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 magnetic 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 publication 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 simulations”, 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

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 (equivalent 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 inductance

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 resistance 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. rad. Power factor angle Ψ Wb turns / p.u. Flux linkage

Ψad p.u. Direct-axis mutual (air-gap) ﬂux linkage Ψaq p.u. Quadrature-axis mutual (air-gap) ﬂux linkage

Ψd p.u. Direct-axis armature winding flux linkage Ψ fd p.u. Field winding flux linkage Ψq p.u. Quadrature-axis armature winding flux linkage

Ψ1d p.u. Direct-axis damper winding ﬂux linkage Ψ1q p.u. Quadrature-axis damper winding ﬂux linkage ω Elect. rad/s Electrical angular frequency

ωm Mech. rad/s Mechanical angular frequency ωms Mech. rad/s Synchronous angular frequency ωs Elect. rad/s Synchronous angular frequency ω0 Mech. rad/s Hunting angular frequency Δω p.u. Angular frequency (speed) deviation

xiv Abbreviations

AC Alternating Current DC Direct Current EC Equivalent Circuit EMF Electromotive Force FE Finite Element FEA Finite Element Analysis FEM Finite Element Method MMF Magnetomotive Force PM Permanent Magnet p.u. Per Unit SiFe Silicon-Iron alloy SMIB Single Machine Inﬁnite Bus THD Total Harmonic Distortion UMP Unbalanced Magnetic Pull

1. Introduction

1.1 Background Large-scale exploitation of hydropower resources in Sweden started in the first decades of the 20th century. The clean and controllable supply of power from hydropower plants was vital for the electrification of the society and the development of the Swedish industry throughout the century. Today, hydropower still remains an essential ingredient in the national energy mix, and accounts for 46%1 of the country’s annual electricity production of 145 TWh [1]. Reli- able and efficient operation of the hydropower plants is crucial, and this calls for safe and professionally designed plant components. The generator is one of the key components of a hydropower plant, since it constitutes the site for the conversion between mechanical and electrical energy. The work presented in this doctoral thesis is a part of a research program devoted to hydropower generator technology at Uppsala University, initiated by The Swedish Hydropower Centre (Svenskt Vattenkraftcentrum, SVC). SVC is a national collaboration platform for power suppliers, manufacturers of hydropower equipment, consulting agencies, The Swedish Energy Agency, The Swedish National Grid Agency and five technical universities. SVC’s vision is to promote the provision of qualified human resources to all branches of the national hydropower industry in order to secure an efficient and safe production of hydro electricity in the future, and to secure a main- tained dam safety2. The scientific aim of the doctoral project was to address subjects associated with electromagnetic analysis of synchronous machines with a particular emphasis on grid-connected operation of hydroelectric generators. Because of the general formulation of scope of the project, the work comprises a set of diversified studies. The field of synchronous machine analysis encompasses both electric, magnetic, thermal and mechanical aspects. As the title of the thesis indicates, the work presented here is largely limited to electric and magnetic phenomena. Electromagnetic analysis is here defined as the study of electric currents, magnetic fields, electric voltages and power flows in an apparatus during steady- state and transient operating conditions. The scope of the work is somewhat

1Calculated average between the years 2000-2008. 2www.svc.nu. Accessed on January 12 2011.

1 extended with a simple model of electromechanical interaction in studies on synchronous machine hunting (see Chap. 7). The studies that are presented in this comprehensive summary can be di- vided into five main subjects. The ten papers, which constitute the founda- tion of the thesis, are in turn subordinate to either of these five subjects. The first main subject will be referred to as applications of permeance models of salient-pole generators. A series of papers (I, II, and III) fall under this subject. The second subject is core loss prediction. A single publication (Paper IV) be- longs to this category. The third subject is entitled form factors of salient pole shoes, and is represented by Paper V. The fourth main subject concerns a non-conventional permanent magnet (PM) generator topology and is labeled analysis of a PM generator with two contra-rotating rotors. Paper VI em- bodies this subject. The final subject is electromechanical transients. Various aspects of this topic are discussed in Papers VII, VIII, and IX. The last paper, Paper X, deals with design considerations for an experimental generator setup and is the only publication not to fall under any of the main subjects. In spite of the diversity of the addressed problems, some studies that belong to different main subjects share common denominators. For instance, the damper winding end-ring connection is discussed both in terms of its impact on the armature voltage waveform distortion (Papers I and X) as well as its mitigating effects on rotor angle oscillations (Papers VII and VIII). Moreover, all studies but one (Paper VI), are concerned with the conventional vertical- axis machine topologies that are typically encountered in large hydropower plants. In the following sections, the five main subjects are briefly introduced and the objectives of the individual studies are stated. The chapter is concluded with a presentation of the outline of the thesis.

1.2 Applications of Permeance Models of Salient-pole Generators The rotating field method determines the air-gap flux density in an electrical machine as the product of a magnetomotive force (MMF) and a permeance function. A calculation scheme that uses this approach to derive the air-gap flux density is referred to as a permeance model. In combination with circuit equations that represent the damper winding, it is possible to determine approximately the full air-gap flux density waveform, including the harmonic contribution of the damper reaction [2]. Different applications of the permeance modeling technique for synchronous generators with salient, laminated poles are explored in Papers I, II, and III. The aim of the study presented in Paper I was to develop a permeance model suitable for the calculation of open-circuit armature voltage harmonics. In the study summarized in Paper II, the objective was to explore the applicability

2 of the model in studies of steady balanced and unbalanced load operation. Finally, in Paper III, the objective was to assess the usefulness of the permeance model in predicting the effects of parallel armature circuits on a steady unbalanced magnetic pull (UMP).

1.3 Core Loss Prediction in Large Hydropower Generators In the conversion between mechanical and electrical energy that takes place in a generator, a certain amount of power is continuously converted to heat through various dissipation mechanisms. This is the loss of the conversion scheme. The term core loss refers to the power loss that is developed in the iron core of the stator. Core losses are fundamentally attributable to the eddy currents that arise in the stator laminations upon exposure of a time-varying magnetic flux. Besides the macroscopic eddy current loss, the internal magnetic domain structure of the soft ferromagnetic steel used in stator laminations gives rise to additional loss components - hysteresis and excess losses - that also add to the core loss. As high machine efficiency is a prioritized objective, iron loss studies continuously generates many scientific papers. Recently addressed problems in this field include improved material modeling [3, 4], the influence of bidirectional magnetic fields (“rotational losses”) [5–8], time-saving analytical loss calculations [9–11], and loss predictions from 3-D magnetic field computa- tions [12]. The goal of the project that resulted in Paper IV was to evaluate the core losses in twelve large hydroelectric generator topologies, using iron loss prediction models of varying complexity. An equally important goal was to assess the importance of the additional loss introduced by bidirectional magnetic fields in these machines.

1.4 Form Factors of Salient Pole Shoes Hydroelectric generators are typically equipped with salient rotor poles, and the shape of the pole shoe directly affects the appearance of the air-gap ﬂux density waveform. In order to determine the inductances of fundamental wave equivalent circuit representations of synchronous machines, the correlation between the fundamental wave amplitude and the maximum wave amplitude is required. To this end, pole shoe form factors are introduced in the mathematical expressions of the different machine inductances. Form factors are deﬁned for three reference cases of magnetic excitation and can be said to characterize the pole shoe shape.

3 In the technical literature, many studies on salient pole shoe design and form factors date from the ﬁrst part of the 20th century [13, 14]. These early studies are founded on analysis techniques that neglect iron saturation and higher order harmonics of the impressed MMF waveforms. The validity of the results for all the practical pole face contour designs that are in use is also unclear. Even so, the results of these studies are usually cited in modern textbooks of synchronous machine design [15]. The primary objective of the work presented in Paper V was to study the effect of iron saturation on pole shoe form factors. The study was however extended to embrace a more general comparison of different pole face contour designs from a form factor perspective. Moreover, the harmonic imprint of different salient pole shoes on the air-gap ﬂux density waveform was considered.

1.5 Analysis of a PM Generator with Two Contra-rotating Rotors Hydraulic turbine concepts with two contra-rotating impellers have been presented both for use in small-scale hydropower plants [16] and in tidal energy conversion schemes [17]. The benefits of employing a turbine with two contra- rotating stages include a near-zero reaction torque on the support structure, near-zero swirl in the wake and high relative rotational speeds. For a complete energy conversion system employing such a turbine, a generator with two contra-rotating rotors and one single stator winding is an interesting, but unexplored machine concept. Caricchi et al. performed one of the rare studies on this particular type of machine topology [18]. Their communication reports of an axial flux motor with two contra-rotating rotors designed to operate in a ship propulsion drive. Motivated by the possible applicability in small-scale hydro schemes as well as the relative sparsity of available information on electrical machines with contra-rotating rotors, a research project aimed at exploring further the operating characteristics of this machine topology was initiated. A selection of findings are reported in Paper VI.

1.6 Electromechanical Transients - Simulation and Experiments During perfect steady-state operation of a grid-connected synchronous generator, the speed of the rotor is identical to the synchronous speed dictated by the mains frequency. The term electromechanical transient will be used here

4 to denote temporary rotor speed excursions around the synchronous speed, and the associated fluctuations in electrical torque. From a physical perspective, the grid-connected generator is in close analogy with a mechanical arrangement consisting of a discrete mass attached to a wall through a spring and a damper. Electric spring and damper action during rotor swings results from the interaction between the rotor and stator circuits, and is described in terms of synchronizing and damping torques. Because of their importance for stable operation of inter-connected power systems, damping and synchronizing torques of synchronous machines have been extensively studied in the past [19–23]. While previous studies have addressed damping and synchronizing torque calculation with analytical formulae, the objective of the study presented here was to determine these machine properties from numerical field simulations. To this end, a coupled field-circuit model of the classical single machine infinite bus (SMIB) system was developed. Papers VII and IX describe the outcome of the numerical experiments performed with this model, while Paper VIII is concerned with the experimental determination of the natural damping properties of a laboratory generator. Particular attention is devoted to the effect of different damper winding configurations.

1.7 Outline of the Thesis Due to the diversity of the research studies, the author has preferred to devote one chapter to each main subject. Each subject chapter contains a description of the method of analysis and a few, selected results. This unconventional outline was deliberately chosen to facilitate for readers who take interest in one particular subject. The first part of Chapter 2 contains a short introduction on the function and the main construction elements of salient-pole synchronous generators. The second part of Chapter 2 discusses equivalent circuit (EC) and finite element (FE) models of synchronous electric machines. The chapter is then concluded with a presentation of the coupled field-circuit models that were used in the different studies. Next, Chapters 3-7 are devoted to the respective main subjects. Permeance model applications are treated in Chapter 3, core losses in Chapter 4, and pole shoe form factors in Chapter 5. Chapter 6 and Chapter 7 are devoted to analysis of a PM generator with two contra-rotating rotors and electromechanical transients respectively. Conclusions are presented in Chap- ter 8 and suggestions for future studies are given Chapter 9.

2. Theory

This chapter is intended to serve two purposes. The first purpose is to provide non-expert readers with some useful notions which will assist digestion of the contents of Chapters 3-7. The second purpose is to provide professional readers with comprehensive mathematical descriptions of the EC and FE models of synchronous generators that have been used in the different studies. In a spirit of compromise between these aims, some general information on EC and FE models of synchronous generators, which the author deemed manda- tory, is also provided. Section 2.1 describes the main construction elements of hydroelectric generators. In Section 2.2, EC models of synchronous generators are briefly discussed. Furthermore, the EC model structure used in Papers VII, VIII and IX is presented. Next, Section 2.3 provides an introduction to FE generator models. Section 2.4 finally presents the mathematical structure of the coupled field-circuit models that have been used in the different studies.

2.1 Salient-pole Synchronous Generators 2.1.1 Main Construction Elements The purpose of a generator is to convert mechanical energy, supplied from a prime mover via a rotating shaft, to electric energy, which is typically fed into the power grid. This electromechanical energy conversion is realized with the magnetic ﬁeld inside the generator acting as an intermediate coupling. Most generators in large hydropower plants are synchronous generators with salient rotor poles. The word “large” here denotes a generator in the MW range. In the past, horizontal-axis units were common, but today, the majority of the hydro generating units are built as vertical-axis machines. The two main parts of a conventional hydroelectric generator are the stator and the rotor. The stator consists of a circular magnetic iron core, constructed from thin silicon steel sheets and supported by a steel frame. The inner stator periphery holds uniformly stamped slots, where a three-phase winding is inserted. This is the armature or stator winding. The winding is typically composed of form-wound copper coils insulated with a high voltage mica-based insulation system. The rotor, or pole wheel, is attached to the rotating shaft. It consists of a frame, an iron ring made from stacked steel sheets, and rotor poles. The rotor

7 Figure 2.1: (a) Axial cross-section of a salient-pole synchronous machine with four poles. 1. Pole body. 2. Pole shoe. 3. Field coils. 4. Stator winding coils. 5. Damper winding. is separated from the stationary stator by an air-gap. The rotor poles, also constructed from laminated steel sheets, hold the field winding, that provides the fundamental magnetic field excitation. Fig. 2.1 shows the axial cross-section of a four-pole synchronous machine with salient poles. The part of the pole which is closest to the air-gap is referred to as the pole shoe. The pole shoes of large synchronous machines typically hold copper or brass bars. This is the amortisseur or damper winding. The bars in adjacent poles can be connected via a short-circuit ring in both machine ends. This configuration is referred to as a complete or a continuous damper winding.1 A damper winding that lacks the inter-pole connection like- wise has many designations in the technical literature. Any of the terms open, incomplete, non-continuous,orgrill damper winding can be used to denote this damper winding configuration. To deal with the asymmetric air-gap produced by the pole saliency, it is convenient to introduce two sets of rotor-fixed reference axes - the direct (d) and quadrature (q) axes (see Fig. 2.1). A d-axis is aligned with the center axis

1Some prefer to refer to this conﬁguration simply as a squirrel cage winding.

8 of a north pole. The q-axes go through inter-polar gaps adjacent to and leading the d-axes.

2.1.2 Grid-connected Operation Most of the global electric energy generation is performed through synchronous generators connected to three-phase alternating current (AC) power grids. The rotational speed, n, of a grid-connected synchronous generator is given by f n = 60 · [rpm], (2.1) Np where f is the grid frequency and Np denotes the number of pole pairs in the generator. n is referred to as the synchronous or rated speed of the unit. During normal load operation, balanced three-phase currents in the armature winding phases produce a magnetic field that rotates at synchronous speed. This field is called the armature reaction. The fundamental waves of the armature reaction and the rotor excitation field have the same number of poles and are at standstill with respect to each other. Through the interaction between these fields, a non-zero synchronous torque is produced which tend to align the fields with each other. During balanced load operation, the angle between the rotor and the armature fields is more or less constant, and the synchronous torque production is manifested as a continuous transfer of power to the AC grid. The steady active and reactive power productions, Pg and Qg, from a synchronous generator are approximately given by 3EU 3 2 1 1 Pg = sinδ + U − sin2δ Xd 2 Xq Xd (2.2) 2δ 2δ 3EU 2 cos sin Qg = cosδ − 3U + . Xd Xd Xq

In the above expressions, the resistive losses in the stator winding are neglected. E is the so called internal EMF (here, an RMS phase quantity in Volts), U is the terminal voltage (RMS phase quantity in Volts), and Xd (Ω) and Xq (Ω) denote the synchronous reactances in the direct- and quadrature axes respectively. δ is the load angle (or rotor angle), and corresponds to the phase angle between the voltages E and U. The function Pg(δ) is called the active power - load angle characteristics of the synchronous generator and is schematically illustrated in Fig. 2.2. During normal operation, the synchronous generator operates at a load angle that is considerably smaller than the critical load angle, δC. The angle δC corresponds to the maximal active power delivery at a given level of excita-

9 Figure 2.2: Active power versus load angle (synchronous generator). tion. The generator is considered to be stable with respect to slow shaft torque 2 or load variations as long as the load angle does not exceed δC [24].

2.2 Equivalent Circuit Generator Model In studies of the electromagnetic interaction between synchronous generators and other electrical equipment, the generators are frequently represented by a set of electrical circuit equations. A long tradition of elaborate refinement and adaptation of such circuit representations to fit almost any problem of interest, makes this the most established and accessible form of generator analysis. A number of factors determine the nature of a generator EC model. Some of the most important factors are briefly discussed in the following.

