International Journal of Power
Elecronics
Technology
Unbalance
andand
Induction
Machines
FJanuary-June 2011, Volume 1, Number 1, pp. 61– 81
F
Unbalance and Induction Machines
V. Ramakrishnan1, and S.K. Srivatsa2
1
Department of EEE, Bharath University, Chennai - 600 073, India.
2
Department of ICE, St. Joseph's College of Engineering, Chennai - 600
119, India
ABSTRACT: An unbalance in a three-phase system can be seen as the
presence of a negative sequence. In induction machines (IM’s), even a small
negative voltage will cause a large amount of negative sequence current. The
negative sequence will cause a synchronous frequency second harmonic
pulsation and unbalanced currents in the system. In induction machine,
unbalanced 3-phase stator voltages cause a number of problems such as over
heating, over current and thrust on the mechanical component from torque
pulsation. Therefore, the machine torque, flux, and reactive power will have
a 100 Hz pulsation. Unbalance condition appear that using active crowbar
protection and direct torque control. The control strategy is validated by
means of simulation. Induction Generator (IG) connected in the power system
under balanced and unbalanced load, symmetrical and unsymmetrical
components of voltages and currents are described. The definition of unbalance
factor, harmonic content and synchronous frame unbalance theory are also
analyzed. Experimental results under balanced condition are presented.
Keywords: Balance and unbalance IM’s, negative and positive and neutral
sequence.
1. INTRODUCTION
Wind power is one of the most promising renewable energy sources
after the progress undergone during the last decade. However, its
integration into power systems has a number of technical challenges
concerning security of supply, in-term of reliability, availability and
power quality. An unbalance of voltage may occur on long distance
power transmission lines as shown in figure 1[1]. It occurs not
infrequently in the case of generating stations supplying mixed
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polyphase and single phase loads. The predetermination of the
performance of three-phase induction motors under such conditions
is a matter of importance that there is a reduction in the maximum
load which an induction motor is capable of carrying safely when
supplied with unbalanced voltages, but the extent of this reduction
has not received the attention which it merits.
It is carried out experiments on the performance of induction
machines on unbalanced votages, but they have assumed that the
load limit of the motor is reached when the current in one phase
reaches its full load value, and hence their experiments underrate
the performance of the motor [2]. It is treated the matter analytically
and derived certain simple formulae to predetermine the output,
but his results have not received conclusive experimental
verification. The various conditions of symmetrical components,
unsymmetrical components, balance and unbalance conditions,
positive, negative, zero sequence and harmonic pulsations, voltage
unbalance theory and factor have been discussed. The output of the
simulation result has been verified by experimental calculation.
Graphical method of resolving an unsymmetrical voltage system
into two symmetrical systems is also described.
Figure 1: Physical Connection in the Power System
2. INDUCTION GENERATOR CONNECTED IN THE
POWER SYSTEM UNDER UNBALANCED LOAD
The unbalancing of the balanced voltages of the alternator was
effected in two ways: (1) by injecting into one of the phases of the
alternator an opposing e.m.f. derived from the secondary of a
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transformer whose primary was connected across the alternator
phase itself, this condition corresponding to pure unbalance of the
star voltages with no phase shift; (2) by injecting into one of the
phases a quadrature e.m.f. obtained from the secondary of a
transformer whose primary was connected across the other two
phases, as shown in Figure 2.2
Figure 2: Diagram Connections, UTr, Connecting Transformers, AA, AB, AC,
Line Ammeters, VAB, VBC, VCA, Line Voltmeter, WA, WB, WC,
Wattmeter’s, VA, VB, VC, Star Voltmeters
Measures were made of the star voltages, the line voltages, the
line currents, and the power in each phase and the output. The
method of procedure was to maintain a constant unbalance and load
the motor gradually, taking all the readings until the currents far
exceeded the normal value. While the unbalanced-phase voltage is
small, large negative-sequence currents can result due to low
negative sequence impedance of an induction generator. These large
currents eventually can cause unbalanced heating (hot spots) in
the machine windings, which can potentially lead to failure.
Unbalanced-voltage operation will also create a pulsating torque
which produces speed pulsation, mechanical vibration, and
consequently, acoustic noise.
