International Journal of Automation and Computing
8(4), November 2011, 429-436
DOI: 10.1007/s11633-011-0600-6
A Reliable and Accurate Calculation of Excitation
Capacitance Value for an Induction Generator Based
on Interval Computation Technique
Rajesh Kumar Thakur1
1
Vivek Agarwal2
Paluri S. V. Nataraj1
Systems and Control Engineering, Interdisciplinary Programme (IDP),Mumbai, 400076, India
2
Department of Electrical Engineering, IIT–Bombay, Powai, Mumbai 400076, India
Abstract: A squirrel cage induction generator (SCIG) offers many advantages for wind energy conversion systems but suffers from
poor voltage regulation under varying operating conditions. The value of excitation capacitance (Cexct ) is very crucial for the selfexcitation and voltage build-up as well as voltage regulation in SCIG. Precise calculation of the value of Cexct is, therefore, of considerable
practical importance. Most of the existing calculation methods make use of the steady-state model of the SCIG in conjunction with
some numerical iterative method to determine the minimum value of Cexct . But this results in over estimation, leading to poor transient
dynamics. This paper presents a novel method, which can precisely calculate the value of Cexct by taking into account the behavior
of the magnetizing inductance during saturation. Interval analysis has been used to solve the equations. In the proposed method, a
range of magnetizing inductance values in the saturation region are included in the calculation of Cexct , required for the self-excitation
of a 3-φ induction generator. Mathematical analysis to derive the basic equation and application of interval method is presented. The
method also yields the magnetizing inductance value in the saturation region which corresponds to an optimum Cexct(min) value. The
proposed method is experimentally tested for a 1.1 kW induction generator and has shown improved results.
Keywords: d-q model, hull consistency, induction generator, interval analysis, interval arithmetic, optimization, range of interval,
realpaver, self-excitation, transients.
1
Introduction
A wind power generation system generates electricity
from wind energy and typically comprises an induction generator coupled to a wind turbine. In a wind power generation system, the mechanical energy of the wind turbine is
converted into electrical energy by the induction generator.
A squirrel cage induction generator (SCIG) is highly suitable to be driven by wind turbine because of its small size
and weight, robust construction and reduced maintenance
cost[1] . In order to initiate voltage generation by the induction generator (self-excitation), a leading reactive power is
provided to the stator windings of the generator by connecting a capacitor bank to the stator windings. The induced
e.m.f. and current in the stator winding starts rising and
attains its steady-state value with frequency dependent on
rotor speed and machine parameters. The generated voltage is sustained at this operating point till reactive power
balance is maintained[2] . The voltage so generated is unstable in the sense that its value changes drastically under various loading conditions. This, in turn, changes the generated
torque and the rotor speed varies causing further changes in
the generated voltage. This leads either to a collapse of the
terminal voltage or building up to an excessively high value
depending upon the values of the magnetizing inductance
and the terminal (excitation) capacitance[2−5] .
The magnetizing inductance value is a function of rotor
speed and is nonlinear[3] . Hence, it is necessary to determine the range of capacitance value to keep the machine in
excitation mode and regulate the generator terminal voltManuscript received March 30, 2010; revised September 26, 2010
age within the limits. Tuning of capacitance is required
with the variation of rotor speed and loading conditions.
This is an impractical task due to the inter-dependence of
the system variables, changing rotor speed and the system0 s
nonlinearity. It is essential that the capacitance value of the
capacitor bank is such that at a given rotor speed, the generated voltage in the stator windings does not undergo large
transients[5, 6] . If the voltage generated in the stator windings is not sustained and smooth, there will be vibrations
in the wind turbine leading to wear and tear of the wind
turbine thereby reducing its life. In order to convert the
wind energy into electrical energy efficiently, it is important that the induction generator operates smoothly, i.e.,
voltage generated in the stator windings is sustained and
is without transients. Therefore, calculation of the capacitance value of the capacitor bank is very critical for the
desired operation of the induction generator.
