Download New techniques for inverter flux control

880
IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 37, NO. 3, MAY/JUNE 2001
New Techniques for Inverter Flux Control
Mukul Chandorkar, Member, IEEE
Abstract—Inverter flux control methods directly control the integral of the inverter output voltage vector, by means of the inverter switching. These methods have found applications in lowand medium-voltage industrial drives, and in inverters with outputs connected to the utility mains. This paper describes a new
method for the direct control of the inverter flux vector, with the
viewpoint of using this vector as the main forcing quantity in a
closed-loop control system. It illustrates the use of the inverter flux
in the control of a utility-connected inverter.
Index Terms—Pulsewidth modulation, three-phase inverters,
utility connection.
I. INTRODUCTION
I
T IS TYPICAL to consider inverter pulsewidth modulation
(PWM) methods under two headings—open loop and closed
loop. The typical open-loop control method is the well-known
sine-triangle intersection method, along with its numerous variations [1]. As the name implies, open-loop methods determine
the switching pattern of the inverter independently of the inverter output voltage or current. Closed-loop control methods,
on the other hand, use feedback of some output quantity to directly determine the next switching state of the inverter. The next
switching state is chosen so as to correct the error between the
desired value of the output quantity, and its actual value. For
example, the output current of an inverter can be controlled by
choosing the next switching state of the inverter to correct the
error between a reference current value, and the measured inverter current [2]. Also under the heading of closed-loop modulation methods are the sigma–delta modulation strategies [3],
often used to control resonant link converters.
Another well-known example of closed-loop inverter control
is the so-called direct torque control (DTC) of induction machines [4]. Typically, closed-loop inverter control methods for
three-phase inverters control two independent output quantities
directly by means of inverter switching. In DTC, the two independent output quantities that are so controlled are the stator
flux magnitude and the torque. However, the torque and the flux
magnitude are not the only independent quantities that can be
controlled directly by the inverter switching.
Paper IPCSD 01–001, presented at the 1999 Industry Applications Society
Annual Meeting, Phoenix, AZ, October 3–7, and approved for publication in
the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Industrial Power
Converter Committee of the IEEE Industry Applications Society. Manuscript
submitted for review October 15, 1999 and released for publication February
27, 2001.
The author is with the Electrical Engineering Department, Indian Institute of
Technology, Mumbai 400076, India (e-mail: [email protected]).
Publisher Item Identifier S 0093-9994(01)03933-0.
Fig. 1.
Flux control by inverter switching.
In this paper, the concern is on the independent closed-loop
control of the two components of the inverter flux vector. The
inverter flux vector is defined as
(1)
is the instantaneous inverter
In this definition,
output voltage vector in the stationary – reference frame. The
task of the flux vector control method is to guide the end of
along a specified path in the –
the inverter flux vector
plane, by independently controlling the two components of the
flux vector, using the inverter switching action. This is shown in
and
are two inverter voltage vectors, and
Fig. 1. In Fig. 1,
is the inverter flux vector. The tip of
is guided along the
desired path by selecting the times and sequence of the voltage
vectors.
An early and comprehensive description of a method to
control the flux vector in the stationary – reference frame,
for induction machine control, was given in [5]. The effects of
constraints on the maximum inverter switching frequency and
minimum switch commutation time—the time between consecutive switchings in an inverter leg—were analyzed in [5]. A
flux-based modulation strategy was developed, which resulted
in minimum machine losses, while meeting the constraints on
maximum switching frequency and minimum commutation
time.
Subsequently, the “fish” method was proposed to analyze and
synthesize inverter switching patterns based on the flux vector
observed in a rotating – reference frame [6]. The flux “fish” is
0093–9994/01$10.00 © 2001 IEEE
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on November 8, 2008 at 01:29 from IEEE Xplore. Restrictions apply.
CHANDORKAR: NEW TECHNIQUES FOR INVERTER FLUX CONTROL
defined as the locus of the flux vector observed in the synchronously rotating reference frame. The PWM pattern is generated
by choosing the inverter states so as to confine the error between
the reference and actual flux vectors within limits along the synchronously rotating and axes.
A space-vector-based flux-vector control technique suitable
for digital processor implementation has been discussed in [7].
