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Citation: Zhang, Y.; Li, H. Research
on Economic Load Dispatch Problem
of Microgrid Based on an Improved
Pelican Optimization Algorithm.
Biomimetics 2024,9, 277. https://
doi.org/10.3390/biomimetics9050277
Academic Editors: Yongquan Zhou,
Huajuan Huang and Guo Zhou
Received: 5 March 2024
Revised: 29 April 2024
Accepted: 29 April 2024
Published: 4 May 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
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Attribution (CC BY) license (https://
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4.0/).
biomimetics
Article
Research on Economic Load Dispatch Problem of Microgrid
Based on an Improved Pelican Optimization Algorithm
Yi Zhang * and Haoxue Li *
College of Electrical and Computer Science, Jilin Jianzhu University, Changchun 130119, China
*Correspondence: zhangyi@jlju.edu.cn (Y.Z.); l17339995562@hotmail.com (H.L.)
Abstract: This paper presents an improved pelican optimization algorithm (IPOA) to solve the
economic load dispatch problem. The vertical crossover operator in the crisscross optimization
algorithm is integrated to expand the diversity of the population in the local search phase. The
optimal individual is also introduced to enhance its ability to guide the whole population and add
disturbance factors to enhance its ability to jump out of the local optimal. The dimensional variation
strategy is adopted to improve the optimal individual and speed up the algorithm’s convergence. The
results of the IPOA showed that coal consumption was reduced by 0.0292%, 2.7273%, and 3.6739%,
respectively, when tested on 10, 40, and 80-dimensional thermal power plant units compared to POA.
The IPOA can significantly reduce the fuel cost of power plants.
Keywords: economic load dispatch; pelican optimization algorithm; crisscross optimization
algorithm; dimensional variation strategy
1. Introduction
The economic load dispatch (ELD) problem is a fundamental problem in power system
control and operation [
1
]. The goal of ELD is to find the best feasible power generation plan
with the lowest fuel cost to meet the generation constraints of the generator set [
2
]. The
power generation system must also comply with various practical limitations due to the
physical constraints associated with the system in addition to meeting the system’s power
needs. These limitations result in the ELD problem being a non-convex, non-continuous,
non-differentiable optimization problem with many equality and inequality constraints [
3
].
There is much literature on the ELD problem, proposing many methods. These algo-
rithms can be mainly divided into two categories. One is traditional optimization methods,
such as the gradient method [
4
], Lambda iterative method [
5
], and quadratic programming
method [
6
]. These methods may not converge to a feasible solution in the solving pro-
cess [
7
,
8
], and it is not easy to get a satisfactory solution in an adequate time [
9
]. The second
category is intelligent optimization algorithms inspired by nature’s physical or biological
behavior [
10
], which have the characteristics of flexible mechanisms, simple operation,
and efficient solutions [
11
,
12
]. They have advantages in solving large-scale and highly
complex optimization problems [
13
,
14
]. Many swarms’ intelligent optimization algorithms
have recently been applied to solve the ELD problem [
15
]. Arman Goudarzi et al. [
16
]
proposed a new algorithm, MGAIPSO, based on an improved genetic algorithm and a
version of particle swarm optimization. Namrata Chopra et al. [
17
] proposed an improved
particle swarm optimization algorithm using the simplex method. Seyedgarmroudi, S.D.
et al. [
18
] proposed an improved pelican optimization algorithm, which benefited from
three motion strategies, predefined knowledge-sharing factors, and a modified dimen-
sion learning-based hunting (DLH). Singh, N. et al. [
19
] utilized a new variant of particle
swarm optimization. Lotfi, H. et al. [
20
] proposed an improved modified grasshopper
optimization algorithm, based on the chaos mechanism. Ismaeel, A.A.K. et al. [
21
] used the
osprey optimization algorithm. Said, M. et al. [
22
] utilized the walrus optimizer. Almalaq,
Biomimetics 2024,9, 277. https://doi.org/10.3390/biomimetics9050277 https://www.mdpi.com/journal/biomimetics
Biomimetics 2024,9, 277 2 of 18
A. et al. [
23
] introduced a new multi-objective optimization technique combining the dif-
ferential evolution (DE) algorithm and chaos theory. Dey, B. et al. [
24
] proposed a new
optimization algorithm combining the greedy JAYA algorithm with an algorithm based
on a crow’s food-seeking approach. Acharya, S. et al. [
25
] proposed the multi-objective
multi-verse optimization (MOMVO) algorithm. These algorithms have been applied to
solve ELD problems and have achieved good results. However, there is still room for
further improvement in the quality and applicability of these algorithms. Therefore, to
solve the ELD problem more effectively, exploring the algorithm with better optimization
ability, higher solution accuracy, and more stable solution results is necessary.
Many excellent meta-heuristics have been proposed in recent years, such as the liver
cancer algorithm (LCA) [
26
], slime mould algorithm (SMA) [
27
], moth search algorithm
(MSA) [
28
], parrot optimizer (PO) [
29
], rime optimization algorithm (RIME) [
30
], and
pelican optimization algorithm (POA) [
31
]. The POA is a new meta-heuristic intelligence
algorithm proposed by Pavel Trojovskýet al. in 2022. The algorithm has the characteristics
of simple theory, easy implementation, and good solving performance, and is suitable
for solving large-scale complex optimization problems, including ELD problems [
32
].
Therefore, many scholars have conducted in-depth research and applied it to different
fields [
33
]. For example, Song, H.M. et al. [
34
] proposed an improved POA based on
chaotic interference factors and essential mathematical functions and applied it to four
engineering design problems. Eluri, R.K. et al. [
35
] proposed a chaotic binary search gecko
optimization algorithm. By converting the basic algorithm into binary and chaotic search
and enhancing the POA’s exploration and development process, Li, J. et al. [
36
] used elite
reverse learning, introduced Levy flight to improve the POA, and applied it to microgrid
scheduling. Xiong, Q. et al. [
37
] improved the POA by introducing fractional order chaotic
sequence and applied it to the memo chaotic system parameter identification. Tuerxun
et al. [
38
] optimized the generalized learning system’s parameters by improving the POA.
Abdelhamid, M. et al. [
39
] proposed an improved pelican optimization algorithm and
applied it to the protection of distributed generators. Chen, X. et al. [
40
] used the pelican
optimization algorithm (POA) to optimize the neural network prediction model, which
significantly improved the model’s accuracy. Zhang, C. et al. [
41
] proposed a symmetric
cross-entropy multilevel threshold image segmentation method (MSIPOA) with a multi-
strategy improved pelican optimization algorithm for global optimization and image
segmentation tasks.
The above improvements have enhanced the application capability of the POA in their
respective fields. However, according to the NFL (no free lunch) [
42
] theorem, there is
no single algorithm that can solve all optimization problems [
43
]. Therefore, there is still
room for further enhancement of the stability of the POA and its suitability for large-scale
complex applications [44].
