This document provides a review of power system state estimation techniques. It discusses both static and dynamic state estimation algorithms. For static state estimation, it covers weighted least squares, decoupled, and robust estimation methods. Weighted least squares is commonly used but can have numerical instability issues. Decoupled state estimation approximates the gain matrix for faster computation. Robust estimation uses M-estimators and other techniques to handle outliers and bad data. Dynamic state estimation applies Kalman filtering, leapfrog algorithms, and other methods to continuously monitor system states over time.

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ACEEE Int. J. on Electrical and Power Engineering , Vol. 5, No. 1, February 2014
Power System State Estimation - A Review
Dudekula Sai Babu1, K Jamuna1, and B.Aryanandiny2
1
2
VIT University, Chennai campus/SELECT, Chennai, India
College of Engineering, Trivandrum/EEE Department, Thiruvananthapuram, India
Email: jamunakamaraj@gmail.com
Abstract— The aim of this article is to provide a comprehensive
survey on power system state estimation techniques. The
algorithms used for finding the system states under both static
and dynamic state estimations are discussed in brief. The
authors are opinion that the scope of pursuing research in the
area of state estimation with PMU and SCADA measurements
is the state of the art and timely.
tion of robust estimation, hierarchical estimation, with and
without the inclusion of current measurements, etc. The SE
uses only volt- age magnitude, real and reactive power injections and flows of SCADA measurements. The inclusion
of branch currents measurements in SE deteriotes the performance of estimators. It also leads to non uniquely observable
which produces more than one state for the given one set of
measurements [6], [7]. Distributed SE for very large power
system has been taken for study since the very beginning.
The computational procedure involved in SE is an optimization function. The optimization function can be first order or
second order of the derivative. The first order methods are
the classical weighted least squares [8], the iteratively reweighted least squares [9] and the linear programming based
on least absolute value estimator. The second order method
involves the evaluation of the Lagrangian Hessian matrix.
The primal-dual interior-point method and Huber M estimator are the solution metholodigies are available in literature to
solve the second order method. The system states are evaluated either by statically or dynamically. The above methods
mentioned are static state estimators. At the given point of
time, the set of measurements are used to estimate the system state at that instant of time. The common method used to
solve the static SE is weighted least square and weighted
least absolute value methods. The heuristic methods are also
applied to find the states. In the dynamic state estimation,
the system states are continuously monitored at the regular
intervals. The DSE uses the Kalman filter, Leapfrog algorithm,
non-linear observer technique and invariant imbedding
method to estimate the system states dynamically.
Index Terms— State Estimation, LAV, WLS, Kalman,
Synchronized phasor measurements, PMU.
I. INTRODUCTION
State Estimation is the process of assigning a value to an
unknown system variables based on the measurements obtained from the system. State estimator has been widely used
as an indispensable tool for online monitoring, analysis and
control of power systems. It is also exploited to filter
redundant data, to eliminate incorrect measurements and to
produce reliable state estimates. Entire power system
measurements are obtained through Remote Terminal Unit
(RTU) of Supervisory Control and Data Acquisition (SCADA)
systems which have both analog and logic measurements.
Logic measurements are used in topology processor to
determine the system configuration.
State estimator uses a set of analog measurements such
as bus voltage magnitude, real power injections, reactive
power injections, active power flow, reactive power flow and
line current flows along with the system configuration. These
measurements are obtained from SCADA systems through
remote terminal unit. The phasor concept was introduced
by Throp [1]. Based on this the new measuring device was
developed, namely Phasor Measurement Unit (PMU). The
PMUs use a navigational satellite system to synchronize
digital sampling at different substations. The Global
Positioning Sys- tem is used to synchronize the measurement
units. The PMU records fifteen different channels of the real
and imaginary parts of the secondary voltages and currents
[2]. The phasor reporting rate is 60 phasors/sec for a voltage,
5 currents, 5 watt measurements, 5 var measurements,
frequency and the rate of change of frequency. The state
estimators are used both measurements together or separately
for estimating system states. The detailed explanations are
given in this review paper.
