Neutral-point-clamped shunt active filter

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(1)Neutral-Point-Clamped Shunt Active Filter Corneluis Erasmus van Greunen. Thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Electronic Engineering at the University of Stellenbosch.. H. du T. Mouton April 2005.

(2) i. Declaration. I, the undersigned, hereby declare that to my knowledge, the work contained in this thesis is my own original work and has not previously in its entirety or in part been submitted at any university for a degree. Ek, die ondergetekende, verklaar hiermee dat volgens my kennis, die werk gedoen in hierdie tesis my eie oorspronklike werk is wat nog nie voorheen gedeeltelik of volledig by enige universiteit vir ’n graad aangebied is nie.. CE van Greunen. Date:.

(3) ii. Summary. This thesis presents the design and implementation of a Neutral-Point-Clamped (NPC) shunt Active Filter (AF) for non-linear loads. A NPC shunt AF is an attractive solution for AFs at the medium volage level, because it eliminates the need for cosly injection transformers. The balance of the capacitors of the direct current (dc) bus of the NPC inverter needs to maintained for stable and satisfactory operation though. Modulation techniques and their associated balancing techniques of the NPC inverter are analysed and discussed in the shunt AF setup. Specifically, the effect of Power Factor (PF), harmonics and unbalance of non-linear loads are considered. The practical design and implementation of the NPC shunt AF is presented. Practical and simulation results are presented which validates the presented theory..

(4) iii. Opsomming. Hierdie skripsie beskryf die ontwerp, analise en implementering van ’n neutrale punt gekonnekteerde parallel aktiewe kompenseerder vir nie-linieˆere laste. Die neutrale punt gekonnekteerde aktiewe filter is ’n aantreklike industrie oplossing omdat dit direk aan medium spanningsvlakke gekonnekteer kan word. Dit benodig nie die konvensionele duur koppel transformators nie. Die balansering tussen die kapasitore van die neutrale punt gekonnekteerde topologie moet egter gehandhaaf word vir stabiele en bevredigende werking. Modulasie tegnieke en hul gepaardgaande balansering tegnieke van die neutrale punt gekonnekteerde omsetter word ondersoek in die aktiewe filter opstelling. Die uitwerking van harmonieke, arbeidsfaktor en onbalans van die las speel ’n merkwaardige rol en word ondersoek. Praktiese ontwerp en implementasie word weergegee. Simulasie sowel as praktiese resultate word weergegee om die teorie te bevestig..

(5) iv. Acknowledgements. I would like to thank the following who made this thesis possible: God Almighty, who gives me power and understanding. Prof. H. Du T. Mouton for his guidance throughout the thesis. The members of the power electronics research group for their help with the implementation of the hardware and software. The foundation for research and development (FRD), ESKOM and the University of Stellenbosch for their financial assistance. My parents for their support throughout my life. Antoinette Le Roux, for her patience, understanding and support the last few years..

(6) v. Glossary : Abbreviations. (3φ) ac AF dc DFT DSP Fig IMC Hz LPF NP NPC NTV PCC PF PI PID PLL PWM PU QPLL RMS SPWM SVPWM THD VCO. Three Phase Alternating Current Active Filter direct current Discrete Fourier Transform Digital Signal Processor Figure Internal Model Control Hertz Low Pass Filter Neutral Point Neutral-Point-Clamped Nearest Three Vectors Point of Common Coupling Power Factor Proportional Integral Proportional Integral Derivative Phase Lock Loop Pulse Width Modulation per unit Quadrature Phase Lock Loop Root Mean Squared Sinusoidal Pulse Width Modulation Space Vector Pulse Width Modulation Total Harmonic Distortion Voltage Controlled Oscillator.

(7) vi. Glossary : Symbols. x∗ (x) xˆ x x˜ |x| xh α β ω φk ζ C C d dX G H m i(t) ia ic iL iS Ik Kx. The * superscript indicates a reference Boldface values normally represent vector values or matrices Hatted variable usually indicates an estimate value Over- lined variables indicate either normalised or average values A tilde variable indicates a nominal or a varying quantity Absolute value An h subscript indicates a quantity presenting harmonics Component in the space vector plane Component in the space vector plane Fundamental frequency in rad/s Phase of kth harmonic Damping ratio Capacitance Clarke or space vector transform Direct component of synchronous quantity duty cycle for vector X Transfer function Transfer function Modulation index Time dependant current Phase current Compensation current Load current Source or utility current Amplitude of kth harmonic (not RMS value) Gain or amount of reactive compensation (constant).

(8) vii. L p q R Sx Ts vδ va vc vd vNP v(t) vS VX Vd VREF ZL ZS. Inductance Instantaneous active power Instantaneous reactive power or quadrant component of synchronous quantity Resistance Switching function for phase x Sample period Error between balanced and unbalanced NP Phase voltage (line to neutral) Compensation voltage DC bus voltage of NPC converter NP voltage of the NPC converter Time dependant voltage Source or utility voltage X indicates a space vector in the space vector plane DC bus voltage of NPC converter Reference Voltage Load impedance Source or utility impedance.

(9) List of Figures 1.1 1.2 1.3 1.4. Electrical grid with PCC . . . . . . . . . . . . . . . . . . . . . . . . Shunt AF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Series AF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Typical voltage and current traces for a thyristor rectifier with DC link inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Phase equivalent circuit of current-source type of harmonic source . . 1.6 Typical voltage and current traces for a diode rectifier with a smoothing capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Phase equivalent circuit of voltage-source type of harmonic source . . 1.8 Shunt AF with Norton equivalent harmonic current source . . . . . . 1.9 Shunt AF with Thevenin equivalent voltage-type harmonic source . . 1.10 Series AF with Norton equivalent current-type harmonic source . . . 1.11 Series AF with Thevenin equivalent voltage-type harmonic source . . 2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 3.3 3.4. 2 5 5 6 6 7 7 8 9 11 12. Power flow for full reactive and harmonic compensation . . . . . . . Space Vector transformation . . . . . . . . . . . . . . . . . . . . . . Directions and phase differences for 5th and 7th harmonics in the stationary frame of reference . . . . . . . . . . . . . . . . . . . . . . . Compensation in the synchronous frame of reference . . . . . . . . . Three dimensional power space . . . . . . . . . . . . . . . . . . . . . Block diagram for compensation using p-q theory . . . . . . . . . . .. 17 18. Three level Neutral-Point-Clamped Topology . . . . . . . . . . . . . Outputs for NPC phase arm . . . . . . . . . . . . . . . . . . . . . . . Current through NP and DC bus connections for vector 14 . . . . . . Zero (black), small (blue), medium (green) and large (red) space vectors produced by 3-phase NPC converter. . . . . . . . . . . . . . . . .. 28 29 29. viii. 20 22 23 25. 30.

(10) LIST OF FIGURES 3.5 3.6 3.7 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20. 5.1 5.2 5.3 5.4 5.5. Symmetrical sector of producible vectors of 3-phase showing triangular regions . . . . . . . . . . . . . . Trigonometric duty cycle derivation in region II . . . Switching instants for Sinusoidal PWM . . . . . . .. ix NPC . . . . . . . . .. converter . . . . . . . . . . . . . . . . . .. Alternating Small Vector Pair Assignment . . . . . . . . . . . . . . . Duty cycles for small vector pairs s0 and s1 . (m = 0.85) . . . . . . . . Duty cycles for medium (dM ) and large (dL ) vectors (m = 0.85). . . . Medium vector NP current contribution (m = 0.85) . . . . . . . . . . Small vector NP current contribution at m = 0.85 using only positive small vectors (solid) and negative small vectors(dashed) . . . . . . . . Medium vector NP current contribution (m = √13 ). . . . . . . . . . . . Small vector pair S0 NP current contribution (m = √13 ). . . . . . . . . Medium vector NP ripple (solid) and controllable NP band (dashed). (m = 0.85, PF=1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimised NP ripple (solid), medium vector NP contribution (dashed) and controllable NP band (dotted) (m = 0.7, PF=0). . . . . . . . . . . Normalised RMS NP ripple for entire range of modulation index and PF. Minimum attainable NP current for even order harmonics n = 2, 4, 8 . Minimum attainable NP current for odd order harmonics n = 5, 7 . . . Regions of voltage polarity change . . . . . . . . . . . . . . . . . . . Neutral Point Clamped Converter . . . . . . . . . . . . . . . . . . . . 3φ balanced load . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3φ Balanced load in α and β parameters . . . . . . . . . . . . . . . . Two-port definition . . . . . . . . . . . . . . . . . . . . . . . . . . . αβ equivalent circuit of the NPC converter . . . . . . . . . . . . . . . Space vector shunt AF setup with coupling inductor . . . . . . . . . . Fundamental compensation voltage vectors for total reactive compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NPC shunt AF with controller . . . . . . . . . . . . . . . . Block diagram of the NPC shunt AF controller . . . . . . . Block diagram of the peak detector . . . . . . . . . . . . . . Block diagram of a simple Phase Lock Loop . . . . . . . . . Block diagram of a three phase Quadrature Phase Lock Loop. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 32 33 34 41 44 44 46 47 48 49 50 50 51 52 52 55 60 62 63 63 63 66 67 71 71 72 74 76.

