Energy and Power Engineering, 2013, 5, 670-676
doi:10.4236/epe.2013.54B130 Published Online July 2013 (http://www.scirp.org/journal/epe)
Optimal Static State Estimation Using hybrid Particle
Swarm-Differential Evolution Based Optimization
Sourav Mallick, S. P. Ghoshal, P. Acharjee, S. S. Thakur
Department of Electrical Engineering, National Institute of Technology, Durgapur, India
Email: sourav.nitdgp2009@gmail.com, spghoshalnitdgp@gmail.com, parimal.acharjee@ee.nitdgp.ac.in,
sst_nit_ee@yahoo.co.in
Received February, 2013
ABSTRACT
In this paper, swarm optimization hybridized with differential evolution (PSO-DE) technique is proposed to solve static
state estimation (SE) problem as a minimization problem. The proposed hybrid method is tested on IEEE 5-bus, 14-bus,
30-bus, 57-bus and 118-bus standard test systems along with 11-bus and 13-bus ill-conditioned test systems under dif-
ferent simulated conditions and the results are compared with the same, obtained using standard weighted least square
state estimation (WLS-SE) technique and general particle swarm optimization (GPSO) based technique. The perform-
ance of the proposed optimization technique for SE, in terms of minimum value of the objective function and standard
deviations of minimum values obtained in 100 runs, is found better as compared to the GPSO based technique. The sta-
tistical error analysis also shows the superiority of the proposed PSO-DE based technique over the other two tech-
niques.
Keywords: Differential Evolution; Ill-conditioned System; Particle Swarm Optimization; State Estimation
1. Introduction
An electric power system can be operated in efficient,
economic and secure manner if the states are known for a
known network topology and loading conditions [1]. The
concept of state estimation (SE) was first introduced by
Schweppe et al. [2] to find the best estimate of the states
by minimizing or maximizing a selected criterion by us-
ing redundant imperfect power system measurements.
Thereafter, the volume of research works on SE has
grown enormously and it has become a basic function in
power system control centers (ECCs), specified by elec-
tric utilities as a mandatory requirement and supplied by
all major control centers as a standard software product.
Although the SE has become a mature, field-proven
workhorse, various aspects of SE like the solution algo-
rithm [3-6], detection and identification of bad data [7-9],
topological error detection [10-12], observability analysis
[13,14] continue to be explored so as to enrich the SE
software used in ECCs. Conventional SE methods as-
sume that the objective function related to SE is differen-
tiable and continuous. However, considering the nonlin-
ear characteristics of the practical equipments, the objec-
tive function is not always differentiable and continuous,
and it is difficult to apply the conventional methods prac-
tically. Therefore, a practical SE method considering the
above-mentioned requirements has been eagerly awaited.
Modern heuristic algorithms are considered as effective
tools for nonlinear optimization problems. The algo-
rithms do not require the objective function to be differ-
entiable and continuous. Particle swarm optimization
(PSO), one of the meta-heuristic algorithms, can be ap-
plied to nonlinear and non-continuous optimization
problems with ntinuous variables such as in SE.
Based on the social behavior of birds’ flocking or fish
schooling, particle swarm optimization (PSO) was de-
veloped by Eberhart and Kennedy in 1995 [15]. PSO is
biologically inspired computational stochastic search
method which requires little memory. PSO has fast con-
verging feature and better global searching ability at the
beginning of the run [16]. But, it has local searching
problem near the end of the run. It suffers from local
optima at the end of execution of a program [17]. In or-
der to overcome this local optima problem, many im-
provisations are adopted by the researchers [16-19].
In 1995, in a pioneer paper, Storn and Prince proposed
an algorithm [20] based on floating point encoded evolu-
tionary technique for global optimization. This algorithm
is termed as DE algorithm because in this algorithm, a
special kind of differential operator is used to create new
off-springs from parent chromosomes instead of classical
crossover or mutation. Here, the target vector is mutated
to find a trial vector using a difference vector which is
obtained as a weighted difference between randomly
selected vectors in the population. T. Hendtlass presented
Copyright © 2013 SciRes. EPE
S. MALLICK ET AL. 671
a new population based algorithm as a hybrid of PSO and
DE [21]. A few variants of this hybridization came later
from various researchers [22,23] for different applica-
tions.
In this paper, to improve both global and local search-
ing performance of PSO and to avoid suboptimal solu-
tions, hybrid particle swarm-differential evolution opti-
mization (PSO-DE) has been proposed to solve SE as an
optimization problem. This also improves the error per-
formance analysis based on statistical indices of SE. The
proposed scheme of SE has been tested on different
standard IEEE test systems and ill-conditioned systems
under different simulated operating conditions and the
results have been compared to those of standard Weighted
Least Square (WLS) technique and general PSO (GPSO)
based technique.
2. Problem Formulation
2.1. Weighted Least Square Estimation
In SE, a power system with m-dimensional measurement
vector z and n-dimensional state vector x may be mod-
eled as,