It is known that the induction generator operates in the
saturated region during the self-excitation phase. It is also
known that the maximum value of magnetizing inductance
(Lmax ) in the saturated region leads to sustained voltage
generation in the stator windings. Therefore, in order to
accurately calculate the capacitance value, it is essential to
know the maximum value of the magnetizing inductance
in the saturated region. But this value is not known and
unavailable. However, it is known that the magnetizing
inductance value in the saturated region lies between zero
and the Lmax value in the unsaturated region. For accurate
calculation of the capacitance value, it is necessary to take
into consideration the magnetizing inductance value for the
saturated region both in the steady state and dynamic state
operating conditions of the induction generator.
430
International Journal of Automation and Computing 8(4), November 2011
Two models are reported in [7–12] for the analysis of selfexcited induction machine. One is based on the per-phase
equivalent approach and the other, the d-q axis model,
is based on the generalized machine theory. Most of the
work is based on per-phase equivalent circuit approach,
which includes loop-impedance method[7−9] and the nodaladmittance method[10, 11] . The d-q method has been used
by Elder et al. and others[2, 4, 6, 12] .
In the per-phase equivalent circuit approach, highly nonlinear simultaneous equations in per unit frequency or magnetizing reactance are solved using iterative methods to obtain minimum and maximum values of the excitation capacitance. Apart from being time consuming, this method
uses the steady-state model and unsaturated magnetizing
inductance value in the calculations. Hence, the minimum value of capacitance so obtained is an over estimation.
This capacitance value is able to initiate self-excitation, but
yields satisfactory results only under steady-state conditions. Higher value of capacitance is not technically viable
as there may be a possibility that the current flowing in the
capacitor might exceed the rated current of the stator, due
to the fact that capacitive reactance reduces as the capacitance value increases[2, 4, 6] . This will cause higher transient
currents during voltage build-up and hence more fluctuation
in the mechanical load on the prime mover which may excite
oscillatory mode of the prime mover. The d-q model based
approach yields better results as it also captures transient
phenomenon of the machine. But this is also an iterative
approach and makes use of unsaturated value of magnetizing inductance. Therefore, this method also yields an over
estimation of capacitance value.
In view of the ongoing discussion, it is clear that there
is a need to calculate the minimum value of capacitance
taking the magnetizing inductance value in the range of
saturated region which ensures self-excitation for a given
induction generator. This paper presents an “interval technique” based novel method for determining the minimum
capacitance (optimum value) required for self-excitation of
a 3-φ induction generator using d-q model of induction machine and using a range of magnetizing inductance values
in saturated region. This method offers the following advantages:
1) It yields all possible values of excitation capacitance.
2) It guarantees that the entire range of excitation capacitance values is found to a user-specified accuracy.
3) The range helps in selection of a suitable excitation
capacitance for wider operating speed variations.
The remaining paper is organized as follows. Section 2
describes the system modeling. Section 3 presents the evolution of the equations needed to be solved to determine the
capacitance value. This section also describes the application procedure of the interval analysis method. Section 4
presents the simulation results. Experimental results and
observations are included in Section 5. Discussion of results and major conclusions of this work are included in
Section 6.
Notations.
Rs , Rr : Per phase stator, rotor (referred to stator)
Ls , Lr : Per phase stator and rotor (referred to stator)
inductance (H)
Lm : Inductance (referred to stator) (H)
C: Per phase terminal excitation capacitive (F)
Iqs , Ids : Stator quadrature and direct axis currents (A)
Iqr , Idr : Rotor quadrature and direct axis currents (A)
Vqs , Vds : Stator quadrature and direct axis voltage (V)
Vqr , Vdr : Rotor quadrature and direct axis voltage (V)
ωr : Angular speed of rotor (rad/s)
ω: Angular speeds of synchronous, reference frame
(rad/s)
p : d/dt, derivative with time.
2
d -qq based system modelling
The parameters of induction machine are obtained by operating the machine as a motor. It has been reported[2, 4]
that per phase resistance and inductance of the stator and
rotor (referred to the stator side) remain nearly constant
over the frequency range 2 Hz–50 Hz. Even if there is a
small variation in the values of these parameters, it has
been observed that there is no significant impact on the system behavior. Fig. 1 shows the laboratory based schematic
diagram of a 3-φ, induction generator connected with 3-φ
capacitor bank and driven by the prime mover (DC motor). The d-q axis model of induction generator based on
the generalized machine theory is shown in Fig. 2.
