University of Seville
Department of Electrical Engineering
Improvement of a fixed-speed wind
turbine soft-starter based on a
sliding-mode controller
Doctoral Thesis
by
Ángel Gaspar González Rodrı́guez
Seville, March 2006
Improvement of a fixed-speed wind turbine
soft-starter based on a sliding-mode controller
Ángel Gaspar González Rodrı́guez
Departamento de Ingenierı́a Electrónica, de Telecomunicación y Automática de la Universidad de Jaén.
Para la obtención del Grado de Doctor por la Universidad de Sevilla
con Mención de Doctorado Europeo.
Directores:
• Dr. D. Manuel Burgos Payán.
Universidad de Sevilla.
• Dr. D. Juan Gómez Ortega.
Universidad de Jaén.
A mi familia
Abstract
This work tackles the problem arising when the induction generator of a
fixed-speed or two-speed wind turbine is connected to the grid. A weak
grid where a local customer and a wind turbine are supplied by the network
by means of a long overhead line has been defined and simulated using
PSCAD/EMTDC and MATLAB. This situation particularly evidences the
impact of switching operations, mainly the start-up or the change between
generator windings.
Since the mechanical parameters defining the performance of the rotor
speed are rarely given by manufacturers, a simplified structural analysis of
a blade has been made in order to estimate the inertia time constant as a
function of the blade length and weight.
The performance and the logic control of the soft-starter gradually connecting the induction generator of the wind turbine to the rotor is also
studied. During this transient, the third order model has been established
as the best model to explain the performance of the induction generator.
Expressions for the real and reactive power has been derived which show
the strong influence of the voltage derivative on the reactive power.
Finally, two closed-loop controllers have been designed that improve the
open-loop linear control of the soft-starter. The former and more basic
structure presents a PI characteristic whose control signals are the supplied
voltage and its derivative. The latter, based on sliding-mode techniques, is
the proposed one and is able to maintain the voltage dropout in a specified
value. For fast connection conditions, the voltage dropout constrain must
be relaxed in order to avoid excessive shaft torques and speed overshoots.
Acknowledgements
Quisiera agradecer a mis directores de tesis, Manolo Burgos y Juan Gómez,
su guı́a y el apoyo incondicional que me han mostrado, ası́ como por la
mezcla de paciencia e insistencia que me ha permitido concluir este trabajo.
A Carlos Izquierdo, al que recuerdo con mucho cariño, y a todos mis
compañeros del Departamento de Ingenierı́a Eléctrica de Sevilla: Antonio
Gómez, Manolo, José Marı́a, Pedro, Jesús, José Luis, José Luis, Paco, Esther, José Antonio, Alicia, Antonio, Reme, Luis y Pilar. Con todos ellos he
vivido momentos muy entrañables.
A mis ya no tan nuevos compañeros de Jaén: Javier Gámez, Silvia, Jesús,
Alejandro y demás compañeros de planta y fatiga.
A Juan, José Juan, Nacho y a los demás amigos de fútbol, válvula de
escape en los muchos momentos de stress.
A mis padres Antonio y Marı́a y a mi hermano Toni, por todo.
Y a mi mujer Marı́a Jesús, y a mis hijos Marı́a Jesús y Gaspar, por el
tiempo que esta tesis les ha robado y por los muchos y buenos momentos
que he pasado y me esperan con ellos. Y porque sı́.
Jaén
24 de Enero de 2006
Ángel Gaspar González Rodrı́guez
Table of Contents
Table of Contents
List of Tables
List of Figures
Index
1 Introduction
1.1 Motivation . .
1.2 Survey . . . . .
1.3 State of the art
1.4 Objectives . . .
1.5 Content . . . .
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1
1
3
5
7
8
2 Start-up transients
11
2.1 Wind generator starting process . . . . . . . . . . . . . . . . . 11
2.2 Voltage fluctuation . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Description of the system
3.1 Turbine-generator mechanical system . . . . .
3.2 Generator electrical model . . . . . . . . . . .
3.2.1 Induction generators in wind turbines
3.2.2 Two-speed induction generators . . . .
3.2.3 PSCAD/EMTDC squirrel cage model
3.2.4 Electrical parameters . . . . . . . . . .
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17
18
22
22
23
24
25
3.2.5
3.3
3.4
PSCAD/EMTDC and MATLAB for simulating
duction generators . . . . . . . . . . . . . . . . .
Soft-starter . . . . . . . . . . . . . . . . . . . . . . . . .
Additional components . . . . . . . . . . . . . . . . . . .
3.4.1 Power Transformer and line to the PCC . . . . .
3.4.2 Local load . . . . . . . . . . . . . . . . . . . . . .
3.4.3 Electrical network and distribution line . . . . .
in. . .
. . .
. . .
. . .
. . .
. . .
4 Estimation of mechanical constants
4.1 Estimation of the inertia time constant . . . . . . . . . . .
4.1.1 Values used for the analysis . . . . . . . . . . . . .
Blade fatigue stresses . . . . . . . . . . . . . . . .
4.1.2 Geometry . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Aerodynamics . . . . . . . . . . . . . . . . . . . . .
4.1.4 Blade weight . . . . . . . . . . . . . . . . . . . . .
4.1.5 Static analysis . . . . . . . . . . . . . . . . . . . .
4.1.6 Inertia Time Constant H . . . . . . . . . . . . . .
Comparison of cumulated mass distributions . . .
4.1.7 Estimating a wind turbine inertia constant . . . .
4.1.8 Estimating H for different wind turbine capacities
4.1.9 Estimating the remaining inertia time constants .
4.2 Other mechanical constants . . . . . . . . . . . . . . . . .
4.2.1 Self damping . . . . . . . . . . . . . . . . . . . . .
4.2.2 Torsional stiffness . . . . . . . . . . . . . . . . . .
4.2.3 Mutual damping . . . . . . . . . . . . . . . . . . .
5 Soft-starter
5.1 Configuration . . . . . . . .
5.2 Thyristor triggering . . . .
5.3 Operation modes . . . . . .
5.4 Wind turbine soft-starter .
5.5 Asymmetrical soft-starter .
5.6 Firing angle control system
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26
27
28
28
30
30
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33
34
37
37
39
41
41
42
49
49
50
51
55
59
59
61
62
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65
65
69
70
73
74
74
6 Induction machine dynamic models
6.1 Fifth order model . . . . . . . . . . . . . . .
6.2 Reduced models for the induction machine .
6.2.1 First approach: third order model .
6.2.2 Second approach: first order model .
6.3 Third order model main equations . . . . .
6.3.1 Validity conditions . . . . . . . . . .
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77
78
81
81
81
82
82
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6.4
6.3.2 Reduced electrical system . . . . . . . . . . . . . . . .
P and Q in the third order model . . . . . . . . . . . . . . .
7 Sliding-mode control to limit voltage dropout
7.1 Voltage dropout in a weak grid . . . . . . . . . . .
7.2 Definition of the sliding trajectory . . . . . . . . .
7.3 Sliding mode controller with integral compensation
7.4 Control law parameters . . . . . . . . . . . . . . .
7.5 Implementation of the proposed controller . . . . .
7.6 Simulation using the proposed controller . . . . . .
7.7 Comparison to other control schemes . . . . . . . .
7.8 Sensitivity to speed and voltage measurements . .
7.9 Influence of the line impedance . . . . . . . . . . .
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85
87
91
94
97
102
107
111
114
117
123
126
8 Conclusions
129
8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.2 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . 132
A Weight and size for different blades
135
B Extended power-diameter table
139
References
143
List of Tables
3.1
3.2
Electrical parameters of wind turbine induction generators. .
Characteristics of different conductors. . . . . . . . . . . . . .
25
29
4.1
4.2
4.3
Data for the estimation of H . . . . . . . . . . . . . . . . . .
Data for the estimation of M and H . . . . . . . . . . . . . .
References including H. . . . . . . . . . . . . . . . . . . . . .
56
57
58
6.1
Electrical parameters for the induction machine. . . . . . . .
−δTr
.. . . . . . . . . . . . . . . . . .
Transfer function Gr =
δωr
δTv
.. . . . . . . . . . . . . . . . . . .
Transfer function Gv =
δVs
82
6.2
6.3
7.1
7.2
83
84
Electrical constants for stability study I. . . . . . . . . . . . . 107
Electrical constants for stability study II. . . . . . . . . . . . 108
A.1 Weight, size and corresponding power for several blades I. . . 135
A.2 Weight, size and corresponding power for several blades II. . 136
A.3 Weight, size and corresponding power for several blades III. . 137
B.1
B.2
B.3
B.4
Diameter
Diameter
Diameter
Diameter
and
and
and
and
power
power
power
power
for
for
for
for
several
several
several
several
wind
wind
wind
wind
turbines
turbines
turbines
turbines
I. . .
II. .
III. .
IV. .
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139
140
141
142
List of Figures
1.1
Market share of wind turbine concepts. . . . . . . . . . . . . .
2
2.1
Flicker curve according to IEC 868. . . . . . . . . . . . . . . .
15
Wind turbine in a weak grid. . . . . . . . . . . . . . . . . . .
PSCAD diagram of the system components. . . . . . . . . . .
Mechanical components in a wind turbine. . . . . . . . . . . .
Multimass and induction machine components. . . . . . . . .
Graphical model for the multimass mechanical dynamics. . .
Two-cage induction machine model. . . . . . . . . . . . . . .
Transient simulations of an induction machine using PSCAD
and MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Real and reactive power dependance on the voltage and its
derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9 Soft-starter power circuit . . . . . . . . . . . . . . . . . . . .
3.10 Normalized voltage evolution for different line impedances. .
3.1 Voltage change vs. line impedance. . . . . . . . . . . . . . . .
18
19
20
21
22
24
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
34
35
35
36
36
37
38
40
3.1
3.2
3.3
3.4
3.5
3.6
3.7
Graphical model for the multimass mechanical dynamics.
Wind turbine blade. . . . . . . . . . . . . . . . . . . . . .
Definitions for a wind turbine blade. . . . . . . . . . . . .
Composition of the blade. . . . . . . . . . . . . . . . . . .
Chord along the span. . . . . . . . . . . . . . . . . . . . .
Relative thickness. . . . . . . . . . . . . . . . . . . . . . .
Fatigue cycles. . . . . . . . . . . . . . . . . . . . . . . . .
Cycle to failure for R = -1, R = 0.1 and R = 10. . . . . .
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26
27
28
30
31
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
4.20
Lift and drag coefficients. . . . . . . . . . . . . . . .
Relationship between weight and length. . . . . . . .
Load-carrying main spar from a wind turbine blade.
Components of the aerodynamic forces. . . . . . . .
Thickness, skin and chord. . . . . . . . . . . . . . . .
Distribution of forces acting at the blade. . . . . . .
Shear web width along the blade span. . . . . . . . .
Typical and calculated cumulated mass. . . . . . . .
Comparison of different weight-length relationships.
Relationship Capacity-Diameter. . . . . . . . . . . .
Inertias for different turbine capacities. . . . . . . . .
Composition of forces exerted on the blade. . . . . .
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41
43
44
44
45
46
49
51
52
54
55
60
5.1
5.2
5.3
5.4
Soft-starter power circuit. . . . . . . . . . . . . . . . . . . . .
Soft starter control circuit and control signal time evolution.
Variation in the supplied voltage by means of a soft-starter. .
Pulses at gates. Separation between forward thyristor triggering pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gate triggering sequence and line currents. . . . . . . . . . .
Conduction periods of the thyristors. Operation with current
interruptions. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conduction periods of the thyristors. Operation without current interruptions. . . . . . . . . . . . . . . . . . . . . . . . .
Asymmetrical pulse sequence at the thyristor gates and phase
current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
67
69
Arbitrary reference frame. . . . . . . . . . . . . . . . . . . . .
Small signal block diagram representation of the induction
generator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Root loci for the mechanical closed-loop function transfer . .
Real and Reactive Power. Comparison of steady state, third
and fifth order models. . . . . . . . . . . . . . . . . . . . . . .
78
5.5
5.6
5.7
5.8
6.1
6.2
6.3
6.4
7.1
7.2
7.3
7.4
7.5
7.6
7.7
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Single wind turbine feeding a consumer in a weak grid.
Voltage in the PCC during the connection process. . .
Sliding trajectories for different αP + βP . . . . . . . .
σ and ∇σ in the phase plane. . . . . . . . . . . . . . .
Example of ~ẋ = (ẋ1 , ẋ2 ) that might not reach σ = 0. .
Simplified performance of the soft-starter. . . . . . . .
1
at different generator voltages and slips. . . . . . .
kss
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70
71
72
73
75
83
84
89
. 94
. 96
. 99
. 100
. 100
. 103
. . . . 109
7.8
7.9
7.10
7.11
7.12
7.13
7.14
7.15
7.16
7.17
7.18
7.19
7.20
7.21
7.22
Sliding-mode controller with integral compensation. . . . . .
Calculation of σ. . . . . . . . . . . . . . . . . . . . . . . . . .
Sliding-mode controller simplified control law. . . . . . . . . .
Overall scheme of the wind turbine feeding a local load. . . .
Performance of the system connected through a soft-starter
fired in accordance with a sliding-mode controller action . . .
Three firing angle control techniques . . . . . . . . . . . . . .
Firing angle controllers used as references. . . . . . . . . . . .
Voltage dropout for different control schemes . . . . . . . . .
Voltage change and shaft torque for different control schemes
Performance of tested control schemes for another generator .
New PSCAD component designed to simulate the presence
of noise in the speed and voltage measurements and the discretization process. . . . . . . . . . . . . . . . . . . . . . . . .
Performance of the sliding-mode controller when noise and
discretization are added to the speed signal. . . . . . . . . .
Controllers’ performance when a random noise in voltage and
speed signals are considered. . . . . . . . . . . . . . . . . . . .
Comparison of tested controllers for different line impedances.
Voltage dropout and voltage change vs. line impedance. . . .
112
113
114
115
116
118
119
121
122
123
124
125
126
127
128
Index
E 0 , 86
Enw , 94
H, inertia time constant, 4, 33, 51,
54, 80
I, momentum of inertia, 43
Ib , base current, 88
J, inertia constant, 34, 50, 51, 54
K, torsional stiffness, 61
L, blade length, 35
Lr , rotor self-inductance, 85
Ls , stator self-inductance, 85
Llr , rotor leakage inductance, 79
Lls , stator leakage inductance, 79
Lms , magnetizing inductance, 79
M , 79
M D, mutual damping, 62
My , flapwise momentum, 43, 45, 46
Mz , chordwise momentum, 43
N c , number of load cycles, 38
Nr , turns of the rotor winding, 79
Ns , turns of the stator winding, 79
Nx , axil force, 43
P CC, point of common coupling, 18,
23, 29, 94, 95, 117
0
Rr , rotor resistance, 23
Rr , rotor resistance, 107
Rs , stator resistance, 107
RF e , equiv. resistance iron losses, 26
Rlin , line resistance, 30, 31, 108, 127
S, complex power, 87, 88
SD, self damping, 59
Sb , base power, 88
Te , mechanical torque, 79
Tr , rotor time constant, 86
Ub , base voltage, 88
UP CC , 94
X, 86
X 0 , transient reactance, 86
Xm , magnetizing reactance, 107
Xr , rotor reactance, 107
Xs , stator reactance, 107
Xcg , center of gravity, 50
Xlin , line reactance, 30, 31, 108, 127
Z, 87
Z 0 , transient impedance, 86, 88
Zb , base impedance, 88
Ω, rotational speed, 45, 62
α, firing angle, 69, 70, 73, 102, 114,
118
α, angle of attack, 41, 59
αP , 98
αQ , 98
βP , 98
βQ , 98
δ, relative wind speed angle, 46
²cap , spar cap width, 35, 43, 50
²skin , skin width, 35, 50
²web , shear web width, 35, 43, 47, 49,
50
²web , 45
λ, flux linkage, 78
λ, tip speed ratio, 35, 39
ωr , generator rotor speed, 78
σ, leakage parameter, 82
σ, stress, 38, 39
σxx , span-wise strength, 43
θ, stator voltage angle, 87, 97, 99
4UL , permitted voltage dropout, 98,
102, 112
cd , aerodynamical drag coefficient, 41,
46, 59
cl , aerodynamical lift coefficient, 35,
41, 46
ch, chord, 35, 59
d, d-axis component, 79
g, gravity acceleration, 45
kperim , perimeter factor, 45
kss , 103
lv, low voltage side of the transf., 95
ngb , gearbox ratio, 52
npp , number of pole pairs, 53, 61, 80
p, derivative operator, 78
q, q-axis component, 79
r, rotor magnitude, 78
s, slip, 68, 82, 85, 111, 116, 120, 123
s, stator magnitude, 78
t, referred to stator, 79
th, thickness, 35
usl , control law, 102
vr , relative wind speed, 46, 59
vtip , tip speed, 53, 59
z, number of blades, 35, 52
aerodynamical drag coefficient, 41, 46,
59
aerodynamical lift coefficient, 35, 41,
46
angle of attack, 41, 59
asymmetrical soft-starter, 74, 120
axil force, 43
axil strength, 42
base current, 88
base impedance, 88
base power, 88
base voltage, 88
blade geometry, 34
blade length, 35
box spar structure, 36
brake, 19
by-pass contactor, 74
cable, 29
capacitor bank, 12, 74, 123
center of gravity, 50
chattering, 117
chord, 35, 45, 59
chordwise momentum, 43
complex power, 87, 88, 92, 97
compressive strength, 39
connection sequence, 11
consumer, 2, 13, 18, 94, 95, 115, 116
contactor, 12
control law, 102, 114, 125
cut-in, 114
cut-in speed, 11, 23
Danish concept, 1, 23
deenergization, 27
deep wound effect, 22, 24
derivative of voltage, 27
derivative operator, 78
discretization, 123
double cage model, 24
drag, 20
efficiency, 23
equiv. resistance iron losses, 26
fatigue, 19, 37
firing angle, 68–71, 73, 102, 114, 118,
126
fixed-speed wind turbine, 14
flapwise momentum, 38, 43, 45, 46
flexible coupling, 19, 21
flicker, 12
flux linkage, 78
momentum of inertia, 43
multimass, 19
mutual damping, 21, 34, 62
noise, 123
number of blades, 35, 52
number of load cycles, 38
number of pole pairs, 53, 61, 80
nw, network, 94
overhead line, 29, 30
per unit system, 33
perimeter factor, 45
gearbox, 18, 20, 33
permitted voltage dropout, 98, 102,
gearbox ratio, 52
112
generator rotor speed, 78
PI-controller, 117, 125
glass reinforced epoxy skin, 36
pitch-controlled, 14
gravity acceleration, 45
point of common coupling, 18, 23, 29,
31, 94–96, 117
hub, 55
power fluctuation, 13
power transformer, 12, 18, 28
in-rush current, 68
PSCAD/EMTDC, 4, 18, 24, 26, 89,
inertia constant, 19, 34, 50, 51, 54
120
inertia time constant, 4, 33, 51, 54,
pulse
sequence,
70
80
pulse train sequence, 69, 120
inrush current, 4
interconnection, 4, 8, 14, 92
iron losses, 23
leakage parameter, 82
leakage reactance, 24
line reactance, 30, 31, 108, 127
line resistance, 30, 31, 108, 127
load cycles, 38
local load, 18, 30, 94, 115, 130
magnetizing inductance, 79
magnetizing reactance, 107
mass distribution, 50
MATLAB, 4, 26, 112
mechanical torque, 79
R-value, 38, 39
reference frame, 78
referred to stator, 79
relative thickness, 35
relative wind speed, 46, 59
relative wind speed angle, 46
relative wind velocity, 20
resolution, 123
rotational speed, 45, 62
rotor leakage inductance, 79
rotor reactance, 107
rotor resistance, 23, 107, 121
rotor self-inductance, 85
rotor time constant, 86
rotor, wind turbine, 18
two-speed induction generator, 12, 23
self damping, 20, 33, 59
shaft spring constant, 21
shaft torque, 121, 122
shear web, 36
shear web width, 35, 43, 47, 49, 50
short circuit power, 30
skin width, 35, 50
sliding mode, 92
slip, 24, 27, 68, 82, 85, 111, 116, 120,
123
slip overshot, 121
soft-starter, 12, 18, 27, 65
span-wise strength, 43
spar cap, 36
spar cap width, 35, 43, 50
speed overshot, 117
squirrel cage induction generator, 24
stall-controlled, 14
start-up sequence, 11
stator leakage inductance, 79
stator reactance, 107
stator resistance, 107
stator self-inductance, 85
stator voltage angle, 87, 97, 99
steady state model, 81, 89
stress, 38, 39
ultimate strength, 39
tensile strenghts, 39
thickness, 35, 47
thyristor, 28
thyristor triggering, 69
tip speed, 53, 59
tip speed ratio, 35, 39
torsional stiffness, 21, 33, 61
transient impedance, 86, 88
transient reactance, 86
turns of the rotor winding, 79
turns of the stator winding, 79
twist angle, 21, 61
voltage change, 31, 96, 121, 122, 125–
127
voltage controller, 66, 67
voltage dropout, 94–96, 120, 121, 125–
127
weak grid, 4, 14, 17, 30, 31, 94, 114,
127, 130
weight, 41
wind farms, 17
wind torque, 19
winding, 12, 23, 114
X/R ratio, 31
Chapter
1
Introduction
1.1
Motivation
Wind turbines for the production of electrical energy have spread all around
the world and have shaped up, together with minihydraulic energy, to be the
main source of renewable energy contributing to the reduction of greenhouse
effect gases.
The declining power electronic device production costs, as well as economies
of scale have introduced two new designs between most accepted wind turbine models: variable-speed and variable-slip turbines. The former typically consist of direct-drive synchronous generators connected to the network through frequency converters. In the latter, one can also find frequency
converters, but connected to the wound rotor of doubly-fed induction generators. In fact these are not novel configurations, and even large variable-speed
wind turbines came prior to fixed (or two) speed Danish concept based on
squirrel cage induction machines. But price reduction of power electronics based on IGBTs and its higher reliability, have lead variable-speed and
variable-slip models to overtake squirrel cage for wind turbines over 1.5 MW
in capacity (see Fig. 1.1 extracted from [1]).
However, for a long time, and mainly for turbines up to 1.5 MW, there
will exist an important quota of constant-speed or two-speed wind turbines.
2
Chap. 1: Introduction
Figure 1.1: Market share of wind turbine concepts.
One of the drawbacks of the induction generators used in these turbines,
derived from its inability to vary the speed except within a narrow range, is
its stiffer performance that can cause disturbances in the electrical grid that
the wind turbine is connected to. In general, two issues greatly determine
the impact of fixed-speed wind turbines:
• Wind speed changes cause fluctuations in the real power delivered to
the distribution network.
• Wind turbine electrical connection gives place to voltage dropout that
could deteriorate the power quality of the nearby consumers.
Both of them cause a decline in the power quality that utilities are responsible for supplying.
Present study addresses the latter issue, the impact of the constant-speed
or two-speed wind turbine start-up on the voltage at the point of common
coupling within the medium voltage electrical distribution network. As a
result of this analysis, a closed-loop controller will be designed in order to
mitigate the undesirable side effect of the resulting transient.
1.2 Survey
1.2
3
Survey
Improvement of the impact of fixed-speed wind turbine.
In comparison to other electrical issues concerning wind turbines, such
as transient stability, self-excited operation or delivered power fluctuations, the impact of electrical connection of wind turbines to the grid
has not been received the same attention. In fact this problem only
arises with constant-speed or two-speed generators (hereafter referred
to as fixed-speed generators), and theoretically, for stall-controlled
wind turbines only. This configuration is not being considered for
new multi-megawatt wind turbines, although it is still used by most
of the main manufacturers (Neg Micon, Bonus, MADE, Ecotecnia) in
their medium-size designs.
A limiting factor for the fixed-speed stall-controlled wind turbines is
the voltage dropout caused by switching operations, mainly during
the start-up. This negative side impact is more marked the weaker
the grid the wind turbine is connected to.
Closed-loop control for soft-starters.
Information about soft-starters can be found in a dispersed and implicit way. Furthermore, this information is basically related to the
motor operation of induction machines, starting from standstill. This
is not the situation in wind turbines, where the induction machine
speed hardly varies during the soft-starter performance, but involves
a shift between motor and generator operation. Since the induction
generator connection must be accomplished for rotational speed very
close to the synchronous speed, applied voltage vs. firing angle characteristics as shown in [2] or [3] are not applicable here.
