaneous power remains zero, or equivalently, that the homopower entering from a network port equals the homopower absorbed by the remaining part of the network as it is seen from that port.
2.5 Instantaneous powers definitions As previously underlined, the definition of instantaneous power terms is not immediate and univocal, so it is important to focus on the power properties which are considered fundamental and that instantaneous quantities are required to fulfil. For the purpose of compensation, as better clarified in the following, the fundamental
44
property is “conservativeness”, i.e. each defined instantaneous power term must fulfil the Tellegen’s Theorem.
2.5.1 Instantaneous real power terms The first considered definition of instantaneous real power is the simple product between voltage and current and reflects the concept of instantaneous power: p1 = u i . (2.23) A different definition of instantaneous real power is: )( u i − ui p2 = . (2.24) 2 It is composed by two addends, the first one coincides with half of the instantaneous power, while the second one, which involves the voltage homointegral and the current homoderivative, is peculiar of our approach. According to a symmetry criterion also the following instantaneous real power definition can be considered, which involves voltage homoderivative and current homointegral: () u i − ui p3 = . (2.25) 2 The meaning and the properties of these different power definitions will be better analyzed and compared in the following. Here it is sufficient to underline that all the proposed definitions are conservative. Moreover under sinusoidal operation, p2 and p3 keep a constant value in each instant of time.
2.5.2 Instantaneous imaginary power terms Also for the instantaneous imaginary power several alternative definitions can be considered. The simplest definition involves the voltage homointegral and the current: ) q1 = u i . (2.26) The dual choice involves the current homointegral and the voltage: ) q 2 = −u i .
(2.27)
The presence of the “minus” sign let the (2.26) and (2.27) have the same average value, as can be seen considering homovariables properties. Another definition, which is obtained combining the two previously defined terms is:
45
) ) u i − ui q3 = . (2.28) 2 Also in this case, the reason for such definition will be better explained in the
following, together with q3 properties, however it is worth to underline that a similar definition is suitable for maintaining an energetic meaning also under nonsinusoidal operations. This can be perceived observing that, from a physical standpoint, power absorption depends on voltages and currents, while energy storage is related to voltage and current integrals, i.e. magnetic flux and electric charge. The instantaneous imaginary power definition which is symmetrical with respect to (2.28) while involving differential quantities instead of integral ones, can be considered too: ( ( u i − ui q4 = . (2.29) 2 Also in this case all the proposed definitions are conservative and, under sinusoidal operation, they keep the same value in each time instant.
2.5.3 Instantaneous complex power The information related to instantaneous real and imaginary power can be lumped up in a single quantity, which is defined as instantaneous complex power: s&(t ) = p i (t ) + jql (t ) (i=1..3, l= 1..4). Obviously the definition of instantaneous complex power depends on which real and imaginary power definitions are adopted. In any case instantaneous complex power is a complex term, whose real part coincides with instantaneous real power, while the imaginary one is given by instantaneous imaginary power.
2.5.4 Extension to multiphase systems All the power terms defined in the previous paragraphs for singlephase networks can be immediately extended to multiphase systems, if vector quantities are substituted to scalar ones. When considering a multiphase system, the only necessary care is to consider as phase voltages the voltages referred to the virtual star point (i.e. the point with respect to whom the sum of phase voltages is zero). This ensures the validity of definitions extension.
46
Under this hypothesis, for a generic Nphase network where the phase voltages and currents are defined by u and i, respectively, the instantaneous real power definitions become:
) ( ( ) u⋅i − u ⋅i u ⋅i − u ⋅ i p2 = , p3 = p1 = u ⋅ i , 2 2 while the instantaneous imaginary powers become: ) ( ( ) ) u ⋅i − u ⋅ i u ⋅i − u ⋅ i ) q3 = q4 = q1 = u ⋅ i , q 2 = −u ⋅ i . 2 2 Similar vector expressions can be applied to all the other definitions. It is worth to
underline that with such definitions, which make use of the instantaneous scalar product, a single instantaneous real/imaginary power is associated to the whole Nphase system, i.e. both p and q are scalar quantities. Also the definition of instantaneous complex power can be easily extended to multiphase systems as follows: N
N
n =1
n =1
s&(t ) = p (t ) + jq (t ) = ∑ p n + ∑ q n . It results a conservative scalar quantity, too. Obviously, it could be possible, if needed, to get the information about instantaneous real and imaginary power of each phase of an Nphase system, simply treating each phase independently and eventually building an Nsized vector, containing all the power information associated to the different system phases. In this case, however, it is worth to note that Tellegen’s Theorem does not generally apply to each single phase, but only to the system as a whole, thus power terms are conservative only if considered in the entire Nphase system.
2.6 Average power definitions In this section attention will be focused on the average values of the power terms which have been previously defined. According to what stated above, the analysis of electric networks working under sinusoidal operations has to be considered a special case of distorted operations. It is fundamental that a theory built up to deal with nonsinusoidal conditions is developed so as to maintain all the consolidated properties which are valid for sinusoidal operations.
47
This especially regards the physical meaning of active and reactive powers, which, as wellknown, are related to the permanent energy flow and energy storage in sinusoidal electric networks.
2.6.1 Active power Under non sinusoidal operation the active power is defined as: P = u, i
(2.30)
It is obvious that this is the same definition which is used under sinusoidal conditions and it maintains the usual meaning, since it represents the permanent energy flow through a given network port. It is also important to note that P represents the average value of all defied instantaneous real power terms p1, p2 and p3. This means that, once averaged, all the different instantaneous real power definitions are completely equivalent. ) ( ( ) u, i − u , i u, i − u , i P = u, i = = 2 2 or equivalently: P = p1 = p 2 = p 3 (2.31) Then it can be also noted that under sinusoidal operation P is constant and coincides at any time with the usual active power UIcosϕ.
2.6.2 Reactive power Under non sinusoidal operation reactive power can be defined as: ) Q = u, i .
(2.32)
Due to the application of homovariables properties, it can be shown also that (2.32) is equivalent to: ) ) u , i − u, i ) Q = − u, i = . 2 This means that at average level expressions in (2.26), (2.27) and (2.28) are completely equivalent, i.e.: Q = q1 = q 2 = q 3 It is worth to note that the average value of q4 differs from Q, however the quantity q4 has been introduced since it can be useful for control purposes, as better clarified in the following.
48
The physical meaning of reactive power and its properties will be discussed in the following paragraphs, however it is worth to note here that, under sinusoidal operation, Q (and also the average value of q4) is constant and coincides at any time with the usual reactive power UIsenϕ.
2.7 Power terms associated to elementary bipoles To better understand the meaning of average active and reactive power it is worth to analyze such terms with reference to elementary linear bipoles. Resistor Basic equations for the bipole are: u = Ri ⇔ i = Gu ) ) ) ) u = Ri ⇔ i =Gu ( ( ( ( u = Ri ⇔ i =Gu 1 where: G = , which is constant under the assumption of linearity. R Active power calculation offers: 2 PR = u , i = R i, i = R i
(2.33)
(2.34)
which is exactly the same expression got under sinusoidal assumption. As regards reactive power, it results: ) ) ) QR = u, i = R i , i = R i , i = 0
(2.35)
where the last equivalence holds due to the properties of homovariables. Inductor Basic equations for the bipole are: ) ( di u u = L =ω Li ⇔ i = dt ωL
(2.36)
Where L is the inductance of the bipole, which is constant under the assumption of linearity. In this case active power calculation offers: ) ) PL = u , i = ω L i , i = ω L i , i = 0
(2.37)
where the last holds due to homovariables properties. For the reactive power we have:
49
) Q L = u , i = ω L i, i = ω L i
2
=ωLI2
(2.38)
It is important to note that, given the instantaneous energy stored in an inductor: 1 wL = L i 2 (2.39) 2 there is a direct relationship between reactive power and inductor energy storage. In fact the average energy stored in the bipole results: T T 1 1 LI 2 Q L 2 W L = wL = ∫ w L (t ) dt = L i (t ) dt = = 2T ∫0 2 2ω T 0
(2.40)
Capacitor Basic equations for the bipole are: ) du i ( i=C =ωCu ⇔ u = dt ωC
(2.41)
where C is the capacitance of the bipole, which is constant under the assumption of linearity. Active power results: ( ( PC = u, i = u , ω Cu = ω C u, u = 0
(2.42)
where the last holds due to properties of homovariables. For the reactive power we have: ) ) ( QC = u , i = u , ω C u = −ω C u
2
= −ω CU 2 .
(2.43)
Being the instantaneous energy stored in a capacitor: 1 wC = C u 2 (2.44) 2 also in this case a precise relationship between reactive power and energy stored in the bipole can be found, i.e.: T T Q 1 1 CU 2 2 WC = wC = ∫ wC (t )dt = C ∫ u (t )dt = =− C T 0 2T 0 2 2ω
(2.45)
2.8 Power absorption in a linear passive network To gain a complete insight in the physical meaning of the average quantities introduced in the previous paragraph, we can consider a passive linear network, comprising H resistors, M inductors and K capacitors. As it has been shown, both active and reactive powers are conservative in every real network, thus, to account for the total network power it is sufficient to sum up all the different contributions.
50
Thus, for the active power we have: F
H
f =1
h =1
P = ∑ u f , i f = ∑ PRh = PRtot ,
(2.46)
F being the number of total network branches. As regards reactive power it results: F M K K M ) Q = ∑ u f , i f = ∑ Q L + ∑ QC = 2 ω ∑ W L − ∑ WC = 2 ω (W L − WC f =1 m =1 k =1 k =1 m =1 m
k
m
k
tot
tot
)(2.47)
As seen before, only resistors contribute to total active power, while reactive power is due to all the reactive elements. As a whole, total average reactive power is proportional to the difference between total average inductive energy and total average capacitive energy. It is worth to underline that both WLtot and WCtot are obtained by a simple sum of the different averaged energy storage contributions of capacitors and inductors, wherever located in the network. The (2.47) offers also an other important contribution in the compensation perspective, since it quantifies the amount of inductive or capacitive energy which is needed to compensate for the reactive power absorbed by the loads. This kind of information has a wider application, in the sense that it is not limited to linear and passive networks. Actually, irrespective of the origin of reactive power, (i.e. also in the case it is generated by non linear loads), it can be always compensated by means of reactive elements, if provided of suitable energy storage elements, as will be better clarified in the following.
2.9 Current terms definition The previous definition of suitable power terms can be used to make a corresponding current decomposition and to guarantee that different current terms maintain a specific physical meaning. Thus, each generic port current can be basically decomposed into three components: active, reactive and void current.
Active current, ia, is defined as the minimum current, i.e. the current with minimum norm, which conveys the total active power P absorbed by the network at a given port. It can be expressed as:
51
ia =
u, i u
2
u=
P u
2
u = Ge u
(2.48)
where Ge is the equivalent port conductance. It is worth to underline that active current is associated to the whole active power and to zero reactive power: Pa = u, i a = u i a = u, i = P ) ) Qa = u , i a = Ge u, u = 0
(2.49)
Reactive current, ir, is defined as the minimum current, i.e. the current with minimum norm, which conveys the total reactive power Q absorbed by the network at a given port. It can be expressed as: ) u, i ) Q ) ) i r = ) 2 u = ) 2 u = Be u (2.50) u u where Be is the equivalent port susceptance. No active power is associated to reactive current, but i r conveys the whole reactive power: ) Pr = u, i r = Be u , u = 0 ) ) ) Qr = u , i r = u i r = u , i = Q
(2.51)
Void current, iv, represents the remaining current component, thus it’s defined as: iv = i − ia − ir . (2.52) Its name is due to the fact that it is associated to neither active nor reactive power. It is: Pv = u, i v = u, i − i a − i r = P − Pa − Pr = 0
Qv = uˆ, i v = uˆ, i − i a − i r = Q − Q a − Q r = 0
(2.53)
To better understand the meaning of void current it is convenient to analyze it in the frequency domain, i.e. turning to Fourier analysis. Let {K i } be the set of harmonic indexes corresponding to the existing harmonics of current i and {K u } the set of harmonics of voltage u. We can write:
i (t ) = u (t ) =
∑i
k ∈{K i }
u ∑ { }
k∈ K u
52
k
(t ) = k
∑
k ∈{K i }
(t ) =
2 I k sin(kωt + β k )
2U ∑ { }
k∈ K u
k
sin(kωt + α k )
(2.54)
It is worth to note that, for the single harmonic: U ) uk (t ) = − 2 k cos(kωt + α k ) , k
(2.55)
thus being: u)k = U k . k
Defining now {K } as the common harmonic set, i.e. the intersection of {K u } and {K i }, we can write: i (t ) = u (t ) =
∑i
k ∈{K }
k
(t ) +
∑ u k (t ) +
k ∈{K }
∑i
k k ∈{K i − K }
(t ) = i h + i g
∑ u k (t ) = u h + u g
(2.56)
k ∈{K u − K }
where ih and u h are the homofrequencial current and voltage terms, while i g and u g are called (independently) generated terms. In this way, the first component of void current can be defined. It is called (independently) generated current, i g , and is composed by those current harmonics that are not present in the voltage spectrum. Now, considering each harmonic component of ih , we can define active and reactive terms according to (2.48) and (2.50): u k , ik Pk i ak = u = u k = Gk u k k 2 U k2 uk ) u k , ik ) k 2 Qk ) ) i rk = ) 2 u k = u k = Bk u k 2 Uk uk
(2.57)
Then, once defined ϕk = α k − β k , we get: Pk = U k I k cos ϕ k , Qk =
U k I k sin ϕ k k
and correspondingly: I kI G k = k cos ϕ k , B k = k sin ϕ k . Uk Uk
(2.58)
(2.59)
Total active and reactive powers are given by the sum of active and reactive powers which are associated to each single harmonic: ∑ Pk = P k ∈{K }
∑ Qk = Q
(2.60)
k ∈{K }
We can then decompose the total current i into:
53
i=
∑i
k ∈{K }
ak
+
∑i
k ∈{K }
rk
+ i g = i ha + i hr + i g
(2.61)
defining iha as the (total) harmonic active current and ihr as (total) harmonic reactive current. It is natural to wonder if total harmonic active current and total harmonic reactive current coincide with active and reactive current defined in (2.48) and (2.50). The answer is negative but it can be verified that: and i a ≤ i ha i r ≤ i hr .
(2.62)
D.2.62 ia ≤ iha We can equivalently write:
P u
It results:
P2 2
u
≤
2
u,
P u
2
u ≤
Pk
∑u k
2
uk ,
k
Pk
∑u k
2
uk
.
k
Pk2
∑u k
2
k
Let’s consider the series at the second member. It is a positive terms series and for every single term Pk2 Pk2 Pk2 Pk2 ≥ . This means that the series: majors and being we can write: 2 2 2 2 uk u k u k uk
∑
∑ k
2 k 2
P u
=
∑P
2 k
k
u
2
≥
P2 u
2
∑
, then (2.62) immediately follows.
Same reasoning can be applied to the inequality involving reactive an (total) harmonic reactive current.
So, we can define the scattering current terms as: isa = iha − ia active scattering component
(2.63)
i sr = i hr − i r
(2.64)
reactive scattering component
Thus, we can observe that void current can be finally decomposed into: iv = i sa + i sr + i g
(2.65)
and this shows, regarding its physical meaning, that scattering current takes account also of the different behaviour that a certain network impedance shows at different frequencies. In other words, scattering current is due to the fact that, generally, G k ≠ Ge and Bk ≠ Be . Susceptance spread over frequencies is something absolutely common, since it’s sufficient to think of inductances and capacitors, but also conductance frequency
54
variations are largely diffused as can be noted considering the wellknown “skin effect”. It is important to point out that the frequency domain approach was used only to clarify the physical meaning of void current, but it is not necessary neither for the development of the theory nor for the elaboration of compensation strategies. Thus the presented theoretical approach can be entirely developed in the time domain. Then it is worth to underline that both the definition of (independently) generated current and the one of active scattering current are in substantial agreement with those proposed by Czarnecki in his fundamental papers [1012]. Finally it is possible to demonstrate that all the current terms which have been defined are orthogonal, i.e.: i
2
= ia
2
+ ir
2
+ iv
2
= ia
2
+ ir
2
+ i sa
2
+ i sr
2
+ ig
2
(2.66)
Demonstration of the orthogonality of the different current components D.2.66 a) Orthogonality between active and reactive current: i a , ir = 0 ) ) ia , ir = Ge u, Be u = Ge Be u , u = 0 b)
Orthogonality between active and void current: ia , iv = 0
i a , iv = i a , i − i a − i r = i a , i − i a , ia − i a , i r = Ge u , i − Ge u, i a − 0 = Ge ( P − P) = 0 c)
i r , iv d)
Orthogonality between reactive and void current: i r , iv = 0 ) ) = i r , i − i a − i r = i r , i − i r , i a − i r , i r = B e u , i − 0 − Be u , i r = B e ( Q − Q ) = 0 Orthogonality between active scattering current and active current: i sa , i a = 0
i sa , i a =
∑
i ak − ia , i a =
k∈{K }
= Ge Gk u k k∈{K }
∑
e)
2
G k u k − Ge u , Ge u = Ge k∈{K }
∑
G k u k , u − Ge u , u k∈{K }
∑
2 − Ge u = Ge ( Pk − P ) = 0 k∈{K }
∑
Orthogonality between active scattering current and reactive current: i sa , i r = 0
i sa , i r =
∑
i ak − i a , i r
k∈{K }
=
) G k u k − G e u , Be u = Be k∈{K }
∑
) ) G k u k , u − G e u, u k∈{K }
∑
) = Be Gk u k , u k − 0 = 0 k∈{K }
∑
f)
Orthogonality between reactive scattering current and reactive current: i sr , ir = 0
55
i sr , i r =
∑
i rk − i r , i r
k∈{K }
) = Be Bk u k k∈{K }
∑
2
) ) ) Bk u k − Be u , Be u = Be k∈{K }
∑
=
) ) ) ) Bk u k , u − Be u , u k∈{K }
∑
) 2 − Be u = B e ( Qk − Q ) = 0 k∈{K }
∑
Orthogonality between reactive scattering current and active current: i sr , i a = 0
g)
i sr , i a =
∑{ }i
rk
∑{ }B u
)
− ir , ia =
k k
k∈ K
k∈ K
) − Be u , Ge u = Ge
∑{ }B u
)
k k
k∈ K
) , u − Be u , u
) = Ge Gk u k , u k − 0 = 0 k∈{K }
∑
Orthogonality between active scattering current and reactive scattering current: i sa , i sr = 0
h)
i sa , i sr =
∑i
k∈{K }
∑{ } i , ∑{ }i ak
k∈ K
i)
ak
rk
k∈ K
− i a ,i sr −
∑{ }i
k∈ K
=
ak , i r
∑i
k∈{K }
=
ak ,i sr
∑ i , ∑{ }i
− i a , i sr =
k∈{K }
∑{ } G u , ∑{ }B u
)
k
k
k∈ K
k
k∈ K
k
−
ak
rk
− ir − 0 =
k∈ K
∑{ }G u k
)
k , Be u
=0
k∈ K
Orthogonality between generated current and all the other current components
Generated current ig is orthogonal to active, reactive and both scattering currents, because it is composed by harmonics of different orders with respect to them all.
2.10 Complex, apparent and distortion power Like commonly done under sinusoidal operation, and extending to average values what previously done for instantaneous quantities, the active and reactive power can be included in a single complex quantity, which is called complex power. It is defined as: S& = P + jQ
(2.67)
It conveys the information about the average power absorption at a specified network port (P) and the one related to the average energy stored in the network (Q), as previously specified. The absolute value of the complex power is obviously given by: S = P2 +Q2
(2.68)
Instead the apparent power is defined as: A= u i .
(2.69)
With reference to the various current components, apparent power can be split in:
56
A = u 2
2
ia
2
+ u
2
ir
2
+ u
2
iv
2
u =P + ) u
2
2
2
Q2 + u
2
iv
2
(2.70)
so apparent power can be correspondingly decomposed into: A = P2 +Q2 + D2 = S 2 + D2 .
(2.71)
This means that distortion power D is given by: 2 )2 u − u 2 2 2 2 2 2 D = A −P −Q = Q 2 + u i v = Du2 + Di2 ) 2 u
(2.72)
where: Du = Q
2 ) u − u )2 u
2
= Qσ u
and
Di = u iv
(2.73)
Du is called voltage distortion power and exists only in presence of reactive power and voltage distortion at the measuring port. If the port voltage is sinusoidal, u is ) equal to u and voltage distortion power vanishes.
Voltage distortion power can be equivalently expressed as the product of reactive power Q and a voltage distortion factor σu. Voltage distortion factor is usually defined as the ratio between total RMS harmonic voltage and fundamental voltage. Whenever a limited voltage distortion is considered, homointegral voltage, which is a lowpass filtered replica of the voltage itself, can be taken equal to voltage ) fundamental with little approximation, i.e. u ≈ u f . Thus: ) u 2 − u 2 Du σu = = (2.74) ) u Q On the other side Di is called current distortion power and is present only with a void current absorption at the considered port, i.e. in presence of useless current distortion. Current distortion factor σi is defined as the ratio between total RMS void current
and total current: iv Di . σi = = A i
(2.75)
Thus, apparent power can be rewritten as: P 2 + Q 2 1 + σ u2 A2 = P 2 + Q 2 + D 2 = . 1 − σ i2
(
)
(2.76)
In a condition of limited voltage distortion, i.e. σ u ≈ 0 , it results:
57
A≈
S
(2.77)
1 − σ i2
thus giving for the power factor: P P λ= ≈ 1 − σ i2 . (2.78) A S If the network is well compensated so that it absorbs a limited void current, current distortion factor approaches zero and the power factor is, as in sinusoidal condition, P λ= S
2.11 Extension to multiphase systems The extension to a generic Nphase system is very important for introducing the analysis of unbalanced networks, symmetrical components and, ultimately, distributed compensation. It recalls most of the definitions already given for single phase systems, but it also introduces peculiar quantities that must be carefully analyzed. Let u and i be respectively voltage and current vector associated to a generic network branch. As already seen, the concepts of active and reactive power can be easily extended: P = u, i active power (2.79) ) ) Q = u , i = − u, i reactive power (2.80) As regards the current decomposition, the definition of active, reactive and void current can be separately applied to the generic phase n of the N phases, thus having: ian =
u n , in
un =
Pn
u = Gn u n 2 n un un ) un , in ) Q ) ) irn = ) 2 u n = ) n 2 u n = Bn u n un un
phase active current
(2.81)
phase reactive current
(2.82)
i vn = i n − i an − i rn
phase void current
(2.83)
2
with the corresponding active and reactive scattering components. Active and reactive terms, however, can be also calculated globally, thus determining the balanced components:
58
N
u, i
i = b a
u
2
u=
P u
2
u=
∑P
n
n =1 N
∑U
) u,i ) Q ) i = )2 u= )2u= u u
∑Q n =1 N
active balanced current vector
(2.84)
2 n
n =1 N
b r
u = Gb u
n
)2
∑U n =1
) ) u = B b u reactive balanced current vector (2.85)
n
By difference, the corresponding unbalance components can be obtained: u b ia = ia −ia active unbalanced current vector (2.86) having: u b i an = i an − i an = Gn − G b u n
(
)
(2.86b)
and: u b ir = ir − ir
reactive unbalanced current vector (2.87)
having: ) i rnu = i rn − i rnb = B n − B b u n .