As in [5] and [6], this technique confines the error between a
reference flux vector and the actual flux vector to be within a
specified tolerance zone. This is done by appropriately choosing
zero and active inverter output voltage vectors. A zero voltage
vector is chosen when the error magnitude becomes less than a
specified tolerance. An active vector is chosen if the error magnitude exceeds the tolerance. The strategy for choosing the active vector is reported to reduce the number of switchings per
output cycle compared to the “fish” method of [6], for the same
total harmonic distortion (THD) in the inverter output.
This paper describes a new method to control the inverter flux
vector in rectangular coordinates. The described method is related to the PWM method of [5], in that the control is performed
in the stationary rectangular – reference frame. The aim is
to develop the flux modulator so that it matches with the outer
loops of the inverter control system, which produce the reference flux vector for the modulator.
This paper also discusses the switching frequency characteristics and linearity of the resulting modulator. It then describes the design and implementation of a closed-loop control
system that uses the flux-vector modulator to effectively control a three-phase inverter connected to the main power system
through a sine-wave filter.
II. INVERTER FLUX-VECTOR CONTROL
Essentially, inverter flux-vector control involves appropriately choosing the inverter voltage vectors, so as to make the
flux follow a reference flux vector within a specified tolerance
band.
Because the modulator does not directly control the load, the
response of the load quantities is not as fast as, for example, the
motor torque with DTC. However, the advantage here is that
it does lead to a general-purpose modulator whose task is to
control the inverter flux vector to whatever value is specified
by the outer control loop (within the limits of the inverter rating
and dc-bus voltage). The outer control loop can then be designed
independently to generate the inverter flux-vector set point for
achieving the desired final result. An example of the use of flux
control for an active filter application can be found in [8].
The well-known vector diagram of a two-level voltage-source
inverter is shown in Fig. 2. In Fig. 2, the length of each vector is
, where
is the inverter dc-bus voltage value.
Six sectors—I … VI—are also defined as shown. Each sector
spans 30 on either side of each inverter vector. The sector of
the inverter flux vector decides the possible voltage vectors that
can be used to correct it.
,
In the following, the reference flux vector is
. The actual flux vector is similarly
where
. The components of both these vectors are observed
in the stationary reference frame. In order to control the inverter
881
Fig. 2.
Voltage-source-inverter vectors.
TABLE I
FLUX ERROR TOLERANCE
flux vector in rectangular coordinates, the two rectangular comand , are controlled.
ponents of the inverter flux vector,
The quantity is the permitted error tolerance in both, the and -axes flux components. The flux controller is expected to
within the permitted tolerance.
keep the error vector
and
are the results of the error comThe binary variables
parison in, respectively, the - and the -axes components of the
flux vector. Table I, in which the subscript represents either
or , shows the specification of the error tolerance. The rules for
determining the output voltage vector are as follows.
• If there is an active voltage vector for the present sector,
that can correct the errors in both the - and the -axes
components simultaneously, then that vector is used.
• Otherwise, a zero vector is used. (Of the two possible
zero vectors, the one selected should result in the smaller
number of switchings.)
These rules result in Table II for the inverter voltage vector .
The flux control method as described in Table II renders itself
very well for a digital signal processor (DSP) implementation.
On the DSP, the inverter flux components are updated at con, using rectangular integration
stant time intervals of
(2)
In this equation, the subscript refers to the sample number, and
and
are, respectively, the - and -axes components of the inverter voltage vector in the previous sampling in-
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on November 8, 2008 at 01:29 from IEEE Xplore. Restrictions apply.
882
IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 37, NO. 3, MAY/JUNE 2001
TABLE II
VOLTAGE-VECTOR SELECTION
Fig. 4.
Inverter output voltages.
Fig. 5.
Linear range calculation.
tant for the use of the inverter flux modulator in a closed-loop
control system. In the linear region, this relationship is
Fig. 3.
Flux-vector locus.
(3)
terval. These components are themselves calculated by the DSP
on the basis of the information of the previous switching state
of the inverter and the dc-bus voltage feedback.
The reference flux-vector components are also made available to the modulator from the outer control loop in the time
. On the basis of this, the error status information
interval
and
are calculated. These bits, along with the sector
bits
information, are then used to determine the new inverter voltage
.
vector
Fig. 3 shows one cycle of the inverter flux vector , for a
V. The angular velocity of the referdc-bus voltage
rad/s , and its magnitude is 0.918
ence flux vector is
V s. The sampling
V s. The flux tolerance value is
s.
interval
Fig. 4 shows the - and -axes voltages that give rise to the
flux-vector locus of Fig. 3.
in which is the magnitude of the fundamental inverter output
is the output angular frequency.
phase voltage, and
is the magnitude of the reference flux vector. The linear region
can be calculated on the basis of Fig. 5.
