Download Consideration of Input Parameter Uncertainties in Load Flow

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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 3, AUGUST 2006
Consideration of Input Parameter Uncertainties in
Load Flow Solution of Three-Phase Unbalanced
Radial Distribution System
Biswarup Das, Member, IEEE
Abstract—In this paper, a technique based on interval arithmetic is presented for considering the uncertainties of the input
parameters in the power flow solution of three-phase unbalanced
radial distribution systems. The uncertainties in both the load
demand and the feeder parameters have been considered. The
results obtained from an interval arithmetic-based power flow
solution have been compared with those obtained from repeated
load flow simulations.
Index Terms—Interval arithmetic, power flow, three-phase distribution system.
I. INTRODUCTION
ECAUSE of its application in many distribution system decision algorithms such as network planning, volt/var control, service restoration, feeder reconfiguration, state estimation, etc., distribution system power flow analysis is nowadays
an integral component of distribution system planning, operation, and control functions. Now, due to various reasons, such
as unbalanced consumer loads, presence of single, double, and
three-phase line sections, existence of asymmetrical line sections etc., present-day distribution systems are primarily unbalanced in nature. As a result, for reliable and accurate solutions, three-phase load flow study of the distribution systems
is required. To cater to this need, substantial effort has been
devoted in the literature for developing efficient and accurate
three-phase load flow algorithms for distribution systems. In the
early endeavors, a direct solution approach using the impedance
Gauss apmatrix of the unbalanced network [1] and the
proach [2] have been suggested. Subsequently, different other
techniques, such as the backward/forward sweep algorithm [3],
the three-phase fast decoupled power flow algorithm [4], [5],
the rectangular Newton–Raphson-based method and its fast decoupled version [6], the current injection method [7], the phase
decoupled method [8], etc., have also been developed. Another
direct approach, which utilizes two matrices developed from the
topological characteristics of the distribution system, has been
introduced in [9]. In a series of papers, Chen et al. have developed the models for three-phase co-generators, transformers,
and loads necessary for three-phase distribution load flow analysis [10]–[12]. Models for voltage control devices have been
B
Manuscript received July 5, 2005; revised December 8, 2005. Paper no.
TPWRS-00399-2005.
The author is with the Department of Electrical Engineering, Indian Institute
of Technology, Roorkee 24 667, India (e-mail: [email protected]).
Digital Object Identifier 10.1109/TPWRS.2006.876698
suggested in [13]. Theoretical aspects of three-phase distribution load flow solution have been studied in [14]. An adaptive
power flow method with improved convergence characteristics
has been introduced in [15]. A method for incorporating transformer nodal admittance matrices into the backward/forward
sweep algorithm has been described in [16].
In all the above works, the analyses have been carried out assuming the input quantities (loads at different buses and feeder
parameters) are known and fixed. However, in real-life situations, the values of these input quantities may contain a significant amount of uncertainties. These uncertainties might occur
due to: 1) error in the calculation or measurement of the feeder
parameters and 2) error in the metered, calculated, or forecasted
values of the demands in the system load buses. In [17], use
of interval arithmetic has been first proposed to incorporate the
uncertainties into the power flow solution of a transmission network. In this work, the authors have used a small five-bus transmission network for illustration.
Motivated by the work of [17], this paper proposes to apply
the interval arithmetic to the power flow algorithm of a threephase unbalanced radial power distribution system to account
for the uncertain input quantities. Basically, in this approach, the
uncertain input quantities are represented as interval numbers
(instead of fixed numbers), and subsequently, interval arithmetic
is used to compute the power flow solution of the distribution
system. This paper is organized as follows. In Section II, the
fundamental concepts of interval arithmetic used in this paper
are discussed. The algorithm for power flow analysis of the
three-phase unbalanced radial distribution system using interval
analysis is described in Section III. Numerical results obtained
for different cases of input parameter uncertainties are presented
in Section IV. Lastly, Section V concludes this paper.
II. INTERVAL ARITHMETIC
The following notations have been used throughout the paper
to describe the fundamental concepts of interval arithmetic and
application of it to the three-phase radial distribution system
power flow algorithm.
A lowercase letter, such as , denotes a scalar (real) number.
