List of Papers
This thesis is based on the following papers, which are referred to in the text
by their Roman numerals.
I
II
III
IV
V
VI
VII
VIII
IX
Ranlöf, M., Perers R. and Lundin U., “On Permeance Modeling of
Large Hydrogenerators With Application to Voltage Harmonics Prediction”, IEEE Trans. on Energy Conversion, vol. 25, pp. 1179-1186, Dec.
2010.
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.
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.
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.
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.
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.
Ranlöf, M. and Lundin U., “Use of a Finite Element Model for the
Determination of Damping and Synchronizing Torques of Hydroelectric Generators”, submitted to The Int. Journal of Electrical Power and
Energy Systems, May 2010.
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.
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 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Applications of Permeance Models of Salient-pole Generators .
1.3 Core Loss Prediction in Large Hydropower Generators . . . . . .
1.4 Form Factors of Salient Pole Shoes . . . . . . . . . . . . . . . . . . . . .
1.5 Analysis of a PM Generator with Two Contra-rotating Rotors .
1.6 Electromechanical Transients - Simulation and Experiments . .
1.7 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Salient-pole Synchronous Generators . . . . . . . . . . . . . . . . . . . .
2.1.1 Main Construction Elements . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Grid-connected Operation . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Equivalent Circuit Generator Model . . . . . . . . . . . . . . . . . . . . .
2.2.1 P.U. Electrical Equations . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Finite Element Generator Model . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Calculation Geometry and Material Property Assignment .
2.3.2 Field Equation Formulation . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Finite Element Discretization . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.5 Calculation of Air-gap Torque and Induced EMF . . . . . . .
2.4 Coupled Field-circuit Models . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Coupling Equations for Circuit-connected Conductors . . .
2.4.2 Rated Voltage No-load Operation Model . . . . . . . . . . . . .
2.4.3 Balanced and Unbalanced Load Models . . . . . . . . . . . . . .
2.4.4 Grid-connected FE Model with Mechanical Equation . . . .
3 Applications of Permeance Models of Salient-pole Generators . . . .
3.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Permeance Model Implementation . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Field and Armature MMF Functions . . . . . . . . . . . . . . . .
3.2.3 Pole Shape Permeance Function . . . . . . . . . . . . . . . . . . . .
3.2.4 Saturation and Stator Slot Permeance Functions . . . . . . . .
3.3 Damper Winding MMF and Circuit Equations . . . . . . . . . . . . .
3.3.1 Flux Density Harmonics . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Unitary Damper Loop MMF Functions . . . . . . . . . . . . . .
3.3.3 Calculation of Damper Loop Currents . . . . . . . . . . . . . . .
vii
1
1
2
3
3
4
4
5
7
7
7
9
10
11
13
13
14
16
17
18
19
19
20
23
25
27
27
28
28
29
31
31
33
34
36
37
3.3.4 Resultant Damper MMF . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Selected Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 THD of the Open-circuit Armature Voltage Waveform . . .
3.4.2 Damper Bar Currents at Rated Load Operation . . . . . . . . .
3.4.3 Reduction of the UMP by Parallel Armature Circuits . . . .
4 Core Loss Prediction in Large Hydroelectric Generators . . . . . . . . .
4.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Iron Loss Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Loss Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Rotational Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Study Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Selected Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Form Factors of Salient Pole Shoes . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Pole Shoe Form Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Study Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Pole Face Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Pole Shoe Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Selected Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Effect of Pole Face Contour . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Linear Models with Saturation Considered . . . . . . . . . . . .
5.4.3 Perspectives on Pole Shoe Shape Selection . . . . . . . . . . . .
6 Analysis of a PM Generator with Two Contra-rotating Rotors . . . . .
6.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Generator Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Dual Contra-rotating Rotor Topology . . . . . . . . . . . . . . . .
6.2.2 Reference Machine Topologies . . . . . . . . . . . . . . . . . . . .
6.3 Selected Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Characterization of the Inter-rotor Cross Coupling . . . . . .
6.3.2 Synchronized Contra-rotating Load Operation . . . . . . . . .
7 Electromechanical Transients - Simulation and Experiments . . . . . .
7.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Rotor Angle Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1 The Swing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.2 Damping and Synchronizing Torques . . . . . . . . . . . . . . . .
7.3 Study Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.1 Torque Coefficient Determination from a Field Model . . .
7.3.2 Experimental Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Selected Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Comparison of Field and Circuit Model Responses . . . . . .