In Sector I (Fig. 2), the smallest tangential velocity with
which the tip of the inverter flux vector can be moved occurs
when the inverter flux vector is aligned with the axis. In
and
have the
this situation, both the voltage vectors
same tangential component, which is in the direction of the
axis. Since the length of each inverter voltage vector is
, the maximum peak fundamental output phase
voltage possible in the linear range is
III. LINEARITY AND FLUX REFERENCE LIMITS
The linearity of the relationship between the fundamental
output voltage magnitude and the output frequency is impor-
(4)
Note that this is also the peak phase voltage which, when
. If the full-load voltage
rectified, results in the dc voltage
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on November 8, 2008 at 01:29 from IEEE Xplore. Restrictions apply.
CHANDORKAR: NEW TECHNIQUES FOR INVERTER FLUX CONTROL
883
Fig. 7. Switching frequency dependence on h.
Fig. 6.
Fundamental output voltage magnitude linearity.
is to be delivered at the nominal output frequency
the nominal value of the reference flux should be
, then
This section presents simulations of the switching frequency behavior for the flux controller described above. For this purpose,
the nominal output frequency and nominal angular velocity are
Hz
(5)
comThis puts a limit on the maximum possible value of
mand that can be given by the outer control loop to the flux
controller, if the latter is to operate in its linear range. This limit
should also account for variations of the dc-bus voltage from its
nominal value. The limit is given by
(6)
is the nominal dc-bus voltage, which gives rise
In this,
to the nominal inverter output voltage at the nominal frequency
. The limit on the
and nominal flux reference magnitude
reference flux magnitude can be calculated in the DSP on the
and the desired output
basis of the dc-bus voltage feedback
frequency . In practice, this limit on the flux reference magnitude can be best implemented if the outer control loop provides
the flux reference vector command in the synchronous reference
frame, rotating at an angular frequency . The reference is limited in the synchronous reference frame, and then transformed
to the stationary reference frame and provided to the flux controller.
Fig. 6 shows the normalized fundamental component output
voltage, as a function of the normalized output frequency. This
was obtained by a simulation of the flux modulator at various
output frequency points, followed by an extraction of the fundamental component magnitude from the resulting switching
pulse pattern. The three curves shown in Fig. 6 represent three
different settings of the flux magnitude reference, as a multiple
[see (5)]. The flux magniof the nominal flux reference
tude reference is further limited according to (6).
IV. SWITCHING FREQUENCY
Predominantly, the output switching frequency is influenced
by the flux error tolerance , the fundamental output angular
, and the flux reference magnitude
.
frequency
Further, the ratio
One switching cycle is considered to be one on and one off
event for any switch. In the simulation, the number of switching
cycles is counted within five fundamental output cycles. The
average value of the total number of switchings for the three
phase legs is calculated over the five fundamental output cycles,
.
and divided by five to give the average switching frequency
as a funcFig. 7 shows the averaged switching frequency
tion of the output frequency , with the flux error tolerance as
the parameter.
For Fig. 7, the integration time step [refer to (2)] is set as
s. The flux reference magnitude is set as
if
if
(7)
and
are calculated by (5) and (6), respecwhere
tively. Equation (7) ensures that the modulator remains in the
linear region, over the entire range of the fundamental frequency
.
In Fig. 7, the tolerance is represented as a fraction of the
. It is important to note that when is specinominal flux
fied in this manner, the curves of Fig. 7 are then independent of
.
the dc-bus voltage
as a function of
Fig. 8 shows the switching frequency
the fundamental output frequency, with the integration time
as the parameter. The tolerance value is fixed at
step
, and the flux magnitude reference is determined
from (7).
, as a function of the
Fig. 9 shows the switching frequency
fundamental output frequency , with the reference flux magnitude as the parameter. The tolerance value is fixed at
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on November 8, 2008 at 01:29 from IEEE Xplore. Restrictions apply.
884
IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 37, NO. 3, MAY/JUNE 2001
Fig. 8. Switching frequency dependence on
1T .
Fig. 9. Switching frequency dependence on
.