(
)
(2.87b)
Consequently, in an Nphase system, currents can be decomposed as follows: b u b b u u i = i + i + iv = ia + ir + ia + ir + iv . (2.88) All the defined terms result mutually orthogonal, thus:
i
2
= i
+ i
b
u 2
+ iv
2
b 2
= ia
b 2
+ ir
u 2
+ ia
u 2
+ ir
+ iv
2
(2.89)
Demonstration of the orthogonality of the different current components D.2.89 Orthogonality between balanced active and unbalanced active current: i ba , i ua = 0
a)
N
(
)
N
i a , i a = ∑ G bu n , Gn − G b u n = ∑ G b Gn u n b
u
n =1
n =1
2
( )
− Gb
2
N
u = Gb ∑ 2
n =1
Pn un
2
un
2
−
P2 u
2
=0
Orthogonality between balanced active and unbalanced reactive current: i ba , i ur = 0
b)
(
)
N ) b u i a , i r = ∑ G b u n , Bn − B b u n = 0 n =1
Orthogonality between balanced active and void current: i ba , i v = 0
c)
N
i a , i v = ∑ G b u n , ivn = 0 b
n =1
d)
Orthogonality between balanced reactive and unbalanced active current: i br , i ua = 0
59
(
)
N ) b u i r , i a = ∑ B b u n , Gn − G b u n = 0 n =1
Orthogonality between balanced reactive and unbalanced reactive current: i br , i ur = 0
e)
(
)
N N ) ) ) b u i r , i r = ∑ B b u n , Bn − B b u n = ∑ B b Bn u n n =1
2
( )
− Bb
n =1
2
N Q ) )2 u = B b ∑ ) n 2 un n =1 u n
2
Q2 − ) 2 =0 u
Orthogonality between balanced reactive and void current: i br , i v = 0
f)
N ) b i r , i v = ∑ B bu n , ivn = 0 n =1
Orthogonality between unbalanced active and unbalanced reactive current: i ua , i ur = 0
g)
(
) (
)
N ) u u i a , i r = ∑ Gn − G b u n , Bn − B b u n = 0 n =1
Orthogonality between unbalanced active and void current: i ua , i v = 0
h)
N
(
)
i a , i v = ∑ Gn − G b u n , ivn = 0 u
n =1
Orthogonality between unbalanced reactive and void current: i ur , i v = 0
i)
(
)
N ) u i r , i v = ∑ Bn − B b u n , ivn = 0 n =1
Orthogonality between balanced and unbalanced current: i b , i u = 0
j) b
i ,i
u
= ia + ir , ia + ir = 0 b
b
u
u
Orthogonality between balanced and void current: i b , i v = 0
k)
i , iv = ia + ir , iv = 0 b
b
b
Orthogonality between unbalanced and void current: i u , i v = 0
l)
i , iv = ia + ir , iv = 0 u
u
u
From the above current terms a new power decomposition can be obtained, that is valid for multiphase systems and that takes into account also the unbalance. Thus, apparent power can be written as: 2 2 2 b 2 b 2 u 2 u A 2 = u i = u i a + i r + i a + i r = P +Q 2
2
u ) u
+ iv
2 u 2
2
2
u 2
+ u ia + u ir + u iv = 424 3 1 424 3 1 424 3 1 2 S au
60
2
2
2
S ru
2
2
Di
2
2
=
(2.90)
) 2 − u 2 2 2 2 = P +Q +Q + S au + S ru + Dv2 = P 2 + Q 2 + S au + S ru + Q 2σ u2 + Di2 )2 1 424 3 123 u S2 Du2 14243 2
2
2
u
2
σ u2
It can be noted that, with respect to single phase systems, two new components are present, i.e.: S au unbalancing active power S ru unbalancing reactive power
which are a consequence of the different (active and reactive) power distribution across the N phases. It can be useful to note that, in a multiphase system, vector quantities are required to define voltages and currents of each single branch, while power information on the N phases are lumped up in single quantities.
2.12 Other properties of the proposed power definitions The following considerations are developed with reference to scalar quantities, but can be easily extended to vector ones. Basic theorems of compensation An important property of imaginary power definition q3 lies in what is called First Theorem of Compensation. It states that if, at a network port, instantaneous imaginary power q3 is zero at any time during the period, then current and voltage at that port are proportional, i.e. q 3 (t ) ≡ 0 ∀ t ⇔ u = R i (2.91) D2.91 Demonstration: ) ) u i − ui i u q3 = =0 ⇒ ) = ) 2 i u Then we have: ) ) u ) ln(i ) + K = ln(u ) ⇒ ln ) = K i
⇒
) ) ) ) 1 di 1 du d i du = ) ⇒ ) = ) ) u ω i dt ω u dt i ) ) ⇒ u = eK i
) ) ⇒ u = Ri
Deriving and dividing by ω each member we finally have: ) 1 d ) 1 d [u ] = [R i ] ⇒ u = R i ω dt ω dt
61
Viceversa:
) ) ) ) ) u i − ui u i ) = ⇒ q3 = =0 u = Ri ⇒ u = Ri ⇒ u i 2
An analogous demonstration holds for instantaneous imaginary power definition q4 . In a similar way, as regards instantaneous real power definition p2, the so called Second Theorem of Compensation affirms that if, at a network port, instantaneous
real power p2 is zero at any time during the period, then current and homointegral voltage are proportional: ) p 2 (t ) ≡ 0 ∀ t ⇔ i = Bu D.2.92 Demonstration: ( )( u i − ui u i p2 = =0 ⇒ ) = 2 u i
(2.92) () ( u i ⇒ )= u i
⇒
) du di ) = u i
Then we have: ) ) u u ) ) ln(u ) = ln(i ) + K ⇒ ln( ) = K ⇒ = eK ⇒ i = Bu i i And viceversa: ) di du ) i = Bu ⇒ =B dt dt
⇒
) ) di i du di du = ) ⇒ = ) dt u dt i u
after multiplying by ω, we can also write: ( )( ) u i − ui ω di ω du u i )( = ) ⇒ ) = ⇒ u i − ui = 0 ⇒ = p2 = 0 i dt u dt u i 2
A similar demonstration holds for instantaneous power definition p3 . As a consequence of the above compensation theorems, in the general case of a current containing both an active and a reactive component, i.e.: ) i (t ) = G e u (t ) + B e u (t ) an equivalent condition in terms of instantaneous real and imaginary power absorption can be given.
Insensitivity to active and reactive current components Instantaneous imaginary power q3 has also the properties of being insensitive to active current components, in the sense that two currents having the same nonactive component but different active components, produce, at the same network port (i.e. with the same voltage u) the same q3 power absorption. This can be easily shown, looking at q3 expression and then decomposing the current into active and non active terms; we get:
62
) ) ) ) ) ) ) ) ) u i − ui u (i a + i na ) − u (ia + ina ) u (Gu + i na ) − u (Gu + i na ) u i na − u i na q3 = = = = . 2 2 2 2 From the compensation standpoint this means that, if we try to base a control strategy
only on imaginary power q3, the active current components remains uncontrolled, with possible detrimental effects in transient conditions. On the other side, instantaneous real power p2 can be shown to be insensitive to reactive current component, thus two currents having the same active and void components but different reactive components, produce, at the same network port (i.e. with the same voltage u) the same p2 power absorption. In fact: ( ) ( ( ( ) ) ( )( u i − u i u (i a + i r + i v ) − u (i a + i r + iv ) u (i a + B u + i v ) − u (i a + Bu + iv ) p2 = = = 2 2 2 ) ( ( u (i a + i v ) − u (i a + iv ) = 2 Thus, in a dual way with respect to what happens for reactive power, a control strategy which is based only on p2 leaves reactive component uncontrolled. In a similar way, when instantaneous real power definition p3 is considered, it can be noted that it is insensitive to whatever current component which is proportional to ( voltage homoderivative, i.e. i x = B x u .
2.13 Comparison between the proposed theory and other ones The last part of this chapter is dedicated to make a comparison between the proposed theory and other fundamental Power Theories. The aim is to underline the differences in the definition of various power and current terms and the possible advantages of the proposed approach. Such comparison will be carried out both with power theories based on average values, as Kusters and Moore Theory and Czarnecki Theory, and with the most relevant instantaneous power theory, i.e. that of Akagi and Nabae.
2.13.1 Comparison with Kusters and Moore Theory The goal of this section is to compare Kusters and Moore Power Theory and the one presented in the previous paragraphs, regarding both current and (average) power terms.
63
Test 1
A simple application example has been chosen, which is constituted by a distorting load (i.e. a single phase diode rectifier, with firing angle α= 60°) directly fed by a voltage source, which can be either sinusoidal (fig. 2.1) or distorted (fig. 2.6). Load current is composed as follows: 1 1 1 i = 1 sen(2π 50 t ) + sen(2π 250 t ) + sen(2π 350 t ) + sen(2π 550 t ) + 5 7 11 1 + sen(2π 650 t ) 13
(2.93)
Sinusoidal voltage is:
π
u = 1 sen(2π 50 t + ) . 3 All the proposed waveforms are taken under steadystate conditions.
(2.94)
As an introduction we can underline as Kusters and Moore Theory defines as reactive current all the nonactive current terms (fig. 2.2) and then it proposes a further decomposition for this reactive term. This decomposition can be done in two ways: the first one involves voltage integral and distinguishes inductive reactive current: T 1 ) u idt T ∫0 I ql = ) u and residual (inductive) reactive current: I qlr = I 2 − I p2 − I ql2 , while the second one involves voltage derivatives and splits the reactive current into capacitive reactive current T 1 ( uidt T ∫0 I qc = ( u and residual (capacitive) reactive current: I qcr = I 2 − I p2 − I qc2 . One of the main disadvantage of Kusters and Moore theory is that current decomposition is not orthogonal, since instantaneous inductive reactive current and instantaneous capacitive reactive current are not orthogonal (and the same holds for the corresponding residual components). The nonorthogonality of the current terms
64
load current, i
1.5
1
[p.u.]
[p.u.] 0
1 1 0.06
0.065
0.5
0.07
0.075
0.08
0.085
0.09
0.095
0.1
0.095
0.1
0.095
0.1
load instantaneous active current, ip 1
[p.u.]
0
0 1 0.06
0.5
0.065
0.07
0.075
0.08
0.085
0.09
load instantaneous reactive current, iq 1
[p.u.]
1
0
1.5
1
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.06
0.1
0.065
0.07
0.075
0.08
0.085
0.09
[s]
[s]
Fig. 2.1 Source sinusoidal voltage and load current waveforms
Fig. 2.2 Load current decomposition into active and reactive current according to K&M theory
can be easily recognized from the proposed example where, due to the sinusoidal voltage, inductive reactive current and capacitive reactive current coincide (thus their internal product is not zero), as can be seen from figs. 2.3a and 2.3b. Obviously also residual inductive reactive current and residual capacitive reactive current are equal. According to Kusters and Moore theory the inductive reactive current is: Iql = 0.612 p.u. while the capacitive reactive current is: Iqc =  0.612 p.u. This leads to question the physical meaning of the Kusters and Moore theory, which considers at the same time both the components, but in the compensation perspective load current, i
load current, i 1
1
[p.u.]
[p.u.]
0
0 1 0.06
[p.u.]
0.095
0.1
0.065
0.07 0.075 0.08 0.085 0.09 load inductive reactive current, iqL
0.095
0.1
1
0.065
0.07 0.075 0.08 0.085 0.09 load residual inductive current, iqLr
0.095
0.1
1
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
0.07 0.075 0.08 0.085 0.09 load instantaneous active current, ip
0.095
0.1
0.065
0.07 0.075 0.08 0.085 0.09 load capacitive reactive current, iqC
0.095
0.1
0.065
0.07 0.075 0.08 0.085 0.09 load residual capacitive current, iqCr
0.095
0.1
0.065
0.07
0.095
0.1
0
1 0
1 0.06
[p.u.]
0
0.065
1
1 0.06
[p.u.]
0
1 0.06
1 0.06
[p.u.]
0
1 0.06
[p.u.]
0.07 0.075 0.08 0.085 0.09 load instantaneous active current, ip
1
1 0.06
[p.u.]
0.065
1 0
1 0.06
0.075
0.08
0.085
0.09
[s]
[s]
Fig. 2.3a Load current decomposition into active, inductive reactive and residual inductive reactive term according to K&M Theory
Fig. 2.3b Load current decomposition into active, capacitive reactive and residual capacitive reactive term according to K&M theory
65
load current, i
load void current, iv
1
[p.u.]
[p.u.]
0
0 1 0.06
0.065
0.07
0.075 0.08 0.085 load active current, ia
0.09
0.095
0.1
1 0.06
[p.u.]
0.065
0.07
0.075 0.08 0.085 load reactive current, ir
0.09
0.095
0.1
1 0.06
0.1
0.065
0.07 0.075 0.08 0.085 0.09 load reactive scattering current, isr
0.095
0.1
0.065
0.07
0.09
0.095
0.1
0.065
0.07
0.09
0.095
0.1
0
0 0.065
0.07
0.075 0.08 0.085 load void current, iv
0.09
0.095
0.1
1 0.06
[p.u.]
[p.u.] 1 0
1 0.06
0.095
[p.u.] 1
1
1 0.06
0.07 0.075 0.08 0.085 0.09 load active scattering current, isa
0
0
[p.u.]
0.065
[p.u.] 1
1
1 0.06
1
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
[s]
Fig. 2.4 Load current decomposition into active, reactive and void terms according to the proposed theory
0.075 0.08 0.085 load generated current, ig
1 0
1 0.06
0.075
0.08
0.085
[s]
Fig. 2.5 Load void current decomposition into scattering active/reactive and generated terms according to the proposed theory
it looks for negative parameters solutions, so that a compensation with capacitors or inductances (of positive value) is feasible. By converse, the here proposed approach univocally identifies the current components which can be compensated by energy storage elements, while having a strict physical meaning. From a comparison with the proposed theory, which instead maintains current components orthogonality, it is clear that, what we define as reactive current, exactly coincides with the inductive reactive current of the Kusters and Moore theory. As a consequence void current can be identified with residual inductive reactive current, as it can be seen comparing figs 2.3a and 2.4. While, however, Kusters and Moore analysis does not introduce further current decompositions, according to the proposed theory, void currents can be theoretically split into other three terms (fig.2.5) whose physical meaning has been already explained. In the case of sinusoidal voltages, however, all the current harmonics contribute to the (independently) generated current, since no corresponding harmonic voltages are present. As a consequence, no active or reactive scattering currents can be detected. As regards power terms, it is interesting to note that, even if Kusters and Moore theory introduces conservative power terms (P, QL and QC), due to nonorthogonality of the corresponding current components they cannot be composed to obtain the apparent power.
66
Average active power term follows the conventional definition, thus it does not need further discussion. About average reactive power definition, it can be noted that Kusters and Moore reactive power comprises everything but the active power, lumping in a single term many different physical phenomena. On the other hand it can be observed that, in presence of sinusoidal voltage, inductive reactive power and capacitive reactive power are equal in amplitude and opposite in sign (fig 2.11): Ql = 0.433 p.u. Qc =  0.433 p.u. In such case, reactive power defined in (2.32) exactly coincides with inductive ) reactive power defined in the Kusters and Moore theory, since u = u . Test 2 As a second example the same distorting load current as in Test 1 is associated to distorted voltage:
π 1 π 1 π u = 1 sen(2π 50 t + ) + sen(2π 250 t + 5 * ) + sen(2π 350 t + 7 * ) . (2.95) 3 5 3 7 3 Once again there is complete correspondence between active current ia and
instantaneous active current ip, defined by Kusters and Moore. In this case this term is not sinusoidal anymore, but it reproduces the distorted voltage waveform. Here, being the voltage distorted, there is no immediate correlation between its integral and derivative, so that inductive reactive current and capacitive reactive currents (as the residual reactive terms) are different, as can be seen from figs 2.8ab. Performing the current decomposition according to the proposed theory, the results in figs 2.92.10 can be found. As regards void current, it can be decomposed into three components that are still orthogonal as previously seen. In this case both active and reactive scattering currents are not zero, while the independently generated one results decreased, since only 11th and 13th current harmonics now contribute to it. As regards active power, it is slightly different from the previous test since, now, also the 5th and 7th current harmonic contribute to produce active power, thus increasing its value.
67
load current, i
1.5
[p.u.]
1
[p.u.] 0
1
1 0.06
0.5
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
0.095
0.1
0.095
0.1
load instantaneous active current, ip 1
[p.u.]
0
0 1
0.5
0.06
0.065
0.07
0.075
0.08
0.085
0.09
load instantaneous reactive current, iq 1
[p.u.]
1
0
1.5
1
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.06
0.065
0.07
0.075
Fig. 2.6 Source distorted voltage and load current waveforms
[p.u.]
0.07 0.075 0.08 0.085 0.09 load instantaneous active current, ip
0.095
0.1
0.065
0.07 0.075 0.08 0.085 0.09 load inductive reactive current, iqL
0.095
0.1
1
0.065
0.07 0.075 0.08 0.085 0.09 load residual inductive current, iqLr
0.095
0.1
0.065
0.07 0.075 0.08 0.085 0.09 load instantaneous active current, ip
0.095
0.1
0.065
0.07 0.075 0.08 0.085 0.09 load capacitive reactive current, iqC
0.095
0.1
0.065
0.07 0.075 0.08 0.085 0.09 load residual capacitive current, iqCr
0.095
0.1
0.065
0.07
0.095
0.1
1 0
1 0.06 1
[p.u.]
0
0
1 0.06
[p.u.] 1
[p.u.] 1
0
0
1 0.06
1 0.06
[p.u.]
0
1 0.06
1 0
0.065
1
1 0.06
0.065
0.07
0.075
0.08
[s]
0.085
0.09
0.095
0.1
Fig. 2.8a Load current decomposition into active, inductive reactive and residual inductive reactive term according to K&M theory
1 0.06
[p.u.]
0
0.065
0.07
0.075 0.08 0.085 load active current, ia
0.09
0.095
0.1
1 0.06
0.065
0.07
0.075 0.08 0.085 load reactive current, ir
0.09
0.095
0.1
0.065
0.07 0.075 0.08 0.085 0.09 load active scattering current, isa
0.095
0.1
0.065
0.07
0.075 0.08 0.085 load generated current, ig
0.09
0.095
0.1
0.065
0.07
0.075
0.09
0.095
0.1
1
[p.u.]
0
0.065
0.07
0.075 0.08 0.085 load reactive current, ir
0.09
0.095
0.1
1
1 0.06
[p.u.]
0 0.065
0.07
0.075 0.08 0.085 load void current, iv
0.09
0.095
0.1
1 0
1 0.06
[p.u.] 1
1
0
0
1 0.06
0.09
load active current, ia
0
1 0.06
0.085
1
[p.u.] 1
[p.u.]
0.08
[s]
[p.u.]
0
1 0.06
0.075
Fig. 2.8b Load current decomposition into active, capacitive reactive and residual capacitive reactive term according to K&M theory
load current, i
[p.u.]1 1 0.06
0.09
load current, i
[p.u.]
0
[p.u.]
0.085
Fig. 2.7 Load current decomposition into active and reactive current according to K&M theory
load current, i 1
[p.u.]
1 0.06
0.08
[s]
[s]
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
1 0.06
0.08
0.085
[s]
[s]
Fig. 2.9 Load current decomposition into active, reactive and void terms according to the proposed theory
Fig. 2.10 Load void current decomposition into scattering active/reactive and generated terms according to the proposed theory
68
active power
active power
reactive powers
0.28
reactive powers
0.28 0.4
[p.u]
[p.u]
Ql = Q
[p.u]
0.27
0.4
[p.u]
0.27
0.3
0.3
zoom 0.2
0.2
0.26
0.26 0.1
P
0.25
0.1
P 0.25
0
0
0.433
Ql
0.4329 0.4328
Q
0.4327 0.1
0.1
0.24
0.4326
0.24 0.2
0.2
0.3
0.23
Qc
0.4 0.22 0.06
0.07
0.08
0.09
0.1
0.06
0.3
0.23
0.07
[s]
0.08
[s]
Fig. 2.11 Active and reactive powers (sinusoidal operations)
Qc
0.4 0.09
0.1
0.22 0.06
0.07
0.08
[s]
0.09
0.1
0.06
0.07
0.08
[s]
0.09
0.1
Fig. 2.12 Active and reactive powers (distorted operations)
Again a deep correspondence can be found between reactive power as defined in (2.32) and Kusters and Moore inductive reactive power, where the only difference ) between these quantities lies in the coefficient u u which is present in the Kusters and Moore definition (fig 2.12). The little difference between Ql and Q that can be detected in fig. 2.12 determines a corresponding difference between the residual reactive power of the Kusters and Moore Theory (Qlr) and the distorting one (D) of the proposed theory. Due to the current nonorthogonality, Kusters and Moore Theory takes also into account the residual capacitive power (Qcr).
2.13.2 Comparison with Czarnecki Theory In this paragraph a brief comparison between the proposed theory and the Currents’ Physical Components (CPC) Theory proposed by Czarnecki is presented. It is worth to recall that Czarnecki’s CPC theory cannot be easily treated in a unified way, since from its very first elaboration [10] to the most recent ones [1517] some differences can be detected, as underlined in the previous chapter. However, as a whole, from the very beginning it has certainly the merit of introducing some new concepts that are useful to gain a better insight into the physical phenomena that determine energy exchanges. In general some important correspondences can be found between the here proposed theory and the first versions of CPC one, and, even if most of the following
69
considerations are limited to the single phase example, some comments on the threephase extension will be reported, too. A first comparison can be done by considering the singlephase nonlinear circuit described in paragraph 2.13.1 fed by the distorted voltage given by (2.95). It can be immediately noted that there is substantial agreement in the definition of the active current, of the independently generated one (which however is not considered in CPC most recent works), and finally of the scattering active terms. While the active current definition is consolidated from Fryze studies, Czarnecki had the merit to firstly introduce the scattered active term, clarifying its physical origin. As can be seen comparing fig. 2.13 with figs 2.92.10, active, scattering active and (independently) generated currents maintain the same trends according to both the analysis. The main difference between the two theories is in the reactive current definition, since in the CPC theory it collects all the harmonic reactive currents (corresponding to total harmonic reactive current defined in (2.61)), giving up to further definitions. Thus CPC Theory only distinguishes active, reactive and active scattering currents in addition to the generated one (fig. 2.13). On the other side, proposed theory maintains a deeper parallelism between active and reactive components and it distinguishes what is defined as reactive current from the reactive scattering current (figs 2.92.10). The deep difference lies in that CPC reactive current has just a mathematical definition, derived from a collection of harmonics. Instead, reactive current as defined by the present theory maintains a physical meaning which is related
load active current, ia
to the energy storage also under
1 0 1 0.06
0.065
0.07
0.075 0.08 0.085 load reactive current, ir
0.09
0.095
0.1
distorted conditions. Other corresponding differences
1 0 1 0.06
0.065
0.07 0.075 0.08 0.085 0.09 load active scattering current, isa
0.095
0.07
0.095
0.1
between
the
two
considered
approaches can be found in the
1 0 1 0.06
0.065
0.075 0.08 0.085 load generated current, ig
0.09
0.1
way they deal with power terms. Czarnecki
1
theory
decomposes
0 1 0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
apparent power into active (P), reactive (Qcz), scattered (Ds) and
Fig. 2.13 Load current decomposition according to CPC Theory
generated (Dg) power, as can be seen from fig. 2.14a. As expected the sum of the squared terms P,
70
0.45 0.6
A
A
0.35
0.5
Qcz
0.4 0.3
0.3
0.3
u ) Q u
0.4
P
Q
0.4
0.6
0.5
P
0.25 0.2
0.2 0.2
0 0.1 0.06
Di
Ds Dg
0.1
0.065
0.07
0.075
0.15
0.1
0.1
0 0.08
0.085
0.09
0.095
0.1
Fig. 2.14a Power decomposition according to Czarnecki Theory
0.1 0.06
0.05
0.07
0.08
0.09
0.1
0 0.06
Du 0.07
0.08
0.09
0.1
Fig. 2.14b Power decomposition according to the proposed theory
Qcz, Ds and Dg gives the whole apparent power. In fig 2.14b, the proposed theory power decomposition is shown. As can be seen, apparent and active powers coincide with Czarnecki ones, as expected, while all the other terms are different.
u It can be observed that the term ) Q can be further decomposed into reactive power u Q and voltage distortion power Du , and the two addends are plotted on the right part of fig 2.14b.
u Obviously the square of ) Q is equal to the sum of Du squared and Q squared. u It can be seen however that Czarnecki reactive power, Qcz, does not coincide with anyone of the proposed power terms, since it comprises part of the contribution u which is included in the ) Q term and part which is inside the current distortion u power Di. By side, it can be observed that if the same test would be done with a sinusoidal voltage, the same results would be obtained both from the proposed theory and from CPC one. Since in this case no scattering (active and reactive) current would be u present: we’d have: Qcz= ) Q , equal to the usual definition of reactive power which u is given under sinusoidal operation. As a consequence, in the Czarnecki analysis the
71
term Ds would disappear, while the term Dg would exactly coincide with the Di of the proposed approach. However it is worth to note that the definition of so many power terms (and related current terms) is quite useless, if it isn’t somehow practically useful. For this reason it is worth to analyze the contribution of the two theories in the field of compensation. As Czarnecki explains for example in [17], the reactive current compensation is possible by the use of passive means, however its definition requires an harmonic approach which is not so straightforward. Moreover in [14] it is shown that also scattering active terms can be compensated by reactive elements, where, however, the complexity of the compensation network makes this approach practically questionable, as recognized by Czarnecki himself. In the end, however, both reactive and scattering active terms are shown to be theoretically compensable by passive means and both require an harmonic approach which however leads to loose the physical meaning. If, for example, a single reactive element would be available, its design would be probably done with reference to the fundamental component, but, again, a mathematical operation with no physical correspondence is required. On the contrary, the here presented definition of reactive current (and reactive power), due to its physical meaning which is related to energy storage, immediately offers an indication of a quite simple passive provision for the power factor improvement, where no Fourier analysis is needed. Moreover it must be underlined that the conservativeness of our reactive power definition ensures that it maintains the energetic information irrespective of the considered network port, while Czarnecki definitions are strongly dependent on the chosen grid terminals. Moreover it is hard to say that CPC Theory is compensation oriented, especially if its attempt to manage also switching compensators (i.e. APFs) is considered. When dealing with threephase circuits to elaborate a compensation strategy for APFs, CPC Theory recurs to the definition of completely new current components (working and detrimental current), while the previous decomposition seems not to provide any useful information. The here proposed theory, on the contrary, manages all the compensation problems in a unified way and the current and power components that it defines are useful to
72
control at the same time Static Var Compensators and Active Power Filters, as will be shown in detail in the following chapters.
