Efficient optimization of static economic load dispatch in electrical power systems using the teaching–learning based optimization algorithm
Introduction. Power grids are considered one of the most critical energy infrastructures in modern societies, and their economic exploitation plays an important role in reducing the costs of generating electrical energy and increasing the efficiency of generation systems. In the meantime, power gene...
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National Technical University "Kharkiv Polytechnic Institute" and Аnatolii Pidhornyi Institute of Power Machines and Systems of NAS of Ukraine
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| author | Hamadneh, T. Alsayyed, O. Al Soudi, M. |
| author_facet | Hamadneh, T. Alsayyed, O. Al Soudi, M. |
| author_institution_txt_mv | [
{
"author": "T. Hamadneh",
"institution": "Al Zaytoonah University of Jordan"
},
{
"author": "O. Alsayyed",
"institution": "The Hashemite University"
},
{
"author": "M. Al Soudi",
"institution": "Applied Science Private University"
}
] |
| author_sort | Hamadneh, T. |
| baseUrl_str | http://eie.khpi.edu.ua/oai |
| collection | OJS |
| datestamp_date | 2026-07-01T21:42:56Z |
| description | Introduction. Power grids are considered one of the most critical energy infrastructures in modern societies, and their economic exploitation plays an important role in reducing the costs of generating electrical energy and increasing the efficiency of generation systems. In the meantime, power generation plants are responsible for providing the required power to the grid, and the optimal distribution of power among them has a direct impact on the final cost of energy generation. For this reason, the economic load dispatch (ELD) problem has been raised as one of the fundamental issues in the optimal operation of power systems. Problem. The static economic load dispatch problem is defined with the aim of determining the amount of power generated by each generator unit in such a way that the total cost of energy generation is minimized, while all operational constraints of the power system, including power balance constraints, transmission network losses, generator production constraints, power rate of change constraints, and prohibited areas, are met. The presence of features such as nonlinear cost function, valve-point effect and nonconvex search space makes solving this problem with classical mathematical methods face serious challenges. Goal. To develop an efficient method for solving the ELD problem and to achieve an optimal production schedule for power system generators with minimum production cost. Methodology. In this study, the metaheuristic algorithm teaching–learning based optimization (TLBO) has been used to solve the ELD problem. The performance evaluation of the algorithm has been carried out on a standard 6-unit power system. Results. The optimization results show that the TLBO algorithm is able to provide an optimal production schedule by observing all system constraints, in which the total production cost reaches $15452.06. To evaluate the performance quality, the results of TLBO were compared with seven well-known metaheuristic algorithms, and the simulation results showed that TLBO provided the best performance by achieving first rank in terms of objective function value, average cost, and performance stability. Scientific novelty. The innovation of this research lies in the effective application of the TLBO algorithm to solve the ELD problem by considering a complete set of operational constraints and providing a comprehensive comparative analysis with several metaheuristic algorithms. Practical value. The findings of this study indicate that the TLBO algorithm can be used as an efficient, stable, and reliable method for solving operation optimization problems in power systems and help reduce the cost of energy generation and increase the economic efficiency of power grids. References 29, tables 4, figures 2. |
| doi_str_mv | 10.20998/2074-272X.2026.4.09 |
| first_indexed | 2026-07-02T01:00:22Z |
| format | Article |
| fulltext |
Power Stations, Grids and Systems
66 Electrical Engineering & Electromechanics, 2026, no. 4
© T. Hamadneh, O. Alsayyed, M. Al Soudi
UDC 621.311 https://doi.org/10.20998/2074-272X.2026.4.09
T. Hamadneh, O. Alsayyed, M. Al Soudi
Efficient optimization of static economic load dispatch in electrical power systems
using the teaching–learning based optimization algorithm
Introduction. Power grids are considered one of the most critical energy infrastructures in modern societies, and their economic
exploitation plays an important role in reducing the costs of generating electrical energy and increasing the efficiency of generation systems.
In the meantime, power generation plants are responsible for providing the required power to the grid, and the optimal distribution of power
among them has a direct impact on the final cost of energy generation. For this reason, the economic load dispatch (ELD) problem has been
raised as one of the fundamental issues in the optimal operation of power systems. Problem. The static economic load dispatch problem is
defined with the aim of determining the amount of power generated by each generator unit in such a way that the total cost of energy
generation is minimized, while all operational constraints of the power system, including power balance constraints, transmission network
losses, generator production constraints, power rate of change constraints, and prohibited areas, are met. The presence of features such as
nonlinear cost function, valve-point effect and nonconvex search space makes solving this problem with classical mathematical methods face
serious challenges. Goal. To develop an efficient method for solving the ELD problem and to achieve an optimal production schedule for
power system generators with minimum production cost. Methodology. In this study, the metaheuristic algorithm teaching–learning based
optimization (TLBO) has been used to solve the ELD problem. The performance evaluation of the algorithm has been carried out on a
standard 6-unit power system. Results. The optimization results show that the TLBO algorithm is able to provide an optimal production
schedule by observing all system constraints, in which the total production cost reaches $15452.06. To evaluate the performance quality, the
results of TLBO were compared with seven well-known metaheuristic algorithms, and the simulation results showed that TLBO provided the
best performance by achieving first rank in terms of objective function value, average cost, and performance stability. Scientific novelty. The
innovation of this research lies in the effective application of the TLBO algorithm to solve the ELD problem by considering a complete set of
operational constraints and providing a comprehensive comparative analysis with several metaheuristic algorithms. Practical value. The
findings of this study indicate that the TLBO algorithm can be used as an efficient, stable, and reliable method for solving operation
optimization problems in power systems and help reduce the cost of energy generation and increase the economic efficiency of power grids.
References 29, tables 4, figures 2.
Key words: economic load dispatch, power system optimization, generation cost minimization, electrical power system
operation, valve-point effect, metaheuristic algorithms, teaching–learning based optimization.
