Power System Stabilizers Layout via Flower Pollination Algorithm
E. S. ALI
Electrical Department, Faculty of Engineering, Jazan University,
Jazan, KINGDOM OF SAUDI ARABIA
Abstract:- In this article, the optimum layout of Power System Stabilizers (PSSs) using Flower Pollination
Algorithm (FPA) is developed in a multimachine environment. The PSSs values tuning problem is turned into
an optimization task which is treated by FPA. FPA is used to check for optimum controller parameters by
reducing an eigenvalues based objective function involving the damping factor, and the damping ratio of the
lightly damped modes. The implementation of the developed FPA based PSSs (FPAPSS) is compared with
Particle Swarm Optimization (PSO) based PSSs (PSOPSS) and the Conventional PSSs (CPSS) for various
loading conditions and disturbances. The results of the developed FPAPSS are confirmed via time domain
analysis, eigenvalues and some indices. Also, the results are introduced to prove the effectiveness of the
developed algorithm over the PSO and conventional one.
Key-Words: Power System Stabilizers; Flower Pollination Algorithm; Particle Swarm Optimization; Power
System Stability.
1. Introduction
Power system stability is one of the modern
considerable issues in the analysis of power
systems, [1]. One of the forcible instances of this is
an interconnected power system. The loaded long
tie-lines could account for a diversity of stability
issues, [2]. This directs to the divergence of the
ultimate investigators towards designing a suitable
PSS. Recently, numerous research missions are
based on an area named “Heuristics from Nature” in
which the analogies of social systems are being
exercised, [3]. These approaches when used in the
research community can demonstrate their
susceptibility of finding optimum solutions of non-
differentiable, multi-model, and compound
objective functions. Numerous new approaches have
been applied for designing a PSS as Differential
Evolution (DE), [4], PSO, [5], Bacterial Swarm
Optimization (BSO, [6], Harmony Algorithm (HA),
[7], Bacteria Foraging (BF) [8], Bat Algorithm (BA)
[9], Water Cycle Algorithm (WCA), [10],
Backtracking Search Algorithm (BSA), [11], Grey
Wolf Algorithm (GWA), [12], Whale Optimization
Approach (WOA) [13], Cuckoo Search Algorithm
(CSA), [14], [15], Genetic Algorithm (GA), [16],
and Kidney-Inspired Algorithm (KIA), [17]. All of
these approaches are based upon Artificial
Intelligence (AI). A novel optimization algorithm
called FPA has been addressed by Yang, [18]. It is
created by the fertilization process of flowering
plants, [19], [20]. It has only one parameter p
(switch probability), that makes the algorithm easier
to apply and quicker to link an optimum solution.
The transferring switch among local and global
fertilization can sponsor escaping from the local
lower solution. Moreover, it examines its efficacy in
another problem as in [21]. Also, it is serene from
the inspection that the purpose of the FPA to settle
the problem of PSS design has not been illustrated.
This supports the utility of the FPA to cure this
problem.
2. Problem Formulation
2.1 Power System Paradigm
The complex nonlinear paradigm related to
n
units
connected power system, can be constituted by a set
of differential equations as:
( , )X f X U
(1)
where
X
is the vector of the state elements and
U
is the vector of input variables.
T
f
V
fd
E
q
E X ],,,,[
and
U
is the product
signals of PSSs in this article.
and
are the rotor
angle and speed, respectively. Also,
q
E
,
fd
E
and
f
V
are the inner, the field, and excitation voltages
respectively.
The linearized models around a point are exercised
in the design of PSS. Thus, the state equation of a
power system with
m
PSSs can be constituted as:
X AX BU
(2)
where
A
is a
matrix of
nn 55
and equals
Xf /
while
B
is a matrix of
mn 5
and equals
Uf /
.
Both
A
and
B
are evaluated at a certain operating
point.
X
is a
vector of
15 n
and
U
is a
1m
input vector.
Received: November 22, 2021. Revised: October 21, 2022. Accepted: November 23, 2022. Published: December 31, 2022.