2.13.3 Comparison with Akagi and Nabae Theory When dealing with the instantaneous part of the proposed theory, it is quite natural to compare it with the AkagiNabae one, which is the reference point of all the studies which have been developed up to now at an instantaneous level. The theory presented in [2122] is aimed at the analysis of threephase systems, which are considered as a whole. For this reason all the examples in the following refer to a threephase fourwire system, to gain wider generality while taking into account also the possible homopolar component. It is worth to note that the approach to the threephase system is slightly different in the two cases, in the sense that the proposed theory considers the neutral wire exactly as other phases and characterizes it through its current and voltage, while in the AkagiNabae approach such information are derived indirectly from the consideration of phase quantities. All the presented waveforms are taken under steady state conditions. The aim of the comparison is to underline the differences among the instantaneous powers p, q and p0, which are defined by Akagi and Nabae and the corresponding quantities p, q (defined according to eqs (2.23) and (2.26)) and p0 introduced by the proposed theory. The analysis will be also extended to the different current components which are considered in the two approaches. Test 1 As a first example the system is considered to be fed by the following distorted and symmetrical voltages (fig. 2.15). π π u a = 1 sen( 2π 50t + ) + 0.05 sen ( 2π 250 + 5 ) 3 3 π π π π u b = 1 sen ( 2π 50t + − 2 ) + 0.05 sen ( 2π 250 + 5 − 5 * 2 ) 3 3 3 3 π π π π u c = 1 sen ( 2π 50t + + 2 ) + 0.05 sen ( 2π 250 + 5 + 5 * 2 ) 3 3 3 3
(2.96)
where the voltages ua,b,c are the phase to neutral ones. The distorted phase currents are given by (fig.2.16):
73
[p.u.]
[p.u.]
1
1
0.5
0.5
0
0
0.5
0.5
1
1
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
0.06
0.065
0.07
0.075
Fig. 2.15 Phase to neutral source voltage of the considered threephase four wire system
i a = 1 sen( 2π 50t +
0.08
0.085
0.09
0.095
0.1
[s]
[s]
π
Fig. 2.16 Phase currents of the considered threephase four wire system
π
π
) + 0.05 sen ( 2π 150 + 3 ) + 0.05 sen ( 2π 250 + 5 ) 3 3 3
π
π
π
π
π
π
ib = 1 sen ( 2π 50t − 2 ) + 0.05 sen ( 2π 150 − 3 * 2 ) + 0.05 sen ( 2π 250 − 5 * 2 ) 3 3 3 ic = 1 sen( 2π 50t + 2 ) + 0.05 sen ( 2π 150 + 3 * 2 ) + 0.05 sen ( 2π 250 + 5 * 2 ) 3 3 3 (2.97)
The presence of the third harmonic current component determines the presence of a nonzero neutral current i0 = i a + ib + ic , i.e. of an homopolar component. At first power quantities considered by the two approaches are taken into account. Since no third harmonic voltage is present, no zerosequence power term can be detected according to both the theories. 2
2
[p.u.]
[p.u.]
1.5
1.5
1
1
0.5
0.5
0
0
0.5
0.5
1
1
1.5
1.5
2 0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
[s]
Fig. 2.17 Instantaneous and average active and reactive powers according to the AkagiNabae Theory
74
2 0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
[s]
Fig. 2.18 Instantaneous and average active and reactive powers according to the proposed theory
[p.u.]
[p.u.]
1
1
0.5
0.5
0
0
0.5
0.5
1
1
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
0.06
0.065
0.07
0.075
[s]
0.08
0.085
0.09
0.095
0.1
[s]
Fig. 2.19 Instantaneous active (phase) current components according to the AkagiNabae theory
Fig. 2.20 Instantaneous active balanced current components according to the proposed theory
In fig. 2.17 instantaneous active and reactive power according to the pq theory are reported, together with their average value. In fig 2.18, corresponding quantities according to the proposed theory are depicted It is immediate to note that, while active (real) power results the same according to both the theories, a substantial difference can be observed regarding reactive (imaginary) power. In particular, Akagi and Nabae reactive power results opposite to the one calculated by the proposed theory (and a similar effect was already pointed out by Czarnecki in [23]). In the proposed theory reactive power has the energetic meaning already pointed out, while no precise meaning can be attributed to instantaneous and average reactive power according to Akagi and Nabae, since such terms have mainly a mathematical origin. [p.u.]
[p.u.]
1
1
0.5
0.5
0
0
0.5
0.5
1
1
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
[s]
[s]
Fig. 2.21 Instantaneous reactive current components according to the AkagiNabae theory
Fig. 2.22 Instantaneous reactive balanced current components according to the proposed theory
75
0.2
0.2
[p.u.]
[p.u.]
0.15
0.15
0.1
0.1
0.05
0.05
0
0
0.05
0.05
0.1
0.1
0.15
0.15
0.2 0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
0.2 0.06
0.065
0.07
0.075
[s]
0.08
0.085
0.09
0.095
0.1
[s]
Fig. 2.23 Instantaneous zero sequence current component according to the AkagiNabae theory
Fig. 2.24 Instantaneous void current components according to the proposed theory
Then it is worth to move to current analysis. While Akagi and Nabae active currents coincide with balanced active current of the proposed study (figs 2.192.20), both reactive (figs 2.212.22) and consequently remaining (zero sequence or void) components (figs 2.232.24) are different. It can be noted that in the proposed theory reactive current waveforms depend on the voltage homointegrals and, as a consequence, void currents lump inside more than the only zerosequence term of the AkagiNabae decomposition. It can also be observed that, since the system is completely balanced, both active and reactive unbalanced components, that are calculated according to the proposed theory, are null (figs 2.252.26).
[p.u.]
[p.u.]
1
1
0.5
0.5
0
0
0.5
0.5
1
1
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
[s]
Fig. 2.25 Instantaneous unbalanced active currents according to the proposed theory
76
0.1
0.06
0.065
0.07
0.075
0.08
[s]
0.085
0.09
0.095
0.1
Fig. 2.26 Instantaneous unbalanced reactive currents according to the proposed theory
[p.u.]
[p.u.]
1
1
0.5
0.5
0
0
0.5
0.5
1
1
0.06
0.065
0.07
0.075
0.08
[s]
0.085
0.09
0.095
0.1
Fig. 2.27 Phase to neutral source voltages of the considered threephase fourwire system
0.06
0.065
0.07
0.075
0.08
[s]
0.085
0.09
0.095
0.1
Fig. 2.28 Phase currents of the considered threephase fourwire system
Test 2 The second example considers a simple sinusoidal system, fed by the following symmetrical voltages: π u a = 1 sen ( 2π 50t + ) 3 π π u b = 1 sen ( 2π 50t + − 2 ) 3 3 π π u c = 1 sen ( 2π 50t + + 2 ) 3 3 A single phase resistive load, whose current absorption results: π i a = 1 sen ( 2π 50t + ) 3 ib = 0
(2.98)
(2.99)
ic = 0
is connected between phase a and the star point and corresponding voltage and current waveforms are reported in figs 2.272.28. As regards power terms, they are reported in figs 2.292.30 and it can be noted that there is a coincidence of the two theories in their definition, since both the analysis reveal the presence of an active power, but not reactive power is detected, since the load is purely resistive. When dealing with current terms, however, major differences can be found. In particular, considering active currents calculated according to the AkagiNabae theory (fig. 2.31) it can be seen that, at first, they are not sinusoidal, in spite of the fact the voltage source is sinusoidal and the load is linear. Moreover active components lose the main property of Fryze currents of being proportional to
77
2
2
[p.u.]
[p.u.]
1.5
1.5
1
1
0.5
0.5
0
0
0.5
0.5
1
1
1.5
1.5
2 0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
2 0.06
0.065
0.07
0.075
[s]
0.08
0.085
0.09
0.095
0.1
[s]
Fig. 2.29 Instantaneous and average active and reactive powers according to the AkagiNabae Theory
Fig. 2.30 Instantaneous and average active and reactive powers according to the proposed theory
the corresponding voltage. To make a comparison with the proposed theory both balanced (fig.2.32) and unbalance (fig. 2.33) components of the active current must be considered. It can be seen that balanced active currents maintain the symmetry of the corresponding voltage, while the whole effect of the unbalance is taken into account by the unbalanced active currents. Then, if reactive current are focused, it can be noted that according to the Akagi and Nabae decomposition they are not null (and are also distorted) even if the load is purely resistive (fig. 2.34). On the contrary, according to the proposed theory, reactive currents (both balanced, fig. 2.35, and unbalanced, fig. 2.36) are null. Finally, while Akagi and Nabae Theory detects a zero sequence current (fig. 2.37), no void current is obtained according to the proposed theory (fig.2.38), since the load is purely resistive, thus it does not [p.u.]
[p.u.]
1
1
0.5
0.5
0
0
0.5
0.5
1
1
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
[s]
Fig. 2.31 Instantaneous active (phase) current components according to the AkagiNabae theory
78
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
[s]
Fig. 2.32 Instantaneous active balanced current components according to the proposed theory
[p.u.]
[p.u.]
1
1
0.5
0.5
0
0
0.5
0.5
1
1
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
[s]
0.06
[p.u.]
1
1
0.5
0.5
0
0
0.5
0.5
1
1
0.07
0.075
0.08
0.075
0.08
0.085
0.09
0.095
0.1
Fig. 2.34 Instantaneous reactive (phase) currents according to the AkagiNabae Theory
[p.u.]
0.065
0.07
[s]
Fig. 2.33 Instantaneous unbalanced active currents according to the proposed theory
0.06
0.065
0.085
0.09
0.095
0.1
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
[s]
[s]
Fig. 2.35 Instantaneous balanced reactive current according to the proposed theory
Fig. 2.36 Instantaneous unbalanced reactive current according to the proposed theory
0.2
[p.u.]1
[p.u.]
0.15 0.8 0.6
0.1
0.4 0.05 0.2 0
0
0.2 0.05 0.4 0.6
0.1
0.8 0.15 1 0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
[s]
Fig. 2.37 Instantaneous zero sequence current component according to the AkagiNabae theory
0.2 0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
[s]
Fig. 2.38 Instantaneous void current according to the proposed theory
79
Time period
Kusters AVG & Moore Czarnecki AVG Akagi & Nabae Proposed theory
INST .& AVG INST & AVG
Physical meaning
Compensa tion oriented
Conservative quantities
Current orthogonality
Power sommability
Number of phases
NO
YES
P, Ql, Qc
NO
NO
SINGLE
PARTIAL
NO
P
YES
YES
SINGLE & THREE
NO
YES
P, p
YES
YES THREE
YES
YES
P,Q,S, p,q,s
YES
YES
SINGLE & THREE
Tab. II.1 Comparative analysis among the bestestablished power theories and the proposed one
generate harmonics, neither its behaviour is frequencydependent. As a consequence of these examples, it can be concluded that Akagi and Nabae theory is substantially a pure mathematical decomposition, thus it fails in attributing a specific physical sense to the various quantities it defines. Moreover, through the application to an unbalanced load, it has been shown that such decomposition does not establish a precise and meaningful correspondence between power and current terms. It has also be underlined how pq theory misinterprets some physical phenomena, introducing current harmonic terms in a purely resistive circuit, losing the main property of Fryze (active) currents and also taking into account reactive terms where clearly no energy storage is possible.
2.14 Conclusions In the present chapter a theoretical approach to the study of networks working under periodic, distorted operations was introduced. It is entirely developed in the timedomain and it has the widest generality. The key point of the proposed approach is that the defined quantities have a specific physical meaning since they are related to power or energy phenomena or to waveform distortion or unbalance, as previously explained. Such quantities can be easily considered the natural extension of the concepts that are commonly applied for sinusoidal operation, since traditional sinusoidal conditions can be considered as a special case of the proposed theory, where it maintains its validity.
80
The proposed study is carried out with reference to both average and instantaneous quantities. Moreover it has the fundamental property of defining conservative power terms, so that the framework for a new compensation strategy was built. Its main advantage stands in making it possible a distributed and coordinated compensation, involving both quasistationary and dynamic units, as better discussed in the following chapters. The extension to multiphase systems, which is the basis for further analysis of specific problems of electric networks, as load unbalance and reactive and harmonic compensation, was presented, too. In the last part of the chapter, a comparison with main wellestablished Power Theories has been introduced, to better underline the advantages of the proposed approach.
81
82
CHAPTER III
COMPENSATION CONTROL PRINCIPLES UNDER STATIONARY AND DYNAMIC CONDITIONS
3.1 Introduction The goal of this chapter is to introduce the fundamentals of compensation. At first the traditional classification which distinguishes reactive and harmonic compensation is presented and also the need for unbalance compensation in threephase systems is described. The importance of clarifying which is the goal of compensation is thus underlined and then the relatively new idea of distributed compensation is introduced. After a brief introduction to point out the terminology, the main approaches available in literature to distributed compensation are briefly summarized, as a starting point to introduce the new idea of cooperative compensation which will be the main focus of this research work. Then it will be shown how our strategy is strictly related to the conventional approach which is based on the evaluation of undesired current component and, at the same time, the advantage due to the introduction of an intermediate power framework to widen the application perspective will be underlined. The use of conservative power terms is the key point of the proposed approach, which makes it particularly suitable for distributed compensation. An other advantage of the proposed strategy is that it can be used for the coordination and control of compensation devices belonging to different categories, from slow compensators, as the SVCs, to fast ones, as the APFs, while implementing a suitable hierarchical approach. In the end of the chapter the most generic structure of a distributed control system is presented, and the different roles of Central Control Unit and Local Control Units
83
1.5
1.5
[p.u.]
[p.u.]
voltage current
1
0.5
0.5
0
0
0.5
0.5
1
1
1.5
0
0.002 0.004 0.006 0.008
0.01
0.012 0.014 0.016 0.018
voltage current
1
0.02
1.5
0
0.002 0.004 0.006 0.008
Fig. 3.1 Single phase sinusoidal system: non ideality due to phase displacement between voltage and current fundamentals
0.01
0.012 0.014 0.016 0.018
0.02
[s]
[s]
Fig. 3.2 Single phase nonsinusoidal system: non ideality due to current harmonics, even if no fundamental displacement is present
will be described, also hinting the need of communication among the different parts. Following, an overview about the functions of the Distributed Control System main components is offered, while a more detailed analysis of most of them, which pays special attention to control strategies, will be discussed in the following chapters, together with the logical path which led to the elaboration of a viable distributed control strategy.
3.2 Fundamentals about compensation According to the traditional approach to the “compensation problem”, a preliminary distinction about what it comprises can be done. The compensation of an electric network involves provisions against at least three main problems, thus the following classification can be introduced:  Reactive compensation: it regards the need of compensation for reactive power. This problem is present also in the simplest case of singlephase electric networks working under sinusoidal conditions, whenever non perfectly resistive loads are connected (fig. 3.1). In the more general case of distorted networks, reactive compensation is somehow related to the presence of phaseshift between the voltage and current fundamentals. However, in this case the definition of reactive power has to be revised. Traditional solutions to manage reactive compensation are especially based on the use of passive devices as capacitors or SVCs, and this can be also
84
1.5
2
[p.u.]
[p.u.]
voltage current
1.5
current ph 1
current ph 2
current ph 3
1
1 0.5
0.5 0
0
0.5 0.5
1 1
1.5 2
0
0.002 0.004 0.006 0.008
0.01
0.012 0.014 0.016 0.018
1.5
0.02
[s]
Fig. 3.3 Single phase nonsinusoidal system: non ideality due to both phase displacement between voltage and current fundamental and to current harmonics
0
0.002 0.004 0.006 0.008
0.01
0.012 0.014 0.016 0.018
0.02
[s]
Fig. 3.4 Threephase sinusoidal system: non ideality due to load current unbalance
extended to the case of distorted conditions, since it is sufficient to have a controllable amount of inductive or capacitive energy storage available. 
Harmonic compensation: it refers to the need of cancellation for current (or voltage) harmonics, to approach the ideal sinusoidal condition. This case can be distinguished from the reactive compensation, since it is possible, for instance, to have voltage and current fundamentals with no phase displacement, when however the presence of current (or voltage) harmonics keeps apart from the ideal situations (fig.3.2). In these cases, harmonic compensation is needed and suitable approaches comprise both passive and active solutions. It is worth to note that in most cases the simultaneous presence of harmonic distortion and fundamental phase shift requires both harmonic and reactive compensation (fig 3.3). 
Unbalance compensation: it can be needed in threephase systems whenever the system behaviour is different from the ideal balanced three phase (fig.3.4), due to the presence of unbalanced loads. In sinusoidal systems it can be shown that such compensation can be performed by means of passive elements, thanks to the Steinmetz theory applied to sequence components. However when dealing with distorted networks a suitable revision of components definitions is needed. In this case the compensation cannot be performed entirely by passive means, but the
85
intervention of active power filters is also required, even if most of the job can be carried out by passive means, too.
3.3 Stationary and dynamic control Reasoning from the load point of view, whenever a reactive or harmonic compensation is needed, the first provision that is adopted is, traditionally, the insertion of passive elements. Up to now, in medium and low power plants, capacitors are the widest choice against reactive power while passive filters are the preferred provision for the harmonic problems. Such solutions are standardized and quite cheap, but they have also serious disadvantages. The main problems related to capacitors and passive filters in general, are, as described in previous chapters, the risk of detrimental interactions with the power grid, and the fact that network parameter changes can severely affect compensation results. The other drawback of such solution is that it offers no regulation at all or, in the case of SVCs, only a slow one, requiring at least one line period to intervene. Because of the increasing number of non linear loads with fast dynamics and of the more and more strict regulation about power quality, such solutions are not satisfactory anymore. A faster compensation is therefore needed. This is the reason for the growing interest in active power filter solutions, that have the advantage of providing an excellent compensation, offering fast dynamic performances and not interacting with the electric grid. Up to now, however, the main disadvantage of these solutions is due to their higher cost. In a tradeoff perspective, hybrid compensators have been considered, too. They make it possible the exploitation of cheaper passive elements as capacitors and inductances for the most of the reactive and harmonic compensation, while reducing the active part rating, which is used only for the residual compensation. Hybrid compensators are obviously an intermediate solution among active and passive ones, both from the performance and from the cost standpoint. Considering the previous distinction between reactive and harmonic compensation, it can be observed that it reflects somehow the traditional frequency approach to electric systems working under distorted operations, even if, in most of the cases, both reactive and harmonic compensation are contemporarily required.
86
A different approach to compensation can be adopted, recurring to a classification based on the type of devices it exploits. Thus the distinction between stationary (or quasistationary) and dynamic compensation can be done. Stationary (or quasistationary) compensation makes it possible the control only of average active (P) and reactive (Q) powers. Such kind of compensation requires the use of energy storage devices, i.e. reactors, capacitors and SVC. Specifically, when reactors and capacitors are used, no regulation is possible (stationary compensation), while the use of Thyristor Switched Capacitors and Thyristor Controlled Reactors makes it possible at least a slow kind of regulation, since each compensating action can be performed only period by period (quasistationary compensation). Dynamic compensation, on the other hand, is needed whenever the control of current waveform during each lineperiod is desired. This kind of compensation requires fast compensation systems as Active Power Filters and it does not involve average power exchange. It can be managed via a power approach based on the instantaneous power terms which were previously introduced.
3.4 Goal of the compensation When approaching the problem of compensation in a generic electrical network, it is fundamental to define the aim of the work, which is generally related to the need to fulfil certain specifications. Quality performances of a generic electrical network can be evaluated through different indexes: Input quality indexes describe the network performance towards the power supply. Frequently used indexes are:  PD, Phase Displacement: PD = cos ϕ1 = cos(α 1 − β1 ) P P  PF or λ, Power Factor: λ = = S UI  THD, Total Harmonic Distortion: THD =
 UF, Unbalance Factor: UF =
Xi Xp
∑X k >1
X 12
(3.1)
2 k
(3.2)
(3.3)
87
or alternatively: UF =
max( X R , X S , X T ) − min( X R , X S , X T ) , X avg
(3.4)
where the numeric subscripts refer to the harmonic order, while capital letters denote RMS and α and β indicate voltage and current phase angles, respectively. Moreover Xi represents the RMS negative phase sequence, Xp the positive one, the subscripts R,S,T indicate the different phases while avg is the average of X.