Nominal or P.U. Representation of Model Variables Model quantities can be represented with physical units (V, A, W e.t.c) or, alternatively, units are eliminated from the calculations by expressing all quantities in terms of fractions of speciﬁed base values. The latter approach is called per unit (p.u.) representation. The p.u. representation is convenient in power systems with many different voltage levels, and also facilitates the comparison of electrical arrangements with dissimilar power ratings.

Winding Representation The armature can be modeled by its three physical stationary armature phases A, B, and C or, alternatively, by means of fictitious rotor-fixed windings. A stator-fixed representation is usually referred to as a phase domain model, while the rotor-fixed representation is called a dq0 or two-axis model.

2This is the static stability of the generator, and is deﬁned as the ability of the generator to remain in synchronism with the power grid when subjected to slow shaft power or load variations.

10 Two-reaction theory, which forms the basis of two-axis representations of synchronous machines, was originally worked out by Blondel [25]. The dq0-representation brings about numerous modeling advantages, such as time-independent circuit inductances and decoupling of the d- and q-axis circuits if iron saturation is neglected. The approach involves the application of the Park transformation to all stator quantities [26]. In power system analysis software, the internal electrical representations of synchronous machines are almost exclusively in dq0-coordinates. Applica- tions of physical armature phase representation in EC models however do exist. A prominent example is the analysis of internal short-circuit faults [27,28]. The representation of the rotor windings primarily concerns the structure of the equivalent damper winding circuits [29, 30]. The level of modeling detail should be adjusted to the problem at hand and the required accuracy of the results.

Consideration of Non-Linear and Harmonic Effects Most dq0-models are fundamental wave models, that is, they only consider the dominating space fundamentals of the magnetic ﬂux density waves inside the generator. Linear EC models either neglect iron saturation or represent the effect by parameter values appropriate at the studied point of operation (“saturated parameters”). Iron saturation can alternatively be accounted for with reﬁned iterative methods [31]. It is also possible to account for some harmonic effects on generator performance [32].

Choice of Independent State Variables The selection of independent state variables depends on the circuit representation [33]. For fundamental parameter circuit representations, winding ﬂux linkages and currents are preferably used. In some applications, a mixed or “hybrid” choice of independent variables may be the best choice [34].

2.2.1 P.U. Electrical Equations EC generator models were used in the studies presented in Papers VII, VIII and IX. The employed circuit model was a dq0-model with one damper circuit in each axis, which is customary for rotor angle stability studies of hydroelectric generating units. The model was represented in the conventional Lad-base reciprocal p.u. system. In Paper IX, the circuit parameters were derived from FE simulations of standard parameter determination tests [35]. In Papers VII and VIII, the parameters were calculated from generator design data. The employed analytic parameter calculation formulae were taken from [15] and [36]. The p.u. electrical equations of the EC model are listed below. All variables, including time, are given in p.u. Zero-sequence equations are omitted, since only balanced generator operation was considered in the studies where a circuit model was used. The system of differential-algebraic equations used to

11 Figure 2.3: Circuit representation of voltage and ﬂux linkage equations. Top: d-axis circuit. Bottom: q-axis circuit. simulate the SMIB systems of Papers VII and IX, can be derived from the expressions below, except for the two equations that describe the grid coupling. These equations are summarized in Paper IX. The listed equations can be represented with the equivalent d- and q-axis circuits shown in Fig. 2.3. The notation follows that used in IEEE Std. 1110- 2002 [29], but for completeness all symbols are also described in the List of Symbols.

Stator voltage equations dΨ e = d − Ψ ω − R i (2.3) d dt q a d dΨ e = q + Ψ ω − R i (2.4) q dt d a q Rotor voltage equations dΨ e = fd + R i (2.5) fd dt fd fd dΨ 0 = 1d + R i (2.6) dt 1d 1d dΨ 0 = 1q + R i (2.7) dt 1q 1q Stator ﬂux linkage equations

Ψd = −(Lad + Ll)id + Ladi fd + Ladi1d (2.8) Ψq = −(Laq + Ll)iq + Laqi1q (2.9)

12 Rotor ﬂux linkage equations

Ψ fd = −Ladid +(Lad + L fd)i fd + Ladi1d (2.10) Ψ1d = −Ladid + Ladi fd +(Lad + L1d)i1d (2.11) Ψ1q = −Laqiq +(Laq + L1q)i1q (2.12)

Air-gap torque Te = Ψadiq − Ψaqid (2.13)

2.3 Finite Element Generator Model In equivalent circuit models, the inherently distributed nature of the electromagnetic interaction inside the generator is “lumped” into a fairly limited set of equations. We here define a field generator model as a model that determines the electrical performance directly from the magnetic field distribution in the active parts (stator, air-gap, rotor) of the generator. The magnetic field distribution is determined from Ampères law, which needs to be appropriately formulated for the application at hand. The problem of solving the field equations by means of digital computing can then be tackled with a variety of numerical methods. For electromagnetic analysis of electrical machines, the Finite Element Method (FEM) has emerged as the most widely applied numerical method. Its popularity is linked to its ability to handle the complicated calculation geometries presented by rotating machinery [37]. FEM was originally used to study problems in structural mechanics. Its employment for the solution of the electromagnetic vector field problems presented by electric machinery became widely diffused in the 1980’s [38, 39]. Today, FE analysis is more or less a standard tool in electrical machine design, and the method can be used to study problems of both electromagnetic, thermal, mechanical and coupled (“multiphysics”) nature. There exists a number of commercial FE software packages specifically designed for analysis of electromagnetic field problems3.

2.3.1 Calculation Geometry and Material Property Assignment The problems addressed in this thesis have been analyzed with a two- dimensional field model. The magnetic field was determined with FEM, and therefore the terms field model and FE model will be used interchangeably to denote this generator modeling approach. The two-dimensionality of the

3http://www.ansys.com/Products/Simulation+Technology/Electromagnetics (accessed on Jan- uary 19 2011) http://www.cedrat.com/en/software-solutions/ﬂux.html (accessed on January 19 2011) http://www.comsol.com/products/acdc/ (accessed on January 19 2011)

13 Iron

Conductor

Air

Figure 2.4: Calculation geometry example (one pole pitch of a hydroelectric generator). model means that it is assumed that the magnetic field in the generator is perfectly parallel to the axial cross-section of the generator. For most problems, symmetry conditions allow for a radical reduction of the region where the magnetic field needs to be evaluated. Fig. 2.4 shows an example of such a reduced calculation geometry, corresponding to one pole pitch of a hydroelectric generator. Lines demarcate different subdomains of the calculation geometry. These regions represent the physical parts of the generator, such as rotor iron core, field winding conductors, stator teeth and stator winding conductors. The subdomains are allocated material properties relevant for the electromagnetic field problem, such as electric conductivity, σ, and relative magnetic permeability, μr. Non-linear ferromagnetic material properties are represented by single-valued B(H)-curves.

2.3.2 Field Equation Formulation The FE code used in the thesis solves Ampère’s law for the magnetic vector potential, A. In the 2-D formulation of the problem, A has only an axial component, denoted Az. Az is related to the Cartesian components of the ﬂux density B according to ∂A B = z (2.14) x ∂y ∂A B = − z (2.15) y ∂x Bz = 0. (2.16)

Hence, there is no axial component of ﬂux density, as dictated by the 2-D nature of the ﬁeld problem formulation. The magnetic vector potential inside

14 the cross-section of the generator is assumed to be governed by the following partial differential equation4:

In conductor subdomains: ∂ ∂ ( , , ) ∂ ∂ ( , , ) ∂ ( , , ) ∂ ( , , ) ν Az x y t + ν Az x y t = σ Az x y t + σ V x y t ∂x ∂x ∂y ∂y ∂t ∂z Elsewhere: ∂ ∂A (x,y,t) ∂ ∂A (x,y,t) ν z + ν z = 0 ∂x ∂x ∂y ∂y (2.17)

Here, 1 ν = , (2.18) μrμ0 denotes the reluctivity, μ0 is the permeability of free space and V is the electric potential. The right-hand side of (2.17) is the total current density. As seen in the equation, only subdomains that correspond to conductors are allowed to have a non-zero current density. The conductor subdomains are therefore referred to as the sources of the ﬁeld problem. The total current density typically depends on the nature of the conductor subdomain and the circuit to which it is connected. Additional coupling equations are typically required to completely specify the ﬁeld problem in a conductor.

Equation (2.17) warrants the following supplementary remarks:

σ ∂V (x,y,t) 1. The term ∂z denotes the applied current density while the term ∂ ( , , ) σ Az x y t ∂t denotes the induced current density. 2. The applied current density plays a key role when one or several conductors are connected in series. In such a situation, the induced current density may not be equal in the different conductors, but the net current must be the same in all conductors. The electric charge distribution introduced by the applied current density term then ensures that this condition is met [41]. The quantity V , which is referred to as the applied voltage, is constant over the conductor subdomain area, and is directly proportional to the potential difference between the (fictitious) ends of the conductor. 3. If the dynamic interaction between the magnetic field and the conductors is to be disregarded, the conductor currents may be specified by pre- determined functional expressions. This is equivalent to connecting the conductors to ideal current sources.

4For a full derivation of this equation see [40] or any textbook on ﬁnite element analysis of electrical machines.

15 ∂Az(x,y,t) 4. The term ∂t only appears explicitly in conductor subdomains treated as solid conductors [42], where eddy currents provoke a non-uniform spatial current distribution in the conductor cross-section. 5. In the FE models used to study the subjects of this thesis, all conductor subdomains have been treated as ﬁlamentary conductors. That is, the current calculated in a given time step is assumed to be uniformly distributed across the subdomain. In the coupled ﬁeld-circuit models to be described subsequently, the induced current density is nevertheless considered on average terms in additional coupling equations.

2.3.3 Finite Element Discretization There exist different techniques to solve (2.17). The starting point for most FE solvers is to reformulate the problem on a variational form. In essence, this means that the problem of finding a function Az(x,y,t) that satisfies (2.17) is transformed into the problem of finding a function Az(x,y,t) which is a stationary point to some functional, F . For the problem at hand, F is typically set to the electromagnetic energy of the system: B F = H · dB − JA dS. (2.19) 0 S

Here, H denotes the magnetic field, J is the current density, and S refers to the area of the calculation geometry. The search for a solution is carried out with ∗ trial functions Az , N ∗ Az(x,y,t) = ∑ A jϑ j(x,y,t), (2.20) j=1 where A j are unknown coefficients and ϑ j are called base functions. The fundamental principle of the finite element method is to subdivide the calculation geometry into many small, non-intersecting elements and make use of base functions that are non-zero only within a single element. If the elements are sufficiently small, the base functions of (2.20) can be very simple, without much loss of computational accuracy. Typically, base functions that are linear or quadratic functions of the spatial coordinates x and y are used. The elements in 2-D FEM are usually shaped as triangles and the vertices of these triangles are referred to as nodes. The complete body of elements is called a mesh. A mesh of triangular elements is illustrated in Fig. 2.5. In the FE formulation of the variational problem, the coefficients A j denote the magnetic vector potentials in the nodes of the mesh. With a trial solution on the form presented in (2.20), it can be shown that the variational problem transforms into a system of ordinary differential-algebraic equations, with the node potentials as the unknown variables. Accordingly, an appropriate numer-

16 Figure 2.5: Triangular mesh in a part of the calculation geometry.

Figure 2.6: Boundary conditions for the example calculation geometry. ical integration method can provide a solution to the original ﬁeld problem in (2.17). If the calculation geometry contains domains with non-linear magnetic properties, the ﬁeld solution in every time step is computed through an iterative procedure that determines the element reluctivities.

2.3.4 Boundary Conditions For the field problem to be completely specified, the outer borders of the calculation geometry need to be assigned with appropriate boundary conditions. Fig. 2.6 exemplifies two boundary conditions that are frequent in finite element analysis (FEA) of electrical machines - the Dirichlet and the periodic boundary condition. A homogeneous Dirichlet boundary condition sets Az to 0. This is equivalent to consider the material external to the boundary to have zero relative

17 permeability (a perfect “magnetic insulator”). The periodic condition exploits the repetitive features of the magnetic ﬁeld inside the machine, and relates the values of Az on two boundaries. In Fig. 2.6, Az on the upper boundary is equal in magnitude but opposite in sign to Az on the lower boundary. Also indicated in Fig. 2.6 is a sliding mesh condition in the middle of the air-gap. This condition is used in time-stepped simulations to mimic rotor motion. In essence, the sliding mesh condition is the intersection between the interfaces of the separately meshed stator and rotor. The potentials of the rotor and stator nodes on the intersection are found through an interpolation procedure. This approach allows for the use of a variable integration time step.

2.3.5 Calculation of Air-gap Torque and Induced EMF It is possible to derive many different electric and magnetic quantities from the ﬁeld solution. Here, the expressions for air-gap torque and induced winding EMF are provided. The air-gap torque of the ﬁeld model is of relevance for Papers VI, VII, VIII, and IX. The induced winding EMF formula was used in the studies summarized in Papers I and VI. The air-gap or electrical braking torque in the generator is given by

Te = ler0 σt dγ, (2.21) Γ0 where le is the effective machine length, Γ0 is an arc in the air-gap, r0 is the arc radius and σt is the tangential stress. σt is given by 1 σt = BrBt , (2.22) μ0 where Br and Bt denote the radial and tangential ﬂux density components respectively. The magnetic ﬂux crossing a surface of effective length le and spanning between the points (x1,y1) and (x2,y2) is

Φ = le · (Az(x1,y1) − Az(x2,y2)). (2.23)

The ﬂux linkage, Ψ, of an arbitrary machine winding can hence be calculated from the 2-D ﬁeld solution as l Ψ = e ∑ A dS − ∑ A dS , (2.24) + z − z S n+ S n− S where n+ and n− are the total number of positively and negatively oriented winding conductors respectively, and S+ and S− are the corresponding conductor areas. It is assumed that S+ = S− = S.

18 The induced winding EMF is derived from the ﬂux linkage as dΨ e = − . (2.25) w dt

2.4 Coupled Field-circuit Models The conductors in a generator field model are inter-connected to form complete windings. The terminals of the field and armature windings are additionally connected to external circuits. As the inclusion of conductor subdomains in windings and circuits affects the conductor currents, additional coupling and circuit equations are required for the field problem to be completely specified in these subdomains. A model where field and circuit equations are solved simultaneously to predict the behavior of an electric apparatus is usually referred to as a coupled field-circuit model. This section provides the circuit equations for the coupled field-circuit models used in the different studies of the thesis. The coupling equations needed to associate a set of conductor subdomains to a winding are also given.

2.4.1 Coupling Equations for Circuit-connected Conductors For a conductor subdomain that is a part of an electric circuit, the field equation (2.17) in that subdomain is supplemented with the following coupling equations dAz σ dS − σψc = 0 (2.26) Sc dt ∂V σψ + S σ c + I = 0, (2.27) c c ∂z where Sc denotes the conductor area, Vc the applied conductor voltage, and I is the current in the conductor. ψc is the induced conductor EMF integrated over the conductor surface. The variables I and Vc needs to be determined from additional circuit equations, to be presented subsequently. The structure of (2.26) and (2.27) is the same for all conductor subdomains that are connected to circuits, regardless if the conductor is a part of the field, damper or armature winding. The exact formulation of the additional circuit equations for the field, damper and armature windings depends on the studied problem, as discussed in the following. Before the circuit equations are introduced, we state the expression for the total electric potential difference across a winding of series-connected con-

19 ductors:

Vw = le ∑ Vc − ∑ Vc . (2.28) c∈C + c∈C −

C + here denotes the set of positively oriented conductors, and C − is the set of negatively oriented conductors in the winding.

2.4.2 Rated Voltage No-load Operation Model Rated voltage no-load operation was studied in Papers I, IV and X. Simulation of rated voltage operation at no-load implies consideration of the requirement e2 + e2 + e2 a b c = U , (2.29) 2 n where ea, eb, and ec denote the induced armature phase EMFs and Un is the rated line-to-line voltage of the generator. The ﬁeld voltage is adjusted such that (2.29) is met. A short numerical transient is to be expected before the problem converges.

