338IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005
Improving Voltage Stability by Reactive
Power Reserve Management
Feng Dong, Student Member, IEEE, Badrul H. Chowdhury, Senior Member, IEEE,
Mariesa L. Crow, Senior Member, IEEE, and Levent Acar, Senior Member, IEEE
Abstract—The amount of reactive reserves at generating sta-
tions is a measure of the degree of voltage stability. With this
perspective, an optimized reactive reserve management scheme
based on the optimal power flow is proposed. Detailed models of
generator limiters, such as those for armature and field current
limiting must be considered in order to utilize the maximum
reactive power capability of generators, so as to meet reactive
power demands during voltage emergencies. Participation factors
for each generator in the management scheme are predetermined
based on the voltage-var (V-Q) curve methodology. The Bender’s
decomposition methodology is applied to the reactive reserve man-
agement problem. The resulting effective reserves and the impact
on voltage stability are studied on a reduced Western Electric
Coordinating Council system. Results prove that the proposed
method can improve both static and dynamic voltage stability.
Index Terms—Bender’s decomposition methodology, optimal
power flow, reactive power management, voltage collapse.
by a disruption, but is usually characterized by shortage of
fast-acting reactive reserves. Voltage collapse often involves
specific areas of the power system, although the entire system
may also be involved. Many system variables may participate
in this phenomenon. However, some physical insight into the
nature of voltage collapse may be gained by examining the
production, transmission and consumption of reactive power.
Voltage collapse is associated with reactive power demands
not being met because of limitations on the production and
transmission of reactive power . Reactive power demand
generally increases with a load increase, motors stalling, or a
change in load composition such as an increased proportion
of compressor load. The fast reactive sources are generators,
synchronous condensers and power electronics-based flexible
ac transmissionsystems(FACTS).During voltageemergencies,
reactive resources should activate to boost transmission voltage
levels. This action reduces reactive losses of transmission
lines and transformers, and increases line charging and shunt
capacitor outputs. In the mid-term time scale, a voltage collapse
usually does not happen until most of the large reactive power
OLTAGE collapse typically occurs on power systems
which are heavily stressed. It may or may not be initiated
Manuscript received May 26, 2004. Paper no. TPWRS-00134-2004.
The authors are with the Electrical and Computer Engineering De-
partment, University of Missouri-Rolla, Rolla, MO 65409 USA (e-mail:
Digital Object Identifier 10.1109/TPWRS.2004.841241
resources reach their respective reactive power limits. Gen-
erally, one or two critical resources reaching their limits can
spawn cascading limiting effects at neighboring units. Hence it
is wise to keep enough reserves in order to improve the voltage
stability. The Western Electric Coordinating Council (WECC)
specifies the voltage stability criteria in terms of real and reac-
tive power margins . The minimum reactive power margin is
determined by the voltage-var (V-Q) curve method. The V-Q
method has been well studied –. Reference  discusses a
mands by installation of properly sized and properly located ca-
pacitor banks which will allow generating units to operate at or
near unity power factor. However, it is a cost-intensive proposi-
tion. Besides, this strategy is not always very effective since not
all the shunt capacitors are fully utilized. Reference  uses the
reactive power margins to evaluate voltage instability problems
reserves of generators and static var compensators (SVCs) that
exhaustreservesinthe processof computinga V-Q curve atany
bus in a coherent group or voltage control area. The authors in
 discuss a hierarchical optimization scheme which optimizes
voltage stability margin. Bender’s decomposition is employed
to handle stressed cases. In , the authors introduce a method-
ology to reschedule the reactive injection from generators and
synchronous condensers with the aim of improving the voltage
stability margin. Their method is formulated based on modal
participations factors and an optimal power flow (OPF) wherein
reduced Jacobian, is maximized by reactive rescheduling. How-
ever, theauthors avoidusing a security-constrained OPF formu-
lation and thus the computed voltage stability margin from the
Jacobianwould nottruly represent the situationunder a stressed
In , the authors employ a security-constrained OPF for
optimal var expansion planning design. The OPF is solved
twice—once with simply the voltage profile constraint and then
with a pre-selected voltage stability margin criterion. The larger
of the two solution is then selected for the combined solution.
