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Electromagnetic Analysis of Hydroelectric Generators

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Electromagnetic Analysis of Hydroelectric Generators List of Papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I Ranlöf, M., Perers R. and Lundin U., “On Permeance Modeling of Large Hydrogenerators With Application to Voltage Harmonics Predic- tion”, IEEE Trans. on Energy Conversion, vol. 25, pp. 1179-1186, Dec. 2010. II Ranlöf, M. and Lundin U., “The Rotating Field Method Applied to Damper Loss Calculation in Large Hydrogenerators”, Proceedings of the XIX Int. Conf. on Electrical Machines (ICEM 2010), Rome, Italy, 6-8 Sept. 2010. III Wallin M., Ranlöf, M. and Lundin U., “Reduction of unbalanced mag- netic pull in synchronous machines due to parallel circuits”, submitted to IEEE Trans. on Magnetics, March 2011. IV Ranlöf, M., Wolfbrandt, A., Lidenholm, J. and Lundin U., “Core Loss Prediction in Large Hydropower Generators: Influence of Rotational Fields”, IEEE Trans. on Magnetics, vol. 45, pp. 3200-3206, Aug. 2009. V Ranlöf, M. and Lundin U., “Form Factors and Harmonic Imprint of Salient Pole Shoes in Large Synchronous Machines”, accepted for pub- lication in Electric Power Components and Systems, Dec. 2010. VI Ranlöf, M. and Lundin U., “Finite Element Analysis of a Permanent Magnet Machine with Two Contra-rotating Rotors”, Electric Power Components and Systems, vol. 37, pp. 1334-1347, Dec. 2009. VII Ranlöf, M. and Lundin U., “Use of a Finite Element Model for the Determination of Damping and Synchronizing Torques of Hydroelec- tric Generators”, submitted to The Int. Journal of Electrical Power and Energy Systems, May 2010. VIII Ranlöf, M., Wallin M. , Bladh J. and Lundin U., “Experimental Study of the Effect of Damper Windings on Synchronous Generator Hunting”, submitted to Electric Power Components and Systems, February 2011. IX Lidenholm J., Ranlöf, M. and Lundin U., “Comparison of field and circuit generator models in single machine infinite bus system simula- tions”, Proceedings of the XIX Int. Conf. on Electrical Machines (ICEM 2010), Rome, Italy, 6-8 Sept. 2010. v X Wallin M., Ranlöf, M. and Lundin U., “Design and construction of a synchronous generator test setup”, Proceedings of the XIX Int. Conf. on Electrical Machines (ICEM 2010), Rome, Italy, 6-8 Sept. 2010. Reprints were made with permission from the publishers. vi Contents 1 Introduction . 1 1.1 Background . 1 1.2 Applications of Permeance Models of Salient-pole Generators . 2 1.3 Core Loss Prediction in Large Hydropower Generators . 3 1.4 Form Factors of Salient Pole Shoes . 3 1.5 Analysis of a PM Generator with Two Contra-rotating Rotors . 4 1.6 Electromechanical Transients - Simulation and Experiments . 4 1.7 Outline of the Thesis . 5 2 Theory . 7 2.1 Salient-pole Synchronous Generators . 7 2.1.1 Main Construction Elements . 7 2.1.2 Grid-connected Operation . 9 2.2 Equivalent Circuit Generator Model . 10 2.2.1 P.U. Electrical Equations . 11 2.3 Finite Element Generator Model . 13 2.3.1 Calculation Geometry and Material Property Assignment . 13 2.3.2 Field Equation Formulation . 14 2.3.3 Finite Element Discretization . 16 2.3.4 Boundary Conditions . 17 2.3.5 Calculation of Air-gap Torque and Induced EMF . 18 2.4 Coupled Field-circuit Models . 19 2.4.1 Coupling Equations for Circuit-connected Conductors . 19 2.4.2 Rated Voltage No-load Operation Model . 20 2.4.3 Balanced and Unbalanced Load Models . 23 2.4.4 Grid-connected FE Model with Mechanical Equation . 25 3 Applications of Permeance Models of Salient-pole Generators . 27 3.1 Previous Work . 27 3.2 Permeance Model Implementation . 28 3.2.1 Coordinate System . 28 3.2.2 Field and Armature MMF Functions . 29 3.2.3 Pole Shape Permeance Function . 31 3.2.4 Saturation and Stator Slot Permeance Functions . 31 3.3 Damper Winding MMF and Circuit Equations . 33 3.3.1 Flux Density Harmonics . 34 3.3.2 Unitary Damper Loop MMF Functions . 36 3.3.3 Calculation of Damper Loop Currents . 37 vii 3.3.4 Resultant Damper MMF . 40 3.4 Selected Results . 41 3.4.1 THD of the Open-circuit Armature Voltage Waveform . 41 3.4.2 Damper Bar Currents at Rated Load Operation . 42 3.4.3 Reduction of the UMP by Parallel Armature Circuits . 43 4 Core Loss Prediction in Large Hydroelectric Generators . 45 4.1 Previous Work . 45 4.2 Iron Loss Estimation . 45 4.2.1 Loss Separation . 45 4.2.2 Rotational Losses . 46 4.3 Study Summary . 48 4.4 Selected Results . 50 5 Form Factors of Salient Pole Shoes . 53 5.1 Background . 53 5.2 Pole Shoe Form Factors . 54 5.3 Study Summary . 55 5.3.1 Pole Face Contours . 56 5.3.2 Pole Shoe Variables . 57 5.4 Selected Results . 58 5.4.1 Effect of Pole Face Contour . 