In order to mitigate this drawback of the fixed-speed wind turbines, a
novel closed-loop control of the soft-starter used to gradually connect
the generator to the distribution network will be presented in this
work. The analysis will be focused on stall-controlled wind turbines
but it can also be applied to pitch-controlled turbines, in the case of
wind gusts suddenly exerting an uncontrolled accelerating torque on
the rotor speed.
Third order model to explain the start-up evolution.
The third order induction machine model has been used to explain the
start-up dynamic process. The steady state model is not suitable as it
4
Chap. 1: Introduction
is not able to explain the reactive power evolution. The value of the
real power also differs from the actual one mainly when its contribution
is of more influence. The fifth order model is the most accurate but it is
difficult to use in order to provide analytical expressions or qualitative
ideas.
As the soft-starter is an electronic device whose performance during
the wind turbine start-up is very poorly modeled, an open-loop control appears as a simple but far from optimum strategy to smooth the
electrical connection of the induction generator. Two closed-loop controls have been simulated: a PI controller and a sliding-mode based
controller. Both of them are based on the qualitative study of the
induction machine performance.
Controlling real and reactive power.
The third order model equations show that reactive power and the
derivative of the generator voltage are closely related. Therefore, the
objective of both closed-loop controllers will not be to limit the inrush
current, as has traditionally been the most important feature of softstarters, but to regulate the reactive power or at least, avoid high
peaks in its value with the aim of decreasing the voltage dropout at the
interconnection. The interconnection is defined in [4] as the electrical
connection between a wind turbine generator system and a network, in
which energy can be transferred from the wind turbine to the network
and vice versa.
Realistic estimation for inertia time constant.
Wind turbine performance during the start-up as well as in transient
stability studies is strongly influenced by the mechanical parameters
of the turbine dynamics: inertia constant, self and mutual damping
or torsional stiffness. These parameters are rarely given by the manufacturer/supplier, and wind turbine modeling is usually forced to use
vague estimated values. Therefore, a discussion about how to obtain
realistic values of the inertia time constant as a function of the blade
mass and length, or the rated wind turbine power will be introduced
in this work. The dependence of the remaining values on the rated
power will be also provided.
PSCAD/EMTDC for electromechanical transients simulations.
A scenario including a weak electrical network using two simulation
transient programmes: MATLAB and PSCAD/EMTDC. The former
1.3 State of the art
5
allows an easier design of new components and it is a more refined and
depurated general purpose program. It also provides a more direct
interface for data management: identification, error analysis, plots...
The latter is quite faster when the system includes a relatively large
number of nodes. One of the reasons is the use of interpolation with
PSCAD/EMTDC in determining the exact switching times that allows the simulation to run at high speed and does not introduce inaccurate results [5]. Another valuable feature of PSCAD/EMTDC is
the accurate model provided for the wind turbine rotors, transformers,
underground cables and other electrical and electronic devices.
1.3
State of the art
Start-up process The electrical connection of wind turbine generators to
the distribution network is a process that has not been well documented in wind energy literature. Furthermore, most of the works
provide data and conclusions regardless of whether the wind turbine
is stall-controlled or pitch-controlled.
Differentiated data for both kind of controls for the start-up and the
shutting down can be found in [6, Larsson]. For the case of stallcontrolled wind turbines, the author vaguely indicates that the generator is connected to the grid when its rotor speed is close to the
synchronous one.
[1, Ackermann], [2, Hansen et al] and [7, Hansen et al] state that
the connection must be initiated when the generator speed reaches or
exceed the synchronous speed. Simulations performed for the case of
stall-controlled wind turbines disagree with this assert, and therefore
the referenced report and article could be referred to pitch-controlled
wind turbines where a control of the applied torque allows to avoid
the over-speed.
In the extreme case of a direct connection without soft-starter [8, Hammons and Lai], the electrical connection is also initiated at a oversynchronous speed.
Soft-Starter Existing information about soft-starters is mainly focused to
induction motors starting from standstill. The objective in this case
is usually to control the firing angle of the soft-starter thyristors in
order to limit the start-up current and to minimize torque pulsations
6
Chap. 1: Introduction
[9, Deleroi et al], [10, Zenginobuz et al], [11, Çadirci et al], [12, Kay
et al] and [13, Prasad and Sastry]. A soft-starter is employed in [14,
Ginart et al] as a discrete frequency inverter to provide a high starting
torque. [15, Gastly and Ahmed] proposes an artificial neural network
to control the speed of induction motor drive systems.
Besides this capability of limiting the current and, in some cases, minimizing the torque pulsations, existing soft-starting for motor applications also include another kind of starting [16, McElveen and Toney],
that is the voltage ramp starting. In this scheme, the voltage is progressively increasing, according to a predetermined evolution of the
firing angle. This second method is the only included by wind turbine
soft-starters. Furthermore, no references to closed-loop strategies, supervising and limiting the current or the voltage dropout, have been
found. In this sense, the uncontrollability of the induction machine at
speeds close to the synchronous one makes no feasible to adapt control
schemes from motor drives to the case of wind turbines.
However, interesting information about the sequence of firing pulses
of the tyristor gates can be found in [9, Deleroi et al], [10, Zenginobuz
et al] or [15, Gastly and Ahmed].
A very interesting and detailed description of the soft-starter conduction modes and patterns for the current and voltage waveforms, can
be found in [9, Deleroi et al], [17, Le and Berg], [18, Barton] or [19,
Murthy and Berg].
Another capability of soft-starters is addressed in [20, Blaabjerg], that
is the performance as a voltage controller reducing the supplied voltage
at low loads, thus reducing the iron losses. The study concludes that
the energy saving is not enough to advise the installation of the softstarter to an ac drive since the payback time will be long.
Calculation of the voltage dropout The voltage dropout experienced
by a local load at the interconnection node has been calculated in
the literature in different ways. For continuous operation it is established that the voltage dropout caused by the wind turbine can be
determined from the components of the complex power and the components of the impedance linking the consumer with the distribution
network. The simplified expression
4U =
P R + QX
U2
(1.3.1)
is usually preferred [21, Thiringer], [22, Saad-Saoud and Jenkins], [23,
1.4 Objectives
7
Brauner and Haidvogl] although a more accurate one is provided [24,
25, Larsson] or [26, Bossanyi, Saad-Saoud and Jenkins].
However, when switching operations are being studied, the voltage
dropout is obtained multiplying the inrush current by the short circuit impedance or the line reactance [6, Larsson], [27, Demoulias and
Dokopoulos], [28, Nevelsteen and Aragon], [12, Kay et al]. Due to
the high value of the derivative of the voltage during the start-up, the
reactive power will be quite higher than the real power during most
of the start-up transient. Nevertheless, in order to more accurate estimate the voltage dropout, both components of power must be taken
into account and the expression 1.3.1 used. A different conclusion
is reached in [8, Hammons and Lai] where induction generators employed in low head hydro schemes are connected without soft-starters,
obtaining that the voltage dip is proportional to inrush current and to
impedance of the supply for the case where impedance of the supply
is predominantly resistive.
Voltage dropout or voltage change Another concern to be taken into
account is that most of papers dealing with the switching operations of
wind turbines, quantifies the start-up impact by means of the voltage
dip or voltage dropout, that is the difference between the voltage prior
the process and the lower voltage during the connection transient.
However, the maximum voltage change during the electrical connection
should be considered [29, Standard CEI 61400-21], which may involve
the final voltage once the electrical connection has been accomplished.
1.4
Objectives
As a result of the literature survey, the following main conclusion can be
derived:
• Most of the technical literature about soft-starters deals with induction
motor performance where the main objective is to reduce the inrush
current following the motor start-up process in order to fulfil technical
and normative regulations. However, regulations and conditions for
the induction generators in wind turbines are quite different.
8
Chap. 1: Introduction
– While induction motors start from standstill, induction generators are connected to the distribution network when the rotor
speed reaches a value close to the synchronism.
– While induction motors regulations limit the inrush current during the start-up process, induction generators regulations limit
the voltage variations at the interconnection to the distribution
network.
In this thesis, the performance of soft-starting devices with wind turbine
induction generators will be addressed in order to cover the lack of work in
this area. Differences between the performance of soft-starters working with
motors and generators will be analyzed.
As a result, a new induction generator soft-starting approach will be proposed, focused on the voltage at the interconnection rather than in the current, as in the previous motor approach. To reach this goal the controller
will limit the reactive power flow and will try to compensate the associated
voltage dropout with the real power injection.
Previously, a simplified structural analysis of a wind turbine blade will be
performed in order to analyze the relationship that links the inertia time
constant of a wind turbine rotor with the blade length and weight. Starting
from this expressions and data extracted from manufacturers catalogues,
approximate laws relating the inertia time constant and the self-damping to
the wind turbine rated capacity will be estimated.
1.5
Content
After the introduction, Chapter 2 describes the electrical connection process,
its impact at the interconnection, and sets the convenience of limiting the
dropout within acceptable values.
Chapter 3 presents the scenario designed to test the proposed solutions to
mitigate the electrical connection impact during the start-up.
Chapter 4 analyzes in depth the mechanical component that models the
wind turbine rotor, as well as its coupling to the generator shaft. A discussion will be introduced to question how to calculate realistic values for the
wind turbine inertia as a function of the mass and length of the blades.
1.5 Content
9
In Chapter 5, an electronic device called soft-starter will be studied. This
component is used to gradually connect the induction machine to the electrical grid where eventually the produced real power will be delivered.
Chapter 6 reviews the main approaches that are used to model induction
machine performance, specially those used for the description of transient
situations. The complete system of differential equations will be simplified
to yield a reduced order model that appears as the most suitable one to
explain the induction machine performance during the start-up.
Chapter 7 will present two improvements to the currently used method to
control the soft-starter, with the aim of alleviating the impact of the wind
turbine connection to the grid.
Finally, in Chapter 8 the main conclusions and the relevant contributions
of this work will be summarized. To conclude, some guidelines and suggestions for future work will be provided.
10
Chap. 1: Introduction
Chapter
2
Start-up transients
2.1
Wind generator starting process
With regard to wind turbines without frequency converters, the electrical
connection of the generator to the network is one of their starting sequence
stages. This sequence begins when a wind speed higher than the cut-in
speed 1 is detected and consists of the following steps:
• The nacelle is positioned for the rotor plane to be perpendicular to
wind direction.
• Slow shaft and/or fast shaft brakes are released.
• Aerodynamical tip brakes are drawn in (fixed-pitch turbines) or blades
are turned to 45o angle (variable pitch turbines). Following that, rotor
will accelerate due to the incoming wind energy.
1
Cut-in wind speed is the lowest wind speed at hub height at which the wind turbine
starts to produce useable power [30]
12
Chap. 2: Start-up transients
• When induction machine rotor speed, equal to the fast shaft speed,
is close to the induction machine synchronous speed, about 0.96-0.98
times this speed, the generator electrical connection to the network
begins [31]. Some reports and research papers state that the softstarter should begin its performance when the generator speed exceeds
the synchronous speed [1][2][7], although simulations show an excessive
voltage dropout and torque overshot in the case of stall-controlled wind
turbines, where the incoming torque exerted by the wind cannot be
controlled.
To accomplish the electrical connection, the network voltage will be
gradually supplied to the machine terminals through an electronic device called soft-starter. It consists in six thyristors whose gates are
excited by trains of pulses initiated at certain firing angles. Controlling these firing angles, the generator voltage can theoretically be
regulated.
• Electrical connection can be considered to be completed when voltage
at the generator terminals presents the same rms value as that of the
network at the low voltage side of the power transformer.
• At this point a contactor will close its poles by-passing the thyristors and the capacitor bank is connected according to compensation
requirements.
Two-speed induction generators possess two stator windings: the former,
used at low wind speed, is the low power one (about five times less) and the
slower as it is generally a three pole-pair winding; the latter is used in the rest
of the wind speed operating range, giving place to the rated real power, and
it is generally a two pole-pair winding. Thus, for normal situations where the
wind speed increases to a high enough value, the connection process must
be repeated for each winding, since the synchronous speed will be different
for each case.
2.2
Voltage fluctuation
The term flicker is derived from the impact of the voltage fluctuation on
lamps such that they are perceived to flicker by the human eye. Flicker
is defined as “an impression of unsteadiness of visual sensation induced by
2.2 Voltage fluctuation
13
a light stimulus, whose luminance or spectral distribution fluctuates with
time”. It can be measured with a flicker meter, where the physiological
process of visual perception is simulated based on voltage measurements
[32]. To be technically correct, voltage fluctuation is an electromagnetic
phenomenon while flicker is an undesirable result of the voltage fluctuation
in some loads. However, the two terms are often linked together in standards
[33], and thus from an electrical point of view, flicker is usually referred to as
a measure of voltage variations which may cause disturbances to consumers.
Voltage variations is one of the main concerns related to the connection
of wind turbines to the network and many investigations have been made
regarding flicker produced by continuous operation [32, 26, 21, 7, 34], by
switching operations [6], or both [35, 36].
Voltage fluctuations under continuous operations is caused by active power
fluctuations which in turn are produced by tower shadow, yaw errors, wind
shear, wind eddies or fluctuations in the control system. Power fluctuations due to wind-speed fluctuations have lower frequencies and thus are
less critical for flicker [1].
On the other hand, switching operations, mainly the start-up, produces
high reactive power transients that cause voltage dropouts.
In some areas where wind farms are being installed, utilities are taking
the issue of flicker induced by operating fixed speed wind turbines seriously,
due to their responsibility to supply a minimum level of quality power to
their customers. They insist on type test results being provided for turbines
proposed for connection to their networks, from which they decide on the
maximum generation capacity which can be connected at the proposed point
of connection.
Therefore voltage fluctuation may be a limiting factor on the size of the
wind farm to be connected. This constraint has been a contributing factor
towards leading some wind turbine manufacturers and wind farm developers to adopt variable-speed turbines, that produce lower voltage fluctuation
than fixed-speed turbines. This lower impact is achieved due to the fact
that power fluctuations and switching operation are smoothed because the
network-side converter of variable-speed turbines can be used to control the
active and reactive flow and hence also the voltage [26].
Indeed, Bossanyi [26] and Fiss [35] state that flicker emissions from fixed
speed wind turbines may prove to be a limiting factor on the capacity of
14
Chap. 2: Start-up transients
wind turbine plants that can be installed in certain areas, particularly with
weak networks.
The problem will tend to be more severe as the unit size of wind turbines
increases, since the magnitude of rotational sampling effects increases as
the rotor size becomes more comparable with the size of turbulent eddies
through which the blades are slicing, and also because the number of turbines
on a wind farm of a given ratio will be smaller, resulting in a smaller degree
of cancelation between uncorrelated fluctuations. Variable speed turbines
generate much lower levels of flicker and are therefore preferred by some
utilities.
This work is mainly focused on the voltage variations due to the startup transient, providing some general and specific ideas of how to improve
the voltage dropout occurring during the electrical connection between the
induction generator and the network. In this case, the term voltage change
is preferred to voltage fluctuation.
If a small number of fixed-speed stall-regulated wind turbines were clustered in a wind park, due to uncontrollable torque during start-up, it will
produce higher voltage dropout, compared to pitch-controlled turbines [6]
where the wind torque can be controlled and the connection to the grid can
be performed in a smoother way (it takes about 1-2 seconds for the connection to be accomplished). In the case of stall-regulated wind turbines,
the accelerating torque cannot be controlled and the generator must be connected quickly, to avoid an excessive over-speed (connection takes 0.3-0.7
seconds) [25]. This gives place to an undesirable high reactive power flow
towards the induction machine.
On the other hand voltage dropout at the interconnection during start
and stop of generators is normally less significant if the wind farm is large,
due to the smoothing effect of the superposition of uncorrelated fluctuations
[37].
Because of these two issues, it is worth investigating how to mitigate the
impact of constant-speed stall-regulated wind turbines during the connection
transient in order to produce less voltage changes in small wind farms or
isolated wind turbines connected to weak grids. Reducing this disturbance
can make the voltage changes caused by switching operations not to be a
limiting factor for constant-speed stall-regulated wind turbine in these small
wind stations, and could decrease the impact of the connection transients
for these kind of turbines in larger wind farms.
2.2 Voltage fluctuation
15
Figure 2.1: Flicker curve according to IEC 868.
According to the IEC 61400-21, measurements have to be taken of the
switching operation during wind turbine cut-in and when switching between
generators, although the latter is more serious as it involves higher values
for the reactive power flow.
The simplest interpretation of the flicker emission caused by switching
operations of wind turbines is the one presented in Standard IEC 868 [38]
used by A. Larsson [25, 24] and E. Bossanyi et al. [26]. In compliance
with this Standard the magnitude of maximum permissible voltage changes
against the number of voltage changes per second is plotted (Fig. 2.1).
According to this, voltage variations occurring every two minutes or more
are allowed to be as large as 3%.
The study of the impact alleviation of the start-up operation must be
done for the whole range of wind speed conditions. At low wind speeds,
the single impact of the connection is expected to be less serious, but wind
turbines may start and stop several times in a few minutes. For high wind
conditions, the connection transient is faster and also the reactive power is
higher, thus increasing the voltage dropout.
16
Chap. 2: Start-up transients
Chapter
3
Description of the system
This chapter will describe the mechanical and electrical components that
take part in the wind turbine electrical connection to the network. Subsequently, the specific values that define the components will be derived in
order to make a realistic simulation.
Standards like IEC 868 indicate the permitted values for the voltage
dropout, and at the same time current regulations (for instance [31] in Spain)
force the manufacturer to introduce devices in order to avoid current surges
during the wind turbine start-up.
Voltage dropout effect due to the wind generator connection to the network is more marked when the generator is connected to a low short-circuit
power network, what is called a weak grid. This is also the case of a reduced
number of wind turbines feeding small centers of population that are connected to the network through long medium voltage (typically 11 or 20 kV)
overhead lines. This case is not usual in many countries like Spain where
wind turbines are mostly grouped in wind farms connected through a 66 kV
line. In this case the effect of uncontrolled and unbalanced consumption of
reactive power can also be felt although but with different characteristics:
voltage dropouts are less severe but more frequent.
The present work mainly analyzes the first case because the different tested
controllers can be more easily compared as the controllers’ effect on the
18
Chap. 3: Description of the system
voltage dropout is more pronounced.
Figure 3.1 shows the main blocks and devices involved in the system to
be studied. From left to right, one can observe the wind turbine rotor, the
gearbox, the induction generator, the soft-starter, the power transformer,
the line linking the transformer to a local load, and another line from the
local load to the utility electrical network. The connection point to the
electrical bus of the site power collection system is called point of common
coupling (PCC ), although in some papers it appears referenced as point of
common connection. PCC is defined in [39] as the point of the public supply
network, electrically nearest to a particular load’s installation, and at which
other loads installations are, or may be, connected.1
Figure 3.1: Wind turbine in a weak grid.
As a part of the study, the system has been simulated by means of an
electromagnetic transient simulator called P SCAD/EM T DC T M , taking
advantage of some components included in the library and creating some
others in Fortran code. Fig. 3.2 shows the main components corresponding
to the electrical and mechanical systems, such as they appear in the program.
Following sections describe each of the components.
3.1
Turbine-generator mechanical system
A three-bladed horizontal axis rotor has been considered, as it is the most
common configuration. Fig. 3.3 shows one of the possible compositions for
the mechanical elements from the blades to the induction generator.
1
The terms consumer, customer or local load will be either used regarding to the
impedance at the PCC
3.1 Turbine-generator mechanical system
19
Figure 3.2: PSCAD diagram of the system components.
There is a flexible coupling to allow for possible misalignments between the
turbine shaft and the generator, and to absorb torque variations, reducing
in this way material fatigue.
In this configuration, there is a brake on the fast shaft, where the braking
torque is lower, but with the drawback of idleness in case of gearbox or
coupling breakage.
For the turbine modeling, a complex PSCAD/EMTDC library component
called Multimass has been used. Fig. 3.4 depicts this component. The
Multimass component simulates the dynamics of up to six masses connected
to a single rotating shaft. In the present work only the two main masses
will be simulated. One mass, as usual, represents the generator rotor. The
other one refers to the wind turbine.2
The mechanical input TL is the wind torque exerted over the wind turbine
due to the lift force component over the rotating plane. This torque is
specified as an external input, and together with the stator voltage system,
the electrical generator and the mechanical turbine-generator parameters
will define the electrical torque and rotor speed evolutions.
Figure 3.5 shows a dynamic model for this component [40][41]. The most
significant components in the dynamics are the generator inertia constant
and above all the turbine inertia, due to the blade length and weight. It is
more useful and usual to translate the inertia into a unit system relatively
2
As the component does not include a gearbox, all slow shaft mechanical values must
be referred to the fast shaft
20
Chap. 3: Description of the system
Figure 3.3: Mechanical components in a wind turbine.
independent of the rated power or speed, and hence inertia values from different turbines can be compared regardless of the rotor size. The expression
used for the system unit change is:
¡
¢2
rpmgen 2π
60
H = nblades · J
2 · Pwatt · n2gb
(3.1.1)
where nblades is the number of blades, rpmgen is the generator rotational
speed, Pwatt is the wind turbine capacity (rated power) expressed in watts
and ngb is the gearbox ratio.
Another important parameter is the self damping of every mass. For the
turbine (SD2 in Fig. 3.5), this parameter accounts for the aerodynamical
drag producing a torque which opposes the wind motor torque. It depends
on the blade length, chord, cleanliness and material, as well as the relative
wind velocity in relation to the blade velocity, specially regarding to the
speed regime (turbulent or laminar).
Torque produced by the aerodynamical drag is proportional to the square
of the speed, although the PSCAD model uses a linear approximation. This
3.1 Turbine-generator mechanical system
21
Figure 3.4: Multimass and induction machine components.
will not lead to significant errors if simulations are performed at almost
constant speeds, as is the case.
Regarding the generator self damping SD1 , it is caused by the friction of
the rotor shaft and to the ventilation losses.
In a simplified way, torque transmission is enabled by the torsion in the
fast and slow shafts. As previously mentioned, a flexible coupling is usually
introduced to allow for misalignments. In this case, torque transmission
takes place due to the coupling twist angle that is inversely proportional
to the Shaft spring constant or Torsional stiffness K12 and directly proportional to the transmitted torque.
When transients in the transmitted torque occur, variations will appear in
the twist angle that are damped by a torque proportional to the difference
of speeds between both rotating masses. The proportionality coefficient is
the mutual damping parameter M D.
Therefore, shaft spring constant and mutual damping values are determined by the coupling between turbine rotor and the gearbox (as appears
in Fig. 3.3), or between gearbox and generator.
22
Chap. 3: Description of the system
Figure 3.5: Graphical model for the multimass mechanical dynamics.
3.2
3.2.1
Generator electrical model
Induction generators in wind turbines
Except for very specific applications, the biggest induction machines are
found as generator units for wind turbines. There are some distinctive features for the induction generators being designed to be part of a wind turbine:
• Since they operate as generators and the rotor is accelerated by the
wind torque, the deep wound effect is not a concern. This effect is
very desirable in the start-up of induction machines acting as motors,
but this is not the case.
3.2 Generator electrical model
23
• For squirrel cage generators, a rotor resistance (referred to stator)
slightly higher than for the case of motors is preferred. This allows a
more flexible performance against changes in the wind torque, and it
softens the start-up. The drawback is an efficiency decrease since it is
related to 1 − Rr0 .
• Iron losses are in general lower than for generators used in other applications [42]. This can be achieved by decreasing the saturation of the
magnetic core. This way, magnetizing current decreases, and hence so
does the no-load stator current, thus allowing a lower cut-in speed.
3.2.2
Two-speed induction generators
Disturbances due to the induction generator connection to the network
are more marked in the case of Danish concept wind turbines (fixed-speed
passive stall-controlled) and recently introduced active stall wind turbines,
equipped with squirrel cage induction machines. Most of these generators
are comprised of two stator windings, with different rated power and number of poles (four and six poles for standard generators). The small six-pole
stator winding is connected when the wind speed is low, and therefore, the
wind torque. As the synchronous speed is lower, this issue allows to diminish the noise at low wind speed (precisely when it is more perceptible). The
power losses also decrease at this wind condition. For higher wind speed,
the generator is switched to the four-pole operation, connecting the stator
winding having the same rated power as the wind turbine capacity.