(11) LIST OF FIGURES. x. 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17. Block diagram of the linearised three phase QPLL . . . . . . . . Discrete block diagram of the linearised three phase QPLL . . . Symmetrical SVPWM for the region of the first sector . . . . . Centre cycle sampling for accurate average compensation current Centre cycle sampling of symmetrical SVPWM . . . . . . . . . Current through DC bus capacitors . . . . . . . . . . . . . . . . Discrete pole locations for the DC Bus regulation controller . . . Block diagram of the IMC current control system . . . . . . . . Modelling of computational delay . . . . . . . . . . . . . . . . αβ representation of coupling inductor . . . . . . . . . . . . . . Block diagram of a trivial phase advance method . . . . . . . . Block diagram of a fast phase advancing method . . . . . . . .. . . . . . . . . . . . .. 77 78 81 82 82 83 84 86 87 90 92 96. 6.1 6.2. Photograph of NPC Shunt Active Filter experimental setup . . . . . . Sinusoidal voltage reference line to line output of the NPC inverter (m = 1, vd = 60V , fs = 6kHz) . . . . . . . . . . . . . . . . . . . . . Sinusoidal voltage reference line to line spectral output of the NPC inverter (m = 1, vd = 60V , fs = 6kHz) . . . . . . . . . . . . . . . . . Phase step response and modulation index m for the IMC and predictive current controllers (vd = 18V ) . . . . . . . . . . . . . . . . . . . Per phase disturbance rejection and corresponding modulation index (m) for: (a) IMC current control and (b) Predictive current control (vsLL = 65V , vd = 100V ) . . . . . . . . . . . . . . . . . . . . . . . . ˆ frequency deviation ω ˆ and phase error vq of the Estimated phase θ, implemented QPLL. (ω = 5Hz) . . . . . . . . . . . . . . . . . . . . Phase outputs (va , vb , vc ) of the NPC converter using symmetrical SVPWM (vd = 60V ) with dead time Tdead = 2µs. . . . . . . . . . . . DC Bus regulator step response for 40V step in reference . . . . . . . Experimental harmonic compensation in the d-q plane. . . . . . . . . Simulated harmonic compensation in the d-q plane. . . . . . . . . . . Harmonic spectra of load (green) and compensated (magenta) current NP voltage variation under harmonic compensation . . . . . . . . . . Current harmonic compensation of a 2nd order harmonic load . . . .. 97. 6.3 6.4 6.5. 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13. . . . . . . . . . . . .. . . . . . . . . . . . .. 98 99 100. 101 102 102 103 104 105 105 106 107.

(12) LIST OF FIGURES. xi. 6.14 Harmonic spectra of phase load current (green) and phase compensated current (magenta) for a 2nd order harmonic load . . . . . . . . . 107 6.15 NP voltage variation under compensation of a 2nd order harmonic load 108 A.1 Space vector representation of a sector of 3-level Neutral-Point-Clamped (NPC) converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118.

(13) List of Tables 3.1 3.2 3.3. 4.1 4.2 4.3 4.4. 6.1 6.2. Normalised space vector components and phase currents for 3 phase NPC converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Space vector PWM duty cycles for the 3-level NPC converter . . . . . Comparison between SVPWM and SPWM for sinusoidal reference outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mapping matrices for medium vectors . . . . . . . . . . . . . . . . . Mapping matrices for small vector pairs s0 . . . . . . . . . . . . . . . Mapping matrices for small vector pairs s1 . . . . . . . . . . . . . . . Normalised vector values for small vectors for small vectors in the first sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38 39 39 43 43 43 58. Parameters used for step response and disturbance rejection tests for the IMC and predictive current controllers . . . . . . . . . . . . . . . 99 Harmonic compensation parameters . . . . . . . . . . . . . . . . . . 104. xii.

(14) Contents 1 Introduction: Active Filters 1.1 The Grid . . . . . . . . . . . . . . . . . . . 1.2 Conforming and Non-Conforming currents 1.3 Active Filters . . . . . . . . . . . . . . . . 1.3.1 Series and Shunt Active Filter . . . 1.3.2 Shunt Active Filter . . . . . . . . . 1.3.3 Series Active Filter . . . . . . . . . 1.4 Summary and comparison . . . . . . . . . 1.5 Outline of Thesis Chapters . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 2 Power Theory and Compensation 2.1 Introduction: Power Flow . . . . . . . . . . . . . . . . . . . 2.2 The Space Vector Plane . . . . . . . . . . . . . . . . . . . . 2.3 Synchronous frame of reference or d-q theory . . . . . . . . 2.3.1 Compensation in the synchronous reference frame . 2.4 Instantaneous Reactive Power Theory for Balanced Systems 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . .. . . . . . . . .. . . . . . .. . . . . . . . .. . . . . . .. . . . . . . . .. . . . . . .. 3 Modulation Techniques 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Space Vector Pulse Width Modulation . . . . . . . . . . . . . . . . 3.2.1 Three level Neutral Point Clamped Converter . . . . . . . . 3.2.2 NPC Space Vector Modulation . . . . . . . . . . . . . . . . 3.2.3 NPC Space Vector Modulation under unbalanced conditions 3.3 Sinusoidal Carrier Pulse Width Modulation . . . . . . . . . . . . . 3.3.1 SPWM under unbalanced conditions . . . . . . . . . . . . .. xiii. . . . . . . . .. 1 1 2 3 5 7 11 14 14. . . . . . .. 16 16 17 19 21 22 26. . . . . . . .. 27 27 27 27 27 34 34 35.

(15) CONTENTS. 3.4. 3.5. 3.3.2 Other Carrier Based Methods . . . . . . . . . . . . . Comparison between Sinusoidal and Space vector modulation 3.4.1 Harmonic output . . . . . . . . . . . . . . . . . . . . 3.4.2 Maximum sinusoidal output . . . . . . . . . . . . . . 3.4.3 Complexity . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Unbalance compensation methods . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xiv . . . . . . .. 4 Neutral Point Voltage Balancing 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Effects of Power Factor and Output Amplitude on NP . . . . . . 4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Current modulation indices . . . . . . . . . . . . . . . . 4.2.3 Unified NP current equation . . . . . . . . . . . . . . . 4.2.4 Effects of different loading conditions . . . . . . . . . . 4.3 Effects of Harmonic Currents and Unbalanced Loads on the NP 4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Harmonic Current Effects . . . . . . . . . . . . . . . . 4.3.3 Effects of Unbalanced Currents on the NP . . . . . . . . 4.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Zero-sequence voltage balancing . . . . . . . . . . . . . . . . . 4.4.1 Neutral Point Control . . . . . . . . . . . . . . . . . . . 4.5 Comparison: Unipolar vs. Bipolar Neutral Point Control . . . . 4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Current Modulation Indices . . . . . . . . . . . . . . . 4.5.3 SVPWM and Bipolar SPWM . . . . . . . . . . . . . . 4.5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Natural Balancing . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Switching Functions . . . . . . . . . . . . . . . . . . . 4.6.3 Space vector representation of balanced three phase load 4.6.4 Model in space vector plane . . . . . . . . . . . . . . . 4.6.5 Frequency domain . . . . . . . . . . . . . . . . . . . . 4.6.6 Summary and Conclusions . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. 35 36 36 36 36 36 37. . . . . . . . . . . . . . . . . . . . . . . . . .. 40 40 40 40 40 42 45 51 51 53 53 53 54 56 57 57 57 58 59 59 59 59 62 63 64 65.

