Measurement 42 (2009) 628–637
Contents lists available at ScienceDirect
Measurement
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / m e a s u r e m e n t
A fuzzy state estimator based on uncertain measurements
A.K. AL-Othman
Electrical Engineering Department, College of Technological Studies, P.O. Box 33198, Alrawda 73452, Kuwait
a r t i c l e
i n f o
a b s t r a c t
Article history:
A fuzzy linear state estimation model is employed, which is based on Tanaka’s fuzzy linear
Received 24 May 2007
regression model, for modeling uncertainty in power system state estimation. The estima-
Received in revised form 27 August 2008
tion process is based on uncertainty measurements as well as uncertainty parametric. The
Accepted 18 October 2008
uncertain measurements and the parameters are expressed as fuzzy numbers with a trian-
Available online 26 October 2008
gular membership function that has middle and spread value reﬂected on the estimated
states. The proposed fuzzy model is formulated as a linear optimization problem, where
the objective is to minimize the sum of the spread of the states, subject to double inequal-
Keywords:
ity constraints on each measurement. Linear programming technique is employed to
Power system state estimation
Fuzzy linear regression
obtain the middle and the symmetric spread for every state variable. The estimated middle
Fuzzy linear state estimator (FLSE)
corresponds to the value of the estimated state, while the symmetric spreads represent the
Measurements uncertainty
tightest uncertainty interval around that estimated states. For illustrative purposes, the
proposed formulation has been applied to various test systems such as, 4-bus, 6-bus, IEEE
30-bus, IEEE 39-bus, IEEE 57-bus and IEEE 118-bus. Furthermore, an assessment of the
time convergence of the proposed method has been carried out to demonstrate the appli-
cability of the proposed estimator as an on-line tool for estimating the uncertainty bounds
in power system state estimation.
Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction
detection of gross errors, detection of invalid topological
information and detection of model parameter errors.
Having an accurate picture of the state of a system is an
If the inaccuracy (or error) in the measurements, for a
important part of the system operations. While a simple
given estimator, is modelled by some random probability
SCADA (Supervisory Control and Data Acquisition) system
distribution function, then the set of feasible estimates
has the ability to provide the system operators with raw
can also be modelled by a probability distribution function.
information about the system operation conditions, only
These estimators are, therefore, probabilistic in nature. In
a state estimator has the ability of ﬁltering the information
fact probability theory is generally utilized to handle inac-
to supply a more accurate picture of the status of the
curacy. Due to the fact that statistics of the measurement
system.
errors are difﬁcult to be probabilistically characterized in
The conventional purpose of state estimation is to re-
practice, imprecision in error modelling cannot be equated
duce the effect of measurement errors by utilizing the
with randomness, [1], and instead can be associated with
redundancy available in the measurement system. In par-
fuzziness [2]. Thus, fuzzy theory can satisfactorily be de-
ticular, the objective is to reduce the variance of the esti-
ployed in such circumstances to overcome this limitation
mates and improve their overall accuracy. The other
and address various uncertainties in the modelling of such
major objectives of state estimation methods include:
statistics. That is particularly due to its ability in handling
uncertainties and vagueness associated with the observa-
tion errors. Generally, in the context of state estimation,
fuzzy estimators are possibilistic in nature. If the observa-
E-mail address: ak.alothman@paaet.edu.kw
tion errors are assumed to be fuzzy due to uncertainty that
0263-2241/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.measurement.2008.10.007