Вступ. Енергетичні мережі вважаються однією з найважливіших інфраструктур енергопостачання в сучасному суспільстві, і їхня
економічна експлуатація відіграє важливу роль у зниженні витрат на виробництво електроенергії та підвищенні ефективності
систем генерації. У той же час електростанції відповідають за забезпечення мережі необхідною потужністю, і оптимальне
розподілення потужності між ними безпосередньо впливає на кінцеву вартість виробництва енергії. З цієї причини завдання
економічного розподілу навантаження (ELD) було поставлено як одну з фундаментальних проблем оптимальної роботи
енергосистем. Проблема. Статичне завдання економічного розподілу навантаження визначається з метою визначення кількості
електроенергії, що виробляється кожним генераторним блоком, таким чином, щоб мінімізувати загальну вартість виробництва
енергії, при цьому дотримувалися всі експлуатаційні обмеження енергосистеми, включаючи обмеження балансу потужності,
втрати в передавальної мережі, обмеження на виробництво електроенергії генераторами. Наявність таких особливостей, як
нелінійна функція вартості, ефект точки включення та невипуклий простір пошуку робить рішення цього завдання класичними
математичними методами серйозною проблемою. Мета. Розробка ефективного методу розв’язання ELD задач та досягнення
оптимального графіка виробництва для генераторів енергосистеми з мінімальними виробничими витратами. Методика. У цьому
дослідженні для розв’язання задачі ELD використовувався метаевристичний алгоритм оптимізації на основі навчання та
викладання (TLBO). Оцінка продуктивності алгоритму проводилася на стандартній шестиблочній енергосистемі. Результати
оптимізації показують, що алгоритм TLBO здатний забезпечити оптимальний графік виробництва з урахуванням усіх системних
обмежень, при цьому загальні виробничі витрати сягають $15452,06. Для оцінки якості роботи результати TLBO порівнювалися з
сімома відомими метаевристичними алгоритмами, і результати моделювання показали, що TLBO забезпечує найкращу
продуктивність, займаючи перше місце за значенням цільової функції, середньої вартості та стабільності роботи. Наукова
новизна. Інновація даного дослідження полягає в ефективному застосуванні алгоритму TLBO для вирішення задачі ELD з
урахуванням повного набору експлуатаційних обмежень та надання всебічного порівняльного аналізу з декількома
метаевристичними алгоритмами. Практична значимість. Результати цього дослідження показують, що алгоритм TLBO може
бути використаний як ефективний, стабільний і надійний метод для вирішення завдань оптимізації роботи енергосистем, сприяючи
зниженню собівартості виробництва енергії та підвищенню економічної ефективності електромереж. Бібл. 29, табл. 4, рис. 2.
Ключові слова: економічний розподіл навантаження, оптимізація енергосистеми, мінімізація собівартості генерації,
експлуатація електроенергетичної системи, ефект перемикання клапанів, метаевристичні алгоритми, оптимізація на
основі навчання та викладання.
Introduction. Electric power networks are
recognized as one of the most important critical
infrastructures in modern societies and play a fundamental
role in providing energy to various industrial, commercial
and domestic sectors. The rapid growth in demand for
electrical energy, the development of industries and the
increasing dependence of societies on electrical systems
have made economical, safe and reliable exploitation of
power networks one of the most important concerns of
engineers and power system planners. In the meantime,
electric power generation plants, as the main production
components in power networks, are responsible for
providing the power needed by consumers. Given the high
costs of electric power generation, the way power is
distributed among generating units has a direct impact on
the final cost of energy production and the economic
efficiency of the power system. Therefore, the development
of efficient methods for optimal allocation of generated
power among power plant units is of particular importance.
Electrical Engineering & Electromechanics, 2026, no. 4 67
One of the fundamental problems in the economic
operation of power systems is the economic load dispatch
(ELD) problem. The main goal of this problem is to
determine the amount of power generated by each
generator unit in such a way that the total cost of generating
electrical energy is minimized while at the same time
observing all operational constraints of the power system
[1]. These constraints include power balance constraints,
transmission network losses, generator generation
constraints, power rate of change constraints, and also
prohibited areas of operation. The ELD problem is actually
one of the most important optimization problems in the
field of power system operation, and its accurate solution
can lead to significant reduction in energy production costs
and increased power plant efficiency [2].
Among the different variants of this problem, static
economic load dispatch (SELD) refers to a case where the
power system conditions are assumed to be constant over
a short period of time – usually 1 hour – and the goal is to
optimally distribute the generated power among the
power plant units [3]. Although the SELD model seems
simpler than its dynamic version, in practice it is still a
complex, nonlinear and nonconvex optimization problem.
The presence of factors such as the valve point loading
effect on the cost function of generators, the consideration
of transmission network losses, power rate constraints and
prohibited operation areas causes the search space of this
problem to have multi-peaked and discontinuous
characteristics [4]. For this reason, despite the successful
performance of classical mathematical methods in solving
many problems [5, 6], the use of classical gradient-based
optimization methods or mathematical programming in
many cases encounters problems such as poor
convergence or getting stuck in local optima [7].
In recent decades, the use of metaheuristic algorithms
as powerful tools for solving complex optimization
problems has attracted widespread attention from
researchers. These algorithms, which are often inspired by
natural phenomena, biological behaviors, or social
processes, are able to effectively explore the response space
without the need for gradient information and using
population search, and provide high-quality answers to
complex optimization problems [8, 9]. Features such as
global search ability, high flexibility, and the ability to
solve nonlinear and nonconvex problems have led to these
algorithms being widely used in various engineering fields,
including power system optimization [10–13].
Numerous studies have also been conducted in the
field of ELD problem using metaheuristic algorithms. For
example, in one of the early studies in this field, an
adaptive particle swarm optimization method was proposed
to solve the ELD problem, which was able to consider
constraints such as network losses, dynamic constraints,
and prohibited areas of operation, and the results showed
that this method performed well compared to conventional
methods in terms of convergence speed and response
quality [14]. In another study, the ELD problem was
modeled as a dynamic programming problem by
considering the spinning reserve constraint, and an efficient
algorithm was presented to solve the sequence of load
dispatch problems in different time intervals [15].
