PSSs Layout using Dandelion Optimization Approach
SAHAR M. ABD-ELAZIM
Computer Science Department, College of Engineering and Computer Science,
Jazan University,
Jizan,
KINGDOM OF SAUDI ARABIA
Abstract: - This paper develops an optimal Power System Stabilizers (PSSs) design employing the Dandelion
Optimization Algorithm (DOA) implemented in a multimachine system. The synthesizing of PSS parameters is
shaped as DOA-addressed optimization matter. An objective equation invoked by eigenvalue, incorporating
lightly damped electromechanical modes, damping ratio, and factor, is utilized for the PSS layout. The
functioning of the suggested DOA-based PSSs (DOAPSS) is evaluated against Differential Evolution-based
PSSs (DEPSS) under various running requirements and disturbances. The supremacy of the DOAPSS is
validated across time-domain analysis, eigenvalues, and functioning indices, demonstrating its superiority over
the DE approach.
Key-Words: - Dandelion Optimization Algorithm, Differential Evolution, Low-Frequency Fluctuations, Power
System Stabilizers, D-shaped, Stability Analysis.
Received: April 28, 2024. Revised: October 30, 2024. Accepted: November 16, 2024. Published: December 30, 2024.
1 Introduction
The stability of the power network remains a critical
issue in modern power system analysis, particularly
in interconnected systems. Long, heavily loaded
transmission lines can lead to various stability
challenges, prompting researchers to focus on
designing effective Power System Stabilizers (PSS),
[1], [2].
Recently, the field of "Heuristics from Nature"
has gained traction, leveraging analogies from
natural and social systems to solve complex
optimization problems. These methods have shown
promise in finding optimal solutions for non-
differentiable, multimodal, and complex objective
functions, [3]. Several heuristic techniques have
been applied to PSS design, including Differential
Evolution (DE) [4], Particle Swarm Optimization
[5], Bacteria Foraging [6], Bacterial Swarm
Optimization [7], [8], Harmony Approach [9], Bat
Algorithm [10], Approach of Water Cycle [11],
Approach of Backtracking Search [12], Approach of
Grey Wolf [13], Cuckoo Search Approach [14],
[15], Genetic Approach [16], Kidney-Inspired
Approach [17], Whale Optimization Approach [18],
[19], [20], [21], Farmland Fertility Approach [22],
Atom Search Approach [23], and Slime Mould
Approach [24]. No matter what, optimization of
PSS invariants remains a significant issue regarding
power system stability.
This paper introduces the Dandelion
Optimization Algorithm (DOA) as a novel method
for determining optimal PSS parameters. Inspired
by the wind-assisted seed propagation of
dandelions, [25], [26], [27], [28], [29], [30], the
DOA is tested and compared with the DE method.
Time-domain simulations are conducted using
MATLAB/Simulink under varying load conditions
to assess the effectiveness of the suggested
approach.
2 Mathematical Issue Pointing
2.1 Power Network Pattern
The complicated nonlinear pattern coupled with
m
generators correlated power grid is constituted by
the following group of differential nonlinear
equalizations:
(1)
X
is the state factors values,
U
is the entry factors
values. while
U
is the
PSS production signs in this research. are
the rotor speed and angle, consecutively. ,
fd
E
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and
q
E
are excitation, field, and the inner potential
consecutively.
The linearized gradual model about a certain
condition is applied commonly to PSS layout.
Consequently, the state framework including
PSSs is created as:
(2)
As
mm 55
matrix called
A
that similar to
Xf /
,
nm5
matrix called
B
that similar to
Uf /
. All matrices are considered at an appointed
working value.
X
vector of
15 m
status and
U
vector of
1n
inner values.
2.2 PSS Frame
Power system benefits still favour the Conventional
PSS (CPSS) model owing to its convenience of
online synthesizing besides the poor stability
guarantee to certain adjustments or variable
architecture techniques. In another way, an
extensive parsing of various CPSS variable's effects
on the aggregate power system dynamic
performance is attained in [1] and [2]. It is depicted
that the occasion pick of CPSS constants enables
acceptable performance through the system
turbulence. The
th
i
PSS model is illustrated as:
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 model is composed of a washout filter, a
magnification factor, a limiter and an active
compensator as appeared in Figure 1. The resulting
sign is inserted as a further entry sign,
i
U
to the
excitation mode regulator. The entry sign
i
is
the speed deflection from the contemporary one.
The stabilizer factor
i
K
is employed to find out the
damping quantity that shall be injected. While a
washout filter fights input sign oscillations to avert
terminal voltage steady-state error. In addition, duo
lead-lag networks are incorporated to eradicate any
lag between the electric torque and the excitation.