Field Circuit Equation The additional circuit equations that complete the problem speciﬁcation in ﬁeld conductor subdomains at rated voltage no-load operation are

u fd0 −Vfd = 0 (2.30) i f + − i f − = 0. (2.31) u fd0 is the field voltage at no-load operation at rated armature voltage and speed and Vfd is the potential drop across the entire field winding. Vfd effectively provides the coupling to (2.17) and (2.26) - (2.27) through (2.28). i f + and i f − denote the currents in conductor subdomains on opposite sides of the pole body. The effects of end winding leakage flux are neglected.

Damper Circuit Equations The damper circuit equations are based on a work by Shen and Meunier [43]. Deﬁnitions of relevant quantities are shown in Fig. 2.7. To state the circuit equations on a compact form, the following column vectors are introduced: T i =[i1 i2 ... in] (2.32) T j =[j1 j2 ... jn] (2.33) T Vb =[Vb1 Vb2 ... Vbn] (2.34) T ve =[ve1 ve2 ... ven] . (2.35)

20 (a)

(b) Figure 2.7: Damper winding equations in the field model. (a) Definition of bar and end-ring currents. (b) Definition of bar potentials and end-ring voltage drops.

21 Here, the integer n denotes the number of damper bars considered in the calculation geometry. For generators with integral slot armature windings, 2N (continuous damper winding) n = d (2.36) Nd (non-continuous damper winding), where Nd denotes the number of damper bars per pole. From Fig. 2.7, the following relations can be established between the bar current vector i, the end-ring current vector j, the bar potential vector Vb and the end-ring voltage vectorve: i = MTj (2.37) MVb = 2ve (2.38) ve = Red j. (2.39)

M denotes the (n × n) matrix ⎡ ⎤ 1 −10...... 0 ⎢ ⎥ ⎢ 01−10... 0 ⎥ ⎢ ⎥ ⎢ − ... ⎥ ⎢ 001 10 ⎥ = ⎢ ⎥ M ⎢ ...... ⎥ (2.40) ⎢ . .01 . . ⎥ ⎢ ...... ⎥ ⎣ ...... ⎦ −10...... 01 and Red denotes the diagonal (n × n) matrix ⎡ ⎤ R 0 ...... ⎢ e1 ⎥ ⎢ ... ⎥ ⎢ 0 Re2 00 ⎥ ⎢ . ⎥ = ⎢ . .. .. ⎥. Red ⎢ .0 . . 0 ⎥ (2.41) ⎢ . . . . ⎥ ⎣ ...... 0 ⎦

0 ...... 0 Ren

From (2.37), (2.38), and (2.39), the following relation between i and Vb can be established: 1 − i = MT R 1 M V (2.42) 2 ed b Equation (2.42) is the circuit equation that complete the ﬁeld problem formulation in damper conductor subdomains. End-ring leakage ﬂux is neglected.

22 Figure 2.8: Illustration of armature circuit equations during balanced load operation.

2.4.3 Balanced and Unbalanced Load Models Field models of balanced and unbalanced load generator operation were used in Paper II.

Field and Damper Circuit Equations At balanced and unbalanced load operation, the structure of the field circuit equation is identical to that of (2.30)-(2.31). In (2.30), the term u fd0 is replaced by the field voltage required to produce rated armature voltage at the prescribed load conditions. The field voltage is determined through an iterative procedure. The initial guess is determined from a magnetostatic field solution, as suggested in [44]. The damper circuit equations during balanced and unbalanced load operation are identical to those presented for rated armature voltage no-load operation.

Armature Circuit Equations The armature circuits at balanced load operation are shown in Fig. 2.8. In the figure, subindices a, b and c are used to denote the three armature phases. Re and Le denote the end-winding resistance and inductance of an armature phase, and RL, LL and CL denote resistance, inductance and capacitance of the load. The latter quantities are calculated from the desired active and reactive power delivery at rated terminal voltage. Further, Rs denotes the resistance of an armature phase and is implicitly modeled inside the field problem. The quantities Va,FE, Vb,FE, and Vc,FE finally denote the electric potentials across the armature phases, and are determined according to(2.28). Note that Va,FE, Vb,FE, and Vc,FE are not the terminal voltages, since they exclude the volt-

23 Figure 2.9: Illustration of armature circuit equations during unbalanced load operation. age drop across the end-windings. The location of the generator terminals are marked in the ﬁgure. The circuit equations can be determined from Kirchoff´s circuit laws as: dia dia 1 V , − R i − L − R i − L − i dt − a FE e a e dt L a L dt C a L (2.43) dib dib 1 −Vb,FE + Reib + Le + RLib + LL + ib dt = 0 dt dt CL dib dib 1 V , − R i − L − R i − L − i dt − b FE e b e dt L b L dt C b L (2.44) dic dic 1 −Vc,FE + Reic + Le + RLic + LL + ic dt = 0 dt dt CL

ia + ib + ic = 0 (2.45)

For simulation of unbalanced load operation, a neutral return is introduced in the circuit, as shown in Fig. 2.9. Equation (2.45) is then modiﬁed according to ia + ib + ic = iN. (2.46)

24 Figure 2.10: Illustration of armature circuits when the generator terminals are connected to an inﬁnite busbar.

Furthermore, the loop equation dic dic 1 V , − R i − L − R i − L − i dt − c FE e c e dt Lc c Lc dt C c Lc (2.47) diN 1 −RNiN − LN − iN dt = 0 dt CN must be added for the problem to be completely speciﬁed. Refer to Fig. 2.9 for the introduced notation.

2.4.4 Grid-connected FE Model with Mechanical Equation Coupled ﬁeld-circuit models of grid-connected generators were used in the studies presented in Papers VII and IX.

Field and Damper Circuit Equations The formulation of ﬁeld and damper winding equations when the generator is connected to an inﬁnite busbar are identical to those outlined for balanced impedance load operation.

Armature Circuit Equations The armature circuits at grid-connected operation are illustrated in Fig. 2.10. The sinusoidal voltage sources uBk (k = a,b,c), represent the inﬁnite bus phase voltages. Transformer and tie-line impedance can be considered by introducing supplementary resistive and inductive voltage drops between the generator terminals and the inﬁnite bus voltage sources. The armature circuit equations with negligible tie-line and transformer impedance are:

25 dia Va,FE − Reia − Le − uBa− dt (2.48) dib −V , + R i + L + u = 0 b FE e b e dt Bb

dib Vb,FE − Reib − Le − uBb− dt (2.49) dic −V , + R i + L + u = 0 c FE e c e dt Bc

ia + ib + ic = 0 (2.50)

Mechanical Equation To study rotor angle oscillations in a SMIB system, an equation that governs rotor motion is needed. To this end, the equation dω 1 m = (T − T ), (2.51) dt J m e is added. In (2.51), ωm denotes mechanical angular speed, J the moment of inertia of the rotor, and Tm is the mechanical (drive) torque. The electrical torque is determined through (2.21).

Problem Initiation A number of different mathematical measures need to be taken in order to initiate the grid-connected generator field model with a prescribed, steady point of operation. The most important actions are: 1. The mechanical equation is “de-activated” during the initial numerical transient by setting J to a very large value. After the field solution has con- verged (typically after ~1-2 electrical periods), J is reset to its actual value. 2. The field current is initially regulated with a proportional controller to quickly obtain the desired power factor. After a few electrical periods, the controller is deactivated and the usual, uncontrolled field winding dynamics is restored.

26 3. Applications of Permeance Models of Salient-pole Generators

This chapter reviews the work presented in Papers I, II and III. As stated earlier, the common denominator of these studies is the use of permeance models of salient-pole synchronous generators. The permeance model code implementation is here described in greater detail and a selection of results is discussed. In the review of results from Paper III, only work that entailed contributions from the author is considered.

3.1 Previous Work The construction of the permeance model presented here was primarily in- spired by the works by Traxler-Samek et al. [2] and Knight et al. [45]. Traxler- Samek et al. developed a semi-analytic permeance model intended for use in design calculations. Among the important features of this model is a stator slot permeance function derived from FE calculations, and the use of an air- gap transformation factor that considers the “bending” of flux tubes of higher- order flux density harmonics [46]. The model presented by Knight et al. also relies on permeance functions determined from FEA. In Paper I, a permeance model is used to determine the effect of the damper winding on the open-circuit armature voltage waveform of salient-pole synchronous generators. The literature holds many studies concerned with this particular subject. Walker [47] presented an elaborate theory on the origin and mitigation of problems with slot ripple harmonics. Rocha et al. [48] used an analytical permeance model combined with damper circuit equations to determine the armature voltage harmonic distortion of a salient-pole generator. Keller et al. [49] used a coupled-circuit model derived from stationary FE calculations to determine armature voltage harmonics of salient-pole generators. In a recent paper, Hargreaves et al. [50] used a coupled-field circuit model to predict the effect of damper bar displacement and pole shoe width on the armature voltage waveform distortion. In both [49] and [50], rotational periodicity was utilized to reduce the computation time. In Paper II, the permeance model is used to predict additional damper winding losses during balanced and unbalanced load operation. Pollard [51] derived an analytical expression for the no-load damper loss. Matsuki et al. [52] measured slot ripple frequency damper currents during steady load operation.

27 Knight et al. [53] studied the impact of axial skew and inter-bar contact resistance on the damper loss during short-circuit test conditions. Traxler-Samek et al. [46] determined the damper loss in a large hydroelectric generator at short-circuit test conditions. In Paper III, the permeance model is used to calculate the UMP in a salient- pole generator with two parallel armature circuits. The use of parallel armature winding paths as a means to reduce the resultant UMP in electrical machines is a topic that has received much attention in the past. A full account of papers on this subject is not provided here. Dorell and Smith [54] used a confor- mal mapping technique to formulate circuit equations that were used to study the effect of parallel phase bands and equalizer connections on the UMP in an induction motor. Oliveira et al. [55] studied the impact of equipotential connections (equalizers) between parallel stator circuits on the UMP in large hydroelectric generators.

3.2 Permeance Model Implementation A permeance model is based on the underlying principle that the air-gap ﬂux density Bδ can be deﬁned as

Bδ (θm,t)=Λ(θm,t) · M(θm,t), (3.1) where Λ denotes an air-gap permeance function and M is the air-gap MMF function. θm denotes the mechanical angular coordinate in a stator ﬁxed reference frame and t denotes the time. In the permeance model studied here, the air-gap permeance function is factorized according to

Λ = μ0ΛPΛsat ΛSslot. (3.2)

ΛP here denotes the pole shape permeance function, Λsat the saturation permeance function and ΛSslot the stator slot permeance function. The air-gap MMF function M is the sum of the ﬁeld (f), the armature (a) and the damper winding (D) MMF: M = Mf + Ma + MD. (3.3)

During no-load generator operation, Ma equals zero.

3.2.1 Coordinate System

The rotor is assumed to move with the mechanical angular speed ωm in the clockwise direction. At t = 0, a rotor-fixed interpolar axis at the trailing end of a “north” field pole coincides with the stator-fixed coordinate reference. Fig. 3.1 illustrates the stator-fixed coordinate system.

28 Figure 3.1: Rotor position with respect to the stator-ﬁxed coordinate system.

3.2.2 Field and Armature MMF Functions The ﬁeld winding MMF is deﬁned by the equation

Mf = Nf If Mfn, (3.4) where N f denotes the number of ﬁeld winding turns per pole and I f is the ﬁeld current. The function M fn denotes a unitary trapezoid function given by 4 1 sinγ M = ∑ f sin(nN (θ − ω t)). (3.5) fn π γ 2 p m m f n=1,3,5,... n

1 The spatial appearance of M fn is shown in Fig. 3.2 , where the angle γ f is also deﬁned. M fn has been plotted versus the fundamental electrical angular coordinate, deﬁned as θ = Npθm. (3.6)

The three-phase armature MMF can be expressed as

Ma = ia · ∑ Mˆ n sin(nNpθm)+ = , , ,... n 1 3 5 2π ib · ∑ Mˆ n sin nNp(θm − ) + = , , ,... 3Np (3.7) n 1 3 5 2π ic · ∑ Mˆ n sin nNp(θm + ) , n=1,3,5,... 3Np

1The illustrated function corresponds to the sum of the 25 ﬁrst non-zero terms in the Fourier series expansion.

29 1.2 1 0.8 0.6 0.4 0.2 γ f 0 −0.2 −0.4 −0.6

Unitary field MMF function −0.8 −1 −1.2 0 30 60 90 120 150 180 210 240 270 300 330 360 Electrical angle, θ (°) Figure 3.2: Unitary trapezoid function (used to model the ﬁeld winding MMF).

1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 Unitary armature MMF function −1 −1.2 0 30 60 90 120 150 180 210 240 270 300 330 360 Electrical angle, θ (°) Figure 3.3: Unitary three-phase armature MMF function.

where ia, ib, and ic denote the currents in the winding phases A, B, and C respectively. The coefﬁcients Mˆ n are given by

4 Ns 1 Mˆ n = kd(n)kp(n)ksl(n), (3.8) π 2Np n where Ns denotes the total number of winding turns per phase circuit. Ex- pressions for the distribution, pitch and slot opening factors (symbols kd(n), kp(n), and ksl(n) respectively), can be found in [56]. Fig. 3.3 shows the spatial appearance of a normalized three-phase armature MMF function for balanced fundamental three-phase current supply.

30 50 )

−1 45 40 35 30 25 20 15 10 5 Pole shape permeance function (m 0 0 30 60 90 120 150 180 210 240 270 300 330 360 Electrical angle, θ (°) Figure 3.4: Pole shape permeance function of Generator I in Paper I.

3.2.3 Pole Shape Permeance Function

The pole shape permeance, ΛP, is determined from a stationary FEA according to the following procedure:

1. The geometry corresponding to one pole pitch of studied generator topology is created in the FE software. 2. Rotor and stator regions are assigned with linear magnetic properties and high relative permeability (μr = 10000). 3. Stator and damper slot regions are assigned with linear magnetic properties and high relative permeability (μr = 10000). 4. The air-gap ﬂux density waveform resulting from ﬁeld winding excitation is sampled along a line in the air-gap. This waveform is denoted Blinear. 5. ΛP is obtained by dividing Blinear with Mf . To avoid division with zero, the functional values of ΛP close to the inter-polar gaps are determined through linear extrapolation.

Fig. 3.4 shows the pole shape permeance function of Generator I from Paper I. Notice the peculiar appearance near the pole shoe tips, located close to the angles 40◦, 140◦, 220◦ and 320◦.

3.2.4 Saturation and Stator Slot Permeance Functions The determination of the saturation and slot permeance functions is a chal- lenging task. The saturation permeance is, in part, a result of the saturation in the stator teeth. Hence, from a physical point of view, it is difﬁcult to motivate a separation of the slot and saturation permeance functions.

31 In the mathematical description of the permeance model, the separation is nevertheless necessary. The reason for this is that the permeance variations due to stator slotting is a stator-fixed phenomenon, while the saturation “profile” should move along with the revolving fields. The mathematical treatment therefore becomes overly complicated unless a factorization according to (3.2) is carried out. The author tested different methods to determine Λsat and ΛSslot. The computational procedure presented next was found to give the best results.

Determination of Λsat 1. The generator geometry is created in a FE software. 2. Rotor and stator regions are assigned with non-linear magnetic properties. 3. The flux density waveform resulting from field excitation (no-load study) or a combination of armature and field excitation (load study) is sampled along a line in the air-gap. This waveform is denoted Breal. 4. The ratio Λcomb = Breal/Blinear is calculated. Blinear here denotes the flux density wave used in the extraction of the pole shape permeance function. Λcomb can be regarded as a relative permeance function that holds the combined effects of saturation and stator slotting. 5. The discrete Fourier series expansion of the function 1/Λcomb is calculated. Contributions from space harmonics of orders 6nq1 ± 1 (n = 1,2...) are then subtracted from this function (q1 denotes the number of stator slots per pole and phase). The result is re-inversed and is denoted Λsat . 6. Linear extrapolation is used to smooth the function near the inter-polar gaps.

Determination of ΛSslot 1. One slot pitch of the function Λcomb close to the pole axis is extracted (see Fig. 3.5). 2. The discrete Fourier series expansion of the extracted portion of Λcomb is used to build the function ΛSslot according to the structure of Eq. (6) in Paper I. 3. ΛSslot is ﬁnally normalized such that its maximum value equals one. Accordingly, Λsat must be multiplied with the same normalization factor in order to preserve the requirement that Λcomb = Λsat ΛSslot.

