Power System State Estimation Using Weighted Least Squares (WLS) and Regularized Weighted Least Squares(RWLS) Method

B. Narsing Rao. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 5, (Part - 3) May 2016, pp.01-06
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Power System State Estimation Using Weighted Least Squares
(WLS) and Regularized Weighted Least Squares(RWLS) Method
B. Narsing Rao1
, Raghavendar Inguva2
1
(PG Scholar, Department of Electrical and Electronics Engineering, Teegala Krishna Reddy Engineering
College, Hyderabad, Telangana, India-500097)
2
(Associate Professor, Departent of Electrical and Electronics Engineering, Teegala Krishna Reddy
Engineering College, Hyderabad, Telangana, India-500097)
ABSTRACT
In this paper, a new formulation for power system state estimation is proposed. The formulation is based on
regularized least squares method which uses the principle of Thikonov’s regularization to overcome the
limitations of conventional state estimation methods. In this approach, the mathematical unfeasibility which
results from the lack of measurements in case of ill-posed problems is eliminated. This paper also deals with
comparison of conventional method of state estimation and proposed formulation. A test procedure based n the
variance of the estimated linearized power flows is proposed to identify the observable islands of the system.
The obtained results are compared with the results obtained by conventional WLS method
Keywords – Covariance matrix, observability, power flow variance, regularized weighted least squares,
weighted least squares, Thikonov regularization
I. INTRODUCTION
The aim of state estimation(SE) is to estimate
the state of a power system accurately based on the
various real time information and measurements
available. The earliest model of state estimation was
developed by Schweppe and Wildes [1]. Since then
SE has been modeled into three major functions:
observability analysis and restoration, state
estimation and gross error detection. These three
functions are usually executed separately but are
related to each other and combination of these three
functions determines the operation and control of the
power system.
The observability analysis is basically a
function that deals with the solvability of the
problem of state estimation. It involves the diagnosis
of whether the available set of measurements is
sufficient to estimate the state of the system. Though
the measurement system is planned to ensure system
observability, some unpredictable situations (such as
failure of the system components, malfunctioning,
accidents etc.) may make the system not completely
observable for different time periods there by
resulting in some unobservable parts in the system
[2]. The identification of these unobservable parts
may help in restoration of system observability by
injecting adequate pseudo measurements (restoration
phase). Various numerical and topological methods
have been developed for observability analysis of
power system as seen in [2]-[3] and [4].
After the observability of system is ensured and
solvability is verified, the state estimator provides
the best estimate of system operating conditions.
Most of the SE programs are formulated as an over-
determined system of non-linear equations and then
solved with normal equations as in [1], [5]-[6].
Finally bad data or gross error analysis is
carried out to detect the measurement errors and
remove/correct the gross errors in the measurements.
Several bad data identification methods are based on
calculation of normalized residuals or normalized
Lagranges multipliers [7].
Considering the advances in state estimation so
far, this paper introduces an improvement in
classical WLS method of state estimation. The aim
has been to develop a mathematical formulation of
state estimation regardless of observability
conditions. This methodology contrasts the classical
methods where observability is carried out as an
initial and separate analysis and eventually
identifying missing measurements to restore
observability before estimating the state of the
system. The unfeasibility caused by lack of
measurements is eliminated in this method by using
regularized least squares model [8], ensuring the
method is able to provide state of the system.
Besides, the observability analysis is carried out by
analysis of variances of the estimated parameters.
RESEARCH ARTICLE OPEN ACCESS