The tests were repeated for various degrees of unbalance. The
practical experimental calculations are given in Experimental Results
chapter. As is well known, an unbalanced three-phase system may
be resolved into two balanced component systems, one of which
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has the same phase sequence or phase rotation as the unbalanced
system, while the other has the opposite phase sequence. For the
sake of brevity, we shall in what follows speak of the first balanced
component as the direct phase or direct rotational system, and of
the second as the reverse phase or counter-rotational system.
The unbalance factor of a system is defined to be the ratio of
the reverse phase voltage (or current) to the direct phase voltage
(or current) and is frequently expressed as a percentage. The
graphical method of relations connecting output and percentage
unbalance is shown in Figure 3.
Figure 3: The Graphical Method of Relations Connecting
Output and Percentage Unbalance
(A) When local losses equal to full load losses
(B) When current in an one phase reaches full load value
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3. TURBINE SPEED CLASSIFIED IN TWO TYPES
The wind turbine technology can mainly be classified into two
sections.
•
Fixed speed wind turbine
•
Variable speed wind turbine
In 2003, 46% of the wind turbines installed were variable speed
wind turbines with double fed induction generator and its market
share is going to increase. It has several advantages over the other
types.
Now days Doubly Fed Induction Generator (DFIG) stator of such
wound rotor machines is directly connected to the electrical grid
and therefore, it is extremely sensitive to voltage disturbances due
to voltage sags [3]. When unbalanced sags occur, the main problem
is that very high current, torque, power oscillations appear at double
the electrical frequency forcing a disconnection. Such oscillation is
provoked by the negative sequence components injected by the
unbalanced disturbance [4]. Unbalance condition appear that using
active crowbar and direct torque control. A method based on a
disturbance refection controller is proposed into compensate the
2*ωe (ωe – electrical angular velocity) oscillation produced by
unbalances, by adding a feed forward component to the current
controllers.
A control strategy is proposed by choosing certain current
reference values in the positive and negative sequences. So that
torque and the DC voltage are kept stable during such unbalanced
sequences. Both rotor and grid side converter are considered;
detailing the control scheme of each converter which considering
the effect of the crowbar protection. The control strategy is validated
by means of simulation.
Unbalance load condition, to control the DFIG, on separating
the positive and negative components of all the current and voltages
for DC /AC converter both the grid side and rotor side[5]. But grid
side converter control is not considered, to keep the DC bus stable
is proposed, based on compensating the rotor power delivered by
the rotor side converter in the grid side converter.
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4. (A) SYMMETRICAL COMPONENTS
Fortes cue method of symmetrical components is used in calculation
in this paper.
1 1 1   X A 
 X0 
 X  = 1 1 a a2   X 
 B 
 +
3 
2
 
 X − 
 1 a a   XC 
(1)
Where XA, XB, XC are the phasors of unbalanced phasor system.
X0, X+ and X_ are phasors of symmetrical components (zero, positive
ands negative sequence respectively and q = 1 < 120º is unit complex
operator.
The level of unbalance is described by current unbalance factor
(more precisely known as sequence current unbalance factor), ρi,
which is given as the modulus of ratio of negative to positive
sequence currents (same as for negative sequence voltage unbalance
factor, ρv) as shown in Figure 4.
Figure 4: One Single-Phase Load Connected to Low Voltage
Moreover, in non-counterweighted systems the zero sequence
current (voltage) unbalance factor εi (εv) is defined as the modulus
of ratio of zero to positive sequence currents (voltages).
εi =
Io
V
× 100, ε υ = o × 100
I+
V+
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The analysis of a three-phase circuit in which phase voltages
and currents are balanced (of equal magnitude in the three phase
and displaced 120º from each other) and in which all circuit elements
in each phases are balanced and symmetrical is relatively single.
Since the treatment of a single phase leads directly to the three phase
solution positive, negative and zero sequence system of three
successive application of ‘a’ will rotate through 360º as shown in
Figure 5 [5].
Figure 5: Symmetrical Components
Positive and negative sequence impedance of a transformer one
equal and zero sequence is also equal to the positive sequence
impedance provided these are a through circuit for the earth currents
and the compensating currents can flow: otherwise the impedance
is infinite.