A lowercase letter with subscript “I,” such as , denotes a real
interval number.
With these notations, the basic concepts of interval arithmetic
are described below.
is the set of real numbers
An interval number
such that
.
and
are known as the lower
limit and upper limit of the interval number, respectively. It is
0885-8950/$20.00 © 2006 IEEE
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to be noted that and are scalar (real) numbers individually.
A rational number is represented as an interval number
. The addition, subtraction, multiplication, and division of
two interval numbers
and
are defined
as follows [18]:
(1)
(2)
(3)
(4)
where
Fig. 1. Typical radial distribution system.
(5)
The distance between two interval numbers is defined as [18]
(6)
A complex number
, where “ ” is the complex
operator, is defined as a complex interval number if both its real
and imaginary parts are interval numbers. The complex conju. For any two
gate of , denoted as , is given by
complex interval numbers
and
,
the addition, subtraction, multiplication, and division operations
are defined as [18]
(7)
(8)
(9)
(10)
where
and
.
It is to be noted that the expressions in (7)–(10) can be evaluated using the fundamental relationships given in equations
(1)–(4).
An interval vector is a vector whose elements are all interval
numbers, and the elements of a complex interval vector are all
complex interval numbers. Similarly, the elements of an interval
matrix and a complex interval matrix are interval numbers and
complex interval numbers, respectively. The addition, subtraction, and multiplication operations of two complex interval vectors (or matrices) obey the same corresponding rules for addition, subtraction, and multiplication of two complex (non-interval) vectors (or matrices), and the resulting expressions can
be evaluated by using (1)–(4) and (7)–(10).
quently, the complex interval arithmetic has been used to compute the power flow solution.
The basic feeder model used in this paper is same as that
depicted in [3, Fig. 1] and hence is not again shown in this paper.
The impedance matrix of a feeder section between nodes and
is given by [3]
(11)
If any particular phase of this feeder section does not exist,
then the elements in the corresponding row and column of
this matrix would all be zero. Now, when no uncertainties in
the feeder parameters are involved, all the matrix elements
etc. in (11) are fixed complex quantities. However,
in the presence of uncertainties, each of these elements would
be represented by complex interval quantities.
Fig. 1 illustrates a typical radial distribution system with
buses and
branches. For the purpose of
power flow analysis, the voltage of the root node is assumed to be known, and a flat voltage profile (equal to the
voltage of the root node) has been assumed for the initial
voltages of all the other nodes of the network. Thus, in a
per unit system, the voltage of the root node and the initial voltages of the other nodes have been assumed to be
p.u.,
p.u.,
and
p.u. for phases a, b, and c,
respectively. With these initial voltages, the following steps are
executed for iterative solution of the distribution system.
Step 1: At iteration “k,” the three-phase nodal current injections at node are calculated as
III. INTERVAL ARITHMETIC-BASED POWER FLOW ANALYSIS
The basic power flow method used in this paper is essentially the backward/forward sweep algorithm described in [3].
However, to account for the uncertainties, the input parameters
(real and reactive power loads at the buses as well as the feeder
parameters) have been treated as interval numbers, and subse-
(12)
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 3, AUGUST 2006
where, in (12),
is the complex interval current for phase a (b, c) at node corresponding to the th
is the constant, pre-specified,
iteration,
complex interval injected power at phase a (b, c) of node
is the complex interval voltage
th iteration, and
of phase a (b, c) of node at the
is the total complex interval shunt admittance
connected at phase a (b, c) of node .
Step 2: At this step, known as the backward sweep, the currents in each branch are calculated starting from the feeder section in the last layer and progressively moving toward the root
node. Thus, with reference to Fig. 1, at iteration “k,” the complex interval current at branch “L” is calculated as
(13)
where, in (13),
is the complex interval current
for phase a (b, c) flowing through feeder section at the th
is the set of feeder section connected to node
iteration, and
.
Step 3: In this step, known as the forward sweep, starting
from the root node and progressively moving toward the last
layer, the node voltages are updated from the knowledge of the
latest updated voltages of the previous nodes. Thus, with referare
ence to Fig. 1, at iteration “k,” the voltages at the node
calculated from the knowledge of the voltages of the node
as
(14)
where, in (14),
is the complex interval
at the th iteration, and
voltage of phase a (b, c) of node
etc. are the complex interval elements of the
impedance matrix of the feeder section .