Figs. 3.6-3.7 illustrate the saturation and stator slot permeance functions of Generator I from Paper I, calculated for no-load operation at rated ﬁeld current.

32 Figure 3.5: Combined saturation and stator slot permeance function of Generator I in Paper I at rated no-load operation. The permeance function is illustrated at t = 0.

1.2

0.8

0.6

0.4

0.2 Saturation permeance function

0 0 30 60 90 120 150 180 210 240 270 300 330 360 Electrical angle, θ (°) Figure 3.6: Saturation permeance function of Generator I in Paper I at rated no-load operation. The permeance function is given at t = 0.

3.3 Damper Winding MMF and Circuit Equations The product of the air-gap permeance and the sum of the field and armature MMFs typically result in waveform with a high harmonic contents. Accord- ing to Lenz’s law, any space harmonic that move with respect to the rotor will induce EMFs in conductors installed on the rotor. If the conductors are part of closed circuits, a flow of electric current will result. These reaction currents introduce an additional MMF component that must be considered in the calculation of the air-gap flux density. The permeance model presented here considers induced currents in the damper winding, but not in the field winding. This simplification is motivated

33 1.2

0.8

0.6

0.4

0.2 Stator slot permeance function

0 0 30 60 90 120 150 180 210 240 270 300 330 360 Electrical angle, θ (°) Figure 3.7: Stator slot permeance function of Generator I in Paper I at rated no-load operation. by the limited depth of penetration of air-gap flux density harmonics into the pole shoe. Since the damper winding is located closer to the air-gap than the field winding, the damper reaction also has a decidedly greater impact on the air-gap flux density waveform. The damper MMF is determined from the flux density waveform set up by the sum of the field and armature MMFs. The damper MMF contribution is then added to the original air-gap flux density wave. The saturation permeance is assumed to be unaffected by the supplementary magnetic flux introduced by the damper MMF.

3.3.1 Flux Density Harmonics Below, air-gap ﬂux density harmonics that introduce damper winding currents and are considered in the permeance model are brieﬂy described.

Slot Harmonic Waves The interaction between the stator slot permeance function and the fundamental MMF wave gives rise to the following series of ﬂux density wave pairs: ˆ+ (( + )β + ω ) − ∑ Bn cos nQs Np m nQs mt n=1,2,... (3.9) ˆ− (( − )β + ω ). ∑ Bn cos nQs Np m nQs mt n=1,2,...

In (3.9), βm = θm − ωmt (3.10)

34 is a rotor-ﬁxed angular coordinate, and Qs denotes the total number of stator slots. The waves of (3.9) travel with linear speeds

6q1n ωτp vn = − (3.11) 6q1n ± 1 π with respect to the rotor, and induce EMFs of angular frequencies

ωn = n6q1ω (n = 1,2,...) (3.12) in the damper winding. Here, ω = Npωm denotes the fundamental electrical angular frequency, q1 is the number of stator slots per pole and phase and τ p denotes the pole pitch.

Armature MMF Space Harmonics In addition to the fundamental wave, a balanced three-phase armature MMF gives rise to the following series of space harmonics: ωmt ∑ Bˆncos(nNp(θm + )+ n=5,11,... n (3.13) ωmt ∑ Bˆncos(nNp(θm − ). n=7,13,... n

The waves travel with linear speeds τ n ± 1 v = − p ω (3.14) n π n with respect to the rotor. The + sign refers to harmonic orders n = 5,11,..., while the − sign refers to harmonic orders n = 7,13,.... It can be shown that these waves induce EMFs of angular frequencies (n + 1)ω n = 5,11,... ωn = (3.15) (n − 1)ω n = 7,13,... in the damper winding. Hence, the wave-pair n = 5,7 induce sinusoidal EMFs of frequency 6ω, the wave-pair n = 11,13 induce EMFs of frequency 12ω, and so forth.

Fundamental Negative Sequence Harmonic Unbalanced steady load operation gives rise to a fundamental space harmonic that rotates backwards. This wave travels with linear speed τ = − p ω v2 2 π (3.16)

35 Figure 3.8: Deﬁnition of a damper loop and the corresponding loop current. with respect to the rotor, and induces EMFs of frequency

ω2 = 2ω (3.17) in the damper winding.

3.3.2 Unitary Damper Loop MMF Functions Each damper bar is considered to be a part of two adjacent damper loops,as illustrated in Fig. 3.8. When the current ik flows in loop k, its effect on the air-gap flux density is considered through a damper loop MMF (θ, )= ( ) (θ, ), MDk t ik t MDk0 t (3.18) where MDk0 denotes the unitary MMF function of loop k. When symmetry conditions allow for a reduction of the calculation geometry to two fundamental pole pitches, the unitary loop MMF function is conventionally modeled as shown in Fig. 3.9. In the figure, the rising and falling flanks of the curve mark the positions of the loop conductors. The unitary loop MMF function of Fig. 3.9 is a normalized square-function with a duty cycle determined by the ratio of the electrical damper loop span and a full fundamental electrical period. The function is shifted upwards such that the condition 2π (θ) θ = MDk0 d 0 (3.19) 0 is fulfilled. For a loop current ik = 0, this conventional unitary loop MMF function predicts uniform air-gap flux density outside the angular span directly in front of the loop. Physically, this is however unrealistic. The only flux lines that actually cross the air-gap, and therefore affects the air-gap flux density waveform, are situated directly in front of the damper loop, as illustrated in Fig. 3.10. Thus, as far as the flux crossing the air-gap radially is concerned, a more appropriate unitary loop MMF function is the one illustrated in Fig. 3.11. In this modified function, the MMF is effectively set to

36 1

0.8

0.6

0.4

0.2

0 Classical unitary loop MMF function −0.2 0 30 60 90 120 150 180 210 240 270 300 330 360 Electrical angle, θ (°) Figure 3.9: The classical unitary damper loop MMF function.

Figure 3.10: Flux line distribution upon excitation of a single damper loop. zero outside the angular span of the damper loop, as this region does contain very few radial flux lines that cross the air-gap. For reasons of symmetry, the currents in damper loops that are identically positioned on adjacent poles approximately become equal in magnitude and 180◦ out of phase. Hence, it may be argued that the condition (3.19) is approximately met even after the adoption of the modified unitary loop MMF definition, provided that the MMF contributions from these loops are considered together. The modified unitary damper loop MMF function was adopted in the studies presented in this thesis.

3.3.3 Calculation of Damper Loop Currents The damper loop currents are calculated from circuit equations derived from Kirchoff’s voltage law. One set of equations is formulated for every angular

37 1

0.8

0.6

0.4

0.2

0 Modified unitary loop MMF function −0.2 0 30 60 90 120 150 180 210 240 270 300 330 360 Electrical angle, θ (°) Figure 3.11: Modiﬁed unitary damper loop MMF function. frequency that exist in the damper winding for a given case study (no-load, balanced load, or unbalanced load operation). The total loop currents are then obtained through addition of the loop current harmonics. For the frequency ωn, the circuit equations can be compactly written as

U = ZlIl + jωnMIl. (3.20)

U here denotes a column vector that holds the induced loop EMFs, Zl is the loop impedance matrix, M is the mutual loop inductance matrix and Il is a column vector whose elements correspond to the loop currents. When generators with integral slot armature windings are analyzed, it is sufﬁcient to study two pole pitches of the air-gap. In this case, U and Il become (2Nd × 1) vectors and Zl and M become (2Nd × 2Nd) matrices.

Induced Loop EMF Vector U contains the induced loop EMFs of frequency ωn. An induced loop EMF is here deﬁned as the sum of the EMFs that are induced in the damper bars that constitute the loop, added with appropriate signs. Complex notation is introduced to represent the loop EMFs according to ⎛ ⎞ ⎛ ⎞ jα U 1 Uˆ1e 1 ⎜ ⎟ ⎜ α ⎟ ⎜ ⎟ ⎜ ˆ j 2 ⎟ ⎜ U 2 ⎟ ⎜ U2e ⎟ U = ⎜ . ⎟ = ⎜ . ⎟. (3.21) ⎝ . ⎠ ⎝ . ⎠ jα U N UˆNe N

N here denotes the total number of damper loops in the calculation geometry. The complex EMFs U k (k = 1,2,...,N) are composed of the vector sum of

38 all the EMFs induced by flux density space harmonics that contribute to the rotor frequency ωn. The amplitude of the bar EMF component introduced by the m-th flux density space harmonic that contribute to the frequency ωn is determined through the flux cutting EMF equation as ˆ b = ˆ . Un,m lb vn,m Bn,m (3.22) lb here denotes the length of the damper bar and vn,m is the linear speed of the m-th space harmonic with respect to the rotor. The flux density amplitude Bˆn,m is estimated from a Fourier series expansion of the flux density wave

Bδ (θm,t = 0)=Λ(θm,t = 0) · [Mf (θm,t = 0)+Ma(θm,t = 0)], (3.23) and is subsequently modulated with a flux reduction coefficient that compen- sates for the decrement in radial harmonic flux from the middle of the air-gap to the damper cage, located below the pole face. The flux reduction coefficients were determined from a series of magnetostatic FE calculations and, seemingly, serve the same purpose as the air-gap transformation factor presented in [46]. Consideration of the location of the positive peaks of the exciting space harmonics with respect to the damper bar positions provide the appropriate phase shifts αk (k = 1,2,...,N).

Loop Impedance Matrix The matrix Zl is given by ⎡ ⎤ Z −Z 0 ... 0 −Z ⎢ 1 b b ⎥ ⎢ . ⎥ ⎢ −Z Z −Z .. ... 0 ⎥ ⎢ b 2 b ⎥ ⎢ . . . . . ⎥ ⎢ 0 ...... ⎥ = ⎢ ⎥. Zl ⎢ . . . ⎥ (3.24) ⎢ . .. −Z Z −Z .. ⎥ ⎢ b k b ⎥ ⎢ . . . . . ⎥ ⎣ 0 ...... ⎦ − .. .. − Zb 0 . . Zb ZN

In the expression above, Zb denotes the bar impedance. Analytical expressions presented in [57] were used to determine the AC resistances and leakage inductances of the damper bars. The impedances on the main diagonal, Z k , are given by = ( + ), Zk 2 Zb Zek (3.25)

39 where Zek denotes the end-ring impedance of the k-th damper loop. End-ring damper impedances were estimated with expressions from [58].

Mutual Loop Inductance Matrix As indicated in Fig. 3.10, the most important flux couplings occur between the closest neighboring damper loops. These mutual flux paths are moreover characterized by the permeances normally associated with slot and tooth tip leakage flux. Based on these observations, it was decided that only the mutual coupling between adjacent damper loops were to be considered in the model. Furthermore, the magnitude of the mutual inductance between two adjacent damper loops was set to MDD = Lso + Ltt, (3.26) where Lso denotes the damper slot opening leakage inductance and Ltt denotes the tooth tip leakage inductance. The self inductance of loop k was similarly defined as Lkk = 2 · (Lso + Ltt)+LmDk, (3.27) where LmDk denotes the main loop inductance. LmDk was determined by aver- aging the air-gap permeance in front of the k-th damper loop. With the introduced notation, the matrix M is formulated as ⎡ ⎤ L −M 0 ... 0 −M ⎢ 11 DD DD ⎥ ⎢ .. ⎥ ⎢ −MDD L22 −MDD . ... 0 ⎥ ⎢ ⎥ ⎢ . . . . . ⎥ ⎢ 0 ...... ⎥ M = ⎢ ⎥. (3.28) ⎢ . .. .. ⎥ ⎢ . . −MDD Lkk −MDD . ⎥ ⎢ ⎥ ⎢ . . . . . ⎥ ⎣ 0 ...... ⎦ .. .. −MDD 0 . . −MDD LNN

3.3.4 Resultant Damper MMF Adopting a cosine reference, a complex damper loop current of frequency ωn and determined through (3.20) is transformed into a real-valued function according to jϕk Ik,n = Iˆke ⇒ ik,n(t)=Iˆkcos(ωnt + ϕk). (3.29)

The resultant damper winding MMF is then calculated as N = ( ) (θ, ). MD ∑ ∑ik,n t MDk0 t (3.30) k=1 n

40 Figure 3.12: THD of the open-circuit armature voltage waveform of Generator I in Paper I. The damper winding is continuous with six damper bars per pole, and is centered around the pole axis.

3.4 Selected Results 3.4.1 THD of the Open-circuit Armature Voltage Waveform In Paper I, the permeance model was used to determine the harmonic contents in the open-circuit armature voltage waveform of large hydroelectric generators. Particular attention was devoted to the impact of the damper winding reaction on the so called slot ripple harmonics. Slot ripple voltage harmonics are produced by the damper MMF induced by the stator slot harmonic ﬂux density wave-pairs in (3.9). They correspond to the harmonic frequencies

ωslot = 6nq1 ± 1 n = 1,2,... (3.31)

Fig. 3.12 shows the calculated Total Harmonic Distortion2 (THD) of the open- circuit armature voltage versus the damper slot pitch (τds) for one of the studied generator topologies. The damper slot pitch is stated as a fraction of the stator slot pitch. In the figure, THD predictions obtained with the permeance model are compared to those obtained with a FE model. The damper winding in the studied unit is a continuous winding with six equidistant damper bars per pole, centered around the pole axis. The THD essentially reflects the stator slot harmonic contents in the voltage waveform, and is seen to be maximal for τds = 1. This is in line with the theory presented in [59]. Moreover, it is observed that the THD predictions of the two models are very close. Further numerical experiments with the permeance model revealed that the THD vs. τds - profile typically is highly susceptible to the resistance of the electrical connection between the damper cages on adjacent poles, Rc. This

2The THD is deﬁned in (19) of Paper I.

41 Figure 3.13: THD versus damper slot pitch for different values of the inter-pole coupling resistance, Rc.

feature is illustrated in Fig. 3.13, where THD vs. τds - profiles are provided for three different Rc-values. The basic damper winding configuration is the same as that described in conjunction with Fig. 3.12 (six equidistant centered bars). A highly resistive inter-pole coupling corresponds to a grill winding, while alowRc-value corresponds to a complete squirrel-cage. Fig. 3.13 indicates that if the objective is to reduce the slot ripple voltage harmonic contents by an appropriate choice of damper slot pitch, the optimal choice will effectively depend on the basic damper winding configuration (grill or complete squirrel- cage).

3.4.2 Damper Bar Currents at Rated Load Operation Fig. 3.14 shows the calculated ﬁrst phase belt harmonic damper bar currents3 in a large hydroelectric generator at rated load operation. The studied damper winding is continuous, with seven bars per pole. Bar 1 is located on the trailing side of the pole. In the ﬁgure, permeance model data is compared with the results from FE calculations. The bar current predictions are in fair, but not excellent agreement. The largest discrepancies are observed in the outermost bars.

3These current harmonics are introduced by MMF armature harmonics of orders 5 and 7.

42 Figure 3.14: Calculated RMS values of ﬁrst phase belt frequency (360 Hz) damper bar currents in the test generator of Paper II.

3.4.3 Reduction of the UMP by Parallel Armature Circuits In permeance models, stationary off-centered rotor operation can be modeled by modulating the air-gap permeance with a function 1 Λecc = , (3.32) 1 − ε · cos(θm − α) where ε denotes the relative rotor eccentricity, and α is the position of the minimum air-gap length with respect to the angular coordinate origin. The uneven air-gap length around the rotor periphery leads to increased flux density levels where the air-gap is shorter, and decreased flux density levels on the side where the air-gap is longer. This uneven air-gap flux density profile gives rise to a magnetic force resultant termed unbalanced magnetic pull (UMP). In synchronous generators with more than one circuit per armature phase, the asymmetric air-gap introduces currents that circulate between the parallel circuits both during no-load and load operation. The MMF set up by these currents counteracts the air-gap flux density wave modulation, and, accordingly, reduces the UMP. In Paper III, a simple extension of the permeance model that allowed for the consideration of currents circulating between parallel armature winding phases was examined. Fig. 3.15 illustrates the air-gap flux density profile in a twelve-pole synchronous machine operating at no-load, before and after addition of the circulating armature current MMF. As can be expected, the calculated armature MMF to some extent “evens out” the air-gap flux density waveform. Thus, the validity of the suggested modeling principle of this phenomenon is, at least qualitatively, confirmed.