However, the small six-pole stator winding will not be taken into account
in the connection study, since it gives place to lower disturbances in relation
to the main stator winding. Three facts support this idea:
• Current flowing through the line impedance between the high voltage
network and the PCC is lower for the secondary winding and hence
voltage dropout will be less serious.
• From (3.1.1) it is clear that the inertia time constant H is higher for
the secondary winding, even when it is slower, since the rated power
is quite lower.
• Rotor resistance is usually higher for the secondary winding.
24
Chap. 3: Description of the system
Figure 3.6: Two-cage induction machine model.
Two last facts give place to a slower connection process, thus alleviating the
impact of the start-up.
3.2.3
PSCAD/EMTDC squirrel cage model
The squirrel cage induction generator model included in the PSCAD/EMTDC
library is a double cage model, that considers the deep wound effect in the
rotor circuit. This configuration assumes two short-circuited bar layers: the
upper bars, having high resistance and low leakage reactance, and the lower
ones, with reduced resistance and higher leakage reactance [40].
At low mechanical speeds, rotor current frequency is high and the currents
flow through the low inductance rotor circuit. In normal operation, the rotor
current frequency is low, and the currents flow through the low resistance
circuit. Therefore, the effective rotor resistance is high for low speed (better
performance for an industry motor start) and low in normal operation (lower
rotor power losses). This effect can be modeled in steady state operation as
two bar sets in parallel as shown in Fig. 3.6.
In any case, the EMTDC routine that simulates the induction machine
does not use a steady state model, but a fifth order dynamic one, whose
parameters can be derived from the double cage induction machine model.
In the case of a wind turbine generator’s electrical connection to the network, slip values are small (in the order of 0.04 or lower) and the rotor
currents frequencies make the lower bars the only ones involved in the machine performance. Since it is not a significant parameter, and, on the other
3.2 Generator electrical model
25
hand it is usually difficult to obtain, the deep wound effect will not be considered [43] and thus, high values to the second cage resistance and reactance
will be introduced.
3.2.4
Electrical parameters
Electrical parameters defining the induction generator whose transient is
to be studied, have been chosen in accordance to values extracted from
several wind turbine catalogues in the range of 350 kW to 1.65 MW. Table
3.1 gathers values for resistances and reactances obtained from different
manufacturers, including some values extracted from some dynamic stability
studies.
Finally, values similar to [44] have been chosen although with a not so
high rotor resistance.
Table 3.1: Electrical parameters of wind turbine induction generators.
Model
P (MW)
rs (p.u.)
xs (p.u.)
rr (p.u.)
xr (p.u.)
xm (p.u.)
Nordex
1
0.0062
0.0787
0.0092
0.0547
3.642
NegMicon
1.5
0.0227
0.0795
0.0156
0.0597
3.755
NegMicon
1
0.0225
0.173
0.008
0.13
3.428
WinWorld
0.6
0.0197
0.1271
0.0089
0.0956
4.667
Bonus
0.6
0.0065
0.0894
0.0093
0.1106
3.887
Bonus
1
0.0062
0.1362
0.0074
0.1123
3.911
Vestas
1.66
0.0077
0.0697
0.0062
0.0834
3.454
Ref. [45]
0.35
0.0064
0.0689
0.0071
0.2105
3.118
Ref. [46]
0.35
0.0063
0.064
0.0066
0.0934
3.306
Ref. [22]
1
0.0076
0.1248
0.0073
0.0884
1.836
Ref. [44]
0.6
0.0059
0.0087
0.019
0.143
4.76
1
0.0059
0.0087
0.01
0.143
4.76
Chosen values
26
Chap. 3: Description of the system
Values for the resistance of equivalent iron losses RF e have not been included since PSCAD/EMTDC, following the American tradition, does not
incorporate this parameter in the induction machine model but in the rotational losses. In order to avoid the deep bar effect, very high values (in
the order of 10 p.u.) have been taken for the second cage resistance and
inductance.
3.2.5
PSCAD/EMTDC and MATLAB for simulating induction generators
Figure 3.7: Transient simulations of an induction machine using PSCAD and MATLAB for s = −0.005: a) voltage at the generator terminals, b) current, c) real power,
d) reactive power.
Figure 3.7 compares simulations made with PSCAD/EMTDC and MATLAB for an induction machine fed by a three-phase voltage system, whose
rms value is the indicated in the upper left box. Some discrepancies can be
appreciated regarding the voltage and current due to the rms values, with a
higher delay and ripple in the case of the PSCAD simulation. An unusually
high value for the inertia has been introduced in order to maintain the slip
constant, specifically in a generator operation where s = −0.005. This helps
3.3 Soft-starter
27
to observe how the current and the real and reactive power depend, not only
on the voltage, but also, and mainly in the case of the reactive power, on its
derivative.
The influence of the derivative of voltage on the system performance will
be explained later on, although Fig. 3.8 already illustrates this fact.
Figure 3.8: Real and reactive power dependance on the voltage and its derivative.
The first graph shows the voltage and real power evolutions, while in the
second one, the derivative of voltage and the reactive power are represented.
The voltage and its derivative are shown with a solid line. Real and reactive
power are shown with dashed line. Powers are considered positive when
flowing towards the induction machine.
Since the slip is negative enough (s = −0.005), real power will always
be negative, although in the graph it appears multiplied by −1. Reactive
power, if the induction machine is analyzed in steady state, should always
be positive. However, negative derivatives of voltage will give place to a
deenergization in the asynchronous machine inductances that can lead to
reactive supplies. This negative reactive power state is difficult to achieve
and will not be pursued, although a control in the voltage derivative will be
introduced in order to avoid excessively high values for the reactive power.
3.3
Soft-starter
The soft-starter is a power converter, which has been introduced as ancillary equipment in fixed speed wind turbines to reduce the transient current
during connection or disconnection of the generator to the grid [2]. The
28
Chap. 3: Description of the system
Figure 3.9: Soft-starter power circuit
soft-starter is a general purpose controller with six thyristors, a back to
back connected pair in each line [3] as shown in Fig. 3.9.
Using firing angle control of the thyristors in the soft-starter, the generator
is smoothly connected to the grid.
The soft starter and the structures to control its performance will be
studied in Chapter 5.
3.4
Additional components
The remaining components being part of the studied system will be presented below.
3.4.1
Power Transformer and line to the PCC
A three-phase power transformer adapts the voltage levels between the induction generator, typically 690 V, and the local network, typically 20 kV.
Most regulations dictate that the transformer must be star-delta configured
(grounded star on the generator side and delta on the network side). The
3.4 Additional components
29
Table 3.2: Characteristics of different conductors.
Conductor
LA
LA
LA
LA
LA
LA
R20o C (Ω/km)
30
56
78
110
145
180
1.075
0.614
0.426
0.307
0.242
0.190
X(Ω/km)
∼ 0.39
Imax (A)
130
185
246
314
330
400
delta winding avoids modifying the zero sequence impedance on the network
side. The grounded star decreases the harmonic distortion.
Cable impedance from the generator to the transformer can be disregarded, even for small or medium size turbines where the transformer is
outside the wind turbine.
The line from the transformer to the point of common coupling can be
an overhead line or underground cable. Characteristics for several overhead
lines obtained from [47] and [48] are presented in Table 3.2. Capacitance
of underground cables is higher, which can help the connection transient
providing reactive power and decreasing the voltage dropout. Instead of
that, a more restrictive situation will be analyzed by including a LA-30
overhead line. The impedance line is:
Ω
km
Ω
= 0.39
km
RLA30 = 1.075
XLA30
As expected, simulations show that this impedance will hardly produce any
effect on the voltage on the customer side. If an underground cable is included in the design, an increase in the voltage level at the interconnection
will be expected at the load node, although from the point of view of the
voltage dropout, almost no difference can be appreciated from the simulations.
30
Chap. 3: Description of the system
Figure 3.10: Normalized voltage evolution for different line impedances.
3.4.2
Local load
It will be considered a situation where a local load exists at the point of
common coupling at the medium voltage level. Simulations will be carried
out for the following values, although as for the previous impedance, these
data are of no influence on the results.
Pcons = 500kW
3.4.3
cos ϕ = 0.85
Electrical network and distribution line
The local load will be connected at an intermediate point of a weak electrical
distribution network modeled as a 500 MVA short circuit power source with
an impedance corresponding to an overhead line 15 km in length. A higher
capacity line (LA-56) is chosen to comply with local regulations [49]. At
20o C, specific line resistance and impedance are
Ω
km
Ω
= 0.39
km
RLA56 = 0.614
XLA56
Short circuit reactance of a 500 MVA short circuit power source must be
added to this impedance although its contribution is negligible.
3.4 Additional components
31
If the voltage dropout is defined as the maximum drop in the rms value
of the voltage at the point of common coupling with respect to the voltage
prior to the connection, its value is determined mainly by the line reactance.
If, instead of it, the term voltage change, defined as the maximum difference in voltage during the connection transient (see Fig. 3.10), is to be
used, then its value is influenced by both the line resistance and reactance.
Fig. 3.1 plots the maximum voltage change, taken as a representative
magnitude of the impact of the start-up process, for different conditions of
the weak grid given by its short circuit power and the Xlin /Rlin ratio. Firing
angles are regulated by means of the open-loop linear controller. In the situations when the voltage before the start-up is higher than the voltage when
the connection transient has finished, dashed lines refers to the difference
between this final voltage and the lowest voltage during this transient.
Figure 3.1: Voltage change vs. line impedance.
The left hand graph is similar to other ones found in [21] or [24], but
the right hand one is more interesting, where the voltage change is plotted against the line resistance and reactance. As can be seen, the impact,
referred to as the maximum voltage change during the connection, is proportional to the line resistance for a fixed reactance. That is because the higher
the Rlin , the higher the difference between the voltage before and after the
connection, which can be translated to a higher voltage change assuming an
approximately constant voltage dropout (Fig. 3.10). This difference can be
obtained from the generated (negative) real power Pwt and the consumed
32
Chap. 3: Description of the system
(positive) reactive power Qwt according to
4U '
Pwt Rlin + Qwt Xlin
.
U2
(3.4.1)
Table 3.2 shows specific resistance and reactance for several conductors of
the same series. As the overhead line capacity increases, specific resistance
decreases with the power of approximately 0.6, and the specific reactance
hardly varies. Thus it is expected that as the number of wind turbines
increases, along with the capacity of the evacuation line, the voltage change
will decrease, although on the other hand the number of switching operations
will increase.
Chapter
4
Estimation of mechanical
constants
Performance of the wind turbine during the start-up as well as in transient
stability studies is strongly influenced by a broad set of mechanical parameters (Fig. 4.1) involved in the turbine dynamics. These parameters are
rarely given by the manufacturer/supplier, and hence only an estimation of
them is usually found in the bibliography.
In this chapter the structure of a wind turbine to estimate its mechanical
parameters is analyzed. Also, an expression that provide a first approach to
a realistic inertia value to be used in transient simulations will be derived.
The parameters to be studied are depicted in Fig. 4.1. As can be seen
the gearbox does not explicitly appear. In fact, simulation programs do not
include this component in their libraries, not even as an ideal torque/speed
transformation. Therefore, it is necessary to refer all values to the fast
speed side (generator side) or rather to directly translate the values of the
mechanical parameters into a per unit system. Expressions to accomplish
this change of unit systems will be provided.
The parameters to be estimated or, at least, bounded are the turbine and
generator inertia time constants H, turbine and generator self damping SD,
torsional stiffness K of the flexible coupling between rotor and generator,
34
Chap. 4: Estimation of mechanical constants
Figure 4.1: Graphical model for the multimass mechanical dynamics.
and mutual damping M D of the changes in the twist angle of this coupling
(Fig.4.1).
4.1
Estimation of the inertia time constant
The inertia constant has a great and direct impact on the transient stability
of wind farms and the start-up of wind turbines, and thereby this value is
referenced continuously in wind farm dynamic performance studies. However, most of these papers only offer an estimation of inertia as suppliers do
not use to facilitate this information [50]. Thus, disregarding papers about
vertical axe wind turbines [51], two-bladed rotors [52, 53] or small size machines [54, 55], very few works indicate numerical values for this constant
[56, 57, 41, 44, 46, 58, 45, 59, 60, 61, 62, 63] most of them being estimated
values, as in [63], which obtains Hturb and Hgen from an experiment where
turbine is abruptly disconnected from the grid, or refer to multi-megawatt
wind turbines. With regard to simulations it is common to find the same
experiment for several inertia values [50, 64, 65, 63].
An estimation of the inertia constant can be obtained starting from the
blade geometry and cross-section, and calculating an approximated mass
distribution along the blade span. A possible blade geometry and some
necessary definitions appear in Fig. 4.2, 4.3 and 4.4. They also show the
blade section perimeter at an intermediate spanwise distance and a possible
inner structure [66, 67].
4.1 Estimation of the inertia time constant
35
Figure 4.2: Wind turbine blade.
Figure 4.3: Definitions for a wind turbine blade.
Chord distribution along the blade span is shown in an approximate way
in figure 4.5, and compared to optimal distribution according to expression
[42]:
chb (r) =
16 π R
1
sµ
¶2
9 · cl λ z
λ
4
r +
L
9
where λ is the tip speed ratio, L is the blade length,
coefficient and z is the number of blades.
(4.1.1)
cl is the lift
Defining thickness as the maximum length in the direction transversal
to the chord line, and relative thickness as this magnitude divided by the
chord (th/chb in Fig. 4.3), the spanwise distribution of this value offers the
approximated distribution of Fig. 4.6 [68][62].
Next to the junction with the hub there is a cylindrical constant-chord
part. The longer and more important part is the farther from the root
and it has an aerodynamical shape as in Fig. 4.3 or 4.4, more elongated
towards the tip and wider towards the root. Between both parts there is a
complicated variable geometry link that fits both kinds of sections.
Starting from the blade geometry, an estimation of the mass distribution,
which in turn determines inertia time constant, can be obtained. A more
formal study of the blade design can be found for example in [67] but in this
36
Chap. 4: Estimation of mechanical constants
Figure 4.4: Composition of the blade.
Figure 4.5: Chord along the span.
work some approximations will be done with the sole aim of establishing an
qualitative idea of the mass distribution:
• As previously indicated, the blade will be divided into three parts
along the span. There is a first cylindrical zone, close to the hub; the
aerodynamical zone, which is the longest, with decreasing chord; and
fitting them the third with more complicated geometry.
• It will be assumed that the mass is due to the glass reinforced epoxy
skin and to the box-spar structure acting as the structural reinforcement for the blade to be more efficient at resisting out-of-plane shear
loads and bending momenta (see Fig.4.4).
• At a certain distance r from the root, skin and shear web widths are
constants.
4.1 Estimation of the inertia time constant
37
Figure 4.6: Relative thickness.
• The only action to be considered will be the out-of-plane bending momentum and the inertia tensor will be considered as a diagonal matrix
(symmetrical cross-section from the point of view of stress distribution). The most exerted points will be assigned a distance from the
flection axes equal to half the thickness.
4.1.1
Values used for the analysis
Blade fatigue stresses
There are two scenarios to be taken into account in the wind turbine blade
design [69]:
• The first one analyzes the extreme loads that a wind turbine can suffer,
which can be estimated using two simplified methods: parked under
extreme winds and an operating gust condition
[70]. The first method calculates the extreme loads with the turbine in
the parked condition in accordance with IEC and Germanisher Lloyd
Class I design recommendations. In the second method the turbine is
considered to be operating at constant speed during a 55 m/s gust.
Both load estimation approaches provide similar results.
• The second scenario refers to the analysis of the cycling loads in normal operation over the blade during the complete life of the wind
turbine which diminishes the maximum strength the material is able
to withstand.
A further knowledge of the methodology for blade design can be found in
[67], but here only a simplification of the second scenario will be taken into
38
Chap. 4: Estimation of mechanical constants
account assuming an amplitude variation of 18% around the mean rated
value. Fatigue load spectra for different numbers of cycles, as explained
in [71] or [72], will not be considered. These variations are due to the
wind shear, to the tower shadow, to gravitational forces, or, with lower
frequencies, to weak wind gusts. Mean value is due to the out-of-plane
flapwise momentum and to the centrifugal forces.
A common parameter in the fatigue behavior is the R-value which for this
fatigue cycle yields (see Fig. 4.7)
R=
σm − σa
σm − 0.18 σm
σmin
=
=
= 0.7
σmax
σm + σa
σm + 0.18 σm
(4.1.2)
where σmin and σmax represent the minimum and maximum stress in a fatigue stress cycle (tension is considered positive and compression is negative)
and σm and σa are the mean value and the amplitude of the fatigue stress
cycle.
Figure 4.7: Fatigue cycles.
The number of load cycles is very high in wind turbines. Assuming an
effective disposability of 80% for a rotational speed of 25 rpm, the number
of load cycles for each blade is:
N c = 0.8 · 20 years 8766
hours 60 min
25 rpm = 2.1 · 108 cycles (4.1.3)
year 1 hour
4.1 Estimation of the inertia time constant
39
Starting from typical data [73] about a composite of glass reinforced epoxy,
typical fatigue characteristic properties can be the ones represented in Fig.
4.8.
It can be noticed from these figures that for always positive tensile
strengths (R = 0.1), the mean permissable tensile strength is in the order
of σm = 100M P a. If compressive strengths are considered at the extrados
or downwind side with R = 10, although the value for static compressive
strength is lower than the static tensile one, at 2.1 · 108 cycles the compressive strength limit is higher (see Fig. 4.8). Furthermore, centrifugal
forces decrease, in absolute value, the compressive strength, and hence this
strength can be considered a less restrictive condition.
For other R-values, [74] proposes the following equation
µ
¶
σmax b
σu − σmax = a σmax
(N c − 1)
σu
(4.1.4)
where σmax is the maximum applied stress, σu is the ultimate tensile
or compressive strength (obtained at a strain rate similar to the 10 Hz
fatigue tests), N c is the number of load cycles and a, b, and c are the fitting
parameters. For an R-value of the fatigue cycle equal to 0.7 the related values
are a = 0.04, b = 2.5, c = 0.45, σu = 360M P a and N c = 2.1 · 108 cycles.
The value of σmax which satisfies (4.1.4) is 124 M P a whose corresponding
σmean is
σmax
σmean =
= 106 M P a
(4.1.5)
1.18
similar, although a bit lower, to 120 − 140 M P a appearing in [67].
4.1.2
Geometry
Data regarding to the blade geometry analyzed as an example has been
derived from tables and graphs included in [68] and correspond to a 26 m
blade from an 850 kW wind turbine.
For a 26 meter blade, having a 1.8 meter hub radius, assuming a rated tip
speed ratio of λ = 7 and a nominal wind speed equal to vwind = 12 m
s , the
rotational speed yields
Ω=
rad
λ · vwind
= 3.02
= 28.83 rpm
Rhub + L
sec
(4.1.6)
40
Chap. 4: Estimation of mechanical constants
Figure 4.8: Cycle to failure for R = -1, R = 0.1 and R = 10.
4.1 Estimation of the inertia time constant
4.1.3
41
Aerodynamics
The lift and drag coefficients have been extracted from Fig.4.9 [42] considering a smooth surface, a Reynolds number of 3 · 106 and an angle of attack
equal to 4o .
cl = 0.7
cd = 0.006
Figure 4.9: Lift and drag coefficients.
According to the line of considered approximations, Reynolds number
variations along the span will not be taken into account. Therefore if blade
twist gives place to a constant angle of attack α, then coefficients cl and cd
are constants along the blade span.
4.1.4
Blade weight
In order to test the validity of the analysis and to fix security factors, the
actual mass of the analyzed blade needs to be obtained.
In the case where blade weight data are not available, an estimation of
the relationship between weight and blade length can be extracted. Hence,
starting from the table at Appendix A obtained from manufacturers’ catalogues and [75], an expression relating both parameters can be derived.
42
Chap. 4: Estimation of mechanical constants
Searching for an expression relating weight and length blade of the type
Mpala = kM · Lαpala
(4.1.7)
the function sum of quadratic errors is minimized
f (kM , α) =
n
X
(Mi − kM · Lα )2
(4.1.8)
i=0
giving
kM = 2.95
α = 2.13
(4.1.9)
These values are quite similar to that of [76] (kM = 1.6 and α = 2.3), and
to that of [77] (kM = 1.50 y α = 2.34). The estimated values provided by
[68] are slightly different (kM = 0.619 and α = 2.63).
Figure 4.10 shows the estimation according to previous parameters and
a representation of some values extracted from manufacturers and other
papers (kM = 2.95, α = 2.13 and diamonds). A close agreement can be seen
for low rotor diameters. The higher dispersion at greater diameters can be
explained by the scarce data available for these turbine sizes.
The α value so close to 2 could suggest to think that skin and structure
reinforcement width does not vary linearly with the blade length.
In accordance to these functions, the weight for a 26 m blade is typically
between 2870 kg and 3260 kg, for example M = 3075 kg.
Another necessary datum is the density of glass reinforced plastic that
kg
depends on the composition of the material. The value of ρgrp = 1700 m
3
will be chosen.
4.1.5
Static analysis
This analysis starts from expressions [78] to calculate axil strengths at a
certain distance r from the rotation axis at a point defined through its
coordinates y and z
σxx (r, y, z) =
My (r)
Nx (r) Mz (r)
+
y+
z
A(r)
Iz (r)
Iy (r)
(4.1.10)
4.1 Estimation of the inertia time constant
43
Figure 4.10: Relationship between weight and length.
where σxx is the span-wise strength, Nx (r) is the force in the same direction
due to centrifugal loads, A(r) is the spar cap and shear web areas, , Mz (r)
and My (r) are the chordwise and flapwise momenta due to the resultant
force component, and finally Iy (r) and Iz (r) are the momenta of inertia of
the spar caps and shear webs structure with respect to the chord line and
to the axis perpendicular to it.
As indicated in [79] and the figure 4.11 extracted from it, the main spar
carries most of the flapwise bending loads whereas the shell carries most of
the edgewise bending loads.
This can be derived by observing Fig. 4.12 which makes evident the higher
value of the z-axis force component in relation to the y-axis one, giving place
to higher values of My in relation to Mz . As a result, the second addend
will not be taken into account in the spar cap thickness analysis.
With regard to the structural reinforcement area, it is obtained by multiplying the section perimeter by the spar width. In fact, this width is not
constant along the box-spar. In [70] the structural shear web was taken to
be 5/3 the thickness of the blade skins, and the spar caps reinforcement is
2/3 of this outer skin. Thus, taking the thickness of the structural shear web
as the base, the area of the main spar can be expressed as the product of the
44
Chap. 4: Estimation of mechanical constants
Figure 4.11: Load-carrying main spar from a wind turbine blade.
Figure 4.12: Components of the aerodynamic forces.
4.1 Estimation of the inertia time constant
45
spar
parameter kperim
, the chord ch(x) and the structural shear web thickness
(Fig. 4.13).
Figure 4.13: Thickness, skin and chord.
spar
S(x) = kperim
(x) · ch(x) · ²web (x)
Hence the expression for the axil force yields
Z L+Rhub
spar
Nxweb (x) = ρGRP Ω2
kperim
(r) · ch(r) ²web (r) r dr
(4.1.11)
(4.1.12)
x
where ρGRP is the reinforced plastic density, Ω is the rotational speed at
rad/s, Rhub is the hub radius, L is the blade length , ²web (x) and ch (x)
are the expressions for the lumped box spar width and the blade chord as a
function of the distance to the rotation axis and g is the gravity acceleration.