This paper proposes an improved pelican optimization algorithm (IPOA) to solve the
ELD problem and improve the POA’s search performance and quality. This IPOA utilizes
the crisscross optimization algorithm and introduces disturbance factors and dimensional
variation strategy. First, in the local search phase, the crisscross optimization algorithm is
integrated to expand the diversity of the population. After that, the optimal individual
is introduced to enhance the guiding ability, accelerate the convergence speed, and add
a disturbance factor to enhance the ability to jump out of the local optimal. Thirdly, the
dimensional variation strategy is adopted to improve the optimal individual and speed up
the algorithm’s convergence. In this paper, the effectiveness of the IPOA is tested on eight
CEC2017 test functions. The experimental results show that the optimization performance
and quality of the IPOA are better than those of the other four algorithms. At the same
time, the IPOA is used to solve the ELD problem. It is applied in different units of 10,
40, and 80 dimensions, respectively. The experimental results show that the IPOA has
good optimization ability and reliability and can effectively solve the problem of the high
operating costs of power systems.
Biomimetics 2024,9, 277 3 of 18
The structure of this paper is as follows: Section 2establishes the mathematical model
of the ELD and introduces the pelican optimization algorithm, including its improved
version (IPOA), which is subsequently tested on the CEC2017 test functions, with the
results analyzed. In Section 3, the IPOA is applied to the ELD problem with 10, 40, and
80 units,
and its ability to solve practical problems is tested. Section 4then summarizes
the findings.
2. Materials and Methods
2.1. Relationship Work
2.1.1. Electric Power Economic Load Dispatch (ELD)
The problem of electric power economic load dispatch (ELD) is an important power
system optimization problem at present. Minimizing the cost is the objective under the
premise of satisfying the equation and inequality constraints. The following objectives and
constraints were considered in the formulation of this paper. The objective function in the
ELD problem can be expressed as:
Min N
∑
i=1
Fi(pi)(1)
In Equation (1), Nis the total number of generator sets,
Fi
is the fuel cost function of
the i
th
generator set, and
pi
is the generation capacity of the i
th
generator set according to
the generation plan. The generator’s cost function is derived from data points acquired
during the “hot run” test. The opening of the steam intake valve changes discontinuously
when the load is adjusted in the thermal generator set. It will cause the efficiency and cost
of the unit to fluctuate. This phenomenon is known as the valve point effect, and it stops
the cost curve from being smooth. Therefore, the valve point effect must be included in the
cost model to represent the power generation cost curve more accurately. Therefore, the
actual output power of the total fuel cost can be expressed as [45]:
F(pi)=aip2
i+bipi+ci+
ei∗sin{fi∗pmin
i−pio
(2)
In Equation (2),
Fi(pi)
represents the fuel cost function of the i
th
unit, and
pi
represents
the generation capacity of the i
th
unit according to the generation plan. The parameters a,b,
and care constants determined by the physical characteristics of the unit, the parameters e
and fare coefficients describing the valve point effect, and
pmin
i
represents the minimum
power output of the ith unit.
The capacity constraints must be met to ensure the safe operation of thermal power
units; the formula is as follows:
pmin
i≤pi≤pmax
i(3)
In Equation (3),
pmin
i
and
pmax
i
represent the minimum and maximum power output
of the i
th
power unit, respectively. The sum of power of each unit needs to be consistent
with the total load because power transmission loss is ignored in this paper, and the load
balance constraint formula is as follows:
N
∑
i=1
pi=pd(4)
In Equation (4), pdrepresents the load demand.
This paper presents a penalty mechanism method to deal with the constraints in
the ELD problem to balance the objective function and constraints and transform the
constrained problem into an unconstrained problem. The solution in the optimization
process is forced to meet all constraints by the introduction of a penalty term into the
Biomimetics 2024,9, 277 4 of 18
objective function. The objective function after the introduction of the penalty term can be
described as:
Min(
N
∑
i=1
fi(pi)+ε∗ |
N
∑
i=1
pi−pd|)(5)
In Equation (5),
N
∑
i=1pi
represents the total generating capacity of all units according to
the power generation plan, and εis the penalty function coefficient.
2.1.2. Pelican Optimization Algorithm
The pelican optimization algorithm is a natural heuristic algorithm proposed by
Pavel Trojovskýet al. in 2022 [
31
]. The model simulates pelicans’ hunting behavior. It
can be divided into two stages: approaching prey (exploration stage) and surface flight
(development stage).
Population initialization
Before hunting, it is necessary to initialize the pelican population, where each member
represents a candidate solution represented by a vector. The mathematical model is shown
in Equation (6):
Xi,j=lj+rand ∗uj−lj,i=1, 2, . . . , N,j=1, 2 . . . , m(6)
In Equation (6),
Xi,j
represents the position of the i
th
pelican in the jdimension, Nis the
population number of pelicans, mis the dimension of the problem, and rand represents the
random number [0, 1].
uj
and
lj
represent the upper and lower bounds of the J
th
dimension
of the problem, respectively.
Exploration phase
In the first stage, the prey positions are randomly generated within the search space,
and the pelicans determine the prey positions. If the objective function value of the pelicans
is less than that of the prey, they move towards the prey; otherwise, they move away from
the prey. Its mathematical model is shown in Equation (7):
XP1
i=Xi+rand ∗(P−I∗Xi),Fp<Fi
Xi+rand ∗(Xi−P),else (7)
In Equation (7),
XP1
i
represents the position of the i
th
pelican after the first stage update,
Irepresent 1 or 2 random integers, Prepresents the position of the prey, rand represents
the random number [0, 1],
Fp
represents the fitness value of the prey, and
Fi
represents the
fitness value of the ith pelican.
The pelican updates its position if the fitness value of the new position is better than
the previous position after the pelican moves toward the prey. Its mathematical model is
shown in Equation (8):
Xi=Xnew
i,Fnew
i<Fi
Xi,else (8)
In Equation (8),
Xnew
i
represents the updated position of the i
th
pelican, and
Fnew
i
represents the fitness value of the updated new position.
Development phase
In the second stage, after the exploration stage is completed, the pelicans enter the
exploitation stage. Upon reaching the water surface, the pelicans capture the prey. During
this stage, the algorithm searches for points within the neighborhood of the pelican’s
position to achieve better convergence. Its mathematical model is shown in Equation (9):
XP2
i=Xi+R∗1−t
T∗(2∗rand −1)∗Xi(9)
Biomimetics 2024,9, 277 5 of 18
In Equation (9),
XP2
i
represents the position of the i
th
pelican after the second stage
update, Ris the constant 0.2, rand represents the random number [0, 1], and tand T
represent the current and maximum iterations, respectively.
In the development phase, the location is updated if the fitness value of the new
location is better than the location before the move after the pelican location is updated as
in the exploration phase. If not, it is left in place.
2.2. Improved Pelican Optimization Algorithm
In this paper, three strategies were introduced to improve the accuracy, convergence
speed, and robustness of the POA.