In summary, the contributions of the paper are
two fold: 1) To propose a new mathematical
formulation of power system state estimation which
can be applied to both observable and unobservable
power systems and its comparison with conventional
WLS method and 2) identification of un-observable
islands of the power system based on variance
analysis of estimated parameters.
This paper is organized as follows: In section II,
the new regularized state estimation model is
presented. In section III, the procedure to find the
observable islands is addressed. Proposed algorithms
are presented in section IV. Tests and results with 3-
bus DC, 3-bus AC, IEEE 14-bus and IEEE-30 bus
system are presented in section V. Finally in section
VI, conclusions are drawn.
II. REGULARIZED STATE ESTIMATOR
Consider the following measurement model:
z = h(x) + w (1)
where z is an m- vector containing
measurements, x is an n-vector containing the true
state, h(.) is an m-vector of non-linear functions
relating measurements to state, and w is the
measurement error vector.
Assuming that measurement vectors are
independent, the covariance matrix Rz, is a diagonal
matrix with variances (σi
2
) in the ith diagonal
position. m is the number of measurements and n is
the number of state variables. The classical state
estimation using weighted least squares (WLS)
formulation obtains the estimate, which minimizes
the index
J(x) = [z - h(x)]‘W[z - h(x)] (2)
Where, W = Rz
-1
. The estimate can be obtained
only if number, the type and the location are enough
to ensure the system observability.
Suppose that voltage magnitudes and
voltage measurements exist in all buses. These
measurements are denoted by u. In this situation, the
problem becomes feasible meaning that the system
is observable. Additionally, let us separate voltage
measurements (real or pseudo) from rest of
measurements (z) in the following way:
𝑧 =
𝑧
𝑢
, ℎ 𝑥 =
ℎ 𝑥
𝑥
, 𝑊 =
𝑊 0
0 𝑆
and
𝛥𝑧 =
𝑧 − ℎ 𝑥
𝑢 − 𝑥
where S is the diagonal weighting matrix associated
to voltage measurements whose entries are inverse
of the measurement variances. The above non linear
problem can be solved by Gauss-Newton method
which results in following iterative procedure:
𝐻′ 𝑊 𝐻 ∆𝑥 𝑣
= 𝐻′
𝑊 ∆𝑧 𝑥 𝑣
𝑥 𝑣+1
= 𝑥 𝑣
+ ∆𝑥 𝑣
(3)
where,
𝐻 =
𝜕ℎ 𝑥
𝜕𝑥
= 𝐻′
𝐼′ ′
and Jacobian matrix of the available measurements
given by
𝐻 = 𝜕ℎ(𝑥)/ 𝜕𝑥
In is an identity n-matrix. The above equation can be
transformed as:
(𝐻′
𝑊𝐻 + 𝑆)∆𝑥 𝑣
= 𝐻′
𝑊 𝑧 − ℎ 𝑥 𝑣
+ 𝑆(𝑢 − 𝑥 𝑣
)
(4)
This is equivalent to particular form of the multi-
objective non-linear least squares problem in a
weighted sum formulation:
Min J + S||u – x|| (5)
Equation (5) is known as Thikonov
Regularization [8], [9] or simply regualarizes least
squares, which is employed for regularization of ill-
posed problems. The diagonal weighting matrix S is
known as Thikonov factor. By definition, S is non-
singular, which makes the system represented by (4)
always feasible as (H’WH + S) has full rank thereby
making the power system always observable.
Thus proper adjustment of S and W to
minimize equation (5) will help in estimating the
state of an unobservable system. Foe example, if the
weighing factors of the real measurements are
assigned by typical values and variances of pseudo
measurements are considered to be small, thereby
making pseudo measurements with large variances,
solution can always be obtained with proposed
model, although it cannot be ensured that state
estimation at un-observable islands is reliable.
The main objective of the new formulation
is to obtain precision of the estimated state on
observable islands. On the other hand, as the
estimated state of un-observable islands may be
imprecise, thus these islands need to be identified
accordingly which is carried out by the observability
analysis proposed in next section.