Bending flapwise momentum My (r) can be calculated by integrating the
force differentials shown in Fig. 4.14 whose expression appears in (4.1.14).
dFL =
dFD =
ρ
ch(x) · vr (x)2 · cl (δ) · dx
2
ρ
ch(x) · vr (x)2 · cd (δ) · dx
2
(4.1.13)
(4.1.14)
where ρ is the air density, ch(x) is the distribution of chord along the blade
46
Chap. 4: Estimation of mechanical constants
Figure 4.14: Distribution of forces acting at the blade.
span, cl and cd are the aerodynamical drag and lift coefficients, and vr (x)
is the resultant relative velocity defined through its direction δ and its
modulus |vr |
³v
´
wind
δ = atan
µ 2Ω · x
¶
2
2 vwind
2
|vr | ' Ω
+x
Ω2
(4.1.15)
(4.1.16)
Thus, flapwise momentum My (x) yields
Ω2
My (x) =
ρ (cd sin(δ)+cl cos(δ))
2
Z
Rhub +L
ch(x) (r−x)
x
µ
¶
2
vwind
2
+r
dr
Ω
(4.1.17)
4.1 Estimation of the inertia time constant
47
for x > lim aerod (see Fig. 4.5) and
Ω2
My (x) =
ρ (cd sin(δ)+cl cos(δ))
2
Z
µ
Rhub +L
ch(x) (r−x)
lim aerod
¶
2
vwind
2
+r
dr
Ω
(4.1.18)
for x ≤ lim aerod.
The momentum of inertia of the box-spar with regard to the chord line
at distance x, Iy (x), is obtained from normalized expressions of the aerodynamical section.
s
µ
¶2
I
d
spar
2
Iy (x) = ²z · ch(x) 1 +
prof ile(x, y) dy
(4.1.19)
dy
which for the sake of clarity, will be expressed as the parameter kIy (x)
multiplied by the shear web thickness and the cube of the chord.
Iyspar (x) = kIy (x) · ²web (x) · ch(x)3
(4.1.20)
In order to take into account the worst scenario, the value of z in (4.1.10)
corresponds to the most stressed point due to the flapwise momentum which
is the farthest point from the median line. A value equal to the half the
thickness is to be considered.
Hence, from (4.1.10), (4.1.12) and (4.1.17) the width of the shear web
yields
Z
ρGRP
σ =
Ω2
x
L+Rhub
spar
r kperim
²web (r) ch(r) dr+
th(x) ²web (x)
Ω2 ρ (cw sinδ + ca cos δ) th(x)
+
4 kIy (x) ²web (x) ch3 (x)
+
Z
Rhub +L
ch(r)(r − x)
min(x,lim aeord)
(4.1.21)
µ
2
vwind
+ r2
Ω2
¶
dr
A variable change will be introduced where S(x) is the new unknown, and
48
Chap. 4: Estimation of mechanical constants
an auxiliary constant will be also defined.
spar web
S(x) = kperim
² (x) · ch(x)
KAcent =
KB(x) =
(4.1.22)
ρGRP Ω2
ks · σ
spar
Ω2 ρ th(x) (cd sin(δ) + cl cos(δ))kperim
(4.1.23)
(4.1.24)
4 kIy (x) ch(x)2 σ
Thus, expression (4.1.22) yields
Z
L+Rhub
S(x) = KAcent
x
Z Rhub +L
+ KB
r S(r) dr
µ
(r − x)
min(x,lim aeord)
2
vwind
+ r2
Ω2
¶
ch(r) dr
(4.1.25)
Deriving (4.1.25), it gives
µ 2
¶
Z Rhub +L
vwind
dS
2
= −KAcent x S(x) − KB(x)
+r
ch(r) dr
dx
Ω2
min(x,lim aeord)
µ 2
¶
Z
vwind
d KB(x) Rhub +L
2
+
(r − x)
+r
ch(r) dr +
dx
Ω2
min(x,lim aeord)
(4.1.26)
KAcent
2
Multiplying both terms by e 2 x and integrating the resulting expression, the equation for the area S(x) along the blade span (the integration
constant has been canceled as derived from (4.1.25)) yields
Z Rhub +L
KAcent 2
KAcent 2
S(x) = −e− 2 x −KAgrav x ·
e 2 p +KAgrav p ·
x
Ã
µ 2
¶
Z Rhub +L
vwind
d KB(p)
2
(r − p)
+r
ch(r) dr
dp
Ω2
min(p,lim aeord)
µ 2
¶ !
Z Rhub +L
vwind
ch(r)
+ r2 dr dp
+ · KB(p)
Ω2
min(p,lim aeord)
(4.1.27)
4.1 Estimation of the inertia time constant
49
Once derived the area S(x), the web width can be obtained.
²web (x) =
S(x)
spar
kperim
(x) ch(x)
(4.1.28)
The shear web width obtained from (4.1.28) follows the distribution represented in 4.15. It can be seen that, due to the low value for the thickness
at the tip, the width of the shear web is considerably larger than the width
closer to the blade root.
Figure 4.15: Shear web width along the blade span.
4.1.6
Inertia Time Constant H
Comparison of cumulated mass distributions
However, difficulties arising during the manufacturing process make that
a constant value for the shear web and the spar cap thickness is usually
preferred. In order to provide for other materials which are also part of the
blade (mainly the balsa core) the value obtained previously is multiplied by
a factor which is greater near the root [80]. In this sense, and in order to
50
Chap. 4: Estimation of mechanical constants
improve the reliability against extreme winds, a 50% increase of the spar
thickness is also applied.
The same analysis can be made for the blade skin. This shell bears most
of the edgewise bending loads which are due to the aerodynamical forces,
but mainly to the weight force when the blade is in a horizontal position.
It also bears the centrifugal forces as tensile ones. Instead of considering a
calculated evolution of the skin width, a constant value for this value will
be applied as well.
As mentioned before, [70] gives values for the approximated ratios between
shear web skin and the outer skin, and also between shear web skin and the
spar caps thickness. Taking into account these relationships, the total mass
will depend on the shear web width. For a 1.9 cm skin (35% over the
maximum calculated in 4.15), a total mass of 3090 kg is obtained.
The cumulated mass distribution along the span is shown in Fig. 4.16 in
comparison to two typical cumulated mass distributions [62, 67]. It shows
that making the value of the skins constant and reinforcing the cylindrical
part of the blade, the cumulated mass distributions fit typical ones. Starting
from calculated skin and spar cap areas along the span and the density of
the glass reinforced plastic, the inertia constant J expressed in kg · m2 and
the center of gravity from the blade root have been calculated
J = 417255 kg · m2
(4.1.29)
Xcg = 7.81 m
The center of gravity is similar to the one given by [68] or [81] for the
same length blade.
Applying the same analysis to different lengths, the values α = 0.552 and
β = 2.645 have been found for the relationship M = α · Lβ . In figure 4.17
this relationship is compared to the other graphics presented previously.
4.1.7
Estimating a wind turbine inertia constant
For geometrically similar blades having a length similar to the previously
analyzed, the inertia constant can be approached from the following expression
J = kJ · mass · L2
(4.1.30)
4.1 Estimation of the inertia time constant
51
Figure 4.16: Typical and calculated cumulated mass.
where kJ is obtained from (4.1.29)
kJ =
417255kg m2
= 0.2
3090kg · 262 m2
(4.1.31)
Similar values are derived by calculating the inertia of the blades whose
cumulated mass distributions are depicted in Fig. 4.16, in relation to [62]
(kJ = 0.184) and [67](kJ = 0.1829).
4.1.8
Estimating H for different wind turbine capacities
Two expressions can be used to relate inertia constants H and J [40]
¡
¢2
rpmrotor 2·π
60
H = z·J
2 · Pwatt
¡
¢2
rpmgen 2π
60
H = z·J
2 · Pwatt · n2gb
(4.1.32)
(4.1.33)
52
Chap. 4: Estimation of mechanical constants
Figure 4.17: Comparison of different weight-length relationships.
where z is the number of blades, rpmrotor and rpmgen are the rotational
speeds of the rotor and the generator, Pwatt 1 is the wind turbine capacity
expressed in watts and ngb is the gearbox ratio. These data are necessary
to estimate the H constant for a wind turbine.
In order to observe the trend of the inertia time constant value along with
the increasing capacity, the P ower − to − length, the W eight − to − length
and the ratiogb − to − length relationships should be introduced in (4.1.33).
Data from Appendix B have been correlated (Fig. 4.18) and the following
relationships estimated.
• Capacity as a function of the rotor diameter
P ' kP · DαP = 310 · D2.01
(4.1.34)
similar to that given in [77](124 · D2.23 ) , and slightly lower than that
given in [82] (195 · D2.155 ).
1
Some simulation programs such as PSCAD/EMTDC consider the apparent power
MVA as the power base
4.1 Estimation of the inertia time constant
53
• Mass of the blade as a function of its length
Mpala ' kM · LαM = 2.95 · L2.13
(4.1.35)
as obtained in (4.1.9).
• Rotor diameter as a function of the blade length
D ' relDL · L ' 2.08 · L
(4.1.36)
• And from Tables 4.1 and 4.2, the gearbox ratio as a function of the
rotor diameter
ngb ' kgb · D = 1.186 · D
(4.1.37)
being the proportionality constant in the form of
50Hz
2π · L
npp
1.186 =
vtip m
s
(4.1.38)
assuming a two pair poled machine npp = 2 and a constant tip speed
of vtip = 63.5 m
s.
In fact, an slightly increasing dependance with power can be found
in the tip speed but is mainly due to the appearance in the statistics
of high power offshore wind turbines, faster as they do not comply a
strict noise constraint.
Starting from (4.1.31), (4.1.8), (4.1.35) and (4.1.8) the expression for the
inertia time constant (4.1.33) turns into
³
H = nblades · kJ · M · L2
=
nblades
· kJ · kM ·
2
µ
f (1+sG )
nppoles
2π
2 · P · n2gb
´2
(4.1.39)
´2
³
f (1+SG )
¶αM µ
¶2
·
2
π
nppoles
D
D
·
2 · D2
relDL
relDL
kP · DαP · kgb
54
Chap. 4: Estimation of mechanical constants
Figure 4.18: Relationship Capacity-Diameter.
which can be expressed as
H(sec) = kH · D(m)αH
µ
kH = nblades · kJ · kM
being
D
relDL
¶αM +2
³
f (1+SG )
nppoles
· 2π
2
kP · kgb
´2
(4.1.40)
= 2.175
αH = αM + 2 − αP − 2 = 0.12
The value for the exponent αH = 0.12 means that the inertia time constant increases slightly as the diameter does. A similar expression can be
estimated for the H − P relationship
H(sec) = 1.544 · P (W )0.0597 .
(4.1.41)
Fig. 4.19 shows this trend in solid blue line according to previous relationship. Green triangles show inertia time constants directly extracted from
the literature, being most of them estimated or assumed values, and not
excessively reliable (Table 4.3). The few red triangles are related to actual
values of J where the inertia time constant can be obtained once the pole
pairs, capacity and gearbox ratio are known. Data regarded to violet diamonds are obtained in the same latter way (from Table 4.1) but estimating
the inertia J from the weight and the blade length as in (4.1.31)2 .
2
Data for Nordex N80 and N90 and DeWind D62 and D64 have been excluded due to
their significatively higher values, over 8.5 sec
4.1 Estimation of the inertia time constant
55
As can be seen, actual or estimated data are mainly in the range of 3 - 5
s, as indicated in [83].
Figure 4.19: Inertias for different turbine capacities.
In accordance with the relationship shown in Fig.
Hturb = 3.5 s will be used as a base for the simulations.
4.1.9
4.19, a value of
Estimating the remaining inertia time constants
With regard to the hub inertia, this device weighs around one third of the
rotor mass (without including shaft nor gearbox), or analogously half the
weight of the three blades. Assuming a maximum radius of 2 meters for a
medium size wind turbine, the hub inertia can be estimated from (4.1.33),
yielding a constant H lower than 0.05 s. The other components of the torque
transmission system (gearbox, brake, fast shaft or slow shaft) have neither
significant inertias, and hence they will be omitted unless direct data are
available.
By comparison, it is easier to find reliable values for the generator inertia.
They show the great influence of the kind of generator on its inertia. For
example, for a generator in the region of 1500 kW, the inertia can vary
from around 75 kg · m2 (generator Weier from Vestas V66-1.65MW, rotor
winding weight 1950 kg, total weight 6473 kg) for a wound rotor and around
56
Chap. 4: Estimation of mechanical constants
Table 4.1: Data for estimating H: rated capacity (kW), blade length (m), gear box
ratio, blade weight (kg) and estimated inertia time constant H (sec).
Turbine
Capacity Blade Gear
length box
Blade Inertia
Weight time
Bazán Bonus MkIV
Dewind Ibérica D46
Dewind Ibérica D48
Ecotecnia 600
MADE AE 46/I
Gamesa V47-660kW
Gamesa G47 Ingecon
Nordex N 50
MADE AE 52
Gamesa G52 850
Gamesa G58 850
DeWind D62
Dewind Ibérica D64
Nordex N62/1.3MW
Nordex N60
Nordex N62
Ecotecnia 62 1300
Neg Micon NM1500/64
Südwinds70/1500
Nordex S70
Nordex S77
Vestas V66-1.65
Ecotecnia 74 1670
Ecotecnia 80 1670
Gamesa G80 2000
Gamesa G87 2000
Gamesa G90 2000
Gamesa G83 2000
Nordex N90
Nordex N80
600
600
600
600
660
660
660
800
800
850
850
1000
1250
1300
1300
1300
1300
1500
1500
1615
1615
1650
1670
1670
2000
2000
2000
2000
2300
2500
1800
1800
1800
2900
2500
1500
1600
3000
3200
1900
2500
4300
4800
4300
4900
4900
5800
6900
5200
5600
6500
3800
5800
6035
6500
6500
7000
9400
10200
8600
19
22,15
23,15
19,1
21
23
23
23,3
25,1
25,3
28,3
29,1
31,1
29
29
29
29
31,2
34
34
37,5
32,15
34
37,3
39
42,3
44
40,5
43,8
38,8
55
45,5
45,5
55,76
59,53
52,63
52,65
63,6
58,34
61,74
61,74
53,5
48,9
79
78,3
78,3
81,8
87,74
95
94,7
104,2
78,8
94,63
94,63
100,5
100,5
100,5
100,5
77
68
2,70
5,37
5,86
4,28
3,56
3,28
3,49
3,80
5,59
2,83
4,67
9,61
11,7
3,37
3,90
3,90
4,23
4,39
3,35
3,38
3,94
2,89
3,39
4,24
3,70
4,35
5,07
5,76
10,8
8,46
4.1 Estimation of the inertia time constant
57
Table 4.2: Data for estimating M and H: rated capacity (kW), rotor diameter (m),
gear box ratio, and estimated inertia time constant H (sec).
Turbine
Capacity Rotor
diameter
Gear box
ratio
Inertia
time
Gamesa G42
Neg Micon 600/43
Gamesa G44
Neg Micon 600/48
Neg Micon 750/44
Neg Micon 750/48
FuhrLander
MADE AE 56
MADE AE 59
Suzlon 950
FuhrLander
Bonus 1MW
Neg Micon 1000/60
Suzlon S60 1MW
Suzlon S62 1MW
Suzlon S64 1MW
DeWind D60
Suzlon S60 1.25MW
DeWind D62
DeWind D64
Suzlon S64 1.25MW
Suzlon S66 1.25MW
Bonus 1,3MW
MADE AE 61
FuhrLander
FuhrLander
Bonus 2MW
DeWind D80
Suzlon 2MW
MADE AE90
Bonus 2,3MW
Ecotecnia 100 3MW
600
600
600
600
750
750
800
800
800
950
1000
1000
1000
1000
1000
1000
1250
1250
1250
1250
1250
1250
1300
1320
1500
1500
2000
2000
2000
2000
2300
3000
44
55,6
45
71,4
55,6
68,2
66
63,02
66,37
89,2
69
69
83,3
67,31
67,31
82,29
46,9
74,92
50,2
53,1
74,92
74,92
78
80,8
94,7
104
89
94
118,1
101
91
126,3
3,66
3,69
3,72
3,81
3,72
3,81
3,81
3,99
4,06
4,16
3,95
3,96
4,08
4,08
4,12
4,16
4,08
4,08
4,12
4,16
4,16
4,20
4,12
4,10
4,27
4,39
4,38
4,45
4,57
4,61
4,49
4,75
42
43
44
48
44
48
48
56
59
64
54
54,2
60
60
62
64
60
60
62
64
64
66
62
61
70
77
76
80
88
90
82,4
100
58
Chap. 4: Estimation of mechanical constants
Table 4.3: References including H.
Reference
[54]
[41]
[84]
[55]
[57]
[45]
[85]
[56]
[60]
[86]
[65]
[1]
[58]
[63]
[63]
[50]
[50]
[50]
[50]
[64]
[87]
Capacity (kW)
H (sec)
180
200
225
225
300
350
400
600
600
2000
2000
2000
2000
2000
2000
2000
2000
2000
2000
2000
6000
3,13
1,93
2,94
2,94
5,3
3,13
5
4,19
3,2
3,71
3,5
4,5
3,52
2,5
4,5
3,5
4,5
5,5
6,5
2,5
4,93
35−50 kg ·m2 for a squirrel cage generator. The expression for the generator
inertia time constant H in seconds is
³
´2
f (1+sG )
2
π
npp
Hgen (s) = Jgen (kg · m2 )
(4.1.42)
2 · P · n2gb
resulting in a value of 0.63 s for the wound rotor and 0.29 − 0.45 s for the
squirrel cage rotor.
These values can be considered to be independent of the rated power [88].
Finally, the value Hgen = 0.4 s has been chosen.
4.2 Other mechanical constants
4.2
59
Other mechanical constants
4.2.1
Self damping
This parameter, as well as the other mechanical transmission system parameters, does not have the same influence as the inertia. In the case of self
damping, since a fixed speed wind turbine is simulated, the effect of the self
damping during the electrical connection to the grid is reduced to an almost
constant antagonistic torque.
In order to estimate how self damping varies with the rated capacity, the
following equation is used [42]:
dFL (r) =
ρ
ch(r) vr (r)2 cd (α) dr
2
(4.2.1)
where dFL is the aerodynamical drag force exerted on a blade differential
at a distance r from the rotor axis, ρ is the air density, ch is the blade
chord, vr is the relative speed with respect to the blade, and cd is the drag
coefficient, that is dependent on the material surface state, the Reynolds
number and mainly the angle of attack α (Fig. 4.9). The component
of this force upon the rotation plane multiplied by the lever arm r gives
place to the differential torque opposing the blade movement (Fig. 4.20).
Integration of these differential momenta for the three blades yields the total
aerodynamical drag torque.
Assuming a tip speed independent of the blade length, then differential
drag force varies with diameter squared, and the upper integration limit also
varies with diameter; hence torque varies with diameter cubed.
Assuming that the rotational speed Ω is inversely proportional to the
diameter and the resistant torque due to the drag can be expressed in the
simplest way as a friction torque (as done by PSCAD/EMTDC) through
µ
¶
N ·m·s
Tdrag = SD
·Ω
(4.2.2)
rad
then self damping is expected to vary with D4 .
Analogously as for the inertia, the resultant value can be transformed from
the international system MKS to per unit p.u. through the expression3 :
3
This expression implicitly consider the self damping in the way indicated by (4.2.1)
60
Chap. 4: Estimation of mechanical constants
Figure 4.20: Composition of forces exerted on the blade.
µ
SD(p.u.) = SD
' SD
Nm · s
rad
¶
³
f (1+sG )
npp
2π
P · n2gb
³
´
¶ f 2π 2
µ
npp
Nm · s
rad
P · n2gb
´2
(4.2.3)
As indicated in (4.1.8) rated power varies practically as the square of the
diameter, and the gear-box ratio varies as diameter does. This can suggest to
assume that self damping in p.u. can be considered as a constant regardless
of the turbine capacity.
SD(p.u.) 6= f (P )
It was assumed that this value depended only on the drag. However it can
be established that it also depends upon the friction and ventilation in the
generator. These effects can be lumped in a single value.
4.2 Other mechanical constants
61
Very few references can be found with estimated or actual values for this
parameter and the range for these values is very wide. For simulation purposes, a self damping of SD = 0.05 p.u. will be finally chosen, close to the
value provided in [41](0.052) or [46] (0.044).
4.2.2
Torsional stiffness
Regarding torsional stiffness, there are some discordances about the expression of this normalized magnitude or in p.u.
PSCAD manuals [40] propose the following equation for the torsional stiffness of the low speed shaft
µ
ls
K (p.u.) = K
ls
N ·m
rad
¶ ¡ 2·π
60
rpmr
P
¢2
µ
=K
ls
N ·m
rad
¶ ¡ 2·π
rpmg
P · n2gb
60
¢2
(4.2.4)
which results in having dimensions of s−1 .
In order to be sure of using the right expression to display torsional stiffness in p.u., an intrinsic analysis of the phenomenon must be made. Thus,
when two magnitudes are linearly related via a proportionality constant K,
the value of this constant in p.u. means the variation in one of the magnitudes when varying the other one. In relation to the torsional stiffness, it
is expected that variations in the real power extracted from the wind give
place to variations in the twist angle of the rotor due to the torque transmission. In this case, the torsional stiffness in p.u. is the constant which
connects both magnitudes
ls
P (p.u.) = Kpu
θ(el.rad.)
(4.2.5)
having dimensions of p.u./el.rad.
Starting from the torque and twist angle in the low speed shaft and operating in electrical radians instead of mechanical ones (related through the
number of pole pairs as el.rad = mec.rad · npp ) the following expression is
obtained
µ
¶
PN · npp · ngb
PN · npp
N ·m
θel
ls
¡ el.rad ¢ =
¡ el.rad ¢ = TN = K
el.rad ngb · npp
f (1 + sN ) 2 π
ΩN
s
s
(4.2.6)
62
Chap. 4: Estimation of mechanical constants
where PN , sN and TN are rated capacity, slip and torque, and ΩN is the
rotational speed at rated conditions.
Comparing previous equations and considering rated conditions (PN = 1),
numerical value of torsional stiffness can be identified
³ p.u. ´
f · (1 + sN ) · 2π
f · 2π
ls
= K ls
Kpu
' K ls
.
(4.2.7)
2
2
el.rad.
P · ngb · npp
P · n2gb · n2pp
For torsional stiffness values related to the high speed shaft
³ p.u. ´
f · (1 + sN ) · 2π
f · 2π
hs
Kpu
= K hs
' K ls
.
2
el.rad.
P · npp
P · n2pp
(4.2.8)
In any case, in order to be used in the PSCAD Unit System 9 that uses
expression (4.2.2), torsional stiffness in p.u. must be multiplied by f · 2π
KP9 SCAD (s−1 ) = Kp.u./el.rad. · f · 2π
(4.2.9)
To consider the torsional stiffness of both shafts, low and high speed, the
equivalent constant results
K ls · K hs
.
(4.2.10)
K ls + K hs
Finally a value of
p.u.
Kpu = 0.3
(4.2.11)
el.rad
was adopted which is similar to those indicated in [63, 64, 54, 44].
K eq =
Influence of this parameter on the system dynamics begins to be noticeable
for lower values than aforementioned ones and when the motor or resistant
torque is suddenly lost, as in the case of a short-circuit with breaker opening. Mechanical parameter evolution is soft enough for the torsional spring
constant influence not to be significant, even in the case of very flexible
couplings (low torsional constant).
4.2.3
Mutual damping
In the same way as for the self damping, in order to translate into per unit,
the following expression must be used
¶2
µ
f
µ
¶
2π
nppoles
Nm · s
M D(p.u.) = M D
(4.2.12)
rad
P · n2gb
4.2 Other mechanical constants
63
where rpm is the rotation speed in revolutions per minute at the place
where the flexible coupling is located. According to the scarce bibliography
[41, 46] the value M D = 25 has been chosen.
Analogously to the stiffness constant, in the case of two flexible couplings
existing, equivalent mutual damping will be the inverse of the sum of the
inverses of the individual mutual dampings.
64
Chap. 4: Estimation of mechanical constants
Chapter
5
Soft-starter
The core of present work is to improve the performance of the soft-starter
connecting the induction machine to the network supply.
This chapter only deals with the configuration and operation of this power
converter, gathering the dispersed information existing regarded to the startup of induction motors, and pointing out the differences for the case wind
turbines where the soft-starters must adapt the induction generator voltage
to that of the network supply.
The improvement in the control structure will be dealt with in next chapter.
5.1
Configuration
The soft-starter is a power electronic converter that is not specific for wind
turbines, but is being introduced more and more frequently in industrial
plants where it is necessary to operate with induction motors controlling
the start currents in a more efficient way than the traditional methods.
A soft-starter device is integrated by 6 thyristors, two per phase, in a back
to back or anti-parallel configuration as can be seen in Fig. 5.1. A snubber
66
Chap. 5: Soft-starter
Figure 5.1: Soft-starter power circuit.