In recent years, various approaches based on
advanced metaheuristic algorithms have been developed
to solve the ELD problem. For example, a hybrid method
based on slime mould algorithm and genetic algorithm
has been proposed to solve the ELD problem in
microgrids, which has been able to provide better
performance than several known algorithms by increasing
the population diversity in the early stages of the search
and enhancing the exploitation power in the final stages
[16]. Also, newer algorithms such as osprey optimization
algorithm have been used to solve ELD and SELD
problems, and their results have shown that these
algorithms have a good ability to reduce the production
cost and improve the stability of the responses [17].
In addition, some research has investigated hybrid
and more advanced approaches. For example, the use of
metaheuristic algorithms in combined economic emission
dispatch planning has been studied considering renewable
energy sources and demand side management [18]. In
another study, a new crowd-learning-based algorithm
called achieving and refining knowledge-based
optimization was introduced to solve the SELD problem,
which uses the crowd-learning process in the population
to improve the search performance [19]. Also, advanced
parameter less algorithms such as self-adaptive multi-
population quadratic approximation guided Jaya have
been used to solve ELD problems in power systems with
different numbers of generating units, which have been
able to significantly reduce the production cost compared
to previous methods [20].
Along with these studies, the use of hybrid
metaheuristic algorithms has also been considered. For
example, a meme version of salp swarm algorithm has
been developed to solve ELD problems with strong
constraints, in which the combination of the global search
of the population algorithm with the local search of the
improver has led to increased accuracy and convergence
speed in finding optimal solutions [21].
Despite significant advances in this area, there are still
several challenges in solving the ELD problem. The high
complexity of the search space, the existence of multiple
operational constraints, the nonlinear and multi-peaked
behavior of the cost function, as well as the need to achieve
stable and reliable responses in limited computational time
are among the factors that necessitate the development of
more efficient optimization methods. In addition, many
metaheuristic algorithms require fine tuning of control
parameters and their performance can be highly dependent
on the choice of these parameters. This issue causes that in
some practical applications, the use of these algorithms is
associated with additional complexities.
In the meantime, the teaching–learning based
optimization (TLBO) algorithm has been proposed as one
of the effective metaheuristic algorithms that are inspired
by the learning process in educational environments. This
algorithm operates based on two main stages teacher phase
and student phase and tries to gradually improve the quality
level of responses by using the interaction between
population members. One of the important features of
TLBO is that, unlike many metaheuristic algorithms, it
does not require complex control parameters and can be
implemented using only the population size and number of
iterations. This feature makes its implementation and use
process relatively simple and has good stability in solving
complex optimization problems.
68 Electrical Engineering & Electromechanics, 2026, no. 4
The goal of this work is to develop an efficient
method for solving the ELD problem and to achieve an
optimal production schedule for power system generators
with minimum production cost.
In this study, the ELD problem is modeled by
considering a set of operational constraints including
power balance constraint, transmission network losses,
generator generation constraints, power rate of change
constraints, and prohibited operation areas. Then, the
TLBO algorithm is used as an optimization tool to
minimize the total cost of electrical energy generation.
The main innovations and contributions of this
research can be summarized as follows:
investigating the SELD problem in a power system
considering a complete set of real operating constraints;
providing an accurate mathematical model of the
ELD problem including the generation cost objective
function and the power system operating constraints;
using the TLBO algorithm as an efficient
optimization method for solving the ELD problem;
conducting simulation studies on a standard 6-unit
power system to evaluate the performance of the
algorithm;
conducting a comprehensive comparative analysis
between the performance of TLBO and seven well-known
metaheuristic algorithms in order to development the
quality of responses, performance stability, and
convergence speed.
The obtained results show that the TLBO algorithm
is able to provide an optimal generation schedule for
generators by observing all power system constraints and
shows very competitive and stable performance compared
to several well-known metaheuristic algorithms.
Problem statement and mathematical modeling.
The ELD problem is one of the fundamental problems in
the optimal operation of power systems. The main
objective in this problem is to determine the amount of
power generated by each generator unit in such a way that
the total cost of generating electrical energy is minimized,
while all operating constraints of the power system are met.
In the static version of this problem, known as
SELD, the operating time period is usually considered to
be a short period such as 1 hour. In this framework, it is
assumed that the system conditions are constant during
this time period and the goal is to distribute the generated
power among the operating units in an optimal manner.
At the same time, it must be ensured that the system load
demand is met and the technical constraints of the
generators such as generation limits, power rate limit and
prohibited operation areas are observed.
In general, the mathematical model of the ELD
problem consists of a production cost objective function
and a set of physical and operational constraints, which
are presented below.
Objective function. In many power system studies,
the fuel cost of the generating units can be approximated
by a quadratic function of the generator output power.
Accordingly, the objective function of the problem is
expressed as the minimization of the total production cost:
GN
i
ii PfFMinimize
1
: , (1)
where F is the total production cost of the system; NG is
the number of generating units in operation; Pi is the
actual output power of generator i (MW); fi(Pi) is the
production cost function of generator i.
The cost function of each generator is modeled as:
fi(Pi) = aiPi
2 + biPi + ci, (2)
where the coefficients ai, bi, ci are the cost coefficients
related to the generating unit i.
In some power plants, due to the valve point loading
effect, the cost function has a nonsmooth and oscillating
behavior. In this case, the cost function is modified as:
iiiiiiiiiii PPfecPbPaPf min2 sin , (3)
where ei, fi are the coefficients related to the valve point
loading effect and cause discontinuity and multi-peaking
of the cost function. The existence of these features makes
the ELD problem a nonlinear and nonconvex optimization
problem.