However it is important to underline that some of these indexes are generally defined only for sinusoidal operation, while a more general definition is needed. Output quality indexes describe the network performance seen from the load
terminal. They relate to voltage waveform purity, symmetry and stability under stationary and dynamic conditions. The problem here is not only to asset the voltage quality, but to identify the source of disturbance. Whenever distortion and instability are caused by the loads, they are charged by proper tariffation/penalization rates. Throughput quality indexes mainly relate to transmission losses and are affected by
non active currents flowing through the distribution networks. All the above described indexes are affected by the presence of compensating units connected to the grid. Thus, controlling reactive, void and unbalanced currents absorbed at various network ports can significantly improve the indexes. Accordingly to what was previously seen, the burden of compensation can be given to the electricity supplier, especially when distortion and asymmetry in the voltage waveform is detected, while it can be attributed to the users when the unbalance, reactive and harmonic pollution responsibility belongs to distorting loads. This kind of problems has the widest impact on electric systems, since they affect the electric grid at various levels and in a diffuse way.
3.5 Distributed, cooperative and delocalized compensation When the compensation problem is considered from a network perspective, rather than from the single load one, peculiar considerations have to be done. This can be useful not only from the electric supplier standpoint, but also for a better management of industrial plants or medium size electrical grids. Moreover, in a scenario where distributed energy generation is gaining increasing diffusion, this new perspective is even more important. In all these cases one is forced to consider many distorting loads, connected to different network ports, which contribute to the total
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Compensator C1 i1
+
u1

iPCC
iL1
+
Power source
+
uL1
Network

π
uPCC
iLK +
uLK 

PCC

uM
Load1
+
LoadK
iM
Compensator CM
Fig. 3.5 Distributed compensation system
reactive power and harmonic pollution, that have to be reduced to fulfil given specifications. In such conditions it often happens that different compensators are already dispersed over the network, as a result of localized compensation provisions targeted on single loads. Most of the compensators, however, have been generally designed in a totally independent way. This means, at first, that they can affect each other behaviour in an undesired and detrimental way and, then, that their functioning is generally not optimized, in the sense of a synergic compensation. Thus, in such a scenario, when an additional compensating action is needed, it would be natural to wonder what kind of intervention is required to achieve the given compensation goal and also, if a better exploitation of the compensators that are already connected to the network can help to reach this goal. Reasoning this way introduces the perspective of distributed, cooperative and delocalized compensation. Thus it is useful to clarify what these definitions are intended to mean, as an introduction to further analysis. Localized compensation: it refers to the common compensation strategy where
certain specifications (in terms of power factor, total harmonic distortion, phase displacement at the fundamental frequency, etc) are given with reference to a defined network point and the compensation systems that have to perform the job are connected to the same port. Delocalized compensation: a different concept is the one of delocalized
compensation. It refers to the fact that desired requirements can be expressed with
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reference to a certain network port, while the compensating intervention is performed at different network ports. This does not necessary involve the presence of more than one compensating unit, but is related only to the fact the point at which specifications are given and the point where compensation is performed, are different. Distributed compensation: this definition has mainly a “topographical” meaning, in
the sense that it points out that the many compensators (stationary, quasistationary or dynamic) that can be used to achieve a goal, are physically connected to different network ports. In this sense, when considering an electrical grid, we deal with compensators that are “distributed” all over the network, and can be quite far from each other. Most of the time, such compensators are designed and driven in an independent way. When, however, an optimized compensation strategy is going to be elaborated, the physical distance among the filtering devices has to be taken into account. This is necessary especially to establish which of them will be involved in the compensation and how they can interact with one another. Cooperative compensation: this definition refers to the fact that, whenever an
optimized compensation strategy is desired, it is necessary to better exploit all the compensators already present in a network, so that they can cooperate to achieve expected results. To obtain such cooperation, however, the elaboration of a communication strategy among the filters is necessary. This means that it has to be established what kind of information they must exchange, but also the communication system they must adopt. Obviously the cooperation is based on the sharing of compensation duty among the involved devices. Also the sharing criteria have to be carefully defined. The idea of cooperation allows also a hierarchical approach to compensation, in the sense that, if various compensators can share the duty, the precedence in application is given to stationary and quasistationary compensators, while active power filters are used only for the residual job. This minimizes the energy loss associated to compensation.
3.6 Advantages of cooperative and distributed compensation When, in a generic network, linear, distorting and time varying loads are present and several compensators are connected to different grid ports, optimized (i.e. distributed and cooperative) compensation aims at:
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 Cost effectiveness: such approach makes use of cheap units whenever possible, thus minimizing the application of expensive compensators. It also avoids redundancy and overrating while fully exploiting existing compensation devices.  Cooperative operation: a cooperative compensators management allows a full exploitation of installed compensation capability and at the same time avoids detrimental interactions among different units, i.e. oscillation due to resonances, negative impedances generated by control, etc.  Synergistic control: the control action is carried out to keep, as a whole, the network behaviour as close as possible to given specifications.
3.7 State of the art of distributed and cooperative compensation As previously underlined the most of commonly proposed compensation solutions are of the localized type, and it is quite recent the idea of distributed and cooperative compensation. Distributed and cooperative control of APFs was developed within the ideal frame of power distribution system. This is quite obvious, since many distorting loads are present and they are widely spread around the network. The main perspective which is taken into account is the electric supplier one, and the problem which is intended to be solved is the possible presence of an high THD of voltage, denoting voltage harmonic propagation. The reason of this situation is the resonance between line inductances and shunt capacitors, which has been reported to be a serious problem especially when networks work in lightload condition [60]. First attempts to solve the problems were aimed at the use of a single APF, which should be conveniently located into the grid [6162]. Such a solution, however, could amplify voltage distortion in some points of the considered network, depending on the magnitude of damping. The idea of exploiting multiple filters installation to achieve voltage THD reduction was proposed in [63]. Here the effectiveness of the cooperation between two APFs both in achieving the THD reduction, and in sharing the duty is shown, but realtime communication among compensating units is required. A different strategy to achieve the same goal was proposed in [6465]. Here the lowering of voltage distortion is achieved without need of communication among the APFs. A droop relationship between the harmonic conductance and the voltampere of each active filter unit is implemented, so that they automatically share the harmonic filtering workload. To improve Distributed Active Power Filters (DAFS) performance the
91
droop functions can be also dynamically tuned, based on the measured voltage THD at the installation point, as proposed in [66]. The approach to a distributed compensation can be related to increasingly diffused Distributed Generation Systems in a quite natural way. It has been shown that the GS droop function approach can be easily added to the droop powerfrequency (Pf)
and reactive powervoltage (QV) that are used to control distributed power generators connection to the electric grid [67].
3.8 Proposed approach to distributed and cooperative compensation All the above presented approaches face the compensation problems from the power distribution system standpoint. This implies, at first, that they focus on the voltage distortion, more than on the current compensation, and that the main goal is to maintain voltage THD lower than a specified values in each point of the grid. We can say that they limit to the damping of harmonic oscillations, operating the APFs as to appear as a pure resistance at every frequency above the fundamental. The proposed compensation approach is different in the basic perspective, since it is not aimed at the control of network voltage distortion, but at the limitation of current harmonic pollution or at the satisfaction of specified requirements expressed in current or power terms. This means also to separate the distortion induced by the load (whose compensation is generally demanded to local units) from that coming from the source (which has to be managed at a central level). Moreover, as effectively expressed in [64], one of the reasons underneath a distributed approach to compensation is that it “offers great filtering capability by integrating several smallcapacity high bandwidth active filter units installed at various locations, and provides superior scalability and flexibility to cope with the growth of harmonic producing loads within the power system”. However in [6366] the integrated approach exploits only Active Power Filters, while no reference is made to the possible contribution of cheaper and more standardized compensators as SVCs. The aim of the proposed research work, instead, is to make it possible the exploitation of both quasistationary and dynamic compensators in a cooperative and optimized way. In this sense the idea of distributed and cooperative approach is maintained and also widen, since the control strategy leads to comprise, not only
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active power filters, but also cheaper compensators as TSCs and TCRs, whenever possible. Thus, a suitable integrated control strategy manages the difference in compensators type, their available compensation capacity and takes also into account their different location and their need of exchanging information. In the above presented literature, deep attention is paid to the choice of the best location for active power filters, to have them achieve their task. This however requires some knowledge of the electric network and the introduction of the system analysis in terms of transmission lines. The cooperation among the filters stands in their automatic workload sharing, which is substantially based on the different ratings of the many units. It is also worth to underline that even if the approach is in terms of distributed compensation, every unit is controlled on the basis of measurements and specifications that are given at the installation point of the compensating unit itself. The proposed approach, on the other hand, makes it possible to connect and control the filtering units to ports that are far from the PCC, since all the control quantities are conservative in every real network.
3.9 Differences in the approach to quasistationary and dynamic compensation As hinted before, one of the peculiarity of the proposed theory is that it makes it possible to manage both SVCs and APFs to achieve the compensation goal. When compensation provisions are needed, an optimized solution can be found considering the problem in a wider perspective, i.e. reasoning at a system level. This kind of procedure requires to think of the compensation, not as a problem of the single distorting load anymore, but as a global network problem. This obviously means also to abandon the idea of a localized action and supports the concept of delocalized and distributed compensation, which is a quite recent proposal for compensation problems. Obviously quasistationary and dynamic compensators are quite different in nature, and this has to be carefully taken into account in the elaboration of the compensation strategy. As regards the use of SVCs, they are devices capable of energy storage, thus they are used to compensate those current components which are related to a specific average
93
power term. Obviously when dealing with TSCs or TCRs, the most important thing is their correct rating, so they need to be carefully designed to achieve the goal. As regards APFs, their compensation duty does not necessarily require energy storage, however their accurate design is not enough, since the best control strategy has to be chosen to have them performing their duty. In this sense an accurate analysis of the best cooperative compensation approach should include some basic problems that are typical of the control of every APF (i.e. the generation of the current reference or the design of a suitable current regulator) but also peculiar features of distributed and cooperative compensation, as the strategy to share the workload among the different units and also the need of communication among the various compensation systems and the central unit). Moreover some aspects that are typical of the proposed compensation strategy have to be also taken into account and require a special attention since they represent the most innovative application of the proposed study. Among them there are: 
use of conservative power terms to build a suitable control frame,

need of systematic correspondence between power approach and current approach,

hierarchical compensation strategy.
3.10 Strategy of compensation of the different current components The main goal of this work is to show the feasibility of a compensation strategy which is based on conservative power terms, in the sense that it uses “power domain” as a framework where working to provide a suitable cooperative control strategy for various type of compensators that are dispersed in electrical networks. Besides, such compensation strategy is required to be suitable to control both quasistationary and dynamic devices. This allows to adopt a hierarchical approach to the compensation problems, in the sense that, when a compensation action is required, whenever possible it will be carried out by SVCs, and only the residual part will be demanded to APFs. To put into practice this principle it is useful to recall the theoretical decomposition of the current into several different terms, to understand in which way each of them can be compensated. In the previous chapter, thanks to the basic theorems of compensation, it has been also shown that control of current waveform during the period is equivalent to the
94
control of specific instantaneous active or reactive powers at the corresponding section. It is also worth to hint here that, when a purely active (and theoretically also a purely reactive) current is to be obtained, the zeroing of a single instantaneous power term (q or correspondingly p) is not sufficient since the amplitude of the active (reactive) component is uncontrolled, and this can compromise the compensated system performances in transient conditions, as will be better clarified in the following. Thus, in the quite general case where the compensation goal is to maintain the active current component, while zeroing all the non active terms, the power approach is analyzed. The strategy to eliminate unwanted current components, i.e. reactive, void and unbalanced currents, is exposed in the following.
Reactive current, ir From eq. (2.50) it can be seen that reactive current ir vanishes if total reactive power absorbed at the considered port is zero. This means that the compensation of reactive currents requires the absorption of a reactive power which is opposite to the one (Q) of the noncompensated network. In the perspective of a distributed compensation, this burden can be shared among several units (here indicated by the apex C) wherever located in the network, i.e.: M
∑Q m =1
C m
= −Q ⇔ ir = 0
(3.5)
Active scattering current, isa As regards active scattering current, from (2.48), (2.572.59) and (2.63), we get: 2 uk Psk P .(3.6) i sa = iha − ia = ∑ (G k − Ge ) u k = ∑ u k , where Psk = Pk − 2 2 u k∈{K } k ∈{K } u k This shows that active scattering compensation requires an harmonic approach, in fact the condition for isa to vanish is that each of its harmonic components is zero. Thus the compensation system has to absorb, for each harmonic k, a compensation power, PkC , opposite to the scattering active term Psk, absorbed by the non compensated network, i.e.: M
PkC = ∑ PkmC = − Psk
(3.7)
m =1
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Reactive scattering current, isr Reactive scattering current, as can be seen from (2.50), (2.572.59) and (2.64), results: ) 2 uk Qsk ) ) isr = ihr − ir = ∑ (Bk − Be ) u k = ∑ u k , where Qsk = Qk − ) 2 Q (3.8) 2 ˆk k ∈{K } k∈{K } u u As for active scattering current, also for the reactive one, the compensation requires an harmonic approach, in that the compensator is required to absorb, for each harmonic k, a compensation power, QkC , opposite to the scattering reactive term Qsk absorbed by the noncompensated network, i.e.: M
C QkC = ∑ Qkm = −Qsk
(3.9)
m =1
Generated current, ig This current component is neither involved with active nor with reactive power, thus it cannot be compensated by means of the control of the harmonic power absorbed by the compensation system. Thus, the compensation of this current component requires an APF and determines an advantage of dynamic control over quasistationary one. If a multiphase system is considered, also the compensation of load unbalance can be required. As regards the compensation of the unbalanced reactive component, it is inherently comprised in the compensation of the reactive currents, which contain both balanced and unbalanced components.
Unbalanced active current can be also compensated by the use of quasistationary compensators, since it is associated to a non homogeneous distribution of active power among the phases. For each phase it is:
i
= i an − i
u an
M
∑P m =1
C mn
b an
= − Pn
= (Gn − G )u n = b
Pun un
u ⇔ ian =0
In the end we can conclude that:
96
2
N
un
n =1
u
u n , where P = ( Pn − ∑ Pn u n
2 2
)u n (3.10)
 Reactive current ir can be effectively compensated through Thyristor Controlled Reactors and Thyristor Switched Capacitors, which are driven based on the average quantity Q, and that do require a suitable energy storage.  Active (isa) and reactive (isr) scattering currents can be compensated based on a quasistationary harmonic frame, without need of energy storage.  On the opposite, generated current ig, which is not involved with neither active nor reactive power, requires a dynamic compensation approach theoretically without need of energy storage (provided that all the harmonics are considered and highly selective filters are available), and this determines the superiority of dynamic compensation with respect to quasistationary one.  Unbalanced (active and reactive) currents can be also compensated by means of Static Var Compensators, since they are associated to a non homogeneous distribution of (active or reactive) average power among the phases. The inherent difference between quasistationary and dynamic compensation strategy may justify the choice of completely different control quantities for the different devices, as will be better clarified in the following chapters.
3.11 Typical control scheme for distributed compensation When considering a distributed compensation approach, the general control scheme for the compensation system is represented in fig. 3.6, with reference to a generic three phase system. The first thing that is important to underline is that, even if the complex power framework is chosen to develop the control strategy, an active power filter is based on current control, in the sense that it requires a current reference to properly operate. Then, as previously underlined, the distributed (delocalized) compensation approach introduces a physical distance between the point the electric grid specifications are refereed to, and the one (or more) where APFs (and possibly SVCs) are connected. This requires also a specific structure in the APF control unit, since it has to be split into a Central Control Unit (CCU) and Local Control Units (LCU). As regards Central Control Unit, its function is to transform the specifications related to the desired network behaviour at the input port into a suitable power command. On the other side, Local Control Units are located wherever, in the network, an Active Power Filter or SVC dedicated to distributed compensation is present; their
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_ Pi
u) i
Central Control Unit
ui CRG
ii* εi + 
EA
i*
x q x p
s&1* &s* s&n* PS * s&N
ui C1 ii +
ui input port 
π
i +
u Cn 
i*
LCU
s&*
CN Fig. 3.6 Generic structure of a distributed control system
role is simply to transform the power command which has been provided to each APF or SCV into a suitable current reference for the compensator itself.
3.11.1 Central Control Unit The Central Control Unit, which represents the centralized intelligence of the distributed compensation system, performs different tasks so it is composed by several subsystems. The first one is the Current Reference Generator (CRG). This is the block which calculates the current reference once given the specifications at input port. This means that required performance, which can be expressed, for instance, in terms of Power Factor (PF) or Total Harmonic Distortion (THD) are elaborated with the data about local measured voltage (ui) and active power absorption (Pi, eventually deprived of the little contribution due to the APF itself) to be converted into a current
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signal reference, ii*. Such block ensures a wide flexibility since the current reference can be derived according to different criteria. Among them: •
Unity power factor: it means that at the input section total current coincides with active current, i.e.: i *i = i a =
•
ui
2
ui =
ui ,ii ui ,ui
ui
Constant input power, i.e.: i *i = i p =
•
pi
pi ui ⋅ui
ui =
ui ,ii ui ⋅ui
ui
Minimum current to get instantaneous power, i.e.: i *i = i p =
u ⋅ ii pi ui = i u ui ⋅ui ui ⋅ui i
Then the measured input port current, ii, is compared to such reference, so that all the undesired current components are detected. As previously shown, in the general case they comprise both unbalanced (active and reactive) components and balanced reactive terms in addition to void current. According to the fact that reactive and unbalanced components can be compensated by means of SVCs, while void current requires the use of an APF, the different contributions to the unwanted current components are separated and each of them is sent to a suitable Error Amplifier. The Error Amplifier block, that in fig. 3.6 comprises both the regulator for the SVC loop and the controller for the APF one, since they have similar functions, is a regulator which processes current error signal ε i (related to reactive + unbalanced current or to void current) to produce the current reference i SVC , i APF to be transformed into a corresponding complex power reference s& SVC , s& APF . The design of the Error Amplifier represents one of the most critical points of the whole implementation. Theoretically it could be designed according to the usual criteria, but in any case it requires the knowledge of the process to be compensated (i.e. the transfer function between the compensator current and the input current), whose identification can be not trivial. Moreover the EA implementation does not allow an instantaneous compensation, due to the unavoidable presence of control delays. In order to mitigate such effects, specific control strategies can be adopted to obtain a precise current tracking provided that load current and supply voltage are periodic or slowly varying: EA can be based on harmonic reference frame [52] or on repetitive control techniques [53]. Also these implementations, which are based on
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the periodic nature of the waveforms, are, however, somehow “slow” and require to know the system transfer function and the amount of the delay, which must be constant during the entire process. Once the current reference i * (i.e. i SVC or i APF ) is obtained, it is multiplied to input voltage and input voltage homointegral so that a corresponding complex power signal is obtained. This signal is then transmitted to the Power Sharing (PS) Unit, whose function is to distribute complex power reference among the various SVCs and APFs involved in the distributed and cooperative compensation. There are several parameters that have to be taken into account to properly distribute the complex power command. Besides the fundamental distinction between quasistationary and dynamic compensators, the first sharing criterion is certainly the compensator power rating, then also the “distance” between the compensator and the input port, since this parameter affects the compensation in terms of response time and attenuation factor of the transfer function between the compensating unit and the input port. Then it is necessary to consider the residual compensation capability of each device, since each of them is generally involved in local compensation, too. It is worth to underline that both the compensator power rating and the physical distance can be established at system set up. On the contrary, information about residual compensation capability requires a realtime communication between compensating system and Power Sharing unit. This peculiar aspect is extremely important for the whole performance of the distributed compensation system and requires specific investigation.
3.11.2 Local Control Unit It represents that part of the control system which is actually distributed, in the sense that a LCU is located wherever each of the units involved in the distributed compensation is connected to the electric grid. The function of LCU is to convert the complex power command which is generated for each compensator by the Power Sharing unit into a corresponding current reference. The current reference is suitable to directly control the APFs, taking into account the measured voltage level at the APF section. As regards the control of the
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SVCs, it can also be elaborated starting from this “local” current reference, as will be diffusely explained in Chapter VII. In general it must be observed that the transformation of a complex power reference into a corresponding current reference is performed in different ways according to what kind of network is considered (i.e. singlephase, threephase threewire, threephase fourwire), so all the different control strategies and related algorithms will be carefully analyzed in the following chapters.
3.12 Conclusions In this chapter some basic concepts related to compensation were introduced, starting from the traditional distinction among reactive, harmonic and unbalance compensation. The importance to clarify the goal of the compensation, in terms of given specifications at a certain network port is underlined, together with the intrinsic difference in the approach depending on the devices that are taken into consideration (quasistationary vs. dynamic compensation) to perform the job. In the central part of the chapter the idea of distributed and cooperative compensation was introduced and the state of the art in this field has been briefly traced. This offered the starting point to present the innovation of the approach to cooperative and distributed compensation which is proposed in this research work, regarding the methodology, the kind of considered compensation means, the elaboration of the control strategy and the introduction of a hierarchical approach. Making reference to the unwanted current components defined in the previous chapter, possible compensation strategies for each of them are introduced and then the general control scheme for a distributed compensation system was presented. Underlining the different tasks belonging to Central Control Unit and Local Control Units, the role of each functional block was briefly described as an introduction to the more detailed analysis which will be presented in the following chapters.
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102
CHAPTER IV
IMPLEMENTATION OF LOCAL CONTROL UNITS IN SINGLEPHASE SYSTEMS
4.1 Introduction The goal of this chapter is to present the first strategies which were elaborated for singlephase applications in the distributed compensation systems perspective. In particular, the convenience of a complex power approach with respect to conventional approaches will be demonstrated. In this section the attention will mainly focus on the Local Control Unit and its capability to convert an instantaneous complex power reference into a current reference as required by Active Power Filters. The starting point is the First Theorem of Compensation which was introduced in Chapter II, and the corresponding need to consider both real and imaginary instantaneous power terms (or equivalently instantaneous complex power) for compensation purposes. Several alternative definitions of complex power can be considered, which correspond to different control algorithms. Advantages and drawbacks of the various solutions will be underlined, while their logical/chronological development will be also hinted to offer a better understanding of some technical choices. Some of the compensation algorithms presented in the following can be also extended to threephase systems, as will be shown in the next chapter. The studies presented in this section converge towards a simplified compensation approach that results the most convenient for singlephase applications. It will be presented in a specific chapter (Ch. VI), together with its extension to three phase systems.
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4.2 Main goal of compensation: first approaches As previously shown, when developing whatever compensation strategy, the first step is to clarify which is the goal, in terms of specifications at the point of common coupling. In our first approach to the elaboration of a control strategy, which took origin from [69], the goal of having a purely active current at the PCC was stated. At first such requirement was translated into the power domain, according to equation (2.29), so that the current reference for the APF is derived from the condition of null imaginary power, i.e: ( ( u i − iu 1 di du q = q4 = = (u − i ) = 0 2 2 ω dt dt
(4.1)
A differential equation in the unknown quantity i is therefore obtained. However the coefficients of such differential equation depend on instantaneous voltage and its derivative and are timevariant. Thus, the solution cannot be determined in the continuous domain for whichever behaviour of power reference q. To overcome this problem a discrete time approach is adopted, where the unknown current is expressed as a function of its value at the previous time step. The solution is: 2ω Ts q~k + (u k − u k −1 )i k i k +1 = i k + uk
(4.2)
Where Ts represents the sampling interval, the sign ~ denotes the average value in the switching period, subscript k indicates that the quantity is evaluated at time instant tk, while subscript k1 corresponds to time instant tk1. The discrete expression of the equation shows that the solution diverges when the voltage value approaches zero, i.e. twice per period. Different provisions can be adopted to overcome this problem. For instance the desired current reference can be found, not directly by solving eq. (4.2), but by minimizing the cost function: ϕ = (q~ − q~ * ) 2 +ν ∆i 2
(4.3)
In (4.3) ∆i represents the difference between ik+1 and ik. and ν is a suitable nonnegative coefficient. Expression (4.3) takes into account both the difference with respect to the reactive power reference and the current increment value and is selected to avoid that ∆i becomes too high. The minimization process gives:
104
∆i =
u k (∆u k +1i k + 2ωTs q~ )
u k2 + ε 2 where ε = 2ωTsν .
or equivalently: i k +1 = i k +
u k (∆u k +1i k + 2ωTs q~ ) u k2 + ε 2
(4.4)
It can be noted that, in (4.4), whenever ε is not null ∆i tends to zero when the voltage is null, thus the problem presented above is avoided. This compensation strategy however shows another important drawback, which needs to be carefully considered: as hinted in Chapter II, q is insensitive to active current component, thus the control strategy based on the solution of (4.1) is not viable, since it is unaffected by the active component of the current, and this means that the current itself is uncontrolled whenever a transient condition occurs. In other words the solution given by (4.4) may include an active current component, which must be identified and removed in order to avoid the absorption of active power by the APF. This however would introduce a control delay, with detrimental effects on dynamic performances, as shown hereafter.