Step 4: For each node
, the distance between its
three-phase voltages of present iteration and those of pre, is calculated at
vious iteration, henceforth denoted by
each iteration “k” by the following procedure. For each
phase, e.g., phase a, the voltages
and
are complex interval numbers, and hence, they can be expressed as
and
. Subsequently,
the distance between
and
is calculated as
. Similarly, the quantities
(distance between
and
and
(distance between
and
are also calculated. Finally,
is calculated
as
. If
(“n” is the number of buses in the system) is less than a specified
tolerance limit, the load flow is considered to be converged;
otherwise, the algorithm goes back to step 1.
IV. SIMULATION RESULTS AND DISCUSSION
The proposed algorithm described in the previous section has
been applied to three different three-phase radial test distribu-
tion systems that have been obtained after making certain simplifications in the original IEEE 13-bus, 34-bus, and 123-bus
radial distribution test feeders [19]. The simplifications adopted
in this paper are essentially the same simplifications followed
in [20]. However, for the sake of completeness, these simplifications are mentioned also in this paper as follows.
1) The transformers, voltage regulators. and the switches are
omitted, and hence, the corresponding feeder sections and
nodes are deleted.
2) The distributed load along any feeder section is lumped and
allocated equally between the two terminal nodes of that
particular feeder section.
3) All the spot loads have been assumed to be constant PQ
load and star connected.
With these three modifications in place, the resulting loading
patterns and feeder parameters in any of these three test systems are henceforth termed as “base loading pattern” and “base
feeder parameter,” respectively, of that particular test system.
For all these three test systems, three different cases have been
considered in this paper: 1) uncertainties only in the load parameters, 2) uncertainties only in the feeder parameters, and 3)
uncertainties both in the load and feeder parameters. However,
due to space limitation, the results obtained only in the IEEE
123-bus system are shown in this paper.
A. Uncertainties in Load Parameters Only
In this case, the feeder parameters have been kept fixed at their
corresponding “base feeder parameters,” and uncertainties are
assumed to be present only in the load parameters. As already
mentioned in Section I, the uncertainties in the load parameters have been taken into account by assuming that the loads
are varying over a certain range or interval. For this purpose,
it has been assumed that for each phase of each node, the uncertainties in its real and reactive loading values are limited to
% variation with respect to the corresponding values at the
“base loading pattern.” Thus, for phase a of node , the real
and reactive loads are assumed to vary over the intervals of
KW and
KVAR, respectively, where
and
are the real and reactive power loads,
respectively, of phase a of node at the “base loading pattern.”
and for all
Similarly, for the remaining two phases of node
the three phases of all the other nodes, the intervals of variation
of load demands have been decided.
When the load demands in a system vary within some intervals, the bus voltages and feeder power flows (for all the three
phases) also vary within certain intervals. These intervals have
been calculated by performing load flow study using the algorithm described in Section III. For further reference, this algorithm would henceforth be termed as “interval load flow method
(ILFM).” For this purpose, the voltage error tolerance limit has
been chosen as 0.000001 p.u. in this paper, and the algorithm
took nine iterations to converge.
For comparison purpose, the intervals of variations of bus
voltages and feeder power flows for all the three phases have
also been calculated by repeated power flow simulations
(RPFS). In this method, the load demands for all the three
phases at any bus have been fixed at some arbitrary value
within their corresponding, pre-specified intervals (the intervals
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Fig. 3. Phase-b voltage profile for load uncertainties only. (a) Magnitude.
(b) Angle (degree).
Fig. 2. Phase-a voltage profile for load uncertainties only. (a) Magnitude.