7.4.2 Experimental Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9 Suggested Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 Summary of Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
40
41
41
42
43
45
45
45
45
46
48
50
53
53
54
55
56
57
58
58
59
60
61
61
61
61
62
63
63
66
69
69
69
70
71
73
73
73
74
74
76
81
83
87
11 Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
93
95
97
List of Symbols and Abbreviations
Fields
Symbol
A
B
H
J
Unit
Tm
T
A/m
A/m2
Definition
Magnetic vector potential
Magnetic flux density / induction
Magnetic field
Current density
Symbol
Az
bp
Bgm
Bmax
ΔBr
ei
Unit
Tm
m
T
T
T
V
ed
p.u.
efd
eq
p.u.
p.u.
E
f
fa
f0
h pp
H
V or p.u.
Hz
Hz
m
s
Definition
Z-component of magnetic vector potential
Pole body width
Peak value of air-gap flux density wave
Peak flux density
Radial flux density distortion
Induced EMF in armature phase i (i =
a, b, c) (field model)
Direct-axis armature voltage (equivalent
circuit model)
Field voltage (equivalent circuit model)
Quadrature-axis armature voltage (equivalent circuit model)
Internal EMF
Electrical frequency
Pole taper
Hunting frequency
Pole shoe height
Inertia constant
Scalars
xi
Scalars (continued)
Symbol
ij
id
ifd
Unit
A
p.u.
p.u.
iq
i1d
i1q
If
J
kc
kd
kE
kf
kH
kq
p.u.
p.u.
p.u.
A
kgm2
Sm4 /kg
Am3 V−0.5 kg−1
Am4 (V s kg )−1
-
Kd
Ks
le
Lad
Laq
Le
Lfd
Ll
L1d
p.u. torque /
(rad/s)
p.u. torque / rad
m
p.u.
p.u.
H
p.u.
p.u.
p.u.
L1q
p.u.
Ma
MD
Mf
A·turns
A·turns
A·turns
Definition
Current in armature phase j ( j = a, b, c)
Direct-axis armature current
Field winding current (equivalent circuit
model)
Quadrature-axis armature current
Direct-axis damper current
Quadrature-axis damper current
Field current
Moment of inertia
Classical loss coefficient
Direct-axis armature pole shoe form factor
Excess loss coefficient
Field winding pole shoe form factor
Hysteresis loss coefficient
Quadrature-axis armature pole shoe form
factor
Damping torque coefficient
Synchronizing torque coefficient
Effective machine length
Direct-axis mutual inductance
Quadrature-axis mutual inductance
Armature end-winding leakage inductance
Field leakage inductance
Armature leakage inductance
Direct-axis damper winding leakage inductance
Quadrature-axis damper winding leakage
inductance
Armature winding MMF
Damper winding MMF
Field winding MMF
xii
Symbol
n
Nd
Nf
Np
q1
ptot
Padd−dyn
Unit
rpm
W/kg
%
Padd−rot
%
Ra
Rc
Re
Rfd
R1d
R1q
p.u.
Ω
Ω
p.u.
p.u.
p.u.
S
Te
ΔTe
Un
V
m2
Nm or p.u.
p.u.
V or p.u.
V
Xd
Xq
Zb
Γ
δ
δ
Δδ
θ
θm
Λ
Λecc
Ω or p.u.
Ω or p.u.
Ω
Elect. rad.
m
Elect. rad.
Elect. rad.
Mech. rad.
Vs/(Am2 )
-
Definition
Rotational speed
Number of damper bars per pole
Number of field winding turns per pole
Pole pair number
Number of stator slots per pole and phase
Total specific iron loss
Fractional loss increase due to rotational
and harmonic fields
Fractional loss increase due to rotational
fields
Armature phase resistance
Inter-pole end-ring resistance
Armature end-winding resistance
Field winding resistance
Direct-axis damper winding resistance
Quadrature-axis damper winding resistance
Conductor area
Electrical torque
Change in electrical torque
Rated terminal voltage (RMS, line-to-line)
Electric potential / applied voltage (field
model)
Direct-axis synchronous reactance
Quadrature-axis synchronous reactance
Damper bar impedance
Degree of rotation
Rotor (load) angle (Chapters 2 and 7)
Air-gap length (Chapter 5)
Rotor angle deviation
Electrical angular coordinate
Mechanical angular coordinate
Air-gap permeance function
Eccentricity permeance function
xiii
Scalars (continued)
Symbol
ΛP
Λsat
ΛSslot
μr
μ0
ν
σ
τD
τds
τp
τ pc
τ pp
τs
φ
Ψ
Ψad
Ψaq
Unit
m−1
Vs/(Am)
m/H
S/m
s
m
m/m/m
Elect. rad.