RC network is usually included in order to limit the rate of change of the
dv
voltage,
, across the thyristors. Figure 5.2 shows the soft-starter control
dt
circuit and the way to obtain the beginning of the excitation of the gates
(firing angle), similar to [89].
With the name of ac voltage controller [90], it can also be found in applications like an electronic breaker, a simple and economic speed controller
for single-phase and three-phase induction machines, in the under load tap
changing for power transformers, in induction heating or in static-var compensators (an analysis of this electronic device, the involved equations and
its performance can be found in [91]). In these cases where soft-starters
gradually vary the voltage at the terminals of an induction motor, they offer many advantages over conventional starters [12, 16], derived from their
ability of working according to three different modes [15]:
• As a proper soft-starter, providing a smooth acceleration, which reduces motor heating and stress on the mechanical drive system due to
high starting torque hence increasing the life and reliability of belts,
5.1 Configuration
67
Figure 5.2: Soft starter control circuit and control signal time evolution.
gear boxes, chain drives, motor bearings, and shafts [16]. By reducing the voltage when an induction machine starts, it also reduces the
high starting current, thus alleviating voltage dips and even eliminating brownout conditions. It also reduces the shock on the driven load
due to high starting torque that can cause a jolt on the conveyor that
damages products, or pump cavitations and water hammers in pipes.
Thus, a fully adjustable acceleration (ramp time) and starting torque
for optimal starting performance, provides enough torque to accelerate
the load while minimizing both mechanical and electrical shock to the
system [10].
• As a solid state voltage, saving energy under lightly loaded conditions
if the load torque requirement can be met with less than rated flux.
This way, core loss and stator copper losses can be reduced [20].
• As a discrete frequency inverter that increases the frequency until the
frequency of the line is reached (50/60 Hz). The discrete frequencies produced are sub-multiples of line-frequency and are generated by
omission or inclusion of line frequency half cycles. [14]
For the two first modes, the ones corresponding to the operation as a
voltage controller, the control of the current or the torque is based on the
68
Chap. 5: Soft-starter
assumption that the terminal voltage is regulated by appropriate adjustment of the firing angles triggering the thyristors. However, the relationship
between voltage and firing angle is highly nonlinear and is also a function
of the power factor and operating conditions of the induction machine. The
power factor depends on the rotor speed, or rather the rotor slip, and as will
be seen in the Chapter 6 describing the third order model, it also depends
on the derivative of voltage. This makes it quite difficult to find the exact
value of the firing angle for any motor speed and torque.
Some methods of optimal soft starting have been presented, as in [15]
that proposes an open loop controller based on artificial neural networks
for the thyristor firing angles or in [13] which detects the phase current and
the voltage across the non conducting thyristor as inputs for a fuzzy logic
based controller. As in other references, the soft-starters’ objective has been
to accelerate induction motors from rest condition with minimized currents
and torque.
In the case of wind turbines, the soft-starter has been introduced to fixed
speed ones to reduce in-rush currents and voltage dropouts. [8] analyzes
phenomena which affect voltage dip and inrush current due to the direct
connection of induction generators running close to synchronous speed to
electrical distribution in low head hydro electric schemes. This is a different
case to that studied in the starting of induction motors, since now slip varies
within a narrow interval around zero, but comprising motor and generator
operation.
This qualitative change in the operation mode gives place to important
numerical changes in the relationship between the firing angle and the voltage at the induction machine terminals. Even if there is not a shift between
motor and generator operation modes, this relationship is also affected by
the power factor of the induction machine which in turn depends on the
voltage derivative as will be shown in Chapter 6.
There is no relevant bibliography related to optimum control of a softstarter for the softened connection of induction machines working as generators.
Fig. 5.3 shows a simplified soft-starter performance for a wind turbine
generator. Basically, its function is to feed the induction machine with
a variable voltage, whose evolution pattern is specified according to some
constraint. This constraint has traditionally been to limit the start-up overcurrent. In the present work, the soft-starter will be controlled in order to
5.2 Thyristor triggering
69
Figure 5.3: Variation in the supplied voltage by means of a soft-starter.
limit the voltage dropout.
In fact thyristor gates are not triggered by means of continuous pulses nor
isolated impulses, but by supplying pulse trains beginning at the desired
firing angle α.
The length of these trains can be shortened [10, 9], to avoid unnecessary
triggering. This last paper shows that adequate operation at each thyristor
can be accomplished with two short trains of triggering pulses separated by
an angle of 60o . Designs where the trains are enlarged for a semi-period can
also be found [11]. This solution is easier to implement, but care must be
taken in not enlarging the trains of pulses beyond a semi-period, in order to
avoid both forward and reverse thyristors being simultaneously triggered.
In general, if the train is not divided, the length of the trains of triggering
pulses must be between 60o and 180o , as indicated in Fig. 5.4.
5.2
Thyristor triggering
Signal Ead in Fig. 5.1 refers to the a phase voltage. Its zero crossing is
the reference taken for the beginning of the pulse train that will excite the
gate of the forward thyristor in phase a. It is more suitable using angles
instead of time instants to refer to the point at which pulse trains begin.
70
Chap. 5: Soft-starter
Figure 5.4: Pulses at gates. Separation between forward thyristor triggering pulses.
In this sense, pulse trains are repeated in the remaining thyristors every
60o . Therefore the firing angle α refers to the angle between the zero
crossing of any voltage phase and the beginning of the train pulses exciting
the corresponding forward thyristor. It determines the voltage supplied to
the connected device.
The triggering pulse sequence is depicted in Fig. 5.5. Arrows indicate the
beginning of the overlapped train of pulses at the corresponding thyristors.
Thus, the first vertical arrow represents the point at which forward thyristor
in phase a and reverse thyristor in phase b are both triggered. Firing angle or
delay angle α is the distance between this point and the rising zero-crossing
of Ua . In order that the thyristors conduct, this first arrow must be at the
left of the intersection of curves Ua and Ub . In Fig. 5.4 the separation angle
between forward thyristor triggering pulses is shown to be 120o . Trigger
signals for forward thyristor gates and the corresponding reverse one are
separated 180o .
5.3
Operation modes
2-0 Mode. If the triggering begins slightly before the voltage curves cross,
it is probable that the current will fade before another thyristor will be
triggered again. This is the case of the interrupted currents shown in
Fig. 5.6, where the conduction intervals appear divided in two. This
is also reflected in the lower graph where it can be seen how the stored
5.3 Operation modes
71
Figure 5.5: Gate triggering sequence and line currents.
energy is not enough to keep the current in phase a and it fades before
another thyristor is fired. There are either two thyristors conducting
or none, thus calling this situation the 2-0 mode. In this mode, the
voltage waveforms fed to the connected device are truncated short.
According to [9] triggering pulses must exist 60o after the firing angle
in the case of operation with current interruptions to allow the second
conduction interval.
3-2 Mode. If the firing angle decreases, a higher voltage difference between
voltage curves exists when the thyristors are triggered, and a higher
energy is supplied to the connected device. This increases the conduction interval until the current in a thyristor (for instance AF) is still
positive when another opposite thyristor is triggered (CR in this case).
This limit in the operation modes also depends on the power factor
of the connected device. Lower firing angle values give place to operation in the 3-2 mode, where there are either three or two thyristors
conducting.
Fig. 5.7 shows the conduction intervals for each thyristor and the
current through phase a. As in the 2-0 mode, the lower firing angle,
the more complete voltage waveforms and the higher rms voltage is
expected at the connected device.
72
Chap. 5: Soft-starter
Figure 5.6: Conduction periods of the thyristors. Operation with current
interruptions.
Uncontrolled Mode. As long as the firing angle is decreased, conduction
intervals extend. Depending on the device’s power factor, there is a
value for the firing angle at which current in one phase fades just before
it begins to flow in the opposite direction. Firing angles lower than
this limit will not produce any change in the supplied voltage. The
soft-starter gets into an uncontrolled operation equivalent to a direct
connection to the grid.
No Conduction Mode. On the other hand, if trains of triggering pulses
begin at the right of the intersection of curves Ua and Ub , the forward
thyristor in phase a will be inversely polarized and will not conduct.
Therefore, the soft-starter will act as an open circuit for firing angles
greater than 150o and no voltage will be supplied at the load terminals. In fact this assertion cannot be made when the connected device
behaves as a generator.
5.4 Wind turbine soft-starter
73
Figure 5.7: Conduction periods of the thyristors. Operation without current
interruptions.
A more detailed description of the conduction modes can be found in
[18, 19], and its current waveforms in [9, 17].
5.4
Wind turbine soft-starter
For firing angles smaller than 150o the relationship between the firing angle
α and the controlled voltage is non-linear and depends additionally on the
power factor of the connected element. The angles that limit operation
modes also depend on the power factor.
The third order induction generator model (Chapter 6) shows that the
power factor depends in turn on the voltage and its derivative, the slip and
74
Chap. 5: Soft-starter
the derivative of the generator voltage angle, not making it feasible to obtain
the supplied voltage versus firing angle characteristic.
Another issue regarding wind generator connection is that when voltages
at both sides of the soft-starter are the same (uncontrolled mode), the generator is completely connected to the grid. At this moment, a contactor
that electrically connects the wind turbine and the low voltage transformer
side is energized, thus by-passing the soft-starter. Finally a capacitor bank
is connected for power factor compensation. The demanded reactive power
will dictate the number of connected capacitors. From the point of view of
decreasing the voltage dropout, it could be desirable to connect the capacitor bank during the start-up. However, the soft-starter produces harmonic
currents that can damage the capacitors, and thus the connection of the
capacitors will not be accomplished until the start-up process has finished
[68].
5.5
Asymmetrical soft-starter
In the induction machine start-up, an uncontrollability problem arises when
slip is close to zero and voltage is close to that of the grid. In order to
increase the soft-starter controllability, an asymmetry has been introduced
in the gate triggering pulses of the thyristors by delaying one of the six
pulses. This gives place to unbalanced voltage and currents systems with
lower rms values. The sequence of pulses supplied to the thyristor gates as
well as the waveform for the current Ia are represented in Fig. 5.8.
As will be seen when different controllers are compared, a slightly lower
dropout is obtained for the same controller when including this asymmetry
in the soft-starter. The improvement is not so significant as to recommend
a modification in the soft-starter hardware.
5.6
Firing angle control system
An adequate control of the firing angle time evolution will decrease overcurrents, and mainly voltage dropouts during start-up. Present systems linearly
decrease the firing angle as a function of the time, that is the beginning of the
train of triggering pulses to the thyristors, in an open-loop control. Therefore, firing angle controllers currently installed in wind turbines perform the
5.6 Firing angle control system
75
Figure 5.8: Asymmetrical pulse sequence at the thyristor gates and phase current.
connection in a predefined number of grid periods [2] and do not provide
any interaction with the external condition that could alleviate electrical
connection effects.
Throughout this work, the causes of voltage dropout will be analyzed
and closed-loop control systems will be introduced in order to improve the
performance of the soft-starter.
76
Chap. 5: Soft-starter
Chapter
6
Induction machine dynamic
models
There are different models to explain the performance of induction machines.
The more simplifications are introduced in the modeling process, the less accurate is the model, but the lower computational cost. But the main feature
of the reduced models is their ability to provide a better understanding of
the system, and the simplicity to derive analytical expressions.
In this chapter, a survey of the fifth order model and the steady state
model will be presented as the most popular ones. The third order model,
of great applicability in transient stability studies, will be introduced for
the first time as the best approach to explain and understand the real and
reactive power evolution during the start-up.
The sufficient conditions for assuring the validity of the third order model
in this process will be presented and checked, and the main equation describing the induction machine performance will be developed.
Finally, additional considerations and simplifications will allow to obtain
expressions for the real and reactive power as a function of the generator
voltage, its derivative, the derivative of the voltage angle and the slip. These
expressions will allow the controller to adjust its performance in order to
decrease the voltage dropout.
78
6.1
Chap. 6: Induction machine dynamic models
Fifth order model
An induction machine can be modeled through an equation system relating voltages, currents and flux linkages and including inductances that are
functions of the rotor speed. A change of variables is often used to reduce
the complexity of these time-varying coefficient differential equations [92].
There are several changes of variables that are used, but all of them are contained in a general transformation that refers machine variables to a frame
of reference that rotates at an arbitrary angular speed (Fig. 6.1).
Figure 6.1: Arbitrary reference frame.
The new equations, together with the mechanical ones, give place to a
fifth order differential equation system. This linear system describes the
induction machine performance with an accuracy that is enough for most
situations. Many computer programs used for transient studies such as
MATLAB and PSCAD/EMTDC have introduced this model in their blocks
or subroutines. These equations are [93][94]:
vds = Rs Ids − ωλqs + pλds
(6.1.1)
vqs = Rs Iqs + ωλds + pλqs
t
t
vdr
= Rrt Idr
− (ω − ωr )λtqr + pλtdr
t
t
vqr
= Rrt Iqr
+ (ω − ωr )λtdr + pλtqr
where λ is the flux linkage, ωr is the rotor speed, ω is the angular speed
of the reference frame, p is the derivative operator and the subscripts s, r,
6.1 Fifth order model
79
d and q stand for the stator, rotor and to d − q axis of the reference system
rotating at ω. Electrical variables in the rotor referred to stator, with the
superscript t, are obtained from the real ones through the ratio of effective
number of turns of the stator Ns and rotor winding Nr :
Ns
Udqr
Nr
Nr
Idqr
Itdqr =
Ns
Ns
λ dqr
λ tdqr =
Nr
µ ¶2
Ns
t
Rr =
Rr .
Nr
Utdqr =
(6.1.2)
Stator and rotor currents give place to the flux linkages according to:
t
λds = Lls Ids + M (Ids + Idr
)
(6.1.3)
t
λqs = Lls Iqs + M (Iqs + Iqr )
t
)
λtdr = Ltlr Idr + M (Ids + Idr
t
λtqr = Ltlr Iqr + M (Iqs + Iqr
)
where Lls is the stator leakage inductance,
M=
3
Lms
2
(6.1.4)
Lms is the magnetizing inductance and Ltlr is the rotor leakage reactance
µ
t
Llr =
Ns
Nr
¶2
Llr .
(6.1.5)
These equations must be completed with the expressions for the mechanical
torque Te and the drive dynamics.
3
t
t
npp M (Iqs Idr
− Ids Iqr
)
2
d ωr
Te − Tl = 2H
dt
Te =
(6.1.6)
(6.1.7)
80
Chap. 6: Induction machine dynamic models
where npp is the number of pole pairs, Tl is the load torque and H is the
inertia time constant.
If the electrical frame angular velocity is the one corresponding to the
fundamental frequency of the power system the induction machine is connected to (ω = ωs )1 , the stationary circuit variables are referred to what is
called the synchronously rotating reference frame. This reference system is
particularly convenient when incorporating the dynamic characteristics of
an induction machine into a digital computer program used to study the
transient and dynamic stability of large power systems [93].
Using the synchronously rotating reference frame avoids sinusoidal components of the stator state variables that appear when referring to other
reference systems [95][96].
On the other hand, using bold typeface for complex variables and naming
Us = Uds + j Uqs
(6.1.8)
Ur = Udr + j Uqr
Is = Ids + j Iqs
Ir = Idr + j Iqr
λ s = λds + j λqs
λ tr = λtdr + j λtqr
equations (6.1.1) and (6.1.3) can be written as:
Us = Rs Is + j ωs λ s + pλ
λs
t
t
t
t
(6.1.9)
t
Ur = Rr Ir + j (ωs − ωr )λ
λr + pλ
λr
t
λ s = (Lls + M ) Is + Ir M
t
t
t
λ r = (Llr + M ) Ir + Is M
(6.1.10)
(6.1.11)
(6.1.12)
Solving this system results in a significant computational effort and, what
is worse, qualitative ideas about the system performance are difficult to
establish. There are several reduced order models that involve equation
system simplifications [95], although two approaches are used most: third
order model and steady state model.
1
Some bibliographies denotes this speed as ωe
6.2 Reduced models for the induction machine
6.2
6.2.1
81
Reduced models for the induction machine
First approach: third order model
This model is derived from the fifth order model by disregarding stator
transients. This means canceling the derivatives of stator flux linkages.
In a synchronous reference system, this approximation is valid for small
slip values since the high frequency component of the flux linkage can be
separated from the low frequency one and has limited effect on the torque
expression. However, for high slip values significant differences exist with
regard to the fifth order model, since high frequency electrical variables are
induced in the rotor and stator giving place to pulsating torques. These
high frequency components appear when solving the fifth order model but
not in the third order model [97][43].
This model is often used in transient stability studies, as in [98] and [99],
that have been taken as a reference in foreseen sections as a first step to
derive expressions for the real and reactive power in a wind turbine induction
generator.
Other works about transient stability starting from this model are [46, 45]
or [61]. In general, third order model will be used in simulations where high
frequency modes are not expected or they are not significant.
6.2.2
Second approach: first order model
This approach is derived neglecting transients in rotor and stator, hence
state variables appear as constant values. This gives place to the steady
state induction machine model. This model has been used in [100] for the
simulation of a fixed speed stall control wind turbine at start-up although
results obtained by means of this model show a disagreement with the observed performance when rotor speed or stator currents or voltages vary
quickly.
82
Chap. 6: Induction machine dynamic models
6.3
Third order model main equations
6.3.1
Validity conditions
In some transients, as shown in [98], derivatives of stator flux linkage will be
neglected, converting the fifth order model in a third order model. According
to [97], this reduced order model is said to be accurate if
σ · (Ltlr + M ) /Rr
Tr0
¡
¢
=
> 0.8
Ts0
σ · Ltlr + M /Rs
(6.3.1)
Rs Rr
2H · sN · Rr
2 < 10 · σ · Lt (1 − s )
(Xs + Xr )
N
lr
(6.3.2)
α=
and
σ being the leakage parameter, Xs and Xrt the stator and rotor (referred to
stator) reactances, and sN the slip speed at operating (rated) conditions.
M2
(M + Ltlr )(M + Lls )
Xs = ωs Lls
σ = 1−
(6.3.3)
Xrt = ωs Ltlr
Xm = ωs M
Drives satisfying the previous conditions can be modeled by third order models having dominant eigenvalues that agree closely with the corresponding
eigenvalues of the full model [97].
A new control structure has been tested for the soft-starter synchronizing
an induction machine with the parameters shown in Table 6.1. Henceforth
Table 6.1: Electrical parameters for the induction machine.
Rs = 0.0059 p.u.
Xs = 0.0087 p.u.
Rr = 0.01 p.u.
Xr = 0.143 p.u.
RF e = ∞
Xm = 4.76 p.u.
6.3 Third order model main equations
Table 6.2: Transfer function Gr =
Zeros:
Poles:
83
−δTr
.
δωr
Full Model
Reduced Model
−20.29
−6.45 ± j 313.7
-20.23
−20.73 ± j 3.95
−12.54 ± j 313.35
−20.68 ± j 3.94
and for sake of clarity, rotor magnitudes will not be added the superscript
t.
The induction machine with previous parameters does not comply with
condition (6.3.1). In fact this condition is a sufficient one to ascertain that
dominant eigenvalues for the reduced and fifth order model are similar to
each other. Therefore, according to the small signal transfer functions shown
in Fig. 6.2 extracted from [97], the determination of zeros and poles must
be made for both the reduced and fifth order models.
Figure 6.2: Small signal block diagram representation of the induction generator.
In a general case of the tested induction machine as a part of an ac drive
where the rotor speed is to be controlled, poles and zeros of Gr must be
found for both models (Table 6.2).
If both root loci are drawn starting from these open-loop zeros and poles
84
Chap. 6: Induction machine dynamic models
Figure 6.3: Root loci for the mechanical closed-loop function transfer for the fifth
order model (in solid line) and the reduced order model (in dashed line).
npp T orqueN
as the varying closed-loop gain (Fig. 6.3), it
2 H ωslipN
can be seen that both models agree closely for low gain values, which corresponds to high inertia time constants. For high gain values, there can appear
complex conjugated dominant poles converting the system into a subdamped
one, and separating the performances for both induction machine models.
and taking K =
However, from the point of view of an induction machine supplied by an
increasing voltage, it is more interesting to analyze the transfer function
δTv
Gv =
since the input variable is the voltage. In this case, poles and
δVs
zeros for the fifth order model and the reduced one can also be obtained
from [97]. Zeros as well as poles corresponding to electrical, not mechanical,
equations are represented in Table 6.3.
Table 6.3: Transfer function Gv =
Zeros:
Poles:
δTv
.
δVs
Full Model
Reduced Model
−29.80
-33.78
21.52 ± j 29.77
34.59
−20.73 ± j 3.95
−12.54 ± j 313.35
−20.68 ± j 3.94
6.3 Third order model main equations
85
If a ramp voltage input is supplied to the transfer function, comparison
of residues indicates that poles at −12.49 ± j 312.94 can be disregarded, and
hence a good agreement between both models is more evident.
Influence of the dynamics of the mechanical system on the feasibility of
reducing to the third order model is given by condition (6.3.2) that is well
satisfied
0.0059 0.01
2 · 3 · 0.01 · 0.01
2 = 0.0025 < 10 · σ · L (1 − s ) = 1.25
(0.0087 + 0.143)
r
N
(6.3.4)
Therefore, a close agreement between fifth and third order model is expected.
6.3.2
Reduced electrical system
If stator and rotor self-inductances are defined as:
Ls = Lls + M
(6.3.5)
Lr = Llr + M
and substituting for Ir from (6.1.12) into (6.1.11) and considering a squirrel
cage induction machine (Ur = 0) yields:
0 = Rr
λr
M
− Rr Is + p λ r + j (ωs − ωr ) λ r
Lr
Lr
Multiplying this equation by
λ 0r = λ r
(6.3.6)
M
and denoting
Lr
M
Lr
(6.3.7)
leads to
p λ 0r =
M 2 Rr
Rr 0
λ − j λ 0r s ws
Is −
2
Lr
Lr r
where s = (ωs − ωr )/ωs is the rotor slip.
(6.3.8)
86
Chap. 6: Induction machine dynamic models
On the other hand, from (6.1.9) and (6.1.12)
µ
¶
M
M2
Us = Rs Is + j ωs Ls Is +
λr −
Is
Lr
Lr
M
= Rs Is + j X 0 Is + j ωs λ r
Lr
(6.3.9)
where X 0 is the transient reactance.
¶
µ
¶
µ
2
M2
Xm
X = Ls −
ωs = Xs + Xm −
Lr
Xr + Xm
¶
µ
Xm · Xr
.
= Xs +
Xr + Xm
0
(6.3.10)
Denoting
Z0 = Rs + j X 0
(6.3.11)
(6.3.9) reduces to
Us = Z0 Is + j ωs λ 0r = E0 + Z0 Is
(6.3.12)
where the voltage behind the transient impedance E0 is defined as
E0 = j ωs λ 0r .
(6.3.13)
Thus, (6.3.8) can be expressed as
p E0 =
¢
1 ¡
j (X − X 0 )Is − E0 − j E0 s ωs
Tr
(6.3.14)
where
Lr
Rr
X = ωs Ls = Xs + Xm
Tr =
It is worth noticing that stator voltages and currents in (6.3.14) are complex quantities defined or expressed according to (6.1.8) and (6.3.12).
6.4 P and Q in the third order model
87
Adopting a synchronously rotating reference frame and assuming a balanced voltage system, voltages Us and currents Is will be the phasors corresponding to the voltage and current at phase a. It is worth noticing that
this voltage phasor cannot be considered a static phase reference. In fact, a
variation in the stator voltage phase angle θ is expected due to firing angle
variations.