A.K. AL-Othman / Measurement 42 (2009) 628–637
629
is inherently present in the system, then the estimates are
for a separate manipulation of the nominal and uncertain
assumed to be a range of possible values. Consequently, in
part. A Linear Matrix Inequalities (LMI) [8] approach is
such situations, it is desirable to provide not just a single
then used to solve the problem to obtain the upper and
‘optimal’ estimate of each state variable but also an uncer-
lower conﬁdence bounds [9].
tainty range within which we can be assured that the ‘true’
In power system state estimation, inequality con-
state variable must lie. This is attainable by utilizing some
straints have been applied in optimization to deal with
a fuzzy function to represent the estimates as fuzzy esti-
uncertainties. In [10], inequality constraints are employed
mates with their associated uncertainty ranges as opposed
in a LAV estimator for handling uncertainty in pseudo-
to crisp estimates (single point only) produced by the con-
measurements, since they are not measured but are known
ventional estimators [3].
to vary within bounded intervals. An inequality con-
The main theme of this paper is to model the uncertain-
strained LAV estimator based on penalty functions, was
ties associated with the measured quantities in a way that
formulated in [11] to estimate states of external systems.
deﬁnes an interval (range) with respect to their nominal
A parameter-bounding model derived from bounded noise
values. The range is governed by the tolerance, of the mea-
measurements was used in [12] with a reformulated con-
suring instrument (a quantiﬁcation of accuracy usually
strained WLS, to handle unmeasured loads in the system.
provided by the manufacturer) and other factors that are
Al-Othman and Irving have introduced in [13–15] dif-
known to have direct effects on network mathematical
ferent methods for estimating the uncertainty interval
model being used in the estimation procedure. By imple-
around the system state variables. One method is based
menting the proposed fuzzy linear techniques the conﬁ-
on using a two-step method is proposed for estimating
dence interval (or bounds) of the state variables can be
the uncertainty interval around the system state variables.
computed. Hence, this study presents an estimator based
The ﬁrst step uses weighted least-squares (WLS) as a point
on fuzzy linear regression formulation for estimating the
estimator to compute the expected values of the state vari-
uncertainty interval around the system state variables.
ables. A linear programming formulation is then utilized to
This estimator is based on Tanaka’s fuzzy linear regression
ﬁnd the tightest possible upper and lower bounds on these
formulation.
The
uncertainty
is
expressed
in
both
estimates [13]. The linear formulation was, however, lim-
measurements and network parameters in a uniﬁed fuzzy
ited to modelling uncertainty only in the measurements
model. The main objective is to minimize the fuzziness in
which was due to meters inaccuracies, when in fact other
the estimated states. This can be achieved by minimize
elements (inaccuracies of the network mathematical mod-
the sum of spreads of all fuzzy states, subject to double
el) can indeed contribute to the uncertainty. As an exten-
inequality constraints on each measurement to guarantee
sion, authors in [14] have introduced another uncertainty
that the original membership is included in the estimated
analysis method in which the uncertainties are expressed
membership. Linear programming has been employed to
in both measurements and network parameters. The
obtain the middle and the symmetric spread of every state
uncertainties in [14] were assumed to be known and
variable. The estimated middle corresponds to the value of
bounded. The problem is formulated as a constrained
the estimated state, whereas the symmetric spreads in the
non-linear optimization problem. To ﬁnd the tightest pos-
membership functions of the state variables represents
sible upper and lower bounds of any state variable, the
the uncertainty interval around that estimated state. Thus,
problem is solved by Sequential Quadratic Programming
the primary goal is to minimize the sums of the uncertain-
(SQP) techniques. In [15] authors have conducted a com-
ties around the states.
parison study of both methodologies presented in [13,14]
in terms of accuracy in estimating the uncertainty interval
2. Uncertainty and state estimation
with various redundancy levels. The study established that
both methods provided almost identical bounds estimates.
Schweppe [4] introduced the concepts of uncertainty in
Also, the study showed that based on CPU execution time
the general context of engineering analysis, estimation and
analysis WLS-LP was found to be faster than the non-linear
optimization. In [4] the concept of unknown-but-bounded
method.
errors for modelling uncertainty in estimation problems
The main drawback in those formulations was the ma-
was introduced. Measurements are assumed to be inexact
jor computational burden of the process which arises from
and have errors that are unknown but fall within a
the need to perform two (LP) or two (SQP), depending on
bounded range.
the formulation used, solutions for every uncertainty inter-
These concepts have been extended and developed re-
val sought. For example, minimizing a particular state var-
cently and have been applied by a number of researchers.
iable of interest, subject to all the measurement inequality
Bargiela and Hainsworth [5] introduced bounds on the
constraints, provides the lower bound on that state vari-
measurements, with the intention to increase the robust-
able. Likewise, maximizing that state variable, again sub-
ness of estimation. Brdys and Chen [6], developed a tech-
ject to all the measurement inequalities, provides the
nique based on bounded states, and they introduced the
upper bound for that state. Consequently, for real world
term Set Bounded State Estimation (SBSE). Nagar et al.
large electrical networks that scenario introduces a signif-
[7] applied concepts from robust control theory and al-
icant amount of computation and CPU time, which may
lowed for uncertainty in both the parameters and the mea-
ultimately question the practicality of those formulations.
surements. The uncertainty is isolated with the use of a
The proposed fuzzy linear state estimator (FLSE) has an
Linear Fractional Transformation (LFT), which enables the
attractive feature that combats the above drawback. The
preservation of the structure of the uncertainty and allows
proposed (FLSE) computes the interval for all states simul-

630
A.K. AL-Othman / Measurement 42 (2009) 628–637
taneously and directly as it converges to the optimal solu-
Therefore, since Ai = (pi, ci), then Eq. (1) may be rewrit-
tion. Unlike the methods presented in [13–15] where
ten as:
uncertainty intervals is determined by the successive solu-
Y
tion of a series of appropriately formulated linear or non-
$ ¼ f ðxÞ ¼ ðp0; c0Þ þ ðp1; c1Þx1
linear optimization problems.
þ ðp2; c2Þx2 þ . . . þ ðpn; cnÞxn
ð2Þ
Fuzzy theory has also been widely used in power sys-
The output membership function is given as:
tem computation. For example, Shahidehpour et al. in
8
P
[16,17] have utilized Fuzzy theory to handle the uncer-
n