A. Weighted Least Square(WLS) Method
In the WLS method, the objective is to minimize the sum
of the squares of the weighted deviations of the estimated
measurements from the actual measurements. The system
states are estimated from the available measurements. The
objective function is expressed as follows,
(1)
where fi (x) and σ2 is the system equation and the variance
of the ith measurement respectively. J(x) is the weighted
residuals. m is the number of measurements and zi is the
ith measured quantity. If fi (x) is the linear function then
the solution of eqn. (1) is a closed form. Usually, the power
flow and power injection equations are described by nonlinear function. Hence the solution leads iterative procedure to determine the state of the system. The steps pro
II. STATIC STATE ESTIMATION
SE is a very useful tool for the economic and secure operation of transmission networks. From early days of
Scheweppe [3]–[5], developments of SE are done as a no
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ACEEE Int. J. on Electrical and Power Engineering , Vol. 5, No. 1, February 2014
cessed in WLS are (1) to find the gradient of J(x) and (2)
force it into zero and solved by Newton’s method. The optimal state estimate is found using eqn. (2).
In the normal equation method, the numerical instability
leads the ill-condition of the linear equation. This can be
solved effectively by Peters-Wilkinson method [16]. The
condition number of a symmetric, positive definite matrix
(G) is defined as whose eigen values are real and positive can
be computed using eqn. (4). It can also be defined as the eqn.
(5).
Ideally the value of κ should be 1 for a well conditioned
system. A very high value of κ indicates that the system is
ill-conditioned. The ill-conditioning occurs due to the
following reasons[17].
 The position of the reference slack bus.
 Existence of the negative line reactance.
Improper choice of line flow measurements.
The network strategy create the first and second
problem. The metering strategy affects the line flow
measurements. The WLS algorithm anticipated to converge
very slowly when the network strategy is wrong and the
metering strategy is correct. The diagonal elements of the
information matrix is increased and solved by LevenbergMarquardt (LM) algorithm, leads to rapid convergence. Both
the metering and network strategy is wrong, the WLS
algorithm will not converge. Even LM algo- rithm provides a
solution but not the right solution. In this case, the
household or orthogonal transformation in conjunction
with LM escorts to the right solution. The Newton’s method
is one of the solutions for the ill-conditioning problem. The
actual WLS technique, the first order gradient of measurement
function only considered and higher order terms are neglected.
The Taylor series expansion of a function g(x) is represented
in eqn (6).
The second order has been included in the measurement
function, and it has solved by Newton’s method, the efficient
gradient convergent technique [18] and the states are
calculated by the eqn.(8). i represents the iteration. The
singular value decomposition is one among the solution for
this case [19].
(2)
(3)
where
R Measurement error covariance matrix
x
System state vector
H Jacobian matrix
z
Measurement vector
The covariance matrix R reflects the relative relation
between the measurements. If there is no interaction
between the various measurements then R will be a
diagonal matrix. The diagonal elements are the variances of
the individual measurements (ri = σ2 ). Recently the SE with
measurement dependencies is solved with WLS technique
[10].
Due to the sparsity of the information matrix (G), the WLS
solution is obtained quickly by triangular factorization
method. This technique is called as normal equation method.
It can also solved by orthogonal transformation method and
combination of these two methods (hybrid method) also
presented in [11]. The steps involved in these methods are
explained in Table. I. The normal equation method fails when
(1) H is in ill- condition and (2) The high measurement
redundancy affects the sparsity. The numerical behavior has
improved in orthogonal transformation method. It has to store
Q1 matrix, hence memory size increases. The hybrid method
solves iteratively the normal equation, where the triangular
factorization is carried out using orthogonal transformations.
It maintains the good sparsity and numerical robustness.
(4)
(5)
TABLE I. WLS ALGORITHM SOLUTION BY TRIANGULAR SOULTION
Normal Eqn.
Orthogonal
Transformation
1. Information matrix
G=HTWH
Form Jacobian
Matrix (H)
Form
Jacobian
Matrix (H)
2. Factorize
G=UTU
Factorize H
Factorize H and Q
is not stored
3.? X=U-1 UT-1(B)
Where, B=HTW? Z
? X=R-1C
C=Q1W0.5? Z
(6)
Hybrid model
? X=R-1R T-1B
B=HTW? Z
(7)
The WLS algorithm effectively provides the system states
under normal operation and identifies the bad data
measurements. The numerical instabilities are also handled
easily. The on line system state estimation is done with WLS
algorithm and implemented in all energy management
systems.