2.2.1. Fusion of Improved Crisscross Optimization Algorithm for Local Search
Crisscross optimization algorithm (CSO) [
46
] is a new search algorithm proposed by
An-bo Meng et al. in 2014. The CSO uses vertical and horizontal crosses to update the
position of individuals in a population, inspired by the cross operation in the Confucian
mean and genetic algorithm. The horizontal crossing is the arithmetic crossing of all
dimensions between two different individuals, whose calculation formula is:
MShc (i,d)=r1∗X(i,d)+(1−r1)∗X(j,d)+C1∗(X(i, d)−X(j, d)) (10)
MShc (j,d)=r2∗X(j,d)+(1−r2)∗X(i,d)+C2∗(X(j, d)−X(i, d)) (11)
In Equations (10) and (11),
X(i,d)
and
X(j,d)
represent the positions of the ddimension
of the i
th
and jindividuals, respectively;
r1
and
r2
represent the random numbers between
0 and 1; and
C1
and
C2
represent the random numbers between
−
1 and 1.
MShc (i,d)
and
MShc (j,d)represent the offspring produced after horizontal crossing.
A vertical crossover is an arithmetic crossover that operates on all individuals between
two different dimensions, calculated by:
MSvc (i,d1)=r∗X(i,d1)+(1−r)∗X(i,d2)(12)
In Equation (12),
X(i,d1)
and
X(i,d2)
represent the positions of the
d1
and
d2
dimen-
sions of the i
th
individual respectively, rrepresents the random number between 0 and 1,
and MSvc (i,d1)represents the offspring produced after vertical crossing.
The POA easily falls into the local optimal because the pelican individual moves within
a small range in the local search process. The CSO is integrated into the local search stage
of the POA to enhance its ability to jump out of the local optimal because of strong global
detection ability and local development ability. In the original POA, the current individual
will be far away from the randomly generated individual when the fitness value of the
randomly generated individual is less than that of the current individual. The randomly
generated individuals are not fully utilized. In this paper, the horizontal crossover in the
CSO is introduced to make full use of the random individuals, guide the pelican individuals
to move to the target position, and enhance the local development ability of the algorithm
and its ability to jump out of the local optimal. The improved formula is as follows:
Xp1
i(i,j)=r1∗X(i,j)+(1−r1)∗P(i,j)+sin(r2)∗(X(i, j)−P(i, j)) (13)
In Equation (13),
X(i,j)
represents the current individual;
P(i,j)
represents the random
individual, i.e., the prey;
r1
represents the random number between 0 and 1; and
r2
represents the random number between 0 and 2π.
2.2.2. Improved Global Search
The pelicans only use their current position to update their positions according to the
POA principle in the global search stage. The position of the optimal individual is not fully
utilized, which makes the development ability of the algorithm insufficient. This paper
introduces the optimal individual in the global search stage of the POA to enhance the
Biomimetics 2024,9, 277 6 of 18
guidance ability of the overall optimization and increase the ability of the algorithm. At
the same time, the adaptive disturbance factor G is introduced to avoid falling into local
optimization, and the improved formula is as follows:
XP2
i=QF ∗Xi+∗(2∗rand −1)∗(Xbest −Xi+sin(r)∗G(14)
QF(t)=2∗rand −1
t(1−T)2(15)
G=2∗1−t
T(16)
In Equations (14), (15), and (16), QF represents the quality function of the balanced
search strategy [
47
], rrepresents the random number from 0 to 2
π
,rand represents the ran-
dom number [0, 1], and tand Trepresent the current and maximum iterations, respectively.
2.2.3. Dimensional Variation Strategy
Like other swarm intelligence algorithms, the POA is prone to local optimality and
slow convergence. The analysis shows that the main reason is that the algorithm does
not make full use of the guiding role of the optimal individual. Therefore, this paper
improves the population diversity by mutating the optimal individual and guiding the
population to evolve to the optimal position to improve its convergence speed. At the
same time, the strategy of dimensional-by-dimension variation is adopted to update the
optimal individual to avoid the problem of inter-dimensional interference in the case of
high dimensions. The calculation formula is as follows:
Xd
new =Xd
best +TD(t)d∗rand (17)
In Equation (17),
Xd
new
represents the position of the optimal individual in the D-
dimension after updating,
Xd
best
represents the position of the optimal individual in the
D-dimension, and TD(t) represents the T-distribution with tdegrees of freedom [
48
]. t
is 25 in this paper.
TD(t)d
represents the random number generated by t-distribution in
the Ddimension. To improve the convergence speed, this paper uses the greedy principle
to choose whether to use the new position instead of the original optimal position. The
specific process is demonstrated in Algorithm 1.
Algorithm 1. Mutates Dimensionally
1: Generate d random numbers of T-distribution with 25 degrees of freedom parameter.
2: for i = 1: d
3 :
The new solution is obtained after calculating the variation according to Equation (17)Xd
new
4: boundary condition procedure
5: if fnew < fbest
6 : Replace the original Xd
best with the new position Xd
new
7 : Calculate the fitness value based on the new positionXbest
8: end if
9: end for
10: Return the best fitness value and the best individual
2.2.4. IPOA Implementation Process
The specific implementation flowchart of the IPOA is shown in Figure 1, based on the
description of the POA improvements in Sections 2.1–2.3.
Biomimetics 2024,9, 277 7 of 18
Biomimetics 2023, 8, x FOR PEER REVIEW 7 of 19
Figure 1. Flowchart of the IPOA.
2.3. IPOA Algorithm Performance Test and Analysis
2.3.1. Experimental Environment and Test Function
Simulation environment: 64-bit Windows 10 operating system, processor Intel(R)
Core (TM) i5-8265U, main frequency 1.80GHz, memory 8GB, programming software
MATLAB R2023b. This paper uses CEC2017 test functions to verify the algorithm. The test
functions are shown in Table 1, where 𝑓1 is a unimodal function, 𝑓2-𝑓4 are simple multi-
modal functions, 𝑓5 and 𝑓6 are mixed mode functions, and 𝑓7 and𝑓8 are combined func-
tions. The algorithm conducted 30 independent experiments on each test function to re-
duce the randomness and contingency of the algorithm.
Table 1. Test functions.
Functions
Best Value
Types
𝑓1
Shifted and Rotated Bent Cigar
100
Unimodal
𝑓2
Shifted and Rotated Rastrigin’s
400
Simple Multimodal
𝑓3
Shifted and Rotate Lunacek Bi_Rastrigin
600
𝑓4
Shifted and Rotated Schwefel’s
900
𝑓5
Hybrid Function 2 (N = 3)
1100
Hybrid
𝑓6
Hybrid Function 6 (N = 5)
1600
𝑓7
Composition Function 1 (N = 3)
2000
Composition
𝑓8
Composition Function 7 (N = 6)
2600
2.3.2. Comparisons with POA, PSO, SSA, and WOA
Four algorithms were selected for comparison with the IPOA to validate its effective-
ness. First is the particle swarm optimization algorithm (PSO) [49], which is a classic op-
timization method, serving as a cornerstone of optimization techniques, and has been
widely applied across various domains since its inception. Its performance in both con-
vergence speed and accuracy is exceptional. Additionally, the IPOA is compared with the
original pelican optimization algorithm (POA), the sparrow search algorithm (SSA) [50],
and the whale algorithm (WOA) [51]. The algorithm parameters were set to the same
START
Initialize parameters
Randomly select an individual
𝐹𝑝< 𝐹𝑖
Update positions by equation (7)
t=T
STOP
Update positions by equation (13)
Update positions by equation (14)
Update the global optimum solution
Use algorithm1 update the global optimum solution
YES NO
YES
NO
t=t+1
Figure 1. Flowchart of the IPOA.
