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A Robust Regularized Hybrid State Estimator for Power Systems

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This article was downloaded by: [Indian Institute of Technology Kanpur]On: 26 May 2014, At: 02:04Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK
Electric Power Components and Systems
Publication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/uemp20
A Robust Regularized Hybrid State Estimator for PowerSystems
Sanjeev Kumar Mallik
a
, Saikat Chakrabarti
a
& Sri Niwas Singh
aa
Department of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur, IndiaPublished online: 24 Apr 2014.
To cite this article:
Sanjeev Kumar Mallik, Saikat Chakrabarti & Sri Niwas Singh (2014) A Robust Regularized Hybrid StateEstimator for Power Systems, Electric Power Components and Systems, 42:7, 671-681, DOI: 10.1080/15325008.2014.890968
To link to this article:
http://dx.doi.org/10.1080/15325008.2014.890968
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Electric Power Components and Systems
, 42(7):671–681, 2014Copyright
C
Taylor & Francis Group, LLCISSN: 1532-5008 print / 1532-5016 onlineDOI: 10.1080/15325008.2014.890968
A Robust Regularized Hybrid State Estimatorfor Power Systems
Sanjeev Kumar Mallik, Saikat Chakrabarti, and Sri Niwas Singh
Department of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur, India
CONTENTS1. Introduction2. Hybrid State-Estimation Model3. Tikhonov Regularization4. Selection of Regularization Parameter by L-Curve Method5. Regularized Hybrid State-Estimation Algorithm6. Results and Discussion7. ConclusionReferences
Keywords: hybrid state estimation, ill-conditioned matrix, condition number,LU factorization, regularization, L-curve, singular value decompositionReceived 14 February 2013; accepted 19 January 2014Address correspondence to Mr. Sanjeev Mallik, Department of ElectricalEngineering, ACES 105, Indian Institute of Technology Kanpur, Kanpur,208016, India. E-mail: skmallik@iitk.ac.itColor versions of one or more of the ﬁgures in the article can be found onlineat www.tandfonline.com/uemp.
Abstract
—Numerical instability is an inherent issue with power sys-tem state estimation, and much effort has been made to overcomethis issue using numerical techniques to ensure stability. This article proposes a regularization-based method for solving ill-conditioned hybrid state-estimation problems in the presence of equality con-straints. The iterative linear state-estimation problem is solved byusing Tikhonov regularization. The trade-off between the residualnorm and the regularized norm is controlled by the regularization parameter in the regularization method. The regularization method employs an L-curve criterion for optimal selection of the regulariza-tion parameter. At the post-estimation stage, the correction in voltage phasor at the zero-injection bus is applied using Kirchhoff’s currentlaw. The proposed method is analyzed and tested for IEEE 14- and 118-bus test systems, as well as for a 13-bus ill-conditioned system.
1. INTRODUCTION
The objective of a power system state estimator (SE) is to provide the best possible estimate of the system states by pro-cessing the collected set of measurements over a time win-dow. The accuracy and speed of the SE are crucial for systemmonitoring and the control functions performed at the controlcenter.Phasormeasurementunits(PMUs)havethepotentialtorev-olutionize the power system state-estimation process throughtheiruniqueabilitytomeasuresynchronizedphasors.WiththeavailabilityofasufﬁcientnumberofPMUs,theSEproblembe-comesasimplelinearestimation[1]withoutanyiteration.But,duetothehighcostandcommunicationinfrastructurerequire-ments associated with PMUs, the number of PMUs installed in a power system is limited. There have been attempts tocombinePMUmeasurementswithconventionalasynchronousmeasurements from the supervisory control and data acquisi-tion (SCADA) system to improve SE performance. The fu-sion of PMU measurement data into SCADA measurementsmay be sequential or integrated; the sequential SE has been preferred by researchers due to the reusability of the energymanagement system (EMS) software [2]. Another approachis to integrate the PMU data into the SCADA measurements
671
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672
Electric Power Components and Systems, Vol. 42 (2014), No. 7