58 5.4.2 Linear Models with Saturation Considered . 59 5.4.3 Perspectives on Pole Shoe Shape Selection . 60 6 Analysis of a PM Generator with Two Contra-rotating Rotors . 61 6.1 Previous Work . 61 6.2 Generator Topology . 61 6.2.1 Dual Contra-rotating Rotor Topology . 61 6.2.2 Reference Machine Topologies . 62 6.3 Selected Results . 63 6.3.1 Characterization of the Inter-rotor Cross Coupling . 63 6.3.2 Synchronized Contra-rotating Load Operation . 66 7 Electromechanical Transients - Simulation and Experiments . 69 7.1 Previous Work . 69 7.2 Rotor Angle Oscillations . 69 7.2.1 The Swing Equation . 70 7.2.2 Damping and Synchronizing Torques . 71 7.3 Study Summary . 73 7.3.1 Torque Coefficient Determination from a Field Model . 73 7.3.2 Experimental Study . 73 7.4 Selected Results . 74 7.4.1 Comparison of Field and Circuit Model Responses . 74 7.4.2 Experimental Study . 76 8 Conclusions . 81 9 Suggested Future Work . 83 10 Summary of Papers . 87 viii 11 Summary in Swedish . 93 Acknowledgment . 95 References . 97 ix List of Symbols and Abbreviations Fields Symbol Unit Definition A Tm Magnetic vector potential B T Magnetic flux density / induction H A/m Magnetic field J A/m2 Current density Scalars Symbol Unit Definition Az Tm Z-component of magnetic vector potential bp m Pole body width Bgm T Peak value of air-gap flux density wave Bmax T Peak flux density ΔBr T Radial flux density distortion ei V Induced EMF in armature phase i (i = a,b,c) (field model) ed p.u. Direct-axis armature voltage (equivalent circuit model) e fd p.u. Field voltage (equivalent circuit model) eq p.u. Quadrature-axis armature voltage (equi- valent circuit model) E V or p.u. Internal EMF f Hz Electrical frequency fa - Pole taper f0 Hz Hunting frequency hpp m Pole shoe height H s Inertia constant xi Scalars (continued) Symbol Unit Definition i j A Current in armature phase j ( j = a,b,c) id p.u. Direct-axis armature current i fd p.u. Field winding current (equivalent circuit model) iq p.u. Quadrature-axis armature current i1d p.u. Direct-axis damper current i1q p.u. Quadrature-axis damper current If A Field current J kgm2 Moment of inertia 4 kc Sm /kg Classical loss coefficient kd - Direct-axis armature pole shoe form factor 3 −0.5 −1 kE Am V kg Excess loss coefficient k f - Field winding pole shoe form factor 4 −1 kH Am (Vskg) Hysteresis loss coefficient kq - Quadrature-axis armature pole shoe form factor Kd p.u. torque / Damping torque coefficient (rad/s) Ks p.u. torque / rad Synchronizing torque coefficient le m Effective machine length Lad p.u. Direct-axis mutual inductance Laq p.u. Quadrature-axis mutual inductance Le H Armature end-winding leakage inductance L fd p.u. Field leakage inductance Ll p.u. Armature leakage inductance L1d p.u. Direct-axis damper winding leakage in- ductance L1q p.u. Quadrature-axis damper winding leakage inductance Ma A·turns Armature winding MMF MD A·turns Damper winding MMF Mf A·turns Field winding MMF xii Symbol Unit Definition n rpm Rotational speed Nd - Number of damper bars per pole Nf - Number of field winding turns per pole Np - Pole pair number q1 - Number of stator slots per pole and phase ptot W/kg Total specific iron loss Padd−dyn % Fractional loss increase due to rotational and harmonic fields Padd−rot % Fractional loss increase due to rotational fields Ra p.u. Armature phase resistance Rc Ω Inter-pole end-ring resistance Re Ω Armature end-winding resistance R fd p.u. Field winding resistance R1d p.u. Direct-axis damper winding resistance R1q p.u. Quadrature-axis damper winding resis- tance S m2 Conductor area Te Nm or p.u. Electrical torque ΔTe p.u. Change in electrical torque Un V or p.u. Rated terminal voltage (RMS, line-to-line) V V Electric potential / applied voltage (field model) Xd Ω or p.u. Direct-axis synchronous reactance Xq Ω or p.u. Quadrature-axis synchronous reactance Zb Ω Damper bar impedance Γ - Degree of rotation δ Elect. rad. Rotor (load) angle (Chapters 2 and 7) δ m Air-gap length (Chapter 5) Δδ Elect. rad. Rotor angle deviation θ Elect. rad. Electrical angular coordinate θm Mech. rad. Mechanical angular coordinate Λ Vs/(Am2) Air-gap permeance function Λecc - Eccentricity permeance function xiii Scalars (continued) Symbol Unit Definition −1 ΛP m Pole-shape permeance function Λsat - Saturation permeance function ΛSslot - Stator slot permeance function μr - Relative magnetic permeability μ0 Vs/(Am) Permeability of free space ν m/H Magnetic reluctivity σ S/m Electric conductivity τD s Damping time constant τds - Damper slot pitch τp m Pole pitch τpc m/- Concentric pole shoe width τpp m/- Pole shoe width τs m Stator slot pitch φ Elect.
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