2.3. IPOA Algorithm Performance Test and Analysis
2.3.1. Experimental Environment and Test Function
Simulation environment: 64-bit Windows 10 operating system, processor Intel(R)
Core (TM) i5-8265U, main frequency 1.80 GHz, memory 8 GB, programming software
MATLAB R2023b. This paper uses CEC2017 test functions to verify the algorithm. The
test functions are shown in Table 1, where
f1
is a unimodal function,
f2
–
f4
are simple
multimodal functions,
f5
and
f6
are mixed mode functions, and
f7and f8
are combined
functions. The algorithm conducted 30 independent experiments on each test function to
reduce the randomness and contingency of the algorithm.
Table 1. Test functions.
Functions Best Value Types
f1Shifted and Rotated Bent Cigar 100 Unimodal
f2Shifted and Rotated Rastrigin’s 400
Simple
Multimodal
f3Shifted and Rotate Lunacek Bi_Rastrigin 600
f4Shifted and Rotated Schwefel’s 900
f5Hybrid Function 2 (N = 3) 1100 Hybrid
f6Hybrid Function 6 (N = 5) 1600
f7Composition Function 1 (N = 3) 2000 Composition
f8Composition Function 7 (N = 6) 2600
2.3.2. Comparisons with POA, PSO, SSA, and WOA
Four algorithms were selected for comparison with the IPOA to validate its effec-
tiveness. First is the particle swarm optimization algorithm (PSO) [
49
], which is a classic
optimization method, serving as a cornerstone of optimization techniques, and has been
widely applied across various domains since its inception. Its performance in both con-
vergence speed and accuracy is exceptional. Additionally, the IPOA is compared with the
original pelican optimization algorithm (POA), the sparrow search algorithm (SSA) [
50
],
Biomimetics 2024,9, 277 8 of 18
and the whale algorithm (WOA) [
51
]. The algorithm parameters were set to the same values
as those in the original literature to ensure the fairness of the comparison. The population
was 30, and the maximum number of iterations was 1000. The optimization performance of
the five algorithms were compared in four respects: best value, worst value, average value,
and standard deviation (see Table 2). The convergence curves of each algorithm on the test
function are shown in Figures 2–9.
Table 2. Evaluation results of test functions.
Function
Index
Algorithm
IPOA POA PSO WOA SSA
f1
Best 100.7108 6097 57,929,542 592,934.6706 133.5179
Worst 1226.4633
1,944,227,684 2,850,258,819
74,471,522.3021
12,381.7875
Mean 4768.0518 219,781,765 792,680,137
9,361,246.7600
4438.0798
Std 1560.1471 495,103,154 831,763,818
15,616,489.4653
3567.2322
f2
Best 400.0002 400.7755 411.1083 400.6303 400.1664
Worst 473.2955 496.9909 825.9439 563.8685 472.0999
Mean 404.1049 418.4436 505.0500 440.8641 404.6719
Std 13.1237 21.6410 102.3337 47.2803 12.7660
f3
Best 600.0000 607.4886 608.6856 610.2782 600.0000
Worst 601.4796 638.7122 635.2207 657.3534 613.0539
Mean 600.1154 621.6511 618.6479 634.4335 602.8292
Std 0.33723 9.2482 6.2520 11.8559 3.8600
f4
Best 900.0000 906.0017 931.4729 921.8618 900.0000
Worst 929.6692 1387.7745 1293.9060 3566.0554 1829.3862
Mean 903.8361 1092.0306 1008.2076 1612.5818 1116.2890
Std 7.1136 141.2868 65.9039 643.7356 308.2101
f5
Best 1100.0366 1109.0200 1171.9598 1123.9527 1103.0719
Worst 1137.6526 1315.8967 1903.2485 1568.3389 1258.4352
Mean 1116.7835 1171.1777 1345.1482 1208.2003 1145.7665
Std 10.4314 49.0942 165.8870 90.6109 41.5469
f6
Best 1600.7438 1607.6178 1636.0000 1622.9391 1601.4464
Worst 1960.8268 1938.6274 2246.0736 2304.1432 2139.5167
Mean 1689.9921 1763.1663 1804.9491 1913.4186 1832.8280
Std 115.7566 106.2803 157.1715 187.8456 138.5551
f7
Best 2000.0000 2024.1393 2057.8897 2043.5482 2005.5991
Worst 2140.3403 2162.7398 2244.1384 2450.8169 2278.7248
Mean 2037.2731 2083.7765 2127.0903 2191.1716 2088.4404
Std 26.1220 40.1675 58.2918 88.8603 67.2939
f8
Best 2600.0043 2608.7520 2967.3881 2628.1123 2800.0000
Worst 3165.7513 3904.5444 4257.0824 4786.9540 3395.5483
Mean 2966.1859 3023.9175 3329.9615 3585.4080 4483.6695
Std 134.8942 241.6267 450.6247 574.6367 532.1813
Biomimetics 2023, 8, x FOR PEER REVIEW 8 of 19
values as those in the original literature to ensure the fairness of the comparison. The pop-
ulation was 30, and the maximum number of iterations was 1000. The optimization per-
formance of the five algorithms were compared in four respects: best value, worst value,
average value, and standard deviation (see Table 2). The convergence curves of each al-
gorithm on the test function are shown in Figures 2–9.
Figure 2. 𝑓1 iteration diagram.
Figure 3. 𝑓2 iteration diagram.
Figure 4. 𝑓3 iteration diagram.
Figure 5. 𝑓4 iteration diagram.
Figure 2. f1iteration diagram.
Biomimetics 2024,9, 277 9 of 18
Biomimetics 2023, 8, x FOR PEER REVIEW 8 of 19
values as those in the original literature to ensure the fairness of the comparison. The pop-
ulation was 30, and the maximum number of iterations was 1000. The optimization per-
formance of the five algorithms were compared in four respects: best value, worst value,
average value, and standard deviation (see Table 2). The convergence curves of each al-
gorithm on the test function are shown in Figures 2–9.
Figure 2. 𝑓1 iteration diagram.
Figure 3. 𝑓2 iteration diagram.
Figure 4. 𝑓3 iteration diagram.
Figure 5. 𝑓4 iteration diagram.
Figure 3. f2iteration diagram.
Biomimetics 2023, 8, x FOR PEER REVIEW 8 of 19
values as those in the original literature to ensure the fairness of the comparison. The pop-
ulation was 30, and the maximum number of iterations was 1000. The optimization per-
formance of the five algorithms were compared in four respects: best value, worst value,
average value, and standard deviation (see Table 2). The convergence curves of each al-
gorithm on the test function are shown in Figures 2–9.