In this paper, a reactive reserve management program
(RRMP) based on an optimal power flow is proposed to
manage critical reactive power reserves. A multi-objective
function specifically for this purpose is introduced. This
0885-8950/$20.00 © 2005 IEEE
DONG et al.: IMPROVING VOLTAGE STABILITY BY REACTIVE POWER RESERVE MANAGEMENT 339
function is used in a mixed-integer nonlinear optimization
formulation. Various generators are assigned different weights
in order to maintain maximum reactive reserves within the
areas that are most vulnerable to voltage instability problem. A
decomposition technique is adopted to formulate subproblems
with various stress levels. The nonlinear interior point method
(NIP) is then used to solve the separate subproblems. The ad-
vantage in this formulation is that the system is able to maintain
a specified voltage stability margin, and at the same time, raise
its level of voltage stability. The proposed method is tested on
a reduced WECC system with 283 buses and 29 generators.
II. OPTIMIZATION FRAMEWORK
A. Reactive Reserves
The reactive power sources consist of synchronous gener-
ators and shunt capacitors and reactors on the transmission
network. During a disturbance, the real power component
of line loadings does not change significantly, whereas the
reactive power flow can change dramatically. The reason is
that the voltage drops resulting from the contingency decreases
the reactive power generation from line charging and shunt
capacitors, thereby increasing reactive power losses. Sufficient
reactive reserves should be available to meet the var changes
following a disturbance. Simply speaking, the reactive power
reserve is the ability of the generators to support bus voltages
under increased load condition or system disturbances. How
much more reactive power the system can deliver depends on
the operating condition and the location of the reserves, as well
as the nature of the impending change.
The reserves of reactive sources can be considered a measure
of the degree of voltage stability. As a matter of fact, BPA—a
transmission provider in the western United States—considers
reactive reserves at major generating plants to be a critical in-
dicator of voltage stability and, consequently, the agency has
installed a reactive power monitor  at critical hydroelectric
power stations that serve its service area.
The available reactive power reserve of a generator is deter-
mined by its capability curves . It is worth noting that for a
by both armature and field heating limits.
The active and reactive power generation output at a syn-
chronous generator may be represented as 
order to write an analytical model to relate the reactive power
limit to the maximum field current, we use a cylindrical rotor
. Thus, from (1) and (2), the maximum
is the terminal voltage of the generator based on the
reactive power with respect to the field current limit may be
Thus, the maximum reactive power
determined by the maximum field current
tionship also shows that the maximum reactive generation is a
function of the terminal voltage. The maximum reactive power
output should also satisfy the armature current limitation as
of the generator is
. The rela-
The reactive power reserve of the th generator is then repre-
and (4) and
ating conditions. A generator’s reactive reserve is calculated by
is lower than. However, if
the reactive reserve is set to zero and
the terminal voltage.
is the smaller of the two values obtained from (3)
is the reactive power output under normal oper-
reaches its limit,
varies as a function of
B. Mathematical Formulation
The RRMP is formulated as an optimal search problem
whose primary objective is to maximize the effective reactive
reserves under normal operating conditions to ensure higher
voltage stability margin. A secondary objective is to minimize
losses subject to various operational constraints. The dual-ob-
jective RRMP is therefore formulated as follows.
be optimized, that is,
the reactive power reserve of the gth generator;
factor of the th generator and is discussed in the following sec-
transmission losses, we pick
The constraints to the above optimization are
is the sum of reactive reserves of the system to
340 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005
voltage magnitude at load bus ;
voltage angle difference between bus
and bus ;
elements of the admittance matrices;
active/reactive power injected into net-
work at bus ;
total number of transformers, generators,
shunt capacitive/reactive installations,
and loads, respectively;
limits of tap setting of transformer
limits of voltage at generator bus ;
compensation capacity limits at bus ;
limits of voltage at load bus ;
reactive power capacity limits at gen bus
The voltage stability constraints may be written as
is the reactive load for the th stressed case in the
, and is the total number of stressed
is the reactive load under normal condition;
is a direction of load increase;
reactive load under normal condition;
reactive margin. For example,
load at bus
with respect to the system load increase, where
A power system can be stressed from the current operating
point to the collapse point by increasing the reactive loads in
several directions. In some directions, the system may have in-
sufficient reactive power margin to guarantee voltage security.
These cases are grouped into one set, which will be called “C”.