yÀ
p

>
> 1 À
i¼1 i xi
>
P
;
x
tainty in decision making and power purchasing in dereg-
n
i – 0
<
c
l
i¼1 i jxi j
ulated environment. In [18,19] authors have applied fuzzy
ðy
ð3Þ
Y$
iÞ ¼
1;
xi ¼ 0; y
>
i ¼ 0
>
set for multi-area generation scheduling and for optimal
>
: 0;
xi ¼ 0; y
reactive power control, respectively. As for state estima-
i – 0
tion the concept of fuzzy-logic has been employed by
The output membership function is depicted in Fig. 2, and
Shabani et al. in [20] to improve the over all performance
the intermediate mathematical derivations leading to Eq.
of the WLS estimator. A hybrid WLS and fuzzy-logic esti-
(3) can be found in [23,24].
mator was developed in [20] to model residual based on
From regression point of view, Eqs. (1)–(3) may be
possibility theory. Shahidehpour et al., on the other, have
applied to m samples where the output can be either
employed fuzzy sets in conjunction with LAV (Least Abso-
non-fuzzy, (certain or exact), in which no assumption of
lute value) estimator and LMS (Least Medium Squares)
ambiguity is associated with the output or fuzzy (uncer-
estimator to robustly eliminate the bad data in [21]. Fur-
tain), where uncertainty in the output is involved due to
thermore, authors in [22] have developed a fuzzy LAV
human judgment or meters imprecision [25]. In this study
estimator based on maximizing the sum of individual
both non fuzzy and fuzzy out will be considered.
memberships. This fuzzy LAV estimator out performed
the standard WLAV in the presence of leverage point.
3.1. Non-fuzzy output model [23]
3. An overview of Tanaka’s fuzzy linear regression
In this model, Tanaka converted regression model into a
linear programming problem [23]. In this case the objec-
Fuzzy linear regression was introduced by Tanaka et al.
tive is to solve for the best parameters, i.e. A*, such that
[23] in 1982. The general from of Tanaka’s formulation is
the fuzzy output set is associated with a membership value
given by:
greater than h as in;
Y
l ðy
$ ¼ f ðxÞ ¼ A0 þ A1x1 þ A2x2 þ . . . þ Anxn ¼ Ax
ð1Þ
Y
jÞ P h;
j ¼ 1; . . . ; m
ð4Þ
$j
where Y$ is the output (dependant fuzzy variable),
where h 2 [0,1] is the degree of the fuzziness and is nor-
{x1, x2, . . . ,xn} is a non fuzzy set of crisp independent param-
mally deﬁned by the user, Y$ ¼ AÃ x
$ i .
eters and {A0, A1, . . . , An} is a fuzzy set of symmetric mem-
Therefore, with Eq. (4) as a condition, the main objec-
bers, unknowns, needs to be estimated. Each fuzzy
tive is to ﬁnd the fuzzy coefﬁcients that minimize the
element in that set may be represented by a symmetrical
spread of all fuzzy output for all data sets. Note that the
triangular membership function, shown in Fig. 1, deﬁned
fuzziness in the output is due to fuzziness assumed in
by a middle and a spread values, pi and ci, respectively.
the system structure A*. Thus, given non-fuzzy data (yi,xi),
The middle is known as the model value and the spread de-
the fuzzy parameters A* = (p,c) may be solve for by the lin-
notes the fuzziness of that model value.
ear programming formulation as:
μ

μ
1.0
A

1.0

0.5

A
h
i

0

y
n
n
0
a
c x
c x
n
i
ij
i
i
ij
i 1
i 1
=
c
c
=
p x
i
p
i
i ij
i
i 1
=
Fig. 1. Membership function of fuzzy coefﬁcient A.
Fig. 2. Membership function of output.
$

A.K. AL-Othman / Measurement 42 (2009) 628–637
631

!
X
m
X
n
μ
Fnon-fuzzy ¼ min
cixij
ð5Þ
j¼1
i¼1
1.0
Subject to:
X
n
X
n
y
p x
c
0.5
j P
i ij À ð1 À hÞ
ixij
ð6Þ
i¼1
i¼1
h
X
n
X
n
y 6
p x
c
j
i ij þ ð1 À hÞ
ixij
ð7Þ
i¼1
i¼1
0
y
e
In the above formulation y
j
j is the jth observation (con-
stant), x
n
ij is a non fuzzy crisp independent parameter,
c x
p
i
ij
i is the ith fuzzy middle and ci is its corresponding sym-
i 1
=
metric spread (both are variables and need to be
estimated).
Fig. 4. Membership function of fuzzy output.
P
Also, note that in (6) and (7),
n
p x
i¼1 i ij, deﬁnes the mid-
P
dle value and
n
c
i¼1 ixij deﬁnes the symmetric spread to the
left, constraint (6), and to the right, constraint (7), as illus-
Subject to:
trated in Fig. 2. As can be seen from the Fig. 2, as hincreases
the fuzziness of the output increases. This is due to the
X
n
X
n
need of a wider spread, ci, to validate the input measured
y
p x
c
j P
i ij À ð1 À hÞ
ixij þ ð1 À hÞej
ð9Þ
value in condition of satisfying higher h [2].
i¼1
i¼1
X
n
X
n
y 6
p x
c
j
i ij þ ð1 À hÞ
ixij À ð1 À hÞej
ð10Þ
3.2. Fuzzy output model [24]
i¼1
i¼1
Due to human error and various other sources of impre-
Note that an additional term, (1 À h)ej, emerged in the for-
cision in the measurements, the output may certainly be
mulation due to the introduction of fuzziness in the mea-
fuzzy. The uncertainty in the measurements is represented
surements. As mentioned, the Eq. (9) represents the yj
by a fuzzy member as Y
when it lies in the interval to the left of the middle value
$j = (yj, ej), where yj is the middle
value and e
with the uncertainty with respect to it added to that inter-
j represents the uncertainty in measurement j
as shown in Fig. 3.
val. In the same manner, Eq. (10) represents the yj when it
Fig. 4 illustrates the overall membership output func-
lies in the interval to the right of the middle value with the
tion that models uncertainty in the regression parameters
uncertainty with respect to it added to that interval. The
along with the output.
proof and detailed derivation for both formulation can be
The objective of fuzzy linear regression is to determine
found in [23,24].
the fuzzy parameters AÃ that minimze the sum of spread as
$
in:
4. Proposed power system linear fuzzy state estimation

