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IEEE TRANSACTIONS ON POWER SYSTEMS 1
Constrained Robust Estimation of Power SystemState Variables and
Transformer Tap Positions
Under Erroneous Zero-InjectionsRobson C. Pires, Member, IEEE,
Lamine Mili, Senior Member, IEEE, and Flavio A. Becon Lemos,
Member, IEEE
AbstractThis paper presents an equally constrained
robustestimator of both the state and the transformer tap positions
ofa power system able to withstand all types of outliers,
includingbad leverage points and erroneous zero-injections. The
statisticalrobustness of the estimator stems from the application
of theSchweppe-type Huber GM estimator (SHGM) while its
numericalrobustness originates from the use of an orthogonal
iterativelyre-weighted least-squares algorithm together with the
Van Loansmethod for processing the equality constraints. The good
per-formance of the new estimator, termed EC-SHGM estimator
forshort, is demonstrated on a small test system and on the
BrazilianSouthern power system with increasing size ranging from
139buses to 1916 buses. It is shown that it exhibits superior
conver-gence properties in all tested cases while the WLS method
maysuffer from numerical instabilities or even divergence
problemswhen large weights are assigned to zero power injections
modelingfalse information.
Index TermsConfident zero bus injection and transformer
tappositions estimates, robust bad data processing of
measurementsand zero power injection.
I. INTRODUCTION
C ONSTRAINED power system state estimation aims atbuilding a
reliable data base for power systems applica-tions such as static
and dynamic security analysis and energymarket pricing system, to
name a few. If the zero injections havetoo large residuals upon
convergence of the state estimation al-gorithm, unacceptable errors
in the power flow solutions mayresult [1], which will invalidate
the contingency analysis andincrease the vulnerability of the power
system to catastrophicfailures.
Manuscript received March 18, 2013; revised June 20, 2013;
accepted July30, 2013. This work was supported in part by the
Brazilian South State Elec-tric Utility (CEEE) under contract
CEEE/2003-No. 9920524 and the NationalScience Foundation under the
prime contract NSF EFRI-0835879, and the Sub-award Agreement
CR-19806-477991. Paper no. TPWRS-00298-2013.R. C. Pires is with the
Power and Energy SystemEngineeringGroupGESis/
ISEE at Federal University of Itajub, Minas Gerais (MG), Brazil
(e-mail:[email protected]).L. Mili is with the Bradley
Department of Electrical and Computer Engi-
neering, Northern Virginia Center, Virginia Tech, Falls Church,
VA 22043 USA(e-mail: [email protected]).F. A. B. Lemos is with the
Department of Electrical Engineering at Federal
University of Rio Grande do SulUFRS, Porto AlegreRio Grande do
Sul(RS), Brazil (e-mail: [email protected]).Color versions of
one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.Digital Object Identifier
10.1109/TPWRS.2013.2284734
In power system state estimation, two methods have beenproposed
to satisfy the equality constraints. These are 1) as-signing large
weights to the zero-injections, which are treatedas
pseudo-measurements, and 2) formulating the problem asa constrained
optimization problem, which is usually solvedvia the Lagrangian
method [2]. Unfortunately, both approachessuffer from convergence
problems in the event where anequality constraint is modeling
incorrect information [3], forexample a wrong status of a load
circuit breaker assumed to beopen whereas it is closed.It turns out
that all the literature dealing with constrained
power system state estimation is assuming perfect zero
injectioninformation when dealing with solution methods.