These critical cases are determined in the third step during as-
signment of weight factors of participating generators, as dis-
cussed at the end of Section II. By satisfying constraint (11),
the system has a minimum reactive margin required for secure
operation. Ensuring all voltage control areas to have the min-
imum reactive margin is equivalent to all stressed cases being
feasible under the operation constraints.
The intent in the above formulation is not only to satisfy
the voltage stability criteria but also to improve voltage sta-
bility. Thus, if (11) is satisfied in the optimization, the system
is guaranteed to be voltage stable and optimizing var reserve in
(6) would not be needed if stability was not to be improved.
Without the aforementioned objective function, it is possible
that some strong areas in the system may rob weaker areas of
the much-needed reactive reserves.
is the total
is the specified
is the relative increase in the
C. Determination of Area Weights
Under an ideal operating condition, all generators operate at
unity power factor so the system possesses the maximum reac-
ment such an operating condition in real power systems. Sec-
ondly, given all generators are equally weighted, maximizing
the reactive reserves for the entire system is both inefficient and
inadequate. Attempting to maximize reactive reserves at gen-
erators as an operational optimization measure with equal par-
ticipation throughout a system may lead to undesirable conse-
quences such as depleting reserves at generators which are crit-
It is well known that the voltage instability phenomenon is
essentially a local problem, wherein some weak areas become
prone to voltage collapse because of shortage of reactive power.
These areas need higher reactive power reserves than others
when subjected to the same voltage emergency. In , the au-
thor discusses the importance of dividing the power network
into voltage control areas because each area exhibits vulnera-
bility to a disturbance in a different way. With the same objec-
tive in mind, we have separated the entire network into voltage
control areas with the basic idea being that the weakest trans-
mission element connected to each bus is identified by a scalar
. The parameter is determined such that the V-Q
curve minima in computing the V-Q curve minimum at a bus
are almost identical for all buses in one voltage control area.
The groups of buses that remain interconnected, after the weak
transmission elements are eliminated, define the voltage con-
trol areas. The reactive power-voltage relationship given by the
Jacobian matrix element
is used to measure electrical dis-
tance between any two nodes. The off diagonal Jacobian ma-
connections or long electrical distances between corresponding
buses and are therefore eliminated from each row until the sum
of the elements eliminated is less than or equal to
. The buses corresponding to the remaining
elements in the Jacobian are grouped into one voltage control
area. Then, the V-Q curve at any bus within each control area is
computed to find the reactive margin of this area.
The bottom of the V-Q curve is the maximum reactive load
that a system can sustain. The distance from this point to the
operating point is usually considered the reactive power margin
at the bus . Using this method, we can reliably claim that the
area that has the smallest reactive margin is the most vulnerable
to voltage instability. At the minima of the V-Q curve, the re-
active reserves of generators that are depleted are the effective
in the area. The amount of effective reserves is a key index in
voltage stability assessment. Therefore, an objective of the pro-
posed RRMP is to maximize the effective reactive reserves in
the system. These reserves associated with each voltage control
area are assigned a weight according to
To summarize, the procedure for weight factors is as follows.
1) Divide the network into voltage control areas as specified
is the reactive margin of the area.
DONG et al.: IMPROVING VOLTAGE STABILITY BY REACTIVE POWER RESERVE MANAGEMENT341
2) Determine the reactive margin for each area by the V-Q
3) Determine the weights for each generator based on the
specific margins in each area.
III. STRESSED CASE SUBPROBLEM
By considering the reactive margin constraints of (11), the
constrained optimization problem (6)–(11) becomes one of re-
active reserve management with dynamic security constraints.
At the optimal point, the system has the maximum reactive
reserves under normal operation condition, and each case in
set C is feasible. It can be hierarchically decomposed into two
parts, a base case subproblem and a stressed case subproblem
that can be solved by the Bender’s cuts decomposition method
–. For some stressed cases, the power flow may be-
come infeasible. However, in order to apply a decomposition
methodology, there is a need to have some measure of this in-
feasibility. Since this kind of infeasibility is caused by reactive
power load increase, fictitious reactive injections are used as
slack variables and added to the constraints corresponding to
the reactive power balance equation at a bus. Therefore, the
objective function for the stressed cases is to minimize the sum
of the fictitious injections of reactive power. If the fictitious
injections are equal to zero, then we claim that the stressed case
is feasible. The stressed case subproblem may then be stated
as shown in the equation at the bottom of the page, where
andare fictitious capacitive and inductive power injection,
respectively, at bus . Although, theoretically, fictitious reactive
power can be injected at any bus, since the set “C” of critical
load directions is given, we select only those buses in this
critical set for a load increase. In the above formulation,
the vector of control variables and
is determined by the base
IV. ALGORITHM IMPLEMENTATION
The procedure for the two level Bender’s decomposition
is presented in Fig. 1. The base case subproblem (master
Fig. 1.Two-level hierarchical structure for optimization.