(16) CONTENTS 4.7. 4.8. xv. Influence of Active Filter Operation on Inverter . . . . . . . . . . . . 4.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Mapping Load and Compensation Characteristics to NPC Inverter Output . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5 Practical Considerations 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Peak Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Quadrature Phase Lock Loop . . . . . . . . . . . . . . . . . . . . 5.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Simple Phase Lock Loop . . . . . . . . . . . . . . . . . . 5.3.3 The Quadrature phase lock loop . . . . . . . . . . . . . . 5.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Losses : Symmetrical PWM and Nearest Three Vectors . . . . . . 5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Nearest Three Vectors . . . . . . . . . . . . . . . . . . . 5.4.3 Symmetrical PWM . . . . . . . . . . . . . . . . . . . . . 5.5 DC Bus Regulation . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 The Controller . . . . . . . . . . . . . . . . . . . . . . . 5.6 Current Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Robust Current Control . . . . . . . . . . . . . . . . . . . 5.7 Predictive Current Control . . . . . . . . . . . . . . . . . . . . . 5.7.1 Current Prediction : Phase Advance . . . . . . . . . . . . 5.7.2 Comparison between Predictive and IMC Current Control 5.7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Experimental Results 6.1 Introduction . . . . . . . . . . 6.2 NPC Inverter . . . . . . . . . 6.3 Current control . . . . . . . . 6.3.1 Step Response . . . . 6.3.2 Disturbance Rejection. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . . .. 65 65 65 68. . . . . . . . . . . . . . . . . . . . . .. 70 70 72 73 73 73 76 79 79 79 80 80 83 83 84 85 85 85 90 91 95 95. . . . . .. 97 97 98 98 99 100.

(17) CONTENTS. 6.4 6.5 6.6 6.7 6.8. xvi. 6.3.3 Summary . . . . . . . . . Phase Lock Loop . . . . . . . . . Symmetrical SVPWM . . . . . . DC Bus Regulator . . . . . . . . . Harmonic compensation . . . . . 6.7.1 2nd Order Harmonic Load Summary . . . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 7 Conclusions and Future Work 7.1 Active Filters . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Power Theory: Generating Reference Compensation Currents 7.3 Modulation Techniques for a NPC converter . . . . . . . . . . 7.4 Balancing Techniques . . . . . . . . . . . . . . . . . . . . . . 7.5 Practical Considerations . . . . . . . . . . . . . . . . . . . . 7.6 Experimental Results . . . . . . . . . . . . . . . . . . . . . . 7.7 Future Work and Recommendations . . . . . . . . . . . . . . 7.7.1 Improvement on Practical Results . . . . . . . . . . . 7.7.2 Balancing properties for Non-linear Loads . . . . . . . 7.7.3 Balancing : Switching Spectra . . . . . . . . . . . . . 7.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Duty Cycle derivation for SVPWM A.1 Region I . . . . . . . . . . . . . A.2 Region II . . . . . . . . . . . . A.3 Region III . . . . . . . . . . . . A.4 Region IV . . . . . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . . . . .. . . . . . . . . . . .. . . . .. . . . . . . .. . . . . . . . . . . .. . . . .. . . . . . . .. . . . . . . . . . . .. . . . .. . . . . . . .. 100 101 101 103 103 106 108. . . . . . . . . . . .. 109 109 109 110 110 111 112 112 112 113 113 113. . . . .. 118 118 120 121 122.

(18) Chapter 1 Introduction: Active Filters This chapter introduces problems experencied with power quality. Active and passive filters are discussed which minimise these power quality problems. A brief overview of shunt and series AF’s are given and their operating conditions are discussed. An outline of the remaining chapters of the thesis follows.. 1.1 The Grid Electric energy is mainly brought to our homes, workplaces, industry etc. via a cabled grid. All the electric suppliers and consumers are connected to one network as shown in Figure 1.1. The place where the consumer is connected to the grid is the Point of Common Coupling (PCC) . The voltage supplied by each supplier (or suppliers in certain countries) must be of such a nature not to be detrimental to the network and to synchronise with it. The same applies to the power drawn by the consumer. Failing to adhere to this could cause the entire network grid to become unstable and possibly collapse (blackout), rendering it useless. Utilities and consumers are thus becoming increasingly concerned about the harmonic distortion (consumer side) and quality of voltage available (supply side). The growing nature of power electronic loads that generate harmonics are posing serious problems to utilities and is detrimental to consumers that have sensitive equipment. Adjustable speed drives, three phase (3φ) diode and thyristor bridges for rectifiers and uninterruptable power supplies are just a few examples of harmonic power electronic loads. Standards for South Africa [19] ([9] for U.S.A) have been defined in an attempt to 1.

(19) C HAPTER 1 — I NTRODUCTION : ACTIVE F ILTERS. 2. CONSUMER SUPPLIER PCC PCC CONSUMER. Figure 1.1: Electrical grid with PCC secure the grid. These standards must be adhered to in an effort to control the stability and quality of the network grid. The fundamental principle in sharing responsibility for harmonic pollution is that the supplier or utility is responsible for the quality of the voltage waveform and the consumer is responsible for the current waveform (which it draws from the grid). If a consumer is supplied with a polluted voltage, the consumer will in effect generate the same harmonics as the voltage in addition to the harmonics it would generate supplied with a pure uncontaminated voltage. The consumer can not be held responsible for these harmonics induced by the supplied voltage. This is where the notion of non- and conforming currents come into place.. 1.2 Conforming and Non-Conforming currents The current waveform drawn by the consumer can be split into two parts. The conforming current has the same shape as the voltage supplied at the PCC. The harmonics present in the conforming current is a scaled replica of the harmonics of the supplied voltage [28]. The distortions in the conforming current are thus only a product of the quality of the supplied voltage and is not the responsibility of the consumer [27]. Large currents drawn by consumers have an effect on the supplied voltage due to line impedances. Utilities should keep line inductances small to minimise harmonics of the supplied voltage. The supplied voltage which helps to define the conforming current should be clarified as the supplied voltage without the particular consumer connected, because large consumer currents can have a noticeable effect on the supplied.

(20) C HAPTER 1 — I NTRODUCTION : ACTIVE F ILTERS. 3. voltage. The difference between the actual current and the conforming current is the nonconforming current. It is the sole responsibility of the customer to eliminate the nonconforming current to a certain extent defined by the regulative standards ([19] [9]). The non-conforming current does not include the fundamental frequency component. The mathematical definition begins with the voltage (v(t)) and current (i(t)) at the PCC in steady state, which can be expressed as a sum of sinusoids: n. v(t) =. ∑ Vk sin (kωt + θk ). (1.1). ∑ Ik sin (kωt + φk ) .. (1.2). k=1 n. i(t) =. k=1. The conforming current represents all the fundamental frequency apparent power: iC (t) =. I1 n ∑ Vk sin [kωt + θk + k(φ1 − θ1)] , V1 k=1. (1.3). or the active conforming current which excludes the reactive power iCactive (t) =. I1 n ∑ Vk sin (kωt + θk ) , V1 k=1. (1.4). and the non conforming current is as previously stated the difference between the actual and conforming current: iNC = i(t) − iC (t).. (1.5). Methods are constantly investigated to effectively suppress harmonic pollution. Passive and active filters are introduced.. 1.3 Active Filters An AF is an actively controlled system which filters certain harmonics present in either the voltage or current at the PCC. It can also be used to force unity power factor and acts as a type of conditioner. It should not be confused with the small signal active filters used in electronics. A more appropriate term would probably be power conditioner, but as the term is used a great deal it is termed an active filter..