Wb turns / p.u.
p.u.
p.u.
Ψd
Ψfd
Ψq
p.u.
p.u.
p.u.
Ψ1d
Ψ1q
p.u.
p.u.
ω
ωm
ωms
ωs
ω0
Δω
Elect. rad/s
Mech. rad/s
Mech. rad/s
Elect. rad/s
Mech. rad/s
p.u.
Definition
Pole-shape permeance function
Saturation permeance function
Stator slot permeance function
Relative magnetic permeability
Permeability of free space
Magnetic reluctivity
Electric conductivity
Damping time constant
Damper slot pitch
Pole pitch
Concentric pole shoe width
Pole shoe width
Stator slot pitch
Power factor angle
Flux linkage
Direct-axis mutual (air-gap) flux linkage
Quadrature-axis mutual (air-gap) flux linkage
Direct-axis armature winding flux linkage
Field winding flux linkage
Quadrature-axis armature winding flux
linkage
Direct-axis damper winding flux linkage
Quadrature-axis damper winding flux linkage
Electrical angular frequency
Mechanical angular frequency
Synchronous angular frequency
Synchronous angular frequency
Hunting angular frequency
Angular frequency (speed) deviation
xiv
Abbreviations
AC
DC
EC
EMF
FE
FEA
FEM
MMF
PM
p.u.
SiFe
SMIB
THD
UMP
Alternating Current
Direct Current
Equivalent Circuit
Electromotive Force
Finite Element
Finite Element Analysis
Finite Element Method
Magnetomotive Force
Permanent Magnet
Per Unit
Silicon-Iron alloy
Single Machine Infinite Bus
Total Harmonic Distortion
Unbalanced Magnetic Pull
xv
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]. Reliable 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 maintained 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 steadystate and transient operating conditions. The scope of the work is somewhat
1 Calculated
2
average between the years 2000-2008.
www.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 divided into five main subjects. The ten papers, which constitute the foundation 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) belongs 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 embodies 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 verticalaxis 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 computations [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 20 th 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 contrarotating 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 Chapter 8 and suggestions for future studies are given Chapter 9.
5
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 mandatory, 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
2.1.1
Salient-pole Synchronous Generators
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 likewise has many designations in the technical literature. Any of the terms open,
incomplete, non-continuous, or grill 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
1 Some
8
prefer to refer to this configuration simply as a squirrel cage winding.
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 N p 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 =
sin2δ
sinδ + U
−
Xd
2
Xq Xd
(2.2)
2
3EU
sin2 δ
2 cos δ
.
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
or load variations as long as the load angle does not exceed δC 2 [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.
2 This
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. Applications 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 L ad -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. 11102002 [29], but for completeness all symbols are also described in the List of
Symbols.
Stator voltage equations
ed =
eq =
dΨd
− Ψq ω − Ra id
dt
dΨq
+ Ψd ω − Ra iq
dt
(2.3)
(2.4)
Rotor voltage equations
efd =
0 =
0 =
dΨ f d
+ R f di f d
dt
dΨ1d
+ R1d i1d
dt
dΨ1q
+ R1q i1q
dt
(2.5)
(2.6)
(2.7)
Stator flux linkage equations
Ψd = −(Lad + Ll )id + Lad i f d + Lad i1d
Ψq = −(Laq + Ll )iq + Laq i1q
12
(2.8)
(2.9)
Rotor flux linkage equations
Ψ f d = −Lad id + (Lad + L f d )i f d + Lad i1d
Ψ1d = −Lad id + Lad i f d + (Lad + L1d )i1d
Ψ1q = −Laq iq + (Laq + L1q )i1q
(2.10)
(2.11)
(2.12)
Te = Ψad iq − Ψaq id
(2.13)
Air-gap torque
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 twodimensional 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
3 http://www.ansys.com/Products/Simulation+Technology/Electromagnetics
(accessed on January 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
∂ Az
∂y
∂ Az
= −
∂x
= 0.
Bx =
(2.14)
By
(2.15)
Bz
(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:
∂
∂
∂ Az (x, y,t)
∂ Az (x, y,t)
∂ Az (x, y,t)
∂ V (x, y,t)
+
=σ
ν
ν
+σ
∂x
∂x
∂y
∂y
∂t
∂z
Elsewhere:
∂
∂
∂ Az (x, y,t)
∂ Az (x, y,t)
+
=0
ν
ν
∂x
∂x
∂y
∂y
(2.17)
Here,
ν=
1
,
μr μ0
(2.18)
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:
1. The term σ ∂ V (x,y,t)
denotes the applied current density while the term
∂z
σ ∂ Az∂(x,y,t)
denotes the induced current density.
t
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 predetermined functional expressions. This is equivalent to connecting the
conductors to ideal current sources.