problem) is the optimization problem given by (6). Thus, the
master problem is formulated as follows:
at the second level minimizes the fictitious reactive injections
for the given values of
and returns to the master problem,
information about the optimal objective function and its sensi-
is briefly discussed below.
An initial guess of control variables
the master problem (19)–(22) with
With the given
, the stressed cases are sequentially formulated
and solved, and the optimal result of each stressed case is used
to update the Bender’s cuts. For example, for the th stressed
case subproblem, thevalue of the objective function
means that the system has sufficient reactive margin required
and that the stressed case is feasible. Alternately,
means that the stressed case is infeasible with the current con-
trol variables. Thus, the more accurate Bender’s cuts
is obtained by solving
342IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005
in the master problem based on the op-
are theLagrangemultipliers as-
timal results are built,where
sociated with the th stressed case. The Bender’s cuts return to
the master problem, the information to remediate infeasibility
of the stressed case. The two-stage process is repeated until a
is found for which all
the fictitious injections of reactive power are equal to zero in all
stressed case subproblems.
Each subproblem is solved through a NIP. The foundation
for interior point methods consist of three building blocks:
barrier method for optimization with inequality constraints,
Lagrange’s method for optimization with equality constraints,
and Newton’s method for solving nonlinear equations. The
master problem is taken as an example to show the application
of the NIP. By using slack variables, the problem (19)–(22) is
. That is,
primary nonnegative slack variables used to transform the in-
equality constraints to equalities. The inequality constraints are
incorporated into the objective function as a logarithmic bar-
rier function. So, barrier functions
introduced with respect to (27). The Lagrange function for the
problem (23)–(27) is defined as
is the vector of state variables, andare the
where , and are the Lagrange multipliers, andand
are the barrier parameter. At the optimal point, barrier pa-
and reach zero. The set of nonlinear equa-
tions are derived following optimality conditions and solved
by Newton’s methods. For the interested reader,  and 
present the interior point method in more detail.
The maximum reactive power generation is calculated to de-
termine whether the limit is reached in each iteration during the
subproblem optimal search. If the limit is exceeded, the corre-
sponding generator node is changed to a PQ node, where the
reactive power generation becomes a function of the terminal
voltage as given by (3)–(4). The generator’s reactive reserve is
made equal to zero.
V. CASE STUDY
A. Reduced WECC System
A reduced WECC system, geographically representing the
western United States, is used to test the RRMP. This system,
REACTIVE MARGIN IN AREAS BEFORE AND AFTER OPTIMIZATION
is divided into several subareas administratively. These are BC,
WA, MT, ID, WY, UT, OR, CO, NM, NV, AZ, and CA. The
heavy load area is located in the CA region. Some subareas
such as WA, MT etc. are responsible for supplying power to
CA via long-distance high-voltage lines. Under normal condi-
tions, there are no operating constraint violations. However, a
preliminary analysis shows that there are several weak areas
prone to voltage instability. The system is divided into sev-
eral different voltage control areas using the method discussed
in Section II-C. A bus in each area is selected for conducting
the V-Q analysis. Table I shows the reactive margins in the
nine areas before optimization. It is noted that there are two
weak areas—areas 8 and 9 with insufficient reactive margin,
vis-à-vis 350 and 200 MVAr, respectively. Both areas are in the
Northern CA area. The required reactive margin is specified as
450 MVAr, which is 5% of the total reactive load Thus, using
MVAr in (12), we determine the weight fac-
tors of effective reserves at each generator as shown in Table I.