Fig. 1 3-φ induction generator with prime mover DC motor and
capacitor bank
Fig. 2
d-q model of SCIG at no load (a) d-axis (b) q-axis
The terminal voltage equation is obtained from the equivalent circuits (see Fig. 2) for a stationary reference frame by
setting ω = 0. The terminal voltage equation obtained is
R. K. Thakur et al. / A Reliable and Accurate Calculation of Excitation Capacitance Value for · · ·
as follows:
"Vqs#  Rs + pLs +
Vds 
0
=
Vqr
Vdr
3
1
pC
pLm
ωr Lm
0
1
Rs + pLs + pC
ωr Lm
pLm
pLm
"
#
iqs
0
0
pLm
Rr + pLr
ωr Lr
ωr Lm
Rr + pLr

ids
iqr
idr
(1)
where the subscripts “s” and “r” are used to represent stator and rotor quantities and “d” and “q” represent direct
and quadrature axis quantities, respectively. For a stationary reference frame (ω = 0) and before excitation, all terminal voltages are considered to be zero. Laplace transform
of (1) in complex variable form “s” results in:
V (s) = A (s) I (s)
(2)
where

Vqs (s)
 V (s)
 ds
V (s) = 
 Vqr (s)
Vdr (s)

Iqs (s)
 I (s)


 ds
 , I (s) = 

 Iqr (s)
Idr (s)






(3)
431
Evolution of equations and interval
analysis
For an induction generator to be self-excited, the machine has to operate in saturation region and one pair of
poles of the machine must lie on the imaginary axis or in
the right half of s-plane as shown in Fig. 3. The minimum
value of capacitance corresponds to location of this pair of
poles on imaginary axis with maximum value of magnetizing inductance. Expression 3 is complex polynomial and if
a pair of poles has to lie on the imaginary axis for a given
capacitance value, then this capacitance value will be minimum for the given set of machine parameters. The real
and imaginary parts of the 6-th order polynomial are generated by replacing the complex variable “s” in the 6-th
order polynomial by jω and substituting the value of Lm as
[0 −Lmax value in the unsaturated region] and value of C
as [0 −∞].
and

1
0
sLm
0


sC


1

0
Rs + sLs +
0
sLm 
A (s)=
.
sC



sLm
ωr Lm
Rr + sLr ωr Lr 
ωr Lm
sLm
ωr Lm Rr + sLr
(4)
Solution of (2) describes the behaviour of the system.
The dynamic behaviour is determined by the location of
the poles of the polynomial or eigen-values of matrix “A”.
The characteristic polynomial of the terminal voltage equation (2) is obtained by taking the determinant of the matrix
“A” in (4).

Rs + sLs +
detA(s) = C 2 X12 s6 + 2C 2 X1 X2 s5 +
Fig. 3 s-plane pole location for Cexct > Cexct(min) and Cexct =
Cexct(min)
(C 2 (X22 + X1 (2Rs Rr + ωr2 X1 ))+
2CL2 X1 )s4 + 2(Rs (Rr X2 + L2 ωr2 X1 )C 2 +
3
(L2 X2 + Rr X1 )C)s + (C
2
The real and imaginary parts are generated as follows:
Rs2 X3 )+
det A(s = jω) = −C 2 X12 ω 6 + 2C 2 X1 X2 jω 5 +
(2C(Rs Rr L2 + Rr X2 + ωr2 L2 X1 )+
L22 ))s2 + 2(Rs CX3 + Rr L2 )s + X3 .
(C 2 (X22 + X1 (2R1 R2 + ωr2 X1 )) + 2CL2 X1 )ω 4 −
(5)
(C 2 R12 X3 + 2C(R1 R2 L2 + R2 X2 + ωr2 L2 X1 ) + L22 )ω 2 +
The 6-th order polynomial obtained is as follows:
2(R1 CX3 + R2 L2 )jω + X3 .