Power balance constraint. The most important
constraint in the ELD problem is the power balance
constraint. According to this constraint, the total power
generated by the generators must be equal to the sum of
the system load demand and the power losses in the
network. This relationship is expressed as:
GN
i
LDi PPP
1
, (4)
where PD is the system load demand power; PL is the
power losses in the transmission network
Power losses are usually calculated using B-factors as:
GG G N
i
ii
N
i
N
j
jijiL BPBPBPP
1
000
1 1
, (5)
where Bij, B0i, B00 are the network loss factors.
Generator production limits. Each generating unit
has a minimum and maximum power that can be
produced, which is determined by the technical
limitations of the equipment. Therefore, the output power
of each generator must be within its allowed range:
Pi
min Pi Pi
max, (6)
where Pi
min is the minimum power output of generator i;
Pi
max is the maximum power output of generator i.
Power rate limit. In practical power plant operation
conditions, generators are not able to change their output
power suddenly. For this reason, the rate of increase or
decrease in their power is limited, which is called ramp
rate limit.
If the generated power increases:
Pi
t – Pi
t–1 URi. (7)
And if the generated power decreases:
Pi
t–1 – Pi
t DRi. (8)
where Pi
t–1 is the generated power of unit i in the previous
time period; URi is the upper limit of the power increase
rate; DRi is the lower limit of the power decrease rate.
As a result, the generator operating range is
modified by considering these limitations as:
max(Pi
min, Pi
t–1 – DRi) Pi
t min(Pi
max, Pi
t–1 + URi). (9)
Prohibited operating zones. In some generators, there
are ranges of generated power in which operation is not
allowed due to mechanical limitations or instability
phenomena. These areas, known as prohibited operating
zones, may be caused by factors such as shaft vibrations,
steam valve problems or mechanical limitations of the
equipment. Therefore, the generator output power must be
outside these areas. Symbolically for unit i:
Electrical Engineering & Electromechanics, 2026, no. 4 69
upper
kii
lower
kii PPPP ,, or , (10)
where lower
kiP , , upper
kiP , are the lower and upper bounds of
the forbidden region k for generator i, respectively.
If the minimum or maximum generator output limits
fall within a forbidden region, these limits must be
modified to prevent the generator operating point from
falling within the forbidden region.
Problem solving method. Due to the presence of
nonlinear, discontinuous, and multi-peaked cost function
as well as numerous operational constraints, the SELD
problem becomes a complex optimization problem that is
difficult to solve with classical mathematical methods in
many cases. For this reason, in recent years, the use of
metaheuristic algorithms to solve this problem has
received widespread attention.
In this paper, the TLBO algorithm is used to solve
the SELD problem. Inspired by the learning process in
educational environments, this algorithm is able to
provide high-quality answers to complex optimization
problems by creating an appropriate balance between
global search and local search. In the following, the
structure and steps of the TLBO algorithm are introduced
and how it is applied to solve the ELD problem in the
power system will be explained.
The TLBO algorithm is one of the efficient
metaheuristic algorithms in the field of optimization,
which was first introduced by Rao R. V. et al [22]. The
main idea of this algorithm is inspired by the learning
process in educational environments. In an educational
environment, the knowledge and scientific level of
students are improved through two main processes: first
learning from the teacher and then learning from
interaction with other students.
The TLBO algorithm, by modeling these two steps,
gradually guides a population of possible answers towards
the optimal answer. In this algorithm, each member of the
population is considered as a learner and the best member
of the population plays the role of teacher. During the
optimization process, students try to improve their
knowledge level by learning from the teacher and also by
interacting with each other.
One of the important features of TLBO is that,
unlike many metaheuristic algorithms, it does not require
complex control parameters and operates only using the
population size and the number of iterations. This feature
makes it simple to implement and has good stability in
solving complex optimization problems.
In the framework of optimization problems, each
student represents an answer vector and the value of the
objective function will indicate the knowledge level or
quality of the answer of that student. In the following, the
mathematical model of the TLBO algorithm is presented in
two main phases, namely teacher phase and student phase.
Representation of answers and initializing the
population. Suppose that the optimization problem has D
decision variables. In the TLBO algorithm, an initial
population of N students is randomly generated in the
search space. Each student is represented as a vector as:
Xi = [xi,1, xi,2, ... , xi,D], i = 1, 2, ... , N, (11)
where Xi is the position of student i in the search space. After
generating the initial population, the value of the objective
function is calculated for all individuals and the best
individual in the population is selected as the class teacher.
Teacher phase. In an educational class, the teacher
tries to increase the knowledge level of the entire class. In the
TLBO algorithm, the teacher also tries to bring the average
knowledge of the class closer to his knowledge level.
First, the population mean is calculated as:
N
i
iX
N
M
1
1
. (12)
Then the position of each student is updated using
the following relationship:
Xi
new = Xi + ri(Xteacher – TFM), (13)
where Xteacher is the best individual in the population
(teacher); ri is the random number in the interval [0, 1];
TF is the teaching factor, which is usually randomly set to
1 or 2, i.e., TF {1, 2}.
After calculating the new position, if the objective
function value for Xi
new is better than Xi, the student’s
position is updated to the new value.
Learner phase. In educational environments,
students learn from interacting and exchanging
information with other students in addition to learning
from the teacher. In the TLBO algorithm, this concept is
modeled in the form of the learner phase.
In this step, for each student Xi, another student Xj is
randomly selected i j. Then, based on the comparison of
the objective function values of these 2 students, the new
position is calculated as:
else,,
;if,
ijii
jijiiinew
i XXrX
XfXfXXrX
X , (14)
where r is the random number in the interval [0, 1].
After calculating the new position, if the objective
function value is better for the new response, the student’s
position is updated. This process increases the diversity of
the population and improves the search power of the
algorithm.
Application of TLBO in solving the ELD problem.
Given the nonlinear, nonconvex and multiple-constrained
nature of the ELD problem, the use of classical methods in
many cases encounters convergence problems and getting
stuck in local optima. The TLBO algorithm is considered a
suitable option for solving such optimization problems due
to its simple structure, appropriate search ability, and no
need to adjust complex parameters.