Assuming a dual objective of maximizing reactive reserves
and minimizing system losses,
and, respectively thereby giving ten times more priority
trol variables include voltage set points of 12 out of 28 gen-
erators, ten ULTCs, and six shunt capacitors. The set “C” in-
cludes five stressed cases. The optimal search starts with the
master problem before consideration of the stressed cases. The
set points are then sent to the subproblems. e.g., one stressed
case is that the total reactive loads in northern CA area are in-
creased by 450 MVAr. The increased reactive demands are pro-
portionally distributed among all loads in this area to each re-
active load under normal condition. The value of the objective
function of this stressed case is 108.3 MVAr, which means that
this stressed case is not feasible. With Bender’s cuts, the base
case is resolved to calculate a set of control variables. After just
one iteration between the master problem and the subproblems,
the overall procedure converges and the new set points ensure
feasibility for all stressed cases.
The specified reactive margin
fect on the results and computation time. An unreasonably high
reactive margin specified may lead to a null solution set for the
optimization problem and necessitate additional reactive com-
margin for a system must be determined carefully. Usually, the
and in (6) are selected as
has a significant ef-
DONG et al.: IMPROVING VOLTAGE STABILITY BY REACTIVE POWER RESERVE MANAGEMENT343
Fig. 2.Voltage profile before and after optimization.
REACTIVE POWER OUTPUT AND VOLTAGE SHCEDULE
SHUNT CAPACTIOR AND ULTC SHCEDULE
NERC-recommended reactive power criteria for area reliability
is a good starting point.
After optimization, the control variables activated include
nine generation voltage settings, two shunt capacitors, and two
ULTCs. In Tables II and III, voltage, shunt capacitors and tap
changer reschedules are listed respectively. In Table II, the
fourth and fifth columns are the reactive power outputs and
voltages at the generator terminal buses after optimization.
Compared with initial conditions, the results show improved
reactive power reserves. The voltage profile of the system is
also improved as shown in Fig. 2 and the transmission losses
are reduced by 45 MW, from 637 MW down to 592 MW.
B. Static Analysis and Dynamic Simulations
In this section, we verify the optimization results from two
aspects—static analysis (V-Q curves) and full dynamic simu-
lations. In Fig. 3, two V-Q curves are plotted. Curve “A” is the
the figure, the reactive margin is 200 MVAr, which is also listed
Fig. 3. V-Q curves at bus 112 before and post optimization.
Fig. 4.Real power margin improvement with RRMP.
in Table I. After running the RRMP optimization, a V-Q anal-
ysis at the same bus reveals that the reactive margin is improved
to 500 MVAr, as shown by curve “B”. Similar conclusions were
drawn by analyzing reactive margins at other voltage control
The improvement in the real power margin after optimization
can be seen in Fig. 4. It shows two PV curves. P is the sum of
344 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005
Fig. 5.Voltage trajectory at bus 118.
condition and curve B corresponds to that after optimization. It
is obvious that the total real power margin has been extended by
3 p.u. or 300 MW.
We now see how the results of the RRMP impacts dynamic
voltage stability. We ran a full time domain simulation of the
system subject to the same disturbances both before and after
OELs installedatGen #118,#148, #138, and#144, 108ULTCs,
as well as ten induction motors in addition to the usual models
associated with synchronous generators (exciters, turbine-gov-
ernors and PSS). In order to reduce simulation computation
time, the OEL time constants were deliberately reduced.
The disturbance consists of a sequence of an outage of Gen
#112 at 10 s, and outages of both circuits of the line 76–82 at
10 and 40 s, respectively. The software used for the dynamic
simulations was EUROSTAG .
In Fig. 5, the voltage trajectories at bus 118 are shown with
respect to the two cases. Curve "b" is with the optimal oper-
ating conditions and curve "a" is with the initial operating con-
ditions. The case with the initial conditions results in a faster
voltage collapse when subjected to the above mentioned distur-
bance. The reasons for the earlier collapse may be summarized
as follows: After the trip-off of the first circuit of the line 76–82
and the generator 112, the OEL at generator 118 is enforced
at 30 s. Hence, this generator loses voltage control. A large in-
duction motor load IND 119, connected to bus 1119—in close
proximity to generator 118, stalls. Then reactive power demand
increases rapidly, which eventually sets the collapse in motion.