Specifically,many papers concentrate either on modeling issues [4],
[5] and[6] or on addressing numerical ill-conditioning problems
whentoo large measurement weights are being assigned [7][10]
and[11]. A few papers focus on improving the statistical
robustnessof the constrained state estimation to non-leverage
outliers inthe measurements using statistical tests on the
residuals [2], [4],[12][14] and [15] or on the Lagrangian
multipliers [16], [17].Therefore, all the proposed methods suffer
from convergenceproblems under erroneous zero injection or bad
leverage points.Recall that a leverage point is a measurement whose
projectionon the space spanned by the row vectors of the Jacobian
matrixis distant from the bulk of the measurements projections
[18].By contrast, the equality constrained state estimation
method
proposed here, which is termed hereafter EC-SHGM for short,is
statistically and numerically robust to three types of
outliers,namely vertical outliers, bad leverage points, and
erroneouszero injections. Its statistical robustness stems from the
use ofthe Schweppe-type Huber GM-estimator [18], the so-calledSHGM,
along with a new techniques inspired by [6] to copewith erroneous
injections. As for its numerical robustness, itoriginates from the
use of an orthogonal iteratively reweightedleast-squares (OIRLS)
algorithm using Givens rotations [19]and [20] together with the
application of the generalizedsingular value decomposition method
initiated by Van Loan[21] to deal with the equality constraints
(see the Appendix).In summary, the proposed OIRLS algorithm that
implementsthe EC-SHGM estimator exhibits the following
interestingproperties:1) It implements the Van Loans method [21]
and the orthog-onal transformations described in Golub [22] in a
unifiedmanner;
2) It assigns, during the first iteration steps, weights of
thesame order of magnitudes to both zero-injections and
0885-8950 2013 IEEE
Lamine MiliTypewritten Text
Lamine MiliTypewritten Text
Lamine MiliTypewritten TextVOL. 29, NO. 3, MAY 2014
Lamine MiliTypewritten Text
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2 IEEE TRANSACTIONS ON POWER SYSTEMS
regular measurements. Then upon convergence of thealgorithm, it
forces the residuals of all the zero-injec-tions flagged as correct
data by the EC-SHGM to verysmall values by applying the Van Loans
method. By thisway, ill-conditioning problems of the Jacobian
matrix areprevented while incorrect zero-injections together
witherroneous measurements are being suppressed.
This paper is organized as follows. Section II presents
theEC-SHGM problem formulation. Section III presents com-putational
issues regarding the algorithm implementation. InSection IV,
numerical results are presented and discussed.Finally, conclusions
are outlined in Section V.
II. PROBLEM FORMULATION
In this section, the EC-SHGM estimator of a power systemstate
and transformer tap positions is developed and solvedusing the
OIRLS algorithm for numerical robustness togetherwith the iterative
refinement weighting method to cope witherroneous zero
injections.
A. Equality Constrained State Estimation Problem
For an power system with on-loadtap changer transformers, the
extended vector has
entries. It is related to the -dimensionalvector containing
tele- and pseudo-measurements through
, where is an -dimensional vector-valuednonlinear function and
is an -dimensional vector assumedto contain independent random
variables with zero mean andknown covariance matrix, . The vectoris
estimated by using an EC-SHGM estimator that minimizes
an objective function given by
(1)
subject to
(2)
Here is the Huber function defined as
forfor
(3)
and its first derivative with respect to , , is expressed as
forfor (4)
In the previous equations, is a standardized residual definedas
, where is the th entry of the residualvector, and . The latteris a
weight that is either equal to one up to a given threshold, ,or to
a decreasing function of the squared projection statistic,
, associated with the th-measurement. The reader is re-ferred to
[18] for a description of an efficient procedure for cal-culating
PS. Note that it is the that make the SHGM robustagainst bad
leverage points. Interestingly, they are calculated of-fline from
the Jacobianmatrix assessed at the flat voltage profile;
they need to be updated only if the measurement
configurationchanges or the topology is modified.In (2), is an
-dimensional vector-valued nonlinear
function and is an -dimensional vector containing con-stant real
and reactive power injections to be exactly satisfied(i.e., zero
values for the zero injections). In our approach, theequality
constraints do not remain the same throughout theiterations.
Indeed, an equality constraint is suppressed or notdepending upon
whether it is identified as outliers or not via theEC-SHGM weight
function, Specifically,if the -function is equal to one during the
initial iterations ofthe OIRLS algorithm, the associated constraint
is deemed to bevalid and, therefore, is enforced in the final
iterative step viathe Van Loans method [21], while if it is smaller
than one, theassociated pseudo-measurement will be downweighted.