The limiter is embedded to prevent the PSS product
sign from leading the excitation process to heavy
fullness, [2]. The PSS and excitation process block
diagram is given in Figure 1.
3 Dandelion Optimization Algorithm
DOA is an invented technique from the plant seeds
motion behaviour. Dandelion plants depend on wind
to disseminate seeds. The two crucial elements that
influence the dandelion seed dissemination are
weather and wind speed. The previous element
practically influences the falling distance of seeds.
Within, weather influences the capability of saplings
In this research, the washout time value
W
T is considered as 10 seconds, the time
constant magnitudes of i
T2 and i
T4 have
steady reasonable values of 0.05 second. So it
is required to maintain the stabilizer time
constants i
T1,i
T3 and gain i
K.
2.3 Tested Grid
The tested grid which is composed of nine
points and three units as obvious in Fig. 2, is
examined here. The system loads and data are
pointed out in [2, and 7].
Fig. 1: CPSS and excitation system block
diagram.
Fig. 2: The tested grid.
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to implant near or far. DOA could be
mathematically modeled in 3 portions, which are the
upward portion, downward portion, and landing
portion. DOA is such as any population-based
technique assuming that any dandelion seed is a
nominated settling, [25].
d
p
x...
p
x
::::
d
x......x
d
x...x
A
...
1
2
1
2
1
...
1
1
(4)
(5)
(6)
(7)
The size of the population space is designated as
𝑝. The size of variables are denominated as 𝑑, while
𝑟𝑎𝑛𝑑 is a random value of [0,1].
3.1 Upward Part
In the rising phase, a windy and sunny mood lifts
dandelion seeds up. On the left side, when it rains
there is no wind. Region model seeking appears in
this part. Flying dandelion seeds are influenced by
wind humidity and speed. Dandelion seeds own the
distinction of being capable of flying away
considering the rising. In this part, two models
depending on weather are namely, [26].
Condition 1: Sunny day circumstances, wind speed
lognormal distribution is considered. In this part,
DOA provides exploration. The wind helps
dandelion seeds to fly wildly to various destinations.
The wind speed specifies the seed height. The
former part is formulated as:
(8)
min
)
minmax
(dim),1( x xxrandrand
s
x
(9)
y if
y if y
y
Y
00
0
2
)(ln
2
2
1
exp(
2
1
ln
(10)
(11)
(12)
(13)
(14)
(15)
The dandelion seed location through iteration is
named as
t
x
. Wildly nominated locations in the
search area via iteration are named as
s
x
. 𝑙𝑛𝑌 is a
lognormal distribution. 𝛼 is the adjusted amount that
adapt step length of the search. The grade of the
dandelion lifting passage because of the separate
eddies vigor are symbolized as
x
v
and
y
v
.
Condition 2: In rainy day circumstances, there is a
dandelion seed growing problem. In these
circumstances, regional exploitation is taking place.
The arithmetic model of part 2 is:
k
t
X
t
X
)1(
(16)
12
2
1
1
12
2
2
12
2
2
TTTT
t
TT
t
q
(17)
qrandk ()1
(18)
else k
t
x
k if randn
t
x
s
xnY
y
v
x
v
t
x
t
X
5.1)(
1
(19)
where 𝑘 is worked out to provide the local searching
range of an agent, 𝑟𝑎𝑛𝑑𝑛 accounted for the wild
values that achieve the fundamental normal
distribution, [27].
3.2 Downward Part
In this part, the exploration phase is employed. The
dandelion seeds locomotion will be certainly
reduced while getting a peak at a specific value. The
average data after the upward part is exercised to
mirror the settlement of the parental offspring. This
is to offer backing for the enhancement of the
aggregate population, [28]. The arithmetic
formulation of this part is:
)
_
(
1t
x
ttmean
x
tt
x
t
x

(20)
pop
ii
xpop
tmean
x
1
1
_
(21)
where
t
indicates the action of Points Brownian. It
is an irregular rate from the standard normal
distribution, [29].
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3.3 Landing Part
The exploitation stage takes place in this part. The
landing location of the dandelion seeds is picked at
stochastic. The dominant location of the most
effective dandelion seed is utilized as the optimum
selection. Elite data is momently exploited in the
region medium to attain inclusive optimum
precision. This attitude could be modelled [29] and
[30].
t
X
elite
Xlevy
elite
X
t
X)()(
1
(22)
t
Slevy
/1
)(
(23)
2/)1(
2
2
1
2
sin)1(

(24)
T
t2
(25)
where
elite
X
mirrors the superior location of the
operator in each iteration. (𝜆) represents the task of
Levy flight, [30].