(21) C HAPTER 1 — I NTRODUCTION : ACTIVE F ILTERS. 4. Previously, passive filters were used to suppress harmonics and maintain unity power factor in a distribution system. Passive filters, in essence, have a great deal of drawbacks: • Resonance between filters placed in closed proximity can cause over voltages and instability in the distribution system. [11] • As the passive filter is composed of passive components, i.e. inductors and capacitors, the component characteristics change over long periods of time, or exposed temperatures, to such an extent that the new characteristics render them useless. • Passive filters can not adapt to dynamically changing loads. A more elegant solution to accommodate dynamic loads and remain stable over a long period of time is of course the AF. Of course, Likewise, AF’s have their drawbacks: • They are more costly than their passive counterparts. Although they are more costly, they can save a lot of money in the long term. If it is either protecting sensitive equipment or eliminating harmonic pollution penalty costs, they can indirectly save money. As technology progresses, costs tend to reduce and AF’s become cheaper to manufacture. It is ironic though that progressing technology (power electronic loads) itself, is the main cause of harmonic pollution. • Internal control tends to be complex. This basically adds up to costs, but then again technological progress increasingly accommodates more complex control. • Harmonics are produced by the AF’s themselves. The core of modern AF’s consist of switching devices which generate harmonics. Proper design of the AF alleviate this problem. • If several AF’s are situated in close proximity, resonance and instability can occur. This can however be solved through appropriate communication and synchronisation. • AF’s have certain ratings and limitations. There is no universal AF which can eliminate all harmonics and maintain precise unity power factor. Different types.

(22) C HAPTER 1 — I NTRODUCTION : ACTIVE F ILTERS Infinite bus. IC. PCC AF. 5. ILOAD Nonlinear Load. INC. Figure 1.2: Shunt AF Infinite bus. IC. AF PCC −+ + − VAF. ILOAD Nonlinear Load. Figure 1.3: Series AF of AF’s handle different types of scenarios, implicating that AF’s cannot entirely accommodate all dynamic loads, but they can certainly handle dynamics to a certain extent which is far greater compared to their passive counterparts. Not being able to handle everything is a common property for all electric phenomena. The different types of AF’s and their applications are henceforth discussed.. 1.3.1 Series and Shunt Active Filter AF’s can be placed in parallel (shunt) or series. The shunt AF is modelled as a current source (Figure 1.2) and the series AF as a voltage source connected in series (Figure 1.3). These two types of AF have different compensation characteristics and are suited to different types of harmonic-source type loads. Harmonic source-type loads classified into current-source and voltage-source type harmonic loads and are henceforth presented. The following discussion basically follows that of [22]. Current-source type harmonic source Figure 1.4 shows the common voltage and current traces of a harmonic current producing thyristor rectifier with sufficient dc inductance. The voltage is a clean sinusoid while the current is harmonic implying the current being less dependant on the alternating current (ac) side. This is a characteristic of a current source and therefore the theoretical presentation of a current-source type of harmonic source (Fig. 1.5)..

(23) C HAPTER 1 — I NTRODUCTION : ACTIVE F ILTERS. 6. 100. 80. VL. 60. 40. 20. 0. IL. −20. −40. −60. −80. −100. 0. 0.005. 0.01. 0.015. 0.02. 0.025. 0.03. 0.035. 0.04. time (s) Figure 1.4: Typical voltage and current traces for a thyristor rectifier with DC link inductance. ZS + VS AC Source. IL −. Current-Source type Harmonic Source. Figure 1.5: Phase equivalent circuit of current-source type of harmonic source.

(24) C HAPTER 1 — I NTRODUCTION : ACTIVE F ILTERS. 150. 7. VLLL. 100. 50. 0. −50. −100. IL −150 0.045. 0.05. 0.055. 0.06. 0.065. 0.07. 0.075. 0.08. time(s) Figure 1.6: Typical voltage and current traces for a diode rectifier with a smoothing capacitor ZS +. VL. VS AC Source. +. −. − Voltage-Source type Harmonic Source. Figure 1.7: Phase equivalent circuit of voltage-source type of harmonic source Voltage-source type harmonic source Figure 1.6 gives the line to line voltage (at the terminals) and current traces of a diode rectifier with a smoothing capacitor. Although the voltage is distorted, it does not depend much on the ac side impedance, but the distorted current is heavily dependant on the ac side impedance. This behaviour is a characteristic of a voltage source and therefore the theoretical definition of a voltage-source type of harmonic source presented in Fig. 1.7.. 1.3.2 Shunt Active Filter A shunt AF is an inverter placed in parallel (therefore termed shunt) between the source and its load (Fig. 1.2). It’s primary objective is to attenuate current harmonics generated by the load. An ideal shunt AF would produce a perfect sinusoidal load current as.

(25) C HAPTER 1 — I NTRODUCTION : ACTIVE F ILTERS ZS. IS. 8. IL. + G · IC. VS. ZL. IL0. − Figure 1.8: Shunt AF with Norton equivalent harmonic current source seen from the source side. The shunt AF extracts load harmonic currents and injects the currents of the opposite direction into the PCC by means of its Pulse Width Modulation (PWM) inverter to eliminate the designated harmonics. The characteristics of the shunt AF is investigated for current-source and voltage-source type harmonic loads. Analysis for current-source type harmonic source Fig. 1.8 shows the shunt AF compensating the harmonic current of the Norton’s equivalent current-source type harmonic source with impedance ZL . It is desirable to compensate for the harmonics and therefore the transfer function of the AF (G) ideally notches the fundamental component. That is, |G| f = 0 for the fundamental and |G|h = 1 for the harmonics. From Fig. 1.8 equations for the currents are obtained:. IC = GIL IS = IL =. ZL ZL ZS + 1−G. ZL 1−G ZL ZS + 1−G. · ILO +. (1.6) VS ZL ZS + 1−G. 1 · ILO + 1−G ·. (1.7). VS ZL ZS + 1−G. When

(26)

(27)

(28) ZL

(29)

(30)

(31)

(32) 1 − G

(33) |ZS |h h. (1.8). (1.9). holds true, the currents can be rewritten : IC = ILh Ish ≈ (1 − G)ILOh + (1 − G) VZshL ILh = ILOh + VZshL .. (1.10) ≈0. (1.11) (1.12).

(34) C HAPTER 1 — I NTRODUCTION : ACTIVE F ILTERS ZS. IS. + −. IL ZL. G · IC. VS. 9. + VL −. Figure 1.9: Shunt AF with Thevenin equivalent voltage-type harmonic source When (1.9) is satisfied, equation (1.11) shows that the source current approaches a sinusoid because |1 − G|h = 0. In order for the shunt AF to cancel harmonics, (1.9) has to be satisfied and is therefore the required operating condition. The source and load impedance ZS and ZL is fixed by the system, so the only designing factor is the transfer function G of the AF. The system’s influence on the compensation characteristics, the source and load impedance ZS and ZL are all fixed, leaving only transfer function of the AF (G). The AF is therefore not solely responsible for the compensation features just like conventional passive filters. Consider the special case where |ZL | |ZS |, (1.7) and (1.9) is respectively reduced to IS ILO. = 1−G. |1 − G|h 1.. (1.13) (1.14). In this special case, i.e. a thyristor rectifier with a large dc inductance, the AF solely determines the compensation characteristics. When a shunt capacitor or passive filter is connected on the side of the thyristor rectifier, ZL will become low and the AF will not have its lone influence anymore. The current flowing into the passive filter or capacitor ILOh = VZShL (from (1.12)) can become harmfully large if ZL is very small (a stiff ac source). The load harmonic currents will still be compensated by the AF but will flow into the passive filter. If both active and passive filters are used in a system, they should be used mutually exclusive to prevent this phenomenon. An AF is better suited to lower order harmonics and passive filters to higher order harmonics, so sharing the responsibility this way is good practice. The combination of a passive and active filter is termed a hybrid AF. The rating of a hybrid AF is greatly reduced compared to a single AF, because the passive filter mainly does reactive compensation and the AF harmonic compensation..

(35) C HAPTER 1 — I NTRODUCTION : ACTIVE F ILTERS. 10. Analysis for voltage-source type harmonic sources The Thevenin equivalent source (VL and ZL ) represents the voltage-source type harmonic source, for which the AF (with transfer function G) compensates harmonics, and is shown in Fig. 1.9. From the figure the current is deduced: IC = G · IL. (1.15). VS −VL ZL ZS + 1−G. (1.16). IS = IL = When. 1 1−G. ·. VS −VL ZL ZS + 1−G. =. VS −VL (1−G)·ZL +ZL ..