!
X
m
X
n
Ffuzzy-output ¼ min
cixij
ð8Þ
For a set of measurement equations the well-known
j¼1
i¼1
state estimation model is:
z ¼ HðxÞ þ e
ð11Þ
where:
z is the (mx1) measurement vector.
μ
H is a vector of non-linear functions that relate the
states to the measurements.
x is an (n x1) state vector to be estimated.
e is an (mx1) measurement error vector.
The measurements are usually obtained from transduc-
ers in the electrical network. For the system to be
observable, it is necessary that m P n and that the m mea-
surements are in locations such that the resulting Jacobian
(sensitivity matrix with respect to the state variables) has
rank n.
0
For a given set of measurements, where m 1 n, x can not
y
e
e
be exactly determined, instead, x can be estimated and it is
j
y
j
j
denoted as ^
x. Eq. (11) is linearized around some operating
point xo using Taylor series expansion, retaining the ﬁrst
Fig. 3. Membership function of fuzzy output.

632
A.K. AL-Othman / Measurement 42 (2009) 628–637
two terms and ignoring the higher order terms. This leads
where h is the degree of the fuzziness and is speciﬁed by
to the following relationship:
the decision maker. In the context of power system state
estimation e
Dz ¼ JðxoÞ Á Dx þ e
ð12Þ
i
may represent the transducer tolerance
which is usually provided by the manufacturer of the me-
where:
ter it self. Both models are linear programming models and
Dz is = z À J(xo).
they can be solved by any linear programming package.
J is the Jacobian of H(xo), i.e. J = @H(xo)/ox
Repeated linearization and solution of (11) then solves
Dx is ¼ ^
x À xo.
the non-linear problem via the Newton–Raphson ap-
The Newton–Raphson method is employed as an itera-
proach. The solution of the power system state estimation
tive method, since it is known that power system models
in equation by the proposed fuzzy linear formulation can
are amenable to solution using the N-R approach. The
be explained as:
dependence on the iteration index is implicitly assumed
Suppose that at iterations k, the state variable is up-
for Dx, J and Dz, where the current state vector is updated
dated by
at each iteration until a stopping criterion is reached.
^
x
The linearized power system in (12) for the Jth mea-
kþ1 ¼ ^
xk þ Dxk
ð22Þ
surement can be rewritten as:
where the incremental change in state variable Dxk, is
Dz
computed by either fuzzy linear models above, Eqs. (16)–
j ¼ Dx1Jj1 þ Dx2Jj2 þ . . . þ DxnJjn
ð13Þ
(18) or, Eqs. (19)–(21), and it can be expressed
If we deﬁne the change in the system state variables, Dx, to
be a fuzzy member having a middle and a spread values, p
Dx
ð23Þ
i
k ¼ ½p1; p2; . . . ; pnTk
and ci, respectively. Then, Eq. (13) can be expressed as:
where pi correspond to the middle value of the incremental
DZj ¼ ðp
change of the system state variables, i.e. (voltage magni-
1; c1ÞJj1 þ ðp2; c2ÞJj2 þ . . . þ ðpn; cnÞJjn
ð14Þ
tudes and phase angles) at the at iterations k.
Note that the modal value pi (i.e. the middle) for a given
Since the optimal spreads represent a quantiﬁed mea-
unknown, represents the value of the change in the system
sure of how uncertain we are about their respective mid-
state variables, Dxi, at the current iteration of the linear-
dles i.e. state variables, and then the interval of
ized model. The spread ci on the other hand, which is sym-
conﬁdence due to uncertainty can be constructed by add-
metric, corresponds to the incremental conﬁdence interval
ing or subtracting the spreads to or from their respective
of that state variable. Therefore, Dx can be deﬁned:
middles. For instance, the lower bound of the incremental
Dx ½ðp
changes at iterations k can be calculated as:
1; c1Þ; ðp2; c2Þ; . . . ðpn; cnÞ
ð15Þ
Tanaka’s fuzzy linear regression models are modiﬁed in or-
DxÀ ¼ Dx
ð24Þ
k
k À ½c1; c2; . . . ; cnT
k
der to be used as state estimator instead. In those linear
And likewise, the upper bound of the incremental changes
fuzzy formulations, the optimal state estimate vector ^
x
at iterations k can be calculated as:
may be determined by minimizing the sum of the spread
of all state variables. In this case the change in state vari-
Dxþ ¼ Dx
ð25Þ
k
k þ ½c1; c2; . . . ; cnT
k
ables, subject to a number of constraint representing mea-
surements can be expressed as:
Ultimately, the lower bound of the interval at iterations k