(b) Angle (degree).
already adopted in ILFM). Similarly, the real and reactive load
demands for all the three phases at all the other nodes have
also been specified at some arbitrary values within their corresponding, pre-specified intervals. Thus, a loading pattern other
than the “base loading pattern” has been generated, and normal
load flow solution has been carried out for this loading pattern
by the method of [3]. Similarly, by fixing the load demands at
other arbitrary values within their corresponding pre-specified
intervals, many other loading conditions for the given system
can be generated, and the corresponding load flow solution can
be obtained. Theoretically, by this method, an infinite number
of loading conditions can be generated. As it is not possible to
perform load flow studies for an infinite number of loading patterns, in this paper, a very large number of loading conditions,
in the range of lacs, has been generated, and the corresponding
load flow solutions have been computed. The minimum and
maximum values of the real and imaginary parts of the complex
bus voltages (for all the three phases) can be obtained from the
results of these 1 million load flow solutions, and these constitute the intervals of variations of the bus voltages obtained from
RPFS. Similarly, the minimum and maximum values of the real
and reactive power flow in the feeders can be determined, and
these would constitute the intervals of variations of the feeder
power flows obtained with RPFS. It is to be noted that, even by
increasing the number of operating points further, the intervals
of variations of bus voltages and feeder power flow (real and
reactive) do not change appreciably.
The intervals of variations of the phase voltages obtained by
both the methods are shown graphically in Figs. 2–4. In these
figures, the legends “uin” and “lin” denote the upper and lower
limits of the voltages, respectively, obtained by ILFM. Similarly,
the legends “urp” and “lrp” denote the upper and lower limits of
the voltages, respectively, obtained by RPFS.
From these figures, an interesting observation can be made.
The intervals obtained by RPFS are always contained within the
intervals depicted by ILFM. In other words, the solutions obtained by ILFM contain all the solutions given by RPFS. Thus,
Fig. 4. Phase-c voltage profile for load uncertainties only. (a) Magnitude.
(b) Angle (degree).
ILFM always suggests little wider intervals than RPFS. However, from these figures, it can be observed that the intervals obtained by these two techniques are actually quite close to each
other. In all these results, the differences in the values suggested
by these two methods start after the decimal point only. The
same pattern has also been observed with the interval of variations of feeder power flows (real and reactive), and hence, these
plots are not given in this paper.
B. Uncertainties in Feeder Parameters Only
In this case, the load parameters have been kept fixed at their
respective values corresponding to the “base loading condition,” and uncertainties are assumed to be present only in the
feeder parameters. The uncertainties in the feeder parameters
have been taken into account by assuming that the feeder
parameters are varying over a certain range or interval. For
this purpose, it has been assumed that for each phase of each
feeder, the uncertainties in its parameters values are limited
% variation with respect to the corresponding values
to
w.r.t. its “base feeder parameter.” For example, for phase a of
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Fig. 5. Phase-a voltage profile for feeder parameter uncertainties only. (a) Magnitude. (b) Angle (degree).
Fig. 6. Phase-b voltage profile for feeder parameter uncertainties only.
(a) Magnitude. (b) Angle (degree).
feeder , the resistance (reactance) values are assumed to vary
over intervals of
ohm
ohm),
are the resistance (reactance) of
respectively, where
phase a of feeder corresponding to the “base feeder parameter.” Similarly, for the remaining two phases of feeder l and
for all the three phases of all the other feeders, the intervals
of variation of feeder parameters have been decided. Subsequently, with these interval feeder parameters, ILFM has been
carried out. For this case also, the voltage error tolerance limit
has been chosen as 0.000001 p.u., and the algorithm took nine
iterations to converge. Moreover, following a procedure similar
to that described in part A of this section, a large number of
operating points (in the range of lacs) has been created by
randomly varying the feeder parameters within the intervals
used in ILFM, and RPFS has been carried out for these 1
million operating points.
The results obtained by these two methods are shown in
Figs. 5–7.
These results also affirm the observations of Figs. 2–4, that
is, the intervals obtained by RPFS are always contained in the
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 3, AUGUST 2006
Fig. 7. Phase-c voltage profile for feeder parameter uncertainties only. (a) Magnitude. (b) Angle (degree).
Fig. 8. Phase-a voltage profile for both feeder parameter and load parameter
uncertainties. (a) Magnitude. (b) Angle (degree).
intervals obtained by ILFM, and the intervals obtained by these
two methods are quite close to each other.