4 For
a full derivation of this equation see [40] or any textbook on finite element analysis of
electrical machines.
15
4. The term ∂ Az∂(x,y,t)
only appears explicitly in conductor subdomains treated
t
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 A z (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)
S
0
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 A∗z ,
Az (x, y,t)∗ =
N
∑ A j ϑ 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 A z 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 = le r0
Γ0
σt dγ ,
(2.21)
where le is the effective machine length, Γ0 is an arc in the air-gap, r 0 is the
arc radius and σt is the tangential stress. σt is given by
σt =
1
Br Bt ,
μ0
(2.22)
where Br and Bt denote the radial and tangential flux density components
respectively.
The magnetic flux crossing a surface of effective length l e 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
le
Az dS − ∑
Az dS ,
(2.24)
Ψ=
+
−
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
ew = −
2.4
dΨ
.
dt
(2.25)
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
dS − σ ψc = 0
Sc dt
∂ Vc
σ ψc + S c σ
+ I = 0,
∂z
σ
(2.26)
(2.27)
where Sc denotes the conductor area, Vc the applied conductor voltage, and I is
the current in the conductor. ψ c 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
c∈C +
−
∑ Vc
c∈C −
.
(2.28)
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
e2a + e2b + e2c
(2.29)
= Un ,
2
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 f d0 −V f d = 0
i f + − i f − = 0.
(2.30)
(2.31)
u f d0 is the field voltage at no-load operation at rated armature voltage and
speed and V f d is the potential drop across the entire field winding. V f d
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:
i = [i1 i2 ... in ]T
j = [ j1 j2 ... jn ]T
Vb = [Vb1 Vb2 ... Vbn ]T
ve = [ve1 ve2 ... ven ]T .
20
(2.32)
(2.33)
(2.34)
(2.35)
(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,
2Nd (continuous damper winding)
(2.36)
n=
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 vector v e :
i = M T j
MVb = 2ve
ve = Red j.
M denotes the (n × n) matrix
⎡
⎤
1 −1 0 . . . . . . 0
⎢ 0
1 −1 0 . . . 0 ⎥
⎢
⎥
⎢
⎥
⎢ 0
0
1 −1 0 . . . ⎥
⎢
⎥
M = ⎢ ..
..
.. .. ⎥
⎢ .
.
. ⎥
.
0
1
⎢
⎥
..
..
⎢ ..
.. .. .. ⎥
.
.
. ⎦
⎣ .
.
.
−1 0 . . . . . . 0 1
and Red denotes the diagonal (n × n) matrix
⎡
Re1 0 . . .
⎢
⎢ 0 Re2 0
⎢ .
..
⎢
.
Red = ⎢ ..
0
⎢ .
..
..
⎢ .
.
.
⎣ .
0
...
⎤
...
⎥
... ⎥
⎥
⎥
0 ⎥.
⎥
..
⎥
. 0 ⎦
. . . 0 Ren
...
0
..
.
(2.37)
(2.38)
(2.39)
(2.40)
(2.41)
From (2.37), (2.38), and (2.39), the following relation between i and Vb can
be established:
i = 1 M T R−1 M Vb
(2.42)
ed
2
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 f d0 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, R s denotes the resistance
of an armature phase and is implicitly modeled inside the field problem. The
quantities Va,FE , Vb,F E , and Vc,F E finally denote the electric potentials across
the armature phases, and are determined according to(2.28). Note that Va,F E ,
Vb,F E , and Vc,F E are not the terminal voltages, since they exclude the volt23
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
ia dt −
− RL ia − LL
−
dt
dt
CL
dib
dib
1
+ Re ib + Le
ib dt = 0
+ RL ib + LL
+
dt
dt
CL
Va,F E − Re ia − Le
−Vb,F E
(2.43)
Vb,FE
−Vc,F E
dib
dib
1
− Re ib − Le
ib dt −
− RL ib − LL
−
dt
dt
CL
dic
dic
1
+ Re ic + Le
ic dt = 0
+ RL ic + LL
+
dt
dt
CL
ia + ib + ic = 0
(2.44)
(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
− RLc ic − LLc
−
dt
dt
CLc
diN
1
iN dt = 0
−RN iN − LN
−
dt
CN
Vc,F E − Re ic − Le
ic dt −
(2.47)
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 u Bk (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
− uBa −
dt
dib
+ Re ib + Le
+ uBb = 0
dt
(2.48)
dib
− uBb −
dt
dic
+ Re ic + Le
+ uBc = 0
dt
(2.49)
Va,F E − Re ia − Le
−Vb,F E
Vb,F E − Re ib − Le
−Vc,F E
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
d ωm 1
= (Tm − Te ),
dt
J
(2.51)
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 converged (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 inspired by the works by Traxler-Samek et al. [2] and Knight et al. [45]. TraxlerSamek 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 airgap transformation factor that considers the “bending” of flux tubes of higherorder 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 salientpole 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 conformal 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 = μ0 ΛP Λsat ΛSslot .