C 2 X12 s6+2C 2 X1 X2 s5 + (C 2 (X22 + X1 (2Rs Rr + ωr2 X1 ))+
2CL2 X1 )s4 + 2(Rs (Rr X2 + L2 ωr2 X1 )C 2 +
(L2 X2 + Rr X1 )C)s3 + (C 2 Rs2 X3 +
The real part on the right half side of s-plane (RHS) in
(8) is
2CL2 X1 )ω 4 − (C 2 R12 X3 + 2C(R1 R2 L2 + R2 X2 +
(6)
ωr2 L2 X1 ) + L22 )ω 2 + X3 .
(9)
The imaginary part on the RHS in (8) is
where
L1 = Ls + Lm ; L2 = Lr + Lm ; X1 = L1 L2 − L2m
X2 = Rs L2 + Rr L1 ; X3 = Rr2 + ωr2 L22 .
(8)
− C 2 X12 ω 6 + (C 2 (X22 + X1 (2R1 R2 + ωr2 X1 ))+
(2C(Rs Rr L2 + Rr X2 + ωr2 L2 X1 ) + L22 ))s2 +
2(Rs CX3 + Rr L2 )s + X3 = 0
(2(R1 (R2 X2 + L2 ωr2 X1 )C 2 + (L2 X2 + R2 X1 )C)jω 3 −
C 2 X1 X2 ω 5 − (R1 (R2 X2 + L2 ωr2 X1 )C 2 +
(7)
(L2 X2 + R2 X1 )C)ω 3 + (R1 CX3 + R2 L2 )ω.
(10)
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International Journal of Automation and Computing 8(4), November 2011
Both (9) and (10) are set to zero and a pair of equations
is obtained in terms of C and Lm as follows:
Fig. 4 (a) and (b), respectively. It is observed that as speed
increases, the capacitance value decreases.
Table 1
− C 2 X12 ω 6 + (C 2 (X22 + X1 (2R1 R2 + ωr2 X1 ))+
2CL2 X1 )ω 4 − (C 2 R12 X3 + 2C(R1 R2 L2 + R2 X2 +
ωr2 L2 X1 ) + L22 )ω 2 + X3 )ω = 0.
(11)
Sr No. Rotor speed Synch speed Range of C C(min) (µF)
(rad/s)
(rad/s)
(µF)
11
377
376
0 – 23
20.1
2
377
370
0 – 25
23.92
25
3
377
360
0 – 34
25.5
34*
4
314
313
0 – 33
31.56
33
5
314
310
0 – 35
33.3
35
6
314
300
0 – 146
42.07
135
7
250
249
0 – 52
49.96
52
8
250
245
0 – 57
54.76
57
9
200
199
0 – 81
78.37
81
10
150
149
0 – 150
140
150
11
100
99
0 – 330
317
320
12
50
49
0 – 1400
1246
1400
13
45
44
0–+∞
1791
2002
14
40
39
0–+∞
No solution
C 2 X1 X2 ω 5 − (R1 (R2 X2 + L2 ωr2 X1 )C 2 +
3
(L2 X2 + R2 X1 )C)ω + (R1 CX3 + R2 L2 )ω = 0.
(12)
Equations (11) and (12) are nonlinear equations of the
order six. It is highly cumbersome to solve such equations
using numerical techniques. In many practical situations in
engineering, data are only known to lie within intervals and
only ranges of values are computed experimentally[13] . In
such cases interval computation yield desired results[14, 15] .
Interval analysis method provides a direct means of solving
the two highly nonlinear equations (11) and (12) to find
the value of capacitance and predicting the behaviour of a
self-excited induction generator under balanced operating
conditions. The coefficients of both the equations vary as
the excitation builds up because of the change in the value
of magnetizing inductance from unsaturated to saturated
region. Hence, these coefficients can only be represented by
interval of values calculated based on interval arithmetic
operation. It is reported that the magnetizing inductance
first increases from its value of unsaturated region and then
starts decreasing while operating in saturated region[2, 4] .
The above equations are solved simultaneously using interval analysis method to obtain all the solutions for C and
Lm . The capacitance value and corresponding magnetizing inductance (Lmax ) value for the saturated region are
extracted from the solutions obtained for C and Lm for
smooth sustained voltage build-up in an induction generator.