zhx

(1)
where is the m-dimensional vector of non-linear
power flow equations and

.h
is the m-dimensional noise
vector with the statistical properties,
0;E
E.
TR



where
E and superscript ‘T’ represent expectation
operator and transposition of a matrix, respectively. ‘R’
is a diagonal matrix and is known as measurement error
co-variance matrix. The WLS SE determines the esti-
mated value of the state vector ˆ
x
minimizing the per-
formance index
 
1
..
T
f
xzhxRzhx

 

1
(2)
Minimization of (2) yields iterative solution as:

1
1TT
x
HRHHR z

 

k
(3)
where 1k
x
xx
 and are known
as the correction vector and mismatch vector, respec-
tively; k being the index of iteration.

k
zzhx 
H
hx x
 is
the Jacobian matrix. Using index notation, (2) can also be
expressed as an optimization problem with the weighted
sum of the squares of the residues as objective or fitness
function .
 
2
1
m
ii ii
i
xwzhx

(4)
In (4), weighting factor 2
1/
ii ii
w
, ii
being the
standard deviation of the meter error.
2.2. Particle Swarm Optimization with
Differential Evolution
GPSO is biologically inspired computational stochastic
search method which requires little memory. GPSO ran-
domly initializes the population (swarm) of individuals
(particles) in the search space. Each particle in GPSO has
a randomized velocity associated to it, which moves
through the space of the problem [15,16]. The particle
velocity is constantly adjusted according to the experi-
ences of the particles and its companions. The velocity
k
j
v and position k
j
h of particle index ‘j’ of kth popula-
tion in the search space are adjusted by (5)-(7).

maxmaxmin *max
cy
www wcy
  (5)


,1 ,,
,,
,
,,
*1*1*
+2*2*
k
kk k
jcy jcyjcy
pbestj cy
k
j
cy
j gbest cy
wcrx
vv x
cr xx
 
(6)
,1 ,,1
kkk
j
cyj cyj cy
xxv
 (7)
where (6) represents the updated value of w with itera-
tion cycle; ,,
k
p
bestj cy
x represents pbest position at cyth
iteration, i.e., the best position of the particle in the cur-
rent iteration; ,,
j
gbest cy denotes the global best position
gbest, i.e., the best position of the particle in the popula-
tion up to the present iteration and maxcy is the maxi-
mum number of iteration cycles. After obtaining the
suboptimal values of fitness function for total population
set, differential evolution (DE) algorithm have been ap-
plied to find the optimal solution. In DE, the initial pop-
ulation is the population obtained from GPSO. The steps
to incorporate DE algorithm with GPSO are shown be-
low as
x
i Initialize population of particles (solutions).Set
GPSO and DE parameters.
ii Calculate fitness values and find gbest and pbest
values.
iii PSO is used to update velocities and positions of
particles using (6) and (7).
iv For the total population set, fitness values are cal-
culated according to (4). These suboptimal fitness values
are termed as .
os GPSO
Ct
v The updated population set is used as the input of
DE. The donor vector is calculated as,
 


12
1
,
,
1*
2*
HJK HJK
kk
donor jjjj
cy k
gbest jj
xxFxx
Fx x
 

(8)
where
1HJK and are indices generated
within the population, to select the two random vectors
within the population.