Thismethod is described next.
B. Unconstrained SE Solution/Outer Loop
In the outer loop, the set of equality constraints are handled
asany other measurements, i.e., the corresponding weights are ofthe
same order of magnitude as those assigned to the measure-ments.
Therefore, it is an unconstrained SHGM-estimator thatis being
solved. It is a root of the necessary condition of opti-matility
derived from (1), , which is expressed as
(5)
The foregoing equation is solved using the iterativelyreweighted
least-squares (IRLS) algorithm because the latteris less prone to
numerical problems than Newtons method[18], [23], [24]. This
algorithm is derived by dividing andmultiplying by in (5) and using
the definition ofyielding
(6)
which, after algebraic simplifications, reduces to
(7)
Now, putting (7) in a matrix form, we get
(8)
where
(9)
Let us replace in (8) by its first Taylor series expansionabout
expressed as
(10)
which gives
(11)
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PIRES et al.: CONSTRAINED ROBUST ESTIMATION OF POWER SYSTEM
STATE VARIABLES 3
or
(12)
where This is equivalent to obtaining thesolution for a linear
system of redundant equations defined as
(13)
where
(14)
and
(15)
Because the weight matrix in (14) and (15) changesthroughout the
iterations, the matrix in (13) must be factoredat each iteration.
This is the price to pay for suppressing outliersduring the updates
of via . The iterationsare stopped when
(16)
where is the infinity norm.In order to gain numerical
robustness, we now apply Givens
rotations as described in [22]. To this end, we redefine the
linearset of equations given by (13) as follows:
(17)
Let us partition (17) into two subsets of equations, one
associ-ated with the measurements and the other one associated
withthe zero-injections flagged as valid, yielding
(18)
Here, and are matrices of dimension and, respectively, while and
are vectors partition of
dimension and , respectively. Also,and are expressed as
(19)
where the parameter is a weight factor to be assigned toand
.
Now, the augmented Jacobian matrix of the OIRLS algorithmis
given by
(20)
which, once factored by applying the Givens rotations
algorithmdescribed in [25] or [26], results in
(21)
The OIRLS outer loop solution is obtained by applying
backsubstitution to the upper part of (21), yielding
(22)
Here, is the unitary upper triangular matrix that stems
fromGivens rotations applied to defined in (20) as proposed by[25].
By contrast, the algorithm proposed in [26] leads to anon-unitary
upper triangular matrix, As for , it is the cor-responding updated
vector in the right-hand-side of (18), oncethe same orthogonal
transformations are applied in (20).In the solution of (13), the
set of non-valid equality con-
straints are identified as those associated with small
diagonalweights of the matrix given in (15), say smaller than
0.099.Note that this outlier identification must be employed over
allthe iterations of the OIRLS outer loop solution while assignedto
the equality constraints is given the same value as the
weightsassigned to the voltage measurements, which is . Bythis way,
only the equality constraints flagged as valid are pro-cessed
through the iterative refinement weighting method that isdescribed
next.
C. Equality Constrained SE Solution/Inner Loop
The iterative refinement weighting method is proposed in [21]for
solvingWLS estimator subject to equality constraints. How-ever, it
is here adapted for solving the EC-SHGM estimatorwhile using Van
Loans algorithm. When using an iterative re-finement to reach a
solution, the augmented Jacobian matrix ex-pressed in (20) does not
require to be refactored. Therefore, onlythe right-hand-side of the
linear system of equations to be solvedin this step is updated.