The least square solution is only optimal when the measurement noise is Gaussian and has well known statistics. In
practical applications, this is not the case. Similarly, WLS
estimator does not account for the random event like the failure in instrument and the communication channels [12]. Data
obtained under these circumstances are named as bad data.
The bad data are eliminated by normalized residual method
[13]–[15].
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B. Decoupled State Estimation
The main drawback in the WLS solution algorithm is the
computational burden that is the calculation and the
triangularization of the gain matrix or the information matrix.
The approximation of the constant gain matrix reduces the
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ACEEE Int. J. on Electrical and Power Engineering , Vol. 5, No. 1, February 2014
computational burden. In WLS solution steps, the elements
of the gain matrix do not significantly change between the flat
start initialization and the converged solution. The sensitivity
of the real (reactive) power equations to changes in the
magnitude (phase angle) of bus voltages is very low,
especially for high voltage transmission systems [20]. These
two observations lead to the fast decoupled formulation of
the state estimation problem. The actual measurement,
Jacobian and covariance matrices are partitioned based on
the real power and reactive power separately. The
measurement equations are represented in eqn. (8).
bad data and outlier are discussed in robust state estimation
but they are closely related. Bad data occurs due to failure
of communication link, meter intermittent fault and incorrect
recoding. An outlier does not contain any error due to
structure of its corresponding equation. The identification
of such kind of outliers is very difficult. Robust estimators
are expected to remain unbiased despite the existence of
different types of outliers. The robustness of an estimator is
evaluated based on the breakdown point. It is the largest
ratio of (s=m/mb ) mb is the number of bad data present in
the measurements.
(8)
(11)
The off diagonal blocks Hqδ and HpV are neglected.
The gain matrix is calculated with these approximations and
the solution vectors are shown in eqs. (9) and (10). The constant decoupled gain matrixes are estimated at the flat start.
(12)
(13)
The maximum bad data is limited by the maximum norm of
the estimated states with and without bad data and it is
expressed in eqn. (11). The largest possible breakdown point
will be limited by the measurement redundancy.
The outlier is represented as leverage point in regression
which is located away from the rest of the measurements. By
the elimination of leverage point, the system observability
will not be affected. It has been identified by finding the hat
matrix (K). The least square state estimate is evaluated using
the eqn. (12). The measurement estimates are represented in
eqn. (13).
If the Kii 2( m ) then the particular measurements
are suspected as leverage. The following condition creates
the leverage measurements.
• An injection measurement placed at a bus which is
incident to a large number of branches.
• An injection measurement placed at a bus which is
incident to branches having very different impedance
values.
• Flow measurements of a branch having different
impedances from other branches in the system.
• Assigning large weights for specific measurements.
The projection statistics is the robust measure of the
effect of the leverage measurements and it is applied to power
system state estimation.
1) M estimator: The robust state estimation is solved
by M estimator which is introduced by Huber. M estimator
is a maximum likelihood estimator as a function of the
measurement residual f(r) subject to the measurement
equation is a constraint. The state estimation problem is
defined as follows.
(9)
(10)
The decoupled SE has been widely accepted in the
industry and implemented in control canters all over the world.
The computation is fast and requires less memory. The
decoupled SE drawbacks are described below.
 The network parameters or operating conditions
violate the stated decoupling assumptions in certain
cases. The decoupled solution produces inaccurate
solutions or may not converge.
 The decoupling properties are not valid for the
branch current magnitude measurements. Hence it
cannot be included in decoupled SE.
C. Robust State Estimation
The major function of the state estimator is to detect,
identify and eliminate errors in the measurements, network
model and parameters. The state estimator is robust when
the estimated states are insensitive to the deviations in a
limited number of redundant measurements. The concept of
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(16)
(17)
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ACEEE Int. J. on Electrical and Power Engineering , Vol. 5, No. 1, February 2014
Programming (LP).