(B) Unsymmetrical Components
Unsymmetrical faults which occur as single line to ground faults,
line to line faults or double to ground fault cause unbalanced currents
to flow in the systems. The absence of a grounded neutral at the
generator does not affect the fault current. It the generator neutral
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is not grounded zero sequence impedance is infinite and zero
sequence voltage is indeterminate, but line to line voltages may be
found since they contains no zero sequence components as shown
in Figure 6.
Figure 6: Unbalanced Current Can be Resolved into under Balanced Condition
(C) Under Balanced Condition
The DFIG is attached to the wind turbine by means of a gearbox.
Stator windings are directly connected to the grid, which the rotor
windings are controlled to a back-to-back converter. The converter
is composed of the grid side converter connected to the grid and the
rotor side converter connected to the wound rotor windings. The
converter set points are established by the so called high level
controller. It user the knowledge of the wind speed and the grid
active and reactive power requirements to determine the optimum
turbine pitch angle and the torque and reactive power set points
referenced to the converter. The rotor side converter controls the
torque and reactive power, which the grid side converter controls
the DC voltage and grid side reactive power. The rotor side
back-to-back converter can control both reactive power injected by
the stator by controlling the rotor currents and the reactive power
injected directly. To the grid with grid side converter, reactive power
to deliver through the stator which keeping a low or null reactive
power set point in the grid side converter.
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(D) 5Under Unbalanced Condition
Unbalanced sags imply negative sequence components in all the
relevant quantities. Therefore, important oscillations appear in
torque, active and reactive power. Such oscillations have a pulsation
of 2*ωe. In order to mitigate such oscillations, an approach taking
into account the negative sequence quantities is required that
approach has been applied to the rotor side converter of a DFIG.
Figure 7: Asymmetrical Components Unbalanced Condition of Voltage Vector
Analyzes of a whole back-to-back converter taking into account
both the positive and negative sequence components and proposes
a technique to control optimally both the DC bus voltage and torque
when unbalanced voltage sags occur [6].
Figure 8: Asymmetrical Components Unbalanced Condition of Current Vector
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The positive and negative sequence components calculation is
done by using the Clarke transformation, rotating either ejωet or
e–jωet, and finally applying a notes-filter of 2*ωe to eliminate the
opposite sequence. PI controller is used, tuned according to internal
mode control. For a time constant T, the parameters obtained yield
Kp = L / T; Ki = R/T. for the rotor side voltages are limited according
to the rotor side voltages can be applied using standard space vector
pulse with modulation technique.
5. PROBLEMS CAUSED BY VOLTAGE UNBALANCE
Three-phase utility unbalance can be a problem in rural areas, where
induction wind generators are likely to be located. In induction
machines, unbalanced three-phase stator voltages cause a number
of problems, such as overheating, over-current, and stress on the
mechanical components from torque pulsations.
Therefore, beyond a certain amount of unbalance, induction
machines must be derated or removed from the network. In the
case of a grid connected induction generator, this can exacerbate
the grid balance. In addition, any time generation is removed, it
means less money for the generation company. In this chapter,
quantification of unbalance and how to analyze unbalanced
induction machines is presented.
6. PER PHASE EQUIVALENT CIRCUIT UNBALANCES THEORY
Using symmetrical component theory [7], a three-phase system can
be represented as the sum of zero, positive and negative sequence
circuits. When the three-phase system is perfectly balanced, only
the positive sequence is present. When the system is unbalanced,
one or both of the zero and negative sequence will be present. For a
star connected machine without a neutral connection, no zero
sequence current will flow. Therefore, for the purpose of
performance analysis of a star connected machine without a neutral
connection, the zero sequence can be ignored.
Induction machines are very susceptible to a stator voltage
unbalance. This can be easily understood by using the per phase
equivalent circuit of a cage-type induction generator for the positive
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and negative sequences, as shown in Figure 9. With any induction
machine, the larger the slip speed between the rotor conductors and
the air-gap flux, the greater voltage and current impressed on the
rotor. This is easily seen in Figure 9(a), as the equivalent rotor
resistance decreases with increasing slip.
Figure 9: Induction Machine Per Phase Circuit for
(a) Positive Sequence; (b) Negative Sequence
The negative sequence voltage is rotating in the opposite
direction of the positive sequence. This gives rise to a flux component
rotating in the negative direction. This negative sequence flux is
rotating counter to the rotor, which appears as a very large slip.