(36)

(37)

(38)

(39) Z L

(40) ZS +

(41) 1

(42) 1 − G

(43) h. (1.17). (1.18). is satisfied, the source current approaches a sinusoid : IC = ILh. (1.19). ISh ≈ 0. (1.20). Lh ILh = VShZ−V . L. (1.21). In order for the AF to compensate for the harmonic voltage source’s harmonics, (1.18) need to be satisfied which is practically difficult. Consider a diode rectifier with a large smoothing dc capacitor, if no series reactor is placed on the ac side of the rectifier, the internal impedance ZL is very low. Equation (1.18) can therefore not be satisfied when |ZL | ≈ 0. A series reactor has to be placed on the ac side of the rectifier to increase ZL in order to satisfy 1.18). With a typical source impedance of 5% (|ZS | = 0.05pu) and a harmonic attenuation of 90% (|1 − G|h = 0.1), a series reactor in the order of 10% will be needed. Even in the case of (1.18) being satisfied, inspection of (1.17) shows that the source impedance is close to zero as seen from the load side. Harmonic compensation current from the AF flows directly into the load (1.19) and harmonics of the source voltage cause large associated harmonics to flow into the load (1.21). Even though the shunt AF is successfull at its task of supplying a sinusoidal load current as seen from the voltage source, undesired large harmonic currents flow into the load. The required rating of the shunt AF is also large when ZL is small..

(44) C HAPTER 1 — I NTRODUCTION : ACTIVE F ILTERS. ZS. IS. + VS. +. VC. 11. −. Series AF. ZL. IL. − Figure 1.10: Series AF with Norton equivalent current-type harmonic source. 1.3.3 Series Active Filter The series AF shown in Fig. 1.3 is connected in series between a load and source and its primary objective is to provide the load with a distortionless source voltage and the source with a sinusoidal current. Equipment sensitive to voltage distortions typically have series AF’s installed. The AF basically isolates the load and source from harmonics, and exhibits a high harmonic impedance, effectively blocking harmonic flow between the source and load. Characteristics for series AF’s compensating for voltage-source and current-source type harmonic sources are investigated. Analysis for current-source type harmonic sources Fig. 1.10 shows the series AF represented by the controlled voltage source VC . The ac side consists of the source VS and it’s impedance ZS , and the Norton equivalent load (IL0 and ZL ). Likewise a transfer function G for the AF which ideally notches the fundamental component is defined. For the fundamental, |G| f = 0 and |G|h = 1 for harmonics. Additionally we define a gain K with the dimension of ohms per unit. The series AF is per definition controlled as VC = K · G · IS .. (1.22). The source current can be deduced as IS =. VS ZL · IL + . ZS + ZL + KG ZS + ZL + KG. (1.23). If the gain K is large enough to satisfy K |ZL |h and K |ZS + ZL |h ,. (1.24). and the fact that the distortion of the ac source (VSh ) is small compared to the current distortion of the load, the source current essentially becomes sinusoidal :.

(45) C HAPTER 1 — I NTRODUCTION : ACTIVE F ILTERS. ZS + VS. IS. +. VC. 12. −. Series AF. −. ZL. + VL −. Figure 1.11: Series AF with Thevenin equivalent voltage-type harmonic source. VC ≈ ZL ILh +VSh. (1.25). ISh ≈ 0. (1.26). The gain, K, should be large and the load side harmonic impedance |ZL |h small enough to satisfy (1.24) which is the operating condition for the series AF to suppress the current-source type harmonic source. For a typical thyristor rectifier, ZL tend to be very large, so the operating condition equation cannot be satisfied. A phase-controlled thyristor rectifier is therefore not a well suited load in a series AF compensation setup. This is the opposite or dual case for the shunt AF. Theoretically, the series AF cannot fully compensate the current-source type harmonic load, because an infinite |ZL |h implies an impractical infinite compensation voltage VC as (1.25) distinctively illustrates. Consider a shunt passive filter to be put across the rectifier, attempting to reduce the impedance seen from the AF (ZL ). This can be practically satisfied, because VC becomes very small and the required operating condition (1.24) is met. This results in a series hybrid AF. Hybrid AF’s are beyond the scope of this writing, but an in depth discussion can be found in [21]. The objective of the series AF is clearly illustrated by equation (1.25), where the voltage distortion (VSh ) is blocked from affecting the load, and load harmonic currents (ILO ) are blocked from flowing to the source side (1.26). Analysis for voltage-source type of harmonic sources The Thevenin equivalent voltage source in Fig. 1.11, series AF VC and the ac source VS with its impedance ZS comprise the series AF compensation of a voltage-source type harmonic load. The definition of the controlled series voltage source is as defined in.

(46) C HAPTER 1 — I NTRODUCTION : ACTIVE F ILTERS. 13. the previous section, i.e. VC = K · G · IS. (1.27). and therefore the source current is IS =. VS −VL ZS + ZL + KG. (1.28). Consider K 1pu,. (1.29). if it is satisfied the following applies: IS ≈ 0. (1.30). VC ≈ VSh −VLh .. (1.31). If we assume the active filter follows it’s reference, the source current is expressed as [22] IS =. VS − (1 − G)VL . ZS + ZL + KG. (1.32). When the ac source has little distortion and |1 − G|h 1. (1.33). is met, the source current becomes sinusoidal: ISh = −(1 − G)VLh ≈ 0. (1.34). even with K = 1pu. The operating condition is expressed by equation (1.33) which shows that the compensation characteristics for a voltage-source type harmonic load depends only on the series AF itself. The compensation is independent of the source and load impedance ZS and ZL respectively as equation (1.34) clearly illustrates. The series AF can therefore effectively compensate for a voltage-source type harmonic load if the source voltage distortion is relatively low. The same applies for a shunt AF compensating for a current-source type harmonic load which is the dual. Comparing the series and shunt AF’s we can draw dualities in terms of compensation characteristics which follows henceforth..

(47) C HAPTER 1 — I NTRODUCTION : ACTIVE F ILTERS. 14. 1.4 Summary and comparison Standards were drawn up and should be adhered to help to supply quality power. Conforming currents are currents a resistive load would produce given a specific supply voltage. Non-conforming currents represent the difference between the actual current and the conforming component. Power quality responsibility should be shared. Consumers should be responsible for current waveform quality and utilities for voltage waveform quality. The shunt AF operates as a current source and the series AF as a voltage source. Both have conditions necessary for proper operation. The shunt AF is not well suited to compensate voltage source type harmonic loads whereas the series AF is not well suited to compensate current source type harmonic loads. Injected current from a shunt AF will flow into the load side for a voltage source type harmonic load overcurrent may result. A low impedance parallel passive filter is needed when the series AF is applied to a current source type harmonic load. Series AF block harmonics between loads and supplies and ‘protects’ the load. Shunt AF attenuate current harmonics caused by loads and ‘protects’ the supply. Series AF usually comprise injection transformers between source and load to act as a voltage source. They are therefore normally large and bulky. Shunt AF are usually smaller and more common. At high voltages, they require costly injection transformers to connect their internal inverters and coupling reactors to the PCC. The next chapter explains this connection ‘problem’ and introduces multilevel topologies. These multilevel topologies enable us to connect the internals of the AF directly to the PCC without the need of costly injection transformers.. 1.5 Outline of Thesis Chapters This chapter introduced the power quality sharing responsibility, the AF and why the NPC inverter is proposed for a shunt AF which will further be presented in this text. The shunt AF injects compensation currents into the PCC to produce the desired compensation. These compensation currents need to be determined depending on compensation demands and load and utility characteristics. Chapter 2 gives two commonly used power theories used today to determine these compensation currents. Block diagrams of compensation current extraction are given for the shunt AF..

(48) C HAPTER 1 — I NTRODUCTION : ACTIVE F ILTERS. 15. Chapter 3 presents the common modulation techniques used to modulate a reference output from the NPC converter and compares their characteristics. Each modulation technique have certain balancing characteristics which is discussed in chapter 4. Additionaly, chapter 4 investigates the effects of the PF, nonlinearity, and unbalance of a load connected to the output of the NPC converter. The effects is then investigated in an AF setup by mapping the load in a compensation setup to that of outputs of the NPC inverter of the AF. Chapter 5 presents the building blocks of an experimental NPC shunt AF. Detailed design of each building block is presented. Finally, chapter 6 gives experimental and some simulation results of the laboratory prototype AF to validate the theory presented in the other chapters. Chapter 7 concludes this thesis and recommends future work..