4.2.1 Compensation based on q zeroing: application example As an example of the compensation strategy presented in the previous paragraph a thyristor rectifier has been considered, fed by a distorted voltage source with a Total Harmonic Distortion (THD) equal to 5%. Fig. 4.1 shows voltage u and current i (in p.u.) before and after compensation, while fig. 4.2 shows instantaneous imaginary power generated by the compensator 1.5
0.5
[p.u.]1
[p.u.]
u
0.5
q4F
0
i
0.5
0
1 1.5 0.04
0.045
0.05
0.055
0.06
0.065
0.07
0.075
0.08
1.5
0.5
[p.u.]1
q4L
u
0.5 0
i
0.5
1
1 1.5 0.04
0.045
0.05
0.055
0.06
[s]
0.065
0.07
0.075
0.08
Fig. 4.1 Load voltage u and current i without compensation (top) and with compensation (bottom)
0.04
0.045
0.05
0.055
0.06
0.065
0.07
0.075
0.08
[s]
Fig. 4.2 Compensation instantaneous imaginary power vs. the opposite of load instantaneous imaginary power
105
2
2
[p.u]
[p.u]
1.5
1.5 1
1
u
0.5
u
0.5
0
i
0
i 0.5
0.5
1
1
1.5
1.5
2 0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
2 0.08
0.09
0.1
0.11
Fig. 4.3 Voltage u and current i at load terminals before compensation in presence of a load current step in t = 0.1 s
0.12
0.13
0.14
0.15
0.16
[s]
[s]
Fig. 4.4 Voltage u and current i at load terminals after compensation in presence of a load current step in t = 0.1 s
compared to the opposite of the instantaneous imaginary power measured at the load terminals. It can be noted that the compensation errors are negligible even around zerocrossing of the line voltage. What previously stated about dynamic properties has been verified throughout a simulation, where the load current was doubled at 100 ms. Figs 4.3 and 4.4 show the voltage and current waveforms during the process, respectively before and after compensation. The comparison of the two figures shows that, when, at t= 0.1 s, both real and imaginary powers are modified, the compensation immediately provides to the q zeroing, as can be seen considering that the current always remains in phase with the corresponding voltage. However, the system dynamic is associated to the active current component: as said before, the compensation strategy cannot intervene directly on the active current component, so it must be following identified and removed from the APF reference, causing one line period delay. The proposed approach shows that it is possible to reconstruct a current reference starting from an instantaneous (imaginary) power one with very good approximation, however the detrimental effect of the need for active current control suggests that the strategy to simply impose the condition of null imaginary power is not viable to guarantee the feasibility of the compensation strategy. Thus a different approach was elaborated, introducing an other condition to fully control the current solution.
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4.3 Introduction of complex power Since instantaneous imaginary power control is not sufficient to directly determine the APF current reference in all the operative conditions, the most immediate solution is to use also instantaneous real power in the control strategy. This means that, to obtain the current reference, both real and imaginary power information (or equivalently complex power information) need to be elaborated. According to what already seen in Chapter II, many different definitions for both real and imaginary instantaneous power can be given and consequently several different control approaches can be elaborated. Here some of them will be analyzed, starting from the one using only current and current homoderivative [70]. This, non only to follow the chronological order of this research work, but mainly because, such complex power expression shows the advantage that it is immediate to calculate, especially in a discrete domain perspective. The reference equation becomes: s = s * = p * + jq *
(4.5)
where :
)( ui − u i p= p = (4.6a) 2 ( ( ui − u i * q=q = (4.6b) 2 Since two different equations are now available, one could be tempted to solve the ( problem considering the two different unknown quantities i and i separately. This *
however would lead to an erroneous result, since the derivation bound between the unknown quantities would be lost. At this point the kernel of the problem can be stated as follows: once a complex power command reference at a certain network port is given, and the voltage (and so its homoderivative) is known, the goal is to determine the current that makes it possible to obtain the given complex power at that port. It can be noted that this introduces the distributed compensation approach, in the sense that, thanks to the conservativeness of power quantities, the network port where the current reference is derived from the complex power reference, does not necessary coincide with the grid section where the power reference was elaborated starting from desired specifications.
107
For the reasons explained in the previous section, the problem is faced in the discrete time domain, assuming a sufficiently high sampling frequency. The discrete time domain is particularly suitable once the practical implementation of an APF is considered, since the inverter modulation inherently implies a discrete time approach. The first step is to express real and imaginary power reference as average values in the switching period Ts. To do this, some assumptions about the voltage waveform are needed. ) If the switching period Ts is conveniently small, both u and u can be approximated by a linear trend, i.e. between sampling instant tk and following sampling instant tk+1 it is assumed: ∆u u (t ) = u k + T t s ) u) (t ) = u) + ∆u t k Ts
t ∈ [0, Ts ]
(4.7)
the average value of complex power during switching period Ts can be expressed as: ~ 1 s& = Ts
Ts
1 ∫0 s&(t ) dt = Ts
Ts
~
~
∫ [ p(t ) + jq(t ) ]dt = p + jq .
(4.8)
0
The real and imaginary powers can be easily derived as: 1 )~ ) ) ) ~ p= 2∆u i − (u k + ∆u k )∆i − ∆u i k (4.9a) 2ωTs 1 ~ q~ = − 2∆u i + (u k + ∆u k )∆i − ∆u i k (4.9b) 2ωT s ~ where i represents the average value of the current in the period Ts, while ∆i is the
[
[
]
]
difference between current value at time tk+1 and current value at tk. From eqs (4.9a) and (4.9b) it can be observed that both real and imaginary power are expressed as a function of average current and current increment in the switching period. The possibility to shape the current waveform needs an indepth examination. ~ If a generic triangular current waveform is considered (fig. 4.5a) i and ∆i are two completely independent parameters, thus it would be possible to solve the above equations to find them. Considering however a two level inverter topology it is quite easy to understand that, ~ to impose both i and ∆i , a variable switching time Ts would be needed, since the
108
i
i d3
d2
d1
d4
d2
d1
∆ik ik
Ts
d2
∆ik
t2 ik+1
t1
d1
ik+2 t Ts
Fig. 4.5a Triangular current waveform in case of fixed switching period and variable current slopes
ik
t2a ik+1
t1a Tsa
∆ik+1
ik+2
t
Tsb
Fig. 4.5b Triangular current waveform in case of variable switching period and fixed current slopes
slope of the current is constant (fig 4.5b) and depends on the voltages and the inverter output inductance. Also with this approach, however, the information which is carried out by the power reference is conveyed by both the average current value and by the current increment, or, in other words, by the current ripple. Going on this way, however, would be extremely misleading, since, once the inverter current is filtered, as always happens for APF applications, the ripple is cancelled and part of the information about complex power is completely lost. This suggests that the whole information about complex power has to be committed to current average value, in the sense that the actual behaviour of current i during the switching period is irrelevant. In fact each term added to its average value contributes only to the highfrequency harmonic content of the current that is then filtered out. In this sense, when shaping the current, only a single parameter (i.e. its average value) is available and can be controlled, and for this reason a simplified approach assuming for the inverter current a linear trend during the switching period does not cause any loss of generality. So we have: T 1 s ∆i i= i (t )dt = i k + ∫ Ts 0 2 and consequently: ( ∆i i = . ω Ts
(4.10a)
(4.10b)
A simplifying hypothesis can be also done about the voltages, since it can be ) ( assumed that u , u , u are nearly constant during the switching period, i.e.:
109
u=
) ) 1 d u 1 ∆u ≅ ω dt ω Ts
and
( 1 du 1 ∆u u= ≅ ω dt ω Ts
Based on these assumptions, the real and imaginary power can be rewritten as: ) 1 ) ) ∆u ~ p= ∆u i 0 − u − (4.11a) ∆i = α r i 0 + β r ∆i 2 ω Ts 2 q~ =
1 2 ω Ts
∆u − ∆u i 0 − u − 2 ∆i = α i i 0 + β i ∆i
(4.11b)
In (4.11a) and (4.11b) ∆u represents the variation of the supply voltage as compared ) to the previous switching period and similarly for integral voltage variation ∆u . In the above equations coefficients αr, βr, αi, βi, depend only on the supply voltage behaviour while i0 is the sampled value at the beginning of the switching period. As a consequence the system made up of (4.11a) and (4.11b) is composed by two equations having a single unknown quantity, i.e. ∆i . For this reason an exact solution cannot be found and a minimization problem has to be formulated.
4.3.1 Minimization problem in terms of power It is now necessary to choose a suitable cost function for the minimization process and it appears reasonable to minimize, in each switching period, the distance between ~ ~ average complex power s& and the corresponding reference s& * , i.e. ϕ = a ( ~p − ~p * ) 2 + b (q~ − q~ * ) 2 (4.12) where a and b are suitable weighting factors, whose choice will be better analyzed later. From the minimization equation
dϕ = 0 the unknown ∆i (or, with equivalent d∆i
notation, i∆ ) can be derived: aβ r ( ~ p * − α r i 0 ) + bβ i (q~ * − α i i 0 ) * i∆ = aβ r2 + bβ i2
(4.13)
with the coefficients:
) ∆u ) ∆u u− u− ) ∆u ∆u 2 . 2 , β =− , αi = − , βr = − αr = i 2ωTs 2ωTs 2ωTs 2ωTs
110
(4.14)
) ) ∆u ) It can be noted that, being u the voltage average value during Ts, then u − = uk , 2 thus β r and β i can be simplified as follows: ) uk u and β i = − k (4.15) βr = − 2ωTs 2ωTs
4.4 Basic principle of a possible implementation Equation (4.13) represents a possible control algorithm whose implementation potentially performs Local Control Unit duty, in the sense that it makes it possible to calculate a current reference starting from a real and imaginary power reference. In particular, since the target is the control of an Active Power Filter, the algorithm which is performed at each time step consists in the calculation of the current increment that is needed to obtain the average current value that minimize the distance between actual complex power and reference complex power as previously explained. The key point of the proposed approach, however, lies in the possibility to use it for a delocalized compensation, so the main goal of the tests will be to verify it. This means that real and imaginary power references have to be created starting from current and voltage at a specific network port and then the control algorithm has to apply such power references somewhere else, where the voltage is different. The goal is to determine a current reference at this second port that produces a complex power absorption as far as possible to the desired one. To prove the validity of the proposed approach, the simple test case represented in fig. 4.6 is considered. In this first step the goal is not to perform a specific compensation to fulfil given specification at the PCC, but simply to calculate the load complex power at the section where uL is present, and to reproduce it, possibly with opposite sign, at a different network port where the Active Power Filter is located and the voltage uF can be detected. In this case a simplifying assumption is that distribution network π absorbs a negligible amount of complex power so that complex power references for the APF nearly coincides with the complex power reference at the PCC. This latter, in turn, is determined to compensate the complex power absorbed by the load. To perform what described above, at least two main blocks are required.
111
is
iL
+
+
us
Distribution network
uL
π


PCC

uF
+
iF
Active Power Filter (APF)
Fig. 4.6 Application example of distributed compensation strategy
The first one is the block that calculates real and imaginary power at the load section, from iL and uL values, transforming them into suitable complex power references. The second block is the one which specifically implements the proposed control algorithm, acquiring real and imaginary power references and calculating the current iF which is needed at the APF section to track as better as possible the given power references. Since this second block represents the original part of this approach few further details are provided about its implementation. The goal of the algorithm is to reproduce the behaviour of an active power filter, so the block works as a triggered system, where the clock frequency coincides with the inverter switching frequency. In this case input data, as real and imaginary power references and voltage, with corresponding homointegral and homoderivative, are acquired with a sampling rate that is about twenty times the inverter switching frequency. Also the controlled variable, i.e. APF current reference (which is immediately derived from current increment ∆i) is released at the same rate, however the control algorithm described in the previous paragraph is performed once per switching period.
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4.4.1 Comparison with the optimization technique in the frequency domain According to what introduced in the previous paragraphs it’s quite clear that the here presented control approach is essentially dynamic, since the algorithm is applied at every switching cycle, so thousands of times per lineperiod. This leads to understand how the minimization process, which is proposed starting from (4.12), is intrinsically an “instantaneous” minimization process. Thus it is natural to wonder if the results that this kind of minimization offers differ from an approach which, instead, is elaborated in the frequency domain and thus is intrinsically “stationary”. For this reason a comparative test was performed, while selecting as a function to be minimized: ϕ = ( ~ p− ~ p * ) 2 + (q~ − q~ * ) 2 (4.16) It is supposed that a sinusoidal voltage: u L = 2 sen(ωt )
(4.17)
is directly applied to a distorting load (fig. 4.7) composed by a thyristor rectifier and an RL impedance, i.e. 2 iL = [ sen(ωt − 25°) + ∑ cos(kωt − 90°)] . 2 k =1, 5, 7 ,11,13
(4.18)
The corresponding complex power absorption is calculated according to eqs (4.11a) and (4.11b). Then it is supposed that desired complex power absorption at the same section is the one associated to a purely active current, thus, by difference, real and imaginary power references are calculated. Then the second voltage, which relates to the network port chosen for the APF 2
2
[p.u.] 1.5
[p.u.]
uL
1.5
iL
1
1 0.5
0.5 0
0
0.5
0.5
1
1
1.5
1.5
2
uF uL
2 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
[s]
[s]
Fig. 4.7 Distorted load current (iL) and sinusoidal applied voltage (uL)
Fig. 4.8 Voltage at the load section (uL) and at the APF section (uF)
113
2
1
[p.u.]
[p.u.]
1.5
0.5
q
0.5
p*
1
p
q*
0
0 0.5 0.5 1 1.5
1
2 2.5
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
1.5
0.04
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
[s]
[s]
Fig. 4.9 Real power reference (p*) and actual real power obtained at the APF section (p)
Fig. 4.10 Imaginary power reference (q*) and actual real power obtained at the APF section (q)
current reconstruction is considered (fig. 4.8): u F = 2 sen(ωt + 30°)
(4.19)
The first test was carried out through the approach proposed in the previous paragraphs, so that the APF current which minimizes ϕ in presence of the modified voltage uF was calculated at each time step. In figs 4.9 and 4.10 real (p*) and imaginary (q*) power references are reported with the red trace, together with the corresponding actual quantities (p and q, blue traces), which can be obtained at the APF section through the current determined from the minimization process. The same test was performed in the frequency domain, in the sense that a 2
1
[p.u.]
[p.u.]
q
1.5
p
1
qstat
0.5
pstat
0.5
0 0 0.5
0.5 1 1.5
1
2 2.5
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
1.5
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
[s]
Fig. 4.11 Real power obtained by our “instantaneous” approach (p) and by the Matlab minimization (pstat)
114
Fig. 4.12 Imaginary power obtained by our “instantaneous” approach (q) and by the Matlab minimization (qstat)
2
[p.u.] 1.5
0.3
iL
1
0.25
iF
0.5
0.2
0
0.15 0.5
0.1
1 1.5
0.05
2 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0
0.04
0
2
4
6
8
10
12
14
16
18
20
[s]
Fig. 4.13 Load current (iL) and APF current (iF) with sinusoidal voltage source
Fig. 4.14 Value of the Fourier coefficients as a function of the harmonic order for the APF current, with sinusoidal voltage source
MATLAB© optimization tool was used to minimize the same cost function, where, in this case, the “distance” is computed in the vector space L2. The results in terms of power references tracking are reported in figs 4.11 and 4.12, where the results obtained by MATLAB© optimization is depicted in red and is directly compared to the solution offered by our “instantaneous” approach (in blue). It can be immediately seen that there is only a minimal difference between the two strategies, which can be quantified through the relative error indexes. The relative errors both for real (pErr) and imaginary (qErr) power are calculated as the RMS value of the difference between the two approaches divided by the RMS value of the corresponding reference quantity. Such errors result: pErr = 0.015 p.u.
qErr = 0.065 p.u.
Both of them are of few percent points, thus showing the substantial equivalence between the two approaches. The main reason of such errors is identified in the sampling delays that are introduced by the dynamic control. In fig 4.13 the APF current iF, which is calculated for the APF together with the load current iL is depicted. In fig 4.14 also the Fourier coefficients of the resulting APF currents are reported. It is quite interesting to note that only harmonics of order 1, 5, 7, 11, 13 are present, i.e. the same current harmonics that were present in the initial load current iL. This is probably due to the presence of a sinusoidal voltage source, however it is interesting to do an equivalent test considering now a distorted voltage source containing a 5%
115
2
[p.u.]
iL
1.5
0.3
1
0.25
iF
0.5
0.2
0
0.15 0.5
0.1
1
0.05
1.5 2 0.04
0.045
0.05
0.055
0.06
0.065
0.07
0.075
0.08
0
0
2
4
6
8
10
12
14
16
18
20
[s]
Fig. 4.15 Load current (iL) and APF current (iF) with distorted voltage source
Fig. 4.16 Value of the Fourier coefficients as a function of the harmonic order for the APF current, with distorted voltage source
of 5th harmonic and a 3% of 7th harmonic. The test results are comparable to the previous case in term of power tracking thus they have not been reported. What is noticeable, instead, is the trend of the APF current (fig. 4.15) and the corresponding spectral analysis (fig. 4.16). Again only 1, 5, 7, 11, 13 harmonics are present. This is important to be underlined because it means that the compensation of a distorted current does not involve harmonics of different orders. This is not obvious because the introduction of the power approach could potentially involve other harmonics contribution due to modulation effects.
4.4.2 Stability analysis The control technique which was presented in the previous paragraphs was elaborated under the assumption of periodic operation, however the resulting control technique looks to be applicable also under transient operation. This however requires to take into account the effects of transient condition so that instability is avoided also in this case. The discrepancy between transient and steady state condition is conveyed by the initial current term i0, in the sense that under transient condition i0 does not generally coincide with the corresponding stationary value i0*. Thus a current error occurs, whose expression is: ε 0 = i 0 − i 0* .
(4.20)
To guarantee the algorithm stability, the way this error propagates from a switching period to the following one must be analyzed. In particular, if the error has a
116
decreasing amplitude while passing from the generic tk to the tk+1, instability is avoided. The reference value of the current increment i∆ can be obtained from (4.13), by substituting i0* to i0, i.e.: aβ r ( ~ p * − α r i 0* ) + bβ i (q~ * − α i i 0* ) * i∆ = . aβ r2 + bβ i2 By converse, during transient operation eq. (4.13) becomes: aβ r [ ~ p * − α r ( i 0* + ε 0 )] + bβ i [q~ * − α i (i 0* + ε 0 )] * i∆ = i∆ + ε ∆ = . aβ r2 + bβ i2
(4.21)
(4.22)
Thus, the error term results: aα β + bα i β i ε ∆ = − r 2r ε0 aβ r + bβ i2 which represents the increment of the error term along the switching period. The stability condition implies that ε0 is lower than ε0+ε∆ , i.e.: ε0 +ε∆ aα r β r + bα i β i 0ϕ u 2 is considered (which ensures that the voltage is not zero)
weight a can be expressed as a function of b as follows: ) ) ) u 2 + u 2 − bu 2 u2 a= = 1 + ( 1 − b ) . u2 u2 )
(6.11a)
In a similar way under the condition u 2 > u 2 , (which now ensures that the voltage homointegral is not zero) it can be written:
148
) ) u 2 + u 2 − au 2 u2 b= = 1 + (1 − a ) ) 2 . ) u2 u
(6.11b)
In both the cases the minimum value of the cost function is given by (6.9). If the attention is now focused on the term Ψ = a b , which is the one that can be modified to optimize the cost function value, the following analysis can be carried ) out: if the condition u 2 > u 2 is considered, it can be written: ) ) ) u2 u2 u2 Ψ = a b = 1 + (1 − b) 2 b = b (1 + 2 ) − b 2 2 . (6.12) u u u If the reference weight (in this case b) is to be maintained in the interval [0,1], the two limit conditions can be taken into account, i.e.  b = 0, corresponding to ψ=0 and
 b = 1, corresponding to ψ=1. As b varies in the interval [0,1] Ψ describes a parabola which degenerates in the ) ) straight line Ψ= b when u = 0 . Moreover, when u 2 = u 2 , it is Ψ= 2bb2, having its maximum for b=1. In figure 6.1, cost function Ψ is sketched as a function of the leading weight, (b in this case). All the possible parabolas are comprised between Ψ= b and Ψ= 2bb2 which are reported in the figure. In fig. 6.2 the weight a is depicted as a function of the leading weight b, for the limit ) ) values of u = u and u = 0 , while for intermediate conditions it would be ) ) u2 u4 a = 1+ 2 − b 2 (6.13) u u 1,4
2,5
1,2 1 Ψ
0,8 0,6 0,4
2
) u =u
) u =u
1,5 a
) u =0
0,2
1
) u =0
0,5 0
0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 1,2 1,3 b
Fig. 6.1 Trend of the function Ψ as a function of ) the weight b in the case of u 2 > u 2
0
0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 b
Fig. 6.2 Trend of the weight a as a function of ) the weight b in the case of u 2 > u 2
149
1,4
1,4
1,2
1,2
a
1
1
0,8
0,8
0,6
b
0,6
b
0,4
a
0,4
0,2
0,2
0
0
0
0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00
0
0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00
) u2 /u2
) u2 /u2
Fig. 6.3 Trend of the weights as a function of ) ) u 2 / u 2 in the case u 2 > u 2
Fig. 6.4 Trend of the weights as a function of ) ) u 2 / u 2 in the case of u 2 > u 2
)
It is worth to note that when the condition u = u is reached, the role of a and b is exchanged. In this case it is a=2b. Then, if the goal is to maintain both a and b inside ) the interval [0,1] when they are leading variables, (i.e. 0 ≤ b ≤ 1 if u 2 > u 2 and ) 0 ≤ a ≤ 1 if u 2 > u 2 ), thus it is Ψ ≤ 1 , a and b must be chosen equal to 1 when ) u =u. Under this position it is: ) ) ) u2 u2 u4 b = 2 ⇒ a = 1 + 2 − 4 if u u u 2 2 u u u4 a = ) 2 ⇒ b = 1 + ) 2 − ) 4 if u u u
) u2 > u 2
(6.14a)
) u2 > u2
(6.14b)
and the corresponding trends as a function in the cases
) u2 u2 and ) are reported in figs u2 u2
6.3 and 6.4, respectively. The above presented choice avoids discontinuity and let the cost function ϕ to be limited:
ϕ min
) (u p * − uq * ) 2 ≤ . ) u2 +u2
(6.15)
As a conclusion it is worth to note that, differently from what seen in Chapter IV, here the choice of the weights is not influenced by stability constraints, but is just done to minimize the tracking error and, in any case, to ensure that it does not diverge. In this sense the here proposed procedure is intrinsically different from all the approaches based on the discrete solution of differential equations.