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2022.4.12
E. S. Ali
E-ISSN: 2769-2507
80
Volume 4, 2022
2.2 PSS Structure
Power system companies recognize CPSS structure
due to the readiness of online settings and the lack
of affirmation of the stability related to some
variable structure techniques. Otherwise, a universal
analysis of the effects of distinct CPSS parameters
on the overall dynamic performance of the power
system is illustrated in [1, 2]. It is clear that the
adequate election of the CPSS parameters leads to
satisfying performance pending the system
disturbances. The structure of the
th
i
PSS is
presented by:
i
Δω
)
i4
ST(1
)
i3
ST(1
)
i2
ST(1
)
i1
ST(1
)
W
ST(1
W
T S
i
K
i
U
(3)
This structure involves a gain, washout filter, a
dynamic compensator and a limiter as it is displayed
in Fig. 1. The product signal is a supplementary
input signal,
i
U
to the organizer of the excitation
system. The input signal
i
is the change in
speed from the synchronous one. The stabilizer gain
i
K
is utilised to locate the value of damping to be
injected. Then, a washout filter makes it just act as a
contra oscillator in the input signal to evade steady
state error in the terminal voltage. Moreover, two
lead-lag circuits are involved to exclude any delay
among the excitation and the electric torque. The
limiter is included to prohibit the product signal of
the PSS from driving the excitation system into
heavy saturation [2].
In this article, the value of the washout time
constant
W
T
is fixed at 10 second, the values of
time constants
i
T2
and
i
T4
are kept at 0.05 second.
The gain
i
K
and time constants
i
T
1
, and
i
T3
are
determined via FPA.
2.3 Test System
A multimachine system involves three units and
nine buses. The system data and loading events are
mentioned in [2, 8].
3. Overview of FPA
FPA was introduced by Yang in 2012 [18]. It is
inspired by the fertilization process of flowers. Real-
world design problems in engineering are
multiobjective [19, 20]. These objectives conflict
with one another. FPA has been adopted to resolve
PSS design problems.
3.1. Characteristics of Flower Pollination
The main purpose of a flower is reproduction via
fertilization. Flower fertilization is correlating with
the transfer of pollen, which is often associated with
pollinators. Some flowers and bees have a very
specialized flower-pollinator sharing [18].
Fertilization can be terminated by self or cross-
fertilization. Also, insects and birds may follow
Lévy flight behavior in which they journey distance
steps obeying a Lévy distribution. In addition,
flower constancy acts as an incremental step
employing the similarity of two flowers [19, 20].
The objective of flower fertilization is the survival
of the fittest and the optimum reproduction of
plants. This can be treated as an optimization task of
plant species. All of these factors created optimum
reproduction of the flowering plants.
3.2. Flower Pollination Algorithm
For FPA, the following four steps are used:
Step 1: Global fertilization represented in biotic and
cross- fertilization processes, as pollen-carrying
pollinators fly following Lévy flight [20].
Step 2: Local fertilization represented in a biotic and
self- fertilization as the process does not require any
pollinators.
Fig. 2. Multimachine test system.
Fig. 1. Block diagram of CPSS with
excitation system.
-
+
+
-
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2022.4.12
E. S. Ali
E-ISSN: 2769-2507
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Volume 4, 2022
Step 3: Flower constancy which can be expanded by
insects, which is on a par with a reproduction
probability that is identical to the identity of two
flowers involved.
Step 4: The interactivity of local and global
pollination is planned by
]1,0[ p
, slightly biased
toward local pollination.
The former steps have to be modified to
conveniently updating equations. For example at the
global pollination step, the pollinators load the
flower pollen gametes, so the pollen can journey
over an extended distance. Consequently, global
pollination and flower constancy step can be
designated by:
))((
1t
i
xgL
t
i
x
t
i
x
(4)
Where
t
i
x
is the pollen
i
, and
g
is the present best
solution found between all solutions at the present
iteration. Here
is a scaling factor controlling the
step size.
In fact,
)(
L
the Lévy flights are based on step size
that corresponds to the intensity of the pollination.
Since long distances can be covered by insects using
diverse distance steps, a Lévy flight can be utilized
to simulate this behaviour. That is,
0L
from a
Lévy distribution.
ss
s
L)0
0
(
1
12)/)sin((
~
(5)
)(
is the standard gamma function, and this
distribution is valid for large steps
0s
.
For the local fertilization, both Step 2 and Step 3
can be represented as
t
k
x
t
j
x
t
i
x
t
i
x)(
1
(6)
where
t
j
x
and
t
k
x
are pollen from distinct flowers of
the same plant species mimicking the flower
constancy in a limited neighborhood. For a local
random walk,
t
j
x
and
t
k
x
comes from the same
species then
is drawn from a uniform distribution
as [0, 1].
The flowchart of FPA is presented in Fig. 3. The
data of FPA is displayed in the appendix.
4. Objective Function
For a multimachine power system the objective
function is modified to include the interaction
between machines. The parameters of the PSS may
be selected to minimize the following objective
function:
np
jij
ij
np
jij
ij
t
J
10
2
)
0
(
10
2
)
0
(
(7)
This will place the system closed loop eigenvalues
in the D-shape sector characterized by
0
ij
and
0
ij
as shown in Fig. 4.
If rand > p
Fig. 3. Flowchart of FPA.