(20)
(21)
Fig.1 Huber Function
The estimators are mainly designed for automatically
detecting measurements with rapidly growing residuals and
sup- pressing their influence on the state estimate. The
measurement location, type and network parameters have a
influence on the creation of the leverage points which have
circuitous influence on the state estimate [26]. The objective
function of the M estimate is chosen according to overcome
the influence of all the above. Hence, the M estimator
objective function should satisfy the following condition and
it is represented in Fig. 1. The objective functions can be
quadratic constant, quadratic linear, square root, schweppeHuber generalized-M, and least absolute valve. The solution
methodology can be done by Newton’s method and
iteratively re weighted least square technique.
 f(r)=0 for r=0.
 f(r) 0 for any r.
 f(r) is monotonically increasing in both +r and -r
directions.
 It should be symmetric around r=0.
Newton’s Method the Newton’s solution methodology is
explained in steps.
1) Compute the gradient f’(r).
2) Compute xk = H -1 f 2 (r).
3) Compute αk using a line search.
4) Update the solution xk+1 = xk +αk xk .
5) Update the Hessian matrix H.
6) Go to step 1.
Iteratively re-weighted least square estimation:
Weighted Least Absolute Value The weighted least absolute
value estimator minimizes the sum of the absolute values of
the residuals and its cost function is given in eqn.(19).
The LP problem can be solved by any well developed LP
solution methods. These formulations have the advantages
of the noise filtering and bad data elimination. Linear
approximation of the power system equations are
accomplished by replacing the non linear equations with their
first order Taylor series approximation expanded at the
operating point. By linearizing the nonlinear functions at each
intermediate solution, a sequence of system states is
generated which under suitable circumstances converges to
an optimal solution of the original non linear programming
problem.
(22)
(23)
(24)
The objective is to minimize the effect of measured noise
in the evaluation of the system states. This method is quite
fast and efficient. The residual should be below the
prescribed value (ëi ). The in equality eqn. (22) is made into
equality by adding the non negative slack variables and the
eqs. (23) and (24) are obtained.
The |r| in the objective function in eqn (49) can be replaced
by ëi . The LAV estimate is given by the solution to the
following linear programming problem,
(25)
(26)
(18)
Interior Point Method The basic steps nvolved in the
interior point method is explained as follows.
1. Formulate the problem
(19)
(27)
2. Introduce the slack variables to make all the inequality
where z is the set of the observed measurements, A is the
constraints into nonnegative.
vector that are linearly related with the measurement vector,
e is the random measurement error, r= z - h(x)”x is the
residual vector, Wi is the reciprocal of the measurement
(28)
error variance. The state estimate of WLAV can be described
as optimization problem and formulated as Linear
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ACEEE Int. J. on Electrical and Power Engineering , Vol. 5, No. 1, February 2014
3. Replace the non-negative constraints with logarithmic
barrier terms in the objective.
(29)
4. Incorporate the equality constraints into the objective
using
(30)
the next sampling time. Prediction capacities of DSE automatically ensure system observability even in the presence
of measurement observable islands. Also predicted values
help to identify topology errors, gross bad data and sudden
change of states. The kalman filter is the best tool for
dynamic state estimation.
A. Kalman Filter based DSE
The power system state and observation equations are
represented in eqs. (42) and (40) respectively.
xk+1 = Axk + ωk
(39)
(40)
The steps involved in Kalman filter is follows.
_
x ˆ˙ = f (xˆ, u, 0) + L(y - yˆ)
zˆ = h(xˆ, u, 0)
5. Set all derivatives to zero.
zk = H xk + υ k
(43)
(44)
(31)
(32)
(33)
6. Rewrite the system
(34)
(35)
(36)
7. Apply Newton’s method to compute the search directions
x, w, y.
(37)
(38)
8. For solving w, use (38). Reduced Karush Khun Tucker
(KKT) system.
9.
III. DYNAMIC STATE ESTIMATION
State estimators are performed by static approaches which
have a single set of measurements to estimate the system
states. The estimator has to be reinitialized for every new set
of measurements without any state prediction from previous
estimations. By the increase in power demand and deregulation market, the system state has to be monitored continuously for the better power system operation. The dynamic
state estimators (DSE) have the capability of tracking the
current system states and also predicting the state vector at
14
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where x is the state vector, u is the input variable, z is the
measured variable vector, ω, ë are the system noise and measurement noise respectively. xˆ is the estimated state vector,
zˆ is the estimated measured vector, L is the observer gain.