4 Objective Function
To assure constancy and achieve further damping at
low frequency of fluctuations, the constants of the
PSSs can be selected to lessen the next equation:
np
jij ij
np
jij ij
t
J
10
2
)
0
(
10
2
)
0
(
(26)
This will locate the eigenvalues of the closed
loop system within the D-shape sector distinguished
by
0
ij
and
0
ij
as indicated in Figure 3.
where
and
are the attenuation rate and real part
of the eigenvalue of an operating point, np is the
working points numbers considered in the layout
process. In this paper,
0
and
0
are chosen to be
0.1 and -0.5 in the given order, [14]. Classic limits
of the optimized factors are from 1 to 100 for
K
,
and from 0.06 to 1 for
i
T1
, and
i
T3
. Optimization
task depends on the equation (26) and it could be
written as: reduce
t
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
(27)
This research depends on the optimal adjusting
of PSS via the DOA approach. The objective of the
optimization is to lower the equation (26) to
enhance the system attitude in terms of overshoots
and settling time for various working events and
lastly, lay a small size controller for successful
operation.
Fig. 3: D-shaped objective function
5 Simulation Outcomes
In this part, the effectiveness of the suggested DOA
approach in PSSs layout compared with optimized
PSSs with DE is evidenced. Figure 4 demonstrates
the variation of equation (26) via two optimization
approaches. The final value of equation (26) is
lessened via DE and DOA iterations. The final value
of equation (26) is zero for every algorithm,
demonstrating that whole modes have been moved
to the assigned D-sector location in the plane and
the suggested target function is attained.
Additionally, DOA converges at a superior rate (33
iterations) compared with DE (43 iterations). The
attenuation ratios and mechanical modes
eigenvalues are specified in Table 1 for three
operating events with duo approaches. It is apparent
that the DOAPSS attenuation factors have been
enhanced to be -1.13, -1.19, and -1.33. Also, the
eigenvalues have been moved to the left side in the
D-shape. Additionally, the DOAPSS attenuation
rates are further than the other controller. Thus,
DOAPSS provides superior attenuation behavior
with respect to DEPSS. The constants of DE and
DOA controllers are reported in Table 2.
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010 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
Iterations
Change of objective function
DE
DOA
Fig. 4: Objective function variations
Table 1.
and modes for three operating events for
both approaches
DEPSS
DOAPSS
Light
load
-1.05 0.66j,0.85
-3.73 6.22j,0.51
-3.61 5.93j,0.52
-1.13 0.63j,0.87
-6.31 6.28j, 0.71
-3.43 5.11j,0.56
Normal
load
-1.13 0.72j,0.84
-4.27 7.00j, 0.52
-4.18 8.02j, 0.46
-1.19 0.68j,0.87
-6.9 6.78j,0.71
-3.38 5.22j,0.54
Heavy
load
-1.16 0.71j, 0.85
-3.5 6.71j,0.46
-3.04 5.2j,0.5
-1.33 0.71j,0.88
-7.98 5.33j,0.83
-4.64 7.26j,0.54
Table 2. Constants of PSSs for both algorithms
DOA
DE
PSS1
K=44.732
T1=0.6364
T3=0.5281
K=29.6446
T1=0.5134
T3=0.7248
PSS2
K=9.1123
T1=0.4637
T3=0.1863
K=7.6338
T1=0.3721
T3=0.2344
PSS3
K=6.4631
T1=0.4241
T3=0.1922
K=4.8271
T1=0.4261
T3=0.2713
5.1 Response via Normal Loading
The ratification of the grid functioning owing to a
20% raise of generator 1 mechanical torque as a
little disturbance is achieved. Figure 5 and Figure 6,
illustrates the outcome of
12
and
13
owing to
this disturbance under normal loading event. The
system with the suggested DOAPSS is more
steadied compared to DEPSS. Also, the average
needed steading time to damp grid fluctuations is
around 1.1 seconds for DOAPSS, and 1.6 seconds
with DEPSS so the nominated controller is eligible
for providing sufficient attenuation to the low-
frequency fluctuations.
0 1 2 3 4 5 6
-2
0
2
4
6
8
10
12
14
16 x 10-5
Time in second
Change in w12 (rad/second)
DOAPSS
DEPSS
Fig. 5: Variations of
12
for normal loading
0 1 2 3 4 5 6
-2
0
2
4
6
8
10
12
14
x 10-5
Tiime in second
Change in w13 (rad/second)
DOAPSS
DEPSS
Fig. 6: Output of
13
for normal loading
5.2 Response via Light Loading
Figure 7 and Figure 8, illustrates the grid
performance via light loading event without
changing the controller constants. It is obvious from
these outputs, that the suggested DOAPSS has better
attenuation behavior on network oscillatory modes,
settling down the grid quickly. Also, the average
steading time of fluctuations is 2.4 and 1.4 seconds
for DEPSS, and DOAPSS consecutively. Hence, the
suggested DOAPSS overcomes the DEPSS
controller in mitigating oscillations efficiently and
diminishing steading time. Therefore, the suggested
DOAPSS increases the stability ceiling of the tested
grid.