Input population size, maximum iteration,
switch probability, number of units, B
matrix, upper and lower limits of units, and
demand.
Start
Initialize a population
solution
Check if the
condition is
satisfied?
Global pollination using Levy
flight
Find the current best solution
Stop
Yes
No
Update current global
best
Output the best solution
Do local
pollination
Evaluate new solutions (outputs of
generating units, cost and losses)
No
Yes
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2022.4.12
E. S. Ali
E-ISSN: 2769-2507
82
Volume 4, 2022
In this article,
0
and
0
are picked to be -0.5 and
0.1 respectively. Typical limits of the optimized
values are [1-100] for
K
and [0.06-1.0] for
i
T
1
and
i
T3
. Optimization process based on the objective
function
J
can be formed as: lower
J
according
to:
max
i
K
i
K
min
i
K
max
1i
T
1i
T
min
1i
T
max
3i
T
3i
T
min
3i
T
(8)
This article focuses on optimum tuning of PSSs
using FPA. The target of the optimization is to
reduce the objective function to enhance the system
behaviour in terms of settling time and overshoots
under distinct loading conditions and then designing
a small order controller for easy application.
5. Results and Simulations
In this section, the sublimity of the developed FPA
algorithm in designing PSS compared with
optimized PSS with PSO and CPSS is illustrated.
The eigenvalues and their damping ratios for
distinct operating conditions and controllers are
displayed in Table 1. Also, the controller parameters
are obtained in Table 2.
5.1 Response under normal load condition
The validation of the behavior under distinct
disturbance is affirmed by applying a 20%
increase of mechanical torque of unit-1. Figs.5-7,
display the response of
23
,
13
and
12
due to this disturbance under normal loading
condition. It can be noted that the system with
the developed FPAPSS is more stabilized than
PSOPSS and CPSS. In addition, the needed
average settling time to alleviate system
oscillations is approximately 1.1 second with
FPAPSS, 1.86 second for PSOPSS, and 8.35
second with CPSS so the developed controller is
competent for provisioning proper damping to
the low frequency oscillations.
Table (1) Mechanical modes and
for distinct
loading events and algorithms.
FPAPSS
PSOPSS
CPSS
Light
load
-1.230.64j,0.89
-6.276.18j, 0.71
-3.055.62j,0.48
-0.620.87j, 0.58
-2.163.91j, 0.48
-3.057.41j,0.38
-0.190.69j,0.26
-2.354.15j, 0.48
-3.245.2j,0.52
Normal
load
-1.270.79j,0.85
-6.026.35j,0.69
-3.115.15j,0.52
-0.740.92j,0.62
-2.234.07j,0.48
-3.648.17j,0.41
-0.240.75j,0.3
-2.414.42j,0.47
-3.325.34j,0.52
Heavy
load
-1.080.86j,0.78
-7.075.02j,0.82
-4.237.44j,0.49
-0.710.79j,0.69
-1.58.72j,0.39
-3.288.01j,0.38
-0.330.89j,0.34
-1.964.32j,0.41
-3.095.25j,0.5
Table (2) Values of controllers for distinct
algorithms.
FPAPSS
PSOPSS
CPSS
PSS1
K=40.7381
T1=0.6326
T3=0.4738
K=29.4632
T1=0.4224
T3=0.6795
K=14.4386
T1=0.2652
T3=0.8952
PSS2
K=9.4541
T1=0.4673
T3=0.1851
K=7.5429
T1=0.6541
T3=0.3441
K=5.1659
T1=0.5242
T3=0.2032
PSS3
K=6.4623
T1=0.4324
T3=0.1971
K=7.2855
T1=0.6358
T3=0.3759
K=8.3287
T1=0.5817
T3=0.4268
shaped sector in the negative half of -Fig. 4. D
.plane the
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DOI: 10.37394/232027.2022.4.12
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E-ISSN: 2769-2507
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5.2 Response under light load condition
Figs. 8-10, show the response of the system under
light loading conditions with fixing the parameters
of the controller. It is clear that the developed
FPAPSS has good alleviation characteristics to
system modes and stabilizes rapidly the system.
Also, the average settling time of oscillations are
s
T
=1.12, 1.78, and 8.32 seconds for FPAPSS,
PSOPSS, and CPSS respectively. Hence, the
developed FPAPSS outlasts effectively PSOPSS
and CPSS in minifying oscillations and attenuating
settling time. Consequently, the developed FPAPSS
expands the limit of power system stability.
Fig. 8. Change in
23
for light load.
0 1 2 3 4 5 6 7 8 9 10
-6
-5
-4
-3
-2
-1
0
1
2
3
4
x 10-5
Time in second
Change in w23 (rad/second)
FPAPSS
PSOPSS
CPSS
Fig. 6. Change in
13
for normal load.