The non linear observer gain is depends on xˆ. The observer gain is shown in eqn. (45). The value od P is found
using Ricctaci equation is represented in (46).
(45)
L = P C T R -1
AP + P AT + GQGT -P C T R -1 C P = 0
(46)
The Q and R are the spectral densities of ω and
respectively. In the optimal state estimate, the observer gain
is also denoted as Kalman gain. The non linear observer is
applied for finding the states of single machine- infinite
busbar (SMIB) power system. In which, the system states are
assumed as machine angle and the frequency deviation. The
system states are estimated with and without observer gain
and it has been compared with the result produced by machine
angle synthesis method. It has been proved that the non
linear observer provides better estimate than those methods.
The discrete non linear observer is also designed for SE. The
iterative sequential observer is also used for estimating the
system states of SMIB power system. In which the observer
gain is evaluated like Kalman gain in Kalman filter.
B. Invariant Imbedding Method:
The non-linear continuous time varying functions are defined as in eqs. (47) and (48).
x(˙t) = f (x(t)) + G(t)ω(t)
(47)
z(t) = h(x(t)) + υ(t)
(48)
The system state estimation is determined by minimizing
the objective functions J as in eqn. (49). The state estimate
of this optimization problem is obtained by applying
Pontryagin Maximum principle.
(49)
(50)
(51)

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ACEEE Int. J. on Electrical and Power Engineering , Vol. 5, No. 1, February 2014
(52)
The Hamiltonian of this optimization is represented in
eqn.(52).
By minimizing the eqn. (52), the Euler-Lagrange function
is applied. The eqns. (51) and (52) are obtained is estimated
and shown in eqn. (55). The initial and final states are unknown. The transvarsatlity condition associated with the
minimization of ë leads to the boundary value condition. Hence
both Λ(t0 ) and Λ(tf ) are made equal to zero. The eqns. (53)
and (54) defines a two point boundary value problem, the
solution
(59)
• Block diagonal constant gain:
(60)
(61)
(62)
(53)
• Full diagonal gain (decoupled δ and V):
(54)
(63)
(55)
In this paper, the single machine infinite busbar power
system model is utilized. The machine state estimates are
obtained in the hunting and out-of step region.
(64)
C. Tracking estimator
Tracking state estimator is the one of the dynamic state
estimation, which identifies the on line states with the changes
occured in the measurements. The LAV based static state
estimator is discussed which have the ability to incorporate
the changes in the measurement set, operating states and
the network configuration. The estimator handles the number
of measurements at a time.
Masiello and Scheweppe is proposed the tracking static
estimator [36]. If the mathematical model of time structure of
the state (x(tn )) and the observation error (υ(tn )) is known
then the estimator could be designed well. The tracking
estimator is started with the flat start (unity bus voltage
magnitude and zero phase angle), which is a recursive
estimator, not an iterative one and the related equation is
shown in eqn. (56).
In digital for, F(x(ˆtn))R-1[z(tn+1) - f(x(ˆtn))] is named as
the feedback correction signal and P(tn) is feedback gain.
The tracker has computed f(x(ˆtn), F(x(ˆtn) and P(tn) at every
instant of time. The state estimate x(ˆtn) is found using eqn.
(56). The major computation time of the recursive estimator is
the finding out the in eqn. (56) and it is a full matrix. By the
properly designing this as sparse matrix, the computation
time has been reduced. The five categories has been proposed
and explained in eqs. (58) - (64).
•
Full diagonal constant gain: Eqn. (64) is
diagonalized with S(x0 )
The discretized state estimation is used for tracking the
system states [37]. Both the bus and branch current
measurements are sampled and the rectangularized formation
was maintained. In addition to the system states, both
observability and bad data analysis has been performed.
D. Leapfrog algorithm
The SE problem is formulated mathematically as follows.
(65)
(66)
(67)
where r(x)=z-h(x). The residuals are represented as quadratic
constraint function in eqn. (65). The weight σi is the
measurement error and wi is the down weights assigned
for leverage measurements. Hence, t h e objective
minimization is done by Gauss-Newton method or leapfrog
method. The leapfrog method is also known as Euler forwardEuler back- ward method.