4
Minimum values of excitation capacitance for
different rotor speeds
23
Simulation and results
RealPaver[16] is a modeling language for numerical constraint solving software. Numerical constraint is a relation
between the unknowns (variables) of a problem which is
defined by an analytic expression:
f (x1 , x2 , · · · , xn )♦0
where ♦ = {6, >, =} and f : Rn → R.
Each domain is represented by a closed interval: [r1 , r2 ] =
{r ∈ R|r1 6 r 6 r2 }
It uses:
A set of real-valued variables, e.g., (x1 , x2 , · · · , xn );
A set of interval domain, e.g., (x1 , x2 , · · · , xn );
A set of numerical constraints, e.g., (c1 , · · · , cm ) over the
given set of variables; Solution is obtained for all the consistent values with respect to all constraints.
Realpaver is used for interval analysis on a 3-φ, Y connected induction generator with machine rating of
220 V, 4.8 A and 60 Hz. Other important machine parameters (in per unit) are, Rs = 0.0946, Rr = 0.0439, Xs =
Xr =0.0865, Xm(max) = 2.12[6] . Results are obtained for
different sets of constants and are summarized in Table 1.
The variations required in the excitation capacitance
w.r.t rotor speed and synchronous speed are shown in
Fig. 4 Graph showing decrease in capacitance as (a) rotor speed
increases and (b) synchronous speed increases
433
R. K. Thakur et al. / A Reliable and Accurate Calculation of Excitation Capacitance Value for · · ·
The result obtained by this method and that mentioned
in [6] is summarized in Table 2. Data at Sr. No. 1 is from
[6], Data at Sr. No. 2 is from the presented work.
Table 2
Sr. No.
Comparison of minimum values of capacitance
Rotor speed
Synchronous speed
Cexct(min)
(rad/s)
(rad/s)
(µF)
1
377
360
84.7
2
377
360
34
It is observed that actual value of minimum capacitance
for the given set of parameters lies in between 25.5 µF and
34 µF which is much smaller than the value obtained by
eigen-value sensitivity method[6] . The simulation result for
Cexct(min) = 84.7 µF is shown in Fig. 5 (a) and the transient
interval therein is marked as “A” whereas that for Cexct(min)
= 34 µF is shown in Fig. 5 (b) with the transient interval
marked as “B”. A comparison of Fig. 5 (a) and (b) reveals
that when operated with Cexct(min) = 34 µF (determined
by the proposed method), the system experiences a smooth
transient interval, which, in turn, reduces the over voltage stress during the transient excitation of the induction
generator and minimizes the adverse effects on the turbine
and associated equipment. It is also observed that with
Cexct(min) = 34 µF, the system can withstand wider variations in operating speeds without pronounced transients as
compared with Cexct(min) = 84.7 µF. In fact, the proposed
method yields a range of all possible values of excitation
capacitance (see Table 1) for the entire operating range.
The range permits trade-off between the specifications on
speed of response and oscillatory nature of voltage build-up.
These aspects are further highlighted in Section 5.
Fig. 5 Simulation results showing self-excitation process of voltage buildup for: (a) C = 84.7 µF, calculated using eigen-value
sensitivity method; (b) C = 34 µF, calculated using interval
method (proposed method)
5
Experimental results and observations
A 3-φ, star connected induction generator coupled with
DC motor is used in this experiment. The DC motor is
used as a prime mover to provide mechanical power at the
induction generator rotor shaft in the form of kinetic energy
which, in turn, is converted into electrical energy. Residual
flux in the induction generator initiates terminal voltage
generation. A 3-φ star connected capacitor bank is used to
provide the requisite reactive power to the induction generator for building up the terminal voltage through selfexcitation process. The capacitor bank, with value determined using the proposed calculation method, is connected
to the generator. The voltage build-up is captured using
Agilent0 s Infiniium digital storage oscilloscope (DSO), with
a multiplying factor of 20 on the vertical axis. The waveforms so obtained are shown in Figs. 6 and 7.