III. OBSERVALBILITY ANALYSIS
The above proposed algorithm solves the
unfeasibility of lack of measurements in estimating
the state of the system. But as the model assumes
pseudo measurements with large variances at all
buses without voltage measurements, it poses a new
problem to identify the un-observable islands in the
system.
As it is assumed that power flows with large
variances are injected as pseudo measurements at all
buses without voltage measurements, the estimated
values at these un-observable islands may be
imprecise. This perspective based on the evaluation
of the variances of estimated parameters is used to
identify the un-observable islands of the system.
Here, the main idea is to calculate the calculate
the confidence interval of the estimated power flow.
The confidence interval is a function of the standard
deviation of estimated power flow on the
corresponding branch and is calculated using the
linearized model of power flow on branch k-m by:
𝑃𝑘𝑚 = (𝜃 𝑘 − 𝜃 𝑚 )
1
𝑥 𝑘𝑚
=
1
𝑥 𝑘𝑚
∈ ′ 𝑘𝑚 𝜃 (6)
where xkm is the branch resistance, is the vector
with estimated voltage angles, and km is a vector
with elements 1 and -1 in positions k and m,
respectively. Thus an un-observable branch will
result in high confidence interval as compared with
observable branch.
The covariance matrix (Ө) of is the inverse of
gain matrix (G) given as follows:
Ө = (𝐻′ 𝑝𝜃 𝑊𝑝𝜃 𝐻𝑝𝜃 )−1
or
Ө = 𝐻′
𝑝𝜃 𝑊𝑝𝜃 𝐻𝑝𝜃 + 𝑆𝑝𝜃
−1
(7)
It can be observed that, this covariance matrix is a
full ranked matrix. However, for calculation of
variances of estimated power flows, only few
elements are necessary. Only elements
corresponding to existing branches plus diagonal
elements are required. Therefore these calculations
can be done effectively with sparse inverse matrix
methods as proposed in [16] for bad data processing.
The sparse inverse matrix can be calculated after or
prior to state estimation and all observable and un-
observable branches can be identified.
The variance of (6) is given by
𝜎𝑃 𝑘𝑚
2
=
1
𝑥 𝑘𝑚
2 𝜖 𝑘𝑚
′
Ө∈ 𝑘𝑚
which may be represented in matrix form by
𝜎𝑃 𝑘𝑚
2
=
1
𝑥 𝑘𝑚
2 Ө 𝑘𝑘 + Ө 𝑚𝑚 − 2Ө 𝑘𝑚
(8)
According to (7) and (8), the variances of the
estimated power flows depend on the weighting
factor. Therefore it is important to verify the
behavior of these estimated variances for different
measurements and pseudo measurement variance
values.
IV. ALGORITHMS FOR STATE ESTIMATION
AND OBSERVABILITY ANALYSIS
Algorithm 1 provides the proposed
methodology for identification of observable islands.
Unlike classical approaches, this can be executed
before or after state estimation.
Algorithm 1: Observability Analysis
1. Calculate the covariance matrix Ө,
according to (7);
2. Calculate the power flow variances for
all branches σ2
, according to (8);
3. If
σ2
< 1, then
branch k-m is observable.
4. Else
branch k-m is unobservable.
5. end if
Usually in cases where the network
topology is symmetrical and the branches are
represented by unitary reactances, it is advantageous
to remove the irrelevant injections before classifying
branches to eliminate their impact on results. The
irrelevant injections consist of the branches which
include atleast one adjacent un-observable branch.
Algorithm 2: RWLS State estimation
1. Initialize all bus voltages (v = 1);
2. Separate voltage measurements from
rest of the measurements;
3. Calculate Thikonov factor S;
4. Compute the weighing matrix W;
5. Calculate ∆xv
according to equation
(4);
6. If max | ∆xv
| ≤ ϵ then
Stop. ∆xv
is the estimated state.
7. else
update ∆xv
to xv + 1
= xv
+ ∆xv
;
v = v + 1; Back to step 5.
8. end if

In Algorithm 2, initially the residual (u - x)
of pseudo measurements is set to zero to eliminate
the effect of pseudo- measurements on the estimated
state and also improve the convergence of the
algorithm. However, if irrelevant injections are
eliminated before the state estimation, zeroing the
pseudo-measurement residuals is not needed.
V. TESTS AND RESULTS
In order to verify the proposed formulation,
tests with 3-bus DC, 3-bus AC, IEEE 14-bus and
IEEE 30-bus system have been set up.
A. 3-bus DC System
Fig.1. Three-bus DC system
The three-bus DC system of Fig.1 is used to test
the proposed formulation and compare it with WLS
system. The results obtained are tabulated in table I.
Table I. Comparison of WLS and RWLS with
3-bus DC system.
The obtained results show that both WLS and
RWLS systems produce similar state estimates for
observable system. But in case of unobservable
system, WLS method of SE fails to estimate the state
of system while proposed RWLS formulation
produces reliable estimates.
B. 3-bus AC System
Simple three-bus AC system used for testing is
shown in Fig.2.
Fig.2. Three-bus AC system
The input data considered for testing the Ac system
in Fig.2 is tabulated in Table II
Table II. Measurement data for 3-bus AC system
Results obtained on testing the AC system with
proposed algorithm are tabulated in Table III
Table III. Comparison of WLS and RWLS with 3-
bus AC system
In this case, the system is tested by both
WLS and RWLS for the available measurement
data. Then an error is created in system by injecting
an error measurement at bus 1-2.
Both WLS and RWLS produce almost
same results in case 1. But on injection of error
measurements in the system, WLS method ceases to
estimate the state of the system.
C. IEEE 14-bus System
Fig.3. IEEE 14-bus system