The state estimation problem considers analogous the
physical dynamical problem of the motion of the unit mass
particle in a n-dimensional conservative method. The
potential energy J of the particle is to be minimized. The
method requires the solution of the particle’s equation of
the subject to the initial position and velocity. Initially total
• Best Gain:
(58)
•
Full Constant Gain:
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energy is the sum of kinetic and potential energy. By
monitoring the kinetic energy, the inferring strategy is
adopted such that potential energy is systemically reduced.
The particle forced to move in the minimum potential energy
position. The advantages of this method are listed below.
• It uses only gradient information. There is no explicit
line search performed.
• It is very robust and it handles the discontinuous and
function steep valley’s and gradients.
• The algorithm searches the low local minimum. Hence
it used it for global minimum optimization.
the linear form of branch current using equation (69), which
relates the line current with both ends of voltage phasor.
When a PMU is placed at the particular bus, the
corresponding element in a first block of matrix H becomes 1,
otherwise zero. PMU Branch current measurements (I) and
their relation with voltage phasor are shown in eqn. (69). A
typical π model of the network is shown in Fig. 2, gij +jbij is
the series branch of the transmission line and gsi + jbsi are
shunt branch of the transmission line.
(70)
IV. PMU BASED STATE ESTIMATION
The concept of voltage phasor for static state estimation
is presented in [1]. Recently the availability of synchro
phasors has made it possible to incorporate phase angle
also into the measurement set. A phasor based state vector
is preferable than SCADA based SE. In many cases, the
number of Phasor Measurement Unit (PMU) requirements
to meet sufficient measurements poses a problem. Due to
the factor of price, technology, communication ability the
PMU cannot be implemented at all buses in the system.
In SCADA based SE the inclusion of power measurements
makes the problem of state estimation highly non-linear,
leading to iterative procedure. Hence it is highly time
consuming and infeasible for on-line implementation. The
Linear State Estimation (LSE) is done with synchronized phasor
measurements which overcomes the drawbacks of SCADA
based SE. In LSE, the current and voltage phasors are used
in the measurement set vector. This makes the problem of
SE linear and hence a non-iterative procedure can be used
for estimating the states. This also helps in ease of on-line
implementation.
The measurement vector z is expressed in (68) formed by
bus voltage and line current phasors collected by PMU.
(71)
The measurement equation is presented in rectangular
form as z = zA + jzB ; x=E+jF, using which it can be rewritten
in matrix notation as,
where Hr , Hm are the real and imaginary parts of
Hessian matrix (Hnew ). The WLS state estimates can be
obtained using eqn. (71). This method is very simple and
needs no iterative procedure as the H matrix is linear. The
matrix S converting the measurements to the state estimate is
constant as long as there is no change in the network
topology. This technique is suitable for online implementation
except for the time delay in communication.
The phasor measurements in SE are implemented in
Sevillana de Electricidad, which shows that the more accuracy
in state estimates [40]. The power system is clustered into
multi area and the PMU measurements are used for SE which
is the replication of a real power system monitoring. Multi
area SE produces the best accurate estimates and bad data
measurements are added to the sufficient numbers of SCADA
measurements, the accuracy of the estimate is much improved
through the iterative procedure. But measurements are
obtained in different time frames. The angle bias correction is
done in order to improve the existing SCADA based SE in
addition with PMU measurements.
(68)
where, H is the matrix of linear equations and e is an error
vector. The eqn. (68) can be expanded as follows,
V. SIMULATION STUDIES
The static state estimations are performed with SCADA
and PMU measurements which are named as traditional and
linear state estimation respectively. The results have been
demonstrated on simple four bus system and IEEE 14, 57 bus
test systems. For each system, the true values of
measurements were generated using load flow. The noise
was added to the true value assuming a Gaussian distribution
with zero mean. Here, the error of TSE measurements follows
the normal distribution with zero mean and standard deviation
of 0.004 for voltage magnitudes, 0.01 for power injections and
0.008 for power flows, while SD of real and imaginary value of
PMU measurements are taken as 0.0001.
4 bus system: A simple 4 bus test system is comprising 2
generators and loads. The PMU is located at nodes 1 and 4.
where, V is the voltage phasors, I is the branch current
phasors. r and im denotes the real and imaginary parts of the
phasors respectively. N is the number of buses and NL is the
number of branches. The notation k1 and k2 obtained from
(69)
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Table 2 gives the comparison of estimated voltage magnitude
and angle using both TSE and LSE. It is observed that the
estimated voltage magnitudes of the LSE method are almost
equal to the actual values. The PMU measurements thus
provide an accurate system state.