at the pre-SE stage, and SE software is then applied on bothmeasurements together. This idea leads to the so-called hybrid SE (HSE), which considers both asynchronous conventionalmeasurements and synchronous PMU measurements.The voltage and current phasor measurements around aPMU bus constitute PMU measurement data, whereas con-ventional SCADA measurements are comprised of power in- jection and power ﬂow (both real and reactive) in additionto the current and voltage magnitudes. The HSE processesthe conventional measurements as well as PMU measure-ments along with the network topology information (in theform of breaker/switch status) for estimating the power sys-tem states. The state-estimation problem is formulated as astandard weighted least squares (WLS) problem and is solved iteratively. Numerical stability is a very critical considerationfor the state-estimation problem. Numerical problems are of-ten reﬂected in the ill-conditioning of the gain matrix in acommonly used WLS SE problem. The numerical behavior of the estimation process is affected by the type of measure-ments and their placement, low measurement redundancy, theweights assigned to the measurements, and network parameter information [3–5].A number of works have been reported to resolve the nu-merical stability issues of the SE. Simoes-Costa and Quintana[6] proposed an orthogonal factorization method to overcomethe issue of an ill-conditioned gain matrix. This method, how-ever, reduces the sparsity of the gain matrix, especially for well-conditioned networks. A new SE algorithm based on Pe-ters and Wilkinson method [7] was developed to overcomethe problem of ill-conditioning without losing matrix sparsity.Further, a clear distinction between a well-conditioned and ill-conditioned system is drawn on the basis of the conditionnumber of the gain matrix. A modiﬁed fast decoupled SE and Croutdecomposition-based SEalgorithmwasproposedin[8].The zero-injection measurements, which are known to be oneof the factors causing numerical ill-conditioning, are oftenincluded in the SE after assigning them very high weights.However, this approach may not completely satisfy the power balance equation at the zero-injection bus. As an alternative,many researchers have proposed to treat the zero-injectionmeasurement as an equality constraint [9]. Insertion of theequality constraint into the normal equation for the SE makesthe coefﬁcient matrix indeﬁnite. An equality-constrained SE based on symbolic optimal ordering and signed Cholesky fac-torization was proposed to improve numerical stabilityin [10].The inclusion of PMU currents in the HSE has been a ma- jor challenge for researchers. There are three possible waysto include the current phasor in the SE problem [11]: (1) acurrent phasor in polar form, (2) a current phasor in rectangu-lar form, or (3) a pseudo measurement in the form of either voltage at the neighboring end or pseudo ﬂow measurementderived from the phasor measurement and network parameter information. It has been established that the rectangular formof the current measurement is the best choice in terms of con-vergence characteristics and estimation accuracy. In [12], anew method was presented to include the PMU current asa pseudo voltage phasor measurement adjacent to a PMU bus, with the pseudo measurement handled by an equalityconstraint.The majority of the existing state-estimation methods are based on inversion and decomposition of the gain matrixin the WLS estimation process. A method based on theLevenberg–Marquardt algorithm was discussed in [4] to re-solve ill-conditioning. In the present article, a new methodol-ogy based on regularization is proposed to solve the numericalinstability of an equality constrained HSE, and the regulariza-tion parameter is evaluated using the L-curve method. At the post-estimationstage,thecorrectioncorrespondingtothezeroinjection is applied to suppress the error due to zero-injection bus voltage. The main advantage of the proposed method isthe minimization of the objective function while maintainingaccuracy.This article is organized as follows. In Section 2, the SEmodel and the numerical ill-conditioning problem are illus-trated. In Sections 3 and 4, the Tikhonov regularization for ill-posed linear systems and the methods for ﬁnding the regu-larization parameter are presented. The proposed regularized state-estimation algorithm is discussed in Section 5. The im- plementation of the regularization method in state estimationand test results are presented in Section 6, and Section 7 con-cludes the article.