0 1 2 3 4 5 6 7 8 9 10
-2
0
2
4
6
8
10
12
14
x 10-5
Tiime in second
Change in w13 (rad/second)
FPAPSS
PSOPSS
CPSS
Fig. 5. Change in
23
for normal load.
0 1 2 3 4 5 6 7 8 9 10
-5
-4
-3
-2
-1
0
1
2
3
4
x 10-5
Time in second
Change in w23 (rad/second)
FPAPSS
PSOPSS
CPSS
Fig. 9. Change in
13
for light load.
0 1 2 3 4 5 6 7 8 9 10
-1
-0.5
0
0.5
1
1.5 x 10-4
Time in second
Change in w13 (rad/second)
FPAPSS
PSOPSS
CPSS
Fig. 7. Change in
12
for normal load.
0 1 2 3 4 5 6 7 8 9 10
-4
-2
0
2
4
6
8
10
12
14
16 x 10-5
Time in second
Change in w12 (rad/second)
FPAPSS
PSOPSS
CPSS
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DOI: 10.37394/232027.2022.4.12
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E-ISSN: 2769-2507
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Volume 4, 2022
5.3 Response under heavy load condition
Figs. 11-13, show the response of the system under
heavy loading conditions. These figures point to the
supremacy of the FPAPSS in lowering the settling
time and suppressing system oscillations. Also, the
average settling time of these oscillations are
s
T
=0.96, 1.26, and 3.76 seconds for FPAPSS,
PSOPSS, and CPSS respectively. Hence, FPAPSS
controller enhances greatly the alleviation
characteristics of power system.
5.4 Performance indices
To estimate the superiority of the developed
FPAPSS, several performance indices: the Integral
of Absolute value of the Error (IAE), and the
Integral of Time multiply Absolute value of the
Error (ITAE) are written as:
IAE
=
20
0132312 dtwww
(9)
ITAE
=
20
0132312 dtwwwt
(10)
Fig. 11. Change in
23
for heavy load.
0 1 2 3 4 5 6 7 8
-10
-8
-6
-4
-2
0
2
4
6
x 10-5
Time in second
Change in w23 (rad/second)
FPAPSS
PSOPSS
CPSS
Fig.12. Change in
13
for heavy load.
0 1 2 3 4 5 6 7 8
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5 x 10-4
Time in second
Change in w13 (rad/second)
FPAPSS
PSOPSS
CPSS
Fig. 10. Change in
12
for light load.
0 1 2 3 4 5 6 7 8 9 10
-1
-0.5
0
0.5
1
1.5
x 10-4
Time in second
Change in w12 (rad/second)
FPAPSS
PSOPSS
CPSS
Fig. 13. Change in
12
for heavy load.
0 1 2 3 4 5 6 7 8
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5 x 10-4
Time in second
Change in w12 (rad/second)
FPAPSS
PSOPSS
CPSS
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DOI: 10.37394/232027.2022.4.12
E. S. Ali
E-ISSN: 2769-2507
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Volume 4, 2022
The weaker the account of indices have, the
higher the system response is. Numeral results of
performance indices for distinct events are
recorded in Table (3). It is manifest that the values
of these indices with the FPAPSS are junior
compared with those of PSOPSS and CPSS. This
believes that the speed deviations of all units,
settling time, and overshoot, are constricted
extremely by setting the developed FPA based
tuned PSSs.
6. Conclusion
FPA is presented in this article for optimum
designing of PSSs parameters. The PSSs parameters
tuning problem is converted as an optimization
problem and FPA is employed to search for
optimum parameters. An eigenvalue based objective
function reflecting the combination of damping
factor and damping ratio is optimized for distinct
operating conditions. Simulation results evidence
the superiority of the developed FPAPSS in
assigning good damping behaviour to system
oscillations for distinct loading events. Also, the
developed FPAPSS affirms its efficacy than
PSOPSS and CPSS through some indices.
Coordination of PSS and FACT devices via FPA is
the future field of this work.
Appendix
Parameters of FPA: Maximum number of iterations
= 500, population size = 20, probability switch =
0.8.
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Table (3) Performance indices for distinct algorithms.
IAE * 10-4
ITAE * 10-4
CPSS
PSO
PSS
FPA
PSS
CPSS
PSO
PSS
FPA
PSS
Light
event
7.21
0.2663
0.0445
23.74
0.4642
0.2734
Normal
event
15.87
0.3973
0.0654
34.05
0.7756
0.5978
Heavy
event
24.79
0.5686
0.100
45.89
0.9729
0.8387
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2022.4.12
E. S. Ali
E-ISSN: 2769-2507
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Volume 4, 2022
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