Power System Stability Enhancement Using Grasshopper
Optimization Approach and PSSs
E. S. ALI1, S. M. ABD ELAZIM2
1Electrical Department, Faculty of Engineering, Jazan University, Jazan, KINGDOM OF SAUDI ARABIA
2Computer Science Department, Faculty of Computer Science and Information System, Jazan University,
Jazan, KINGDOM OF SAUDI ARABIA
Abstract: A new meta-heuristic algorithm namely Grasshopper Optimization Approach (GOA) for Power System
Stabilizer (PSS) design problem is investigated in this paper. The parameters of PSSs are optimized by GOA to
minimize the time domain objective function. The performance of the designed GOA based PSSs (GOAPSS) has
been has been compared with Differential Evolution (DE) based PSSs (DEPSS) and the Particle Swarm
Optimization (PSO) based PSSs (PSOPSS) under various loading events. The results of the proposed GOAPSS
are confirmed via eigenvalues, damping ratio, time domain analysis, and performance indices. Moreover, the
robustness of the GOA in getting good damping characteristics is verified.
KeyWords: Power System Stabilizers; Grasshopper Optimization Algorithm; Particle Swarm Optimization;
Differential Evolution; Power System Stability; Low Frequency Oscillations.
Received: August 11, 2022. Revised: July 24, 2023. Accepted: August 27, 2023. Published: October 3, 2023.
1. Introduction
Power system stability is one of the recent significant
issues in the analysis of power systems [1]. One of
the compulsory instances of this is an interconnected
power system. The heavily loaded long tie-lines
could account for a variety of stability issues [2]. This
leads to the inclination of most researchers towards
designing a suitable Power System Stabilizer (PSS).
Recently, a lot of research work is based on an area
called “Heuristics from Nature” in which the
analogies of nature or social systems are being
utilized [3]. These techniques when used in the
research community can prove their capability of
finding optimal solutions of multi-model, non-
differentiable and complex objective functions.
Various new algorithms have been used for designing
a PSS as Differential Evolution (DE) [4], Particle
Swarm Optimization (PSO) [5], Bacterial Swarm
Optimization (BSO) [6-7], Harmony Search
Approach (HSA) [8-9], Bacterial Foraging (BF) [10-
11], Bat Algorithm (BA) [12-13], Water Cycle
Approach (WCA) [14], Backtracking Search
Approach (BSA) [15-16], Grey Wolf Approach
(GWA) [17], Whale Optimization Approach (WOA)
[18], Cuckoo Search Approach (CSA) [19-20],
Flower Pollination Approach (FPA) [21], Genetic
Approach (GA) [22], Kidney-Inspired Approach
(KIA) [23], etc. All of these algorithms are based
upon Artificial Intelligence (AI).
A new nature-inspired technique inspired from social
activities of grasshoppers is introduced by Mirjalili.
The technique is termed as Grasshopper
Optimization Approach (GOA) [24]. Because of its
simplicity, avoiding the high local optimum value as
well as gradient-free mechanism, and inspiration by
nature, it has been commonly implemented these
days. Therefore, the effectiveness of implementing
the proposed approach to handle real-life issues is
evaluated. The solutions must be upgraded in nature-
inspired algorithms until the end criterion is met.
Alongside this the optimization procedure partitioned
in two stages named exploration and exploitation.
Exploration relates to the algorithm's tendency to
have randomized behaviour to change the solutions.
Large variations in solutions lead to more search
space exploration and subsequently discovery of its
promising areas. However, as an approach tends to
exploit, solutions usually encounter smaller-scale
variations and tend to search locally. An appropriate
exploration and exploitation balance can lead to the
search for the global optimum of a specified
optimization problem. It is evident from [24] that the
GOA method gives improved results as compared
with several optimization techniques. Previous works
clearly reflect the growing interest of the researchers
in designing PSS when it comes to stability
improvement. Further, the GOA technique has not
been used.
2. Mathematical Problem Formulation
2.1 Power System Model
Generally, a power system can be established by a
group of nonlinear differential equations as:
f
(1)
Where and are the vectors of the state variables
and of input variables. In this study,
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DOI: 10.37394/232016.2023.18.14
E. S. Ali, S. M. Abd Elazim
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and is the output of PSSs.
,
and are the internal, the field, and excitation
voltages respectively. Also, and are the rotor
angle and speed, respectively.
In the design of PSS, the state equation of a power
system can be formalized as:
(2)
2.2 PSS Structure
Due to the ease of online tuning, power system
companies prefer the structure of conventional PSS
(CPSS). The appropriate selection of the CPSS
parameters results in satisfactory performance
during the system disturbances. The CPSS can be
modeled 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)
Fig. 1. shows the block diagram of CPSS and
excitation system. The model of CPSS contains a
limiter, a gain, a dynamic compensator and washout
filter. To avoid the delay between the excitation and
the electric torque, two lead-lag circuits are included
[1, 2]. In this paper, the time constants
i
T1
, and
i
T3
,
and the gain
i
K
are optimized by GOA to reduce a
time domain objective function.