(49) Chapter 2 Power Theory and Compensation 2.1 Introduction: Power Flow The previous chapter introduced the concept of conforming and non-conforming currents in section 1.2. The shunt AF injects currents into the PCC to enforce conforming current seen from the utility side. To achieve perfect conforming currents, both harmonic and reactive power of the load need to be compensated for. This is not always the practical case: Reactive compensation need no energy storage [1], but demand a high power rating of the AF. Some systems only do partial reactive compensation. The bandwidth of AF’s are limited due to bandwidth of devices incorporated and computational limits (section 5.7.1). For these reasons, mostly the dominant harmonics are compensated for to comply with specified standards [9][19]. A practical shunt AF normally does a specified amount of harmonic compensation and a specified amount of reactive power compensation. These amounts can vary from no compensation to full compensation. Power flows between the AF and the non-linear load, and similarly power flows between the AF and the utility. The type of power flow depends on the compensation configuration. Fig. 2.1 shows the power flow where full harmonic and reactive compensation is applied. The only power that flows from the utility is the conforming power which is pure active power with no harmonics. This active power comprises that of the load and the losses of the shunt AF. The non-harmonic active load power flows between the utility and the load and the non-harmonic power due to losses flows between the utility and the AF. Harmonic power cycles between the AF and the nonlinear load if not cancelled between phases. Harmonic power varies in a cycle and 16.

(50) C HAPTER 2 — P OWER T HEORY AND C OMPENSATION non-harmonic load active power. 17. Non-Linear Load. PCC. 3∼ AF power losses AF. reactive and harmonic load power. Figure 2.1: Power flow for full reactive and harmonic compensation therefore energy storage is needed for harmonic compensation. Harmonic powers add up to zero in a cycle however. Reactive power cycles between the AF and the load in addition to the harmonic power. It should be clarified that reactive power can also include harmonics (harmonic reactive power). The power theories developed over the years help us to identify these powers and generate current references for the AF to necessitate the desired compensation. In the following theories, the 3φ system is assumed to be a balanced three wire system.. 2.2 The Space Vector Plane The space vector plane is a scaled transformation from the representation of 3φ quantity vectors to a single vector quantity on a two dimensional plane as seen in Figure 2.2. One vector thus represents three voltages. Assuming a balanced system, one vector would represent three unique voltages. The transformed phase quantities are then:   " # r " # van 1 1 −2 vα 2 1 −   √2 √ (2.1) =  vbn  3 3 3 0 2 − 2 vβ vcn p and is known as the Clarke transform. The reason for the 2/3 constant is to obtain correct power quantities in the p-q theory detailed in section 2.4. The transformation with this vector is hence referred to as the p-q space vector transformation. The space vector transform used for d-q theory (section 2.3) uses the constant 32 which results in a space vector amplitude equivalent to that of one of the phases for a balanced system..

(51) C HAPTER 2 — P OWER T HEORY AND C OMPENSATION. 18. β q 2 3b. bβ q 2 3a. bα cα. q 2 3c. aα. α. cβ. Figure 2.2: Space Vector transformation The two quantities produced are orthogonal, therefore the size of the vector is the sum of the two quantities |v|2 = |vα |2 + |vβ |2 .. (2.2). If we assume a balanced 3φ system, i.e. van + vbn + vcn = 0, a simplified version of the Clarke transform holds: " " #  q # 3 0 vα v an 2 = . (2.3) √  √1 vβ vbn 2 2. The inverse of the transform has a unique solution " #  q # " 2 0 van v α 3  = . vbn vβ − √1 √1 6. (2.4). 2. The same holds for the currents of the system   " # r " # ia 1 − 12 iα 2 1 −   √2 √ =  ib  , 3 3 3 0 2 − 2 iβ ic. (2.5).

(52) C HAPTER 2 — P OWER T HEORY AND C OMPENSATION. 19. and in the case of a balanced 3φ system (ia + ib + ic = 0) we have simplified versions of the transform and its inverse for the currents also: " " #  q # 3 0 i iα a 2 = , (2.6) √  1 √ iβ ib 2 2 " " #  q # 2 0 ia i 3  α . = (2.7) 1 1 ib iβ −√ √ 6. 2. 2.3 Synchronous frame of reference or d-q theory Consider 3φ balanced positive sequence voltages defined as     va cos (ωt)      vb  = E  cos ωt − 2π 3  cos ωt + 2π vc 3. (2.8). and harmonic currents of order n, fundamental phase φ1 and phase φn defined as     ian cos [n (ωt + φ1 ) + φn ]     (2.9) + φ + φ  ibn  = In  cos n ωt − 2π . n 1 3 2π cos n ωt + 3 + φ1 + φn icn If the currents in (2.9) are transformed to the space vector plane by means of the d-q space vector transform (using the constant 23 ) it yields sinusoidal quantities ( iβ = In. iα = In cos [n (ωt + φ1 ) + φn ]. (2.10). sin [n (ωt + φ1 ) + φn ] , n = 3m + 1 − sin [n (ωt + φ1 ) + φn ] , n = 3m − 1. (2.11). where m is a positive integer. When the α and β components correspond to real and imaginary components respectively, (2.11) can be represented as a rotating vector in the complex plane ( e j[n(ωt+φ1 )+φn ] n = 3m + 1 in = In , (2.12) e− j[n(ωt+φ1 )+φn ] n = 3m − 1 for which vectors of n = 3m + 1 i.e. n = 1, 4, 7, . . . rotate in the positive or counterclockwise direction, and vectors of n = 2, 5, 8, . . . rotate in the negative or counterclockwise direction at the angular speed of nω. This is graphically depicted in Fig..

(53) C HAPTER 2 — P OWER T HEORY AND C OMPENSATION. 20. Im{in } 7ω θ7Ts. I7 I5 θ5Ts. 0. 5ω. Re{in }. Figure 2.3: Directions and phase differences for 5th and 7th harmonics in the stationary frame of reference 2.3 for the 5th and 7th harmonic. The figure also shows that for a certain period (Ts ), higher harmonics have larger phase differences. It can also be noticed that currents of n = 3m + 1 cause positive sequences of currents and harmonics currents of n = 3m − 1 cause negative sequences of currents and are referred to as positive and negative sequence harmonics respectively. Harmonics of n = 3m cause zero sequences currents which flow in the fourth wire. The shunt AF considered in this writing is assumed to be a 3-wire balanced system and therefore these harmonics do not exist. A new frame of reference is introduced that rotates with the fundamental frequency (i.e. at a frequency ω and positive or counter clockwise). This is referred to as the synchronous frame of reference. A fundamental frequency vector appears as a stationary vector in the synchronous frame of reference. This vector is further divided up its the direct (d) and quadrant (q) components. A fundamental frequency can thus be represented as 2 constants. This poses a lot of advantages as Proportional Integral Derivative (PID) type controllers, low pass filters, etc. are well suited to dc quantities or constants. The d-q transformation is given as " # " #" # iα id cos θ sin θ = , (2.13) iq − sin θ cos θ iβ.

(54) C HAPTER 2 — P OWER T HEORY AND C OMPENSATION. 21. where the over line represents a normalised quantity. The direct and quadrant currents of (2.9) can then be represented as id = I1 cos φ1 +. ∑. idk. (2.14). ∑. iqk. (2.15). k=3m. iq = I1 sin φ1 +. k=3m. where m is a positive integer. The 2nd and 4th harmonics show up as a 3rd harmonic, and likewise the 5th and 7th harmonics show up as a 6th harmonic in the synchronous frame of reference, and so on. The constant component of id corresponds to the steady state fundamental active current amplitude and the constant component of iq corresponds to the steady state fundamental reactive current amplitude. This makes extraction of components for compensation easy and is henceforth be discussed.. 2.3.1 Compensation in the synchronous reference frame Compensation is made easy in the synchronous frame as the current can be split up into active and reactive components in which the steady state fundamentals show up as constants. These slow varying signals are easy to manipulate with low pass filters and Proportional Integral (PI) controllers with minimal computational effort. We define a factor K where 0 ≤ K ≤ 1 which defines the amount of reactive current to compensate. For K = 1 we will have total reactive compensation. For harmonic and reactive compensation, the compensation currents are derived from (2.14) and (2.15) where the unwanted currents are extracted. This yields the active i∗dc and reactive i∗qc compensations currents: i∗dc = −. ∑. idk ,. (2.16). k=3m. i∗qc = −KI1 sin φ1 −. ∑. iqk. (2.17). k=3m. where the * denotes reference currents. Figure 2.4 shows a block diagram which will aid in the practical extraction implementation. The Low Pass Filter (LPF)’s normally consist of Butterworth type filters with a 20 Hertz (Hz) cut-off frequency..