Figure 2. 𝑓1 iteration diagram.
Figure 3. 𝑓2 iteration diagram.
Figure 4. 𝑓3 iteration diagram.
Figure 5. 𝑓4 iteration diagram.
Figure 4. f3iteration diagram.
Biomimetics 2023, 8, x FOR PEER REVIEW 8 of 19
values as those in the original literature to ensure the fairness of the comparison. The pop-
ulation was 30, and the maximum number of iterations was 1000. The optimization per-
formance of the five algorithms were compared in four respects: best value, worst value,
average value, and standard deviation (see Table 2). The convergence curves of each al-
gorithm on the test function are shown in Figures 2–9.
Figure 2. 𝑓1 iteration diagram.
Figure 3. 𝑓2 iteration diagram.
Figure 4. 𝑓3 iteration diagram.
Figure 5. 𝑓4 iteration diagram.
Figure 5. f4iteration diagram.
Biomimetics 2023, 8, x FOR PEER REVIEW 9 of 19
Figure 6. 𝑓5 iteration diagram.
Figure 7. 𝑓6 iteration diagram.
Figure 8. 𝑓7 iteration diagram.
Figure 9. 𝑓8 iteration diagram.
The optimization results of the IPOA in eight different tests are superior to those of
the POA, PSO, SSA, and WOA, according to the experimental results in Table 2. The IPOA
can simultaneously find the theoretical optimal values of the functions 𝑓3, 𝑓4, and 𝑓7, re-
spectively. It is very close to the theoretical optimal values when compared with the other
algorithms. Among them, 𝑓1 is a unimodal function with no local minimum and only a
Figure 6. f5iteration diagram.
Biomimetics 2024,9, 277 10 of 18
Biomimetics 2023, 8, x FOR PEER REVIEW 9 of 19
Figure 6. 𝑓5 iteration diagram.
Figure 7. 𝑓6 iteration diagram.
Figure 8. 𝑓7 iteration diagram.
Figure 9. 𝑓8 iteration diagram.
The optimization results of the IPOA in eight different tests are superior to those of
the POA, PSO, SSA, and WOA, according to the experimental results in Table 2. The IPOA
can simultaneously find the theoretical optimal values of the functions 𝑓3, 𝑓4, and 𝑓7, re-
spectively. It is very close to the theoretical optimal values when compared with the other
algorithms. Among them, 𝑓1 is a unimodal function with no local minimum and only a
Figure 7. f6iteration diagram.
Biomimetics 2023, 8, x FOR PEER REVIEW 9 of 19
Figure 6. 𝑓5 iteration diagram.
Figure 7. 𝑓6 iteration diagram.
Figure 8. 𝑓7 iteration diagram.
Figure 9. 𝑓8 iteration diagram.
The optimization results of the IPOA in eight different tests are superior to those of
the POA, PSO, SSA, and WOA, according to the experimental results in Table 2. The IPOA
can simultaneously find the theoretical optimal values of the functions 𝑓3, 𝑓4, and 𝑓7, re-
spectively. It is very close to the theoretical optimal values when compared with the other
algorithms. Among them, 𝑓1 is a unimodal function with no local minimum and only a
Figure 8. f7iteration diagram.
Biomimetics 2023, 8, x FOR PEER REVIEW 9 of 19
Figure 6. 𝑓5 iteration diagram.
Figure 7. 𝑓6 iteration diagram.
Figure 8. 𝑓7 iteration diagram.
Figure 9. 𝑓8 iteration diagram.
The optimization results of the IPOA in eight different tests are superior to those of
the POA, PSO, SSA, and WOA, according to the experimental results in Table 2. The IPOA
can simultaneously find the theoretical optimal values of the functions 𝑓3, 𝑓4, and 𝑓7, re-
spectively. It is very close to the theoretical optimal values when compared with the other
algorithms. Among them, 𝑓1 is a unimodal function with no local minimum and only a
Figure 9. f8iteration diagram.
The optimization results of the IPOA in eight different tests are superior to those
of the POA, PSO, SSA, and WOA, according to the experimental results in Table 2. The
IPOA can simultaneously find the theoretical optimal values of the functions
f3
,
f4
, and
f7
,
respectively. It is very close to the theoretical optimal values when compared with the other
algorithms. Among them,
f1
is a unimodal function with no local minimum and only a
global minimum. In comparison with other algorithms, the IPOA demonstrates significant
advantages. As shown in Figure 2, both the SSA and the IPOA exhibit fast convergence
speeds, but the IPOA achieves higher accuracy, indicating its strong global convergence
capability.
f2
,
f3
, and
f4
are multi-modal functions with local minimums. The standard
deviation of the IPOA is more stable, though the IPOA and the SSA can both find optimal
values at
f2
,
f3
, and
f4
in Table 2. From Figures 4and 5, it can be observed that the IPOA not
only can find the optimal solution but also has a faster convergence speed. From Figure 3,
it can be seen that the convergence speed and accuracy of these algorithms are very close,
but the IPOA has higher accuracy. This reflects the IPOA’s stronger ability to escape the
local optimal compared to the other algorithms.
f5
and
f6
are mixed functions of random
subfunctions. The hybrid function comprises three or more CEC2017 reference functions,
Biomimetics 2024,9, 277 11 of 18
rotated and shifted. Each subfunction is assigned a corresponding weight, which increases
the difficulty of the algorithm in finding the optimal solution.
f7
and
f8
are composite
functions consisting of at least three mixed functions or CEC2017 reference functions after
rotation and shifting. Each subfunction has an additional deviation value and is then
assigned a weight. These combined functions further increase the optimization difficulty
of the algorithm. From Figures 5–9, it can be seen that the IPOA converges significantly
faster compared to the other four algorithms, and the accuracy of the solutions is also
higher. Whether observed horizontally or vertically, the IPOA outperforms the other four
algorithms, indicating its strong convergence performance and excellent exploration ability.
The improvements incorporated the crisscross optimization algorithm in the local
search stage to improve the diversity of the population; at the same time, the optimal
individual is introduced in the global search stage to enhance the guiding ability of the
whole population, and the disturbance factor is added to increase the ability to jump out
of the local optimal; finally, the optimal individual is adopted by the dimensional-by-
dimension variation strategy to guide the evolution to the optimal position better, thereby
improving the convergence speed of the algorithm, leading to better performance of the
IPOA compared to the other algorithms.
2.4. IPOA Solves the Problem of Economic Dispatch
Firstly, the relevant parameters of the IPOA algorithm need to be adjusted in the
process of solving the ELD problem. The population is randomly generated with the upper
and lower limits of the power load as constraints, and the population represents the unit
output. The objective function proposed after considering the penalty coefficient is taken as
the fitness function, and the number of units is taken as the solution dimension. Secondly,
the IPOA is used to update the population and find the individual that can minimize the
fitness function. Then, Formula (1) is used to calculate the minimum cost. Finally, the
optimal load distribution and coal consumption of each unit are outputted. The specific
process is demonstrated in Algorithm 2.