PMUs located at the various nodes. The comparison is made
between TSE and LSE in terms of estimated bus voltage
magnitude and angle from the true value. Fig. 2 explains the
bus voltage magnitude error. LSE produces very little error in
bus voltage magnitude.
TABLE II. C OMPARISON OF ESTIMATED SYSTEM STATES FOR 4 BUS SYSTEM
Bus
No
TSE Approach
V(pu)
1
2
3
4
True Value
V
(pu)
0.9988
0.9397
0.9420
0.9454
1.0000
0.9414
0.9423
0.9515
Ang
(deg)
0.0000
-0.533
-1.631
2.6365
Ang
(deg)
0
-0.522
-1.612
-2.631
LSE
Approach
V(pu)
Ang
(deg)
0.9982
0
0.9402
-0.532
0.9422
-1.613
0.9520
2.624
IEEE 14 bus system: The single line diagram and bus/branch
details are given in Appendix A. In TSE, the measurement set
consists of voltage magnitudes, forward/reverse real power
flows, forward/reverse reactive power flows and real/reactive
power injections. All branch current and few bus voltage
phasor measurements are considered as measurements in LSE.
The PMU located nodes are 2,6,7,9,11,12,14. Table 3 shows
the comparison among the TSE and LSE in terms of estimated
system states. The simulation results show the estimated
bus voltage magnitude and angle along with the true value.
The TSE have some differences in magnitude and angle. But
LSE exhibits very minute error in both magnitude and angle.
The TSE usually chooses the slack bus as a reference bus for
the entire system. Synchro phasor measurements may have a
different reference, which is determined by the instant
synchronized sampling initiated. The results would be
different if the reference problem is not considered. In order
to adjust the angle of the slack bus (bus no.1 for IEEE 14), a
PMU is installed in the slack bus, which is taken as reference
for the TSE. Once the PMU is installed in SB, the particular
estimated angle can be directly deducted from all buses.
T ABLE III. C OMPARISON
OF ESTIMATED SYSTEM STATES FOR
IEEE 14
Fig. 2
Errors in the estimated bus voltage magnitude error for
IEEE 57 bus systems
Fig.3 presents bus spread angle error for IEEE 57 bus
system. The angle error for LSE is negligible. Hence more
accurate estimate or value of system states is obtained with
synchronized phasor measurements.
BUS
SYSTEM
Bus
No
TSE Approach
V(pu)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
True Value
V
(pu)
1.046
1.032
1.010
1.002
1.003
1.063
1.051
1.082
1.046
1.042
1.046
1.045
1.044
1.029
1.060
1.045
1.010
1.018
1.020
1.070
1.061
1.090
1.054
1.049
1.056
1.055
1.050
1.034
Ang
(deg)
0
-4.981
-12.71
-10.32
-8.781
-14.24
-13.35
-13.35
-14.93
-15.09
-14.80
-15.09
-15.17
-16.04
Ang
(deg)
0
-4.834
-11.07
-10.21
-8.953
-15.47
-14.77
-15.25
-16.04
-16.48
-16.38
-16.56
-16.52
-17.46
LSE
Approach
V(pu)
Ang
(deg)
1.060
0
1.045
-4.979
1.010
-12.71
1.018
-10.32
1.020
-8.780
1.070
-14.24
1.061
-13.35
1.090
-13.35
1.054
-14.93
1.049
-15.09
1.056
-14.80
1.055
-15.09
1.050
-15.17
1.034
-16.03
Fig. 3
CONCLUSIONS
IEEE 57 bus system: To demonstrate both traditional and
linear state estimation, the results are displayed for the lower
order systems. IEEE 57 test system is chosen to ensure
effective working of the higher order systems. There are 25
© 2014 ACEEE
DOI: 01.IJEPE.5.1.12
Errors in the estimated bus angle error for IEEE 57 bus systems
The article presented a classified list of algorithms for
state estimation in approximately 40 years of extensive
17

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© 2014 ACEEE
DOI: 01.IJEPE.5.1.12
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