150
6.3.3 Threephase fourwire units When dealing with threephase fourwire systems a special caution is required. In fact, at a generic grid section, the knowledge of power terms is no longer sufficient to determine the phase currents given the phase voltages. This can be easily verified by choosing the neutral wire as voltage reference (i.e. u0 = 0, with subscript 0 denoting the neutral wire): with this assumption, the current in the fourth wire, i.e. the homopolar current, does not affect the power balance. Moreover, the equations describing the threephase fourwire system are: 3 ∑ u ' no in = p ' u ' 20 u ' 30 i1 p ' u '10 n =31 ) ) ) ) u '10 u ' 20 u '30 i 2 = q ' ∑ u ' no in = q ' ⇒ n =13 1 3 1 3 1 3 i3 − i0 1 44424443 { 123 i = − i i n 0 A b ∑ n =1
(6.16)
where uno is the voltage of phase n referred to the neutral wire and apex ‘ denotes normalized quantities, as described in 6.3.1. The difference with respect to threephase threewire systems (eq. 6.4) lies in the contribution of current i0, which causes an homopolar component to appear in currents i1, i2, i3. The above equation system can be rewritten as: u ' 20 u '30 i1 p' 0 u '10 ) ) u) ' u ' u '30 i2 = q' + 0 . 10 20 1 3 1 3 1 3 i3 0 − i0
(6.17)
Recall now that, while power terms are conservative in electric networks, this is not true for current terms. In a distributed compensation perspective this means that, in a threephase fourwire system, the power information can effectively be transmitted among the different grid sections, however the LCU cannot convert univocally a power command into a current reference. The homopolar component, which is not influencing the power transfer, must be set apart, according to different criteria. To calculate the phase current reference, system equation (6.17) is solved by matrix inversion, giving: i1* i1*s i10* * * * i 2 = i2 s + i20 * i3* i3*s i30
(6.18)
151
1 2 3 0
i2s i3s
i1s
Powersensitive currents
i10 i20 i30
i0
Homopolar related currents
Fig. 6.5 Ideal representation of a threephase fourwire compensator
where: i1*s u '1 u ' 2 ) ) * i s = i2*s = u '1 u ' 2 i3*s 1 1
u '3 ) u '3 1
−1
p '* * q' 0
(6.19)
0 0 − i0
(6.20)
and:
i10* u '1 u ' 2 * ) ) * i 0 = i 20 = u '1 u ' 2 * i30 1 1
u '3 ) u '3 1
−1
The phase current references include therefore two contributions: the first one, is*, is associated to the complex power command; the second one, i0*, relates to the homopolar current component. A shown in fig. 6.5 this corresponds to ideally split the compensator action (represented by current generators) into two different parts, one associated to the “powersensitive” current components (whose determination is the same as in threephase threewire systems) and the other one related to the homopolar component. This corresponds to have two different control loops, the first one working in the power frame, as previously described, and the second one which is a traditional current loop. The two control loops are decoupled according to equations (6.1718). References i10*, i20*, i30* can be set by detecting the unwanted homopolar components at the grid section and then, through a feedback loop based on a current regulator,
152
acting at the compensator section by modifying current i0. Obviously this kind of regulation is affected by the network transfer function and the current regulator must be adjusted for the specific application.
6.4 Basic principle of implementation While Chapter IV focused especially on LCU implementations, the goal of this chapter is to show the feasibility of the proposed control technique in a wider perspective. For this reason also computer simulations are aimed to analyze the problem of distributed compensation paying attention, not only to LCU, but also to the implementation of the Central Control Unit. To properly analyze the implementation of a distributed control approach it is worth to refer to fig. 6.6. It can be seen that this scheme resembles the generic one depicted in fig. 3.6 except for the absence of the Power Sharing Unit, since, at first, here the goal is to verify the feasibility of a delocalized compensation, irrespective of the specific criteria for the compensation burden sharing in the case of several APFs or SVCs. However it can be noted that this scheme also differs from all the previous simulations for many crucial aspects. At first it implements a closedloop distributed APF control, so that, at the PCC section, the error between reference current and actual current is measured and it feeds an Error Amplifier, which produces the control signal to be converted into a complex power reference for transmission to a different network port.
Central Control Unit PPCC
i*PCC ε i CRG + uPCC iPCC
EA
iεcomp
x p x q
scomp
LCU
icomp
ucomp
Fig. 6.6 Generic implementation of a distributed compensation system based on instantaneous complex power
153
This duty is carried out by the Central Control Unit, which provides an instantaneous complex power signal as an output. Under the assumption of working with a distribution network which absorbs a negligible amount of complex power, this complex power signal can be directly transmitted to another network port (i.e. a port having a different voltage), where the Local Control Unit reconverts it into a suitable current reference, according to one of the algorithms analyzed above. It is worth to note that the most immediate application of such idea is to control APFs, as specifically presented in the following part of this chapter. However the same implementation scheme (CCU+LCU, i.e. conversion current/power/current) can be also usefully applied to SVCs control as better shown in Chapter VII. Here, to effectively simulate the network behaviour it is necessary to inject the APF current obtained in this way into the real network, thus emulating the insertion of an ideal APF, to properly evaluate its effect on the current at the PCC. It is also worth to note that an higher realism than previous simulation is obtained, since the Local Control Unit uses the actual voltage at the APF section to convert the complex power reference into the corresponding current one. This means that the simulation takes into account the perturbative effect that the APF insertion has on the value of the APF voltage. It is also important to underline the meaning of the hypothesis of having a network absorbing a negligible amount of complex power. When reasoning in voltage terms, it can be noted that voltage at the PCC and at the APF section can be quite different, since line impedance may cause voltage drops, resulting in phase shift and increased voltage distortion and transformers may provoke level shift and phase rotation. When considering the power frame, however, it can be seen that power references are only marginally affected by line impedances and transformers, whose contribution to the total power absorption is due only to parasitic elements. Differently from what was done in Chapter IV, here the implementation of the Local Control Unit is not built up as a triggered subsystem where the current increment is derived from the discrete resolution of two differential equations, but is based on a continuous system where the control algorithm derived from eqs (6.5), (6.8) or (6.186.20) directly offers the average value of the state variable in the ideally short switching period.
154
u1 ii1
Ls
i1APF
u2 ii2
Ls
i2APF
ii3
Ls
i3APF
u3
i1svc i2svc i3svc
iL1
iL2
iL3
RL
RL
RL
LL
LL
LL
SVC
iC1
iC2
CF
APF
iC3
CF
CF
Fig. 6.7 Application example: three phase network
6.4.1 Threephase threewire systems: application example The compensation algorithm presented in 6.3.1 was tested through a computer simulation. The chosen test case was the one depicted in fig. 6.7, which represents a threephase network including a wye/delta transformer feeding an ohmicinductive load RLLL (with ZL = 1 p.u. and cosϕ =0.9). The line inductance is represented by Ls, where corresponding impedance is Zs = 0.05 p.u. at the fundamental frequency. The three phase system includes also the following compensation units: a SVC composed by a resonant filter tuned on the 5th harmonic and a TCR, which is connected at the input section, and an APF, connected in parallel to the load, acting as a controlled current source with highfrequency filter capacitors CF. The goal of the compensation is to make the network appearing as purely resistive at the input terminal by proper control of SVC and APF. In particular, the SVC is designed and driven to compensate for the reactive power absorbed by the load, while injecting limited current harmonics. The APF performs the rest of the job, by compensating the remaining reactive and void current components. To gain a better insight on the selected test case, some further details about the implementation are provided. As regards the Central Control Unit, the Current Reference Generator calculates the active power which is associated to the load and, through the evaluation of the equivalent conductance, it determines the desired active current trend at the input
155
instantaneous real powers
instantaneous imaginary powers
1.5
2
1
[p.u.]
[p.u.]
[p.u.]
ui1
1
1
pAPF
0
pRL
0
0.5
0.5
0.5
0
1
1
0.5
1.5
0
0.02
0.04
qRL
0.5
qAPF
ii1
0.5
pπ
1.5
0.06
0.08
0.1
0.12
0.14
0.16
1
[s]
Fig. 6.8 Three phase threewire application: phase 1 input current (plotted with a 0.8 scale factor) and voltage
0.15
0.155
qSVC
1.5
pSVC 0.145
qπ
0.16
2
[s]
0.145
0.15
0.155
0.16
[s]
Fig. 6.9 Threephase application: instantaneous real and imaginary power in steady state conditions
section. The error signal, which comes from the comparison between the reference input current and the actual one, feeds the regulator (EA) which, in this case, is a simple PI controller. The Central Control Unit also calculates the instantaneous complex power according to eq. (6.3), thus lumping the information about the different phases into a single quantity. Then the Local Control Unit performs its job according to what explained in paragraph 6.3.1 and the so obtained current is injected into the grid. It is worth to note that the complex power which is associated to the APF current passing through Ls, is not taken into account in the power balance, however its contribution (and the corresponding compensation error) is expected to be small. Regarding the simulation, three balanced and distorted (THD = 3.6% due to 5th and 7th harmonic) voltages (u1, u2, u3) feed the network. Initially all the compensators are off and only the load current appears at the input port. At t = 60 ms the SVC is switched on and compensates for the reactive current of the load at the expense of the increased input current distortion. Then at t = 100 ms, the APF is switched on, too, and after a rampup, it brings the system to a full
compensation. Fig. 6.8 shows voltage and current of phase 1 at the input section, where, the current is plotted with a 0.8 scale factor. Fig. 6.9 represents corresponding real and imaginary power absorption of load, SVC, APF and total network. The active power is due to the RLLL load, while SVC and APF do not contribute to such power term. On the other hand, they contribute to the
156
absorption of fluctuating components of real power, which are associated to harmonics and void current. It can be seen that the load reactive current is compensated by the SVC, while the APF absorbs an imaginary power fluctuation term, which relates to harmonic and void current compensation. Provided test case is an example of cooperative and distributed compensation, since it can be noted that the system behaves, with very good approximation, as a resistive load and that the SVC and APF perform cooperatively although connected to different network ports.
6.4.2 Singlephase systems: application example The test case which is chosen to prove the validity of the single phase approach to distributed compensation is not very different from the previous one. In this case the network is the one of fig. 6.10. As in the previous case an RLLL load is present, having ZL = 1 p.u. and cosϕ =0.9. Instead of the SVC, a capacitor bank with Zc= 2.3 p.u. is introduced, whose function is to compensate for the load reactive power. To better show the capability of the proposed compensation algorithm it is assumed that the line inductance is very high, i.e. Ls= 0.2 p.u., causing a significant voltage difference between the input and the APF terminals (fig 6.11). In this case the supply voltage has a much more severe distortion, due to the 5th and 7th harmonic, having THD = 11.2%. Ls
ii iC
Electrical grid
iRL
iAPF
RL
ui
C +
LL
PCC
Fig. 6.10 Application example: single phase network
157
2
2
[p.u.]
[p.u.]
1.5
1.5
ui
1
ii
0.5
0.5 0
0
0.5
0.5
1
1
1.5
1.5
2
ui
1
uAPF
2 0
0.02
0.04
0.06
0.08
0.1
0.12
0
0.02
0.04
[s]
Fig. 6.11 Singlephase application: input voltage and APF voltage before and after compensation
0.06
0.08
0.1
0.12
[s]
Fig. 6.12 Singlephase application: input voltage and current before and after compensation
The simulative implementation is quite similar to the one described for the threephase threewire case, except for the fact that, in this case, Central Control Unit uses a PID regulator as an Error Amplifier. At the beginning of the simulation the APF is switched off and the input waveforms are due only to the RLLL load and the compensation capacitor. Then, at t = 70 ms the APF is switched on and performs its compensation job. In fig. 6.12 input voltage and current, before and after compensation, are reported. The analysis of the proposed waveforms shows that, in spite of the inherent compensation error introduced by the minimization algorithm of (6.8), the input current tracks the input voltage with good accuracy, so that the goal of having a purely active input current is almost reached, even if a heavy line inductance effect is present.
6.5 Conclusions The goal of this chapter is to introduce a simplified compensation strategy which is based on instantaneous complex power and which can be easily applied to both singlephase and threephase systems (with or without neutral wire). Suitable control power terms, which are conservative, simple to calculate and suitable for a closed loop control are defined. Then the different algorithms which need to be implemented by the Local Control Unit, according to the kind of APF structure, are presented.
158
While threephase threewire approach determines an univocal correspondence between complex power and currents absorbed at a network port, thus providing an exact solution for the LCU, when a singlephase system is considered, a minimization strategy should be adopted. In this case real and imaginary part of the power reference set two independent conditions which cannot be met by the single current solution. For this reason the goal of the LCU becomes to determine the current reference which minimizes the squared difference between real and imaginary actual powers and corresponding references. The validity of both threephase threewire and singlephase approaches is shown throughout dynamical computer simulations, which implement a distributed and cooperative compensation scenario. Both the simulations are performed in closedloop conditions and show the possibility to drive both quasistationary compensators and APFs according to the proposed theory. The different tasks of Central Control Unit and Local Control Unit are underlined and implementation details are also provided. To gain wider generality, the proposed simulations take also account of the perturbation due to the APF insertion on the local voltages and the loop configuration is studied to guarantee network stability in spite of this effect. Both the provided test cases show the validity of the proposed approach. It was also explained as a similar control strategy can be applied to threephase fourwire systems, by a separate control of the current homopolar term whose control loop is decoupled from the power based one, which works exactly as in the threephase threewire configuration. Even if, in this section, the analysis was limited to distorted and balanced threephase systems, in the next chapter the study will be completed showing as the proposed technique can be extended with good results also to unbalanced threephase applications.
159
160
CHAPTER VII
POWER COMPENSATION STRATEGIES FOR THREEPHASE UNBALANCED AND ASYMMETRICAL SYSTEMS
7.1 Introduction The goal of this chapter is to focus on those systems which present a different behaviour among the phases, due to the presence of load unbalance or voltage asymmetry. It is ideally divided into two subsections: the first one shows how the already presented control techniques can be extended to these systems, while the second part of the chapter is dedicated to the definition (and application) of generalized symmetrical components to distorted systems. At first the simplified approach to distributed compensation based on instantaneous complex power, which was developed in the previous chapter, will be extended to deal also with the problem of unbalance. The first step will be to show that, if only dynamic compensators are used to manage also load unbalance, the control algorithm which was presented in the previous chapter is still effective. The following step will be to face the problem using both quasistationary and dynamic compensators. In particular, the main goal will be to extend the Steinmetz approach to asymmetrical and distorted networks, so that the compensation of load unbalance can be performed by variable reactances, while more expensive active power filters can be used only for a marginal compensation job. Such an integrated and hierarchical strategy is fundamental in the distributed compensation perspective. Thus, the feasibility of a synergic distributed approach involving both SVCs and APFs will be proved throughout simulations.
161
The last part of the chapter will be dedicated to the extension of the symmetrical components to periodic, non sinusoidal threephase systems. The main properties that such quantities are required to maintain also under distorted operation are focused and clarified by examples.
7.2 Compensation of load unbalance using APFs: application example The first step in the analysis of unbalanced systems will be to show that the simplified instantaneous control strategy, which was presented in the previous chapter, is intrinsically capable of taking into account also of this nonideality. Specifically, it can be shown that the LCU algorithms which were presented in 6.3.16.3.3 can be applied to unbalanced systems with no modification. To prove the effectiveness of the method a threephase threewire system is considered, which is supplied by a symmetrical distorted voltage (10% of 5th harmonic, 5% of 7th harmonic) and where an ohmicinductive load (equivalent impedance ZL = 1 p.u, cosϕ = 0.9) is connected between phase 1 and phase 2 on the secondary side of a wye/delta transformer (with Ls = 0.05 p.u. at the fundamental frequency). In this case the goal is to obtain a purely active current at the input section, despite load unbalance, by simply using an APF (provided of ripple filtering capacitors CF) connected to the load section (fig. 7.1). As it can be seen from fig. 7.2, in the first time interval, when the APF is switched u1 ii1
Ls
i1APF
u2 ii2
Ls
i2APF
u3 ii3
Ls
i3APF iL1
iL2
iC1
iC2
iC3
RL LL
CF
CF
CF
Fig. 7.1 Application example: threephase network, with a single phase load.
162
APF
2.5
2.5
[p.u.]
2
ii1 ii2 ii3
2
1.5
1.5
1
1
0.5
0.5
0
0
0.5
0.5
1
i1APF i3APF
1
1.5
1.5
2 2
i2APF
2.5 0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
2.5
0.16
[s]
Fig. 7.2 Currents of the three phases at the input section
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Fig. 7.3 Currents of the three phases of the Active Power Filter
off, the effect of the singlephase load causes, also at the primary side of the transformer, a deep phase currents unbalance. On the other hand, when at t = 60 ms the APF starts operating according to the control strategy described in par. 6.3.1, the current at the input section appears almost immediately balanced, due to the different APF phase currents contributions at the secondaryside of the transformer (fig. 7.3). The provided example shows that the simplified control strategy which was previously presented is useful to solve also unbalance problems, while no further control algorithm modifications are required.
7.3 Use of SVCs for unbalance compensation In the perspective of a cooperative and distributed compensation it is interesting to explore the possibilities deriving from the use of both dynamic and quasistationary compensation systems, in the compensation of load unbalance. As it was shown in the previous paragraph, the APFs insertion does not present any peculiar concern, however it is now meaningful to investigate the possible use of SVCs for load unbalance compensation. This is even more important since, under symmetrical and sinusoidal condition, the singlephase loads balancing can be easily performed by the use of variable reactances, according to the Steinmetz method [73]. Thus, the first crucial point is to understand if the Steinmetz network can be also used in distorted and even unbalanced systems, which is the goal of the analysis of the following section.
163
u1 i1
iL1
u 2 i2
iL2
u 3 i3
iL3
PCC
i1C
L12
i2C
C23
C12
LOAD
i3C
L23
L31
C31 Fig. 7.4 Steinmetz compensator
Obviously the theoretical elaboration, which is developed for nonideal conditions, must provide the traditional results once applied to sinusoidal systems. It is worth to note from the very beginning that, due to the quasistationary nature of SVCs, they are capable of compensating only for unbalance and reactive power terms which are averaged over a line period, while, as seen in the previous paragraph, APFs can be used also for the compensation of unwanted harmonic current components.
7.3.1 Extension of the Steinmetz method to nonsinusoidal systems The extension of the Steinmetz method to the compensation of load unbalance is initially based on the topology presented in fig. 7.4, where variable reactances, arranged in a delta connection, are connected in parallel to the unbalanced load. To be more coherent with the common practical implementation of this kind of reactive compensators, in the following analysis it will be supposed that the Steinmetz compensator comprises three variable inductances and three equal capacitors (i.e. C12= C23= C31). At first the reactances contribution to the power balance is considered. It is easy to see that the capacitors bank does not contribute to the active power absorption on the different phases, since, as can be seen for phase 1:
164
( ( ( P1C = u1 , i1C = u1 , i12C + i13C = B C u1 , u12 + u13 = B C u1 ,3u1 = 0
(7.1)
In this case the apex “C” indicates the capacitor and an expression similar to (7.1) can be written also for phase 2 and 3. Consequently it’s apparent that the capacitor bank does not participate in the load balancing. It just contributes to the reactive power absorption, since: ) ) ) ( ( ) ( Q1C = u1 , i1C = u1 , i12C + i13C = B C u1 , u12 + u13 = B C u1 ,3u1 = −3 B C u1
2
(7.2)
and similarly for the other phases. Thus, using the vector notation for the power absorption of the capacitor bank, it is: u1 2 0 2 C C P = 0 Q = −3 B C u 2 . u 2 0 3
(7.3)
As a whole, the capacitive bank absorbs the reactive power: 2 Q C = −3B C u .
(7.4)
As regards the variable inductances, the active power absorbed by phase 1 is: ) ) P1L = u1 , i1L = u1 , i12L + i13L = u1 , B12L u12 + B31L u13 (7.5) ) ) = B12L u1 , u 2 − B31L u 3 , u1 and similar expressions hold for the other phases. Reactive power is: ) ) Q1L = u1 , i1L = u1 , i12L + i13L = ) ) ) 2 + B31L u1 , u13 = B12L u1 −
(
) ) ) ) ) u1 , B12L u12 + B31L u13 = B12L u1 , u12 ) ) ) 2 ) ) u1 , u 2 + B31L u1 − u 3 , u1
)
(
)
Thus, according to the threephase vector notation, once introduced: B12L x = B23L , it can be written: B31L P1L Q1L L L P = P2L = A x , Q = Q2L = B x , P3L Q3L where matrices A and B are: ) ) u1 , u 2 0 − u 3 , u1 ) ) A = − u1 , u 2 u 2 , u3 0 ) ) 0 − u2 , u3 u 3 , u1
(7.6)
(7.7)
(7.8a)
165
u)1 2 ) 2 B = u2 0
0 ) u2 ) u3
2 2
) 2 u1 0 − ) 2 u 3
) ) u1 , u 2 ) ) u1 , u 2 0
0 ) ) u 2 , u3 ) ) u 2 , u3
) ) u 3 , u1 0 . ) ) u 3 , u1
(7.8b)
It is worth to note that the active power sum across the phases is null, due to the reactive nature of the compensator.
7.3.1.1 Symmetrical voltage source At first the case of distorted but symmetrical voltage source is considered, which determines a simplification in the matrices A and B, due to the fact that: U = u = u1 = u 2 = u3 . (7.9) So, once assumed: ) ) ) a = u1 , u 2 = u 2 , u3 = u3 , u1 )2 ) ) )2 ) ) ) b = u − u1 , u 2 = u − u 2 , u 3 = u it can be written: 0 − 1 1 A = a − 1 1 0 0 − 1 1
(7.10a) 2
) ) − u 3 , u1
1 0 1 B = b 1 1 0 . 0 1 1
(7.10b)
(7.11)
If the active and reactive power are lumped into a single vector y, it follows: 0 − 1 1 − 1 1 P 0 0 −1 1 a y= =Cx C= (7.12) . 1 0 1 Q 1 1 0 b 1 0 1 It can be immediately noted that the rank of matrix C is 3, thus the dimension of the system can be reduced, by eliminating, for instance, the 3rd , 5th and 6th row. So the reduced system in (7.13a) can be obtained: P1L a 1 0 − 1 L y ' = P2 a y' = C ' x C ' = − 1 1 0 . Q1L b 1 0 1 And it is also:
166
(7.13a)
P3L a y ' ' = Q2L b Q3L b
0 − 1 1 C ' ' = 1 1 0 0 1 1
y '' = C ' ' x
(7.13b)
The matrix C’ has full rank and can be inverted, thus, once assigned the power references y’* corresponding reactances can be obtained from: P1* a y '* = P2* a x = C ' −1 y '* y" = C " x = y"* Q1* b
(7.14)
7.3.1.2 Asymmetrical voltage source If the voltage source is not only distorted, but also asymmetrical, the procedure is more complex. In fact matrix C cannot be reduced and the problem is overconstrained, since there are five equations (regarding two active powers and three reactive powers), but only three unknown quantities (the inductive susceptances). Thus an exact solution, as in the case of symmetrical source voltages, cannot be found and an approximate one must be selected. Obviously the choice about the cost function to be minimized determines different possible approaches. In the following the two most meaningful ones are presented. Minimization of the error on active and reactive powers
The basic idea, in this case, is to find the values of susceptances that minimize the distance between the reference and the actual values of active and reactive powers. Thus the minimum of the following cost function must be found:
(
ϕ = P − P*
) ⋅ (P − P ) + (Q − Q ) ⋅ (Q − Q ) = T
* T
*
= P ⋅P −2P ⋅P + P T
T
*
The minimization implies
*T
*
⋅ P + Q ⋅Q − 2Q ⋅Q + Q *
T
T
*
*T
⋅Q
*
(7.15)
∂ϕ = 0 , resulting in: ∂x
T ∂ϕ ∂ P ∂ϕ ∂ Q ∂ϕ T * * = ⋅ + ⋅ = 2 A P − P + 2 BT Q − Q = ∂x ∂x d P ∂ x ∂Q T
(
)
(
)
T * * * T T T = 2 A A + B B ⋅ x − 2 A P + B Q = 2 D ⋅ x −ν = 0 14 42 44 3 1 4 4 2 4 4 3 D ν*
(
)
(
)
(7.16)
167
Due to the fact that matrix D is symmetrical, it can be diagonalized and inverted, so the solution of the minimization problem is given by: * x = D −1 ⋅ν
(7.17)
Minimization of the error on reactive powers, bounding active power values
In some practical cases it can be required a perfect balance of active powers (which otherwise can produce parasitic torques in electric generators), while an error on reactive power reference tracking can be accepted. In this case the problem can be faced by minimizing the reactive power error, i.e.:
(
ϕ = Q − Q*
) ⋅ (Q − Q ) = Q T
*
T
⋅Q − 2Q ⋅Q + Q T
*
*T
⋅Q , *
with the constraint: ψ = P '− P '* = A '⋅ x − P '* = 0 .