Therefore, the equivalent rotor resistance for the negative
sequence is much smaller than the equivalent rotor resistance for
the positive sequence, shown in Figure 9 (b). Even a small amount
of negative sequence voltage will give rise to a large amount of
negative sequence current. This extra negative sequence current can
cause over-heating, and it unbalances the stator currents.
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In addition to the unbalanced currents, there will also be a
periodic pulsation in the torque and reactive power. This pulsation
will occur at twice the synchronous frequency. This can be
understood better using more advanced tools: space vectors and dq
theory.
7. DEFINITION OF UNBALANCE FACTOR
There are many standards for defining the amount of unbalance for
a three-phase system. Some methods focus on ease of calculation in
the field (usually ignoring phase shift), others are more
mathematically rigorous. The National Electrical Manufacturers
Association (NEMA), defines line voltage unbalance (LVUR) [8] as
(
max Vab − VLavg , Vbc − VLavg , Vca − VLavg
where,
VLavg =
VLavg
Vab + Vbc − Vca
3
)
(3)
(4)
This method has the advantage of being easily calculated in the
field, as it depends only on phase-phase voltage magnitudes.
However, it does not account phase shift. In this research, the amount
of unbalanced for a three-phase system is defined as the magnitude
of the negative sequence over the magnitude of positive sequence.
This is called the “unbalance factor”2 and for voltage is denoted as
voltage unbalance factor( VUF) and for current as current unbalance
factor( IUF) Referring to Figure 9:
VUF =
Vs 2
Vs 1
(5)
The advantage of this definition of unbalanced is that it accounts
for both a magnitude unbalance (where one of three-phases is off
nominal in magnitude) and a phase unbalance (where one of the
phases is not 120 degrees separated). It should be noted that not all
unbalances are created equal. A 5 % VUF for one system with an
off-nominal voltage (like a voltage sag in one phase) and a five
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percent VUF for another system with a phase-shifted phase can affect
a machine quite differently. However, the definition of is chosen as
the best overall method of quantifying unbalance. As an interesting
side note, for a system with perfect 120 phase delay between phases,
and in the case where there is an over or under-voltage condition in
a single phase, then LVUR is approximately twice VUF.
8. SYNCHRONOUS FRAME UNBALANCE THEORY
A three-phase unbalance will add a double synchronous frequency
harmonic to synchronous frame quantities. For example, a voltage
unbalance (either by magnitude, phase, or in combination) in a
50 Hz system will add a 100 Hz harmonic to all the synchronous
frame voltages and currents. This in turn adds a 100 Hz harmonics
to the machine torque and power.
A space vector is defined as
Vs(t) = Va(t) + Vb(t)ej120 + Vc(t)ej240
(6)
These variables are shown as a function of time to emphasize
that they are instantaneous values. Each phase quantity can be said
to be equivalent to the sum of a zero, positive, and negative sequence
component, represented by the subscripts 0, 1 and 2 respectively.
Va(t) = Vs0 cos (ωsyn t + θ0) + Vs1 cos (ωsyn t + θ1)
+ Vs2 cos (ωsynt + θ2)
(7)
Vb(t) = Vs0 cos (ωsynt + θ0) + Vs1 cos (ωsynt + θ1)
+ Vs2 cos(ωsynt + θ2 + 120)
(8)
Vc(t) = Vs0 cos(ωsynt + θ0) Vs1 cos (ωsynt + θ1)
+ Vs2 cos(ωsynt + θ2 – 120)
(9)
Substituting (7) through (9) in to (6), we get
Vs (t ) =
3 ˆ j ( ω synt +θ1 ) ˆ j ( ω synt +θ2 )
Vs 1 e
+ Vs 2 e
2
(
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(10)
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Here then it is seen that a space vector can be represented as a
sum, of two synchronously rotating vector: one proportional to the
positive sequence component, rotating in the positive direction, and
another proportional to the negative sequence component, rotating
in the negative direction. These two vectors will add constructively
and destructively twice per revolution, thus causing the double
frequency harmonic in torque and power.