(3.2)
ΛP here denotes the pole shape permeance function, Λ sat 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 = M f + 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 ω m 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
M f = N f I f M f n,
(3.4)
where N f denotes the number of field winding turns per pole and I f is the field
current. The function M f n denotes a unitary trapezoid function given by
Mf n =
4 1
π γf
sin γ f
sin(nN p (θm − ωmt)).
2
n=1,3,5,... n
∑
(3.5)
The spatial appearance of M f n is shown in Fig. 3.21 , where the angle γ f is
also defined. M f n has been plotted versus the fundamental electrical angular
coordinate, defined as
θ = N p θm .
(3.6)
The three-phase armature MMF can be expressed as
Ma = ia ·
∑
M̂n sin(nN p θm ) +
n=1,3,5,...
2π
) +
ib · ∑ M̂n sin nN p (θm −
3N p
n=1,3,5,...
2π
) ,
ic · ∑ M̂n sin nN p (θm +
3N p
n=1,3,5,...
(3.7)
1 The
illustrated function corresponds to the sum of the 25 first non-zero terms in the Fourier
series expansion.
29
Unitary field MMF function
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1.2
0
γ
f
30
60
90
120 150 180 210 240 270 300 330 360
Electrical angle, θ (°)
Unitary armature MMF function
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
−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 M̂n are given by
M̂n =
4 Ns 1
kd (n)k p (n)ksl (n),
π 2N p n
(3.8)
where Ns denotes the total number of winding turns per phase circuit. Expressions for the distribution, pitch and slot opening factors (symbols k d (n),
k p (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
−1
Pole shape permeance function (m )
50
45
40
35
30
25
20
15
10
5
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, Λ P , 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 B linear .
5. ΛP is obtained by dividing Blinear with M f . 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 challenging 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 Λ sat 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 B real .
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/Λ comb is calculated.
Contributions from space harmonics of orders 6nq 1 ± 1 (n = 1, 2 . . .) are
then subtracted from this function (q 1 denotes the number of stator slots
per pole and phase). The result is re-inversed and is denoted Λ sat .
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 Λ comb 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 Λ comb = 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.
Saturation permeance function
1.2
1
0.8
0.6
0.4
0.2
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. According 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
Stator slot permeance function
1.2
1
0.8
0.6
0.4
0.2
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:
∑
B̂+
n cos((nQs + N p )βm + nQs ωmt) −
∑
B̂−
n cos((nQs − N p )βm + nQs ωmt).
n=1,2,...
(3.9)
n=1,2,...
In (3.9),
34
βm = θm − ωmt
(3.10)
is a rotor-fixed angular coordinate, and Q s denotes the total number of stator
slots. The waves of (3.9) travel with linear speeds
vn = −
6q1 n ωτ p
6q1 n ± 1 π
(3.11)
with respect to the rotor, and induce EMFs of angular frequencies
ωn = n 6q1 ω
(n = 1, 2, . . .)
(3.12)
in the damper winding. Here, ω = N p ωm denotes the fundamental electrical
angular frequency, q 1 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:
ωm t
∑ B̂n cos(nNp(θm + n ) +
n=5,11,...
(3.13)
ωm t
B̂
cos(nN
(
θ
−
).
p m
∑ n
n
n=7,13,...
The waves travel with linear speeds
vn = −
τp n ± 1
ω
π n
(3.14)
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, . . .
(3.15)
ωn =
(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
τp
v2 = −2 ω
(3.16)
π
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
MDk (θ ,t) = ik (t)MDk0 (θ ,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
0
MDk0 (θ )dθ = 0
(3.19)
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
Classical unitary loop MMF function
1
0.8
0.6
0.4
0.2
0
−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
Modified unitary loop MMF function
1
0.8
0.6
0.4
0.2
0
−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 ω n , the circuit equations can be compactly written as
U = Zl Il + jωn MIl .
(3.20)
U here denotes a column vector that holds the induced loop EMFs, Z l 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 I l become (2Nd × 1)
vectors and Zl and M become (2Nd × 2Nd ) matrices.