2. HYBRID STATE-ESTIMATION MODEL
The measurement model for the state estimation is given as
z
=
h
(
x
)
+
e
,
(1)where
z
isthevectorof
m
measurementsconsistingofSCADAmeasurements as well as PMU measurements;
h
(
x
) is the re-lationship between measurement vector
z
and state vector
x
,which consists of voltage magnitudes and phase angles at the buses of size
n
; and
e
is the error vector for the
m
measure-ments. In this article, power ﬂow as well as power injection(both realand reactive) are considered tobe SCADA measure-ments. PMU measurements include the voltage phasors at themonitored buses and current phasors through the monitored lines. PMU voltage phasor measurements are collected and considered in polar forms. PMU current phasor measurementsare in polar form, but they are considered in rectangular form
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Mallik et al.: A Robust Regularized Hybrid State Estimator for Power Systems
673
for SE formulation, and their weights are evaluated by usingclassical theory of propagation of uncertainty [11].The state-estimation problem in the presence of equalityconstraints can be formulated asminimize
J
(
x
)
=
[
z
−
h
(
x
)]
T
W
[
z
−
h
(
x
)] (2)subject to
c
(
x
)
=
0
,
(3)where
W
is the diagonal weight matrix (assuming the mea-surements to be independent), and
c
(
x
) represents the zero-injection power measurements (both real and reactive power injection).Thestate-estimationsolutionisachievedbyiterativelysolv-ing the following system of linear equations:
G C
T
C
0
x
λ
L
=
H
T
W
[
z
−
h
(
x
)]
−
c
(
x
)
,
(4)where
λ
L
isthevectorofLagrangemultipliers,
H
=
∂
h
(
x
)
∂
x
and
C
=
∂
c
(
x
)
∂
x
are the measurement Jacobian matrices,and
G
=
H
T
WH
is the gain matrix corresponding to the WLSSE:
Ey
=
b
,
(5)where
y
=
[
x
λ
L
]
T
, and
b
is the right-hand side of Eq. (4).The coefﬁcient matrix
E
is deﬁned as
E
=
G C
T
C
0
.
(6)Due to severe numerical ill-conditioning of the matrix
E
and the error in the right-hand side vector
b
, the solution of Eq. (5) may not be achieved accurately by simply invertingthe matrix. The operating condition of the coefﬁcient matrixis evaluated by its condition number, which may be deﬁned asfollows [
13
]:Cond(
E
)
= ||
E
||||
E
−
1
||
,
(7)where
·
represents the Euclidian norm throughout this arti-cle. An ill-conditioned matrix has a large condition number.The inversion of a matrix having a large condition number is prone to numerical problems. For example, if the conditionnumber of the matrix
E
is 10
k
, then at least
k
digits of preci-sion are lost while solving the linear system of Eq. (5) [13].The IEEE standard double precision numbers [14] have 16decimal digits of accuracy, and if the solution loses
k
digits of accuracy, the accuracy achieved by the solution of Eq. (5) is(16 –
k
). A numerically robust technique is, therefore, needed to handle the ill-conditioning issues. In this article, a method-ology based on Tikhonov regularization is proposed to handlethe ill-conditioning of the state-estimation problem. The pro- posed methodology and the steps to implement the algorithmare discussed in detail in the following sections.
3. TIKHONOV REGULARIZATION
A substantial amount of research has been done in the ﬁeld of regularization and optimization of inverse ill-posed problems.The least squares method is one of the methods to solve the problem of minimization of
E y
–
b
2
. The least squaresolutionforEq.(5)isalsoknownasapseudo-inverse solution.The output of least squares optimization is affected by the dataerror and rounding error [15]. Regularization methods can beused to damp out these errors in solving the inverse problems.Tikhonov regularization is a widely used method for solvingill-conditioned problems and is described below in the contextof solving Eq. (5) [16]:min
y
Ey
−
b
2
+
λ
2
L
(
y
−
y
0
)
2
,
(8)where
L
is referred to as a regularization operator, and
λ
isa regularization parameter. The common choice of
L
is theidentity matrix
I
. The vector
y
0
is an
a priori
estimate of
y
. Inthe absence of
a priori
information,
y
0
is assumed to be zero.With these assumptions applied in the proposed work, Eq. (8)is simpliﬁed asmin
y
Ey
−
b
2
+
λ
2
y
2
.
(9)The ﬁrst term
Ey
−
b
2
is known as the least square or residual norm, and
y
2
is the regularized or solution norm.From Eq. (9), it is clear that the estimate of
y
depends on theregularization parameter. The un-regularized solution (
λ
=
0)of Eq. (9) corresponds to the pseudo-inverse solution. The pseudo-inverse solution only considers the idea of the curveﬁt,
i.e.
, minimization of the least square norm. It does not con-sider the solution norm,
i.e.