6.4
P and Q in the third order model
An alternative expression for (6.3.14) is
d Us d Is 0
1
−
Z = −j ωs (Us − Is Z0 ) −
(Us − Is Z)
dt
dt
Tr
(6.4.1)
Z = Rs + j X
(6.4.2)
where
Once this simplification is made, if the conjugate of (6.4.1) is multiplied by
U, it yields
Us
1
d U∗s dI∗s
∗
−
Us Z0∗ = j ωs (Us2 − S · Z0 ) −
(Us2 − S · Z∗ ) (6.4.3)
dt
dt
Tr
where S is the single-phase complex power. Solving this equation for S,
gives
µ
¶
dI∗s
Rr
dU∗s
0∗
2
Us
−
U Z + Us ωs
−js
dt
dt
Xr + Xm
µ
¶
(6.4.4)
S1f ase '
Rr
∗
ωs
Z∗ − j s Z0
Xr + Xm
Assuming the complex voltage and current in the form U = U ej θ and
I = I ej θ+ϕ the first two addends in the numerator are
³ ´ dU e−j θ
dUs
dθ
dU∗s
s
= Us
− j Us2
(6.4.5)
Us
= U s ej θ
dt
dt
dt
dt
µ
¶
³ ´ dI e−j (θ+ϕ)
dI∗
dIs −j ϕ
dθ dϕ
s
Us s = Us ej θ
= Us
e
−jI ·U
+
e−j ϕ
dt
dt
dt
dt
dt
88
Chap. 6: Induction machine dynamic models
and hence complex power S, expressed in p.u. yield
µ
¶
rr
dus
dθ
2
us
+ us ωs
−js−j
dt
xr + xm
dt
µ
¶
p+j·q '
+
rr
∗
0∗
ωs
z −jsz
xr + xm
µ
µ
¶¶
dθ dϕ
s
us di
−
j
i
u
+
z0∗ e−jϕ
s s
dt
dt
dt
¶
µ
+
rr
∗
0∗
z −jsz
ωs
xr + xm
(6.4.6)
where phase voltage and one third of the generator rate capacity have been
taken as base magnitudes.
UL
Ub = √ typically 398 V
3
M V Aturbine
Sb =
3
Ub2
Zb =
Sb
Sb
Ib =
.
Ub
Since
p + j · q = u · i · e−jφ ,
(6.4.7)
then (6.4.6) can also be expressed as
µ
¶
dus
rr
dθ
dis 0∗ −jϕ
2
us
+ us ωs
−js−j
+ us
z e
dt
xr + xm
dt
dt
µ
¶
µ
¶
p+j·q '
(6.4.8)
rr
dθ dϕ
∗
ωs
z∗ − j s z0 + j is us
+
z0∗
xr + xm
dt
dt
For large induction machines as used to be installed in wind turbines,
the value for Z0 is small and current is indirectly controlled such that its
evolution will not be too fast. Specifically, for the tested induction machine
whose parameters are indicated in Table 6.1, Z0 = 0.0059 + j 0.1475 Ω.
6.4 P and Q in the third order model
89
Figure 6.4: Real and Reactive Power. Comparison of steady state, third and fifth
order models.
Therefore, the derivative of the current term in (6.4.6) can be neglected
without a significant error, giving
µ
¶
dus
rr
dθ
2
us
+ us ωs
−js−j
dt
xr + xm
dt
µ
¶
p+j·q '
.
(6.4.9)
rr
∗
0∗
ωs
z −jsz
xr + xm
Fig. 6.4 visualizes the close agreement between the fifth and third order
models and the validity of disregarding the derivative of currents. Unacceptable values are obtained from the steady state model, whose evolution
is far from the fifth and third order models. The power values calculated by
PSCAD/EMTDC using a complete model appear as Pgen and Qgen . The
other two curves correspond to the reduced order model, either including or
disregarding the derivative of currents.
90
Chap. 6: Induction machine dynamic models
Chapter
7
Sliding-mode control to limit
voltage dropout
Voltage dropout at a given node in a power system depends on the real and
reactive power flowing from the network towards that node.
In order to keep the voltage within the regulated limit, the proposed softstarter controller must be able to estimate these power components and
impose a suitable action. Controllers based on the variable structure
system theory have received much attention in recent years to design robust
state feedback systems, mainly for controlling dc and ac servo drives [101,
102, 103].
A variable structure control system based on sliding-mode techniques can
be switched between two distinct control structures, constraining the system
state trajectory to a region known as a switching surface or in general,
switching hyperplane [101, 104].
In general, there are two basic steps in the design of the variable structure
controller: the design of the switching phase and the design of the reaching
or switching control.
With regard to the design of the sliding hyperplane (a line in a twodimension case), it is solely defined by parameters that are independent of
92
Chap. 7: Sliding-mode control to limit voltage dropout
the plant model, at least in an explicit way. Therefore, once the controlled
systems states enter the sliding mode, the choice of sliding hyperplanes determines the dynamics of the system which is provided insensitivity to bounded
plant parameter changes, external disturbance rejection and fast dynamic
response. In the present study the sliding line has been chosen in order to
reduce the voltage dropout to a limited value.
In relation to the design of the law control, the parameters involved in
it have to be chosen in order to guarantee that the system must reach the
switching surface (hitting phase). When all the state variables of the controlled system are constrained to lie in a switching hyperplane, the closed
loop dynamics are said to be in a sliding mode, or an sliding mode occurs
(sliding phase).
The advantages of the sliding-mode control have been employed to control
the position and speed of ac servo systems, where the main difficulty is to
precisely measure or accurately estimate, due to the noisy environment, the
discrete resolution of speed transducers or inaccurate system parameters.
With regard to the electrical magnitudes involved in the electrical connection process of a wind turbine, for the definition of the variable structure
controller, the following two issues must be taken into account:
• an expression for the voltage dropout will be determined starting from
the complex power flowing from the wind turbine generator and
• the generator voltage and its derivative are the main magnitudes influencing the complex power delivered by the induction generator, and
in turn the complex power flowing from the wind turbine.
According to the first item, a variable structure control strategy will be
used to control the relationship between real and reactive power in the high
voltage side of a wind turbine interconnection. Along the sliding-line, the
system will describe a trajectory defined by a desired relationship between
state variables. Therefore, according to the second item, forcing the system
to follow a sliding trajectory given by a suitable relationship between generator voltage and its derivative will, theoretically, produce the desired voltage
dropout. In fact, in order to make the system asymptotically convergent to
the sliding trajectory, the voltage at the generator terminals and its derivative will not be chosen as state variables, but some one to one function of
them.
93
Once the system has reached the switching hyperplane (a line in a twodimensional problem) the controller will constrain the system state trajectory to a band around it, thus limiting the voltage dropout around the
desired value. When the connection process is to be finally fulfilled, the system will naturally separate from the sliding trajectory, and the voltage at
the point common coupling begins to recover, thus completing the start-up
process.
The reference value for the voltage dropout will be determined as a function of the estimated rotor acceleration and must always be sufficiently inferior to the permitted voltage dropout set by local regulations.
Therefore, a sliding-mode based controller has been considered to be the
more suitable controller for this task due to its robustness and because its
control action fits closely with the aim of taking the system to a state where
a determined variable (the voltage dropout in this case) is kept within a
narrow interval.
In the first section of this chapter the expressions linking the voltage
dropout at the interconnection and the real and reactive power flowing towards the induction machine are explained.
In section two the sliding or switching trajectory in order to keep the
voltage dropout close to the prefixed value is established.
Section three will present the theory of design of the proposed controller.
An analysis of a specific system will be presented in section four, and
the control law parameters to guarantee system stability will be derived.
This means that the system will always be directed to the sliding trajectory,
where theoretically the voltage dropout is close to the desired one.
Particular details of the implementation of the sliding-mode controllers
are described in section five.
Start-up simulations when controlling the soft-starter by means of several
controllers and a comparison of the obtained results are presented in sections
six and seven.
More simulation results will be shown in section eight, but focused on
providing some ideas about the influence of line impedance on the voltage
evolution.
94
Chap. 7: Sliding-mode control to limit voltage dropout
Figure 7.1: Single wind turbine feeding a consumer in a weak grid.
7.1
Voltage dropout in a weak grid
The voltage dropout controller has been designed and tested considering
an electrical system with a single wind turbine feeding a consumer that is
connected to a weak grid (Fig. 7.1).
The voltage modulus difference in per unit between Thevenin voltage
source and the point of common coupling (PCC) takes the form (Fig. 7.1)
Enw − UP CC ' Pnw Rlin + Qnw Xlin .
(7.1.1)
It should be noticed that all magnitudes are real values, not complex ones
(not in bold type).
Before the connection process the real and reactive power components
transferred from the network to the PCC are the only power components
determining the voltage dropout at the local load node
Pwt = Qwt = 0 ⇒ Pcons = Pnw ,
Enw −
UP0 CC
Qcons = Qnw ⇒
' Pcons Rlin + Qcons Xlin
(7.1.2)
but once the wind turbine begins its electrical connection
Pwt , Qwt 6= 0 ⇒ Enw − UP CC ' Pnw Rlin + Qnw Xlin =
(Pcons + Pwt ) Rlin + (Qcons + Qwt ) Xlin
(7.1.3)
where Pwt and Qwt are the real and reactive power entering the wind turbine and hence are considered positive when flowing towards the generator
7.1 Voltage dropout in a weak grid
95
(motor convention). Neglecting real (power losses) and reactive power taken
by the transformer and in the medium voltage line conductors linking the
wind turbine to the PCC, Pwt and Qwt can be obtained from the powers
consumed/generated by the induction machine as
Pwt ' Pgen
s
µ 2
¶
Ulv
Ulv2
2 +P2
Q
−
1
Qwt '
gen
gen
2
2
Ugen
Ugen
(7.1.4)
(7.1.5)
where Pgen and Qgen are the real and reactive power taken by the induction machine and Ulv is the voltage at the low voltage side of the power
transformer.
Previous equations lead to
4U = UP0 CC − UP CC ' Pgen Rlin + Qwt Xlin ⇒
s
¶
µ 2
Ulv2
Ulv
2
2
4U ' Pgen Rlin + Xlin
Qgen + Pgen
−1
2
2
Ugen
Ugen
(7.1.6)
This means that there will be a voltage dropout with respect to voltage
previous to the connection process, that depends on the real and reactive
power as shown in (7.1.6). To observe regulations currently in force in many
countries, the voltage dropout must be below a certain limit of around 23%. Regarding the short term flicker value Pst , at low frequencies, up to 3%
voltage variation is acceptable [26]. However, the conditions under which
this dropout must be measured are not well defined.
Indeed, during the electrical connection of the wind generator to the network, there will be a first stage in which the turbine induction machine behaves as a motor, extracting real power from the network the wind turbine
is connected to. In Fig. 7.1 this network is represented as its single-phase
equivalent source and an impedance. With no interruption and slightly after
the synchronous speed is reached, the induction machine will behave as a
generator, transferring power to the consumer or to the network.
Therefore, it is expected that before initiating the wind generator connec0 will be greater than the voltage
tion, the voltage at the consumer point Upcc
during most of the connection process but lower than the voltage when the
f
connection is completely accomplished, Upcc
(Fig. 7.2). Papers dealing with
96
Chap. 7: Sliding-mode control to limit voltage dropout
Figure 7.2: Voltage in the PCC during the connection process.
the switching operation impact calculate or estimate the voltage change
starting the inrush current. Therefore, they implicitly equate the voltage
change with the voltage dropout. However the maximum voltage change
during the electrical connection should be considered [29, Standard CEI
61400-21], which may involve the final voltage once the electrical connection
has been accomplished.
According to Fig. 7.2 the voltage dropout strictly speaking was defined
in section 3.4.3 as the maximum drop in the rms value of the voltage at the
point of common coupling with respect to the voltage prior to the connection.
Voltage change was denoted as the maximum difference in voltage during
the connection transient.
Since the final value of the voltage cannot be determined during the startup, in order to have a reference to adjust and compare different controllers,
the voltage dropout instead of voltage change will be the magnitude to be
optimized. In any case, since the final value is independent of the start-up
process, minimizing voltage dropout means optimizing voltage change.
Thus, with the objective of decreasing the voltage dropout, a variable
structure control scheme will be designed starting from (7.1.6).
7.2 Definition of the sliding trajectory
7.2
97
Definition of the sliding trajectory
Neglecting the derivatives of current, the complex power (in a phase or in
per unit) takes the form:
¶
µ
dU∗gen
Rr
2
Ugen
− js
+ Ugen ωs
dt
Xr + Xm
µ
¶
Sgen '
(7.2.1)
Rr
∗
0∗
ωs
Z − jsZ
Xr + Xm
where
¡
¢
Z = Rs + j Xs + Xm
¡
Xr Xm ¢
Z 0 = Rs + j Xs +
Xr + Xm
(7.2.2)
Real and reactive powers are considered positive when they are flowing towards the generator. A moving frame in which the stator voltage angle θ
is always zero can be considered. However, its derivative cannot be disregarded.
´
dU∗gen
dUgen
d ³
dθ
2
Ugen
= Ugen ej θ
− j Ugen
ej θ
Ugen e−j θ = Ugen
dt
dt
dt
dt
2
1 dUgen
2 dθ
=
− j Ugen
(7.2.3)
2 dt
dt
and considering that Xm À Xr , Rr and s ' 0, which implies
¶
µ
Rr
ωs
Z∗ − j s Z 0∗ ' −j ωs Rr
Xr + Xm
(7.2.4)
then approximated values for the real and reactive power can be derived
2
Pgen ' Ugen
Qgen ' Ugen
= Ugen
s
1 dθ
2
2
2
+ Ugen
= Ugen
αP (s) + Ugen
βP (θ̇)
Rr
Rr ωs dt
dUgen 1
1
2
+ Ugen
dt ωs Rr
Xr + Xm
dUgen
2
αQ + Ugen
βQ
dt
(7.2.5)
(7.2.6)
98
Chap. 7: Sliding-mode control to limit voltage dropout
where, logically
s
Rr
1 dθ
βP (θ̇) =
Rr ωs dt
1
αQ =
ωs Rr
1
βQ =
Xr + Xm
αP (s) =
(7.2.7)
Equations (7.2.5) and (7.2.6) provide approximative expressions for real
and reactive power as functions of the generator voltage and its derivative.
Equation (7.1.6) links both components of complex power in a expression
for the voltage dropout. Thus, a sliding trajectory can be defined in which
the voltage dropout equals its limit value 4UL 1
2
{σ = 0 ⇔ 4U = 4UL } ⇒ σ = Qwt
−
(4UL − Pgen Rlin )2
.
2
Xlin
(7.2.8)
Pgen and Qnw are functions of Ugen and its derivative but instead of them,
it is more convenient using

 

µ ¶
Ulv − Ugen
Ulv − Ugen
x1

x=
=
(7.2.9)
dUgen  '  d
x2
(Ulv − Ugen )
−
dt
dt
as the components of the phase plane where the sliding mode will be defined.
For actual connection transients, real power will present a close-to-zero
positive value (motor) or negative values (generator). Therefore, positive
values of σ means that the voltage dropout is higher than 4UL and thus,
the system should be steered towards the sliding trajectory, to decrease this
voltage dropout. Negative values mean that the system is not surpassing the
permitted limit, but if the value is too low or is kept low enough for a relatively long time, the transient would delay too much over the synchronous
speed and the shaft torque would reach an excessive value.
1
henceforth, electrical magnitudes will be expressed in p.u., although uppercase letters
will be maintained for sake of clarity
7.2 Definition of the sliding trajectory
99
Figure 7.3: Sliding trajectories for different αP + βP .
Thus, from (7.1.5),(7.2.5), (7.2.6) and (7.2.8)
σ =−
2
¡
¢
(4UL )2
2 Rlin
4
2 Rlin
−
U
(α
+
β
)
+ 2 Ugen
4UL αP + βP +
P
P
gen
2
2
2
Xlin
Xlin
Xlin
2
+x22 Ulv2 αQ2 + Ugen
Ulv2 βQ2 − 2 x2 Ugen αQ βQ Ulv2 +
¡
¢2
¡
¢2
2
4
+Ugen
Ulv2 αP + βP − Ugen
αP + βP
(7.2.10)
For the sake of clarity, Ulv − x1 has not been substituted for Ugen .
Besides being a function of x1 and x2 , σ also depends on the slip s and the
derivative of stator voltage angle θ̇. In the phase plane, there is a family of
curves σ = 0 as long as the connection progresses (see Fig.7.3).
If the following generalized Lyapunov function is introduced
V (x1 , x2 ) =
1 2
σ (x1 , x2 )
4
(7.2.11)
and if αP , βP , αQ and βQ could be considered as constants or slow variables,
a sliding mode will be present if
dV (x1 , x2 )
1
< 0 ⇔ σ (∇σ · ẋ) < 0
dt
2
(7.2.12)
This condition can be geometrically understood taking into account that a
sliding mode is guaranteed if vector (x1 , x2 ) points toward the sliding line
at every instant in the connection transient. Fig. 7.4 shows a curve σ = 0
Chap. 7: Sliding-mode control to limit voltage dropout
Sliding trajectory and its gradient
0
−1
σ<0
· ·
x2(p.u./s) = − dUgen/dt
(x1,x2)
−2
(will cross)
−3
· ·
σ=0
(x1,x2)
∇σ
−4
(will not cross)
−5
σ>0
−6
−7
0
0.2
0.4
0.6
0.8
1
x1(p.u.) = Ulv − Ugen
Figure 7.4: σ and ∇σ in the phase plane.
1
0
σ = 0 (αΡ + βΡ = 0.5)
−1
x2(p.u./s) = −dUgen/dt
100
might not cross
sliding trajectory
−2
∇σ
.
x
−3
−4
−5
−6
−7
0
σ = 0 (αΡ + βΡ = −0.5)
0.2
0.4
0.6
0.8
1
x1(p.u.) = Ulv − Ugen
Figure 7.5: Example of ~ẋ = (ẋ1 , ẋ2 ) that might not reach σ = 0.
7.2 Definition of the sliding trajectory
101
−→
for given values of αP and βP , and the arrows show the direction of ∇σ
pointing to increasing values of σ. For positive values of σ, the system will
−
→
be directed to the sliding trajectory if vector ẋ = (ẋ1 , ẋ2 ) has the opposite
−→
−→ −
→
direction to ∇σ , that is ∇σ · ẋ < 0. For negative values of σ, the vector
−→
−→ −
→
−
→
ẋ = (ẋ1 , ẋ2 ) and ∇σ should have the same direction (∇σ · ẋ > 0). This
means that, in order to reach the voltage dropout and not exceed it, then
)
−→ −
→
−→ −
→
if σ > 0 → and ∇σ · ẋ < 0
⇔ σ ∇σ · ẋ < 0
−→ −
→
if σ < 0 → and ∇σ · ẋ > 0
(7.2.13)
If αP , βP , αQ and βQ cannot be considered constants or slow variables, the
condition in (7.2.12) would not be a sufficient one to assure that the system
head for the sliding trajectory. Fig. (7.5) visualizes this situation.
~ · ~ẋ < 0
In the represented example, a vector ~ẋ = (ẋ1 , ẋ2 ) that satisfies σ ∇σ
might not cross the sliding trajectory if αP + βP moves too fast.
To be precise, previous conditions 7.2.12 or 7.2.13 turn into
µ
¶
−→ −
1
dσ dαP
dσ dβP
→
σ ∇σ · ẋ +
+
<0
2
dαP dt
dβP dt
(7.2.14)
From 7.2.10 and 7.2.14
¡
¢
R2
1 dσ
Rlin
3
σ
= 2σUgen
x2 (αP + βP )2 lin
−
2σ
U
x
4U
α
+
β
+
gen
2
L
P
P
2
2
2 dt
Xlin
Xlin
¡
¢
+ σx2 ẋ2 Ulv2 αQ2 − σUgen x2 Ulv2 βQ2 + σ x22 − Ugen ẋ2 αQ βQ Ulv2 +
¡
¢2
¡
¢2
3
− σUgen x2 Ulv2 αP + βP + 2 σUgen
x2 αP + βP +
µ
µ
³
´
2 ¶¶
Rlin
2
2
4
+ σ Ugen Ulv − Ugen 1 + 2
(αP + βP ) α̇P + β˙P
Xlin
³
´
2 Rlin
˙
+ σUgen
4U
(7.2.15)
α̇
+
β
L
P
P <0
2
Xlin
102
Chap. 7: Sliding-mode control to limit voltage dropout
Grouping, it yields
·
2
¡
¢
Rlin
Rlin
− 2 2 4UL αP + βP +
2
Xlin
Xlin
¡
¡
¢
¢2 i
2
2 2
2
2
− Ulv βQ − Ulv αP + βP + 2 Ugen αP + βP
+
2
σUgen x2 2Ugen
(αP + βP )2
+ σx22 αQ βQ Ulv2 +
+ σx2 ẋ2 Ulv2 αQ2 +
− σUgen ẋ2 αQ βQ Ulv2 +
µ
¾
³
´ ½·
2 ¶¸
R
R
lin
2
2
2
lin
+ σ Ugen α̇P + β˙P
Ulv − Ugen 1 + 2
(αP + βP ) + 2 4UL
Xlin
Xlin
<0
(7.2.16)
7.3
Sliding mode controller with integral compensation
In this section and in the following, the gains involved in the variable structure of the controllers will be chosen in order to assure the stability of the
system, referred to as the ability of keeping the voltage dropout within a
certain value 4UL .
A sliding mode controller with integral compensation [105, 106] will be
used in which the control law usl is integrated to give place to the control
signal α. The firing angle α is the angle, starting from the ascending zerocrossing of the phase voltage (see figure 7.6), that controls the rms value of
the voltage applied to the induction machine. Firing angle is scaled so as
α = 1 corresponds to half a period.
Z t
α(t) =
usl dτ
(7.3.1)
0
A desirable control law usl will take the form
usl (x1 , x2 , σ) = Ψ1 (σ, x1 ) · x1 + Ψ2 (σ, x2 ) · x2
(7.3.2)
and will be limited in the firing angle variation rate
lim
−ulim
sl < usl < usl
(7.3.3)
7.3 Sliding mode controller with integral compensation
103
Figure 7.6: Simplified performance of the soft-starter.
Finally, to bring the analysis to a close, it is necessary to find the system response to the control signal α. It has been finally observed that a
relationship of the type
0
ẋ2 = kss (x1 , s) · usl + kss
(x1 , s)
kss (x1 , s),
0
kss
(x1 , s) > 0
(7.3.4)
can be found between the second derivative of the generator voltage and
the derivative of the firing angle.
104
Chap. 7: Sliding-mode control to limit voltage dropout
Thus, (7.2.16) can be rewritten as
· 2
½
¸
¡
¢
Rlin
2 Rlin
2
σUgen x2 2Ugen (αP + βP )
+
1
−
2
4U
α
+
β
+
L
P
P
2
2
Xlin
Xlin
¡
¢2 o
− Ulv2 βQ2 − Ulv2 αP + βP
+
+σ x22 αQ βQ Ulv2 +
£ 0
¤¡
¢
+σ kss
+ kss (Ψ1 (s) · x1 + Ψ2 (s) · x2 ) x2 Ulv2 αQ2 − Ugen Ulv2 αQ βQ +
µ
¾
´ ½·
³
2 ¶¸
Rlin
Rlin
2
2
2
˙
+σ Ugen α̇P + βP
Ulv − Ugen 1 + 2
(αP + βP ) + 2 4UL
Xlin
Xlin
<0
(7.3.5)
which is composed of several addends. A sufficient condition for the inequality (7.3.5) to be true is that all these addends are negative.
2
σUgen
½³
α̇P + β˙P
´ ·µ
Ulv2
−
2
Ugen
µ
¸
2 ¶¶
Rlin
Rlin
1+ 2
(αP + βP ) + 2 4UL +
Xlin
Xlin
¾
¢
αQ βQ Ulv2 ¡
0
−
kss Ψ1 x1 + kss
<0
Ugen
(7.3.6)
½
·
µ
µ 2
¶
¶
¡
¢
Rlin
Rlin
2
2
2
σ x2 Ugen (αP + βP ) 2Ugen
+ 1 − Ulv − 2 2 4UL αP + βP +
2
Xlin
Xlin
¤
ª
0
2
− Ulv2 βQ2 − kss Ψ2 Ulv2 αQ βQ + Ulv2 αQ2 kss
+ σ kss Ψ1 x1 x2 Ulv2 αQ
<0
(7.3.7)
¡
¢
2
σx22 αQ βQ Ulv2 + kss Ψ2 Ulv2 αQ
<0
(7.3.8)
In order to make these inequalities true, a switching in the values of the
control law parameters can be forced if:
• the sign of σ changes and/or
• the sign of x2 changes
7.3 Sliding mode controller with integral compensation
105
The first state variable x1 = Ulv − Ugen will always be positive.
Previous equations can be developed, but prior to that, it is worth noticing
that αQ , βQ and kss > 0.