Consequently, the sequence of Givens ro-tations are stored instead
of the orthogonal matrix .The iterative refinement weighting method
aims at min-
imizing the Euclidean norm of residual vector
partitioncorresponding to the equality constraints. Although the
min-imization process is performed through an iterative scheme,the
algorithm presents fast convergence rate at each
outer-loopiteration. Once (22) is solved while the convergence
criteriongiven by (16) is not satisfied, the inner loop starts to
applythe Van Loans Method[21]. To minimize the
zero-injectiondeviations provided by the iterative refinement
weighting algo-rithm , the value previously assigned to the
parameter shouldbe replaced by another one, since it is lower than
the highestrecommended value, i.e., , where is processoraccuracy
[27]. For instance, in our work is pickedin double precision
calculations. Essentially, this algorithm canbe summarized as
follows:1) After the inner loop counter has been initialized,
i.e.,
, set equal to and update the parameter .2) Check the
convergence criterion given by
(23)
where the tolerance can be the same one used in the outerloop
and where and are the Euclidean and Infinity
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4 IEEE TRANSACTIONS ON POWER SYSTEMS
norms, respectively. If the criterion is satisfied, go to step7.
Otherwise, go to step 3.
3) Calculate4) While applying Givens rotations, solve for
thatsatisfies
(24)
which is the solution of
(25)
where is an orthonormal matrix that stemsfrom the factorization
of the Jacobian matrix defined in(18) as: , and is the residual
vectorrelated to the non-violated equality constraints.
5) Calculate6) Increment inner loop counter: and return tostep
2.
7) Increment outer loop counter: and performnext step.
8) Update as and return to the begin-ning of the iterative outer
loop.
III. COMPUTATIONAL ISSUES
A. Ordering Schemes
When solving the EC-SHGM, the Jacobian matrix has to befactored
at each iteration, except in the inner loop. Techniquesemployed for
compact storage and row and column orderingschemes are essential
for an enhanced computational perfor-mance of Givens method [26].
The row ordering proposedin [28] aims at minimizing the
intermediary fill-in while thecolumn ordering [29] aims at
minimizing the fill-in in theunitary upper triangular matrix, that
is obtained fromthe orthogonal Givens rotations [25]. Additionally,
it is wellproven that the matrix has the same pattern and entries
asthe matrix that stems from the Cholesky decomposition whenboth
matrices result from the WLS solution [24]. This is themain reason
for applying the above ordering schemes on thegain matrix, , so it
can be symbolically factored. Thesestrategies decrease the
computing time during the numericalfactorization of the augmented
Jacobian Matrix [30], as it isshown in (20).
B. Iteration Strategy for Convergence
The iterative process implementing the EC-SHGM estimatorrequires
a starting point that is not too far from a reliable solu-tion. For
instance, if the iterative solution begins from the flatstart
condition, then most of the corresponding residual mag-nitudes may
be greater than the break-even-point, . Thus, therelated
measurements will be strongly downweighted. The rec-ommended
strategy is to perform the first iteration using a WLSestimator,
and then switch to the EC-SHGM estimator. This pro-cedure takes 3
to 5 iterations to attain convergence. However,special care should
be taken with the weights assigned to the
pseudo-tap-position measurements placed on an
on-load-tap-changer (OLTC) transformer whose tap position is
modeled asan unknown parameter. This important issue is discussed
next.
C. Estimating the Tap Position of the OLTC Transformers
The OLTC model implemented in this work follows that pre-sented
in [31]. Regarding the choice of the number and theplacement of the
measurements for successfully estimating thetap positions of OLTC
transformers, the reader is referred to[32]. To enhance the
measurement redundancy and improve theconvergence rate of the
algorithm, phasor measurement units(PMUs) may be used [33]. As for
the iterative procedure, theimplementation of the following
heuristics is recommended:1) Take as an initial value the nominal
tap position for eachOLTC. This procedure allows us to reduce the
magnitudeof the residuals of the associated voltage and active
andreactive power flow measurements;
2) To avoid instability of the iterative process, place
pseudo-measurements on the tap positions to be estimated, set
themequal to the nominal values and pick their standard devia-tions
equal to one-third of the tap range.