, the accuracy of the estimate of
y
.Tikhonov regularization provides the solution by minimizingthe least square norm as well as the solution norm, and the so-lution is controlled by the regularization parameter. The nextstep is to analyze the effect of regularization on the solution.If too much regularization is imposed on the solution, it willnot ﬁt the given data
b
. On the other hand, if very small regu-larization is imposed, the ﬁt will be good, but the solution will be dominated by the contribution from the data errors. Hence,there is a trade-off between the size of the regularized solu-tion and the quantity of the ﬁt. This trade-off is controlled bythe selection of a proper regularization parameter. Some of thewell-knownmethodsforselectingtheregularizationparameter include
•
the discrepancy principle,
•
the generalized cross-validation (GCV) method, and
•
the L-curve method.
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674
Electric Power Components and Systems, Vol. 42 (2014), No. 7
According to the discrepancy principle, the regularization parameter
λ
is chosen so that the discrepancy is satisﬁed as-suming accurate estimation in
ε
[17]:
b
−
Ey
=
ε,
(10)where
ε
is the norm of the error in
b
. The discrepancy prin-ciple faces difﬁculty in the absence of the information on
ε
.The GCV method is based on cross-validation of each data point, as reported in [18]. The GCV method is not satisfactorywhen the errors are highly correlated. The L-curve method is robust and has the ability to handle perturbations caused by correlation noise. Hence, the L-curve method for selectionof the regularization parameter has gained attention in recentyears and is used in the present work. In the next section, theL-curve method of regularization is discussed in detail.
4. SELECTION OF REGULARIZATIONPARAMETER BY L-CURVE METHOD
An L-curve is a log-log plot between the squared norm of theregularized solution and the squared norm of the residual for a range of values of the regularization parameter. The name isderived from the fact that this plot closely resembles the letter “L.” The following notations are considered:
η
=
y
2
, ρ
=
Ey
−
b
2
,
(11)ˆ
η
=
log
η,
ˆ
ρ
=
log
ρ.
(12)TheL-curveistheplotof ˆ
η/
2versus ˆ
ρ/
2.Withthenotationsexpressed in Eqs. (11) and (12), the solution norm as well asleast square norm depend on
y
, which is the solution of Eq.(9), expressed as
y
=
E
T
E
+
λ
2
I
−
1
E
T
b
.
(13)The sparsity of the matrix (
E
T
E
+
λ
2
I
) used in Eq. (13) isreduced, and the evaluation of
y
is computationally expensive.Hence, a computationally efﬁcient tool is required to simplifythe calculation of two norms associated with
y
. In this article,singular value decomposition (SVD) is used for the evaluationof both the norms.The SVD component of matrix
E
can be expressed as
E
=
ns
i
=
1
u
i
σ
i
v
T i
,
(14)
σ
1
≥
σ
2
≥
σ
3
≥ ··· ≥
σ
ns
≥
0
,
(15)where
u
i
and
v
i
are the left and right singular vectors for thecorrespondingsingularvalue
σ
i
,respectively;
ns
isthenumber of non-zero singular values of
E
. After inserting the value of the SVD component of
E
into Eq. (13), the Tikhonov solutionis given as
y
=
ns
i
=
1
f
i
u
T i
b
σ
i
v
i
,
(16)
f
i
=
σ
2
i
σ
2
i
+
λ
2
,
(17)where
f
1
,
f
2
, ...,
f
ns
are the Tikhonov ﬁlter factors.With this SVD formulation, the solution and residual normcan be expressed as
η
=
y
2
=
ns
i
=
1
f
i
u
T i
b
σ
i
2
,
(18)
ρ
=
Ey
−
b
2
=
ns
i
=
1
(1
−
f
i
)
u
T i
b
2
.
(19)For different values of
λ
, the solution norm as well as leastsquare norm based on Eqs. (18) and (19) may be plotted on alogarithmic scale. A typical L-curve for the regularized state-estimation problem of the IEEE 14-bus system, used for thecase studies having two PMUs at buses 1 and 9, is shownin Figure 1. The indicated values in the plot correspond todifferent values of the regularization parameter.Vector
b
in Eq. (5) can be expressed as
b
=
¯
b
+
e
b
,
(20)
FIGURE1.
L-curveduring2nditerationof550thMonteCarlosample of IEEE 14-bus system for different values of
λ
.
D o w n l o a d e d b y [ I n d i a n I n s t i t u t e o f T e c h n o l o g y K a n p u r ] a t 0 2 : 0 4 2 6 M a y 2 0 1 4

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