!
of all states can be computed:
X
m
X
n
F
^
xÀ
non-fuzzy ¼ min
ci J
¼ ^x
ð26Þ
ij
ð16Þ
kþ1
k þ DxÀ
k
j¼1
i¼1
And the similarly, the upper bound of that interval is com-
Subject to:
puted as:
X
n
X
n
y
p J
c
^
xþ
¼ ^xk þ Dxþ
ð27Þ
j P
i ij À ð1 À hÞ
i Jij
ð17Þ
kþ1
k
i¼1
i¼1
Upon choosing an appropriate initial guess xo, an arbitrary
X
n
X
n
y 6
p J
c
initial guess of considered state variables, N-R should iter-
j
i ij þ ð1 À hÞ
i Jij
ð18Þ
i¼1
i¼1
ate until the stopping criterion is reached. Thus the non-
linear problem is solved and eventually not only the states
Similarly, the fuzzy output model may be given as:
are computed by the fuzzy linear estimator but also, an

!
X
m
X
n
uncertainty range of the state variables (voltage magni-
F
tudes and phase angles) are contracted within which we
fuzzy-output ¼ min
ci Jij
ð19Þ
j¼1
i¼1
can be assured that the ‘‘true” state may lie with high
conﬁdence.
Subject to:
It is important to mention that the fundamental concept
of power system state estimation is to determine the esti-
X
n
X
n
mated ^
x which best ﬁts the redundant set of measurements
y
p J
c
j P
i ij À ð1 À hÞ
i Jij þ ð1 À hÞej
ð20Þ
z. The proposed fuzzy formulation provides the set of esti-
i¼1
i¼1
X
n
X
n
mates ^
x (middle values) along with an upper bound of ^
xþ
y 6
p J
c
and lower bound ^
xÀ for the estimated middle values.
j
i ij þ ð1 À hÞ
i Jij À ð1 À hÞej
ð21Þ
i¼1
i¼1
Determination of estimated middle values is extremely

A.K. AL-Othman / Measurement 42 (2009) 628–637
633
crucial. Upper and lower bounds (which are computed
adopted formulation by the ei coefﬁcient. These coefﬁ-
with the help of the spread) are added features that gener-
cients correspond to the overall accuracy of the meter,
ally offer two relevant indications:
(such as ±3%), and can usually be provided by the manufac-
turer. Nonetheless, different values for the elements of po-
An extended conﬁdence in the particular system states
sitive and negative tolerances are permissible so that a
estimates.
transducer can be speciﬁed to have asymmetric accuracy
A possible violation of some operation limits or system
if required (e.g. an accuracy of À3% to +5% of the nominal
closeness to dangerous states, i.e. in case when the mid-
value) [33]. In fact, in all test cases the meters accuracies
dle estimate looks acceptable, but the spread is already
were obtained by generating normally distributed values
approaching the system limits.
multiplied by symmetric meter tolerances and are there-
fore approximately modelling unknown uncertainty in a
given reading of measurement, but it is bounded between
4.1. Implementation of case studies
+ or À the value of the meter tolerance.
This assumption corresponds to real-life situation
This section presents some typical results obtained by
where acquired measurement values are not exact but
applying the proposed algorithms to the 4-bus system
are contained within the range speciﬁed by the accuracy
from [26], 6-bus test system from [27], IEEE 30-bus, IEEE
of meters. It is important to mention that the transducer
39-bus, IEEE 57-bus and IEEE 118-bus test network data
tolerances ei are assumed to be known and ﬁxed. In re-
from [28]. A set of MATLABTM ﬁles has been developed to
alty the instrument inaccuracies will increase as the
facilitate the computation of all state variables to illustrate
instruments age under the action of various processes
the concepts. The LP problems have been solved by the
and as the instruments may not be recalibrated. It
function linprog() incorporated in the MATLABTM optimiza-
should be noted that measurement recalibration is rarely
tion toolbox [29].
carried out in a systematic manner by utilities [34,35],
Selected measurements, (i.e. active and reactive power
mainly due to the fact that large numbers of measure-
injections, active and reactive power ﬂow and current
ments exist in a power network and the time and exper-
magnitudes with redundancy levels %1.8 to 2.2), have been
tise required to check each individual transducer would
acquired from load-ﬂow solution for all test cases. To sim-
be expensive.
ulate parametric uncertainty, elements of the admittance
matrix have been perturbed by adding uniformly distrib-
4.2. Application of FLSE on 4-bus test system
uted random values to the nominal values of those ele-
ments over an interval (for example [À1%, 1%]) and
Table 1 present typical results obtained by the proposed
therefore, approximately representing typical inaccuracies
FLSE, when applied to the 4-bus network from [26] and
related to the acquisition and computation of network
shown in Fig. 5. The transducer tolerance is assumed to
transmission lines resistances R, reactances X and the total
line charging values B (susceptance). As a matter of fact
variation or, (ambiguity), in the network parameters is
mainly a function of line loading and other factors like
ambient temperature and wind speed [30–32]. While mea-
surements used for in all the test cases were acquired from
base case load-ﬂow, a 1–5% uncertainty in the parameters
seems to be appropriate for the small and medium size test
cases. A relatively larger uncertainty range, i.e. 7–10%, is
acceptable for the IEEE 57 & IEEE118 test cases due to
the increase in transmission lines and, therefore leading
to an increase in the overall degree of the parameters
uncertainty. Hence, all implementation based on those
ranges of parameters uncertainty have been carried out
and presented.
The fuzziness in the output (uncertainty in the mea-
surements) due to meters inaccuracies is modelled in the
Fig. 5. Single-line diagram of 4-bus system.
Table 1
Estimated state variables and uncertainty bounds for the four-bus network with h = 0.5.
Busi
WLS
Fuzzy-LPÀ
Fuzzy-LP middle
Fuzzy-LP+
jVij
hi
jVij
hi
jVij
hi
jVij
hi
1
0.9666
0
0.9557
0
0.9666
0
0.9776
0
2
0.9476
À0.9349
0.9362
À0.9503
0.9476
À0.9349
0.9591
À0.9196
3
0.9354
À1.9639
0.9244
À1.9951
0.9354
À1.9639
0.9464
À1.9326
4
0.9870
1.7687
0.9761
1.6823
0.9870
1.7687
0.9978
1.8550