2.3 Test System
A multimachine system that consists of three
generators and nine buses is considered here. The
system data and loading events are given in [2, 25].
3. Grasshopper Optimization
Approach
GOA is an intelligence approach presented by
Mirjalili [24]. It is a population based method which
imitates grasshopper swarming behaviour. It is an
insect pest since its destructive effect on crops. Its
life has two stages, nymph and adulthood. For the
nymph stage, the insects have no wings so they
move slowly but after growing up they become
adults with wings that allow them to move very fast
covering a large scale area. Grasshopper swarming
might be considered as the largest one among all
creatures as it is a nightmare for farmers [26].
In the swarming process, there is a larval phase
which is characterized by slow movement with
small grasshopper steps but for adults long-rang and
abrupt movements. In the food seeking process,
grasshoppers follow two strategies, exploration and
exploitation. Each grasshopper represents a
solution, the next position Xj is influenced by the
social interaction between grasshopper and the other
one Sj, gravity force Gj and wind advection Aj as
shown in the following equation:
(4)
Social interaction can be calculated by the following
equation:
(5)
(6)
Where N is no. of grasshoppers, dik is the distance
from grasshopper k to grasshopper i and s is the
strength of attraction and repulsion forces between
grasshoppers. Since repulsion force appears when
distance between grasshoppers between zero and
2.079 units, while at a distance of 2.079 neither
repulsion or attraction force as it is a comfortable
zone. Attraction force increases at a distance greater
than 2.079 until reach 4 then it decreases and after
10 there will be no forces. Form the previous, the
interval should be from 1 to 4 and s calculated as
following:
(7)
Where a is the intensity of attraction and is the
attractive length scale.
Gravity force can be calculated by the following
equation:
(8)
Where is a gravitational constant and is the
center of the earth unit vector.
Wind advection force Aj can be determined by the
following equation:
(9)
Where is a drift constant and is the wind
direction unit vector.
Fig. 1. Block diagram of CPSS with
excitation system.
-
+
+
-
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Equation (4) will be represented as following:
(10)
To avoid comfortable zone and global optimum, the
grasshopper position will be
(11)
Where ubd and lbd represent upper and lower bounds
respectively in Dth dimension,
is the target value
assuming wind direction tends towards target and c
is decreasing constant to minimize all zones
neglecting gravity [26].
(12)
is the current iteration, cmin=10-5 , cmax= 1 and
L is the maximum number of iterations. Fig. 2 shows
the flowchart of GOA. In addition, other recent
applications for GOA can be found in [27- 28].
4. Objective Function
An Integral Time Absolute Error (ITAE) of the
speed deviation of a generator is considered as the
proposed objective function. It can be written as:
t
sim
t
d t J
0132312
(13)
The lower and upper limits of the stabilizer gain are
[1- 50]. Also, these limits are [0.06 -1.0] for
i
T1
and
i
T3
. Other time constants
i
T2
and
i
T4
are fixed at
0.05 second. GOA searches for the optimal
parameters of PSSs to enhance the damping
behaviour and reduce the overshoot and settling
time of the system response.
5. Results and Analysis
The eigenvalues and their damping ratios of
mechanical modes are given in Table (1) for three
various loading conditions and different approaches.
It is obvious that the damping factors corresponding
to GOAPSS have improved to be (
=-1.12,- 1.19,-
1.32) and the eigenvalues have been shifted to the
left of S plane. Moreover, the damping ratios related
to GOAPSS are greater than other controllers. Thus,
GOAPSS gives better damping performance
compared with DEPSS and PSOPSS. Also, the
parameters of each controller using GOA, DE and
PSO are shown in Table (2).
5.1 Response for light load event:
The effectiveness of the decided controller is proved
by setting a 3 phase fault near bus 7 of 6 cycle at 1
second. The system response is shown in Fig. 3 for
light load event. It is obvious that the system
responses with the decided GOAPSS are better than
PSOPSS and DEPSS. Also, the settling times are
2.2, 3.2, and 3.5 second with GOAPSS, DEPSS, and
PSOPSS respectively. The decided controller is
competent to assign appropriate damping
characteristics compared with DEPSS and PSOPSS.
start
Initialize the GOA parameters, set
iter=1 and maximum iteration (N)
Generate initial population and
calculate the fitness of each search
agent
Determine the best search
agent
Update c
Normalize distance between
grasshoppers for each agent
Update position of current
search agent
If iter < N
Is the updated
position in the
search
boundary?