IV. NUMERICAL RESULTS
The proposed methodology is implemented in a program de-veloped
in C++ Builder Version 6.0. It makes use of an ob-ject-oriented
framework [34][36]. Besides the demonstrationperformed on an
example system depicted in Fig. 1, the OIRLSalgorithm is executed
on a real-life system of about 150 buses.Also, its performance is
evaluated on the Brazilian Southernpower systems with increasing
size of 340, 730 and 1916 buses.In all the simulations carried out
on these systems, standard de-viations of 0.1% and 1% are
respectively assigned to the volt-ages and the power measurements.
To model the uncertainty inthe measurements, Gaussian errors with
zero mean and givenvariances are added to the corresponding true
values obtainedfrom load flow calculations. Gross measurements and
erroneouszero injections are included in the measurement set by
replacingthe associated good values with large values.
A. EC-SHGM Estimator Applied to an Example System
The one-line diagram of an example system, including
themeasurement configuration identified by means of bulletsplaced
at each measurement point, is shown in Fig. 1, whilethe
corresponding network parameters in pu are presented inTable IIn
this example, two OLTC transformers are included, whose
models are given in [31]. Therefore, there are 9 states
variablesand 2 tap positions to be estimated. There are also four
equalityconstraints associated with zero power injections
(active/reac-tive) on Buses #3 and 4. Two bad data are included in
the mea-surement set. Firstly, a reactive power injection is set to
zero atBus #3 whereas in the field it has a non-zero value, which
re-sults in one topological error. Secondly, a gross error is
addedby changing the sign of the reactive power injection
measure-ment at Bus #5. This case simulates an inversion of the
cur-rent winding polarity following a measurement calibration
pro-cedure.
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PIRES et al.: CONSTRAINED ROBUST ESTIMATION OF POWER SYSTEM
STATE VARIABLES 5
Fig. 1. Online diagram of the 5-bus system.
TABLE IPARAMETER VALUES OF THE 5-BUS SYSTEM
TABLE IIESTIMATION RESULTS FOR THE 5-BUS SYSTEM
Table II presents the estimated variables throughout the
iter-ations, except for the first one where the WLS iterative
solutionbegins from the flat state condition and the tap setting
state vari-able starts from the nominal position. The OIRLS
outer/innerloop counter, viz. , are indicated in the headings of
Table II.Table III presents measured and estimated values along
with
the weighting factors and the chi-squared indices at eachstep of
the iterative process. All underlined measurements areleverage
points. As observed, both bad data, and , which
are bad leverage points, are correctly identified and
suppressed.Regardless whether the reactive power injection at Bus
#3 iswrongly set to zero or the reactive power injection at Bus #5
isincorrectly assigned a negative value, the state estimator
solu-tion points out that there is actually a shunt capacitor
connectedto Bus #3, and that Bus #5 is actually supplying the
networkwith reactive power. These bad data are severely
downweightedduring the iterative solution; they are the main cause
for thelarge number of iterations performed in the inner loop.
Usually,if only true information is modeled as equality
constraints, theOIRLS outer/inner loop algorithms require on
average 3/1 itera-tions to converge. Moreover, the deviations of
the active powerinjections at Buses #3 and 4 and on the reactive
power injectionat Bus #4 are reduced to very small values. These
results showthe effectiveness of EC-SHGM estimator.Note that for
the estimation results shown in Table II, the
tap positions are modeled as unknown variables while thoseshown
in Table III, they are modeled as pseudo-measurements.In both
cases, the standard-deviation assigned to them complieswith
heuristic #1 and 2 and their weights remain unchangedthroughout the
iterations.
B. EC-SHGM Estimator Applied to Power Systems NetworksTable IV
summarizes the monitoring features of large-scale
power system networks used for evaluating the EC-SHGM
es-timator.In this work, a network of about 150 buses is used as a
bench-
mark system. The corresponding historical data are snapshots
ofstored digital and analog measurements taken from the
networkevery minute from January 2003 to July 2007. For instance,
inTable IV the data of this system correspond to the snapshot
takenat 04:00 PM on July 4, 2007. The other 340-, 730-, and
1916-bussystems are used to evaluate the computational performance
ofthe implemented algorithm.The benchmark system consists of two
interconnected asyn-
chronous subsystems belonging to neighboring countries,
onesubsystem stretches in Argentina and Uruguay and operates at50
Hz and the other one stretches in Brazil and operates at 60Hz.