634
A.K. AL-Othman / Measurement 42 (2009) 628–637
be of ±3% of nominal values, while parametric uncertain-
assumed to be of ±3% of nominal values, while parametric
ties are bounded by ±1%. For comparison purposes, the
uncertainties are bounded by ±1%. The algorithm con-
Weighted Least Squares (WLS) estimates have been ob-
verged in 3 iterations, with execution time 0.6474 s see
tained for the perturbed set of measurements. Note that
Table 4.
the estimated center points of the interval, (Fuzzy-LP Mid-
dle), obtained by the FLSE are identical to those obtained
4.5. Application of FLSE on IEEE 39-bus test system
by WLS estimates. This particular outcome is expected
since the FLSE aims to minimize the sum of spreads of all
The proposed FLSE has been applied to the IEEE39-bus
fuzzy parameters and states which is analogous to the least
network, from [28], where the transducer tolerance is as-
squares criterion [2]. It must be stressed that no general-
sumed to be of ±5% of nominal values, while parametric
ization may be made based upon this outcome, particularly
uncertainties
are
bounded
by
±3%.
The
algorithm
if WLS assumptions (normality and homogeneity of error
converged in 3 iterations with execution time 0.7111 s,
terms) are violated [2].
see Table 4.
As for the uncertainty interval, the estimated upper and
lower bound are shown in Table 1, where the estimated cen-
4.6. Application of FLSE to IEEE 57-bus and 118-bus test
ter points appear to be fuzzy, (uncertain) or non crisp (a crisp
systems
estimate occurs when its corresponding spread or width is
0), since during convergence a spread was produced and
The proposed Fuzzy LP algorithm has been applied on
therefore an upper bound and a lower bound have been
the IEEE57-bus and IEEE118-bus systems. The CPU time
eventually produced by Eqs. (24–27). In this particular test,
as well as the number of iterations required for conver-
if the supplied set of measurements is noise-free, the FLSE
gence of the IEEE57-bus and IEEE118-bus systems is
spread will deﬁnitely be zero indicating no uncertainty in
shown in Table 4.
the estimates. It is also apparent that the estimated center
Note that in this study it is found that the degree of
points lie exactly in the middle of the conﬁdence interval.
fuzziness h seems to have no signiﬁcant effect on the com-
This particular outcome is expected since a symmetric
putation of spreads ci, which appears to be rather counter
spread was adopted by FLSE to model the uncertainties.
intuitive. One reason is due to the fact that having to esti-
The FLSE has been found to perform reliably, with con-
mate incremental changes of state variable (in the linear-
vergence occurring in 4 iterations. This is consistent with
ized domain) that are relatively very small. Had there
the behaviour of the Newton Raphson process in solving
been any change in the value of the degree of fuzziness h
other types of power system state estimation problems. Fur-
prior the estimation at any given iteration, this would yield
thermore, the execution time was found to be 0.4490 s, see
a very small change in the values of spreads ci, (to the order
Table 4. With the same initial guess the state estimation
of 10-5). This small change is really insigniﬁcant and is
problem was solved for the 4-bus system by (WLS). Table 4
likely to be trivial in the computation of the ﬁnal incre-
shows that WLS required 5 iterations to converge (with
mental changes of the spreads ci.
the same tolerance of 10À7) and considerably less CPU time.
4.7. Discussion and results analysis
4.3. Application of FLSE on 6-bus test system
Based on the time performance shown in the previous
Table 2 shows the fuzzy state estimates for the 6-bus
section in Table 4, the proposed fuzzy LP estimator was
network from [27] and shown in Fig. 6, where the trans-
found to converge in either one less or an equal number
ducer tolerance is assumed to be of ±3% of nominal values,
of the conventional WLS. On the other hand, the CPU exe-
while parametric uncertainties are bounded by ±1%. The
cution time of the fuzzy LP estimator required for conver-
algorithm converged in 3 iterations, with execution time
gence is relatively higher that WLS estimator. This slightly
0.4106 s, see Table 4.
more CPU time of the fuzzy LP may be attributed to having
to solve a constrained state estimation linear programming
4.4. Application of FLSE on IEEE 30-bus test system
problem, where each measurement considered in the esti-
mation process is represented by two constraints in the
Table 3 shows the fuzzy state estimates for the IEEE30-
fuzzy domain. This in turn leads to the construction of
bus network from [28], where the transducer tolerance is
2m constraints leading to a slightly more computational
Table 2
Estimated state variables and uncertainty bounds for the six-bus network with h = 0.5.
Busi
WLS
Fuzzy-LPÀ
Fuzzy-LP middle
Fuzzy-LP+
jVij
hi
jVij
hi
jVij
hi
jVij
hi
1
0.9922
0
0.9735
0
0.9922
0
1.0110
0
2
0.9901
À3.8625
0.9706
À3.9269
0.9901
À3.8625
1.0096
À3.7980
3
1.0120
À4.5438
0.9930
À4.6370
1.0120
À4.5438
1.0309
À4.4507
4
0.9258
À4.5463
0.9050
À4.6692
0.9258
À4.5463
0.9466
À4.4233
5
0.9217
À5.7330
0.9009
À5.8938
0.9217
À5.7330
0.9426
À5.5722
6
0.9383
À6.4765
0.9165
À6.6631
0.9383
À6.4765
0.9600
À6.2898