(55) C HAPTER 2 — P OWER T HEORY AND C OMPENSATION. 22. LPF iLa iLb. ". # 1 0 1 − √3 − √23. iLα iLβ. ". cos θ sin θ − sin θ cos θ. #. iLd. +. iLq. + LPF. –. –. 1-K. Figure 2.4: Compensation in the synchronous frame of reference. 2.4 Instantaneous Reactive Power Theory for Balanced Systems The instantaneous reactive power theory was introduced in [1] and is similarly discussed in this section. The instantaneous active power in a 3φ balanced system is p = van ia + vbn ib + vcn ic .. (2.18). Using (2.4) and (2.7) the instantaneous active power for a balanced system can be expressed in terms of its α and β quantities p = vα iα + vβ iβ .. (2.19). Consider the previously defined space vector plane where the real plane is defined at a ”height” of zero, depicted in Figure (2.5). Consider also a new instantaneous imaginary power quantity q defined as the cross product between the opposite voltages and currents, q = vα × iβ + vβ × iα ,. (2.20). and conforming to the conventional right hand rule the scalar value is obtained: q = vα iβ − vβ iα . The instantaneous powers can be written in matrix form " # " #" # p vα vβ iα = , q −vβ vα iβ. (2.21). (2.22). i∗d i∗q.

(56) C HAPTER 2 — P OWER T HEORY AND C OMPENSATION. 23. q(imaginary) α. vα × iβ. iβ vβ. vα. β. iα Real plane (p). vβ × iα Figure 2.5: Three dimensional power space and the currents as the inverse " # " #−1 " # iα vα vβ p . = q iβ −vβ vα. (2.23). The determinant of (2.23) is nonzero and finite, resulting in a unique solution for the space vector currents. We wish to express these currents in terms of components dependent on the active power p and our imaginary defined power q. The two powers are orthogonal enabling us to express the currents in independent components. If we have currents dependent on instantaneous active power and currents dependent on our defined imaginary power, we can compensate for the imaginary power leaving only the active power currents behind. If we only have active power left, we have successfully compensated for instantaneous reactive power. The power q should not be confused as instantaneous reactive power at this stage. The power q is just an imaginary power defined in the space vector space enabling us to extract the instantaneous active power. In a balanced 3φ system however, it is the only remaining power, and q is therefore the instantaneous reactive power. In 4-wire systems, q is a mixture of unbalanced powers and reactive power and is not then necessarily the instantaneous reactive power [23]. The power p however is the instantaneous active power proven in equations (2.18) and (2.19) as the space vector quantity equals the phase equivalent..

(57) C HAPTER 2 — P OWER T HEORY AND C OMPENSATION Manipulating (2.23) to two components p and q yield " # " #−1 " # " #−1 " # iα vα vβ p vα vβ 0 = + iβ −vβ vα 0 −vβ vα q " # " # iαp iαq = + . iβp iβq. 24. (2.24). Solving the currents the following is obtained: iα p = iβ p =. vα p v2α +v2β vβ p v2α +v2β. iαq =. −vβ 2 vα +v2β. iβq =. vα q v2α +v2β. q (2.25). which clearly illustrates the exclusive dependence on each instantaneous power. The instantaneous active powers pα and pβ are defined and expressed from the above currents (2.25): " # " # " # " # pα vα iα vα iα p vα iαq = = + . (2.26) pβ vβ iβ vβ iβ p vβ iβq Substituting the currents of (2.25) into (2.26) and using (2.19) the active power is obtained: 2. p = pα + pβ. vβ vα vβ vα vβ v2 = 2 α 2 p+ 2 p + q − q. vα + vβ vα + v2β v2α + v2β v2α + v2β. (2.27). The last two terms in (2.27) add up to zero and correspond to the α and β instantaneous reactive power components respectively. This is noteworthy because no instantaneous power flow, and therefore no energy storage devices, would be needed in a system to compensate for instantaneous reactive power. The active power can then be expressed only as the active power from the α and β active powers: p = p α p + p βq =. v2β v2α p + p, v2α + v2β v2α + v2β. and the reactive α and β power components add up to zero: vα vβ vα vβ q − q = 0. pα p + pβq = 2 vα + v2β v2α + v2β. (2.28). (2.29).

(58) C HAPTER 2 — P OWER T HEORY AND C OMPENSATION. va. iα. vα. ab. –. q. q. LPF. q∗. –. 1−K. +. +. αβ v β. vb. vβ. 25. iβ vα. iLa iLb. iα. iα. ab αβ. vα. +. p. p. LPF. p∗. – +. + iβ. iβ. vβ p∗. vα. + –. q∗ vβ. · v2α +v2β. i∗α. p∗. vβ. + +. q∗. · v2α +v2β. i∗β. vα. Figure 2.6: Block diagram for compensation using p-q theory The instantaneous active and reactive powers can further be divided into their average and alternating components which correspond to their average and harmonic components: p = p + p˜ q = q + q, ˜. (2.30). where the over line and the tilde correspond to the average and alternating value respectively. The average parts can be extracted by low pass filtering and the remaining component correspond to the alternating component. These power components are exactly what we need to compensate for harmonic and reactive power. The block diagram in Fig. 2.6 gives a practical way to implement p–q theory compensation where the constant K is defined in section 2.3.1. The * superscript denote the compensation values. The p-q compensation theory followed in this text basically follows that of [1] and is valid for balanced 3φ systems. A generalised theory which extends to 4-wire systems can be found in [23]..

(59) C HAPTER 2 — P OWER T HEORY AND C OMPENSATION. 26. 2.5 Summary This chapter has introduced the concept of power flow between a compensator and its load and the necessary power theories to extract current compensation references. A shunt AF normally does selective filtering on certain harmonic components and optionally a specified amount of reactive power compensation. The space vector plane represents a 3φ quantities as a single rotating vector. The synchronous frame of reference is synchronised with a rotating vector, resulting in active or direct (d), and reactive or quadrant (q) components. These quantities are easily processed with LPF’s and PID type controllers to extract the necessary compensation currents. Additionally the p-q theory is introduced where the instantaneous active and reactive powers are extracted. The average and harmonic powers can further be extracted by LPF’s. The compensation currents can then be derived from these values. Energy storage devices are not needed for reactive power compensation. The extracted compensation currents need to be injected to necessitate compensation. The following chapter explores methods to realise the compensating currents through the switching devices of the converter..

(60) Chapter 3 Modulation Techniques 3.1 Introduction A modulation technique is the method of producing a reference output voltage with the available converter. Two main modulation techniques and their variants are introduced for the NPC converter. Their characteristics for a sinusoidal output reference are explored and compared.. 3.2 Space Vector Pulse Width Modulation 3.2.1 Three level Neutral Point Clamped Converter The NPC converter first appeared in 1981 by [2]. Figure 3.1 shows the Neutral Point Clamped Converter with 3 phase arms each capable of producing 3 levels of output from the dc bus Vd and vice versa.. 3.2.2 NPC Space Vector Modulation Vectors produced by 3 phase NPC inverter Figure 3.2 shows that each phase has three different outputs. For 3 phases we have 33 = 27 different outputs for the entire converter. As each different phase leg is connected to either − V2d , + V2d or the Neutral Point (NP), different currents flow through the NP, the upper and lower dc bus denoted iNP , i p and in respectively. If we take vector 14 (Table 3.1) where phase C is connected to + V2d and phases A and B are connected to the 27.

(61) C HAPTER 3 — M ODULATION T ECHNIQUES. 28. ip Sa0. Sb0. Sc0. Sa1. Sb1. Sc1. + vCHI −. Vd + −. NP. A. iNP. B. C. Sa2. Sb2. Sc2. Sa3. Sb3. Sc3. + vCLO −. in Figure 3.1: Three level Neutral-Point-Clamped Topology NP, the neutral point current would be −ic . No current flows through the unconnected lower dc bus connection. Figure 3.3 shows the case for vector 14 and Table 3.1 gives the currents and normalised space vector components for all the 27 possible vectors. We can represent these 27 vectors in the p-q space vector plane (see section 2.2) reflected in Fig. 3.4. The numbering system in the figure is chosen for ease of practical implementation and will be explained later. Looking at the currents and space vector representation the following observations can be perceived: • Zero Vectors : Three redundant vectors giving no output voltage or any neutral point or dc bus currents. • Positive small vectors : Vectors 3–8. Only one phase is connected to the NP resulting in a corresponding NP current. The same current with opposite phase flows either through the upper or lower dc bus connection depending on the.