2HJK
vi The fitness values are evaluated within the popula-
tion using (4) and is termed as os
D
onor
Ct
Copyright © 2013 SciRes. EPE
S. MALLICK ET AL.
672
vii Trial vectors Trial
x
are formed by random cross-
over of elements of donor vectors and target vectors de-
pending on random number generated within ([0,1]),
greater or less than a fixed probabilistic crossover ratio
value (CRR = 0.3, in this case). If CRR is less than the
random number, is assigned to
,Trialj
x,
D
onor j
x; otherwise
is assigned to
,Trialj
x
j
x
.
viii The fitness function is evaluated for each jth trial
vector using (4) and termed as.
os Trial
Ct
ix In the Selection stage, either ,
D
onorj or ,Trialjis
selected (
x x
s
elect
x
os
) depending on the minimum value be-
tween
D
ono
Ct r
x Using the selected vectors ,
and . os Trial
Ct
Select
xx and x
p
bestgbest
are updated.
xi Check whether the maximum iteration cycle is
reached, if yes, then x
g
best is the optimal solution vector.
Otherwise, go to step iii with the vectors as the
input vectors of the GPSO.
Select
x
2.3. Bad Data Analysis
For detection and identification of bad data, scheme
proposed by N.G. Bretas et al. [9] has been adopted in
this paper for its high efficiency. The idempotent matrix
is formed as

1
PTT1
H
HRHHR
z
(9)
The measurement residuals are expressed as

ˆIM
rzzzPzI PzS    (10)
Here, S(=IM
I
P) is called the residual sensitivity
matrix.
I
M
I
is the identity matrix with dimension equal
to the length of Y. e is the complex noise vector. This S
matrix is the operator that projects
I
onto measure-
ment Jacobean space (R(Y) ).
Now for the ith measurement vector, that is, i
m
ii
M
with and i
[0....1 ....0]
i
i
T
M
is the magnitude
of the measurement i, the two components of measure-
ments are found to be
 
R( )R( )
P and (P)
i YiiIMii
iY
MMMIM
.
Therefore, the innovative index (II) is calculated as
R( )
R( )
i
iY W
W
II MM
iY
(11)
The largest element (Nth) in II is compared against a
statistical threshold, , to decide on the exis-
tence of bad data. The index value of the largest element
gives the index of bad data of measurement. As the
presence of bad data is detected and indentified, the
measurements should be recovered from errors. The cor-
rected normalized measurement error is computed using
the following equation as suggested in [9].
0.250
222
11
i
m
eII
where is the measurement ith residual measurement.
i
In a power system, a sudden large change of load may
occur. Therefore, it is very important to discriminate be-
tween sudden large change of load and the presence of
bad data in measurements. For this discrimination, an
index, called asymmetry index (AI) [8], has been used.
AI is defined as
r
3
3, kk
AI M
(13)
where 3,k
M
is the third moment of the discrimination at
time k and k
t
is the standard deviation of the distri-
bution at . If AI is greater than a pre defined value
(here
k
t
max
), then measurements are with gross errors
and if AI is less than max
, large load change occurred
is considered.
3. Simulation Details
The simulation study has been carried over a period of 30
time samples by linearly varying the load at each bus
from 70% to 120%. In addition, the system jitter is rep-
resented by a normally distributed random fluctuation
with a zero mean and a standard deviation of 2% of the
trend component. As load variation is not possible for
ill-conditioned systems, it is not done. The power factor
is assumed to be constant, so that the reactive power fol-
lowed the active counterpart. The change in total load
has been distributed among the generators according to
their participation factors. The true values of active and
reactive powers are evaluated by successive load flows.
For ill-conditioned systems, the method of Incremental
power flow [24] has been used for obvious reasons. The
simulated measurements are obtained by adding a nor-
mally distributed error function with zero mean and a
standard deviation of 2% of the true values. Also, simu-
lated bad data of magnitude 15
for the active and
reactive line flows at the 20th time step for the different
test systems are as shown in Table 1. Flat voltage start
has been used for both proposed schemes and the toler-
ance value
is set at 0.00001. The statistical threshold
to find the existence of bad data is set at 3. For each
optimization technique, the maximum cycles (maxcy)
have been set to 500. The control parameters for the
GPSO and the PSO-DE are as shown, respectively, in
Table 2. These parameters are found to be the most suit-
able to get the minimum value of fitness function used in
the work.
Table 1. Details of events simulation.
Test System Time sample Bad data Measurements at
IEEE 5-bus 20 P5,P2-3,Q2-3
IEEE 14-bus 20 Q10,P4-9,Q4-9
IEEE 30-bus 20 P15-18, P2-5,Q2-5
IEEE 57-bus 20 P12, Q12, P14-46,Q14-46
IEEE 118-bus 20 Q15, P89-92,Q89-92
ii
r (12)
Copyright © 2013 SciRes. EPE
S. MALLICK ET AL.
Copyright © 2013 SciRes. EPE
673
Table 2. Control parameters of gpso and pso-de tec hnique s.
Optimization technique Cognitive Acceleration Factor (C1)Social Acceleration Factor (C2)wmaxwminF1 F2 Crossover Ratio (CRR)
GPSO 2.05 2.05 0.8 0.4 - - -
PSO-DE 1.6 1.6 0.8 0.4 0.2 0.4 0.3
Performance Assessment
The performances of the proposed SE techniques have
been assessed under both the normal operation and bad
data measurement condition by using different perform-
ance indices and compared with the same of WLS tech-
nique.
The average absolute state error (AASE) is calculated
as
(2* 1)
1
1
()() ())
ˆ
(2*1)(
NB t
i
i
i
A
ASE kkk
x
x
NB