Induced Loop EMF Vector
U contains the induced loop EMFs of frequency ω n . 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
⎛
⎞ ⎛
⎞
U1
Û1 e jα1
⎜
⎟ ⎜
⎟
⎜ U 2 ⎟ ⎜ Û2 e jα2 ⎟
⎟ ⎜
⎟.
U=⎜
(3.21)
..
⎜ .. ⎟ = ⎜
⎟
.
⎝ . ⎠ ⎝
⎠
UN
ÛN e jαN
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 ω n .
The amplitude of the bar EMF component introduced by the m-th flux density space harmonic that contribute to the frequency ω n is determined through
the flux cutting EMF equation as
b
Ûn,m
= lb vn,m B̂n,m .
(3.22)
lb here denotes the length of the damper bar and v n,m is the linear speed of the
m-th space harmonic with respect to the rotor. The flux density amplitude B̂n,m
is estimated from a Fourier series expansion of the flux density wave
Bδ (θm ,t = 0) = Λ(θm ,t = 0) · [M f (θm ,t = 0) + Ma (θm ,t = 0)],
(3.23)
and is subsequently modulated with a flux reduction coefficient that compensates 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
⎡
Z1
⎢
⎢ −Z
b
⎢
⎢
⎢ 0
⎢
Zl = ⎢ .
⎢ ..
⎢
⎢
⎢ 0
⎣
−Z b
−Z b
0
...
..
.
..
.
0
−Z b
...
..
.
0
..
.
Z2
..
.
−Z b
..
.
..
.
..
.
−Z b Z k −Z b
..
..
..
.
.
.
..
..
.
. −Z b
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
.
.. ⎥
. ⎥
⎥
.. ⎥
. ⎥
⎦
(3.24)
ZN
In the expression above, Z b denotes the bar impedance. Analytical expressions
presented in [57] were used to determine the AC resistances and leakage in
ductances of the damper bars. The impedances on the main diagonal, Z k , are
given by
Z k = 2(Z b + Z ek ),
(3.25)
39
where Z ek 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 L tt 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 averaging the air-gap permeance in front of the k-th damper loop.
With the introduced notation, the matrix M is formulated as
⎡
L11
⎢
⎢ −MDD
⎢
⎢
⎢
0
⎢
M=⎢
..
⎢
.
⎢
⎢
⎢
0
⎣
−MDD
3.3.4
−MDD
0
L22
..
.
−MDD
..
.
..
.
..
.
−MDD Lkk −MDD
..
..
..
.
.
.
..
..
.
. −MDD
0
...
..
.
..
.
0
−MDD
...
..
.
0
..
.
..
.
..
.
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥.
⎥
⎥
⎥
⎥
⎦
(3.28)
LNN
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
I k,n = Iˆk e jϕk ⇒ ik,n (t) = Iˆk cos(ωnt + ϕk ).
(3.29)
The resultant damper winding MMF is then calculated as
N MD = ∑ ∑ ik,n (t) MDk0 (θ ,t).
k=1
40
n
(3.30)
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 opencircuit armature voltage versus the damper slot pitch (τ ds ) 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 τ ds = 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, R c . This
2 The
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 R c -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
a low Rc -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 squirrelcage).
3.4.2
Damper Bar Currents at Rated Load Operation
Fig. 3.14 shows the calculated first phase belt harmonic damper bar currents 3
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.
3 These
42
current harmonics are introduced by MMF armature harmonics of orders 5 and 7.
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
Λecc =
1
,
1 − ε · cos(θm − α )
(3.32)
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 postprocessing 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
4.2.1
Iron Loss Estimation
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 B max and frequency f is conventionally modeled as
ptot = kH f B2max + kc f 2 B2max + kE f 1.5 B1.5
max .
(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 k c is the classical
loss coefficient
π 2σ d2
kc =
,
(4.2)
6ρm
where σ denotes the electric conductivity, d the thickness, and ρ m the mass
density of the steel laminate. The loss coefficients k H 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
kc T dB 2
kE T dB 1.5
2
ptot = kH f Bmax +
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
kc =
and
kc
,
2π 2
kE
k E = √ 2π
,
2π 0 |cosx|1.5 dx
(4.4)
(4.5)
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
Bmin
Γ=
,
(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
1.5
ptot = (1 + δ Γ) · (kH B2max f + kc B2max f 2 + kE B1.5
),
max 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]
kc T dB 2
2
ptot = ((1 − Γ) + Γ · RH (Bmax ))kH Bmax f +
dt
T 0 dt kE T dB 1.5
dt.