In this study, each student in the TLBO algorithm
represents a generator power vector in the ELD problem,
and the objective function will be the total system
production cost as defined in the previous section. During
the optimization process, the TLBO algorithm tries to
minimize the value of the objective function by gradually
updating the power vectors and at the same time satisfy
all the constraints of the power system.
In the next section of the article, the modeling and
implementation of the TLBO algorithm for solving the
SELD problem will be presented in detail and the results
of the optimization will be examined.
Simulation studies and results analysis. In this
section, the performance of the TLBO algorithm in
solving the SELD problem is investigated and evaluated.
The main objective of these simulation studies is to
development the ability of the TLBO algorithm to
70 Electrical Engineering & Electromechanics, 2026, no. 4
determine the optimal generation schedule of power
plants in order to minimize the total cost of electrical
energy production in a real power system, taking into
account all operational constraints.
Due to the nonlinear, nonconvex, and multi-peaked
nature of the ELD problem – especially in the presence of
the valve point loading effect, power rate limitation, and
forbidden operation zones – solving this problem using
classical optimization methods is very difficult. Hence,
the use of advanced metaheuristic algorithms can be an
effective solution to achieve high-quality answers.
In this regard, in this section, the ELD problem is
investigated using the TLBO algorithm. First, the case study
and data of the power system used are introduced. Then, the
results obtained from implementing the TLBO algorithm on
this problem are presented and analyzed. In the following,
comparative studies between TLBO and several well-known
metaheuristic algorithms are conducted to accurately
evaluate the performance quality of this algorithm.
Introduction to the studied system and its data. In
order to evaluate the performance of the proposed
algorithm, the ELD problem is investigated on a power
system consisting of 6 generating units (generators). This
system is used as a standard case study in many power
system optimization studies and includes a set of realistic
operational constraints including network losses, power
rate of change constraints, and prohibited operation areas.
In this system, the total network load demand is
equal to PD = 1263 MW. And the goal is to distribute the
generated power among the 6 generators in such a way
that the total cost of production is minimized while at the
same time respecting all system constraints.
The transmission network losses in this system are
modeled using the B coefficient matrix, which is defined as:
510
152.08.06.01.02.0
2.09.126.01.06.05.0
8.06.024.001.01.0
6.01.001.39.07.0
1.06.01.09.04.12.1
2.05.01.07.02.17.1
B .
Also, the coefficient vector and the loss constant are
defined as:
B0i = [–0.3908, –0.1297, 0.7047, 0.0591, 0.2161, –0.6635]10–5;
B00 = 0.005610–2.
The technical characteristics of the generating units,
including minimum and maximum generation power, cost
function coefficients, and valve-point effect coefficients,
are presented in Table 1. These data model the actual
behavior of the power plant units and create a nonlinear
and discontinuous cost function.
Also, information related to initial generation power,
power increase and decrease rate constraints, and forbidden
operation areas is reported in Table 2. The existence of
these constraints complicates the search space of the
problem and turns it into an optimization problem with
multiple constraints and discrete response space.
Consequently, finding an optimal generation
schedule for this system requires the use of a powerful
optimization algorithm with the ability to search
efficiently in the response space.
Table 1
Generators data
Unit Pmin, MW Pmax, MW a, $/MW2 b, $/MW c, $ e f
1 100 500 0.007 7 240 300 0.035
2 50 200 0.0095 10 200 200 0.042
3 80 300 0.009 8.5 220 150 0.042
4 50 150 0.009 11 200 120 0.063
5 50 200 0.008 10.5 220 150 0.063
6 50 120 0.0075 12 190 100 0.063
Table 2
Initial generations P0, ramp rate and operating zone constraints
Unit
P0,
MW
UR,
MW/h
DR,
MW/h
Zone 1
min
Zone 1
max
Zone 2
min
Zone 2
max
1 440 80 120 210 240 350 380
2 170 50 90 90 110 140 160
3 200 65 100 150 170 210 240
4 150 50 90 80 90 110 120
5 190 50 90 90 110 140 150
6 150 50 90 75 85 100 105
Optimization of the ELD problem using TLBO.
In this section, the TLBO algorithm is implemented to
solve the ELD problem of the studied system. In the
framework of this algorithm, each student represents a
vector containing the generation power of 6 generators:
X = [P1, P2, P3, P4, P5, P6]. And the value of the objective
function is equal to the total production cost of the
system, which must be minimized.
The results obtained from the implementation of the
TLBO algorithm are presented in Table 3, which shows
the optimal generation schedule for each of the power
plant units.
Table 3
Generation schedule of generators
Gen. 1 452.7068 MW
Gen. 2 171.6543 MW
Gen. 3 254.0167 MW
Gen. 4 135.0149 MW
Gen. 5 155.2349 MW
Gen. 6 106.9519 MW
Generation 1275.58 MW
Demand 1263 MW
Loss 12.58 MW
Objective function $15452.06
Based on these results:
total power generated: 1275.58 MW;
system demand load:1263 MW;
network losses: 12.58 MW;
total production cost: $15452.06.
It is observed that the total power generated by the
generators is equal to the sum of demand load and
network losses, which indicates exact implementation of
the power balance constraint in the optimal response.
In order to examine the convergence behavior of the
algorithm, the TLBO convergence curve is shown in Fig. 1.
Analysis of this curve shows that the TLBO
algorithm experiences a very rapid decrease in the value
of the objective function in the initial stages of the search.
This indicates the high ability of the algorithm in
Exploration of the search space. After the first few
iterations, the objective function decreases gradually and
the algorithm enters the Exploitation phase, during which
the answers gradually converge around the best answer.
Electrical Engineering & Electromechanics, 2026, no. 4 71
Iterations
Fig. 1. TLBO convergence curve in solving the ELD problem
Around the middle iterations, the objective function
value approaches a nearly constant value and very small
fluctuations are observed. This behavior indicates that the
algorithm has been able to achieve a stable optimal region
in the search space.