(62) C HAPTER 3 — M ODULATION T ECHNIQUES. I phase. + V2d. 29. + V2d. + V2d. I phase. NP. Phase. NP. − V2d. Phase. NP. − V2d. (a) Positive output. Phase. I phase. (b) Neutral point output. − V2d. (c) Negative output. Figure 3.2: Outputs for NPC phase arm. + V2d NP. i p = ic. ic. C. iNP = ia + ib = −ic. − V2d unconnected ⇒ in = 0. ia A. ib B. Figure 3.3: Current through NP and DC bus connections for vector 14.

(63) C HAPTER 3 — M ODULATION T ECHNIQUES. 30. vβ 25. 19. 5. 4 11. 12 6 13. 26. 24. 18. 0, 1, 2. 7 20. VREF 3 10. 23. vα. 8 14. 27. 17. 15. 21. 22. 28. Figure 3.4: Zero (black), small (blue), medium (green) and large (red) space vectors produced by 3-phase NPC converter. vector. • Negative small vectors : Vectors 10–15. Two phases are connected to the NP resulting in the other phase current with opposite polarity flowing through the NP. The inverted NP current flows through either the upper or lower dc bus connection depending on the vector. • Small vectors are also redundant and come in pairs of positive and negative vectors giving the same output voltage. Positive vectors are defined so that if the phase current for that vector (Table 3.1) is positive, the corresponding NP current would also be positive. The same applies for negative small vectors, where the currents are negative. For each pair of positive and negative vectors the upper and lower dc bus currents alternate. No vector pair have both upper dc bus connection currents or vice versa. For any small vector, a NP current flows and the opposite current flows through either the upper or lower dc bus connection depending on the vector. • Medium vectors : 17-22. Second largest amplitude output with no redundancy..

(64) C HAPTER 3 — M ODULATION T ECHNIQUES. 31. Each phase leg of the converter is connected to a different dc point (+ V2d , − V2d , 0). This results in each dc connection point having that particular phase current. • Large vectors : 23-28. Largest amplitude output with no redundancy. Two phases are either connecter to the upper dc bus connection or the lower dc bus connection with the other phase connected to the opposite dc bus connection. This results in those connection points having the sum of those phase currents and dc bus connection points having opposite currents. No neutral point current and therefore no influence on the balancing of the capacitors. • Redundancy decreases from the midpoint (3 zero voltage vectors) to 2 at the small vectors and none at the medium and large voltage vector level. Higher level NPC converters exhibit a higher level of redundancy likewise decreasing from the midpoint outwards. Pulse Width Modulation The converter can only output 19 different output voltages (27 switching states taking redundancy into account), but in practical applications we require intermediate voltages. This can be accomplished by spending a certain amount of time in each switching state. Pulse width modulation is the pulsing of different voltage values with different widths to modulate an average reference voltage over a certain period of time. A fixed number of vectors or voltages are applied in a period of time known as the switching period. The ratio of the pulse width or on time of each vector used, to that of the switching period is defined as that vector’s duty cycle [15]. The voltages or vectors and duty cycles determine the average value in the switching period, while the switching period in turn determines the ‘resolution’ of the produced average value. The vectors in Fig. 3.4 exhibit a 60° symmetry defined as a sector, presented in Fig. 3.5. The reference vector is the vector we want to produce and will fall in one of the six sectors. Only one sector has to be considered due to the symmetrical aspect and the results can be applied to the particular sector in which the reference vector lies. The three closest vectors are used to produce the reference vector in multilevel converters to minimise switching losses [14]. The sector is further subdivided into regions (I–IV shown in fig. 3.5) to aid in recognising the closest three vectors. The re-.

(65) C HAPTER 3 — M ODULATION T ECHNIQUES. 32. vβ VF. VC. IV. VE. III VA. I. II. VD. VB. vα. Figure 3.5: Symmetrical sector of producible vectors of 3-phase NPC converter showing triangular regions gion’s boundaries in which the reference vector lies clearly illustrate the closest vectors involved in producing that particular reference vector. A sinusoidal reference vector with results in a rotating reference vector (section 2.3). The largest sinusoidal vector amplitude producible without going into the overVd . This corresponds to a maximum modulation region is that of the medium vector : √ 2 line-line sinusoidal amplitude of Vd . It is convenient to define a quantity from 0 to 1 which corresponds to the range from minimum to maximum producible amplitude. This quantity is defined as the modulation index m. The reference vector can therefore be represented as Vd VREF = VREF e jθ = m √ e jθ , 2. (0 ≤ m ≤ 1).. (3.1). Once we have the region and thus the involved vectors, their corresponding duty cycles are derived trigonometrically by taking into account that the reference vector is the sum of the three dutiable vectors VREF = d0 V0 + d1 V1 + d2 V2 ,. (3.2). and that the duty cycles add up to unity (i.e. they must fill the switching period): d0 + d1 + d2 = 1.. (3.3). Taking region II as an example we have VB , VD and VE as the three closest vectors to the reference vector VREF (fig. 3.6). Taking the largest vector amplitude as unity (large.

(66) C HAPTER 3 — M ODULATION T ECHNIQUES vβ. 33. VE VREF. VA. VB. θ. VD. vα. Figure 3.6: Trigonometric duty cycle derivation in region II vectors) and representing the space vector plane as the complex plane we have VREF = dB VB + dD VD + dE VE √ 1 3 jπ = dB + dD + dE e 6. 2 2. (3.4) (3.5). Writing out in trigonometric form: √ π i 3 h π 1 + j sin , VREF [cos(θ) + j sin(θ)] = dB + dD + dE cos 2 2 6 6. (3.6). splitting into real and imaginary parts and simplifying: 1 3 dB + dD + dE 2 4 √ 3 VREF sin(θ) = dE , 4. VREF cos(θ) =. (3.7) (3.8). together with the fact that all duty cycles should sum up to unity dB + dD + dE = 1. (3.9). we have three equations with three solvable unknowns. Solving the unknowns we have derived the duty cycles for region II: π , (3.10) dB = 2 − 2m sin θ + π 3 dD = 2m sin −θ , (3.11) 3 dE = 2m sin(θ). (3.12) Table 3.2 summarises the duty cycles for all regions while Appendix A gives a more detailed version of all the duty cycle derivations..

(67) C HAPTER 3 — M ODULATION T ECHNIQUES. 34. 3.2.3 NPC Space Vector Modulation under unbalanced conditions An unbalanced condition exists when the two capacitors dividing the dc bus in Fig. 3.1 have different voltages. Under unbalanced conditions the medium vectors in Fig. 3.4 undergo a phase shift in the space vector plane. One of the small vectors in a small vector pair decrease in magnitude and the other increases in magnitude. This gives the effect of the vector pair splitting up. The large vectors remain unchanged, resulting in the same attainable maximum sinusoidal line-line amplitude output as the balanced condition [3]. The duty cycles for the medium and small vectors can be adjusted to compensate for the unbalance of the NP [24]. If the ripple of the NP is very small, the corresponding duty cycle adjustment is negligible. [3]. 3.3 Sinusoidal Carrier Pulse Width Modulation 1. K p 0.5 vREF 0. Kn−0.5 −1 0.028. 0.0285. 0.029. 0.0295. 0.0285. 0.029. 0.0295. 0.03. 0.0305. 0.031. 0.0315. 0.032. 0.03. 0.0305. 0.031. 0.0315. 0.032. 1 0.5. v phase. 0. −0.5 −1 0.028. t (s) Figure 3.7: Switching instants for Sinusoidal PWM A simple method to generate PWM signals is using triangular carriers and a modulation reference referred to as Sinusoidal Pulse Width Modulation (SPWM). Figure 3.7 reflects the basic operating principle behind SPWM. The sinusoidal waveform vREF is the reference. The phase output v phase is normalised to V2d . The triangular waveforms or carriers are at the switching frequency and their amplitudes K p and K p represent a.

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