Algorithm 2. IPOA for ELD
1: Input: Population size, Dimension, variable bounds Maximum failure count
2: Initialization: Initialize population X and Calculate fitness value using Equation (5)
3: for i = 1: Max_iterations
4: for j = 1: N
5: Randomly select an individual
6: if fit(p) < fit(j)
7: Update positions by Equation (7)
8: else
9: Update positions by Equation (13)
10: end if
11: Update positions by Equation (14)
12: Use algorithm1 update the global optimum solution
13: Handling boundary conditions
14: Calculating individual fitness values using Equation (5)
15: Update the global optimum solution
16: end for
17: end for
18: Calculate fuel cost using Equation (1)
19: Output: Optimal cost, Unit’s output
3. Experimental Results and Discussion
In this paper, 10 small power plants and 40 medium power plants were selected
to evaluate the effectiveness of the IPOA algorithm. The test results of the IPOA were
compared with those of the POA, the Harris hawk’s optimization (HHO) [
52
], the SSA, and
the WOA to verify the solving ability of the IPOA more comprehensively. The parameters
Biomimetics 2024,9, 277 12 of 18
of the algorithm were set to the same values as in their respective original literature in
order to ensure the fairness of comparison, and the number of runs, population size,
spatial dimension, and maximum number of iterations were made consistent. That is, the
population was 30, the maximum number of iterations was 1000, and the algorithm was
run independently 30 times.
3.1. 10 Units
In this case study, the ELD system was composed of 10 generator sets; the coal
consumption characteristic parameters of the unit and the upper and lower limits of the
unit load [
53
] are shown in Table 3. Unit data for the 10 generating units power system
with VPE. The total load demand was 2700 MW.
Table 3. Unit data for the 10 generating units power system with VPE.
Units Pmin Pmax a b c e f
1 100 250 0.002176 −0.3975 26.97 0.02697 −3.975
2 50 230 0.004194 −1.269 118.4 0.1184 −12.69
3 200 500 0.00001176 0.4864 −95.14 −0.05914 4.864
4 99 265 0.005935 −2.338 266.8 0.2668 −23.38
5 190 490 0.0001498 0.4462 −53.99 −0.05399 4.462
6 85 265 0.005935 −2.338 266.8 0.2668 −23.38
7 200 500 0.0002454 0.3559 −43.35 −0.04335 3.559
8 99 265 0.005935 −2.338 266.8 0.2668 −23.38
9 130 440 0.0006121 −0.0182 14.23 0.01423 −0.1817
10 200 490 0.0000416 0.5084 −61.13 −0.06113 5.084
Different algorithms cannot generate feasible solutions meeting the constraint condi-
tions simultaneously because of the same penalty function coefficient. Different penalty
function coefficients were applied to the different algorithms based on the experimental
results. For the IPOA it was 0.500, for the SSA, WOA, HHO it was 0.531, and for the POA it
was 0.610. After 30 independent experiments of each algorithm, the optimal output of each
unit is shown in Table 4. The total fuel cost of the IPOA was the lowest at 651.8784 ($/h),
as seen in Table 5. Compared with the traditional POA algorithm, the coal consumption
was reduced by 0.1903 ($/h), a decrease of 0.0292%. Compared with the whale algorithm
(WOA), the coal consumption was reduced by 1.6003 ($/h), a decrease of 0.2455%. And
it can be seen from Figure 10 that the IPOA demonstrated faster convergence speed and
better convergence results. Reducing total fuel cost can improve the efficiency of a power
plant and reduce its economic costs. Compared with the other four algorithms, the stan-
dard deviation of the IPOA was the smallest, which indicates that the improved pelican
optimization algorithm has good development ability and strong stability in dealing with
ELD problems.
Table 4. Optimal dispatch for the 10 generating units power system.
Units
Algorithms
IPOA POA HHO SSA WOA
P1203.5350 202.8740 211.5970 202.7439 220.4145
P2210.4219 210.4247 215.8357 210.9169 207.4651
P3200.6466 200.0152 206.2645 200.0000 224.0729
P4238.8801 237.3994 238.8798 239.5520 242.8912
P5185.0712 194.4705 215.2235 190.0000 200.0707
P6236.0326 238.9872 238.5807 238.3172 235.6278
P7273.2652 269.0280 267.1062 282.0928 226.3169
P8238.3423 238.6122 245.7352 237.8052 239.6864
P9423.9302 418.0496 405.2114 408.7595 413.4965
P10 489.8126 489.9749 454.6139 489.8125 489.9581
Biomimetics 2024,9, 277 13 of 18
Table 5. Fuel cost ($/h) for the 10 generating unit power system.
Algorithms Statistics
Min Cost Max Cost Mean Cost SD
IPOA 651.8784 655.5161 652.6444 1.0014
POA 652.0687 654.4392 659.458 1.7685
HHO 653.4787 662.7219 679.2167 6.3263
SSA 651.9516 653.2228 656.5612 1.614
WOA 653.7402 672.8395 699.5087 12.5738
Biomimetics 2023, 8, x FOR PEER REVIEW 12 of 19
the WOA to verify the solving ability of the IPOA more comprehensively. The parameters
of the algorithm were set to the same values as in their respective original literature in
order to ensure the fairness of comparison, and the number of runs, population size, spa-
tial dimension, and maximum number of iterations were made consistent. That is, the
population was 30, the maximum number of iterations was 1000, and the algorithm was
run independently 30 times.
3.1. 10 Units
In this case study, the ELD system was composed of 10 generator sets; the coal con-
sumption characteristic parameters of the unit and the upper and lower limits of the unit
load [53] are shown in Table 3. Unit data for the 10 generating units power system with
VPE. The total load demand was 2700 MW.
Different algorithms cannot generate feasible solutions meeting the constraint condi-
tions simultaneously because of the same penalty function coefficient. Different penalty
function coefficients were applied to the different algorithms based on the experimental
results. For the IPOA it was 0.500, for the SSA, WOA, HHO it was 0.531, and for the POA
it was 0.610. After 30 independent experiments of each algorithm, the optimal output of
each unit is shown in Table 4. The total fuel cost of the IPOA was the lowest at 651.8784
($/h), as seen in Table 5. Compared with the traditional POA algorithm, the coal consump-
tion was reduced by 0.1903 ($/h), a decrease of 0.0292%. Compared with the whale algo-
rithm (WOA), the coal consumption was reduced by 1.6003 ($/h), a decrease of 0.2455%.
And it can be seen from Figure 10 that the IPOA demonstrated faster convergence speed
and better convergence results. Reducing total fuel cost can improve the efficiency of a
power plant and reduce its economic costs. Compared with the other four algorithms, the
standard deviation of the IPOA was the smallest, which indicates that the improved peli-
can optimization algorithm has good development ability and strong stability in dealing
with ELD problems.
Figure 10. Convergence curve of unit 10.
Table 3. Unit data for the 10 generating units power system with VPE.