(4.8)
+ ((1 − Γ) + Γ · RE (Bmax ))
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 k H 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 ·
[%].
(4.9)
A
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
,
A
and the loss Ptot ,
C
P − PA
[%],
(4.10)
Padd−dyn = 100 · tot A tot
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 philosophy, 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 Generator 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
X jm ∝ k j Xm
( j = d, q, f ) ,
(5.1)
where Xm denotes the main armature inductance of a machine with constant
air-gap length, and k j is the form factor for excitation type i1 . 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:
1
Some authors also refer to these constants as flux distribution coefficients.
53
Figure 5.1: Air-gap flux density waveform, B gd (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 k d 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
kd =
Bgd1
,
Bgdm
(5.2)
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, B gq (θ ) set up by q-axis armature excitation. Bgq1 (θ ) is the fundamental wave.
form factor is here defined as 2
kq =
Bgq1
,
Bgd1
(5.3)
where Bgq1 denotes the amplitude of the fundamental of the waveform B gq (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
kf =
Bg f 1
.
Bg f m
(5.4)
Bg f 1 is the amplitude of the fundamental and Bg f m is the peak value of the
resulting wave.
5.3
Study Summary
The pole shoe form factors k d , 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, B g f (θ ), produced by the field winding.
Bg f 1 (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 (τ p ), pole shoe width (τ pp ), pole width (b p ),
pole shoe length (h pp ), 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=
56
δ0
,
sin θ
(5.5)
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 δ max 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 τ pp /τ p , δmin /τ p , and δmax /δmin are
introduced. The ratio δmax /δmin , also referred to as the pole taper, is henceforth
denoted fa . The ratio τ pp /τ p will be denoted τ pp for short. It is understood that
variable τ pp refers to the pole shoe width expressed as a fraction of the pole
width.
The pole taper f a 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 τ pc , 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 B gm .
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 B gm 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
τ pp
δmin /τ p
0.6 - 0.75
0.025 - 0.040
fa = δmax /δmin
τ pc
1.5 - 2.5
1
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 /τ p = 0.030. The results are
derived from non-linear FEAs (iron saturation considered). Concentric/tapered data
corresponds to calculations with τ pc = 0.46.
5.4
5.4.1
Selected Results
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 /τ p = 0.030. The abscissa of the plot holds the
pole taper fa , and the pole shoe width τ pp 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 τ pc = 0.46.
Fig. 5.5 indicates that two pole shoes with identical values of the parameters τ pp and fa , but manufactured with different pole face contours, may ex58
Table 5.2: Linear Model Coefficients
kd
Inv. Sine
Elliptic
Conc./Tap.
kq
Inv. Sine
Elliptic
Conc./Tap.
kf
Inv. Sine
Elliptic
Conc./Tap.
kd0
0.708
0.693
0.632
kq0
0.051
-0.142
-0.158
kf0
0.771
0.715
0.646
β1
0.147
0.137
0.124
β1
0.108
0.1125
0.126
β1
0.110
0.083
0.0052
β2
0.0437
0.424
0.211
β2
0.486
1.052
0.880
β2
0.165
0.735
0.521
β3
0.055
-0.522
0.197
β3
1.826
1.444
0.197
β3
0.427
-0.159
0.570
β4
-0.088
-0.032
β4
-0.074
1.529
β4
-0.117
-0.062
β5
0.24
β5
-0.071
β5
0.38
R2
0.93
0.99
0.96
R2
0.97
0.98
0.97
R2
0.92
0.99
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 cannot 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 f a
and τ pp , 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
k j = k j0 + β1 Bgm + β2 τ pp + β3
δmin
+ β4 fa + β5 τ pc
τp
( j = d, q, f ),
(5.6)
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 /τ p = 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 radialflux PM motor.
6.2
6.2.1
Generator Topology
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 remanent 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
ω
3
1
i
4
2
ω
7
6
o
5
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
τ
p
Outer
air gap
Inner
air gap
A
C’
B
A’
C
B’
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. τ p = 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
C
ωi
ba
α
ror
rir
c
a)
b)
ωo
g
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. Before 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, ω i ,
is equal, but opposite in sign, to the angular velocity of the outer rotor, ω o . 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
60º
B
N
c
ωo
ωi
P
mo
a
mi
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 ) ,
64
(6.1)
Stationary standing wave distortion
P
a
N
b
P
c/N
Magnetic direction along the inner air−gap
P
Figure 6.6: Position of the standing wave disturbance caused by the rotor crosscoupling 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 B r,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 dualrotor 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 ΔB ra , 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 cos 60 ◦ ≈ 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 μ T. Moreover, the phase-shifts between the signals ΔB ra ,
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, ΔB r , 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. ΔB r 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 5 th and 7th armature space harmonics and are
66
Figure 6.8: Calculated air gap torques in the generator during synchronized contrarotating 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.