From (7.3.6) the following inequalities must be satisfied
σΨ1 x1 >
³
·
´
Ulv2
σUgen α̇P + β̇P
−
2
Ugen
µ
2 ¶¸
Rlin
Rlin
1+ 2
(αP + βP ) + 2 4UL
0
Xlin
Xlin
kss
−
σ
kss
kss αQ βQ Ulv2
if σ > 0 ⇒
Ψ+
1 >
·
³
´
α̇P + β̇P Ugen
(7.3.9)
¶¸
µ
2
R
Rlin
2
Ulv2 − Ugen
(αP + βP ) + 2 4UL
1 + lin
2
0
Xlin
Xlin
kss
−
kss x1
x1 kss αQ βQ Ulv2
if σ < 0 ⇒
Ψ−
1 <
·
³
´
α̇P + β̇P Ugen
(7.3.10)
µ
¶¸
R2
Rlin
2
Ulv2 − Ugen
1 + lin
(αP + βP ) + 2 4UL
2
0
Xlin
Xlin
kss
−
kss x1
x1 kss αQ βQ Ulv2
at each moment of the connection process.
Figures 7.3 and 7.4 show that, for actual (αp + βp ) values, then
σ > 0 ⇒ x2 < 0.
(7.3.11)
Therefore the combination σ > 0, x2 > 0 is not feasible. On the other
hand, the combination σ < 0, x2 > 0 is feasible, but a logical connection
transient implies that generator voltage increases continuously until the end
of the process, which means x2 = −d Ugen /d t < 0. In fact, in the case
where x2 is close to zero values during the connection transient, the sliding
controller will perform a correcting control that will tend to steer the system
towards the sliding trajectory. When considering positive values of x2 more
106
Chap. 7: Sliding-mode control to limit voltage dropout
variants are introduced that complicate the analysis. Instead of that, the
case where x2 is positive, will be treated separately. Under those conditions,
sgn(σ) = −sgn(σx2 )
From (7.3.8)
i f σ > 0 ⇒ kss Ψ−
2 <−
βQ
1
−ωs Rr
ωs Rr
=
⇒ Ψ−
2 <−
min
αQ
Xr + Xm
Xr + Xm kss
i f σ < 0 ⇒ kss Ψ+
2 >−
βQ
ωs Rr
1
−ωs Rr
⇒ Ψ+
=
2 >−
αQ
Xr + Xm
Xr + Xm kssmax
(7.3.12)
In a similar way, from (7.3.7)
if σ x2 > 0 ⇒
0
αQ x1
αQ
kss
+
+
(7.3.13)
βQ Ugen kss Ugen βQ
´
´
³
³ 2
Rlin
2 + 2 Rlin 4U (α + β ) + U 2 β 2
2
+
1
+
U
(αP + βP )2 −2Ugen
L
P
P
2
lv
lv Q
X
X2
Ψ+
2 > Ψ1
−
lin
lin
kss Ulv2 αQ βQ
if σ x2 < 0 ⇒
0
αQ x1
αQ
kss
+
+
(7.3.14)
βQ Ugen kss Ugen βQ
´
´
³
³ 2
Rlin
2 + 2 Rlin 4U (α + β ) + U 2 β 2
2
(αP + βP )2 −2Ugen
+
1
+
U
L
P
P
2
lv
lv Q
X
X2
Ψ−
2 < Ψ1
−
lin
lin
kss Ulv2 αQ βQ
−
Eqs. (7.3.10) and (7.3.11) evidence that Ψ+
1 > Ψ1 . If finally they are
+
−
chosen such that Ψ1 > 0 > Ψ1 , which implies σΨ1 > 0, and assuming
x2 < 0 which in turn implies σ x2 > 0 ⇔ σ < 0, then (7.3.13) and (7.3.14)
must be rewritten as
0
αQ
kss
+
(7.3.15)
kss Ugen βQ
³
³ 2
´
´
Rlin
2
2 + 2 Rlin 4U (α + β ) + U 2 β 2
(αP + βP )2 −2Ugen
+
1
+
U
L
P
P
2
lv
lv Q
X
X2
Ψ+
2 >
−
lin
lin
kss Ulv2 αQ βQ
7.4 Control law parameters
107
0
αQ
kss
+
(7.3.16)
kss Ugen βQ
³
³ 2
´
´
Rlin
2
2 + 2 Rlin 4U (α + β ) + U 2 β 2
(αP + βP )2 −2Ugen
+
1
+
U
L
P
P
2
lv
lv Q
X
X2
Ψ−
2 <
−
7.4
lin
lin
kss Ulv2 αQ βQ
Control law parameters
At this point, few more analytical manipulations
At this point, some more analysis regarding the boundary control law
−
+
−
parameters Ψ+
1 , Ψ1 , Ψ2 and Ψ2 will be presented. The actual induction
generator constants and performance must be introduced in order to bound
these parameters.
With respect to electrical induction machine constants, Table 7.4 summarizes the steady state electrical constants for the analyzed wind generator.
Table 7.1: Electrical constants for stability study I.
Induction machine steady state constants
Rs
Rr
Xs
Xr
Xm
Stator Resistance
Rotor Resistance
Stator Reactance
Rotor Reactance
Magnetizing Reactance
0.0059 p.u.
0.010 p.u.
0.0087 p.u.
0.143 p.u.
4.76 p.u.
Table 7.4 other independent values which are also needed for the boundary
of law control parameters.
The values of αQ and βQ appearing in previous inequalities are derived
from (7.2.7)
1
= 0.2122s/rad
ωs Rr
1
=
= 0.2094.
Xr + Xm
αQ =
βQ
(7.4.1)
108
Chap. 7: Sliding-mode control to limit voltage dropout
Table 7.2: Electrical constants for stability study II.
Additional independent constants
ωs
Rlin
Xlin
4U0
Electrical angular speed
Line Resistance
Line Reactance
Maximum voltage dropout
314.1592 rad/s
0.02441 p.u.
0.0176 p.u.
0.015 p.u.
Unfortunately, there are many other terms in previous inequalities that
are not constants. Furthermore, some of them are poorly modeled, as β̇P ,
which makes it difficult for Ψ1 (and consequently Ψ2 ) to be bounded. Thus,
an analysis has to be made of every term of (7.3.10) or (7.3.11) in order to
bound the range of possible values for these expressions. In that sense some
sensitivity tests have been performed in order to demonstrate that βP does
not have a significant influence on the expressions.
Two issues must be pointed out before bounding Ψ1 :
• as seen in Fig. 7.7 the boundary is not valid in the area in which kss = 0
(in the referenced figure, this is the lower right area); fortunately this
is the last stage of the connection process and the generator voltage
smoothly develops to its uncontrolled final value and the real power
is high enough to compensate the effect of the reactive power in the
voltage dropout (see (7.1.1))
• the analysis has been made avoiding dependences on the type of control; only the range for β̇P (and subsequently β̇P + α̇P ) has been obtained from analyzing several kinds of control in different conditions
and extracting the extreme values.
In this regard, the series of performed simulations yield the following range
for β̇P
7.4 Control law parameters
109
Relationship 1/(d2U/dt − d alpha/dt)
0.02
0.8
0.015
0.6
slip (p.u.)
0.01
Motor
0.4
0.2
0.005
0
0
−0.2
−0.4
−0.005
Generator
−0.6
−0.01
−0.8
0.2
Figure 7.7:
0.4
0.6
Generator voltage (p.u.)
0.8
1
at different generator voltages and slips.
kss
p.u.
d2 θ
p.u.
< 50 2 ⇒
<
2
2
s
dt
s
1
dβP
1 d2 θ
p.u.
−10.61 <
=
< 6.79
.
2
s
dt
Rr ωs dt
s
− 32
(7.4.2)
With regard to α̇P = ṡ/Rr , this value can be derived from the applied torque
Tw , the inner electromechanical torque Tmi and the frictional term
ds
= Twind − Tmi − Bωr < Twind 5 1 in p.u.
dt
which leads to
ds 1
1
p.u.
0 = α̇P =
=−
= −12.65
dt Rr
2HRr
s
−2H
(7.4.3)
(7.4.4)
Therefore
−23.26
p.u.
p.u.
< α̇P + β̇P < 5.86
s
s
(7.4.5)
110
Chap. 7: Sliding-mode control to limit voltage dropout
This will lead to excessively high gains for the law control which is not desirable. Instead of these values, (7.2.5) can be used to give place to a narrower
range for α̇P + β̇P . Thus, a series of simulations in different conditions and
with different kinds of controls show that
−9
d Pgen
p.u.
p.u.
<
' α̇P + β̇P < 4.6
2
s
dt Ugen
s
(7.4.6)
Within the valid region of Fig. 7.7, and making a sweep for α̇P + β̇P in the
range of [−9, 4.6], it is obtained that the expression
·
µ
2 ¶¸
Rlin
Rlin
2
2
Ulv − Ugen 1 + 2
(αP + βP ) + 2 4UL
³
´
0
Xlin
Xlin
kss
α̇P + β̇P Ugen
−
kss x1
x1 kss αQ βQ Ulv2
(7.4.7)
is delimited by -4 and 20. As referred in (7.3.10) and (7.3.11) these values
−
will be assigned to Ψ+
1 and Ψ1 of the control law
Ψ+
1 = 4
1
s
1
Ψ−
1 = −20 .
s
(7.4.8)
With regard to Ψ2 , expressions (7.3.15) and (7.3.16) must be taken into
account. All the terms in these expressions, except βp , are constants or
values related to the generator voltage x1 and/or the slip s. From a series
of simulations in different conditions and controllers,
−0.52 < βp =
θ̇
< 0.7
Rr ωs
(7.4.9)
Within the valid region of Fig. 7.7, and making a sweep for βP in the range
of [−0.52, 0.7], the following delimitation can be obtained.
−6<
−
−
³
³ 2
´
´
Rlin
2
2
(αP + βP )2 −2Ugen
+
1
+
U
2
lv
X
lin
kss Ulv2 αQ βQ
Rlin
2 2
2X
2 4UL (αP + βP ) + Ulv βQ
lin
kss Ulv2 αQ βQ
+
+
0 α
kss
Q
< 5.5
kss x1 βQ
(7.4.10)
7.5 Implementation of the proposed controller
111
In order to minimize the control effort, the control law parameters regarding
to Ψ2 results in
Ψ+
2 = 5.5
Ψ−
2
(7.4.11)
= −6.0.
It does satisfy (7.3.12) since
− 9.2 = Ψ−
2 <−
ωs Rr
1
314.1592 · 0.01
=−
= −0.021
min
Xr + Xm kss
4.9 · 10
8.1 = Ψ+
2 >−
1
314.1592 · 0.01
ωs Rr
=−
= −0.026
max
Xr + Xm kss
4.9 · 37
(7.4.12)
The case where x2 > 0 means that the generator voltage is decreasing. The
sliding trajectory varies as long as αP + βP varies, but it is always found to
be below abscises axis (x2 = 0). Therefore, a suitable control action for the
case where x2 > 0 is to decrease the firing angle which will eventually make
Ugen increase, thus approaching the vector state to the sliding line.
if x2 > 0 ⇒ usl =
7.5
1 lim
u
2 sl
(7.4.13)
Implementation of the proposed controller
A sliding-mode controller has been calculated starting from the characteristics extracted from the induction generator connected through a soft-starter.
The proposed sliding-mode controller presents an integral compensation
(SLMCIC) as seen in Fig. 7.8, which gives place to a more continuous
performance of the system since the SLMCIC acquires a PI characteristic.
Apart from the generator voltage and its derivative, there are some other
inputs to the sliding-mode controller module:
Ulv Voltage at the network side of the soft-starter, although a not significant error is made if U lv = 1 p.u. is considered.
slip The difference in per unit between the synchronous speed and the rotor
speed. This is required for the calculation of σ.
112
Chap. 7: Sliding-mode control to limit voltage dropout
Figure 7.8: Sliding-mode controller with integral compensation.
dThU Derivative of the voltage generator angle. At each moment, the angle
can be considered equal to zero, but not its derivative. Neglecting it
would give place to unacceptable errors in the calculation of σ, that
will deteriorate the controller performance. MATLAB Toolbox for
System Identification has been used to estimate this from Ugen , the
slip, and their derivatives.
The value for σ, whose sign dictates the shift in the control law parameters,
is obtained from the following expressions previously presented.
2
Pgen ' Ugen
s
1 dθ
2
+ Ugen
Rr
Rr ωs dt
Qgen ' Ugen
dUgen 1
1
2
+ Ugen
dt ωs Rr
Xr + Xm
Pwt ' Pgen
s
µ 2
¶
Ulv2
Ulv
2
2
Qwt '
Qgen + Pgen
−1
2
2
Ugen
Ugen
(7.5.1)
and
σ=
2
Qwt
(4UL − Pwt Rlin )2
−
2
Xlin
(7.5.2)
As seen in Fig. 7.9 and eq. (7.5.2) the permitted voltage dropout appears
in the calculation of σ, and the expected derivative of voltage, in turn,
7.5 Implementation of the proposed controller
113
Figure 7.9: Calculation of σ.
determines the permitted voltage dropout. This is due to the fact that when
the connection transient is fast, then keeping a low value for the voltage
dropout will make the turbine overpass its final speed, and so will do the
shaft torque. This is a consequence of the mechanical system equation
Pgen
.
(7.5.3)
ω
Disregarding the sign of Twind and Pgen , it can be found that a high wind
torque must be compensated with a high inner mechanical torque from the
generator in order to avoid an excessive speed increase. And for the real
power to be significant it is necessary that the generator voltage reaches
a high value. For a high wind torque, these circumstances appear a short
time after the connection process begins, which means that the voltage must
reach a high value in a short time, giving place to a high derivative, which
in turn produces a high reactive power and voltage dropout.
Twind = J ω̇ + Bω +
For low inertia values, low rotor resistance or high wind torque, there is a
trade-off between a low dropout and a low speed overshot. Therefore, it is
convenient to relax the voltage dropout limit the controller should maintain,
and allow a higher value in order to decrease the speed overshot. This is
accomplished by a component which estimates an optimum voltage dropout
from the rotor resistance and the initial derivative of slip, which is related
to the wind turbine inertia and the wind torque.
This is another feature of the sliding-mode controller: the capability of
accomplishing an optimized dropout performance, an optimized overshot
114
Chap. 7: Sliding-mode control to limit voltage dropout
Figure 7.10: Sliding-mode controller simplified control law.
one or a trade-off performance, depending respectively on whether a low
dropout value is chosen in the calculation of σ, a high one, or a variable one
according to the connection transient speed.
Once σ is obtained, the control law output is derived from x1 = Ulv − Ugen
and x2 = ẋ1 . Integration of this output will give place to the firing angle
value α being introduced in the soft-starter module. The control law is
presented in a simplified way in Fig. 7.10.
7.6
Simulation using the proposed controller
An overall scheme of the wind turbine connected to a weak grid is depicted
in Fig. 7.11. A weak grid system is the best test bench to check the firing
angle controller validity and to compare it to other kinds of controllers.
According to IEC 61400-21, switching operations have to be measured
during the cut-in of a wind turbine and for switching operations between
generators, that are only relevant for wind turbines with more than one
generator or a generator with two windings. However, simulations will only
be plotted for the main generator since its connection gives places to a more
serious transient. There are several factors to consider:
7.6 Simulation using the proposed controller
115
Figure 7.11: Overall scheme of the wind turbine feeding a local load.
• From (4.1.33) it is clear that the inertia time constant H is lower for the
main generator, even being faster, as the rated power is significantly
higher
• Since the power flow is higher for the main winding, voltage dropout
is also more marked
• Rotor resistance is usually higher for the secondary winding.
Taking into account the values for the main winding of the induction
machine and the values for the Thevenin impedance at the point of common
coupling (at the consumer), the control law parameters have been calculated
in order to comply with the stability restraints and tuned to achieve to an
optimum performance.
The initial rotor speed at which the connection process should be initiated
has been selected as 0.98 p.u. For the sliding-mode controller, as well as for
the rest of the controllers, the connection starts at a rotor speed that has
been chosen after a sweep within the range [0.96 - 0.99]. It has been checked
that for low initial rotor speeds (high slips), the closed-loop controller must
stop the generator voltage until the rotor speed approaches the synchronous
one. The resulting performance is more unpredictable and generally worse.
For open-loop controllers the result is definitely worse for initial rotor speeds
lower than 0.98.
It has also been checked that for low wind torque, a higher initial rotor
speed (0.99) gives place to a better performance of the system. However this
feature has not been included because starting so close to the synchronous
116
Chap. 7: Sliding-mode control to limit voltage dropout
speed will make the system behavior too sensitive to inaccurate rotor speed
measurements and to changes in wind torque.
The result of the connection transient for Twind = −0.5 p.u. is depicted in
7.12. It can be seen that:
Figure 7.12: Performance of the system connected through a soft-starter fired in
accordance with a sliding-mode controller action: voltage in p.u. seen by the consumer (upper left), induction generator slip in % (upper right), voltage generator
in p.u. (lower left) and σ (lower right).
• The load voltage seen by the consumer is well over 0.97 p.u. taking
into account that the initial and final values, which are independent
of the connection transient, are 0.983 p.u. and 0.989 p.u. respectively.
• In a first stage, the rotor slip decrease is almost linear, due to the
fact that the real power is low in relation to the wind torque. In a
7.7 Comparison to other control schemes
117
second stage, the generator voltage is high enough, which gives places
to a higher reactive power, but by contrast the real power turns out to
present such a value as to brake the turbine rotor and to counterbalance the reactive power effect over the voltage dropout. However, for
low inertia, low rotor resistance or high wind torque situations, the real
power will not be high enough until the last instants of the connection
transient, and consequently it is expected that a speed overshot will
occur. The torque will also experience an overshot.
• The sliding-mode philosophy imposes that the system must be directed towards the sliding trajectory, and once the system has reached
it, it should follow this trajectory and be led to an equilibrium situation. Since there are some delays and dead zones in the soft-starter
electronics, a chattering - small fluctuations around the sliding trajectory - is inevitable, as shown in the lower left hand plot. Finally,
when the connection is accomplished, the real power largely surpasses
the reactive power and the dropout is rather negative in relation to
the initial voltage at the PCC. This is the equilibrium state, when
x1 = Ulv − Ugen = 0 and x2 = ẋ1 = 0, and marks the end of the
connection process.
7.7
Comparison to other control schemes
In order to test the quality and robustness of the proposed sliding-mode
based controller, an open-loop linear controller and a PI-controller2 have
also been simulated, and the results compared. Both of them have been
tuned for the following case: H = 3.9 s, Rr = 0.015 p.u. and Tw = −0.5 p.u.
Fig. 7.13 shows the voltage at the PCC and slip results for the considered
wind turbine for the three time-firing angle strategies previously mentioned.
All of them show a good performance since the voltage dropout is well
below 1.5% referred to the voltage before the connection process. The speed
overshot is not significant in either case.
As can be seen, the proposed sliding-mode controller is displayed to give
place to a lower dropout than the PI controller which in turn is slightly
better than the open-loop linear evolution scheme.
2
Although there is no feedback from the voltage error, and hence it is not a PI controller
strictly speaking, for sake of clarity this denomination will be kept
118
Chap. 7: Sliding-mode control to limit voltage dropout
Voltage at Pcc (p.u.)
Linear evolution (·), PI controller (x), SLMC (o)
0.99
0.985
0.98
0.975
0.97
0.2
0.3
0.4
0.5
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
2
Slip (%)
1.5
1
0.5
0
−0.5
−1
0.2
time (s)
Figure 7.13: Three firing angle control techniques: linear (blue and dot), PI (red
and x-mark) and sliding-mode controller (black and circle).
Fig. 7.14 depicts the block diagram used for the latter controllers. The
last component is an integrator which presents a lower limit for the firing
angle in α = 0.2 p.u. and starts from α0 = 0.63 p.u. This initial value for α
corresponds to an angle equal to 113.4o , and in order to improve the pero
formance of these controllers, it has been chosen instead of 150
180o = 0.83 p.u.,
that is the decreasing zero cross point for voltage Uab . A similar value can
also be found in [89] and in some manufacturers’ catalogues. For the slidingmode controller, a initial value of α0 = 0.75 p.u. has been established from
simulation results.
However, the robustness is the main feature of this controller, as have
been fully tested in three broad scenarios:
Different conditions. The controller is adaptable to different conditions
such as variable or different wind torques.
7.7 Comparison to other control schemes
119
Figure 7.14: Firing angle controllers used as references.
Parameter sensitivity. The controller shows a good performance for a
wide range of variable parameters. For example, the rotor resistance
value is one the parameters that more strongly determines the induction machine response. The rotor resistance value (and other parameters) cannot only vary from a wind turbine to other, but it is always
known with a high degree of uncertainty related to:
• The standard tests to identify the induction machine parameters
(no-load and locked-rotor) are performed with voltages, currents
or speed values far away from those related to full load
• Changes in rotor temperature related to variable load or ambient
temperature
• Changes in the load which are translated to changes in the slip,
rotor frequency, currents or rotor current density
Wind turbine inertia. The value of the inertia time constant probably is
the parameter that more strongly influences the whole wind turbine
dynamics, but this information is usually difficult to known (usually,
manufactures do not provided this data), so an estimated value must
be considered. If the actual value of the inertia time constant is different from the considered value, the controller is able to adjust the
voltage dropout reference in order to avoid excessive speed overshot,
or to keep the voltage dropout restraint, at the expense of a higher
speed overshot.
120
Chap. 7: Sliding-mode control to limit voltage dropout
Fig. 7.15 depicts the voltage dropout at the consumer side over thirty six
situations.
The voltage dropout is calculated as the maximum difference between the
voltage before the connection process and the voltage during the transient at
the consumer side. The thirty six situations have been obtained by varying
the turbine inertia time constant H, the rotor resistance Rr and the wind
torque within the following values:
H = {3.0 s, 3.5 s, 4.0 s}
Rr = {0.007 p.u., 0.010 p.u., 0.015 p.u.}
Twind = {−0.25 p.u., −0.5 p.u., −0.75 p.u., −1 p.u.}
It is worth noticing that, for PSCAD/EMTDC a torque in p.u. equal to
−0.9 p.u. is approximately equivalent to the rated torque TN , as they are
related through the expression
P SCAD
Tp.u.
=
T
T
cos ϕN
' 0.9
ηmec (1 − sN ) TN
TN
(7.7.1)
where ηmec is the mechanical efficiency after extracting all mechanical losses
(aerodynamical drag, ventilation, friction...), sN is the rated slip (sN < 0
for a generator), and cos ϕN is the rated power factor.
It can be seen how the linearly ascending firing angle scheme gives place
to a good performance for a reduced number of situations. They correspond
to the values of the parameters similar to the used to tune the controller.
That is Rr = 0.015 and Twind = −0.3 p.u., −0.5 p.u.
Taking the open-loop linear scheme as a reference, a PI-controller improves
the soft-starter performance, as the voltage dropout is lower.
The best performance however corresponds to the proposed sliding-mode
controller, which is shown controlling two different soft-starters. The first is
a symmetrical soft-starter identical in hardware to that tested for the openloop and PI controllers. In the second case, an asymmetrical soft-starter
has been used in order to decrease the rms value of the voltage generated
by means of the soft-starter. Its pulse train sequence and a representation
of the phase current waveforms was previously presented in Chapter 5.
As to the voltage dropout refers, both sliding-mode based controllers are
definitely better than the other controllers. And in turn, for the same firing angle control an asymmetrical soft-starter is slightly better than the
7.7 Comparison to other control schemes
121
Sorted
2.5
2
2
Voltage dropout (%)
Voltage dropout (%)
Unsorted
2.5
1.5
1
∗ Linear
º PI
SLMC sm
◊ SLMC as
0.5
1.5
1
∗ Linear
º PI
SLMC sm
◊ SLMC as
0.5
·
0
10
20
30
·
0
10
20
30
# Situation(Rr,H,Twind)
Figure 7.15: Voltage dropout for different control schemes: linear (black and asterisk), PI (magenta and circle), sliding-mode controller (red and dot) and slidingmode controller with asymmetrical soft-starter (blue and diamond).
symmetrical one, although it presents a higher shaft torque due to a higher
overshot in the slip. This is a repetitive issue in the performance of several firing angle controllers based in sliding-mode techniques. The more
restrained voltage dropout, the higher the overshot and the higher the shaft
torque, mainly for low inertia, low rotor resistance or high wind torque situations. The presented controllers are the product of a trade-off between a
lower dropout and an acceptable shaft torque.