Units
Pmin
Pmax
a
b
c
e
f
1
100
250
0.002176
−0.3975
26.97
0.02697
−3.975
2
50
230
0.004194
−1.269
118.4
0.1184
−12.69
3
200
500
0.00001176
0.4864
−95.14
−0.05914
4.864
4
99
265
0.005935
−2.338
266.8
0.2668
−23.38
5
190
490
0.0001498
0.4462
−53.99
−0.05399
4.462
6
85
265
0.005935
−2.338
266.8
0.2668
−23.38
7
200
500
0.0002454
0.3559
−43.35
−0.04335
3.559
Figure 10. Convergence curve of unit 10.
3.2. 40 Units
In this section, a medium-sized power plant of 40 units is taken as an example, with a
load demand of 10,500 MW. The coal consumption characteristic parameters of the unit and
the upper and lower limits of the unit load [
54
] are shown in Table 6. The penalty function
coefficients of the IPOA, WOA and SSA were 17.5, HHO was 16, and POA was 21.5. The
optimal output of each unit is shown in Table 7after 30 independent experiments.
The total fuel cost of the IPOA was the lowest at 121,591.3068 ($/h), as shown in Table 8.
The coal consumption was reduced by 3316.1208 ($/h) compared with the traditional POA
algorithm, a decrease of 2.7273%, and the effect was more significant than that of the WOA.
The coal consumption decreased by 4288.7396 ($/h), or 3.5272%. The standard deviation of
the IPOA was the smallest among the five algorithms, the convergence speed of the IPOA
in the early stage was second only to the SSA, as seen in Figure 11, and the convergence
speed in the later stage was the fastest, all of which indicate that the IPOA has faster
convergence speed and better convergence results. This is mainly because the IPOA adopts
a dimensional-by-dimension variation strategy to avoid the problem of inter-dimensional
interference in the case of high dimensions, which allows it to perform well in practical
problems in high dimensions. The longitudinal crossover strategy was introduced to
improve the diversity of the population and the stability of the algorithm. In the local
development stage, the optimal individual and disturbance factor were introduced to
improve the convergence ability of the algorithm.
Biomimetics 2024,9, 277 14 of 18
Table 6. Unit data for the 40 generating units power system with VPE.
Units Pmin Pmax a b c e f
1 36 114 0.00690 6.73 94.705 100 0.084
2 36 114 0.00690 6.73 94.705 100 0.084
3 60 120 0.02028 7.07 309.540 100 0.084
4 80 190 0.00942 8.18 369.030 150 0.063
5 47 97 0.01140 5.35 148.890 120 0.077
6 68 140 0.01142 8.05 222.330 100 0.084
7 110 300 0.00357 8.03 287.710 200 0.042
8 135 300 0.00492 6.99 391.980 200 0.042
9 135 300 0.00573 6.60 455.760 200 0.042
10 130 300 0.00605 12.90 722.820 200 0.042
11 94 375 0.00515 12.90 635.200 200 0.042
12 94 375 0.00569 12.80 654.690 200 0.042
13 125 500 0.00421 12.50 913.400 300 0.035
14 125 500 0.00752 8.84 1760.400 300 0.035
15 125 500 0.00708 9.15 1728.300 300 0.035
16 125 500 0.00708 9.15 1728.300 300 0.035
17 220 500 0.00313 7.97 647.850 300 0.035
18 220 500 0.00313 7.95 649.690 300 0.035
19 242 550 0.00313 7.97 647.830 300 0.035
20 242 550 0.00313 7.97 647.810 300 0.035
21 254 550 0.00298 6.63 785.960 300 0.035
22 254 550 0.00298 6.63 785.960 300 0.035
23 254 550 0.00284 6.66 794.530 300 0.035
24 254 550 0.00284 6.66 794.530 300 0.035
25 254 550 0.00277 7.10 801.320 300 0.035
26 254 550 0.00277 7.10 801.320 300 0.035
27 10 150 0.52124 3.33 1055.100 120 0.077
28 10 150 0.52124 3.33 1055.100 120 0.077
29 10 150 0.52124 3.33 1055.100 120 0.077
30 47 97 0.01140 5.35 148.890 120 0.077
31 60 190 0.00160 6.43 222.920 150 0.063
32 60 190 0.00160 6.43 222.920 150 0.063
33 60 190 0.00160 6.43 222.920 150 0.063
34 90 200 0.00010 8.95 107.870 200 0.042
35 90 200 0.00010 8.62 116.580 200 0.042
36 90 200 0.00010 8.62 116.580 200 0.042
37 25 110 0.01610 5.88 307.450 80 0.098
38 25 110 0.01610 5.88 307.450 80 0.098
39 25 110 0.01610 5.88 307.450 80 0.098
40 242 550 0.00313 7.97 647.830 300 0.035
Table 7. Optimal dispatch of IPOA for the 40 generating units power system.
Units Outputs Unit Outputs Unit Outputs Unit Outputs
P1113.1496 P11 243.6059 P21 523.2740 P31 190.0000
P2114.0000 P12 94.00949 P22 523.2890 P32 190.0000
P397.40526 P13 304.5174 P23 523.2808 P33 190.0000
P4179.7357 P14 304.5203 P24 523.2905 P34 200.0000
P594.50869 P15 304.5219 P25 523.2831 P35 167.4762
P6140.0000 P16 304.5212 P26 523.2792 P36 200.0000
P7259.6008 P17 489.2985 P27 10.00649 P37 110.0000
P8284.6023 P18 489.2820 P28 10.00295 P38 110.0000
P9284.6312 P19 511.2877 P29 10.00000 P39 110.0000
P10 130.0066 P20 511.2906 P30 97.00000 P40 511.2834
Biomimetics 2024,9, 277 15 of 18
Table 8. Fuel cost ($/h) for the 40 generating unit power system.
Algorithms Statistics
Min Cost Max Cost Mean Cost SD
IPOA 121,591.3068 123,933.37 122,659.9709 654.9886
POA 124,907.4276 129,260.1887 126,473.6095 937.3753
HHO 123,387.6705 128,381.2468 125,532.5618 1075.9279
SSA 122,693.0062 127,500.1279 124,321.0393 1124.6677
WOA 125,880.0464 134,779.6761 129,308.2354 1817.6021
Biomimetics 2023, 8, x FOR PEER REVIEW 15 of 19
𝑃9
284.6312
𝑃19
511.2877
𝑃29
10.00000
𝑃39
110.0000
𝑃10
130.0066
𝑃20
511.2906
𝑃30
97.00000
𝑃40
511.2834
Table 8. Fuel cost ($/h) for the 40 generating unit power system.
Algorithms
Statistics
Min cost
Max cost
Mean cost
SD
IPOA
121591.3068
123933.37
122659.9709
654.9886
POA
124907.4276
129260.1887
126473.6095
937.3753
HHO
123387.6705
128381.2468
125532.5618
1075.9279
SSA
122693.0062
127500.1279
124321.0393
1124.6677
WOA
125880.0464
134779.6761
129308.2354
1817.6021
Figure 11. Convergence curve of unit 40.