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0 1 2 3 4 5 6
-2
0
2
4
6
8
10
12
14
16 x 10-5
Time in second
Change in w12 (rad/second)
DOAPSS
DEPSS
Fig. 7: Variations of
12
for light loading
0 1 2 3 4 5 6
-3
-2
-1
0
1
2
3
4
x 10-5
Time in second
Change in w23 (rad/second)
DOAPSS
DEPSS
Fig. 8: Variations of
23
for light loading
5.3 Response via Heavy Loading
Figure 9 and Figure 10, illustrates the system
performance via heavy loading event. These outputs
denote the supremacy of the DOAPSS in lessening
the steading time and repressing power grid
fluctuations. Additionally, the average settlement
time of these fluctuations is 1.47 and 1.1 seconds for
DEPSS, and DOAPSS consecutively. Thus, the
DOAPSS controller highly alleviates the attenuation
behavior of the tested grid. Moreover, the settlement
time of the suggested PSSs is lower than that in [6]
and [8].
0 1 2 3 4 5 6
-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)
DOAPSS
DEPSS
Fig. 9: Variations of
12
for heavy loading
0 1 2 3 4 5 6
-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)
DOAPSS
DEPSS
Fig. 10: Variations of
13
for heavy loading
5.4 Response for Large Perturbation
Figure 11 and Figure 12, illustrates the performance
of
12
and
13
via serious perturbation. It is
emphasized by the fulfillment of a three-phase fault
of 6 interval duration at 1.0 seconds near node 7.
From these figures, the DOAPSS utilizing the
suggested objective function offers convenient
damping and gains powerful behavior compared
with the other methods.
0 1 2 3 4 5 6
-2
-1.5
-1
-0.5
0
0.5
1x 10-3
Time in second
Change in w12 (rad/second)
DOAPSS
DEPSS
Fig. 11: Variations of
12
under severe
disturbance
0 1 2 3 4 5 6
-1
-0.5
0
0.5
1
1.5
2x 10-3
Time in second
Change in w13 (rad/second)
DOAPSS
DEPSS
Fig. 12: Variations of
13
under severe
perturbation
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5.5 Durability and Performance Indices
To prove the durability of the suggested controller,
certain performance indices: Integral of Time
multiplied Absolute amount of the Error (ITAE) and
Integral of Absolute amount of the Error (IAE) are
being employed:
ITAE
=
0132312 dtwwwt
(28)
IAE
=
0132312 dtwww
(29)
It is trustworthy that the values of these indices
are lessened a system better functioning in terms of
time domain behavior [8] and [31]. Digital results of
robust realization for whole events are tabulated in
Table 3. The values of these system indices for
DOAPSS are lower compared to DEPSS. This
signalizes that the overshoot, steading time, and
speed deflections of all units are highly reduced by
employing the suggested PSSs via DOA tuning.
Also, the values of these pointers are lower than
those gained in [10] and [14].
Table 3. Behaviour pointers for both algorithms
loading
IAE *10-4
ITAE *10-4
DEPSS
DOAPSS
DEPSS
DOAPSS
Light
0.1484
0.0442
0.4148
0.2704
Normal
0.2648
0.0648
0.7551
0.5916
Heavy
0.4126
0.1000
0.9406
0.8397
6 Conclusions
A modern optimization approach named as DOA
search approach, for optimum modeling of PSS
constants is suggested in this research. The PSS
constants synthesizing trouble is elaborated as an
optimizing one and DOA is utilized to obtain the
optimum constants. An objective function basis
eigenvalue mirroring the composition of attenuation
agents and attenuation ratio which is optimized for
different working events. Simulation results assure
the validity of the suggested controller to provide
better attenuation behavior for grid fluctuations over
a large domain of loading events. Also, the network
behavior with regard to the ‘ITAE’ and ‘IAE’
indices shows that the suggested DOAPSS
illustrates its supremacy over DEPSS. Application
of such suggested algorithm tuned via new
optimization approaches to highly level power
systems is the outlook domain of this paper.
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APPENDIX
a) The constants of DOA are given as: Population
size = 50; Maximum generation = 100.
b) The constants of DE are given as: Mutation
probabilities = 0.5; Crossover probabilities =
0.5; and count of population = 100.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Prof. Sahar M. Abd Elazim carried out completely
this paper.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on ELECTRONICS
DOI: 10.37394/232017.2024.15.18
Sahar M. Abd-Elazim
E-ISSN: 2415-1513
165