Unlike other applications, power flows over back-to-backDC links
are modeled as equality constraints of power injec-tions because DC
link primary control mode is a constant powercontrol [37].
Specifically, the interconnections between the Ar-gentinian and the
Uruguayan grid and between the Argentinianand the Brazilian grid
are made through a frequency-converterstation with a maximum
scheduled interchanged power around50 MW. As for the
interconnection between the Brazilianand the Uruguayan grids, it is
made through the Livramentofrequency-converter station with a
maximum scheduled inter-changed power around 70 MW. Power
interchanged via theseconverters are used to meet local load
seasonally.The performance of the EC-SHGM estimator was also
evalu-
ated on a real-life power system of about 150 buses of an
indus-trial district located in the city of Porto Alegre in Rio
Grande doSul, a Brazilian Southern State. Fig. 2 displays the
measured andestimated values of the voltage magnitude and real and
reactivepower at Bus #1258 over 288 snapshots recorded every 5
minfrom 00:00 AM to 11:55 PM on July 4, 2007. The infinite normsof
the residual vectors for , , and are 0.01356, 0.00988 and
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6 IEEE TRANSACTIONS ON POWER SYSTEMS
TABLE IIIMETERED VALUES AND ESTIMATES FOR THE 5-BUS SYSTEM
TABLE IVCHARACTERISTICS OF THE BRAZILIAN POWER SYSTEMS
, respectively. All these values are within stan-dard deviation
of the residuals.Table V shows various performance indices of the
OIRLS
algorithm evaluated for the Brazilian 159-, 340-, 730-,
and1916-bus power systems. These indices are the numbers
ofiterations of the outer and inner loop along with the averageand
the maximum values of the active and reactive power
TABLE VPERFORMANCE INDICES FOR THE BRAZILIAN POWER SYSTEMS
injection absolute deviations,and , at selected buses. The bus
where the maximumdeviation has occurred is also indicated. The
largest deviationvalue showed in the table is the reactive power
injection atSTA-ESUL 230-kV bus. It was induced by a gross error
onSTA 525-kV voltage, which is a neighboring bus where anequality
constraint is imposed as zero reactive power injection(AL2 230-kV)
whereas in the field, actually, an equivalentshunt reactor
compensator of 77.1 MVAr (3 25 MVAr) is
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PIRES et al.: CONSTRAINED ROBUST ESTIMATION OF POWER SYSTEM
STATE VARIABLES 7
Fig. 2. Measured (in blue) and estimated (in red) values
recorded over one day of (a) the voltage, (b) the real power
injection, (c) the reactive power injection atBus #1258 of the
power system of the industrial district located in the city of
Porto Alegre/Rio Grande Do Sul.
connected.The main reason for the large number of
iterationsrequired to reach the solution is the existence of a
large numberof gross errors to be suppressed. Note that if an
unconstrainedSHGM is carried out, the convergence is not
attained.
Finally, Table VI presents the computing times of the
OIRLSalgorithm for the four test systems. Each of these times is
aver-aged over different hardware platforms. Also, the table
indicatesoverhead times for up and down allocated memory. As it can
be
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8 IEEE TRANSACTIONS ON POWER SYSTEMS
TABLE VINUMBER OF ITERATIONS OF THE OUTER/INNER LOOP AND
COMPUTING TIMES IN SECONDS
observed, the computing times of the OIRLS algorithm that
im-plements an EC-SHGM estimator are compatible with
real-timeapplications, even for very large systems. Notice that it
may alsobe applied in study mode with additional features that
requiresheavier computing times.