3.3. 80 Units
In this section, the system was built by repeating the 40-unit system twice, with a
load requirement of 21,000 MW, by taking a large power plant with 80 units as an exam-
ple. The local minima of the solutions increase as the number of solutions increases. There-
fore, the solution algorithm needs a stronger global search ability to overcome the preco-
cious convergence problem. The penalty function coefficients of each algorithm were as
follows: 17.5 for the IPOA, HHO, WOA and SSA, and 20.5 for the POA. Each algorithm
underwent 30 independent experiments.
The total fuel cost of the IPOA was 244105.2816 ($/h), as shown in Table 9; the coal
consumption was reduced by 8968.1651 ($/h) compared with the traditional POA algo-
rithm, a reduction of 3.6739%, and by 2427.0296 ($/h) compared with the second-best spar-
row algorithm (SSA), a decrease of 0.9943%. The application results of the IPOA in large
units were better than those in small and medium-sized units, indicating that IPOA has
significant advantages in dealing with high-dimensional problems.
Figure 11. Convergence curve of unit 40.
3.3. 80 Units
In this section, the system was built by repeating the 40-unit system twice, with a
load requirement of 21,000 MW, by taking a large power plant with 80 units as an example.
The local minima of the solutions increase as the number of solutions increases. Therefore,
the solution algorithm needs a stronger global search ability to overcome the precocious
convergence problem. The penalty function coefficients of each algorithm were as follows:
17.5 for the IPOA, HHO, WOA and SSA, and 20.5 for the POA. Each algorithm underwent
30 independent experiments.
The total fuel cost of the IPOA was 244,105.2816 ($/h), as shown in Table 9; the
coal consumption was reduced by 8968.1651 ($/h) compared with the traditional POA
algorithm, a reduction of 3.6739%, and by 2427.0296 ($/h) compared with the second-best
sparrow algorithm (SSA), a decrease of 0.9943%. The application results of the IPOA in
large units were better than those in small and medium-sized units, indicating that IPOA
has significant advantages in dealing with high-dimensional problems.
Table 9. Fuel cost ($/h) for the 80 generating unit power system.
Algorithms Statistics
Min Cost Max Cost Mean Cost SD
IPOA 244,105.2816 249,955.5348 247,043.7003 1493.4631
POA 253,073.4467 258,114.9399 255,577.6569 1279.7300
HHO 249,554.8627 257,240.0592 252,846.8087 1948.0503
SSA 246,532.3112 251,589.9268 248,650.6662 1167.5877
WOA 258,734.1637 271,925.7694 263,327.7722 3230.8254
4. Conclusions
This paper proposes an improved pelican optimization algorithm (IPOA) to optimize
the original POA by using longitudinal crossover and dimensional variation strategies
and introducing optimal individuals and disturbance factors in the global phase. In this
Biomimetics 2024,9, 277 16 of 18
paper, the IPOA was tested on eight CEC2017 test functions, and the test results show
that the algorithm can jump out of the local area. Secondly, the IPOA was applied to the
economic scheduling problem of thermal power units with multiple practical constraints,
and its effectiveness was verified with
10 units,
40 units, and 80 units, respectively. In the
case of low-dimension 10 units, the coal consumption was reduced by 0.0292% compared
with the original POA. In the case of 40 units, it was reduced by 2.7273% compared with
the original POA. In the case of high-dimension 80 units, it was reduced by 3.6739%
compared with the POA; from Figure 12, it can be observed that compared to the cases with
10 units
and 40 units, the IPOA exhibited a more significant advantage in both convergence
speed and convergence accuracy, indicating that it has a significant advantage in solving
high-dimensional problems. The IPOA showed that the improved method has good
performance in solving the complex non-convex ELD problem, which can significantly
reduce coal consumption and improve the economic benefit of power plants. The algorithm
is promising and can be applied to other complex practical problems. In the follow-up
study, we will apply the IPOA to the economic scheduling problem of multi-fuel and
multi-constrained thermal power units to better verify the algorithm’s performance.
Biomimetics 2023, 8, x FOR PEER REVIEW 16 of 19
Table 9. Fuel cost ($ / h) for the 80 generating unit power system.
Algorithms
Statistics
Min Cost
Max Cost
Mean Cost
SD
IPOA
244105.2816
249955.5348
247043.7003
1493.4631
POA
253073.4467
258114.9399
255577.6569
1279.7300
HHO
249554.8627
257240.0592
252846.8087
1948.0503
SSA
246532.3112
251589.9268
248650.6662
1167.5877
WOA
258734.1637
271925.7694
263327.7722
3230.8254
4. Conclusions
This paper proposes an improved pelican optimization algorithm (IPOA) to optimize
the original POA by using longitudinal crossover and dimensional variation strategies
and introducing optimal individuals and disturbance factors in the global phase. In this
paper, the IPOA was tested on eight CEC2017 test functions, and the test results show that
the algorithm can jump out of the local area. Secondly, the IPOA was applied to the eco-
nomic scheduling problem of thermal power units with multiple practical constraints, and
its effectiveness was verified with 10 units, 40 units, and 80 units, respectively. In the case
of low-dimension 10 units, the coal consumption was reduced by 0.0292% compared with
the original POA. In the case of 40 units, it was reduced by 2.7273% compared with the
original POA. In the case of high-dimension 80 units, it was reduced by 3.6739% compared
with the POA; from Figure 12, it can be observed that compared to the cases with 10 units
and 40 units, the IPOA exhibited a more significant advantage in both convergence speed
and convergence accuracy, indicating that it has a significant advantage in solving high-
dimensional problems. The IPOA showed that the improved method has good perfor-
mance in solving the complex non-convex ELD problem, which can significantly reduce
coal consumption and improve the economic benefit of power plants. The algorithm is
promising and can be applied to other complex practical problems. In the follow-up study,
we will apply the IPOA to the economic scheduling problem of multi-fuel and multi-con-
strained thermal power units to better verify the algorithm’s performance.
Figure 12. Convergence curve of unit 80.
Author Contributions: Conceptualization, Y.Z. and H.L.; methodology, Y.Z.; software, H.L.; vali-
dation, H.L.; formal analysis, H.L.; investigation, H.L.; resources, H.L.; data curation, H.L.; writ-
ing—original draft preparation, Y.Z. and H.L.; writing—review and editing, Y.Z.; supervision, Y.Z.
and H.L.; funding acquisition, Y.Z. All authors have read and agreed to the published version of the
manuscript.
Figure 12. Convergence curve of unit 80.
Author Contributions: Conceptualization, Y.Z. and H.L.; methodology, Y.Z.; software, H.L.; validation,
H.L.; formal analysis, H.L.; investigation, H.L.; resources, H.L.; data curation, H.L.;
writing—original
draft preparation, Y.Z. and H.L.; writing—review and editing, Y.Z.; supervision, Y.Z. and H.L.;
funding acquisition, Y.Z. All authors have read and agreed to the published version of the manuscript.
Funding: This work is supported by the fund of the Science and Technology Development Project
of Jilin Province No. 20220203190SF and the fund of the education department of Jilin province No.
JJKH20210257KJ.
Data Availability Statement: Dataset available on request from the authors.
Conflicts of Interest: The authors declare no conflict of interest.
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