(7.18) (7.19)
In (7.19), P’ indicates the (N1)sized vector (where N is the number of phases) of independent active powers. A’ is the corresponding matrix, which can be obtained by A through the elimination of one row. The solution is obtained by solving: ∂ ϕ ∂ψ T λ=0 + ∂x ∂x ψ = 0
(7.20)
Where λ represents Lagrange multipliers. The solving system becomes: ∂ ϕ ∂ψ T λ = 2 D ⋅ x − ν * + A 'T ⋅λ = 0 + ∂x ∂x ψ = A '⋅ x − P '* = 0
(
)
(7.21)
Since D is symmetrical and can be generally inverted, from the first equation of (7.21) it follows: * 1 x = D −1 ν − A 'T ⋅λ 2
(7.22)
whose substitution in the second equation gives: 1 * 1 * A ' x = A '⋅ D −1 ν − A 'T ⋅λ = 1 A '42 ⋅ D −1 4 ⋅ν3 − 1 A '⋅42 D −1 ⋅4A3 'T ⋅ λ 2 2 F * χ
1 * = χ − F ⋅ λ = P '* 2
168
(7.23)
Due to the fact that F is symmetrical and generally invertible, it is: λ = 2 F −1 χ * − P '* .
(
)
(7.24)
Solving (7.24) and then (7.22) the values of the wanted susceptances can be found. It is useful to note that both the presented minimization approaches constitute essentially a mathematical calculation, thus they can offer both positive and negative values for the susceptances in x. Since however, they represent the values of the needed variable inductances, an allpositive solution is wanted. To have this, a common term can be added to each of the susceptances. This does not affect the active power balancing, but only the reactive power distribution. Once assumed: B12L + B0L x ' = B 23L + B0L = x + B0L ⋅ 1 , B31L + B0L it can be written: P ' = A ⋅ x ' = A ⋅ x + B0L { A ⋅1 = P 0 . L Q ' = B ⋅ x ' = B ⋅ x + B0 B ⋅ 1 = Q + Q0
(7.25)
(7.26)
In (7.26) Q0 represents the additional amount of reactive power, due to the insertion of the susceptances B0, i.e.: u)1 2 ) 2 Q 0 = 3 B 0L u 2 . u) 2 3
(7.27)
7.3.2 Generation of active and reactive power references In the previous paragraphs it has be shown how the three variable inductances can be chosen in order to minimize the distance from desired power references. Thus the initial problem can be brought back to determine the suitable power references that the SVC must follow to produce load balancing. If Pn0 indicates the active powers absorbed by the phase n of the unbalanced load and 3
P 0 = ∑ Pn0 is the load total active power, the effect of balancing must be to n =1
determine, for each phase, the absorption of an active power corresponding to the only active balanced current component:
169
b ian = Gb u n
(7.28)
where: 2
Gb = P 0 u .
(7.29)
Correspondingly the active power of each phase results: P = Gb u n b an
2
=P
0
un u
2 2
.
As a consequence, the compensation duty is to absorb, for each phase: Pn* = Panb − Pn0 . 0 n
Moreover, if Q
(7.30)
(7.31)
denotes the phase reactive power absorbed by the load and
3
Q 0 = ∑ Q n0 is the total load reactive power, the effect of balancing is to cause, for n =1
each phase, the absorption of the reactive current component, i.e.: ) irn = Be u n
(7.32)
where:
)2 Be = Q 0 u . Correspondingly, each phase reactive power results: ) 2 un ) 2 0 Qrn = Be u n = Q ) 2 . u Consequently, the compensator must absorb, for each phase: Qn* = Qrn − Qn0
(7.33)
(7.34)
(7.35)
Obviously, if a complete reactive compensation is required it is sufficient to put in eq. (7.35): Qrn = 0
7.3.3 The distributed compensation approach applied to load balancing Since the focus of the present research work is distributed and cooperative compensation, it is useful to point out how the load unbalance compensation strategy, which was presented in the previous paragraphs with reference to SVCs and APFs separately, can be transformed into a distributed and cooperative approach. At first it is worth to recall the differences in the use of quasistationary and dynamic compensators. In particular it must be noted that in the distributed approach the full exploitation of cheaper compensators is generally preferred; thus, where possible, the
170
SVCs are charged of the most of the compensation duty, according to their possibility. It is however important to underline that, as previously hinted, they can intervene only on those undesired current components that are associated to active and reactive power terms (i.e. to average powers), while, APFs can perform also the compensation of those current components that are related to zero average power terms. As a consequence, if the current decomposition presented in paragraph 2.11 is here recalled, it is quite natural to observe that SVCs can be used for the compensation of reactive and unbalanced active current components, while APFs can be used to account for the residual void components. This is the basis of a cooperative (and hierarchical) sharing of the compensation burden between quasistationary and dynamic units. At the input section, where the supply voltages ui are supposed to be known, all the useless current components can be conveniently calculated and separated, according to what explained in par. 2.11 and 3.10. Specifically, unbalanced active components can be calculated for each phase as follows:
i
u an
= i an − i
b an
= (Gn − G )u in = b
Pun u in
2
N
u in
n =1
ui
u in (where P = ( Pn − ∑ Pn u n
2 2
)u in ) (7.36)
and then arranged into the vector form: u b ia = ia − ia
(7.37)
while reactive terms (both balanced and unbalanced) are given, for each phase, by: ) u in , i ) Q ) ) irn = Bn u in = ) 2 u in = ) n2 u in (7.38) U in u in and can be lumped in the vector i r . Finally the void component results: iv = i − ia − ir
(7.39)
Regarding, instead, the second aspect of the desired compensation, (i.e. the distributed nature of the compensators along the grid), as it was explained in the previous chapters, the basis of distributed compensation is the transmission along the electric networks of instantaneous complex power references.
171
Since SVCs are driven to compensate for unbalance and reactive current terms, while APFs perform the remaining duty, i.e. void current compensation, two different instantaneous power references must be generated: ) ) b u b u u * s&SVC = − ui + j ui ⋅ i r + i = − ui + j u i ⋅ i r + i a + i r ) s& *APF = −(u i + j u i ) ⋅ i v
(
)(
) (
)(
)
(7.40)
Accordingly to what explained in Charter III and with reference to the distributed control scheme of fig. 3.6, s& * are the complex power references which enter the Power Sharing unit to be then distributed among the different suitable compensators. Then each Local Control Unit can calculate the correct current reference, taking account of the local voltages, according to the algorithms presented in the previous chapter. Once the compensator current references are known, the procedure described sections 7.3.1 and 7.3.2 can be applied to determine the correct susceptance values of the Steinmetz network.
7.3.4 Application example: cooperative compensation of load unbalance The distributed and cooperative control strategy described in the previous paragraph has been tested throughout computer simulation. The threephase network depicted in fig. 7.5 has been considered. It includes an ohmicinductive unbalanced load connected to the secondary of a wye/delta transformer. A capacitor bank is connected to the input section, while a TCR and an APF are connected to the secondary side of a wye/wye transformer, which causes a 15° phase shift of the secondary voltages with respect to the primary ones. The system is supplied at the PCC by a distorted and asymmetric voltage given by:
u1 = 1cos(ωt ) + 0.03 cos(5ωt ) + 0.02 cos(7ωt ) 2π π 2π π 2π π u 2 = 1cos(ωt − − ) + 0.03 cos(5ωt − 5 − 5 ) + 0.02 cos(7ωt − 7 −7 ) 3 20 3 20 3 20 u 3 = −u1 − u 2 (7.41) The goal of the compensation is to finally obtain, at the input section, only the balanced active current.
172
Ls1
i1L
Ls1
i2L
Ls1
i3L
LL
Wye/ Wye
RL
Singlephase load
u1
i1
Ls2
i1APF
u2
i2
Ls2
i2APF
u3
i3
Ls2
i3APF
PCC
i1c
i2c
i3c
i1TCR i2TCR i3TCR
Wye/ Delta
L12F
CF
CF
APF
L23F
TCR
CF
L13F Fig. 7.5 Application example
At the beginning no compensators are switched on, thus the input current is only due the contribution of the RL load. At t = 20 ms the capacitor bank connected to the input section is inserted and this alters the phase shift between currents and corresponding voltages, while the current waveforms severely worsen, due to the harmonic injection of the capacitors supplied by distorted voltages. Then at t = 60 ms the TCRs are switched on, which are controlled as described in par. 7.3.1.2 due to the supply voltage asymmetry. Moreover, the control strategy which privileges the active power balancing, while minimizing the error on reactive current is applied. Practically, the current of the input section is measured and unbalanced active currents and reactive currents are detected. Then such current signals are processed by a PD regulator and the current error is converted into a corresponding instantaneous complex power signal through the multiplication for
173
i1
[pu] 2 u1
[pu] 2
0 2
2 0
[pu]
ic1
0
50
2
100
150
200
250
300
i2
0
[pu]
0 2
0
100
150
200
250
300
0
[pu] 2
250
300
50
100
150
200
250
300
150
200
250
300
ic3
0 2
i3 t0 t1
200
4
0
0
150
2 50
[pu] 2 u3 2
100
ic2
0
u2
50
2
4 50
t2
100
150
[ms]
200
250
t3
300
t4
Fig. 7.6 Currents of the three phases at the PCC section
0
t0 t1
50
t2
100
[ms]
t3
t4
Fig. 7.7 Phase currents (ic1, ic2, ic3) provided by the TCR + APF compensators
input voltage and voltage homointegral. Such complex power reference signal is then sent to the TCR which is controlled according to the strategy of 7.3.1. It is worth to note that due to the fact the quasistationary compensation is based on a minimization process the complete elimination of reactive and unbalanced active current components cannot be expected. It can be noted from figs 7.6 and 7.7 that even if the TCR requires several periods to settle, its effect is evident from the beginning in reducing the amplitude and phase shifts of the phase currents towards the desired values. Finally at t = 200 ms also the APF which is connected to the same section of the TCR is switched on and it is used for the compensation of void currents. In fact void components in the input current are detected and processed by a simple proportional regulator, before being transformed into a complex power reference. Such reference is transmitted to the local control unit which transforms it into a local current reference for APF control. Thus, in the final time interval (t= 200300 ms) all the compensators are on an the line current waveforms approach a purely resistive behaviour. It is worth to note that the accuracy of the results depends on the TCR functioning. This at first because in the case of asymmetrical voltages a minimization process is involved, but also because the APF provides only void current compensation while it does not take into account the residual current deriving from the non complete compensation of reactive and active unbalanced components.
174
If the APF would receive also the information about the effective compensation job of the SVC, current tracking would certainly be improved, but this requires a coordinate control strategy with a superior degree of complexity.
7.4 Generalization of symmetrical components for nonsinusoidal systems The second part of this chapter deals with another important aspect of threephase systems analysis, i.e. the possibility to extend the symmetrical components theory which was usefully introduced for sinusoidal conditions, to periodic distorted operations [74]. The use of symmetrical components is a wellestablished theory for the analysis of sinusoidal three phase power systems with unbalanced sinusoidal currents or voltages and, in the case of balanced systems, the waveforms decomposition into positive, negative and zero sequence also allows to analyze the threephase system as a singlephase system. The goal of the following paragraphs will be to shown how the concept of symmetrical components can be generalized to distorted operations and to underline the peculiarities with respect to traditional decomposition. The first step will be the definition of symmetrical components in nonsinusoidal systems, then it will be shown how these components can be mathematically derived and their main properties will be explained. Analogies and differences with respect to the traditional sinusoidal case will be underlined and simulation examples will be used to better clarify the whole analysis.
7.4.1 Definition of symmetrical components under distorted operation If a distorted threephase system (with or without neutral wire) is considered, it is theoretically possible to decompose the phase signals into a Fourier series of periodic functions and then to apply the traditional symmetrical components decomposition, to each single harmonic separately. Once that the single kth harmonic is considered over the threephases, the traditional positive sequence, negative sequence and zero sequence can be detected. It is however natural to wonder whether a simple decomposition can be obtained without recurring to the frequency domain. Practically the goal of the analysis is to propose a symmetrical components decomposition which is completely developed in the time domain and which leads to the definition of the socalled “generalized symmetrical components”.
175
The basic idea is to maintain the symmetry properties of symmetrical components under sinusoidal condition, while keeping also the fundamental requirement of orthogonality. Thus:  the generalized zero sequence component is such that the current (or voltage) in phase 2 and 3 are the same as in phase 1;  the generalized positive sequence component is a component such that the current or voltage in phase 2 is the same as in phase 1, but lagging over T/3, and the current or voltage in phase 3 is the same as in 1 but lagging over 2T/3;  the generalized negative sequence component is a component such that the current or voltage in phase 2 is the same as in phase 1, but leading over T/3, and the current or voltage in phase 3 is the same as in 1 but leading over 2T/3. The mathematical derivation of the generalized components can be derived as follows:  The component with zero sequence symmetry is: f 10 (t ) f 0 (t ) 0 f = f 20 (t ) = f 0 (t ) , f 30 (t ) f 0 (t ) where the generalized zero sequence component is: 1 f 0 (t ) = [ f1 (t ) + f 2 (t ) + f 3 (t )] . 3  The component with positive sequence symmetry is: f 1 p (t ) f p (t ) p p p f = f 2 (t ) = f (t − T / 3) f 3p (t ) f p (t − 2T / 3)
(7.42)
(7.43)
(7.44)
with the generalized positive sequence component: 1 f p (t ) = [ f 1 (t ) − f 0 (t ) + f 2 (t + T / 3) − f 0 (t + T / 3) + f 3 (t + 2T / 3) − f 0 (t + 2T / 3)] 3 (7.45)  The component with negative sequence symmetry is: f n1 (t ) f n (t ) 2 n n f = f n (t ) = f (t + T / 3) f n3 (t ) f n (t + 2T / 3) with the generalized negative sequence component:
176
(7.46)
1 f n (t ) = [ f1 (t ) − f 0 (t ) + f 2 (t − T / 3) − f 0 (t − T / 3) + f 3 (t − 2T / 3) − f 0 (t − 2T / 3)] 3 (7.47) For the simplification of the following analysis it is also worth to define heteropolar components, which corresponds to the phase quantities, once the homopolar term has been subtracted: f 1h (t ) f 1 (t ) − f 0 (t ) h f = f 2h (t ) = f 2 (t ) − f 0 (t ) . f 3h (t ) f 3 (t ) − f 0 (t )
(7.48)
Using the above notation, positive and negative sequence components can be expressed as follows: 1 f p (t ) = [ f 1h (t ) + f 2h (t + T / 3) + f 3h (t + 2T / 3)] (7.49a) 3 1 f n (t ) = [ f1h (t ) + f 2h (t − T / 3) + f 3h (t − 2T / 3)] , (7.49b) 3 and it can be noted that they are formally the same as in the sinusoidal case. To better understand the meaning of the so defined generalized symmetrical components, it is worth to note that the positive sequence component is obtained by shifting the second waveform ahead by 1/3 of period T and the third by 2/3 of the period T, then computing the mean value of the three superimposed variables. Similar considerations hold for the negative sequence components. It is a known property that the mean value of a number of quantities has the minimum RMS distance from this set of quantities. The expressions found for the generalized symmetrical components are highly similar to classical expressions valid for sinusoidal condition. It is however important to underline a fundamental difference, since, under distorted operation, the zerosequence component must be subtracted from the total quantity, before calculating the positive sequence component and the negative sequence component. It is quite natural to wonder if the zerosequence, the positive sequence and the negativesequence components of the phase quantities add up to the phase quantity. The conclusion is that this is not the case, differently from the sinusoidal situation. Therefore a residual component must be defined, which takes account of the remaining term:
177
[ [ [
] ] ]
1 h f 1 (t ) + f1h (t + T / 3) + f 1h (t + 2T / 3) r 3 f 1 (t ) r 1 h h h f ( t ) = f ( t ) + f ( t + T / 3 ) + f ( t + 2 T / 3 ) 2 2 2 2 3 f 3r (t ) 1 f h (t ) + f h (t + T / 3) + f h (t + 2T / 3) 3 3 3 3
(7.50)
From the above expression it can be seen that residual components are periodic with period T/3. The condition for residual component vanishing is: f 1h (t ) + f1h (t + T / 3) + f 1h (t + 2T / 3) = f 2h (t ) + f 2h (t + T / 3) + f 2h (t + 2T / 3) =
f 3h (t ) + f 3h (t + T / 3) + f 3h (t + 2T / 3) = f 1h (t ) + f 2h (t ) + f 3h (t ) (7.51) in each instant of the period, which is always true under sinusoidal conditions. Thus, it can be concluded that the periodic threephase sinusoidal quantities can be exactly decomposed into the zerosequence components, the generalized positive sequence components, the generalized negative sequence components and the residual components, i.e.: f p (t ) + f n (t ) + f 0 (t ) + f 1r (t ) f 1 (t ) f (t ) = f p (t − T / 3) + f n (t + T / 3) + f 0 (t ) + f r (t ) 2 2 f 3 (t ) f p (t − 2T / 3) + f n (t + 2T / 3) + f 0 (t ) + f 3r (t )
(7.52)
The decomposition into generalized symmetrical components represents an extension to distorted operations of the classical theory which is developed for sinusoidal conditions. The main differences, that represent peculiar features of nonsinusoidal systems, are in the presence of the residual component and in the necessity of eliminating the zerosequence term before deriving positive and negative ones.
7.4.2 The relation to the Fourier series expansion As previously hinted an other approach to the analysis of non sinusoidal threephase systems consists in the decomposition of the periodic functions into Fourier series and to apply the usual decomposition into symmetrical components to each harmonic separately. Now it is interesting to underline how the two approaches are related. To do this, it is at first worth to recall some basics on Fourier expansion and sequence components.
178
A generic periodical phase variable fn(t) can be expanded in Fourier series in the form:
(
∞
)
(
∞
f n (t ) = ∑ Fnk (t ) = ℜ F&nk e jωk t = ∑ 2 Fnk cos ω k t + ϕ nk k =1 k n
k =1
)
(7.53)
where f (t ) is the kth harmonic term, which is sinusoidal with angular frequency
ω k = k ω , RMS value Fnk and phase ϕ nk . In the above relation, complex variable:
(
k F&nk = 2 Fnk e jϕ n = 2 Fnk cos ϕ nk + j sin ϕ nk
)
(7.54)
is the harmonic phasor associated to sinusoidal variable f nk (t ) . Considering now the threephase variables f 1k , f 2k , f 3k , which are the generic harmonic terms of variables f at angular frequency ω k , we can apply the usual definition of zero, positive and negative sequence components which holds for sinusoidal variables. The corresponding harmonic sequence phasors are: k & k F&1k + F&2k + F&3k = 2 Fok e jϕo Fo = 3 k k & k F&1 + α& F&2k + α& 2 F&3k (7.55) = 2 F+k e jϕ + F+ = 3 k F&1k + α& 2 F&2k + α& F&3k k = 2 F−k e jϕ − F&− = 3
with: 2π 4π −j j 1 3 3 3 & α = e = e =− + j 2 2 4π 2π j −j 1 3 2 α& = e 3 = e 3 = − − j 2 2 6π 3 j j0 α& = e 3 = e = 1
(7.56)
while the corresponding timedomain harmonic sequence components are: f 1k 0 (t ) = ℜ F& k 0 e jωk t = 2 F k 0 cos(ω k t + ϕ ok ) k 0 jω t k0 k0 k (7.57) f 2 (t ) = ℜ F& e k = 2 F cos(ω k t + ϕ o ) k0 j ω t k 0 k 0 k k = 2 F cos(ω k t + ϕ o ) f 3 (t ) = ℜ F& e
( ( (
) ) )
179
( (t ) = ℜ(α&
)
f1 k + (t ) = ℜ F& k + e jωk t = 2 F f 2k +
2
F&
(
f 3 k + (t ) = ℜ α& F&
( (t ) = ℜ(α& F&
k+
k+
k+
cos(ω k t + ϕ +k )
2π ). 3 4π = 2 F k + cos(ω k t + ϕ +k − ) 3
)
e jωk t = 2 F k + cos(ω k t + ϕ +k −
e jωk t
)
(7.57b)
)
f1k − (t ) = ℜ F& k − e jωk t = 2 F k − cos(ω k t + ϕ −k ) f 2k −
(
k−
)
2π ) 3 4π = 2 F k − cos(ω k t + ϕ −k + ) 3
e jωk t = 2 F k − cos(ω k t + ϕ −k +
f 3k − (t ) = ℜ α& 2 F& k − e jωk t
)
(7.57c)
Considering that reverse transformation from harmonic sequence phasors into harmonic phasors is: k F& k 0 + F& k + + F& k − F&1 = 3 k 0 & F + α& 2 F& k + + α F& k − k & F2 = 3 k0 & & k F + α& F k + + α 2 F& k − F&3 = 3
(7.58)
it is easily verified that, by adding the harmonic sequence components at a given angular frequency ω k , we reconstruct the corresponding harmonic terms, i.e.: f nk (t ) = f nok (t ) + f nk+ (t ) + f nk− (t )
∀n = 1 ÷ 3, ∀k = 1 ÷ ∞ .