(14)
where
.
x
is the state vector, containing the magni-
tudes and phase angles of complex bus voltages. ˆ()
x
k
and t
i
x
are the estimated and the true values of the state
vector at kth time step, respectively.
The performance index ()
J
k is calculated as Figure1. Comparison of convergence characteristics of
GPSO and PSO-DE.
1
1
() ()
()
() ()
ˆ
mt
ii
imt
ii
i
kk
Jk
kk
zz
zz
(15)
each technique has been run 100 times and the values of
500th optimization cycle are noted. The results are pre-
sented in Table 3. The total range of these values is se-
lected as the difference of maximum values and mini-
mum values. The total range is sub-divided into four
equal small sub-ranges viz. Range-1, Range-2, Range-3
and Range-4. The ranges of sub-ranges are shown in Ta-
ble 3. The comparative study of standard deviations
clearly indicates the superiority of the proposed PSO-DE
based optimal SE technique. Hence, it can be stated that
the PSO-DE based SE has better optimization character-
istics of the objective function than the GPSO based SE.
Frequency of occurrence (FO) indicates the occurrence
of the fitness values in the sub-ranges at the end of 500th
optimization cycle. Therefore, the higher FO in the
Range-1 indicates the superiority of the algorithm. In
Figures 2(a)-(g), the FO values have been plotted against
sub-ranges for each test case. From the figure, it is clear
that the PSO-DE based SE has higher FO in the Range-1
and lesser FO in other sub-ranges than the same of the
GPSO based SE technique. This clearly proves the supe-
riority of the PSO-DE based optimal SE method.
where and represent estimated, meas-
ured and true values of the measurements , respectively,
and m represents the number of measurements used.
ˆ(),zk ()zk ()
t
zk
4. Results and Discussions
The optimized estimators have been tested on all five
standard and two ill-conditioned test systems extensively
under different normal and bad data measurement condi-
tions. The choice of explicit results to present is difficult
as the number of interesting outputs is very large. For the
sake of brevity the performance of the proposed SE me-
thod has been presented for some of the important results.
The results, presented here, can be divided in two distinct
categories; one is based on the optimization characteris-
tics of the algorithms and the other is based on the per-
formance characteristics of SE techniques.
4.1. The Optimization Characteristics of
Algorithms
In Figure 1, the convergence characteristics for IEEE
118-bus test system have been presented for GPSO and
PSO-DE based SE. The optimal value of the fitness func-
tion of PSO-DE is much less than that of GPSO.
4.2. Performance Characteristics of the SE
Techniques
In Figure 3, AASE(k) and J(k) indices for 118-bus test
system are presented. From the figure, it is clear that the
AASE graph of PSO-DE based technique is the closest to
In order to check the robustness of the optimization
algorithms applied to solve state estimation problem,
S. MALLICK ET AL.
674
Table 3. Description of ranges of minimum values of objective function and standard deviation of the minimum fitness func-
tion values for different test systems for 100 runs of the algorithms.
GPSO PSO-DE
Test Bus
system Minimum
value of objective
function
Maximum
value of objective
function
Range of
sub-range
values
Standard
deviation
Minimum
value of objective