Sample IEEE 14-bus system on which the proposed
algorithm is tested to verify the accuracy in case of
larger systems is shown in Fig.3.
With the available measurement data, the
results obtained with proposed model are tabulated
in Table IV.
Table IV. State estimation of IEEE 14-bus system
A similar approach as specified in case of
3-bus AC system by injecting error measurement has
been carried out on IEEE 14-bus and IEEE 30-bus
system and results asre tabulated in Table IV and V
respectively.
Table V. State Estimation of IEEE 30-bus system
It is observed from the above results that, as
the size of the system increases, the accuracy of
estimation by WLS decreases. On the other hand,
RWLS method of state estimation produces reliable
outputs larger systems for both observable and
unobservable systems.
The test results for Algorithm 1
(observability) for 3-bus DC, 3-bus AC, IEEE 14-
bus and IEEE-30 bus systems are as shown in Fig. 4,
Fig.5, Fig. 6 and Fig. 7 respectively.
Fig.4. Three-bus DC System
Fig.5. Three-bus AC system
From the above graphical representation; it
can be observed that the variance of unobservable
islands of the power system are greater than the
threshold values and are depicted by long bar
graphs. On the other hand; the observable parts of
the system have variances within the threshold and
are almost equal to zero.
VI. CONCLUSION
This paper has presented a new model for power
system state estimation based on Thikonov
regularization and its comparison with conventional
WLS method. This new formulation has the property
of regularizing the non-linear system to linear
system and hence can be applied to both observable

and un-observable systems. This feature in-turn
eliminates the problem of lack of measurement data
which leads to ill-posed problems. A new approach
of observability analysis of a power system based on
variance of estimated power flows has also been
formulated.
Several tests have been conducted and has been
observed that SE with RWLS is far more
advantageous than conventional WLS method of
state estimation as it eliminates the problem of ill-
posed problems, provides accurate and reliable
outputs for both observable and un-observable
systems, and maintains the same range of accuracy
for small and large systems.
A major contribution of the proposed model is,
it simplified the process of state estimation. It also
transforms the problem of observability analysis
from being impediment to calculation to part of
problem analysis, i.e., this method eliminates the
need of observability analysis before state estimation
of system, Now, with this new formulation,
observability analysis is only a part of system
monitoring and may be carried out before of after
state estimation without any impact on the estimated
parameters.
REFERENCES
[1] F.C Schweppe and J. Wildes, “Power
system state estimation, part I: Exact
model,” IEEE Trans. Power App. Syst..,
vol. PAS-89, no.1, pp. 120-125, Jan 1970.
[2] A. Monticelli and F.F. Wu, “Network
Observability: Theory,” IEEE Trans. Power
App. Syst.., vol. PAS-104, no.5, pp. 1042-
1048, May, 1985.
[3] B. Gou, “Jacobian matrix based
observability analysis for state estimation,”
IEEE Trans. Power Syst., vol. 21, no. 1, pp.
348-356, Feb. 2006.
[4] E. Castillo, A. J. Conejo, R. E. Pruneda,
and C. Solares, “Observability analysis in
state estimation: A unified numerical
approach,” IEEE Trans. Power Syst., vol.
21, no. 2, pp. 877-886, May 2006.
[5] A. Garcia, A. Montcelli, and P. Abreu,
“Fast Decoupled state estimation and bad
data processing,” IEEE Trans. Power App.
Syst., vol. 13, no. 3, pp. 1645-1652,
Sep./Oct. 1979.
[6] L. Holten, A. Gjeslswik, S. Aam, F. Wu
and W-H Liu, “Comparison of different
methods of state estimation,” IEEE Trans.
Power Syst., vol. 3, no. 4, pp. 1798-1806,
Nov. 1988.
[7] K. A. Clements and A. Simoes-Costa,
“Topology error identification using
normalized Lagranges multipliers,” IEEE
Trans. Power Syst., vol. 13, no. 2, pp. 347-
353, May 1998.
[8] A. N. Thikonov and V. Arsenin, Solutions
of Ill-Posed Problems(Scripta Series in
Mathematics). Washington, DC: Winston,
1977.
[9] P. C. Hansen, “Rank-deficient and discrete
ill-posed problems: Numerical aspects of
linear inversion,” in SIAM Mononographs
on Mathematical Modeling and
Computations. Philadelphia, PA: SIAM,
1998.
[10] Madson C. de Almeida, Ariovoldo V.
Garcia, Eduardo N. Asada, “Regularized
least squares power system state
estimation,” IEEE Trans. Power Syst., vol.
29, no. 1, Feb 2012.
[11] F. C. Schweppe, D. B. Rom, “Power
system static state estimation, part II
Approximate model,” IEEE Trans. Power
App. Syst., vol. 89, no. 1, Jan 1970.
[12] A. Monticelli, State estimation in electric
power system, A generalized approach
(Kluwer Academic Publishers, 1999).
[13] Ali Abur, Antonio Gomez, Power system
state estimation theory and implementation
(Marcel Dekker New York, 2004)
[14] Hadi Saadat, Power system analysis (TMH
Ltd., New Delhi, 1999).

Power System State Estimation Using Weighted Least Squares (WLS) and Regularized Weighted Least Squares(RWLS) Method

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