Overall, the convergence curve shows that the
TLBO algorithm has a suitable convergence speed and
high stability in solving the ELD problem.
Comparative studies with competing algorithms. In
order to accurately evaluate the performance quality of the
TLBO algorithm, the results obtained have been compared
with the performance of seven well-known metaheuristic
algorithms: genetic algorithm (GA) [23], particle swarm
optimization (PSO) [24], gravitational search algorithm
(GSA) [25], whale optimization algorithm (WOA) [26],
multi-verse optimizer (MVO) [27], tunicate swarm
algorithm (TSA) [28] and grey wolf optimizer (GWO) [29].
The statistical results obtained from the implementation of
these algorithms are presented in Table 4. Examining the
mean objective function value shows that TLBO, with a
value of 15467.22, has the lowest average production cost
among all the algorithms. This indicates the high stability
of the TLBO algorithm in multiple implementations.
Table 4
Statistical results of TLBO and competitor algorithms applied to
the ELD problem
SI TLBO GA PSO GSA
Mean 15467.22 22037.65 15535.48 189616.1
Best 15452.06 15615.83 15486.05 145612.4
Worst 15484 40320.06 15599.81 234290
Std 14.69466 12196.11 48.77296 36358.59
Median 15466.4 16107.36 15528.04 189281
Rank 1 7 5 8
SI MVO WOA GWO TSA
Mean 15502.55 15589.42 15504.69 15509.38
Best 15472.31 15457.59 15469.45 15485.58
Worst 15557.81 15725.69 15524.75 15520.32
Std 38.14957 125.157 25.18478 16.01018
Median 15490.04 15587.2 15512.27 15515.8
Rank 2 6 3 4
In terms of best value, TLBO also performed very
competitively with a value of 15452.06 and in many cases
better than other algorithms.
Also, the standard deviation (std) for TLBO is 14.69,
which is a very small value compared to most other
algorithms. This shows that the TLBO algorithm has very
stable and reliable behavior in different implementations.
In contrast, algorithms such as GA and GSA have
very large standard deviations, which indicate lack of
stability and high sensitivity to initial conditions.
Finally, the ranking presented in Table 4 shows that
TLBO has achieved the 1st rank among all algorithms.
To provide a visual analysis of the results, Boxplot
graphs related to the performance of the algorithms are
displayed in Fig. 2.
Fig. 2. Boxplot diagram of the performance of the algorithms on
the ELD problem
Analysis of these diagrams shows that:
TLBO has the lowest dispersion of results;
the range of changes in the responses in TLBO is
very limited;
the median value in TLBO is very close to the
optimal value.
These features indicate that TLBO is able to achieve
responses very close to the optimal response in most
implementations.
Discussion. The results of simulation studies show
that the TLBO algorithm exhibits very effective
performance in solving the SELD problem. This
algorithm has been able to provide an optimal generation
schedule for generators, which leads to minimizing the
total cost of energy production, while respecting all power
system constraints.
Comparative studies also show that TLBO has
superior performance compared to known metaheuristic
algorithms. In particular, this algorithm has achieved the
lowest objective function value, lowest standard
deviation, and best performance rank among the
algorithms studied.
One of the most important advantages of TLBO is
the no need for complex control parameters. Unlike many
metaheuristic algorithms such as GA or PSO that require
fine-tuning of parameters such as mutation rate, crossover
rate, or acceleration coefficients, TLBO operates only
using the population size and number of iterations. This
feature makes its implementation and use process much
simpler and more reliable.
In addition, the two-stage structure of TLBO
including teacher phase and student phase creates a good
balance between global search and local search in the
response space. This feature plays an important role in
preventing the algorithm from getting stuck in local optima.
Despite these advantages, the use of TLBO also has
some limitations. For example, in very large-scale
problems, the convergence speed of the algorithm may
decrease. Also, in some problems with very complex
search spaces, there may be a need to combine TLBO
with other improvement techniques.
72 Electrical Engineering & Electromechanics, 2026, no. 4
However, the results of this study show that TLBO
is an efficient, stable, and reliable method for solving the
ELD problem in power systems.
Finally, the most important finding of this study is that
the TLBO algorithm is able to provide high-quality answers
to the SELD problem with appropriate convergence speed
and high stability and has superior performance compared
to many known metaheuristic algorithms. This indicates that
TLBO can be used as an effective tool in power system
operation optimization problems.
Conclusions. Power systems, as one of the vital
infrastructures of modern societies, play a fundamental role
in providing the electrical energy needed by the industrial,
commercial and domestic sectors. The economic and
reliable operation of these systems largely depends on the
proper operation of power generation plants and the
optimal distribution of generated power among them. In the
meantime, the ELD problem has been raised as one of the
fundamental problems in the operation of power networks,
the main goal of which is to determine the optimal
generation schedule of power plant units in such a way that
the total cost of electrical energy generation is minimized
while all system operating constraints are met.
Due to the nonlinear, nonconvex and multi-peaked
characteristics of the ELD problem – especially in the
presence of valve point loading effect, power rate
constraints and prohibited operation areas – solving this
problem using classical mathematical methods has faced
serious challenges in many cases. For this reason, in recent
years, the use of metaheuristic algorithms as powerful tools
for solving complex optimization problems in power
systems has received widespread attention from
researchers.
In this study, the SELD problem was investigated in
a power system consisting of 6 generating units, and the
TLBO algorithm was used as a problem-solving method.
In this framework, each member of the TLBO algorithm
population was considered to represent a generator power
vector in the power system, and the objective function
was defined as minimizing the total cost of energy
production. Also, all system operating constraints,
including power balance constraints, transmission
network losses, generator generation constraints, power
rate of change constraints, and prohibited operation areas
were included in the optimization model.