The Cexct(min) values, for different (example) rotor speeds
of the SCIG, are obtained with the proposed method.
Experiments are conducted with each example operating
speeds and the performance of the SCIG is tested with the
various values of Cexct as summarized in Table 3. Key waveforms corresponding to each experiment are captured using
the DSO and are shown in Fig. 6. These are discussed as
follows.
Table 3
Example
Minimum capacitance values corresponding to the
various example rotor speeds
IM rotor speed
Synchronous
Minimum capacitance
(rad/s)
speed (rad/s)
value (µF)
1
329
314
62
2
301
288
67
3
270
258
84
4
245
233
103
The capacitance value by proposed method is found to
be 62 µF at a rotor speed of 329 rad/s and synchronous
speed of 314 rad/s. On performing the experiment with a
capacitance value of 60 µF at a rotor speed of 314 rad/s,
no initiation of self-excitation was observed. Next, a capacitance value = 70 µF is used and self-excitation process
is observed, as shown in Fig. 6 (a). Also, the minimum
capacitance value for a rotor speed of 301 rad/s and synchronous speed of 288 rad/s is found to be 67 µF. When
the experiment was performed at this operating point with
capacitance value = 60 µF, no initiation of self-excitation
was observed. The next capacitance value used was 70 µF
and the self-excitation process was observed as shown in
Fig. 6 (b). Similarly, at rotor speeds of 270 rad/s and 245
rad/s, Cexct(min) = 84 µF and 103 µF, respectively, were used
and it was observed that the machine failed to excite with
capacitance value of 80 µF at rotor speed 270 rad/s and
capacitance value of 100 µF at rotor speed of 245 rad/s,
whereas self-excitation took place with capacitance values
of 90 µF and 110 µF, respectively, as shown in Fig. 6 (c) and
(d). Thus, it is experimentally verified that the capacitance
value calculated by using the proposed interval method is
accurate and reliable. Further investigations were carried
out by performing experiments with various capacitance
434
International Journal of Automation and Computing 8(4), November 2011
values at different operating conditions to analyze its effect on the transients of the induced voltage through cases
1–6 as summarized in Table 4.
Table 4
Various case studies with different combinations of
the capacitance value and rotor speed
Sr. No.
Case No.
Cexct , (µF)
Rotor speed (rad/s)
1
Case 1
90
301
2
Case 2
90
329
3
Case 3
110
251
4
Case 4
110
257.6
5
Case 5
110
270
6
Case 6
110
282
The following observations were made:
1) As the value of rotor speed was increased from
301 rad/s to 329 rad/s with a capacitance value of 70 µF,
a significant increase in voltage fluctuation was observed
(Fig. 6 (a) and (b)). But the speed of response increased.
2) As the value of rotor speed was increased through
270 rad/s, 301 rad/s and 329 rad/s with a capacitance
value of 90 µF, a significant increase in voltage fluctuation
(Figs. 6 (c), 7 (a), and 7 (b)) took place. The speed of response also increased considerably.
3) As the value of rotor speed is increased through
245 rad/s, 251 rad/s, 257.6 rad/s, 270 rad/s and 282.7 rad/s
with a capacitance value of 110 µF, there was significant increase in voltage fluctuation (Figs. 6 (d), 7 (c)–7 (f)). Once
again, the speed of response was found to have increased.
Thus, it was experimentally verified that if the value of
capacitance used is large, there are large transients during
the self-excitation process, which can have adverse effects
on the turbines. Thus, it may be concluded that the proposed method gives more suitable/accurate and optimum
value of the excitation capacitance.
6
Fig. 6 Experimental results showing voltage building at the terminals of the induction generator: (a) Rotor speed = 329 rad/s;
Cexct(min) = 62 µF; (b) Rotor speed = 301 rad/s; Cexct(min) =
67 µF; (c) Rotor speed = 270 rad/s; Cexct(min) = 84 µF; (d) Rotor
speed = 245 rad/s; Cexct(min) = 103 µF.
Discussion of results and conclusions
A direct method based on interval computation technique has been proposed to calculate minimum capacitance
value required for self-excitation of an induction generator.