In the other case, the OEL at the same generator 118 delays ac-
tivation until 35 s due to a higher system voltage profile. The
system is strong enough to survive from transient voltage in-
stability as well as the ensuing outage of the second circuit of
the line 76–82. After a time delay, ULTCs start to operate to
restore voltages on the load side and the system is stressed fur-
generation from shunt capacitors and transmission lines reduce
drastically. Finally the system collapses at 130 s. The longer pe-
riod until collapse leaves some time for operator intervention
so as to initiate some control strategies to prevent the system
Fig. 6. Reduced WECC System.
have been to block tap changers. By blocking tap changers, the
system is voltage stable, shown as curve "c" of Fig. 5. The com-
parison shows that reactive reserves are beneficial to improve
and 1.0, respectively, thereby ignoring the reactive reserves and
minimizing the losses. The constraints were kept the same, and
the reactive power margin was still 450 MVAr. In this case, the
global optimization problem failed to converge after ten itera-
tions. Two out of five stressed cases were found to be infeasible.
The first infeasible stressed case was one where we increased
reactive power demand in area 8 by 450 MVAr uniformly dis-
tributing the load increase at area buses 113,114, 115, and so
on. The second infeasible stressed case was where we increased
reactive power demand in area 9 by 450 MVAr, uniformly dis-
tributing the load increase at area buses 13, 142, 150, 151, and
At this point, the required reactive margin was decreased
from 450 to 300 MVAr and the global optimization problem
converged. With the reduced reactive margin, power losses
decreased from 637 to 573 MW, which was to be expected.
However, three generators (buses 140, 148, and 159) from out-
side the weak area received additional reactive reserves, while
two generators (buses 116 and 118) in the weak area were left
unchanged. In addition, two additional ULTCs (
) that were outside the weak area were op-
timized and the output of one new shunt capacitor outside the
weak area was reduced. Additionally, the real power margin
was reduced to 120 MW.
in (6) were then selected as 0
DONG et al.: IMPROVING VOLTAGE STABILITY BY REACTIVE POWER RESERVE MANAGEMENT345 Download full-text
This paper discusses the management of dynamic reactive
is based onoptimal powerflow and theBender’s decomposition
technique. The proposed RRMP is decomposed into two parts
giving it a hierarchical structure that gives the optimal problem
troduced into the optimization problem so as to make the model
more accurate for mid-term voltage stability analysis. The ro-
bust interior point method is utilized to solve the problem. De-
tailed generator models were considered in the optimization.
Since synchronous generators contain the most effective dy-
namic reactive reserves, a weighting scheme is designed to se-
lect the set of participating generators that have the most influ-
by V-Q curves. Both static and dynamic voltage stability mar-
gins are improved by managing the reactive reserves at these
are beneficial for maintaining and improving voltage stability.
stems from the fact that when dealing with var optimization,
most optimization procedures described in the literature focus
upon reactive power rescheduling and not reactive reserve
margin optimization. In the realm of mid-term voltage stability,
the reactive reserve margin is extremely important at genera-
tors because it gives an advance indication of how close the
generator might be to its OEL becoming active. Conceivably,
a weak area could become weaker if a generator’s OEL would
operate in a stressed situation because of the simple reason that
its reactive capability would be exhausted. In turn, this would
lead to the area importing reactive power from neighboring
areas, thus creating an onerous situation for a collapse to occur.
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Feng Dong (S’03) received the B.S. and M.S. degrees in electrical engineering
from Hohai University, Nanjing, China, in 1995 and 1999, respectively. He is
currently pursuing the Ph.D. degree in the Department of Electrical and Com-
puter Engineering, University of Missouri-Rolla.
Badrul H. Chowdhury (S’86–M’87–SM’92) received the M.S. and Ph.D. de-
grees in electrical engineering from Virginia Tech, Blacksburg, VA, in 1983 and
He is currently a Professor in the Electrical and Computer Engineering De-
partment, University of Missouri-Rolla.
Mariesa L. Crow (S’83–M’90–SM’94) received the B.S.E.E. degree from the
University of Michigan, Ann Arbor, and the Ph.D. degree from the University
of Illinois at Urbana–Champaign in 1989, both in electrical engineering.
She is currently Associate Dean for Graduate Studies and Research and a
Levent Acar (M’86–SM’88) received the M.S. and Ph.D. degrees in electrical
engineering from The Ohio State University, Columbus, OH, in 1984 and 1988,
He is currently an Associate Professor in the Electrical and Computer Engi-
neering Department, University of Missouri-Rolla.