67
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 stabilizing 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 convenience
d ωm 1
(7.1)
= (Tm − Te ).
dt
J
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]
dΔωm
1
=
(Tm − Te ).
dt
2H
(7.2)
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
2
1 J ωms
H=
,
(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
(7.5)
= ωs Δωm ,
dt
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
δ (t) = δ0 + ωs
t
0
Δωm dt,
(7.6)
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 (courtesy 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 = Ks Δδ + Kd Δω .
(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 K s
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
0
ΔTe (t)Δδ (t)dt = Ks
+ Kd
nT
0
ΔTe (t)Δω (t)dt = Ks
+ Kd
nT
0
nT
0
nT
0
nT
0
(Δδ (t))2 dt
(7.8)
Δω (t)Δδ (t)dt
Δδ (t)Δω (t)dt
(7.9)
(Δω (t)) dt,
2
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 δ 0 . It can
be written on operational form as
ΔTe
(s) =
Δδ
N1 (s)N2(s) + N3 (s)N4(s)
.
D(s)
(7.10)
Here,
N1 (s)
N2 (s)
N3 (s)
N4 (s)
D(s)
=
=
=
=
=
Ψd0 + id0 Xq (s)
(Usin δ0 + Ψd0 s)Zd (s) + (Ucos δ0 + Ψq0 s)Xd (s)
Ψq0 + iq0 Xd (s)
(Ucos δ0 + Ψq0 s)Zq (s) − (Usin δ0 + Ψd0 s)Xq (s)
Zd (s)Zq(s) + Xd (s)Xq (s).
(7.11)
(7.12)
(7.13)
(7.14)
(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. Z d (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 ω 0 (p.u.), s can be replaced by jω 0 , and
Ks and Kd can be identified from the real and imaginary parts of the resulting
expression according to
ΔTe
( jω0 ) = Ks + jω0 Kd .
Δδ
72
(7.16)
Figure 7.2: Schematic overview of the experimental setup.
7.3
7.3.1
Study Summary
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
eq = UBq ,
(7.17)
(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)
Outer stator diameter (mm)
872
75
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
7.4.1
Selected Results
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 noncontinuous 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. Nevertheless, 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 miscalculations 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 L 1q 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 significantly 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 overestimated 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 (τ D ),
the oscillation frequency ( f 0 ) 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 noncontinuous 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
Sustained Power Oscillation
Amplitude (p.u.)
0.25
Non−continuous damper
No damper
Continuous damper
0.2
0.15
0.1
0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Active Power Output (p.u.)
0.4
0.45
Instantaneous
power
(p.u.)
Instantaneous
power
(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
0.3
0.2
0.1
0
0
1
2
3
Time (s)
(a)
4
5
6
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
Time (s)
(b)
4
5
6
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. τ D = 3.1 s,
f0 = 2.60 Hz, %-overshoot = 55%.
78
Instantaneous
power
(p.u.)
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
0
2
4
6
8
10
Instantaneous
power
(p.u.)
Time (s)
(a)
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
0
1
2
3
4
5
6
Time (s)
(b)
7
8
9
10
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, f 0 = 2.37 Hz. (b) Simulated response. τ D = -34 s, f0 = 2.60 Hz.
79
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. Measurements 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 Response 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.
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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 currently 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 applicationdependent.
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 B max -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 constructional 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 correctness of the predicted effect of squirrel-cage damper windings is needed.
With today’s effective controlled damping through Power System Stabilizers, 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.
85
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 magnitudes are however seen for the outermost bars. The advantages of permeance models in design studies, such as computational speed and model transparency, 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. 32003206, 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 peerreview 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 beräkning 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 uppkommer till följd av bidirektionella magnetflöden i statorkärnan. I kombination med vissa dynamiska effekter befanns rotationsförlusterna typiskt öka
den totala järnförlustskattningen med ca 28%. Beräkningsresultaten påvisade
även en korrelation mellan rotationsförlusternas storlek och statorspårens dimensioner.
Avhandlingen presenterar även linjära modeller för beräkning av formfaktorer för utpräglade polskor av godtycklig geometri och mättnadsgrad. En
översikt av hur polplattan bör väljas att få önskad luftgapsflödestäthetsvågform 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.
96
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