43 Figure 3.15: No-load air-gap ﬂux density waveforms in the test generator of Paper III with 24% relative static eccentricity, calculated with the permeance model. The rotor is displaced toward the “middle point” of one phase circuit group and the applied ﬁeld current equals 30 A. The stator slot permeance was omitted from the analysis.

44 4. Core Loss Prediction in Large Hydroelectric Generators

This chapter reviews the work presented in Paper IV. A brief background on practical iron loss estimation is also provided.

4.1 Previous Work The technical literature contains a vast amount of papers on the subject of iron losses in electrical machines. Here, only a very limited number of works with emphasis on iron loss modeling and rotational core losses are mentioned. Fiorillo and Novikov [60] derived formulae for the calculation of the average iron loss in magnetic steel laminations, applicable for arbitrary periodic flux density waveforms. Moses [61] measured the locus of the magnetic flux density vector in the core of rotating machines and identified regions where bidirectional flux is prominent. Stranges and Findlay [6] tested different loss prediction schemes on the field solutions obtained from FEA of induction machines. Experimental loss curves for various axis ratios were used to account for flux density bidirectionality. Bottauscio et al. [3] compared post- processing and vector hysteresis iron loss evaluation techniques on field solutions obtained from FEA. Díaz et al. [10] presented analytical formulae for rotational loss prediction in induction machines. The study concluded that it was sufficient to consider rotational losses only in a region near the stator tooth roots.

4.2 Iron Loss Estimation 4.2.1 Loss Separation The total instantaneous speciﬁc power loss in steel laminations is usually considered to be constituted of three parts; the hysteresis loss (1), the classical eddy current loss (2) and the excess loss (3). The decomposition of the loss into three distinct terms is called loss separation. Physically, this concept emerges as a result of magnetization reversal processes that occur on different spatial scales [62].

45 The time average of the total speciﬁc power loss in magnetic steel sheets exposed to sinusoidal ﬂux density of peak value Bmax and frequency f is conventionally modeled as = 2 + 2 2 + 1.5 1.5 . ptot kH fBmax kc f Bmax kE f Bmax (4.1)

Here, the first term represents the hysteresis loss, the second term is the classical loss, and the last term is the excess loss. The coefficient kc is the classical loss coefficient π2σd2 kc = , (4.2) 6ρm where σ denotes the electric conductivity, d the thickness, and ρm the mass density of the steel laminate. The loss coefficients kH and kE are typically determined through curve fitting of measured loss data. Equation (4.1) corresponds to a frequency domain model, and rigorously only applies for sinusoidal excitation. The model can be re-formulated as a time domain model according to T 2 T 1.5 = 2 + kc dB + kE dB , ptot kH fBmax dt dt (4.3) T 0 dt T 0 dt where B denotes the flux density vector and T = 1/ f is the excitation period. Equation (4.3) can be used to estimate the iron loss of any periodic flux density waveform, provided its frequency is close to f . In some situations, empirical corrections for the loss associated with minor loop flux reversals can be motivated [63]. The modified loss coefficients kc and kE are equal to k k = c , (4.4) c 2π2 and k k = √ E , (4.5) E π 2π | |1.5 2 0 cosx dx and are determined such that (4.3) yields the same result as (4.1) for the special case of sinusoidal excitation.

4.2.2 Rotational Losses The loss separation formulae introduced in the previous section are valid if the time-varying flux density oscillates in one distinct direction. Such a flux density vector is here referred to as an alternating quantity. In electrical machines, the flux density in some regions of the stator core is however strongly bidirectional. That is, the locus traced by the tip of the flux density vector

46 Figure 4.1: Deﬁnition of Bmax and Bmin in the spatial locus of a time-periodic ﬂux density vector. xy is an arbitrary Cartesian coordinate system.

Figure 4.2: Measured ratio between the power loss at purely rotational conditions and the power loss at purely alternating conditions. The curve was derived from data presented in [6]. The test material was a 0.47 mm SiFe steel with 2.7% silicon content. during a full electrical period resembles an ellipse rather than a straight line. Regions with bidirectional flux density are said to be exposed to rotational flux. The degree of rotation of the flux density waveform is determined by the fundamental axis ratio, Γ, of the locus traced by the tip of the flux density vector during a complete excitation period. Here, the fundamental axis ratio of an arbitrary locus is approximated by B Γ = min , (4.6) Bmax where the reader is referred to Fig. 4.1 for a definition of Bmin and Bmax. For identical excitation frequency and amplitude, the power loss associated with exposure to flux density of a non-zero degree of rotation typically is higher than the loss at purely alternating conditions. The ratio between the power loss at purely rotating conditions (Γ = 1) and purely alternating conditions (Γ = 0) typically depends on Bmax, as illustrated in Fig. 4.2.

47 Iron loss prediction models may be adjusted for the influence of these “rotational losses” in various ways. The frequency domain model (4.1) may for instance be modified according to =( + δΓ) · ( 2 + 2 2 + 1.5 1.5), ptot 1 kHBmax f kcBmax f kEBmax f (4.7) where the factor δ is a weighting factor that determines the specific loss increase attributable to flux rotation. Equation (4.7) effectively determines the specific loss associated with an arbitrary elliptical flux density locus through linear interpolation between a purely alternating and a purely rotating loss. The modification was originally proposed by Ma et al. [5]. For the time-domain model (4.3), a detailed consideration of rotational effects results in the following modified expression [64] T 2 =((− Γ)+Γ · ( )) 2 + kc dB ptot 1 RH Bmax kHBmax f dt T 0 dt T 1.5 kE dB +((1 − Γ)+Γ · RE(Bmax)) dt. (4.8) T 0 dt

Again, a weighted interpolation technique between purely alternating and purely rotating losses is adopted, but contrary to (4.7), the interpolation is performed on the individual loss components. In (4.8), the function RH (Bmax) represents the ratio between the purely rotational and the purely alternating hysteresis loss and RE (Bmax) is the ratio between the purely rotational and the purely alternating excess loss. These functions can be obtained by applying a three-term loss decomposition scheme on measured speciﬁc loss data [64]. The functions RH (Bmax) and RE(Bmax) typically display strong qualitative resemblance [60,64]. For instance, they both decrease monotonically and drop to zero at high ﬂux density levels.

4.3 Study Summary No-load operation of twelve large hydropower generators denoted I - XII was simulated with a 2-D time-stepped FE model. From the calculated magnetic ﬂux density distributions, the no-load core losses were estimated with three different loss prediction models, denoted A, B and C. Information about the studied units can be found in Table II, Paper IV.

Model A Model A, referred to as the alternating loss model, evaluated the specific core loss with (4.1). The constant loss coefficients kH and kE were obtained from multivariate curve fitting of the lamination manufacturer’s loss data recorded

48 Figure 4.3: Assumed ratio between the rotational and alternating hysteresis loss in the core materials. The curve is derived from data presented by Bottauscio et al. [3]. at 50, 100 and 200 Hz.

Model B Model B assessed the specific core loss through (4.7). The weighting factor δ was set to the constant value 0.6. This implies that the model predicts a specific loss density that is 60 % higher for Γ = 1 compared to when Γ = 0, independently of Bmax. This is a fair estimate for flux density levels on the order of 1-1.6 Tesla, but will lead to an underestimation of the additional rotational loss at low flux density levels [6]. The loss coefficients are the same as those used in Model A.

Model C Model C assessed the speciﬁc core loss through (4.8). Because rotational loss data were not available for the studied core materials, a rotational hysteresis loss curve considered “typical” for non-oriented SiFe-laminations used in electrical machines was used to model the function RH . RE was further assumed to be identical to RH . The employed curve was derived from [3] and is provided in Fig. 4.3.

In order to assess the influence of flux bidirectionality on the calculated core loss, the total core loss figures obtained with Models B and C were compared with the total core loss figure obtained with Model A. If the total core loss A B estimate obtained with Model A is termed Ptot, and Ptot denotes the core loss obtained with Model B, then the fractional core loss increase attributable to rotational effects as predicted by Model B can be evaluated as B − A = · Ptot Ptot [ ]. Padd−rot 100 A % (4.9) Ptot

49 Figure 4.4: Core losses predicted by Models A, B, and C. The loss ﬁgures are presented in % of the measured electromagnetic no-load loss.

C Similarly, the difference between the core loss calculated with Model C, Ptot, and the loss PA , tot C − A = · Ptot Ptot [ ], Padd−dyn 100 A % (4.10) Ptot is a measure of the combined importance of harmonics and rotational effects on the total core loss estimate.

4.4 Selected Results The calculated core losses for the twelve generators, as predicted by Models A, B, and C are shown in Fig. 4.4. Observe that the lines between the data points merely serve as “guides for the eye”. The loss figures are presented in % of the measured electromagnetic no-load loss. Model A consistently yielded the smallest loss predictions, with an the average of about 51% of the measured loss. Model C, that takes the influence of harmonics and rotational effects into account, yielded the highest loss predictions, with an average of 65% of the measured loss. Hence, additional stray losses, model inaccuracies and measurement errors are indirectly predicted to account for 35% of the total electromagnetic no-load loss. The significant spread in the discrepancy between calculated and measured loss figures between the different machines suggests that differences in machine design phi- losophy, which in turn determine the magnitude of the stray losses, have a decisive impact on this type of loss comparisons.

50 Figure 4.5: Degree of rotation in the core of Generator XII during no-load operation. Γ = 0 (blue) signifies purely alternating flux while Γ = 1 means purely rotational flux.

Figure 4.6: Padd−dyn = loss increase attributable to dynamic effects (ﬂux rotation + harmonic distortion), as predicted by Model C. Padd−rot = loss increase attributable to ﬂux rotation, as predicted by Model B.

Fig. 4.5 shows the calculated degree of rotation in the core of Generator XII during no-load operation. The highest degree of rotation is found in the stator teeth roots and typically amounted to about 0.7-0.8 in the studied generators. More than 50% of the yoke is exposed to fields with a degree of rotation higher than 0.3. The flux in the teeth is nearly purely alternating. The effect of harmonics and flux rotation on the calculated core losses is illustrated in Fig. 4.6. The rotational loss correction predicted by Model B (Padd−rot) varied between 10 and 18% for the studied generators, the average being 13%. The fractional loss increase Padd−dyn, which takes both harmonic and rotational effects into account, varied between 11 and 50% and was 28% on the average. The major part of this loss increase is attributable to rotational effects.

51 The exceptionally high loss increase caused by dynamic effects in Gener- ator I is due to low ﬂux density levels in the over-dimensioned stator core of this machine. At low ﬂux density levels, the rotational loss correction is substantial (see Fig 4.3), and hence the fractional loss increase Padd−dyn becomes very pronounced.

52 5. Form Factors of Salient Pole Shoes

This chapter reviews the work presented in Paper V.

5.1 Background The study on salient pole shoes started in parallel with the author’s elaboration of a computer program for the analytic determination of the main inductances of salient-pole synchronous machines. In analytic calculations, the direct (d), quadrature (q) and ﬁeld ( f ) magnetizing inductances are determined as

Xjm ∝ k j Xm ( j = d,q, f ) , (5.1) where Xm denotes the main armature inductance of a machine with constant 1 air-gap length, and k j is the form factor for excitation type i . The form factors are scalars that take the combined effects of the air-gap permeance and MMF waveforms into account. To assist in the determination of pole shoe form factors, many textbooks cite a classical paper by Wieseman [14]. In that study, a graphical method is used to characterize a large number of pole shoe shapes. Curves for the determination of form factors of arbitrary pole shoe geometries are also presented. In order to assess the accuracy of the curves presented in [14], we compared the output from Wieseman’s form factor formulae to data extracted from FE calculations. It was deduced that for certain pole shoe geometries and excitation levels, Weiseman’s form factors deviated with about 10-20% from the FE data. The work presented in Paper V had two main objectives. The ﬁrst objective was to derive accurate form factors for the salient-pole generator topologies studied in Papers VII-VIII. The second objective was to conduct a comprehensive study on the subject of pole shoe form factors, as a modern review of Wieseman’s work. The intention with the latter study was to provide updated data that can assist machine designers in the selection of the pole shoe shape.

With Wieseman’s study as a point of departure, two aspects were given special attention:

1Some authors also refer to these constants as ﬂux distribution coefﬁcients.

53 Figure 5.1: Air-gap ﬂux density waveform, Bgd(θ), set up by armature excitation along the pole (d-) axis. Bgd1(θ) is the fundamental wave.

1. The effect of iron saturation on the pole shoe form factors. Weiseman’s study is based on the assumption of inﬁnite relative permeability in the rotor and stator. The impact of saturation on the form factors is therefore of interest. 2. The extent to which the details of the pole face contour affect the form factors. Weiseman’s form factors are determined through a very limited set of parameters that characterizes the geometry of the pole shoe.

5.2 Pole Shoe Form Factors Direct Axis Armature Pole Shoe Form Factor The direct axis armature pole shoe form factor kd is determined from the air-gap ﬂux density waveform produced by a sinusoidal armature MMF acting directly in front of the pole axis. This excitation results in a slightly peaked waveform, as shown in Fig. 5.1. kd is deﬁned as

Bgd1 kd = , (5.2) Bgdm where Bgd1 is the amplitude of the fundamental and Bgdm is the peak value of the ﬂux density waveform.

Quadrature Axis Armature Pole Shoe Form Factor The impression of sinusoidal armature MMF in front of the pole-gap (quadrature) axis, produces an air-gap ﬂux density wave whose qualitative appearance is illustrated in Fig. 5.2. The quadrature axis armature pole shoe

54 Figure 5.2: Air-gap ﬂux density waveform, Bgq(θ) set up by q-axis armature excitation. Bgq1(θ) is the fundamental wave. form factor is here deﬁned as2 Bgq1 kq = , (5.3) Bgd1 where Bgq1 denotes the amplitude of the fundamental of the waveform Bgq(θ) and Bgd1 is the fundamental of the waveform set up by excitation along the direct axis. kq is a direct measure of the ratio Xqm/Xdm.

Field Winding Pole Shoe Form Factor Field winding excitation results in a flat-topped air-gap flux density waveform, as illustrated in Fig. 5.3. In analogy with the previous definitions, the field winding pole shoe form factor k f is defined as

Bgf1 k f = . (5.4) Bgfm

Bgf1 is the amplitude of the fundamental and Bgfm is the peak value of the resulting wave.

5.3 Study Summary

The pole shoe form factors kd, kq, and k f of a large number of salient poles were determined from air-gap flux density waves obtained in 2-D magnetostatic FE calculations. Additionally, the THD of the air-gap flux density waves produced by field winding excitation was determined, since this is a traditional measure of the harmonic “imprint” of the pole shoe.

2 The deﬁnition is different from the one used in [14]. The employment of kq in inductance calculations is therefore slightly modiﬁed.

55 Figure 5.3: Air-gap ﬂux density waveform, Bgf(θ), produced by the ﬁeld winding. Bgf1(θ) denotes the fundamental wave.

Figure 5.4: Deﬁnition of geometrical pole shoe variables. A large stator diameter is assumed.

5.3.1 Pole Face Contours A general salient pole shoe geometry is shown in Fig. 2.4 and deﬁnitions of the geometric variables pole pitch (τp), pole shoe width (τpp), pole width (bp), pole shoe length (hpp), minimum air-gap length (δmin), and maximum air-gap length (δmax) are provided. The study was limited to pole shoes belonging to any of the following three pole face contour categories:

1. Inverse Sine Pole Shoes The inverse sine pole shoe is a classical pole face design based on the idea that the air-gap length δ in front of the pole should vary as δ δ = 0 , (5.5) sinθ

56 where δ0 is the air-gap length directly in front of the pole, and θ denotes an electrical angle measured from the inter-pole axis. This pole shape is known to give low harmonic contents in the air-gap flux density waveform. 2. Concentric / Tapered Pole Shoes The faces of pole shoes in this category have of a central part that is concentric with the inner stator periphery. On the sides of the center arc, the pole face is cut such that a specified value of δmax is obtained at the pole tips. The two off-centered cuts are sometimes slightly curved. Concentric/tapered pole shoes are “bulkier” than their inverse sine counterparts, and exhibit a higher mean air-gap permeance. This results in higher fundamental air-gap flux, and, accordingly, higher form factors values. 3. Elliptic Pole Shoes For pole shoes in this category, the curved path between the mid-point of the pole face and the pole tip is shaped as one quadrant of an ellipse. The design is one of several possible polynomial pole face contours - the higher the polynomial order, the bulkier the pole face. The elliptic pole shoe represents a “shape average” of the other two designs; it is bulkier than the inverse sine pole shoe, and smoother than the concentric/tapered pole shoe. In contrast to the other two pole face contour categories, the elliptic pole shoe is a theoretical reference case that is not used commercially.