A.K. AL-Othman / Measurement 42 (2009) 628–637
635
Fig. 6. Single-line diagram of 6-bus system.
effort than the conventional WLS estimator. Nonetheless,
the installation or upgrading of an online state estimator.
looking at the CPU time required for execution of test sys-
In addition with the introduction of parametric variation
tems considered in this study, it can be said that the
in the formulation, a more realistic and accurate uncer-
proposed fuzzy estimator posses no signiﬁcant computa-
tainty range is attainable now about the different system
tional burden. Furthermore, for improved computational
quantities.
efﬁciency, particularly for large actual systems, the dual
Some other advantages of FLSE are (as apposed to other
formulation may be employed [36]. Tanaka et al., in [24],
conventional estimators):
derived the dual formulation of their primal formulation
presented in Eqs. 7,8,9 or 12,13,14, where the number of
It has the ability to provide ‘‘interval” estimation rather
equations is related to the number variables, n, as opposed
than ‘‘point” estimation.
to the number of constraints, i.e. 2m.
Execution time is reasonable, which makes the pro-
posed FLSE efﬁcient and applicable to large electric
4.8. Advantages and practicalities
networks.
The FLSE is able to provide direct estimates along with
The availability of the upper and lower bounds on state
their bound without having to relay on any other esti-
estimates can have practical advantages for the power sys-
mator i.e. WLS or LAV as an intermediate stage.
tem operator. For critical quantities, such as a power ﬂow
The FLSE is more suitable for uncertainty modelling and
which is close to its thermal, stability or contractual limit,
analysis due to its possibilistic nature.
the operator can gain conﬁdence that the true value is not
Parametric uncertainty may be introduced and modeled
exceeding the constraint provided that the state estimate
along with the measurement uncertainty in a combined
and both bounds are all within the limit. The uncertainty
framework. As a consequence, computation of more
range on the estimate also gives a useful indication of the
realistic analysis of the bounds is possible.
quality of the metering conﬁguration for the relevant part
of the power system. For example, where a voltage level of-
However, the FLSE may have the following disadantages:
ten has a wide estimated uncertainty range, this would
suggest that the metering in that area is insufﬁcient. This
Linearization is needed. Therefore, the construction of
type of additional information could be very useful during
Jacobian is required in every iteration of N-R.