Update search agent
Return best search agent
iter = iter +
1
Push
search
agent into
the search
boundary
Stop
Yes
Yes
N
o
N
o
Fig. 2. Flow chart of the GOA.
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5.2 Response for normal load event:
The system response under normal loading is given
in Fig. 4. From this response, the damping behaviour
has been improved by the decided GOAPSS. The
settling times of these responses are
s
T
=2.4, 3.1,
and 3.2 second for GOAPSS, DEPSS, and PSOPSS
respectively. Also, the decided GOAPSS outlasts
DEPSS and PSOPSS in mitigating oscillations and
shortening settling time. Hence, the decided
GOAPSS expands the system stability limit.
5.3 Response for heavy load event:
Fig. 5, gives the response for heavy loading event.
The superiority of the GOAPSS in attenuating
system oscillations and minimizing the settling time
are indicated. Also, the settling times of these
oscillations are
s
T
=2.5, 3.1, and 3.3 second for
GOAPSS, DEPSS, and PSOPSS respectively.
Hence, the GOAPSS controller largely develops the
system stability and increases the damping
behaviour of the power system. Moreover, the
settling times of the decided GOAPSS are shorter
than these in [5, 12, and 19].
Table (1) Mechanical modes and for various
loading events and approaches.
PSO
PSS
DE
PSS
GOA
PSS
Light
load
-0.22
0.67j, 0.31
-2.43
4.01j, 0.51
-3.45
7.1j,0.44
-1.06
0.66j,0.85
-3.75
6.23j,0.51
-3.65
5.94j,0.52
-1.12
0.64j,0.87
-6.3
6.34j, 0.70
-3.33
5.12j,0.54
Normal
load
-0.36
0.72j,0.37
-2.41
4.32j,0.48
-3.64
8.17j,0.41
-1.12
0.68j,0.85
-4.29
7.0j,0.52
-4.21
8.02j,0.46
-1.19
0.69j,0.87
-6.9
6.88j,0.71
-3.37
5.24j,0.54
Heavy
load
-0.35
0.89j,0.36
-1.99
4.31j,0.42
-3.8
8.9j,0.39
-1.19
0.71j,0.86
-3.52
6.7j,0.47
-3.06
5.15j,0.51
-1.32
0.72j,0.88
-7.99
5.34j,0.83
-4.65
7.29j,0.54
Table (2) Parameters of controllers for several
approaches.
GOA DE PSO
PSS1 K=42.128
T1=0.5436
T3=0.428
K=27.4566
T1=0.5264
T3=0.7578
K=17.4736
T1=0.4224
T3=0.7853
PSS2 K=9.4211
T1=0.4723
T3=0.1643
K=7.9983
T1=0.3108
T3=0.1469
K=6.3649
T1=0.5542
T3=0.3231
PSS3 K=5.2641
T1=0.3234
T3=0.1861
K=4.7541
T1=0.5361
T3=0.3931
K=7.8875
T1=0.5668
T3=0.4567
Fig. 3. Change of for light load event.
Fig. 4. Change of for normal load event.
Fig. 5. Change of for heavy load event.
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5.4 Response under small disturbance
The responses of are given in Fig. 6 due to 0.2
step increase in mechanical torque of machine 1 like
a small disturbance. It is clear from these figures,
GOAPSS presents supreme damping and acquires
the best behaviour compared with DEPSS and
PSOPSS.
5.5 Performance indices
To assign the superiority of the decided GOAPSS,
some performance indices: the Integral of Absolute
value of the Error (IAE), and ITAE are considered
as:
IAE
=
20
0132312 dtwww
(14)
ITAE
=
20
0132312 dtwwwt
(15)
The weaker the value of indices have, the more
supreme the system response is. Numeral results of
performance indices for various events are given in
Table (3). It is obvious that the values of these
indices with the GOAPSS are junior compared with
those of DEPSS and PSOPSS. This asserts that the
speed deviations of all generators, settling time, and
overshoot, are extremely diminished by setting the
decided GOA based tuned PSSs.
6. Conclusions
GOA is introduced in this paper for optimal
designing of PSSs parameters as minimizing the
proposed time domain objective function. An ITAE
of the generator speed is considered as the objective
function to enhance the system stability. Simulation
results evidence the superiority of the decided
GOAPSS in assigning good damping behaviour to
system oscillations for several loading events.
Moreover, the decided GOAPSS affirms its efficacy
than PSOPSS and DEPSS through some indices.
Coordination of PSS and FACT controller with
GOA is the future scope of this work.
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that are relevant to the content of this article.
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WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2023.18.14
E. S. Ali, S. M. Abd Elazim
E-ISSN: 2224-350X
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Volume 18, 2023