The proposed method is fast and can be directly used for
transient analysis investigation of the generator voltage. A
salient feature of the proposed calculation method is that
it also yields a range of capacitance values which will not
be able to induce self-excitation in the generator for the
given operating conditions. The use of the interval (range)
of magnetizing inductance values leads to an accurate prediction of whether or not self-excitation will occur for various capacitance values. The minimum capacitance value
calculated by the interval analysis method, for different rotor speeds, is found to be very close to the capacitance value
at which the induction machine initiates the self-excitation
process, rendering an accurate value of the self-excitation
capacitance.
R. K. Thakur et al. / A Reliable and Accurate Calculation of Excitation Capacitance Value for · · ·
435
Fig. 7 Experimental results showing voltage building at the terminals of the induction generator. (a) Case 1: Capacitance value =
90 µ, rotor speed =301 rad/s; (b) Case 2: Capacitance value = 90 µ, rotor speed = 329 rad/s; (c) Case 3: Capacitance value = 110 µF,
rotor speed = 251 rad/s; (d) Case 4: Capacitance value = 110 µ, rotor speed = 257.6 rad/s; (e) Case 5: Capacitance value = 110 µF,
rotor speed = 270 rad/s; (f) Case 6: Capacitance value = 110 µF, rotor speed = 282.7 rad/s
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Rajesh Kumar Thakur received the
Bachelor0 s degree in electrical engineering from Muzaffarpur Institute of Technology, Bihar University, Muzaffarpur, India, Master0 s degree in electrical engineering (control systems) from the Institute
of Tecnology, Banaras Hindu University,
Varanasi, India, and pursuing for Ph. D.
degree in systems and control engineering
from the Indian Institute of Technology
Bombay, India. In 1992, he joined the Department of Instrumentation & Control Engineering, Regional Engineering College
Jalandhar, India, where he served as lecturer till 2000. He is
currently serving as associate professor at College of Military
Engineering, Pune, India. He is a life member of the Institution
of Engineers.
His research interests include modeling and simulation of large
systems, robust stability and control, nonlinear system analysis
and control, and conditioning of energy from non-conventional
sources.
E-mail: rkthakur@sc.iitb.ac.in (Corresponding author)
Vivek Agarwal received the Bachelor’s
degree in physics from St.
Stephen0 s
College, Delhi University, Delhi, India,
Master0 s degree in electrical engineering
from the Indian Institute of Science, Bangalore, India, and the Ph. D. degree in electrical and computer engineering from the University of Victoria, Victoria, BC, Canada.
Subsequently, he worked as a research engineer with Statpower Technologies, Burnaby, BC, Canada. In 1995, he joined the Department of Electrical
Engineering, Indian Institute of Technology–Bombay, Mumbai,
India, where he is currently a professor. He is a fellow of the Indian National Academy of Engineering, a fellow of the Institute
of Electronics and Telecommunication Engineers (IETE), and a
life member of the Indian Society for Technical Education. He
is a senior member of IEEE and serves on the editorial boards
of IEEE Transactions on Power Electronics and Smart Grid. He
has received multiples awards and honors for his contributions
towards research in various areas.
His research interests include modeling and simulation of new
power converter configurations, intelligent and hybrid control of
power electronic systems, power quality issues, electromagnetic
interference (EMI)/electromagnetic compatibility (EMC) issues,
and conditioning of energy from nonconventional sources.
E-mail: agarwal@ee.iitb.ac.in
Paluri S. V. Nataraj is a professor of
Systems and Control Engg Group at IIT
Bombay. He obtained his Ph. D. from IIT
Madras in process dynamics and control in
1987. He then worked in the CAD center at
IIT Bombay, India for about one and half
years before joining the faculty of the Systems and Control Engineering Group at IIT
Bombay in 1988. He is an associate editor
of International Journal of Systems Assurance Engineering and Management (Springer), and editor of two
international journals– International Journal of Automation and
Control (Inderscience) and Opsearch (Springer).
His research interests include chemical process control, global
optimization, robust stability and control, nonlinear system analysis and control, and reliable computing.
E-mail: nataraj@sc.iitb.ac.in