5.3.2 Pole Shoe Variables Given a pole face contour category, a number of additional parameters need to be assigned with values in order for the salient pole-shoe geometry to be completely specified. To this end, the three ratios τpp/τp, δmin/τp, and δmax/δmin are introduced. The ratio δmax/δmin, also referred to as the pole taper, is henceforth denoted fa. The ratio τpp/τp will be denoted τpp for short. It is understood that variable τpp refers to the pole shoe width expressed as a fraction of the pole width. The pole taper fa is not needed to specify a pole shoe with an inverse sine pole face contour, since the air-gap length at the pole tips is given by (5.5). Moreover, it is necessary to introduce the additional variable τpc, which denotes the width of the concentric part of the pole shoe, to fully specify the geometry of pole shoes with concentric/tapered pole face contours. In order to assess the effects of iron saturation on pole shoe form factors, different levels of excitation were tested. To this end, the excitation current in the magnetostatic calculation was adjusted so that the peak value of the air-gap flux density met a pre-specified value Bgm. Table 5.1 provides the range of pole shoe variable values that were examined for each pole face contour category. The listed values are typical for large hydroelectric generators. The variable Bgm is treated like any other variable, and is here considered to be a measure of the level of saturation.

57 Table 5.1: Examined Pole Shoe Variable Values Variable Values

τpp 0.6 - 0.75

δmin/τp 0.025 - 0.040

fa = δmax/δmin 1.5 - 2.5 1 τpc 0.42 - 0.50

Bgm 0.8 - 1.0 T

1 This variable only applies for pole shoes with a concentric/tapered pole face.

Figure 5.5: Selected results from the analysis of the form factor k f . The presented values correspond to the settings Bgm = 0.8 and δmin/τp = 0.030. The results are derived from non-linear FEAs (iron saturation considered). Concentric/tapered data corresponds to calculations with τpc = 0.46.

5.4 Selected Results 5.4.1 Effect of Pole Face Contour

Fig. 5.5 shows calculated values of k f for different pole face contour categories at Bgm = 0.8 T and δmin/τp = 0.030. The abscissa of the plot holds the pole taper fa, and the pole shoe width τpp is a parameter in the curve families. k f -values for two pole shoes with inverse sine pole face contours are indicated with straight lines. k f -data for concentric/tapered pole shoes corresponds to pole face contours with τpc = 0.46. Fig. 5.5 indicates that two pole shoes with identical values of the parameters τpp and fa, but manufactured with different pole face contours, may ex-

58 Table 5.2: Linear Model Coefﬁcients

2 kd kd0 β1 β2 β3 β4 β5 R Inv. Sine 0.708 0.147 0.0437 0.055 - - 0.93 Elliptic 0.693 0.137 0.424 -0.522 -0.088 - 0.99 Conc./Tap. 0.632 0.124 0.211 0.197 -0.032 0.24 0.96 2 kq kq0 β1 β2 β3 β4 β5 R Inv. Sine 0.051 0.108 0.486 1.826 - - 0.97 Elliptic -0.142 0.1125 1.052 1.444 -0.074 - 0.98 Conc./Tap. -0.158 0.126 0.880 0.197 1.529 -0.071 0.97 2 k f k f 0 β1 β2 β3 β4 β5 R Inv. Sine 0.771 0.110 0.165 0.427 - - 0.92 Elliptic 0.715 0.083 0.735 -0.159 -0.117 - 0.99 Conc./Tap. 0.646 0.0052 0.521 0.570 -0.062 0.38 0.96

hibit quite diverse k f -values. Thus, a direct employment of Wieseman’s form factor formulae, regardless of the details of the pole face contour, clearly can- not be recommended. The slim design of inverse sine pole shoes leads to comparably low k f - values, while the bulky design of concentric/tapered pole shoes eases the transmission of more fundamental ﬂux across the air-gap. Accordingly, k f - values are generally quite high for this pole face contour category. Pole shoes with elliptic pole faces are seen to be very susceptible to both variations in fa and τpp, and is therefore a rather ﬂexible design.

5.4.2 Linear Models with Saturation Considered As indicated in Fig. 5.5, the form factor variations for pole shoe geometries that are considered in practice are both predictable and fairly limited. It was therefore possible to establish linear models on the form

δmin k j = k j0 + β1Bgm + β2τpp + β3 + β4 fa + β5τpc ( j = d,q, f ), (5.6) τp for the evaluation of form factors of pole shoes that belong to a given pole face category. The model coefficients k j0 ( j = d,q, f ) and βi (i = 1,...,5) were derived from a linear regression scheme applied to the calculated FE data. The coefficients are compiled in Table 5.2. It should be noted that the high values of the coefficients of determination (R2) seen in Table 5.2 are a direct result of the inclusion of the explanatory variable Bgm, which considers form factor variations introduced by saturation.

59 Figure 5.6: kf vs. THD for the complete set of tested pole shoes.

In absolute terms, the effect of saturation is however quite small (see Figs. 7 and 9 in Paper V).

5.4.3 Perspectives on Pole Shoe Shape Selection Fig. 5.6 shows a concentrated overview of the harmonic imprint and magnetic performance in terms of k f for all the examined pole shoes. Every data point represents a unique pole shoe defined by its pole face contour, pole shoe width and pole taper. The abscissa and ordinate of Fig. 5.6 hold k f and the THD of the air-gap flux density waveform produced by field excitation respectively. The plotted data correspond to calculations with δmin/τp = 0.030. The excellent performance of inverse-sine pole shoes in terms of THD is clearly seen. The low harmonic contents however comes to the price of fairly low k f -values. This implies that a higher magnetization current is needed to obtain a specified voltage level. In essence, the selection of pole shoe shape typically is a compromise between the contradictory requirements of low harmonic imprint and high mean air-gap permeance. The former requirement is given priority if low surface harmonic losses and a high-quality armature voltage shape are considered to be essential design features. Similarly, a high mean air-gap permeance is given priority if it is desirable to minimize the magnetization losses.

60 6. Analysis of a PM Generator with Two Contra-rotating Rotors

This chapter reviews the work presented in Paper VI.

6.1 Previous Work Caricchi et al. [18] described and studied a dual-rotor axial-flux machine topology, characterized by synchronous counter-rotation of the two rotors. Clarke et al. [65] demonstrated the operation of an axial-flux machine with two contra-rotating stators in a tidal energy conversion scheme. Yeh et al. [66] demonstrated asynchronous rotor operation of a dual-rotor radial-flux motor. Danilevic et al. [67] calculated the performance of a slotless dual-rotor radial- flux PM motor.

6.2 Generator Topology 6.2.1 Dual Contra-rotating Rotor Topology Fig. 6.1 provides an exploded-view drawing of the active parts of the studied dual-rotor generator topology. The central features are the two concentric contra-rotating rotors that operate on opposite sides of a central stator core. Each rotor is equipped with surface-mounted NdFeB-magnets with a rema- nent polarization level of 1.2 T. Stationary coils are positioned both along the inner and outer stator peripheries. Coils placed along the outer core periphery, facing the outer rotor, constitute an outer stator winding section. Similarly, coils positioned along the inner core periphery constitute an inner stator winding section. The stator coils can be connected into winding phases according to the three-phase winding arrangement shown in Fig. 6.2. The inner and outer winding sections can be connected in series, or, alternatively, each of the winding sections can be connected to a separate voltage supply. When the winding sections are connected in series and supplied by a three-phase voltage source, the two air-gap MMFs will rotate in opposite directions. Accordingly, the outer winding section becomes a negative sequence arrangement (A-C-B) relative to the direction of inner rotor movement. Simi-

61 ωi 4 1 3 2 7 ωo 6

Figure 6.1: Exploded-view drawing of the contra-rotating generator topology (only active materials - iron, copper and PMs - are shown). 1. Outer rotor. 2. Outer rotor PMs. 3. Outer winding section. 4. Stator core. 5. Inner winding section. 6. Inner rotor. 7. Inner rotor PMs.

Direction of rotation, outer rotor

τp Outer AC’BCA’ B’ air gap

Inner air gap B’ C A’ B C’ A

Direction of rotation, inner rotor Figure 6.2: Three-phase winding arrangement for the contra-rotating generator. The three phases are designated A, B, and C. Primed letters indicate negative conductor orientation. τp = pole pitch. larly, the inner winding section is a negative sequence arrangement in relation to the direction for outer rotor movement.

6.2.2 Reference Machine Topologies A laboratory-sized 50 Hz dual-rotor generator geometry with dimensions speciﬁed in Paper VI was created in a FE software. In order to assess the nature and magnitude of cross-coupling phenomena between the rotors, two reference machine geometries were also implemented. The reference geometries, correspond to the “inner” and “outer” machines of the full contra-rotating generator topology, and are shown in Fig. 6.3. The magnetic axes of the inner (a, b, c) and outer (A, B, C) winding sections are also marked in the ﬁgure.

62 A dPM B

dst α C ωi ror

rir b a c

g ωo a) b) c)

Figure 6.3: a) Inner reference machine geometry (outer rotor removed). a, b, and c denote the magnetic axes of the inner winding section. b) Outer reference machine geometry (inner rotor removed). A, B, and B denote the magnetic axes of the outer winding section. c) Full dual-rotor contra-rotating generator topology.

6.3 Selected Results The time-resolved performance of the dual contra-rotating rotor generator topology was assessed via a sequence of stationary 2-D FE calculations. Be- fore each new calculation, the rotors were redrawn in new positions to simulate rotor motion. Time-stepped FEA could not be employed since the used FE software only allowed for the use of a single sliding mesh boundary condition.

6.3.1 Characterization of the Inter-rotor Cross Coupling The contra-rotating movements of the magnetized rotors were found to give rise to a periodic cross-coupling distortion. The space phasor diagram shown in Fig. 6.4 provides a basis for the understanding of the nature of this distortion. The figure applies to synchronized contra-rotating operation of the rotors and zero armature current. Hence, the angular velocity of the inner rotor, ωi, is equal, but opposite in sign, to the angular velocity of the outer rotor, ωo. mi and mo denote the inner and outer rotor MMFs respectively. If the initial position of the inner rotor with respect to the inner winding (a, b, c) is identical with the position of the outer rotor with respect to the outer winding (A, B, C), then the rotor magnets will always align along the same spatial directions. These spatial directions are here denoted N and P and correspond to the rotor configurations shown in Fig. 6.5. A cross-coupling distortion comes about as a result of the interaction between the two contra-rotating field waves. Since the sum of two waves that travel in opposite direction is a standing wave, it may be postulated that this is what the nature of the fundamental cross-coupling distortion should be. Furthermore, the nodes and antinodes of the standing wave are likely located along the P-axes and N-axes, as indicated in Fig. 6.6.

63 b C

N P 60º B 60º ωi a mi P ωo N mo

c A

Figure 6.4: Orientation of antinode-axes N and node-axes P with respect to the magnetic phase axes. mi and mo are the inner and outer rotor MMFs respectively. a, b, and c are the magnetic axes of the inner winding section. A, B, and C are the magnetic axes of the outer winding section.

P N P N

(a) (b)

Figure 6.5: Field lines in the dual-rotor generator when the magnets on the inner and outer rotors are radially aligned. The stator winding is open-circuited. a) PMs of the same polarity face each other (coupling distortion equals zero). If the rotors are operated at the same speed and electrically in phase, this alignment always occurs along the stationary axis P. b) PMs of different polarity face each other (maximum coupling distortion). The alignment occurs along the axis N.

In order to conﬁrm the existence of a standing wave distortion along an arbitrary direction θ in the air-gap, the radial ﬂux distortion

ΔBr(θ)=Br(θ) − Br,0(θ) , (6.1) 64 Stationary standing wave distortion

P a N b P c/N P Magnetic direction along the inner air−gap Figure 6.6: Position of the standing wave disturbance caused by the rotor cross- coupling relative to the magnetic axes of the inner winding section.

was determined. Br here represents the radial air-gap flux density in one of the air-gaps of the dual-rotor generator and Br,0 is the radial air-gap flux density in the corresponding reference machine. Fig. 6.7 shows the quantity ΔBr versus time in the inner air-gap of a dual- rotor generator topology for three different stator arrangements. Fig. 6.7a corresponds to a topology with an air-cored stator, and Fig. 6.7b-c to machines with iron cores. ΔBr was calculated along the axes a, b, and c and the corresponding signals are denoted ΔBra, ΔBrb, and ΔBrc. In Fig. 6.7a, it can be observed that the peak flux distortion is greater along the axis c than along the axes a and b. Moreover, the distortions along a and b are in phase and their peak values are about cos60◦ ≈ 0.5 times the peak distortion along the c-axis. These observations are in agreement with the postulated spatial position of the standing wave distortion in Fig. 6.6. The introduction of an iron core between the rotors leads to an efficient decoupling of the rotors, and, hence, a decreased standing wave distortion. In Fig. 6.7b, a 20mm iron core has been introduced between the rotors. It is seen that the peak flux distortion is reduced from 30 mT in the case of an air-cored stator, to about 60 μT. Moreover, the phase-shifts between the signals ΔBra, ΔBrb, and ΔBrc are modified. In Fig. 6.7c, a 30mm central iron core is used. The radial flux peak distortions are reduced even further and the standing wave distortion caused by the rotor coupling is now effectively eliminated. The successful reduction of the standing wave reveals a weak background negative sequence flux distortion.

65 Figure 6.7: Radial ﬂux density distortion, ΔBr, vs. time in the inner air gap of the dual-rotor generator for different core layouts. a) Air core, 2 mm thick. b) Iron core, 20 mm thick. c) Iron core, 30 mm thick. No-load operation is assumed. ΔBr is plotted along axes a (ΔBra), b (ΔBrb), and c (ΔBrc). The nominal ﬂux density level is 0.4 T.

6.3.2 Synchronized Contra-rotating Load Operation Fig. 6.8 shows calculated air gap torques at 2.5 A load current for synchronized operation of the two rotors. The distinct 6th harmonic torque pulsations result from poorly suppressed 5th and 7th armature space harmonics and are

66 Figure 6.8: Calculated air gap torques in the generator during synchronized contra- rotating load operation at different power factors. a) Outer air gap. b) Inner air gap. not caused by disturbances owing to magnetic coupling between the rotors. The results suggest that acceptable machine performance could be achieved in this operational mode.

7. Electromechanical Transients - Simulation and Experiments

This chapter reviews the work presented in Papers VII, VIII and IX.

7.1 Previous Work Electromechanical transients and rotor angle stability are important subjects both in power systems engineering and in electrical machine engineering. Consequently, the topics are frequently addressed both in the specialized power systems literature as well as in works devoted to synchronous machine analysis. The work presented in this thesis is intended to address the topic from a generator perspective. The list of cited works, which is not intended to be comprehensive, should reflect this perspective. Park [26] derived an analytical expression for the electrical torque in synchronous machines during small rotor oscillations. Concordia [21] used Park’s equation to study the effects of tie-line impedance, armature resistance and damper winding parameters on the damping and synchronizing torques of a generator connected to an infinite bus. Simplified analytical expressions for the damping and synchronizing torque contributions from individual rotor windings were derived by Shepherd [22]. Schleif et al. [68] demonstrated transmission line stabilization by means of additional damping torque production in a hydro generating unit. DeMello and Concordia [69] analyzed the sta- bilizing actions of excitation systems and voltage regulators with a block diagram model. Alden and Shaltout [70] presented a method to estimate damping and synchronizing torques from transient response signals. Escalera et al. [71] presented a coupled field-circuit model of a generator connected to an infinite busbar.