, and the integration time step is
s. The
parameter for the plots in Fig. 9 is the flux reference magnitude,
[see (5)]. Specifically, the flux refgiven as a multiple of
erence magnitude for Fig. 9 is set as
if
if
(8)
, as shown in Fig. 9. With the
where is the multiplier of
flux reference magnitude set from (8), the modulator is linear
over the entire fundamental frequency range of Fig. 9.
Some comments regarding the switching frequency characteristics of Figs. 7–9 are in order here. All these figures are in. Furthermore, if both the dependent of the dc-bus voltage
and -axes coordinates for these figures are normalized by the
, these characteristics are indenominal output frequency
as well.
pendent of
A further point to be noted is that the curves in Figs. 7 and 8
show a maximum switching frequency at a fundamental output
. This is a consefrequency of approximately
quence of the fact that, for such tolerance band controllers,
the characteristic of the switching frequency normalized by
the output frequency, as a function of the normalized output
frequency, is approximately a straight line with a negative slope
. This is shown in Fig. 10, with the
in the range
tolerance as the parameter for the curves, and the integration
s.
time step
In Fig. 10, the normalized switching frequency characteristics
can be approximated by equations of straight lines of the form
(9)
The values of the constants and are determined by
in Fig. 8.
Fig. 7, and by
From (9), it is easily seen that the maximum value of
occurs for
Fig. 10.
Normalized switching frequency.
This maximum value of
is given by
The fact that the normalized switching frequency curves are approximated by straight lines, as in Fig. 10, can be very useful in
developing a switching frequency limiting method for the modulator and can be of importance to high-power converters which
have strong average switching frequency constraints. Specifically, the constant , which determines the tolerance value
as a fraction of
(Fig. 7), can be determined on the basis of
, as a
(9). This enables the easy determination of
function of the desired inverter switching frequency.
The following relationships can be determined empirically
from Fig. 10:
in
When these relationships are used with (9), the following
equation results:
(10)
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on November 8, 2008 at 01:29 from IEEE Xplore. Restrictions apply.
CHANDORKAR: NEW TECHNIQUES FOR INVERTER FLUX CONTROL
Fig. 11.
Utility-connected three-phase inverter.
885
either by the use of a phase-locked loop (PLL) or by the use
of a power-frequency droop characteristic as in [10], [11]. In
either case, in the steady state, the – frame angular frequency
equals the mains frequency, and its position relative to the
mains voltage determines the sharing of the total load power
between the inverter and the utility mains.
The inverter controller is required to align the filtered output
voltage vector with the rotating axis of the reference frame.
Inherent to this requirement is the need to actively damp the
and . As shown
natural oscillations of the filter formed by
below, the inverter flux vector can be used as a very effective
forcing quantity to achieve this.
Further, being a continuous quantity, it is very convenient to
use the flux vector to define the power angle, which essentially
determines the flow of real power from the inverter to the load
bus [10]. In Fig. 12, this is the angle between the vectors and
, where
is the flux vector associated with the load bus
voltage .
For the purpose of the flux-based control of the inverter interface, one of the state variables used to model the interface is the
output voltage vector on the filter capacitor. The other state
, instead of the
variable is chosen to be the filter flux vector
conventionally used inductor current in .
In the reference frame rotating with the angular velocity ,
the inverter and filter are modeled, in terms of the flux vectors,
by the following vector differential equation:
(11)
Fig. 12.
Flux-vector diagram of inverter interface.
Similarly to the curves of Fig. 7, (10) is independent of the
, and of the nominal frequency
. However,
dc voltage
s. Using
it is valid only for an integration time step
(10), it is possible to approximately calculate the value of the
tolerance , which results in a specified switching frequency
.
is the filter natural freIn this equation,
is the current fed by the inverter and filter into
quency, and
the load bus. The inverter flux vector is the forcing quantity,
used to align the filtered voltage vector with the rotating
axis, and ensure that it has the desired magnitude. It is readily
seen that the two eigenvalues of the complex system matrix of
(11) are
V. UTILITY-CONNECTED INVERTER CONTROL
Inverter flux-vector control can be used very effectively for
the control of inverters which have their outputs connected to the
main utility system [9]. This section describes the flux control
of a three-phase inverter connected to the utility system through
a sine-wave output filter.
Fig. 11 shows the schematic diagram of a three-phase inverter
connected to the mains, and together with the mains, feeding a
load, in the manner of a line-interactive uninterruptible power
supply (UPS) system.