C. Uncertainties in Both Feeder Parameter and
Load Parameter
In this case, it is assumed that the uncertainties are present
both in the feeder parameters and load parameters simultaneously. For this purpose, similar to the cases considered in parts
A and B, the uncertainties both in the load parameters and the
%
feeder parameters have been assumed to be limited to
variation with respect to the corresponding values at “base
loading condition” and “base feeder parameter,” respectively.
For a voltage limit tolerance limit of 0.000001 p.u., the ILFM
method took ten iterations to converge in this case. Again, similar to the studies carried out in parts A and B, RPFS has been
carried out for a large number of (in the range of lacs) operating
points, which had been created by randomly varying the load
parameters and the feeder parameters within the intervals used
in ILFM.
The results obtained by these two methods are shown in
Figs. 8–10.
DAS: CONSIDERATION OF INPUT PARAMETER UNCERTAINTIES IN LOAD FLOW SOLUTION
Fig. 9. Phase-b voltage profile for both feeder parameter and load parameter
uncertainties. (a) Magnitude. (b) Angle (degree).
Fig. 10. Phase-c voltage profile for both feeder parameter and load parameter
uncertainties. (a) Magnitude. (b) Angle (degree).
Again, these results affirm the observations of Figs. 2–7, that
is, although the intervals obtained by these two methods are
quite close to each other, the intervals obtained by ILFM always
contain the intervals obtained by RPFS.
The above observation has two important implications for operational and planning studies of a distribution system. First,
while considering uncertainties, ILFM produces the most pessimistic results, although these most pessimistic results are quite
close to the results obtained by RPFS.
Therefore, any strategy adopted for improving the system performance, taken on the basis of the results obtained from ILFM,
would be able to handle all possible scenarios in the distribution
network arising out of the uncertainties in the system parameters. Second, in a planning and design studies, where a large
number of operating conditions need to be considered, ILFM
can be used effectively. As ILFM encompasses all the solutions
of RPFS, in the initial stages (of planning or design studies), use
of ILFM instead of RPFS (for obtaining the outer bounds of all
possible situations) can save a lot of time, effort, and resources.
Although in the final stage, after converging to a particular planning or design strategy, repeated load flow simulations may still
be necessary before reaching the ultimate decision, ILFM can be
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used in the initial stages effectively to save on the time, effort,
and resources required. However, before practical implementation, several related issues need to be considered.
1) In this paper, as mentioned earlier, the transformers have
been omitted. However, their inclusion in this method is
quite straightforward. As any three-phase transformer can
be represented as an equivalent 3 3 impedance matrix,
it can be included in the interval calculation in the same
manner as shown in (14).
2) The developed method can also easily accommodate the
model of distributed generators. In most countries, according to the grid codes, any distributed generator (DG)
is generally required to supply a contracted amount of real
and reactive power to the local distribution grid. Therefore,
for modeling purposes, any such DG can be represented as
a constant PQ source (or as a constant negative PQ load),
and its effect can be included in the interval calculation in
the same manner as in (12) and (13).
3) The proposed interval arithmetic-based method can take
care of any load characteristics. Essentially, in a given time
period, the minimum and maximum power consumed by
any load is governed by its characteristics. As the proposed
method needs only these two extreme (minimum and maximum) values for calculation, it can be said that the proposed method incorporates the load characteristics in the
load flow computation implicitly.
4) In this paper, only 10% variation in the parameters has
been taken for illustration purposes. However, the developed method is generalized enough to be able to handle any
amount of uncertainties that might occur in the distribution
system. Now, the proposed method does not take into account the diversity of the loads. As a result, it can be argued that use of intervals for modeling different loads may
lead to an over-conservative design. Indeed, if the maximum and minimum power consumptions (during the 24-h
period) of every load are utilized in the interval calculation, then the results would surely be over-conservative.
However, if the time window (for determining the maximum and minimum power consumptions) is shortened depending upon the diversity of loads, then the over-conservative nature of the results would reduce to a large extent.
A simple, hypothetical example would probably help to illustrate this point.
Let, in a given power distribution system, the loads be
grouped into three distinct classes, and also let these three
different classes follow distinct load curves (the peak loads
of these three load curves occur at different periods during
the 24-h period). Also assume that the peak loads for these
three classes occur during the periods of 09–12 hours,
13–16 hours, and 17–20 hours, respectively. During the
remaining periods (other than these three peak loading
periods), the maximum and minimum loading of all the
three classes follow more or less the same pattern. Table I
shows the maximum and minimum real power loads (in
KW) of the three classes, depending upon the time periods.