(7.59)
Let’s now analyze the relationships between symmetrical components and harmonic sequence components.  Homopolar component, f 0 . According to equation (7.43), and taking into account definitions (7.57) and property (7.59), it can be written: 3 ∞ 3 ∞ 3 ∞ f f k 0 + f nk + + f nk − f k0 f 0 = ∑ n = ∑∑ n = ∑∑ n = ∑ f 3 n =1 3 k =1 n =1 k =1 n =1 3 k =1
k0
(7.60)
0
Thus, the homopolar component f of vector f is entirely due to zerosequence harmonic components.  Positive symmetry component, f p. From equations (7.59) and (7.60) we also find:
180
∞
(
f nh = f n − f 0 = ∑ f nk − f k =1
)= ∑(f ∞
k0
k =1
k+ n
+ f nk −
)
(7.61)
Thus, according to equations (7.44), we can write: f 1h (t ) + f 2h (t + T ) + f 3h (t + 2T ) 3 3 = f1 = 3 k+ k+ T ) + f 3k + (t + 2T ) ∞ f 1k − (t ) + f 2k − (t + T ) + f 3k − (t + 2T ) ∞ f 1 (t ) + f 2 (t + 3 3 + 3 3 = =∑ ∑ 3 3 k =1 k =1 p
∞
=∑ k =1
∞
+∑ k =1
∞
=∑ k =1
∞
+∑ k =1
2π 2π 4π 4π k 2 F k + cos ω k t + ϕ +k + cos ω k t + ϕ +k − +k +k + cos ω k t + ϕ + − 3 3 3 3 + 3
(
)
2π 2π 4π 4π k 2 F k − cos ω k t + ϕ −k + cos ω k t + ϕ −k + +k +k + cos ω k t + ϕ − + 3 3 3 3 = 3
(
2F k + 3
)
3
∑ cos ω t + ϕ k
n =1
2F 3
k−
3
k +
∑ cos ω t + ϕ n =1
k
+ (n − 1)(k − 1) k −
2π 3
+ (n − 1)(k + 1)
(7.62)
2π 3
This equation shows that: positivesequence harmonic components contribute to positive symmetry component f p if and only if: k − 1 = 3l ∀l = 1 ÷ ∞ negativesequence harmonic components contribute to positive symmetry component f p if and only if: k + 1 = 3l ∀l = 1 ÷ ∞ With similar considerations it can be shown that: positivesequence harmonic components contribute to negative symmetry component f n if and only if: k + 1 = 3l ∀l = 1 ÷ ∞ negativesequence harmonic components contribute component f n if and only if: k − 1 = 3l ∀l = 1 ÷ ∞
to
negative
symmetry
By difference, it can be seen that remaining sequence harmonic components add together to form the residual component f r of vector f. In particular: positive and negativesequence harmonic components contribute to residual component f r if and only if: k = 3l ∀l = 1 ÷ ∞
181
7.4.3 Properties of the generalized symmetrical components The generalized symmetrical components defined in the previous paragraph have some noticeable properties. Variables f p have direct symmetry and variables f n have inverse symmetry. Moreover homopolar components are instantaneously orthogonal to all the other symmetrical sequence components: 0 h f ⋅ f =0
(7.63a)
f ⋅ f =0
(7.63b)
f ⋅ f =0
(7.63c)
f ⋅ f =0
(7.63d)
0
p
0
n
0
r
D.7.63a
Instantaneous orthogonality between homopolar and heteropolar component
By definition heteropolar components sum zero at any time, i.e.: 3
∑f n =1
= ( f 1 − f 0 ) + ( f 2 − f 0 ) + ( f 3 − f 0 ) = f 1 + f 2 + f 3 − 3 f 0 = 0 ∀t ∈ [0, T ] ,
h n
consequently
f ⋅f 0
h
3
= ∑ f n0 f nh = f n =1
D.7. 63b
3
0
∑f n =1
h n
=0
Instantaneous orthogonality between homopolar and positive sequence component
Being derived from heteropolar components, positive sequence components sum zero at any time, i.e.: 3
∑f n =1
= f 1 p + f 2p + f 3p =
p n
[
]
1 h f 1 (t ) + f 2h (t ) + f 3h (t ) + 3
[
]
1 h f 1 (t + T / 3) + f 2h (t + T / 3) + f 3h (t + T / 3) + , 3 1 + f 1h (t + 2T / 3) + f 2h (t + 2T / 3) + f 3h (t + 2T / 3) = 0 ∀t ∈ [0, T ] 3
+
[
]
consequently
f ⋅f 0
p
3
= ∑ f n0 f np = f n =1
D.7. 63c
3
0
∑f n =1
p n
=0
Instantaneous orthogonality between homopolar and negative sequence component
Being derived from heteropolar components, negative sequence components sum zero at any time, i.e.:
182
3
∑f n =1
n n
= f1n + f 2n + f 3n =
[
]
1 h f 1 (t ) + f 2h (t ) + f 3h (t ) + 3
[
]
1 h f 1 (t + T / 3) + f 2h (t + T / 3) + f 3h (t + T / 3) + , 3 1 + f 1h (t + 2T / 3) + f 2h (t + 2T / 3) + f 3h (t + 2T / 3) = 0 ∀t ∈ [0, T ] 3
+
[
]
consequently
f ⋅f 0
n
3
= ∑ f n0 f nn = f
3
0
n =1
D.7. 63d
∑f n =1
n n
=0
Instantaneous orthogonality between homopolar and residual sequence component
Being derived from heteropolar components, residual sequence components sum zero at any time, i.e.: 3
∑f n =1
r n
= f 1r + f 2r + f 3r =
[
]
1 h f 1 (t ) + f 2h (t ) + f 3h (t ) + 3
[
]
1 h f 1 (t + T / 3) + f 2h (t + T / 3) + f 3h (t + T / 3) + , 3 1 + f 1h (t + 2T / 3) + f 2h (t + 2T / 3) + f 3h (t + 2T / 3) = 0 ∀t ∈ [0, T ] 3
+
[
]
consequently 3
f ⋅ f = ∑ f n0 f nr = f 0
r
n =1
3
0
∑f n =1
r n
=0
It must be also noted that positive, negative and residual components are mutually orthogonal (at average level), i.e.: p n f ,f =0
(7.64a)
p
r
=0
(7.64b)
n
r
=0
(7.64c)
f ,f f ,f
D.7. 64a
Orthogonality between positive and negative sequence components:
As it was shown in the previous paragraph, only positive sequence harmonic components f nk+ of order
k = 3l + 1 and the negative sequence harmonic components f nk− of order k = 3l − 1 contribute to the positive sequence components f p , thus: ∞
(
)
f np = ∑ x n(3l +1) + + x n( 3l −1) − . l =1
(7.65)
183
Instead the negative sequence component, f n , is due to positive sequence harmonic component
f nk+ of order k = 3l − 1 and the negative sequence harmonic components f nk− of order k = 3l − 1 , thus: ∞
(
)
f nn = ∑ f n( 3l −1) + + f n(3l +1) − . l =1
(7.66)
The internal product results: p
f ,g
n
3
1 n =1 T
=∑
∫
T
0
3
∞
1 n =1 l =1 T
f np g nn dt = ∑∑
∫ (f T
0
( 3 l −1) − n
)
⋅ g n(3l −1) + + f n(3l +1) + ⋅ g n(3l +1) − dt (7.67)
The above expression involves only harmonic components with the same harmonic order. The calculation of the generic internal product term offers: 3
1
∑T ∫ n =1
T
0
=
f nk − ⋅ g nk + dt =
2 T k− k+ 2π 2π F G cos ϑ −k ⋅ cosψ +k + cosϑ−k + ⋅ cosψ +k − + ∫ T 0 3 3
2 T 4π k 4π + ∫ F k − G k + cosϑ−k + ⋅ cosψ + − = 0 T 3 3 4π = F k − G k + cos ϕ k + cos ϕ k + 3
2π + cos ϕ k + 3
(7.68)
= 0
where it is assumed:
2π 2π k k f nk − = 2 F k − cos ω k t + α −k + ( n − 1) = 2 F− cosϑ − + ( n − 1) 3 3 2π 2π k k g nk + = 2 G k + cos ω k t + β +k − ( n − 1) = 2 G + cosψ + − ( n − 1) 3 3
(7.69)
ϕ k = ϑ−k − ψ +k = α −k − β +k Since each term of (7.67) is null, (7.64a) is immediately derived.
D.7. 64b7.64c
Orthogonality between residual and positive/negative sequence components:
Since residual component is due only to harmonic components of order k = 3l , while positive and negative sequence components include only components of order k = 3l ± 1 , no terms of the same harmonic order appear in the internal product, which is consequently null.
According to the orthogonality properties derived in the present paragraph, it can be written:
184
f
2
= f, f = f
o 2
+ f
p 2
+ f
n 2
+ f
r 2
(7.70)
which states the independence of the generalized symmetrical components, so that, exactly as in sinusoidal case, active powers and square of the RMS value can be computed as the sum of the corresponding values of the sequence components.
7.4.4 Generalized symmetrical components: application example The generalized symmetrical components decomposition that has been presented in the previous paragraph is here clarified through a simulation example. The threephase line to neutral voltages are considered, which are depicted in fig. 7.8: 1 1 1 1 u1 (t ) = 1 sen(ωt ) + sen(3ωt ) + sen(5ωt ) + sen(ωt ) + 3 5 10 10 2 1 2 1 2 1 1 u 2 (t ) = 1 sen(ωt − π ) + sen(3ωt + π ) + sen(5ωt − π ) + sen(ωt ) + 10 3 3 3 5 3 10 2 1 2 1 2 1 1 u 3 (t ) = 1 sen(ωt + π ) + sen(3ωt − π ) + sen(5ωt + π ) + sen(ωt ) + 3 3 3 5 3 10 10 (7.71) At first the homopolar component is detected (fig.7.9), which corresponds to the component that is the same on the three phases. By difference the heteropolar terms are derived (fig 7.10). They are the starting point for the derivation of both generalized positive sequence components, reported in fig. 7.11, and generalized negative sequence components, depicted in fig. 7.12.
1.5
0.5
[p.u.]
[p.u.]
0.4
u1
1
0.3 0.2
0.5
0.1 0
0 0.1
0.5 0.2 0.3
1
1.5
u3
u2 0
2
4
6
0.4 8
10
12
14
16
18
[ms]
Fig. 7.8 Line (to neutral) voltages of the considered example
20
0.5
0
2
4
6
8
10
[ms]
12
14
16
18
20
Fig. 7.9 Homopolar sequence component
185
1.5
1.5
[p.u.]
[p.u.] u1h
1
0.5
0.5
0
0
0.5
0.5
1
1
u2h 1.5
u1p
1
0
2
4
6
8
10
[ms]
12
14
16
18
20
Fig. 7.10 Generalized heteropolar sequence components
1.5
1.5
0
2
4
6
8
10
[ms]
12
14
16
18
20
Fig. 7.11 Generalized positive sequence components
1.5
[p.u.]
[p.u.]
1
1
0.5
u1n
u3r
u 1r
0.5
u3n
0
0
u2n 0.5
0.5
1
1
1.5
u3p
u2p
u3h
0
2
4
6
8
10
[ms]
12
14
16
18
20
Fig. 7.12 Generalized negative sequence components
1.5
u 2r
0
2
4
6
8
10
12
14
16
18
20
[ms]
Fig. 7.13 Generalized residual components
It can be observed that generalized positive sequence components take into account the fundamental component of each phase, while the generalized negative sequence components comprise the 5th harmonic components. To complete the decomposition, also the residual components must be considered, which are reported in fig. 7.13 and are composed by the third harmonic components. It is worth to note that both the generalized positive components and the generalized negative ones maintain the symmetry properties that are typical of symmetrical decomposition under sinusoidal conditions.
7.4.5 Comparison with Depenbrock decomposition The goal of this paragraph is to make a brief comparison between the proposed decomposition into symmetrical components and the one which was proposed by
186
Depenbrock in [75]. In his contribution Depenbrock claims that for zerosum three phase quantities the whole decomposition includes only the positive and negative sequence components, while according to the proposed theory the residual component is necessary, too. To underline the difference between the two approaches it is useful to apply Depenbrock decomposition to the test case which was analyzed in the previous paragraph. To apply Depenbrock decomposition the line to line voltages corresponding to the line to neutral voltages of eq. (7.71) must be at first derived. The homopolar component (fig. 7.15) derived by Depenbrock clearly corresponds to the one detected by the proposed decomposition, however it can be seen from fig. 2.5
0.5
[p.u.]
[p.u.]
u12
2
0.4
1.5
0.3
1
0.2
0.5
0.1
0
0
0.5
0.1
1
0.2
1.5
u23
0.3
2 2.5
0.4
u31 0
2
4
6
8
10
[ms]
12
14
16
18
20
Fig. 7.14 Line to line voltages corresponding to the line to neutral voltages of the proposed example 1.5
0.5
2
4
6
8
10
[ms]
12
14
16
18
20
Fig. 7.15 Homopolar sequence component
1.5
[p.u.]
[p.u.] u1d
1
1
0.5
0.5
0
0
0.5
0.5
1
1
u2d 1.5
0
0
2
u3d 4
6
8
10
[ms]
12
14
16
Fig. 7.16 Generalized direct sequence components according to Depenbrock decomposition
18
20
1.5
0
2
4
6
8
10
[ms]
12
14
16
18
20
Fig. 7.17 Generalized inverse sequence components according to Depenbrock decomposition
187
7.16 that the direct sequence components (which correspond to the positive one of the proposed theory) take into account the whole currents. As a consequence no indirect sequence component is detected according to Depenbrock approach. It is clear, however, that the three phase direct sequence components do not have positive sequence symmetry, since the voltages in phase 2 and 3 cannot be obtained from the voltages in phase 1 by lagging over respectively T/3 and 2T/3.
7.5 Conclusions In this chapter it was shown how the simplified control technique which is based on instantaneous complex power is suitable to be applied also to unbalanced and asymmetrical systems. Moreover, attention was paid to the possibility of using passive compensation to perform load balancing; specifically, the Steinmetz method which is valid under symmetrical and sinusoidal operation, was extended to comprise also the case of voltage asymmetry and waveforms distortion. It was shown that if the source symmetry is maintained, an exact solution for the Steinmetz network parameters can be determined, while in the case of voltage asymmetry, only an approximate solution can be found. The employment of quasi stationary compensators for load balancing is fundamental to generalize the concept of cooperative and distributed compensation. It was shown that, to perform the compensation job, different units can be conveniently used, according to a hierarchical approach. Specifically, quasistationary devices can be used for reactive and unbalance compensation, while APFs can be applied to the elimination of the remaining void currents. It must be noted that, if such division among the current components to be eliminated is adopted, a perfect compensation is generally not possible in the case of voltage asymmetry, since the APF does not perform also the residual reactive and unbalance compensation which is neglected due to the best fit approach included in the SVC control algorithm. The second part of this chapter faces the problem of system unbalance from a different and more theoretical standpoint, since the goal is to extend the definition of symmetrical components also to distorted operation. The basic idea is that symmetrical components must have the same symmetry properties that they have under sinusoidal conditions.
188
It was shown that the threephase considered voltages or currents cannot be generally decomposed simply into a generalized positive, a generalized negative and a homopolar component: a residual term is also required, which vanishes under sinusoidal operation. The residual term is the only one that has not symmetry on the three phases. It is also worth to recall that, differently from the sinusoidal case, the zerosequence term must be eliminated before calculating positive and negative sequence components. A comparison to the Depenbrock approach to symmetrical components decomposition under distorted operation was also proposed to underline the advantages of the proposed strategy.
189
190
CHAPTER VIII
CONCLUSIONS AND FUTURE WORK
The goal of this work was to present an innovative approach to reactive, harmonic and unbalance compensation in the distributed perspective. This required, at first, to establish a suitable theoretical background to deal with electric systems working under non sinusoidal conditions, but especially to focus on specific power terms, which are conservative in every real network and maintain a physical meaning also under distorted operation. Such power frame is the ideal context to build a brand new distributed and cooperative control strategy. The core of the work was devoted to the development of control techniques aimed at optimizing use and exploitation of compensators connected to different points of the electric grid. Control algorithms were developed taking into account the availability of compensation devices of different types (Static VAR Compensators, Active Power Filters, Power Electronics Interfaces), the various aspects of network optimization (limitation of reactive power, attenuation of harmonic pollution, elimination of load unbalance) and the applicability to singlephase and multiphase systems. A control structure of general validity was devised and its main operational blocks were analyzed in detail from a theoretical point of view and then designed and tested by extensive simulation. The new perspective depicted in the present work involves the capability to approach and solve the compensation problem at a system level, i.e. considering, in a comprehensive analysis, the presence and the effects of nonlinear and unbalanced loads, distributed power sources and compensators of various types and ratings. The goal is the optimization of system operation at various levels, including control of power flow, transmission efficiency, power quality and stability.
191
Future research lines on distributed and cooperative control implementation will need to face the problem of network identification, since optimal design and control of compensators requires characterization of the electric network at various ports and in a wide frequency range. Another future development relates with plug & play control algorithms that make it possible proper operation of individual compensators at local level without requiring any knowledge on the surrounding network. In addition, distributed compensators can make available their residual compensation capability to help global network operation: this requires the development of suitable communication and control platforms aimed at global network optimization. Finally it will also be fundamental, in the implementation perspective, the in depth study of communication techniques along electric networks, since the choice of communication channel and protocol can severely impact on the system response and, consequently, on the whole compensation performances.
192
APPENDIX A
BASICS OF VECTOR ALGEBRA
The goal of this section is to recall basic definitions and concepts related to the space of geometrical vectors RN and to the space L2[0,T] of those functions which are squaresommable. The importance of the geometric approach in the field of “Power Theories” lies in that identifying currents and voltages as elements of a suitable vector space, the introduction of concepts as RMS value, active power, apparent power etc. can be generalized. Moreover it allows the introduction of the idea of a measure for the quantities under investigation. For instance, it can be noted that the idea of compensation itself, in requiring the identification of active and non active current components needs to measure those currents associated to the active power transfer; such measure is obtained from the norm of a specific quantity, that, in specific vector spaces can be directly brought back to the concept of RMS value1.
Vector space RN Given N generic quantities x1, x2, ..., xN, they can be considered as the components of the generic Ndimensional geometric vector x: x1 x x= 2 (A1) ... xN If x and y are two generic elements of RN the scalar product of these two vectors results:
1
The norm used in this dissertation is always the Euclidean norm, which makes it possible to naturally extend concepts related to the RMS value of periodic quantities.
193
N
x ⋅ y = x y = ∑ xn y n T
(A2)
n =1
while the norm of vector x (i.e. the instantaneous norm) is:
x = x⋅x =
N
∑x n =1
2 n
The CauchySchwartz inequality states that: x⋅y ≤ x y
(A3)
(A4)
where the equal sign holds if, and only if, the two vectors are linearly dependent. If, instead, they are orthogonal it is also: 2
x+y = x + y 2
2
(A5)
As an example of application of the above presented concepts, it can be observed that, if u indicates the voltage that supplies a generic multiphase load and i is the corresponding current that passes through it, the instantaneous power absorbed by the multiphase load can be expressed as: N
p = u ⋅ i = ∑ u n in
(A6)
n =1
Vector space L2 Now the squaresommable functions are considered, i.e. those functions xn(t) defined in the time interval [0,T] so that: T
∫ x (t )dt 2
n
(A7)
0
exists and is finite. If x is a vector function, it belongs to L2 if each of its components x1(t), x2(t), ...xN(t) is squaresommable. If x and y indicate two elements of L2, their scalar product is indicated as: T 1 N x , y = ∫ x ⋅ y = ∫ ∑ x n (t ) y n (t )dt (A8) T 0 n =1 The norm (indicated with x ) which is associated to the scalar product is:
x =
x, x =
∫x⋅x .
The CauchySchwartz inequality is x⋅y ≤ x y
194
(A9)
(A10)
where the equal sign holds if, and only if, the two vectors are linearly dependent. In the case they are orthogonal it is also: x+y
2
= x + y 2
2
(A11)
In a Nphase system, where u is the load supplying voltage and i the corresponding current, the absorbed active power can be expressed as: 1 N P = u , i = ∫ ∑ u n in T n=1 With such definitions the RMS value of a scalar function results:
∫x
Xn =
(A12)
T
2 n
1 2 = x n (t )dt ∫ T 0
(A13)
while the RMS value of a vector function is: N
X =
x N
=
x, x N
=
∑ ∫ xn n =1
N
N
2
=
∑X n =1
N
2 n
(A14)
If the vector x(t) is an Ndimension set of symmetrical voltages or of balanced currents it is also: 1 N 2 2 2 2 (A15) X 2 = ∑ X n = X 1 = X 2 = ... = X N . N n =1 From this definition also the definitions of RMS value of a scalar quantity and of RMS collective value of a vector quantity are derived, even if with different notations.
195
196
APPENDIX B
BASICS OF MATHEMATICAL OPTIMIZATION
The goal of this section is to recall some fundamental principles for the determination of a maximum or minimum of a function both in the case it is unconstrained and in the case it is subject to constraints.
Optimum points of functions without constraints If an nfunction f ( x1 , x 2 ,..., x n ) 2 is considered and its optimum values are looked for, the first step is to find stationary points using the gradient method and then checking all stationary and boundary points to find optimum values. A necessary condition for a point P(x1,x2,...,xn) to be an (internal) maximum o minimum for the function f is that grad ( f ) = 0 , i.e. ∂f ( P) ∂f ( P) ∂f ( P) = 0; = 0; ....; = 0; (B1) ∂x1 ∂x 2 ∂x n The so obtained points are the stationary points for the function. A common method to determine whether or not a stationary point is also an extreme point for a function is to evaluate the hessian of the function at that point, i.e.: ∂2 f ∂2 f ∂2 f ... ∂x1∂x 2 ∂x1∂x n ∂ 2 x1 2 2 ∂ f ∂ f ... ... . H ( f ) = ∂x ∂x (B2) ∂ 2 x2 2 1 ... ... ... ... ∂2 f ∂2 f ∂2 f ... ∂x n ∂x1 ∂ 2 x n ∂x 2 ∂ 2 xn Such method is going to be exemplified in the case of n=2. In this case, under the hypothesis: 2
The function is considered to be defined in an open interval I of Rn and belonging to C2(I)
197
∂ 2 f ( P) > 0; ∂x 2
∂ 2 f ( P) ∂ 2 f ( P) ∂ 2 f ( P) 2 −( ) > 0; ∂x∂y ∂x 2 ∂y 2
the point P is a point of internal relative minimum. If, instead, it is: ∂ 2 f ( P) ∂ 2 f ( P) ∂ 2 f ( P) 2 ∂ 2 f ( P) < 0 ; −( ) > 0; ∂x 2 ∂x∂y ∂x 2 ∂y 2 the point P is a point of internal relative maximum. If, finally, it is: ∂ 2 f ( P) ∂ 2 f ( P) ∂ 2 f ( P) 2 −( ) < 0; ∂x∂y ∂x 2 ∂y 2
(B3)
(B4)
(B5)
certainly the point P is neither a maximum or a minimum. It can be noted that the second conditions of (B3 and B4) correspond to study the sign of the hessian of the function, and this is the base for the extension of the method to the generic ndimensional case.
Optimum points of constrained functions In the case a constrained optimization problem is considered, it is extremely useful the Lagrange to adopt multipliers method, which states a necessary condition for the presence of constrained maximum/minimum. Basically the method solves the constrained optimization problem by transforming it into a nonconstrained one. The method is here presented at first in the twodimensional case, and following extended to the multidimensional one through matrix notation. Let’s consider the function of two variables f ( x, y ) to be maximized [minimized] and the constraint expressed by g ( x, y ) = c The minimization process is done by introducing a new variable, λ, called a Lagrange multiplier, and studying the Lagrange function defined by: Λ ( x, y , λ ) = f ( x, y ) − λ ( g ( x, y ) − c ) .
(B6)
From the solution of the system: ∇ x , y ,λ Λ ( x, y , λ ) = 0
(B7)
the stationary points are found. In the general multidimensional case the function to be minimized can be expressed as: ϕ = xT Ax
(B8)
While the constraint can be written as: Ψ = Bx − c = 0
(B9)
198
The solving system becomes: ∂ϕ ∂ϕ +λ = 2 AT x + B T λ ∂x ∂x B x − c = 0
(B10)
From the first equation it can be obtained: −1 1 x = − ( AT B T λ ) (B11) 2 whose substitution in the second one gives the value of the Lagrange multiplier vector: −1
λ = 2( BAT B T ) −1 c
(B12)
which can be then substituted in the (B11) to obtain the stationary point vector. Consequently it is necessary to determine if those points corresponds to a maximum or minimum of the function f (or ϕ) under the constraint. This can be done, for example, using the method of the hessian which was hinted in the previous paragraph.
199
200
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