7.2 Rotor Angle Oscillations In Chapter 1, the term electromechanical transient was deﬁned as a rotor speed excursion around the synchronous speed and the associated ﬂuctuations in electrical torque. Electromechanical oscillations here refer to transients of oscillatory nature which, in the presence of net positive damping, diminish in

69 amplitude. In the literature, the terms rotor angle oscillations and hunting are also used to denote the same phenomenon.

7.2.1 The Swing Equation Rotor angle oscillations governed by (2.51), which is repeated here for conve- nience dω 1 m = (T − T ). (7.1) dt J m e In (7.1), the rotating assembly (shaft, rotor and prime mover) is modeled as a single mass. If torsional modes are to be considered, additional mechanical equations need to be added [72]. In p.u. EC models, (7.1) is usually reformulated as [73] dΔω 1 m = (T − T ). (7.2) dt 2H m e

In (7.2), the mechanical and electrical torques, Tm and Te, are expressed in p.u., and ωm − ωms Δωm = . (7.3) ωms

ωms here denotes synchronous mechanical speed. The inertia constant H in (7.3) is given by 1 Jω2 H = ms , (7.4) 2 Sbase where Sbase denotes the MVA base of the studied system. The rotor angle dynamics in the p.u. model discussed in Section 2.2 is governed by the equation dδ = ω Δω , (7.5) dt s m where ωs is the synchronous electrical angular velocity. The two first order differential equations (7.1) and (7.5) are together referred to as the Swing Equation. In a coupled field-circuit model, the rotor angle is not a natural state variable. The load angle at time t can nevertheless be determined as t δ(t)=δ0 + ωs Δωm dt, (7.6) 0 where δ0 denotes an initial rotor angle that can be estimated from a magnetostatic field solution [44].

70 Figure 7.1: Stretched spring analogy of a generator connected to a strong grid (cour- tesy of Mr. J. Bladh). The torque coefﬁcients Ks and Kd are represented by a mechanical spring and a dashpot respectively.

7.2.2 Damping and Synchronizing Torques Rotor angle oscillations of grid-connected synchronous generators are associated with changes in electrical torque. For small oscillations, the associated change in electrical torque is traditionally assumed to consist of one part in time phase with the angular frequency deviation, Δω, and one part in time phase with the rotor angle deviation, Δδ. Mathematically, this is expressed as

ΔTe = KsΔδ + KdΔω. (7.7)

The coefficients Ks and Kd are referred to as the synchronizing and damping torque coefficients respectively. In p.u., the electrical angular frequency deviation Δω is numerically equal to the mechanical angular frequency deviation, Δωm. Ks and Kd can, in simple terms, be thought of as the “spring and damping constants” of the rotor assembly in a synchronous reference frame. The rotor angle can in turn be regarded as a measure of the spring displacement. The analogy is illustrated in Fig. 7.1. The stretched spring analogy is complicated by the fact that both Ks and Kd depend on the active and reactive load as well as on the electric parameters of the generator and the power system to which the unit is connected. For stable operation, Ks and Kd both need to be positive. The synchronizing and damping torque coefficients can be determined in different ways. Alden and Shaltout [70] presented a method to calculate Ks and Kd from the time response signals ΔTe, Δδ and Δω following a minor system disturbance. The method is based on a least-square principle and leads

71 to the system of equations nT nT 2 ΔTe(t)Δδ(t)dt = Ks (Δδ(t)) dt 0 0 nT (7.8) + Kd Δω(t)Δδ(t)dt 0 nT nT ΔTe(t)Δω(t)dt = Ks Δδ(t)Δω(t)dt 0 0 nT (7.9) 2 + Kd (Δω(t)) dt, 0 where n is a positive integer and T denotes the oscillation period. If the time response signals are not available, it is also possible to estimate Ks and Kd from analytical formulae, like Park’s electrical torque equation [20]. This equation provides the ratio between change in electrical torque and change in rotor angle of a synchronous machine connected to an inﬁnite bus, and subject to rotor oscillations around an average rotor angle δ0.Itcan be written on operational form as ΔT N (s)N (s)+N (s)N (s) e (s)= 1 2 3 4 . (7.10) Δδ D(s)

Here,

N1(s)=Ψd0 + id0Xq(s) (7.11) N2(s)=(Usinδ0 + Ψd0s)Zd(s)+(Ucosδ0 + Ψq0s)Xd(s) (7.12) N3(s)=Ψq0 + iq0Xd(s) (7.13) N4(s)=(Ucosδ0 + Ψq0s)Zq(s) − (Usinδ0 + Ψd0s)Xq(s) (7.14) D(s)=Zd(s)Zq(s)+Xd(s)Xq(s). (7.15)

Ψd0, Ψq0, id0, iq0 here denote steady-state values of ﬂux and current in the two axes, and s is the complex operator. Zd(s), Zq(s), Xd(s) and Xq(s) denote operational impedances and reactances of the direct- and quadrature axis respectively, and U is the terminal voltage. For steady pulsations of frequency ω0 (p.u.), s can be replaced by jω0, and Ks and Kd can be identiﬁed from the real and imaginary parts of the resulting expression according to Δ Te ( ω )= + ω . Δδ j 0 Ks j 0Kd (7.16)

72 Figure 7.2: Schematic overview of the experimental setup.

7.3 Study Summary 7.3.1 Torque Coefficient Determination from a Field Model In Papers VII and IX, the electromechanical properties of coupled field-circuit models of hydrogenerators connected to infinite busbars were analyzed. To get a quantitative assessment of the model features, the torque coefficients Ks and Kd were derived from oscillatory responses initiated by small system disturbances. The results were compared to the torque coefficients derived from two-axis model simulations of the same units. The electrical dynamics of the employed two-axis model was governed by differential and algebraic equations that can be derived from (2.3)-(2.12). To this end, network transient and stator transformer voltage terms were neglected according to the recommended practice [74]. The infinite busbar coupling was considered in the additional equations

ed = UBd (7.17) eq = UBq, (7.18) where UBd and UBq denote the d- and q-axis components of the inﬁnite bus voltage.

7.3.2 Experimental Study In Paper VIII, the effect of damper windings on the electromechanical damping capability of a laboratory generator was assessed. The generator was installed in the experimental setup illustrated in Fig. 7.2. The central part of the installation is a vertical-axis three-phase salient-pole synchronous generator. Shaft torque is provided by a DTC induction motor drive through an intermediate gearbox. Ratings and dimensions of the test generator are given in Table 7.1. The laminated pole shoes have three slots where it is possible to insert damper bars. The center slot is located in the middle of the pole face and the outer slots are located a distance τs and 1.2τs from the center slot respectively (τs = stator slot pitch). The damper winding used in the experiments consisted of insulated copper bars. To form a closed squirrel cage, a copper end-ring

73 Table 7.1: Test Generator Data Rated power (kVA) 75 Air-gap (mm) 8.3 Rated voltage (V) 156 Length (mm) 303 Frequency (Hz) 50 Rotor weight (kg) 900 Speed (rpm) 500 Inertia constant (s) 1.37 Inner stator diameter (mm) 725 Drive motor power (kW) 75 Outer stator diameter (mm) 872

Table 7.2: Torque Coefﬁcients and Oscillation Frequency FE Model Circuit Model

Kd (p.u torque/(rad/s)) 0.14 0.090

Ks (p.u torque/rad) 5.6 3.3 Frequency (Hz) 2.60 2.03 connection can be installed between the pole damper cages with bolted joints. Fig. 7.3 shows a collection of photos of the experimental setup. A disturbance was initiated by a step change in the drive torque. The system damping for different damper winding conﬁgurations was quantiﬁed with a damping time constant, τD. The time constant was determined from the rate of decrement of the response in instantaneous power.

7.4 Selected Results 7.4.1 Comparison of Field and Circuit Model Responses Table 7.2 shows the damping and synchronizing torque coefﬁcients calculated for rated operation of Generator I in Paper VII. The calculated fundamental mode oscillation frequency is also provided. There is a striking discrepancy between the damping and synchronizing torques extracted from the FE model and those obtained in the two-axis model simulations. The FE model is seen to be much stiffer (higher Ks) and also exhibits higher inherent damping. Further investigations revealed that the introduction of the inter-pole end-ring connection in the damper circuit equations (2.42) accounted for an important part of the synchronizing and damping torque production in the FE model. This fact is highlighted in Table 7.3, where Kd and Ks of the FE model are shown for both a continuous and non- continuous damper conﬁguration. With the inter-pole connection removed, the stiffness of the FE model is reduced with almost 40 % and the inherent electromagnetic damping is reduced to zero.

74 Figure 7.3: Photos of the experimental setup. (a) Stator frame. (b) Rotor poles, slip rings, brushes. (c) Synchronization equipment (left) and frequency converter (right). (d) Midway opening of an armature winding phase. This feature is introduced to operate the generator with two parallel circuits per armature phase. (e) Data acquisition system. (f) Transformer.

The model discrepancies seen in Table 7.2 represent an extreme case. Nev- ertheless, the typical agreement between FE models and two-axis models with respect to electromechanical transient performance was also found to be poor. One plausible reason for this could be that the employed two-axis model parameter sets lacked sufﬁcient accuracy for the investigation at hand. However, to produce the stiffness and damping levels seen in the FE models, severe mis- calculations of a number of key parameters are required. This is illustrated in Fig. 7.4, where the dependency of Kd and Ks of Generator I in Paper VII on the q-axis damper parameters L1q and R1q is shown. The uppermost curve in

75 Table 7.3: Torque Coefﬁcient Dependency on Damper Winding Type Damper winding Continuous Non-continuous No damper

Kd (p.u torque/(rad/s)) 0.14 0.004 0.0001

Ks (p.u torque/rad) 5.64 3.55 3.42 Frequency (Hz) 2.6 2.1 2.1 each subfigure illustrates a case with a very efficient damper in combination with an armature leakage inductance that is smaller than the one used in the simulations. The curves were obtained using (7.10). All the tested coupled field-circuit SMIB models were found to exhibit sig- nificantly higher stiffness and damping properties compared to their two-axis model equivalents when a low-impedance inter-pole coupling was present in the damper winding. Additional research is however needed to decide whether the predicted effect of the inter-pole coupling is accurate or if it is overesti- mated in the coupled field-circuit model.

7.4.2 Experimental Study In a first series of tests, the appearance of the instantaneous power delivered by test generator to the grid was observed for different damper winding configurations. Typically, the instantaneous power of grid-connected generators consists of a mean value, dictated by the prime mover, modulated by power pulsations at the natural oscillation frequency of the system. Fig. 7.5 shows the measured power pulsation amplitudes versus mean active power output for different damper winding configurations. It is clearly seen that the introduction of a continuous damper results in a more stable power output (lower pulsation amplitude). The effect is most pronounced when the mean power output is small. It is furthermore observed that the problem with power pulsations is worse when a non-continuous damper winding is installed compared to when the generator has no damper winding at all. Figs. 7.6 and 7.7 show measured and simulated responses to a step change in the drive torque for the continuous and non-continuous damper configurations respectively. The figure captions state the damping time constant (τD), the oscillation frequency ( f0) and the %-overshoot of the respective signals. The simulated responses were obtained from a system model set up in the MATLAB SIMULINK simulation environment. The measured damping time constant for the generator with a continuous damper winding was 3.0 seconds. This was in good agreement with the simulated response (τD = 3.1 s). The measured damping time constant for a non- continuous damper configuration was found to be 13.8 seconds. The corre-

76 Figure 7.4: Dependency of the damping and synchronizing torque coefficients on the parameters L1q and R1q. The employed base parameter set corresponds to Generator I in Paper VII. (a) Synchronizing torque coefficient. (b) Damping torque coefficient. The normal settings are L1q = 0.066, R1q = 0.011, Ll = 0.15. Black crosses mark the position of the corresponding values of Kd and Ks. These values are also presented in Table 7.2.

sponding simulation predicted weak negative damping (τD =-34 s) at the studied point of operation.

77 0.25 Non−continuous damper No damper 0.2 Continuous damper

0.15

0.1 Amplitude (p.u.)

0.05 Sustained Power Oscillation

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Active Power Output (p.u.) Figure 7.5: Sustained oscillation amplitude at different points of operation. The ﬁeld current equals 13 A.

0.8 0.7 0.6 0.5 0.4 (p.u.)

power 0.3 0.2 Instantaneous 0.1 0 0 1 2 3 4 5 6 Time (s) (a) 0.8 0.7 0.6 0.5 0.4 (p.u.)

power 0.3 0.2 Instantaneous 0.1 0 0 1 2 3 4 5 6 Time (s) (b) Figure 7.6: Measured and simulated response in instantaneous power to a drive torque step change 0 → 0.4 p.u (continuous damper winding). (a) Measured response. τD = 3.0 s, f0 = 2.42 Hz, %-overshoot = 48%. (b) Simulated response. τD = 3.1 s, f0 = 2.60 Hz, %-overshoot = 55%.

78 1.2 1 0.8 0.6 0.4 (p.u.) power 0.2

Instantaneous 0 −0.2 −0.4 0 2 4 6 8 10 Time (s) (a) 1.2 1 0.8 0.6 0.4 (p.u.) power 0.2

Instantaneous 0 −0.2 −0.4 0 1 2 3 4 5 6 7 8 9 10 Time (s) (b) Figure 7.7: Measured and simulated response in instantaneous power to a drive torque step change 0 → 0.4 p.u (non-continuous damper winding). (a) Measured response. τD = 13.8 s, f0 = 2.37 Hz. (b) Simulated response. τD = -34 s, f0 = 2.60 Hz.

8. Conclusions

Permeance Models of Salient-pole Generators A permeance model of a salient-pole synchronous generator was developed. During the process of model development, special attention was devoted to the calculation of the damper reaction. This resulted both in a simpliﬁed treatment of the mutual damper loop coupling as well as the introduction of a new unitary damper loop MMF function. The permeance model was in good agreement with a 2-D FE model in terms of armature voltage harmonics and damper current distributions at open-circuit conditions. At rated load operation, the agreement in terms of damper current distribution was still fair. Mea- surements are required to get a conclusive validation of the simpliﬁed damper reaction model. An additional set of circuit equations were introduced in the permeance model to account for the effect of parallel armature circuit currents on the UMP during steady eccentric conditions. The permeance model correctly predicted a reduced force resultant when parallel armature circuits were considered. It was furthermore found that model was incapable of reproducing the details of the UMP, but that the agreement between the measured and predicted average radial force resultants was acceptable.

Core Loss Prediction in Large Hydroelectric Generators A three-term loss model corrected for rotational effects was found to typically yield core loss estimates on the order of 65% of the measured total electromagnetic loss. The spread in the ratio between calculated core loss and measured total electromagnetic loss was however substantial in the set of twelve investigated generators. The discrepancies between the measured loss figures and the calculated core losses are attributable to stray no-load losses and modeling inaccuracies. A time-domain iron loss model corrected for rotational effects on the average yielded a core loss estimate that was 28% higher than the loss figure predicted by a classical frequency domain model. It was finally suggested that the average degree of flux rotation in the stator core, and hence the additional rotational loss, is correlated to the stator teeth dimensions.

Form Factors of Salient Pole Shoes Air-gap ﬂux density waveforms in salient-pole synchronous machines with large air-gap diameters were characterized in terms of pole shoe form

81 factors and THD. The flux density waveforms were obtained with 2-D FEA, and hence the influence of high-order flux density harmonics and iron saturation were appropriately considered. The design of the pole face contour was found to have a significant impact on the form factors, and on the form factors’ susceptibility to changes in basic geometrical parameters. Linear models for the calculation of form factors of arbitrary pole shoe geometries were derived. Models of high accuracy could only be established if the distributed effect of iron saturation on the flux density waveform was considered.

Analysis of a PM Generator with Two Contra-rotating Rotors A finite element model of a radial flux PM generator topology with two contra-rotating rotors was realized and studied. Synchronized speed operation was found to give acceptable operational characteristics while asynchronous rotor speed operation resulted in significant torque pulsations. It is therefore concluded that the proposed generator is not a suitable choice in energy conversion schemes where the two stages of the contra-rotating prime mover operate at different speeds. The nature and magnitude of the inter-rotor cross coupling disturbance in this type of electrical machines was also studied. At synchronized rotor operation, a standing flux density wave that upsets the three-phase symmetry was discovered. The introduction of a central iron core was found to effectively eliminate the standing wave disturbance.