function
Maximum
value of objective
function
Range of
sub-range
values
Standard
deviation
IEEE 5-Bus 0.3453 0.5630 0.0551 0.0417 0.0059 0.0350 0.0072 0.0046
IEEE 14-Bus 0.3745 0.8372 0.1541 0.1239 0.0099 0.0282 0.0045 0.0029
IEEE 30-Bus 0.4161 1.5096 0.2733 0.2295 0.0272 0.0414 0.0035 0.0031
IEEE 57-Bus 0.9642 3.1213 0.5392 0.3413 0.0842 0.2881 0.0509 0.0270
IEEE 118-Bus 15.4566 32.1037 4.1617 3.7259 1.8485 3.7460 0.4743 0.3110
11-Bus 5.9385 9.4668 0.8820 0.5958 0.3470 0.8692 0.1305 0.1127
13-Bus 0.1684 0.2427 0.0185 0.0114 0.0024 0.0035 0.0002 0.0002
Figure 2. Comparison of F.O. of the optimal values of objective function for 100th run of the algorithms the GPSO and the
PSO-DE based SE.
Copyright © 2013 SciRes. EPE
S. MALLICK ET AL. 675
Figure 3. Comparison of AASE(k) a n d J(k) obtained by WLS,
GPSO and PSO-DE for IEEE 118-Bus test system.
Figure 4. Comparative voltage magnitudes of bus number
92 of IEEE 118-Bus test system obtained by WLS, GPSO
and PSO-DE.
zero among the three techniques. So, it is obvious that
the PSO-DE based technique is more accurate than the
GPSO based technique and the WLS-SE technique. WLS
has the J(k) values close to 0.8, whereas the GPSO based
SE and the PSO-DE based SE provide J(k) values almost
constant and parallel to x-axis though the load is varied
from 70% to 120%. The effect of inclusion of bad data
measurement is overcome in PSO-DE based SE whereas
both WLS and the GPSO based estimators cannot elimi-
nate the effect of inclusion of bad data measurement.
This clears the superiority of the PSO-DE based SE over
the other two techniques.
The true values of bus voltage magnitudes obtained
using the standard WLS technique, the GPSO based
technique and the PSO-DE based technique for load var-
iation of bus number 92 of IEEE 118-bus test system are
compared and presented in Figure 4. The PSO-DE based
SE estimates the voltage magnitudes more accurately
than the other two techniques. The GPSO based SE pre-
dicts the voltage magnitudes slightly better than the
standard WLS technique.
5. Conclusions
In this paper, hybrid PSO-DE based SE algorithm has
been proposed to find the minimum value of fitness
function of the SE problem. The proposed method has
been tested on IEEE 5-bus, 14-bus, 30-bus, 57-bus and
118-bus standard test systems and 11-bus and 13-bus
ill-conditioned test systems extensively for different
normal operating conditions and various combinations of
bad-data measurement conditions to verify their efficien-
cies. The results are compared with the same of the stan-
dard WLS technique and the GPSO method. From the
comparison of results, it has been observed that (i) the
PSO-DE based state estimator minimizes the fitness
function far better than both GPSO based estimator
and WLS based estimator; (ii) the frequency of occur-
rence of the minimum value near the mean value of the
solutions for 100 runs of each algorithm is more in case
of the PSO-DE based SE than the GPSO based SE and
WLS-SE and (iii) the error analysis study among the
three techniques, using AASE(k) and J(k) index, proves
the superiority of the PSO-DE based technique over the
other two. Comparing all performances, it may thus be
concluded that the PSO-DE based state estimation tech-
nique shows the best efficiency in state estimation analy-
sis with high accuracy.
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