The left hand graph of Fig. 7.16 shows a comparison of the voltage change
that different controllers produce, taken as a reference in this case to the final
voltage at the consumer side. Once the connection process is accomplished,
a negative (generated) real power will raise the voltage at the network connection point, and this is the voltage taken into account in the voltage
122
Chap. 7: Sliding-mode control to limit voltage dropout
Sorted
Sorted
4
0
∗ Linear
º PI
SLMC sm
◊ SLMC as
3.5
·
Shaft Torque (%)
Voltage change (%)
3
2.5
2
1.5
1
−100
∗ Linear
º PI
SLMC sm
◊ SLMC as
0.5
0
−50
·
10
20
30
−150
10
20
30
Figure 7.16: Voltage change and shaft torque for different control schemes: linear
(black and asterisk), PI (magenta and circle), sliding-mode controller (red and dot)
and sliding-mode controller with asymmetrical soft-starter (blue and diamond).
dropout calculation3 . The graph at the right side shows the maximum shaft
torque during the start-up process. A difference with respect to PI and
linear controllers can be observed for both sliding-mode based controllers.
However, for higher wind torques giving place to higher final shaft torques,
the behavior of the symmetrical sliding-mode controller agrees closely with
the linear and PI-controller. The torque with the asymmetrical controller is
7% higher. For lower wind torques, a torque lower than the rated value is
exerted on the shaft, and hence a slight torque overshot is not a concern.
Fig. 7.17 shows a comparison of the performance of the same controllers
for a generator having different electrical parameters (Rs = 0.01p.u. Xs =
0.1p.u. Rr = 0.01p.u. Xs = 0.1p.u. Xm = 3.7p.u.). As can be seen, both
sliding-mode based controllers give place to lower voltage dropouts.
7.8 Sensitivity to speed and voltage measurements
−20
Shaft Torque (%)
Voltage dropout (%)
1.8
1.6
1.4
1.2
1
0.8
123
0.4
0.6
Torque (p.u)
0.8
−40
−60
−80
−100
−120
0.4
0.6
0.8
Torque (p.u.)
Figure 7.17: Performance of tested control schemes for another generator: linear
(black and asterisk), PI (magenta and circle), sliding-mode controller (red and dot)
and sliding-mode controller with asymmetrical soft-starter (blue and diamond).
7.8
Sensitivity to speed and voltage measurements
The input values for the sliding-mode based controller are the induction
generator voltage and the slip. The derivatives of these magnitudes are also
needed in the σ calculation and in the control law.
For a real system acquiring the generator voltage and the induction machine slip, it is expected that a noise in the measurements and a discretization in the analog to digital conversion will be introduced in the input values. The discretization can also be understood as a non-zero resolution of
the measuring device. In order to take into account these practical effects,
a new PSCAD component has been designed (Fig. 7.18) that adds a zeromean random noise to the voltage or the speed and simulates a discretization
process in the distorted signal.
For the induction machine speed, random values in the range of ±0.005 p.u.
are added to the pure value, which means ±7.5 rpm for a four pole induction
machine (50 Hz). A small variation in the speed will give place to a high
variation in the slip within the connection/operation range as can be seen
in Fig. 7.19. After noise addition, a resolution of 1 rpm is supposed for the
speed measuring device.
3
Final voltage is even higher after the capacitor bank connection
124
Chap. 7: Sliding-mode control to limit voltage dropout
Figure 7.18: New PSCAD component designed to simulate the presence of noise in
the speed and voltage measurements and the discretization process.
With regard to the generator voltage, a random noise of the same ±0.005 p.u.
value is considered, which yields ±3.5 V for a 690 V induction generator
rated voltage. Regarding to the discretization process, the voltmeter resolution has been fixed to 0.69 V.
The same simulations as in the non-distorted situations have been performed considering noise in the generator voltage and speed signals. In the
case of the open-loop linear controller it makes no sense performing new
simulations as there are no inputs for the controller. In the case of the PI
controller, noise in the generator voltage signal has been considered. In the
case of the proposed sliding mode controller, noisy voltage signal as well as
speed noisy signal situations have been tested.
Fig. 7.20 shows the voltage dropout and shaft torque for the tested controllers.
The original PI controller performance is represented by the black dotted
line. The black dots correspond to the same controller where the voltage
is affected by a random noise. As can be seen, for the PI controller, in the
tested range there is no performance degradation when noise and discretization are considered in voltage signals.
The SLMC performance is depicted with a solid red line. Plus (+) marks
correspond to simulations where the noise and discretization in the generator voltage signal are considered. It can be seen how the proposed controller
7.8 Sensitivity to speed and voltage measurements
125
Figure 7.19: Performance of the sliding-mode controller when noise and discretization are added to the speed signal.
is more sensitive to voltage noise. However, the performance of the sliding
mode based controller is better than the performance of the linear and PI
controllers in any situation. Different simulations have shown that the controllers are hardly sensitive to the tested discretization when it is applied
without noise. With regard to simulations considering noise in the speed
signals, diamond (♦) marks represent these situations showing a satisfactory insensitivity to random variations around the actual value in the order
of ±7.5 rpm.
To sum up, a closed-loop in the firing angle control of the trains of pulses
exciting the thyristors gates will reduce the impact of the electrical connection of the wind turbine to the network since the voltage dropout decreases.
This can be achieved with a PI-controller, integrating a control law output
that involves the generator voltage and its derivative. The improvement
achieved is higher for the more problematic conditions (high wind torque,
low rotor inertia and low rotor resistance), when the voltage dropout can
126
Chap. 7: Sliding-mode control to limit voltage dropout
Sorted
Sorted
4
0
−−PI
· PI U noise
3.5
− SLMC
+ SLMC U noise
◊ SLMC s noise
Shaft Torque (%)
Voltage dropout (%)
3
2.5
2
1.5
−−PI
1
0.5
0
−50
−100
· PI U noise
− SLMC
+ SLMC U noise
◊ SLMC s noise
10
20
30
−150
10
20
30
# Situation(Rr,H,Twind)
Figure 7.20: Controllers’ performance when a random noise in voltage and speed
signals are considered.
be expected to be higher. If the slip measurement is included in the control unit, a sliding-mode based controller can replace the PI controller to
calculate the firing angle at each moment. This gives places to the best performance since an eventual consumer in the network connection point will
suffer lower voltage dropouts at its terminals. Voltage change, as defined in
section 3.4.3 is improved in a parallel way.
7.9
Influence of the line impedance
Fig. 7.21 shows a comparison among the three tested structures for different line impedance ratios and different short-circuit ratios, defined as the
7.9 Influence of the line impedance
127
quotient between the short-circuit power and the wind turbine capacity. As
can be seen the sliding-mode based controller shows the best result4 .
Figure 7.21: Comparison of tested controllers for different line impedances.
Fig. 7.22 is similar, but only the results of the sliding-based controller
with the line resistance and reactance characteristic of the weak grid are
considered. This figure plots the voltage dropout and the [maximum] voltage
change, such as defined in section 7.1, against the line impedance. The line
resistance does not influence the voltage dropout too much, and thus, as long
as the line reactance remains constant, a higher capacity of the evacuation
line will not decrease this value. However, if the start-up impact is referred
to the maximum voltage change, the right hand plot indicates that lower
line resistances, corresponding to higher capacity lines, give place to lower
voltage changes. This is due to the fact that a higher resistance will increase
the voltage at the end of the connection transient, since it is mainly related
to the real power delivered by the induction generator of the wind turbine
according to
4UP CC = Pgen Rlin + Qgen Xlin
4
(7.9.1)
The component that calculates the permitted voltage dropout (Fig. 7.9) has been
added the line resistance and reactance as new inputs
128
Chap. 7: Sliding-mode control to limit voltage dropout
10
Voltage change (%)
Voltage dropout (%)
10
8
Xlin = 25
Xlin = 20
6
Xlin = 15
4
Xlin = 10
2
0
Xlin = 5
5
10
15
Rlin (Ohm)
20
25
Xlin = 25
Xlin = 20
8
Xlin = 15
6
Xlin = 10
Xlin = 5
4
2
0
5
10
15
20
25
Rlin (Ohm)
Figure 7.22: Voltage dropout and voltage change vs. line impedance.
where 4UP CC is the difference between the voltage before and after the softstarter operation and Pgen and Qgen are the real and reactive power delivered
by the wind turbine induction generator once the electrical connection is
accomplished (Pgen is negative and Qgen is positive).
Chapter
8
Conclusions
This final chapter is integrated by to sections: Conclusions and Future work.
The first section compiles the more relevant features of the proposed controller and some ideas derived from the analysis of the mechanical and electrical components of a wind turbine. The main contributions of this work
and the conclusions derived from the theoretical analysis and the simulation results are also listed in this section. The second section offers some
guides and suggestions for future research work that may continue the issues
tackled in this thesis.
8.1
Conclusions
The concern about the negative impact of wind turbines on the power quality
that utilities are responsible to supply is one of the limiting factors taken
into account when selecting a wind turbine model to be placed at a specific
site. The impact can be quantified and measured by means of the voltage
change at the point of common coupling. Wind turbine switching operations,
mainly the start-up, is one of the causes of this power quality decline.
Voltage changes due to the starting transient are higher for stall-controlled
wind turbines, since the accelerating wind torque is not controlled and the
130
Chap. 8: Conclusions
connection transients give place to higher reactive power demand and hence
higher voltage dropouts. The voltage change generated by switching operations is even higher for isolated wind turbines or small wind farms placed far
away from the electrical distribution network. This electrical configuration,
where a local load or a small power station is connected to the distribution
network through a rather long line or a high short-circuit impedance value
is usually referred to as a weak grid. This is a very demanding situation
since real power fluctuations or reactive power demand is translated into
amplified voltage fluctuations at the interconnection node.
The main components of a weak grid have been analyzed and parameterized in order to design a scenario where a wind turbine start-up transient
could be analyzed in a realistic way.
The impact of stall-controlled wind turbines starting transient on the
power quality at the point of common coupling has been analyzed, simulating the designed weak grid in different conditions.
A simplified structural analysis of a wind turbine blade has been
performed. The analysis was mainly focused on the relationship between
the inertia time constant of a wind turbine rotor and the blade length and
weight. As a result, an approximate expression relating the inertia time
constant and the wind turbine rated capacity has been derived. A similar
trend expression for the self damping has also been presented. In order
to estimate this statistical expressions, different MS Excel sheets with a
wide number of wind turbine records have been developed, which are freely
available at the author´s research page.
Dispersed information about soft-starters has been gathered and classified, putting special emphasis on the triggering of the thyristors. Most
of the technical literature about soft-starters deals with induction motor
performance, where the main objective is to reduce the inrush current following the motor start-up process in order to fulfil technical and normative
regulations. However, conditions and regulations regarding to induction generators are quite different. While induction motors starts from standstill,
induction generators are connected to distribution network when the rotor
speed is close to the synchronous value. While induction motor regulations
limit the inrush current during the start-up process, induction generator
regulations limit the voltage variations at the point of common coupling to
the distribution network1 .
1
The inrush current must also be limited but this is a less restricted condition
8.1 Conclusions
131
In this thesis, the performance of soft-starting devices with wind turbine
induction generators has been addressed in order to cover the lack of work in
this area. Differences between the regulations and the performance of softstarters working with motors and generators have been thoroughly analyzed.
As a result, a new approach to the design of the soft-starter controller has
been proposed, focused on the voltage at the point of common coupling
rather than in the inrush current. To reach this goal the controller limits
the reactive power flow and compensates, as far as possible, the associated
voltage dropout with the real power injection.
Apart from the design of a new control strategy to regulate the firing
angles in order to maintain the voltage dropout, modifications to the logic
control of the triggering of the thyristor gates have also been presented. One
of the modifications means an asymmetry in the currents flowing through the
soft-starter that provides an improved control of the rms voltage supplied
to the induction generator of the wind turbine.
The main goal of the present work has been to proof that the third order
model of the induction machine is the most suitable model to understand
the start-up transients. A small signal linear model of the induction generator and a comparison based on the root loci techniques have been used
to test and proof the feasibility of this reduced induction generator model.
Simplified, but accurate enough, analytical expressions for real and reactive
power components during the connection transients have been derived from
the third order model, showing the influence of the derivative of the voltage
on these magnitudes.
A soft-starter connected to a specific generator has been simulated by
means of the Electromagnetic Transient Simulator (EMTDC), whose graphical user interface is called PSCAD. Same simulations have also been performed with the MATLAB power system toolbox, but it has been rejected
due to the fact that simulations are carried out in a significant higher time.
Thus, PSCAD/EMTDC package has been chosen to simulate a weak
grid whose component has been parameterized. In order to include the
power electronic circuit, the logic control circuit for the thyristor triggering
and the tested controllers, some PSCAD/EMTDC simulation components
have been customized and more than fifty new specific modules have been
designed, tested, tuned and, finally, integrated into the simulation cases.
Starting from the performance of the soft-starter derived from these simulations, a firing angle controller has been designed based on sliding-mode
132
Chap. 8: Conclusions
techniques. The gains involved in the variable structure of the controllers
have been chosen in order to assure the stability of the system, referred to
as the ability to maintain the voltage dropout within a certain value.
The performance of the new closed-loop control strategy has been thoroughly tested in a broad variety of simulation scenarios. As a result, the
improvement of the induction generator soft starting has been fully demonstrated. The new soft-starter controller performance has been superior and
favorable compared with the classical open-loop controller.
For a comparison purpose, a PI-controller based on the integration of a
signal function of the generator voltage and its derivative has also been
designed, tuned and simulated, resulting in an intermediate performance
between the open-loop controller and the proposed sliding-mode based controller.
Real practical effects distorting the signals, such as the presence of noise
in the speed and voltage signals or errors associated to the discretization
process performed by analog-to-digital converters have also been taken into
account. The performance of both closed-loop controllers, PI and slidingmode based controllers, has no significant variation when introducing these
simulated distortions in the measurements.
Robustness is the main feature of the new proposed sliding-mode controller, as have been fully tested, especially in three main areas: adaptability
to different conditions (such as variable or different wind torques), parameter sensitivity (as in the case of the rotor resistance), and capability to
manage estimated values of wind turbine inertia time constant (a piece of
information that is not usually provided by manufactures).
The proposed soft-starting controller needs no more hardware, since voltage, current or speed transducers already installed in wind turbines could
be used. Therefore the changes to implement the new approach are limited
to the software control (fast and cheap), even for previously installed wind
turbines.
8.2
Future research
A sliding-mode based controller has been designed and tested for a single
wind turbine in a weak grid. This configuration is a good test bench to
8.2 Future research
133
check the validity of the controller. However this is not an usual situation in
countries like Spain. One aspect of this work to be given further consideration is the simulation of a wind turbine start-up being part of a wind farm
where the higher-capacity overhead line will make the grid stronger and the
interconnection stiffer.
The inputs of the sliding-mode based controller are the voltage and its
derivative. Gains for the control law have been derived starting from the
simulation of the performance of the soft-starter under different conditions.
The resulting gains are quite high, which makes it necessary to saturate
the output to be integrated. Including a suitable third input (αP + βP in
(7.2.7)) might provide lower gains thus smoothing the performance of the
soft-starter.
Another issue that could be furthered is how to derive more generically
the voltage dropout that the controller should try to maintain as a function
of the rotor acceleration, the rotor resistance and the line impedance. Fuzzy
logic seems a good choice, although four inputs are too many for a fuzzylogic controller, and thus one or two heuristic combinations of them should
be attempted in order to reduce this number.
A very simplified structural analysis has been made with the aim of encouraging deeper studies by mechanical experts that could provide more
realistic estimations of the inertia time constants as a function of the blade
characteristics.
Simulations performed for different values of the electrical parameters of
the induction generator have shown that a better performance is obtained
by reducing the rotor reactance to a 60% of its actual value. It is worth
investigating if this improvement is generalized for all squirrel cage induction
generators and if so, study its cause.
Exploitation of the soft-starter in continuous operation at low load has
been studied for typical Weibull distribution and power-wind speed characteristics. An increase in the efficiency up to 4% can be achieved by controlling the voltage at the induction generator terminals, not only during the
start-up, but also in continuous operation at low wind speeds. Capacitors
banks should then be provided with resonant filters to draw the harmonics
current, mainly for the fifth, seventh, eleventh and thirteenth harmonics.
And finally, although simulations have been done pursuing the closest
agreement with real conditions, the controllers should be tested in a real
134
Chap. 8: Conclusions
wind turbine or, if not available, connected to a large induction machine
driven by a dc machine with the possibility of controlling its armature current.
Appendix
A
Weight and size for different
blades
Table A.1: Weight (kg), size (m) and corresponding power (kW) for several blades
I.
Blade type
NOI 16.2
NOI 16.5
LM 17.2
Bazán Bonus 600 kW Mk IV
LM 19.1
Ecotecnia 600
NOI 19.3
Length
Weight
16,2
16,5
17,15
19
19,04
19,1
19,3
750
800
1620
1800
1960
2900
1650
Diameter
Power
300
37.3
44
44
44
600
600
500
136
Chap. A: Weight and size for different blades
Table A.2: Weight (kg), size (m) and corresponding power (kW) for several blades
II.
Blade type
NOI 20.0
LM 21.0
LM 21.5
NOI 21.8
NOI 22.1
Gamesa V47-660kW
Gamesa G47-660 Ingecon-W
LM 23.0 P
Dewind Ibérica D48-600 kW
LM 23.2
LM 23.3
NOI 23.3
NOI 24.0
EU50.1250-3
EU50.1250-2
NOI 25.0
LM 25.1 P
EU53.1400-1
NOI 26.0
LM 26.1 P
LM 26.1
NOI 26.9
NOI 27.1
EU56.1400-2
NOI 28.5
Nordex N62/1.3 MW
LM 29.0 P
LM 29.0
EU60.1400-3
NOI 29.1
Length
Weight
20,8
21
21,5
21,9
22,15
23
23
23
23,1
23,2
23,3
23,36
24,2
24,3
24,3
25
25,1
25,65
26
26,04
26,04
26,9
27,1
27,5
28,5
29
29
29
29,05
29,1
1800
2200
2700
1670
1720
1500
1600
3000
1800
2990
2990
1970
2200
1900
2100
2400
3100
2800
2600
4250
4350
3050
3150
2850
3250
4300
4550
4850
3200
3500
Diameter
Power
47
48
600
47
47
48
48
48,4
50
660
660
600
50
50
750
750
750
52
53
850
54
54
1000
56,8
850
62
62
62
60
1300
1000
137
Table A.3: Weight (kg), size (m) and corresponding power (kW) for several blades
III.
Blade type
Dewind Ibérica D62-1000 kW
LM 29.1 P
LM 29.1
Dewind Ibérica D64-1250 kW
Neg Micon NM1500/64
EU65.1600-3
Vestas V66-1.65
UM70
SSP 34
UM70-2
Südwind s70/1500
NOI 34.0
LM 34.0 P
EU70.1800-2
LM 36.8 P
LM 37.3 P
EU77.1800-3
NOI 37.5
UM77
NOI 38,0
LM 38.8 P
EU80.1800-3
EU80.2000-1
NOI 39.0
LM 43.8 P
EU90.2300-1
NOI 44.0
EU90.2300-2
LM 44.8 P
NOI 46.0
NOI 48.0
EU100.2300-3
EU100.2300-2
LM 54.0 P
LM 61.5 P
Length
Weight
Diameter
Power
29,1
29,15
29,15
31,1
31,2
31,7
32,15
34
34
34
34
34
34
34,5
36,8
37,25
37,5
37,5
37,5
38
38,8
39
39
39
43,8
44
44
44
44,8
46
48
48,8
48,8
54
61,5
4300
4175
4900
4800
6900
3550
3800
4550
4600
4650
5200
5450
5600
5100
9125
6035
5500
5800
6100
5400
8650
6000
6600
7400
10100
8900
9900
10500
9980
11500
12700
11500
11900
13500
17740
62
62,3
62,3
64
64
63
66
70
70
70
70
1000
70,5
70
76
77
77
77
80
80
80
90
90
90
92
100
100
110.8
126.3
1250
1500
1200
1650
1500
1500
2000
1500
1500
1500
1500
1500
1500
2000
2500
3000
3000
3000
3000
3000
138
Chap. A: Weight and size for different blades
Appendix
B
Extended power-diameter
table
Table B.1: Diameter and power for several wind turbines I.
Blade type
Enercon E33
Suzlon 350
WindWorld W37/550
Gamesa G39
Vestas V39
Ecotecnia 40/500
Enercon EN-40
Gamesa G42
Diameter (m)
Power (kW)
33,4
33,4
37
39
39
40
40,3
42
330
350
550
500
500
500
500
600
140
Chap. B: Extended power-diameter table
Table B.2: Diameter and power for several wind turbines II.
Blade type
WindWorld W42/600
Neg Micon 600/43
Nordex N43/600
Nordtank
Bazán Bonus Mk IV
Ecotecnia 600
Gamesa G44
Neg Micon 750/44
DESA
Genesis 600
Dewind Ibérica D46
MADE AE 46/I
Gamesa G47-660
Gamesa V47-660kW
Dewind Ibérica D48
Neg Micon 600
Ecotecnia 750
Jeumont 750
Neg Micon 750/48
WindWorld W48/750
Enercon E48
FuhrLander
FuhrLander
Nordex N 50
WindWorld W52/750
MADE AE 52
Gamesa G52 850
Vestas V52 850
NEG Micon NM 900
Fuhrlander FL 800
FuhrLander FL 1000
Nordex N54
Diameter (m)
Power (kW)
42
43
43
43
44
44
44
44
45
45,9
46
46
47
47
48
48
48
48
48
48
48
48
50
50
52
52
52
52
52,2
52,7
54
54
600
600
600
600
600
600
600
750
650
600
600
660
660
660
600
600
750
750
750
750
800
800
600
800
750
800
850
850
900
800
1000
1000
141
Table B.3: Diameter and power for several wind turbines III.
Blade type
Nordic 1000
Bonus 1MW
Enron Wind 900s
MADE AE 56
MWT 1000
Wind Wind 56
Frisia F 56/850 kW
Gamesa G58 850
Enercon E-58
FuhrLander
MADE AE 59
Neg Micon 1000/60
Suzlon S60 1MW
Wind Wind 60
DeWind D60
Suzlon S60 1.25MW
Nordex N60
MADE AE 61
DeWind D62
Suzlon S62 1MW
DeWind D62
Bonus 1,3MW
Ecotecnia 62 1300
Nordex N62
Suzlon 950
Suzlon S64 1MW
Dewind Ibérica D64
Suzlon S64 1.25MW
Neg Micon NM1500
Suzlon S66 1.25MW
Diameter (m)
Power (kW)
54
54,2
55
56
56
56
56,3
58
58
58
59
60
60
60
60
60
60
61
62
62
62
62
62
62
64
64
64
64
64
66
1000
1000
900
800
1000
1000
850
850
1000
1000
800
1000
1000
1000
1250
1250
1300
1320
1000
1000
1250
1300
1300
1300
950
1000
1250
1250
1500
1250
142
Chap. B: Extended power-diameter table
Table B.4: Diameter and power for several wind turbines IV.
Blade type
Enercon E-66/15,66
PWE 1566
Vestas V66-1.65
BWU/Jacobs MD 70
FuhrLander MD 70
Südwind s70/1500
Nordex S70
Enercon E-66/18,70
Enron EW 1,5s
GE Wind Energy 1.5sl
Jacobs MD 70
Enercon E70
Mtorres 72
TWT 1750
NEG Micon NM 2000
Ecotecnia 74 1670
AN Bonus 2 MW/76
Bonus 2MW
BWU/Jacobs MD 77
Fuhrlander MD 77
Sudwind S-77
Nordex S77
Vestas V80/2,0 MW
Nordex N-80
Diameter (m)
Power (kW)
66
66
66
70
70
70
70
70
70,5
70,5
70,5
71
72
72
72
74
76
76
77
77
77
77
80
80
1500
1500
1650
1500
1500
1500
1615
1800
1500
1500
1500
2000
1500
1750
2000
1670
2000
2000
1500
1500
1500
1615
2000
2500
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