636
A.K. AL-Othman / Measurement 42 (2009) 628–637
Table 3
Estimated state variables and uncertainty bounds for the 30-bus network with h = 0.5.
Busi
WLS
Fuzzy-LPÀ
Fuzzy-LP middle
Fuzzy-LP+
jVij
hi
jVij
hi
jVij
hi
jVij
hi
1
1.0696
0
1.0405
0
1.0696
0
1.0986
0
2
1.0719
À0.0943
1.0422
À0.1871
1.0719
À0.0943
1.1017
À0.0014
3
1.0601
À0.9809
1.0288
À1.1479
1.0601
À0.9809
1.0914
À0.8139
4
1.0567
À1.2286
1.0255
À1.4056
1.0567
À1.2286
1.0879
À1.0517
5
1.0504
À1.0305
1.0218
À1.2822
1.0504
À1.0305
1.0791
À0.7789
6
1.0493
À1.5273
1.0182
À1.7576
1.0493
À1.5273
1.0805
À1.2970
7
1.0407
À1.7776
1.0104
À2.0500
1.0407
À1.7776
1.0710
À1.5052
8
1.0382
À1.9067
1.0066
À2.1644
1.0382
À1.9067
1.0698
À1.6489
9
1.0575
À2.3387
1.0263
À2.5542
1.0575
À2.3387
1.0888
À2.1232
10
1.0585
À2.9667
1.0282
À3.1171
1.0585
À2.9667
1.0889
À2.8162
11
1.0670
À2.2262
1.0329
À2.4723
1.0670
À2.2262
1.1011
À1.9802
12
1.0651
À1.5697
1.0331
À1.5770
1.0651
À1.5697
1.0971
À1.5625
13
1.0799
0.8656
1.0479
0.6788
1.0799
0.8656
1.1119
1.0524
14
1.0564
À1.9683
1.0243
À2.0865
1.0564
À1.9683
1.0884
À1.8501
15
1.0585
À2.4010
1.0268
À2.4020
1.0585
À2.4010
1.0901
À2.4000
16
1.0559
À2.3555
1.0242
À2.4643
1.0559
À2.3555
1.0876
À2.2467
17
1.0516
À2.9974
1.0210
À3.1444
1.0516
À2.9974
1.0821
À2.8504
81
1.0459
À3.7323
1.0144
À3.7621
1.0459
À3.7323
1.0773
À3.7024
19
1.0436
À4.3441
1.0119
À4.4041
1.0436
À4.3441
1.0753
À4.2841
20
1.0480
À4.2075
1.0162
À4.2551
1.0480
À4.2075
1.0798
À4.1598
21
1.0643
À2.9044
1.0350
À3.1048
1.0643
À2.9044
1.0936
À2.7039
22
1.0708
À2.8385
1.0416
À3.0295
1.0708
À2.8385
1.1000
À2.6475
23
1.0793
À1.8738
1.0475
À1.9351
1.0793
À1.8738
1.1112
À1.8124
24
1.0665
À2.3494
1.0351
À2.4557
1.0665
À2.3494
1.0980
À2.2431
25
1.0762
À0.7157
1.0424
À0.9989
1.0762
À0.7157
1.1100
À0.4325
26
1.0638
À1.2811
1.0282
À1.5384
1.0638
À1.2811
1.0993
À1.0238
27
1.0868
0.3060
1.0528
À0.0102
1.0868
0.3060
1.1208
0.6223
28
1.0515
À1.4348
1.0202
À1.6898
1.0515
À1.4348
1.0828
À1.1799
29
1.0827
À0.6183
1.0438
À1.0490
1.0827
À0.6183
1.1216
À0.1876
30
1.0785
À1.5834
1.0373
À2.0106
1.0785
À1.5834
1.1197
À1.1562
Table 4
introduced by Donoho and Huber in [38]. In theory the high-
CPU and execution time (CPU: Pentium 4, 1.7 GHZ).
est breakdown point one can achieve is 0.5 (or 50%) because
for any higher contamination level, one is not guaranteed to
Test system
Fuzzy LP
WLS
be able to distinguish the good points from the bad.
# iterations
CPU time (s)
# iterations
CPU time (s)
Based on experimentation, FLSE was found to fail with a
Four-bus
4
0.4490
5
0.21252
single outlier in the measurement set and therefore leading
Six-bus
3
0.4106
3
0.17996
to having a 0% Breakdown Point (Note that both the Least
IEEE 30-bus
4
0.6474
4
0.28292
IEEE 39-bus
3
0.7111
4
0.27158
squares (LS) and the Least absolute value (LAV) also have
IEEE 57-bus
3
0.8595
3
0.16113
0% Breakdown Point [39] and may fail with a single outlier).
IEEE 118-bus
3
1.7449
3
0.36153
That breakdown percentage clearly shows how vulnerable
the FLSE is to outliers. Nonetheless, this weakness may be
overcome by using a high breakdown point static estima-
Bounds are always symmetric which is due to the trian-
tor, (such as Least Median Squares (LMS) or Least Trimmed
gular membership functions adopted by the FLSE. Rep-
Squares (LTS) [39–41]), where any outliers would be iden-
resenting
the
uncertainties
by
an
asymmetric
tiﬁed and eliminated from the measurement set, prior to
membership function would provide a better estimation
the estimation process for the uncertainty bounds.
of the bounds.
The FLSE is unable to handle faulty measurements and
5. Conclusion
outliers. The FLSE is very sensitive to outliers [1,37].
Based on this fact, it is expected that the FLSE would
An analysis of uncertainty in power system state estima-
produce deceptive estimation of the center points and
tion is presented in this paper. The uncertainty is modelled
their respective upper and lower bounds.
and is assumed to be present in the system parameters and
in the measurements which take into account known meter
In general, the Breakdown Point1 is a known concept
accuracies. A Fuzzy linear estimator was employed to esti-
used to quantify the robustness of an estimator which was
mate the both the states and their respective upper and
lower bounds. The provision of bounds by the proposed
1
FLSE offers useful additional information to the power sys-
Breakdown Point may be deﬁned as the smallest fraction of contami-
tem operator. By examining bounds on the estimates one
nations that critically offsets the estimator from the true measurements
[1,5].
can infer the quality of the metering conﬁguration and

A.K. AL-Othman / Measurement 42 (2009) 628–637
637
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Document Outline

A fuzzy state estimator based on uncertain measurements