Simulation studies were conducted on a 6-unit power
system in which the total network load demand was
considered to be 1263 MW. The optimization results showed
that the TLBO algorithm was able to provide an optimal
generation schedule for the generators such that the total
generated power was 1275.58 MW and after considering
12.58 MW of network losses, the system power balance
constraint was accurately established. The total generation
cost obtained was also reported to be $15452.06, which
indicates the appropriate performance of the algorithm in
achieving quality responses.
In order to accurately evaluate the performance
quality of the TLBO algorithm, the results obtained from
this algorithm were compared with the performance of
seven well-known metaheuristic algorithms GA, PSO,
GSA, WOA, MVO, TSA, and GWO. The results of the
comparative studies showed that the TLBO algorithm was
able to provide the best performance among all the
algorithms studied. Specifically, this algorithm achieved
the lowest objective function value, the lowest average
production cost, and the lowest standard deviation in
different runs. Based on the presented statistical results,
TLBO was introduced as the most efficient method for
solving the ELD problem in this study, ranking first
among the eight algorithms studied.
The analysis of the simulation results also showed that
the TLBO algorithm has appropriate convergence speed,
high stability in multiple runs, and effective ability to search
the response space. One of the most important advantages
of this algorithm is the lack of need to adjust complex
control parameters, which simplifies its implementation and
increases its reliability in practical applications. In addition,
the two-stage structure of the algorithm, including teacher
phase and student phase, creates an effective balance
between global search and local search and reduces the
possibility of getting stuck in local optima.
Based on the results obtained, the findings of this study
showed that the TLBO algorithm is an efficient, stable and
competitive method for solving the ELD problem in power
systems and can be used as a suitable tool in power grid
operation optimization problems. The use of this algorithm
can lead to reducing the cost of electric energy generation,
increasing the efficiency of power plants and improving the
economic performance of the power system.
Despite the successful results obtained in this study,
there are still several areas for the development and
expansion of this study in future research. Some of the most
important research suggestions for future work include:
1. Developing the proposed model for solving the ELD
problem, in which time variations of the load and
dynamic constraints of the power system are considered.
2. Investigating the performance of the TLBO
algorithm in large-scale power systems with a larger
number of generators and comparing it with other
advanced metaheuristic algorithms.
3. Combining the TLBO algorithm with other
optimization methods or artificial intelligence techniques
to create hybrid algorithms with stronger search
capabilities and higher convergence speed.
4. Extending the ELD problem model by considering
renewable energy sources such as wind and solar power
plants and investigating the impact of the uncertainty of the
production of these sources on the optimization process.
5. Development of multi-objective versions of the
TLBO algorithm to solve power system operation
problems in which, in addition to the production cost,
indicators such as pollutant emissions, system reliability,
and network stability are simultaneously optimized.
Overall, the results of this research showed that using
the TLBO algorithm can be an effective approach to solving
complex optimization problems in power systems and
provide a suitable path for the development of future research
in the field of economic operation of power networks.
Conflict of interest. The authors declare that they
have no conflicts of interest.
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Received 02.01.2026
Accepted 18.03.2026
Published 02.07.2026
T. Hamadneh 1, PhD, Associate Professor,
O. Alsayyed 2, PhD, Professor,
M. Al Soudi 3, PhD, Assistant Professor,
1 Department of Mathematics,
Al Zaytoonah University of Jordan, Amman 11733, Jordan,
e-mail: t.hamadneh@zuj.edu.jo (Corresponding Author)
2 Department of Mathematics, Faculty of Science,
The Hashemite University, p.o. box 330127, Zarqa 13133, Jordan.
3 Department of Basic Scientific Sciences,
Applied Science Private University, Amman 11931, Jordan.
How to cite this article:
Hamadneh T., Alsayyed O., Al Soudi M. Efficient optimization of static economic load dispatch in electrical power systems using the
teaching–learning based optimization algorithm. Electrical Engineering & Electromechanics, 2026, no. 4, pp. 66-73. doi:
https://doi.org/10.20998/2074-272X.2026.4.09
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| id | eiekhpieduua-article-366078 |
| institution | Electrical Engineering & Electromechanics |
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| publishDate | 2026 |
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| spelling | eiekhpieduua-article-3660782026-07-01T21:42:56Z Efficient optimization of static economic load dispatch in electrical power systems using the teaching–learning based optimization algorithm Efficient optimization of static economic load dispatch in electrical power systems using the teaching–learning based optimization algorithm Hamadneh, T. Alsayyed, O. Al Soudi, M. економічний розподіл навантаження оптимізація енергосистеми мінімізація собівартості генерації експлуатація електроенергетичної системи ефект перемикання клапанів метаевристичні алгоритми оптимізація на основі навчання та викладання economic load dispatch power system optimization generation cost minimization electrical power system operation valve-point effect metaheuristic algorithms teaching–learning based optimization Introduction. Power grids are considered one of the most critical energy infrastructures in modern societies, and their economic exploitation plays an important role in reducing the costs of generating electrical energy and increasing the efficiency of generation systems. In the meantime, power generation plants are responsible for providing the required power to the grid, and the optimal distribution of power among them has a direct impact on the final cost of energy generation. For this reason, the economic load dispatch (ELD) problem has been raised as one of the fundamental issues in the optimal operation of power systems. Problem. The static economic load dispatch problem is defined with the aim of determining the amount of power generated by each generator unit in such a way that the total cost of energy generation is minimized, while all operational constraints of the power system, including power balance constraints, transmission network losses, generator production constraints, power rate of change constraints, and prohibited areas, are met. The presence of features such as nonlinear cost function, valve-point effect and nonconvex search space makes