Fig. 12 shows the vector diagram of the inverter interface,
in a reference frame rotating with the angular velocity . The
axes – are stationary, and the axes – rotate as shown,
counterclockwise with an angular velocity .
The instantaneous position of the rotating – reference
frame axes is determined relative to the utility mains voltage,
To force the filtered voltage vector to assume the desired
can be generated
value , the inverter flux-vector reference
by a proportional–integral (PI) controller acting on the voltage
, in the rotating reference frame
vector error
(12)
and
In this equation, the complex constants
are the gains of the rotating frame vector PI
controller.
For the purpose of controller design, it is assumed that
the inverter, as controlled by the flux modulator, is capable
of producing the commanded flux vector at its output with
is
negligible delay. That is, the assumption that
always valid. With this assumption, the controller action can be
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on November 8, 2008 at 01:29 from IEEE Xplore. Restrictions apply.
886
Fig. 13.
IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 37, NO. 3, MAY/JUNE 2001
Closed-oop system based on flux-vector control.
easily included in the system description of (11). The resulting
closed-loop system is as follows:
Fig. 14.
(13)
Fig. 13 shows the schematic diagram of the closed-loop
voltage control system, based on inverter flux control. Equation
, in which the
(13) is in the classical form
definitions of the matrices , , and are readily apparent.
The complex system of (13) can be expanded into a system
of six real first-order differential equations, by expanding all
into their components, and setting the
vectors
and
.
complex gains to be
Three of the six eigenvalues of the real system matrix are the
of (13), and the
three eigenvalues of the complex matrix
other three are their complex conjugates.
and
can be selected to ensure the
The complex gains
desired steady-state value for the filtered voltage vector , and
to shape its transient response. The characteristic polynomial of
the system matrix [see (13)] is found to be
(14)
and
affects the roots of
, which
The choice of
are the eigenvalues of the closed-loop system matrix of (13).
and
of
Conversely, the specification of two distinct roots
results in two complex linear simultaneous equations of
the type
These two equations can be used to solve for unique values
and . These values are then used to calculate the third
of
of
, which must be checked for acceptability. If it
root
is not acceptable, the process of calculating the complex gains
must be repeated.
are determined as described above,
When the roots of
the active damping of the filter natural oscillations is guaranteed,
in addition to the regulation of the voltage vector .
No-load startup of inverter and filter.
It is important to note that only a simple PI regulator is sufficient in this case to stabilize the filter, and there is no need for a
derivative term or high-pass filtering. This is a consequence of
the use of the inverter and filter flux to model the system. If, instead, the traditional model with voltages and currents is used,
a PI regulator can never stabilize the filter, even in the rotating
reference frame.
A. Experimental Results
The closed-loop flux-based voltage control method described
above was implemented for a 2.25-kW 110-V three-phase
MOSFET inverter feeding a load as shown in Fig. 11, together
with the 110-V three-phase mains. The system component
values were:
mH
F
H
The complex controller gains were fixed at
The frequency of the rotating – reference frame of the
inverter was determined according to a power-frequency droop
characteristic as in [11]. The control system implementation including the flux modulator was on a Motorola 56000 fixed-point
DSP system operating on a 27-MHz clock.
Fig. 14 shows the startup on no load of the isolated inverter.
The upper trace of the oscillogram shows the line–neutral
voltage across the filter capacitor. The lower trace shows the
inverter output line current. The inverter and filter rapidly
reach steady state, and the filter natural oscillations are quickly
damped.
Fig. 15 shows the inverter and filter performance immediately
following the connection of the utility mains to the load bus.
Prior to the mains connection, the inverter was supplying the
load by itself. After mains connection, the utility picks up the
load, as is indicated by the increase in the utility current (upper
trace). The lower trace shows the line–neutral voltage across the
filter capacitor. The middle trace shows the line–neutral voltage,
expanded tenfold in time, around the instant of the utility connection. There are no disturbances on the filter output voltage
before or after the point of utility connection. This is achieved
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on November 8, 2008 at 01:29 from IEEE Xplore. Restrictions apply.
CHANDORKAR: NEW TECHNIQUES FOR INVERTER FLUX CONTROL
887
Electronics Consortium (WEMPEC), University of Wisconsin,
Madison. The author wishes to thank ABB Corporate Research
Ltd., Switzerland, for extending support in carrying out this
work.
REFERENCES
Fig. 15.
Utility connection to load bus.
by the tight control of the voltage by inverter switching due to
flux-vector control.