The d and q components of the vs space vector are defined as
the scaled reflections of the space vector on to the dq reference frame
(see Figure 7)
Vsdq =
Vsd + jVsq = 2 /3(Va + Vb e j 120
(11)
+ Vc e j 240 )e jθd
Substituting (7) through (9) into (10):
Vsdq =
Vsd + jVsq = 3/2(Vs 1 e
+ Vs 2 e
j ( ω synt +θ1 )
j ( ω synt +θ 2 )
(12)
) e jθd
For a dq frame synchronized to the stator voltage space vector
(“grid flux oriented”, see Chapter 4, θd simply the angle of vs.
(
(
 img Vs 1 e j ( ωsynt +θ1 ) + Vˆs 2 e j ( ωsynt +θ2 )
Θd = tan 
 real Vˆs 1 e j ( ωsynt +θ1 ) + Vˆs 2 e j ( ωsynt +θ2 )

−1
) 
) 
(13)
 Vˆs 1 sin(ω syn t + θ 1 ) − Vˆs 2 sin(ω syn t + θ 2 ) 
Θd = tan −1 
 (14)
 Vˆs 1 cos(ω syn t + θ 1 ) + Vˆs 2 cos(ω syn t + θ 2 ) 
This equation becomes quite complex to analyze when there is
an unbalance (i.e., vs2 ≠ 0). Because the dq axes are aligned to analyze
with the stator voltage space vector, at all times. The d-axis voltage is
Vsd =
2 /3 Vs = 3/2
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Vs21 + Vs22 + (2Vs 1Vs 2
cos(2ω syn t + θ1 + θ 2 ))
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It is interesting to note here that contains mostly a DC and second
harmonic component, but due to the non-linearity of (13), there will
be higher order harmonics as well. Figure 10 shows vsd for a system
with a 0.05 stator voltage balance factor (VUF), and a 50 Hz
synchronous frequency. Figure 11 shows the harmonic content.
Clearly, it can be seen that while there are higher order harmonics,
the second harmonic (at twice the synchronous frequency, 100 Hz)
is dominant. Similar results are shown for the rotational speed ?d
as shown in Figure 12. and 13.
Figure 10: Vsd for 0.05 Stator Voltage VUF
Figure 11: Vsd Harmonic Content
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Looking at Figure 10 through Figure 12, it is then easy to see
how this second harmonic propagates through the entire circuit,
causing second harmonic pulsations in the reactive power and
torque (active power).
Figure 12:
d
for 0.05 Stator Voltage VUF
Figure 13:
d
Harmonic Content
It’s shown then that higher order harmonics resulting from
instantaneous alignment can likely be safely neglected. Also, it may
be that the positive sequence alignment strategy may result in less
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of the second harmonic disturbance, since it does not add any noise
through ωd. However, this may be offset by the fact positive sequence
alignment has a second harmonic in vsq. All things considered, the
instantaneous alignment is effective and simpler to implement. For
more discussion on unbalance for positive sequence alignment, see
[9] and [10]
9. EXPERIMENTAL RESULTS
The test consisted in determining how far the induction motor is
capable of correcting the unbalance in the supply voltage. Induction
motor on unbalanced supply through Lab Experimental IM’s values
is given below.
A three phase star connected 440 Volt (line to line), 10 HP 50 Hz
six pole induction motor has the following constants in ohms per
phase. R1 = 0.30 Ω/phase, R2 = 0.14 Ω/phase, Rm = 120 Ω/phase,
X1 = X2 = 0.35 Ω/phase, Xm = 13.2 Ω/phase. For a slip s = 0.025
(operation as a motor), compute I, VA, Im, I2, speed in r/min, Total
output torque and power, power factor, Total three-phase losses
and efficiency.