Electromechanical Transients A coupled field-circuit model of a grid-connected hydroelectric generator was realized and the damping and synchronizing torques generated during rotor angle oscillations were studied. The introduction of a low-impedance connection between the pole damper cages (i.e. a short-circuit ring) was found to have a very strong impact on the damping and synchronizing torques of the field model. For generators with continuous damper winding configurations, large deviations between the electromechanical responses of the field and two-axis models were typically observed. Further research and additional numerical comparisons with two-axis models derived from a Standstill Frequency Re- sponse Test data are needed to confirm the findings. The importance of the damper inter-pole coupling for the damping of rotor angle oscillations was also established experimentally.

82 9. Suggested Future Work

Permeance Models of Salient-pole Generators The validity of the presented semi-analytic permeance model needs to be verified with experimental data. Preparations for this upcoming work are cur- rently in progress. There is ample room both for improvements and simplifications of the model. For instance, the process of determining the different permeance functions could be simplified in some cases. A pure analytical approach was tested, but was found not to yield sufficiently accurate results in the armature voltage harmonics prediction application. It is nevertheless likely that permeance functions determined from analytical formulae would provide reasonable accuracy in other applications, such as UMP calculations. In conclusion, the degree of required modeling refinement should be anticipated to be application- dependent. The author at present does not consider the “elimination” of the FE-step from the program to be a prioritized concern, because of the relatively small computational burden associated with 2-D magnetostatic field solutions. Furthermore, FE software is more and more becoming an integrated part of the modern machine designer’s toolbox.

Core Loss Prediction in Large Hydroelectric Generators The degree of model refinement should be increased to check how this affects the results. A first step could be to introduce Bmax-dependent loss coefficients to obtain better fits with measured loss data. The results presented in this thesis have indicated that there is a correlation between the core loss attributable to flux rotation and the dimensions of the stator teeth. This trace could be followed a little further by systematically varying the slot dimensions of a single test generator, and then examining the corresponding variation of the additional rotational loss. Depending on the outcome of such a study, a prediction model for the additional rotational loss could perhaps be worked out using standard regression methods. Even though iron losses is an interesting subject, improved methods for stray loss prediction have, from a scientific point of view, better prospects of generating relevant results. This topic is also in line with the present interest from the industry.

83 Form Factors of Salient Pole Shoes Manufacturers of salient-pole synchronous machines usually employ a number of standard pole shoe geometries with well-known magnetic properties. In the presented study, the author wished to provide a new perspective on pole shoe shape selection and explore the machine designer’s possibilities if the constraints in a new design (be they thermal or mechanical) do not allow for any of the standard pole shoes to be used. In such a situation, the linear prediction models can perhaps be useful. In a future study on the shape of salient pole-shoes, the author would like to see the subject of pole shoe shape selection for optimal operation at different load conditions be addressed.

Analysis of a PM Generator with Two Contra-rotating Rotors The practical interest in realizing a prototype radial flux generator with two contra-rotating rotors is most likely small. The design implies numerous con- structional challenges, for instance the mounting of the central core and the stator winding. Moreover, the connection between the stator winding and stationary external terminals would most likely be tedious to realize. Finally, maintenance operations are expected to be laborious, due to the “in-built” nature of the machine. The findings related to the magnetic inter-rotor cross coupling are relevant also to an axial flux machine topology, which is a superior design in contra-rotating applications. A foremost concern in subsequent studies is the assessment of possible technological and economical benefits of using a single contra-rotating electrical machine instead of two conventional machines in a contra-rotating drive train.

Electromechanical Transients The coupled field-circuit model of a grid-connected generator is not intended for use in power system studies. However, the model might find applications in detailed diagnosis of phenomena related to generator-grid interaction, since it provides the internal generator operating conditions. Additional efforts must however be made to reduce the computational burden. Moreover, model validation with test data is crucial. In particular, a check of the correct- ness of the predicted effect of squirrel-cage damper windings is needed. With today’s effective controlled damping through Power System Stabiliz- ers, the role of the damper winding during hunting is of somewhat secondary importance. The author therefore suggests that future studies related to damper winding design should address the effectiveness of the winding’s supplementary functions (field winding overvoltage protection, flux density harmonic reduction, subtransient saliency ratio and so forth). Future experimental work concerned with grid-connected operation should be devoted to studies which involve excitation control, since this is a more realistic system configuration. The damper currents during various transients

84 should also be monitored. From a purely academic perspective, a projection of the measured damper bar currents on the direct and quadrature equivalent damper windings would provide for an interesting assessment of the ability of different two-axis model structures to correctly predict the damper reaction.

10. Summary of Papers

In this chapter, short summaries of the contents of the papers are presented and the author’s contribution to each paper is speciﬁed.

Paper I On Permeance Modeling of Large Hydrogenerators With Application to Voltage Harmonics Prediction A semi-analytical permeance model is used to calculate the THD of the rated open-circuit armature voltage waveform of hydroelectric generators with integral slot windings. The appearance of the damper loop MMF waveform is modified following observations of the radial flux distribution set up by a single damper loop current. A simplified method to handle mutual couplings in the damper network equations is also introduced. Results from permeance model calculations are shown to be in fair agreement with results obtained with transient finite element analysis. The author developed the semi-analytical computer model, analyzed calculation data and is the main author of the paper. The paper is published in IEEE Transactions on Energy Conversion, vol. 25, pp. 1179-1186, Dec. 2010.

87 Paper II The Rotating Field Method Applied to Damper Loss Calculation in Large Hydrogenerators A permeance model is used to calculate damper bar currents and the associated ohmic losses during balanced and unbalanced load operation of a large hydroelectric generator. The agreement between calculated damper bar currents and bar currents obtained from coupled ﬁeld-circuit simulations are in fair agreement for balanced load operation, considering the simplicity of the model. For unbalanced load operation, large deviations in the current mag- nitudes are however seen for the outermost bars. The advantages of permeance models in design studies, such as computational speed and model trans- parency, are emphasized. The author extended the permeance model discussed in Paper I. He carried out all calculations and the work associated with data analysis. He is the main author of the paper. The paper was presented at the XIX Int. Conf. on Electrical Machines, Rome, Italy, Sept. 6-8 2010.

Paper III Reduction of unbalanced magnetic pull in synchronous machines due to parallel circuits The impact of currents circulating between parallel armature circuits on the UMP in synchronous machines with off-centered rotors is assessed in a series of experiments. Two calculation schemes are also used to determine the UMP, a sophisticated transient ﬁnite element model and a simple linear permeance model. Both models were found to give accurate predictions of the radial UMP reduction. When switching from one to two parallel circuits per stator phase, the maximal reduction of the radial UMP was found to be on the order of 60%. The author adapted the permeance model discussed in Paper I such that it could be used study to the problem at hand. He wrote a section of the paper. The paper was submitted to IEEE Transactions on Magnetics for peer-review on March 14, 2011.

88 Paper IV Core Loss Prediction in Large Hydrogenerators: Influence of Rotational Fields The accuracy of three-term loss prediction schemes corrected for flux bidirectionality when used for core loss estimation in large hydropower generators is discussed. Core loss estimates obtained from the field distribution predicted by transient 2-D finite element analysis were typically on the order of 65% of the measured electromagnetic no-load loss. The study suggested that the additional loss attributable to rotational flux is influenced by the stator slot (tooth) dimensions. The author suggested and prepared the studied iron loss models. He also carried out the major part of the work associated with computer simulations and data analysis. He is the main author of the paper. The paper is published in IEEE Transactions on Magnetics, vol. 45, pp. 3200- 3206, Aug. 2009.

Paper V Form Factors and Harmonic Imprint of Salient Pole Shoes in Large Synchronous Machines The paper discusses the form factors that are commonly used to model saliency effects in electrical machine design codes. Pole shoes with different pole face contour designs are studied in detail with ﬁnite element analysis. The harmonic imprint of the pole shoe shape on the air-gap ﬂux density waveform is also considered. Form factor dependencies on different geometrical quantities as well as the level of iron saturation are studied. Linear models for the calculation of form factors are derived. The prediction models typically exhibit excellent accuracy if a variable that considers the level of saturation is included. The author did most of the work associated with this study and is the main author of the paper. The paper was accepted for publication in Electric Power Components and Systems on Dec. 2, 2010.

89 Paper VI Finite Element Analysis of a Permanent Magnet Machine with Two Contra-rotating Rotors The paper is concerned with basic no-load and load operational characteristics of a PM generator with two contra-rotating rotors. Particular attention is devoted to a pulsating inter-rotor ﬂux distortion that is introduced via common core paths. It is shown that the distortion will be negligible if the stator core is sufﬁciently wide. Load simulations of a slotless air-gap wound generator appropriate for laboratory experiments indicated acceptable machine performance during identical speed rotor operation. The author was responsible for computer simulations, data analysis, and is the main author of the paper. The paper is published in Electric Power Components and Systems, vol. 37, pp. 1334-1347, Dec. 2009.

Paper VII Use of a Finite Element Model for the Determination of Damping and Synchronizing Torques of Hydroelectric Generators Damping and synchronizing torque coefficients are derived from time-stepped finite element simulations of a hydroelectric generator connected to an infinite busbar. Torque coefficients are also derived from equivalent circuit simulations, and a comparison between the results of the two methods is made. Particular attention is devoted to the impact of the damper winding type (continuous or non-continuous) on the transient electromechanical response. Finite element models are found to exhibit both higher damping and higher synchronizing properties compared to equivalent circuit models of the studied machine type. The author assisted in the development of the finite element model code, wrote the equivalent circuit simulation program and the parameter calculation script, and carried out data analysis. He is the main author of the paper. The paper was submitted to The International Journal of Electrical Power and Energy Systems for peer-review on May 11 2010.

90 Paper VIII Experimental Study of the Effect of Damper Windings on Synchronous Generator Hunting The damping properties of a 75 kVA vertical-axis laboratory synchronous generator with respect to electromechanical oscillations are determined experimentally. Damping time constants are derived from the oscillatory response in electrical generator power initiated by step changes in the drive torque. The experimental responses are further compared with calculated responses, and the predictive precision of the used system model is assessed. The damping in the tested unit is found to be highly susceptible to the impedance of the electrical connection between the damper cages on adjacent poles. In two-axis circuit terminology, this corresponds to the presence or absence of an effective q-axis damper. The author installed the synchronization unit needed to achieve grid-connected generator operation, as well as voltage and current metering devices. He also assisted in the construction of the damper cage and performed the experimental work and data analysis. He is the main author of the paper. The paper was submitted to Electric Power Components and Systems for peer- review on Feb. 3 2011.

Paper IX Comparison of field and circuit generator models in single machine infinite bus system simulations The paper compares the transient electromechanical response of a coupled field-circuit model of a single machine infinite bus system to that of a model where the generator is represented by equivalent circuits. The characteristics of the two models are made equal as far as possible by using the finite element model for the estimation of circuit parameters. The finite element model is found to exhibit higher stiffness and higher damping. The differences in model response are believed to be attributable to the diverse representations of the rotor circuits. The author wrote parts of the equivalent circuit simulation program and contributed with ideas in the development process of the coupled field-circuit model. He also wrote a short section of the paper. The paper was presented at the XIX Int. Conf. on Electrical Machines, Rome, Italy, Sept. 6-8 2010.

91 Paper X Design and construction of a synchronous generator test setup The paper describes practical design considerations for a synchronous generator test setup, to be used in studies of off-centered rotor operation. Advantages and disadvantages of mechanical and instrumentation solutions are discussed. The slot harmonic amplitudes in the open-circuit armature voltage waveform for two different damper winding conﬁgurations are provided as a ﬁrst example of measurements. The author contributed to the design, construction and installation of the magnetization equipment, the generator terminal enclosure and various measurement transducers. He performed the open-circuit voltage waveform analysis and wrote a short section of the paper. The paper was presented at the XIX Int. Conf. on Electrical Machines, Rome, Italy, Sept. 6-8 2010.

92 11. Summary in Swedish

Elektromagnetisk analys av vattenkraftgeneratorer Vattenkraften bibehåller sin position som världens viktigaste förnybara energislag. Tekniken är efter mer än hundra års utveckling både mogen och tillförlitlig och verkningsgraden i storskaliga vattenkraftverk är mycket hög. Medan vattenkraftsutbyggnaden ännu fortgår i Asien och Sydamerika, så genomgår de flesta europeiska länder med vattenkraftsresurser en fas av omfattande uppgradering och förnyelse av den befintliga maskinparken. I Sverige står vattenkraftindustrin inför utmaningar i form av kompetensöverföring till kommande generationer samt anpassning av de uppgraderade stationerna till förändrade driftförhållanden. Datoriserade hjälpmedel har i grunden förändrat det ingenjörsmässiga design- och analysarbete som är förknippat med konstruktionen av ett vattenkraftverk och dess huvudkomponenter. Den här doktorsavhandlingen behandlar en av vattenkraftverkets nyckelkomponenter - generatorn - samt hur en rad designaspekter av elektromagnetisk natur kan hanteras med moderna beräkningsmetoder. I synnerhet så demonstreras en rad tillämpningar av finita elementmodeller samt roterande fältmodeller. I en första studie presenteras en roterande fältmodell för noggrann beräkn- ing av den magnetiska luftgapsflödestäthetens vågform. En förenklad metod för att beräkna dämplindningens magnetiska reaktionsflöde förevisas också. Modellen har med framgång använts för att beräkna spårtoner i en generators tomgångsspänningskurvform, samt strömmar i dämplindningen vid såväl tomgång som vid last. En annan studie har tillägnats de magnetiska rotationsförluster som upp- kommer till följd av bidirektionella magnetflöden i statorkärnan. I kombina- tion med vissa dynamiska effekter befanns rotationsförlusterna typiskt öka den totala järnförlustskattningen med ca 28%. Beräkningsresultaten påvisade även en korrelation mellan rotationsförlusternas storlek och statorspårens di- mensioner. Avhandlingen presenterar även linjära modeller för beräkning av formfak- torer för utpräglade polskor av godtycklig geometri och mättnadsgrad. En översikt av hur polplattan bör väljas att få önskad luftgapsflödestäthetsvåg- form ges. Slutligen redovisas en numerisk studie av de elektromekaniska egenskaperna hos finita elementmodeller av nätanslutna vattenkraftgenerator. Kortslutningsringens betydelse för modellens dämpande egenskaper vid

93 rotorvinkelpendlingar betonas särskilt. Slutsatserna från denna studie veriﬁerades i en serie experiment, där rotorvinkelpendlingar initierades med kontrollerade momentstötar.

94 Acknowledgments

The research presented in this thesis was carried out as a part of The Swedish Hydropower Centre (Svenskt Vattenkraftcentrum, SVC). SVC was established by The Swedish Energy Agency, Elforsk, The Swedish National Grid Agency together with Luleå University of Technology, The Royal Institute of Technology, Chalmers University of Technology and Uppsala University.

All members of the SVC steering committee for research in the ﬁeld of turbines and generators are acknowledged for guidance and advice.

Anders Hagnestål, Simon Tyrberg and Katarina Yuen-Lasson, Uppsala University, are acknowledged for their help with proof-reading.

The author additionally would like to express gratitude to these persons:

Niklas Dahlbäck, Vattenfall Vattenkraft, Göran Franzén, BEVI AB, Thomas Götschl, Uppsala University, Gunnel Ivarsson, Uppsala University, Dr. Thommy Karlsson, Vattenfall Power Consultant, Peter Ljung, Vattenfall Vattenkraft, Gunilla Ries-Jende, Vattenfall Power Consultant, Ulf Ring, Uppsala University, Richard Perers, VG Power / Voith Siemens, Dr. Anna Wolfbrandt, E-ON ES, and Dr. Arne Wolfbrandt, Uppsala University.

Finally, a special thanks is addressed to the following persons:

My colleague Johan Bladh, Vattenfall Research and Development, for his friendship, good advice and support during these four years of joint efforts.

My colleague Mattias Wallin, Uppsala University, for rewarding discussions. The author is also indebted to Mr. Wallin for his untiring efforts with the test generator.

My supervisor Dr. Urban Lundin, Uppsala University, for his guidance and support throughout the project.

95 My assistant supervisor Prof. Mats Leijon, Uppsala University, for inspiration and support.

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