solving this problem with classical mathematical methods face serious challenges. Goal. To develop an efficient method for solving the ELD problem and to achieve an optimal production schedule for power system generators with minimum production cost. Methodology. In this study, the metaheuristic algorithm teaching–learning based optimization (TLBO) has been used to solve the ELD problem. The performance evaluation of the algorithm has been carried out on a standard 6-unit power system. Results. The optimization results show that the TLBO algorithm is able to provide an optimal production schedule by observing all system constraints, in which the total production cost reaches $15452.06. To evaluate the performance quality, the results of TLBO were compared with seven well-known metaheuristic algorithms, and the simulation results showed that TLBO provided the best performance by achieving first rank in terms of objective function value, average cost, and performance stability. Scientific novelty. The innovation of this research lies in the effective application of the TLBO algorithm to solve the ELD problem by considering a complete set of operational constraints and providing a comprehensive comparative analysis with several metaheuristic algorithms. Practical value. The findings of this study indicate that the TLBO algorithm can be used as an efficient, stable, and reliable method for solving operation optimization problems in power systems and help reduce the cost of energy generation and increase the economic efficiency of power grids. References 29, tables 4, figures 2. Вступ. Енергетичні мережі вважаються однією з найважливіших інфраструктур енергопостачання в сучасному суспільстві, і їхня економічна експлуатація відіграє важливу роль у зниженні витрат на виробництво електроенергії та підвищенні ефективності систем генерації. У той же час електростанції відповідають за забезпечення мережі необхідною потужністю, і оптимальне розподілення потужності між ними безпосередньо впливає на кінцеву вартість виробництва енергії. З цієї причини завдання економічного розподілу навантаження (ELD) було поставлено як одну з фундаментальних проблем оптимальної роботи енергосистем. Проблема. Статичне завдання економічного розподілу навантаження визначається з метою визначення кількості електроенергії, що виробляється кожним генераторним блоком, таким чином, щоб мінімізувати загальну вартість виробництва енергії, при цьому дотримувалися всі експлуатаційні обмеження енергосистеми, включаючи обмеження балансу потужності, втрати в передавальної мережі, обмеження на виробництво електроенергії генераторами. Наявність таких особливостей, як нелінійна функція вартості, ефект точки включення та невипуклий простір пошуку робить рішення цього завдання класичними математичними методами серйозною проблемою. Мета. Розробка ефективного методу розв’язання ELD задач та досягнення оптимального графіка виробництва для генераторів енергосистеми з мінімальними виробничими витратами. Методика. У цьому дослідженні для розв’язання задачі ELD використовувався метаевристичний алгоритм оптимізації на основі навчання та викладання (TLBO). Оцінка продуктивності алгоритму проводилася на стандартній шестиблочній енергосистемі. Результати оптимізації показують, що алгоритм TLBO здатний забезпечити оптимальний графік виробництва з урахуванням усіх системних обмежень, при цьому загальні виробничі витрати сягають $15452,06. Для оцінки якості роботи результати TLBO порівнювалися з сімома відомими метаевристичними алгоритмами, і результати моделювання показали, що TLBO забезпечує найкращу продуктивність, займаючи перше місце за значенням цільової функції, середньої вартості та стабільності роботи. Наукова новизна. Інновація даного дослідження полягає в ефективному застосуванні алгоритму TLBO для вирішення задачі ELD з урахуванням повного набору експлуатаційних обмежень та надання всебічного порівняльного аналізу з декількома метаевристичними алгоритмами. Практична значимість. Результати цього дослідження показують, що алгоритм TLBO може бути використаний як ефективний, стабільний і надійний метод для вирішення завдань оптимізації роботи енергосистем, сприяючи зниженню собівартості виробництва енергії та підвищенню економічної ефективності електромереж. Бібл. 29, табл. 4, рис. 2. National Technical University "Kharkiv Polytechnic Institute" and Аnatolii Pidhornyi Institute of Power Machines and Systems of NAS of Ukraine 2026-07-02 Article Article application/pdf https://eie.khpi.edu.ua/article/view/366078 10.20998/2074-272X.2026.4.09 Electrical Engineering & Electromechanics; No. 4 (2026); 66-73 Электротехника и Электромеханика; № 4 (2026); 66-73 Електротехніка і Електромеханіка; № 4 (2026); 66-73 2309-3404 2074-272X en https://eie.khpi.edu.ua/article/view/366078/351647 Copyright (c) 2026 T. Hamadneh, O. Alsayyed, M. Al Soudi http://creativecommons.org/licenses/by-nc/4.0 |
| spellingShingle | economic load dispatch power system optimization generation cost minimization electrical power system operation valve-point effect metaheuristic algorithms teaching–learning based optimization Hamadneh, T. Alsayyed, O. Al Soudi, M. Efficient optimization of static economic load dispatch in electrical power systems using the teaching–learning based optimization algorithm |
| title | Efficient optimization of static economic load dispatch in electrical power systems using the teaching–learning based optimization algorithm |
| title_alt | Efficient optimization of static economic load dispatch in electrical power systems using the teaching–learning based optimization algorithm |
| title_full | Efficient optimization of static economic load dispatch in electrical power systems using the teaching–learning based optimization algorithm |
| title_fullStr | Efficient optimization of static economic load dispatch in electrical power systems using the teaching–learning based optimization algorithm |
| title_full_unstemmed | Efficient optimization of static economic load dispatch in electrical power systems using the teaching–learning based optimization algorithm |
| title_short | Efficient optimization of static economic load dispatch in electrical power systems using the teaching–learning based optimization algorithm |
| title_sort | efficient optimization of static economic load dispatch in electrical power systems using the teaching–learning based optimization algorithm |
| topic | economic load dispatch power system optimization generation cost minimization electrical power system operation valve-point effect metaheuristic algorithms teaching–learning based optimization |
| topic_facet | економічний розподіл навантаження оптимізація енергосистеми мінімізація собівартості генерації експлуатація електроенергетичної системи ефект перемикання клапанів метаевристичні алгоритми оптимізація на основі навчання та викладання economic load dispatch power system optimization generation cost minimization electrical power system operation valve-point effect metaheuristic algorithms teaching–learning based optimization |
| url | https://eie.khpi.edu.ua/article/view/366078 |
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