V. CONCLUSION
A reliable orthogonal iterative algorithm for solving a
robustequality-constrained state estimator has been proposed. It
es-timates power system state variables and transformer tap
po-sitions under erroneous zero-power injections. Simulations
onlarge-scale systems showed that the proposed method has
theability to suppress gross errors corrupting both measurementsand
zero injections, be they in position of leverage or not. More-over,
it presents high convergences rates with low number ofiterations
and small computing times. These features can befurther enhanced if
a fast decoupled OIRLS version is imple-mented.
APPENDIXPRINCIPLES OF THE VAN LOANS METHOD
One way to satisfy the equality constraints in power systemstate
estimation is to assign large weights to the zero-injections,which
are treated as pseudo-measurements. An approximatedsolution for
this problem can be reached by solving the fol-lowing WLS
unconstrained problem:
(26)
where and are defined as in (18) and and are expressedas in
(19).Applying a generalized singular value decomposition as
sug-
gested in [21], we get
(27)
where ,and As shown in [27], the exact solutionto the WLS
problem given by (1) and (2) is expressed as
(28)
while the solution using WLS estimator in the problem ex-pressed
in (26) is given by
(29)
and when Obviously, dependson the value assigned to .To get
small errors for zero-power injections, must be as-
signed large values. Following the recommendations made in[26],
in this applications is set around while themeasurement weights are
set around . Note that forother parameter values, numerical
instabilities may occur.
ACKNOWLEDGMENT
Prof. R.C. Pires gratefully acknowledges the contributionsof N.
Dallocchio for his work on the original version of theVDTap program
that provides estimates of both the state vari-ables and the
transformer tap positions of a power system. Thisprogram was
developed as part of Dallocchios Bachelor andMS research work [36]
carried out when he was enrolled in theUNIFEI Electrical
Engineering program at ISEE/GESis.
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Robson C. Pires (S98A99M02) received theB.Sc. degree in 1983,
the M.Sc. degree in 1989,and the D.Sc. degree in 1998, all in
electricalengineering, from Fluminense Federal University(UFF)RJ;
Federal University of Itajub (UNIFEI)MG, and Federal University of
Santa Catarina(UFSC) SC, respectively, all in Brazil.In 1996
(January to July), he did part of his
graduate program at The Bradley Department ofElectrical and
Computer Engineering VTech,Blacksburg, VA, USA. Since 1987 he has
been
with the Power System and Energy Institute (ISEE) at UNIFEI,
where he iscurrently an Associate Professor. His research interests
include electromagneticcompatibility (EMC) issues, analysis and
control of large power systems androbust state estimation network
application.
Lamine Mili (SM90) received the electrical engi-neering diploma
from EPFL, Lausanne, Switzerland,in 1976, and the Ph.D. degree from
the University ofLiege, Liege, Belgium, in 1987.He is a Professor
of electrical and computer
engineering at Virginia Tech, Blacksburg, VA, USA.His research
interests include robust statistics, powersystem dynamics and
control and risk managementof critical infrastructures. He has five
years ofindustrial experience with the electric power utility,STEG,
where he worked as an engineer in the
planning department and the Test and Metering Laboratory from
1976 until1981. He is co-founder and co-editor of the International
Journal of CriticalInfrastructures,
http://www.inderscience.com/jhome.php?jcode=ijcis.
Flavio A. Becon Lemos (S94M01) received theB.Sc. degree in
electrical engineering in 1988 fromthe Federal University of Santa
Maria UFSM, andthe M.Sc. degree in 1994 and the Ph.D. degree in2000
in electrical engineering from the Federal Uni-versity of Santa
Catarina UFSC, respectively.During 19961997, he was a fellow
research
at Brunel University, U.K. He was with the StateElectrical
Utility in Porto Alegre-RS, from 1988 to1992. Currently, he is an
Adjunct Professor in theDepartment of Electrical Engineering at
Federal
University of Rio Grande do Sul UFRS, Porto Alegre, Brazil. His
mainresearch interests include power systems dynamics, voltage
coordinated controland stability, and power system operation.