Now, with reference to the above table, if the interval
values of loading are taken covering the entire 24-h period, then, for these three classes, the interval loading
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 3, AUGUST 2006
TABLE I
LOADING IN THE HYPOTHETICAL SYSTEM
values would be [25 500] KW, [30 400] KW, and [40 600]
KW, respectively. Let these three intervals be termed as
“global” intervals. If these three “global” interval quantities are used for load flow calculation, the results indeed
would be over-conservative. However, if four different
time windows are chosen (corresponding to the first
column of Table I), then each row of this table would
give the corresponding values of the interval loading for
the three different classes. As the intervals represented in
each row are quite narrow in comparison with the “global”
intervals, the corresponding interval power flow solutions
would also be quite narrow in comparison with the power
flow solution obtained with the “global” intervals. Of
course, in this case, four different interval load flow solutions need to be obtained, but this is a small price to pay
for reducing the width of the solution intervals.
The above example has been set up to be extremely simple
just to illustrate the main idea. The actual load variations
as well as the load curves in any practical power distribution system would be much more complex (as compared
to the hypothetical case presented in Table I). However,
following the same principle as presented above, the time
windows can be chosen properly to narrow down the intervals of the load variations. By this process, one may
have to compute several interval load flow computations,
but this number (of load flow computations) would surely
be much less than the number of repeated load flow calculations needed to obtain the intervals of variations of different quantities.
5) In the literature, fuzzy logic has been proposed to deal with
the uncertainties, although to date, to the best of the knowledge of the author, no paper has used fuzzy logic to compute the three-phase power flow analysis of the radial distribution system. Now, in fuzzy set, each uncertain variable needs to be defined in terms of a suitable membership function within its corresponding maximum and minimum values. Generally, the choice of the fuzzy membership function is often subjective as there is no concrete algorithm (or guideline) for choosing the membership function. As a result, depending upon the choice of fuzzy membership functions (even within the same maximum and
minimum values), different solutions can be obtained for
the power flow analysis problem. On the other hand, the
proposed interval arithmetic-based technique does not assume any specific variation of the uncertain quantities (it
only works with the maximum and minimum values) and
therefore can be considered as more general method as
compared to fuzzy set theory.
6) In modern power distribution systems, different voltage
control devices, such as voltage regulators, etc., are
placed at strategic locations to improve/control the overall
voltage profile of the system. Therefore, the models of
these voltage control devices also need to be considered
in the interval arithmetic-based method. However, in the
course of our work, it was found that the convergence
property of the basic interval arithmetic-based method
described in this paper deteriorates substantially when the
models of the voltage control devices are incorporated in
the calculation. Therefore, it is felt that advanced interval
calculation techniques need to be used for accommodating
the control device models. We are working on it presently,
and we will report it as soon as we get acceptable results
in a separate paper.
V. CONCLUSION
In this paper, a method for considering the uncertainties of the
input parameters in the power flow solution of three-phase unbalanced radial distribution systems has been presented. Based
on interval arithmetic, the proposed methodology can consider
the uncertainties in both the load demand and the feeder parameters successfully. The solutions obtained from the interval arithmetic-based power flow method encompass all the solutions obtained from repeated load flow simulations. Consequently, any
strategy for system improvement taken on the basis of interval
power flow study would hopefully be effective over all the possible situations. Moreover, in the initial stages of planning and
design studies, the proposed technique can be a useful tool to
save on the time, effort, and resources required.
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Biswarup Das (M’02) received the B.E.E. (Hons.) and M.E. degrees from
Jadavpur University, Calcutta, India, in 1989 and 1991, respectively, and the
Ph.D. degree in electrical engineering from IIT Kanpur, Kanpur, India, in 1998,
with specialization in electric power systems.
Since 1998, he has been with Department of Electrical Engineering, IIT
Roorkee, Roorkee, India, where he is presently an Associate Professor.
His current research interests are in the area of FACTS, distribution automation, distributed generation, and renewable energy sources.