The applied voltage to neutral is
V1 =
440
 Voltage 
= 254.034 0o V/Phase; 


3
3 
As per U.S.A Standard value adopted in calculation,
Line to Line Voltage = 220V,
Phase Voltage value =
Z2 =
220
= 127 0º V / phase
3
R2
0.14
+ jX 2 =
+ j 0.35
S
0.025
= 5.60 + j 0.35 = 5.61 3.58ºΩ
Zm =
jRm Xm
j(120)(13.2)
=
= 13.2 83.72º Ω
Rm + jXm 120 + j 13.2
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Zin = R1 + jX1 +
Zm Z2
Zm + Z2
= 0.30 + j0.35 +
(13.2 83.72º)(5.61 3.58º)
13.2 83.72º + 5.61 3.58º
= 5.29 27.08º Ω
I1 =
V1
127 0 o
=
= 24.01 −27.08º A
Zin 5.29 27.08o
VA = V1 – I1(R1 + jX1)
= 127 0º − (24.01 −27.08º)(0.30 + j0.35)
= 116.84 −2.06º V
Im = =
I2 =
116.84 −2.06º
= 8.91 −85.78º A
13.2 83.72º
VA 116.84 −2.06º
=
= 20.83 −5.64º A
Z2
5.61 3.58º
The speed (ns) =
7200
= 1200 r/m
6
The Total Torque = Tm =
=
90[ I 2 ]2 R2
πnsS
90(20.83)2 (0.14)
= 58.01 N-m/red
π(1200) (0.025)
Total Mechanical Power
2 πTm
2π(1170)(58.01)
=
60
60
= 7107 Watts
Pm = ω n * Tm =
Hoarse Power =
7107
= 9.53 HP
746
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Power Factor (p.f) = cos θ;
where,
θ = ∠V – ∠I rad, ∠XC =
Q = V I sin θ var,
Q = VI = I 2 X =
1
ωC
P = VI = I 2 R =
V2
Watts,
R
V2
var
X
(θ) Lead indicates that θ is negative i.e., p.f. = 0.8 lead
Complex Power S = P + jQ + |S| ∠θ VA
1
π ; X L = ωLπ
ωC
Z = R + jX; XC =
Total losses in the motor
PLoss =
3 VA
Rm
2
2
2
+ 3 I 1 R1 + 3 I 2 R2 Watts
3 (116.84 )
+3 (24.01)2 (0.30) + 3(20.83)2 (0.14) watts
120
= 1042 Watts
=
Efficiency = ηm =
2
Pm
7107
=
Pm + PLoss 7107 + 1042
= 0.872, 87.2% (Induction motor)
The efficiency of the 3-phase induction motor has been
determined under balanced conditions. If any fault occurs in one of
the phases, unbalance condition arises and hence the efficiency can
be determined only after bringing to balance condition. It was
considered desirable to observe experimentally what the resulting
balance voltage is when the unbalanced supply is corrected by means
of a reverse phase series booster, and how far this value agrees with
the direct phase component as obtained from the vector analysis of
the unbalanced voltages.
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10. CONCLUSION
Thus, a controller of doubly fed induction generator used in wind
turbine has to withstand disturbances, have capabilities discussed
earlier, and keep turbine operational. It is also desirable to have
controller, which can guarantee to perform despite known variations
in system parameters and wind.
DFIG Unbalance Compensation Control Design presents the
modifications to the basic control that are designed to handle the
unbalanced voltage. An unbalance in a 3-phase system can be seen
as the presence of a negative sequence. In induction machines, even
a small negative sequence voltage will cause a large amount of
negative sequence current. The negative sequence will cause a
synchronous frequency second harmonic pulsation in the system
and cause unbalance current.
The amount of torque pulsation and reactive power pulsation
are almost linear function of voltage unbalance factor. This is another
advantage of using VUF to quantify unbalance but it was observed
that it is important for both reactive power and active power to be
compensated. If only one at the loop is compensated, the stator
current will become very much distorted. Also it was observed that
the compensation loops tend to decrease the total harmonic
distortion slightly. It may be worth further investigation.
First and foremost would be the grid side rotor converter. It may
be advantageous to implement a special control on this converter to
deal with the extra harmonics that will be active on the DC link.
Also, it may be possible to unbalance the current drawn from the
converter to complement the unbalance stator current to effectively
completely balance the current into the system though this may again
impact the DC link voltage. It is an area worth investigating. In
addition, it should be possible to extend the compensation control
to dq control used in power systems, such as three-phase
synchronous rectifiers.
The induction motor is tested & verified at laboratory as per
USA standard and it gives the result as follows: Total Torque-58.01
N-m/radian, Total Mechanical Power – 9.53 HP, Pf – 0.8 lead, Total
Losses in the Motor - 1042 Watts, Efficiency of the Motor - 82.2%.
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Unbalance and Induction Machines presents and explains the
problems caused in induction machines when connected to an
unbalanced supply. The mathematical impact of unbalance is
explained, tested in the lab, and the amount of unbalance is
quantified.
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Transactions on Power Delivery, 16, No. 4, October 2001.
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