VI. CONCLUSION
This paper has presented the direct control of the inverter
output flux vector by means of the inverter switching. The two
components of the flux vector are independently controlled so
as to guide the vector along a desired trajectory. The method
lends itself very well to a digital implementation on a DSP.
The linearity between the fundamental output voltage magnitude and the output frequency is examined. The limits on the
inverter flux reference magnitude, needed to ensure operation in
the linear region, are presented.
The dependence of the switching frequency on various
modulator parameters is examined. Of particular practical
importance is the dependence of the switching frequency on
the error tolerance band value . An approximate relationship
between and the switching frequency is presented, which
is independent of the inverter dc voltage, and of the nominal
output frequency. This would prove useful for designing a
switching frequency limiting scheme, especially for high-power
inverters having strong switching frequency limitations.
The design and implementation of a flux-based controller for
a three-phase inverter connected to the utility mains was presented. This illustrates the advantages of using the inverter flux
vector as a forcing variable in the control.
ACKNOWLEDGMENT
The work reported on utility-connected inverter control was
carried out in the Wisconsin Electric Machines and Power
[1] A. M. Hava, R. J. Kerkman, and T. A. Lipo, “A high performance generalized discontinuous PWM algorithm,” in Proc. IEEE APEC’97, 1997,
pp. 886–894.
[2] D. M. Brod and D. W. Novotny, “Current control of VSI-PWM
inverters,” IEEE Trans. Ind. Applicat., vol. 21, pp. 562–570, May/June
1985.
[3] A. Mertens, “Performance analysis of three-phase inverters controlled
by synchronous delta-modulation systems,” IEEE Trans. Ind. Applicat.,
vol. 30, pp. 1016–1027, July/Aug. 1994.
[4] I. Takahashi and T. Noguchi, “A new quick-response and high-efficiency
control strategy of an induction motor,” IEEE Trans. Ind. Applicat., vol.
22, pp. 820–827, Sept./Oct. 1986.
[5] V. G. Török, “Near-optimum on-line modulation of PWM inverters,”
in Proc. IFAC Conf. Control in Power Electronics and Electric Drives,
Lausanne, Switzerland, 1983, pp. 247–254.
[6] A. Veltman, P. P. J. van den Bosch, and R. J. A. Gorter, “On-line optimal
switching for 2-level and 3-level inverters using the fish method,” in
Proc. IEEE PESC’93, 1993, pp. 1061–1067.
[7] A. M. Trzynadlowski, M. M. Bech, F. Blaabjerg, and J. K. Pedersen,
“An integral space-vector PWM technique for DSP-controlled voltagesource inverters,” in Conf. Rec. IEEE-IAS Annu. Meeting, 1998, pp.
1215–1221.
[8] S. Bhattacharya, A. Veltman, D. Divan, and R. Lorenz, “Flux based controller for active filter,” IEEE Trans. Ind. Applicat., vol. 32, pp. 491–502,
May/June 1996.
[9] L. Ängquist and L. Lindberg, “Inner phase angle control of voltage
source converter in high power applications,” in Proc. IEEE PESC’91,
1991, pp. 293–298.
[10] M. C. Chandorkar, D. M. Divan, and R. Adapa, “Control of parallel
connected inverters in stand-alone AC supply systems,” IEEE Trans.
Ind. Applicat., vol. 29, pp. 136–143, Jan./Feb. 1993.
[11] M. C. Chandorkar, D. M. Divan, and B. Banerjee, “Control of distributed
UPS systems,” in Proc. IEEE PESC’94, 1994, pp. 197–204.
Mukul Chandorkar (M’84) received the B.Tech.
degree from Indian Institute of Technology, Bombay,
India, the M. Tech. degree from Indian Institute of
Technology, Madras, India, and the Ph.D. degree
from the University of Wisconsin, Madison, in
1984, 1987, and 1995, respectively, all in electrical
engineering.
He has several years of experience in the power
electronics industry in India, Europe, and the U.S.
During 1996–1999, he was with ABB Corporate Research Ltd., Baden-Dättwil, Switzerland. He is currently an Associate Professor in the Electrical Engineering Department, Indian
Institute of Technology, Mumbai, India. His technical interests include uninterruptible power supplies, drives, real-time simulation of power electronic systems, and the measurement and analysis of power quality.
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on November 8, 2008 at 01:29 from IEEE Xplore. Restrictions apply.