1. Introduction
Economic load dispatch (ELD) is considered the most elementary and computationally heavy optimization problem in electricity industry studies. ELD enables the power system to analyze the committed generating units to allocate the optimum level of active power output to achieve the total power demands at the cheapest operating cost while satisfying several physical and operational constraints [
1]. Over the last few years, various mathematical programming techniques and optimization methods have been applied to solve ELD problems, such as the Newton method [
2], the gradient method [
3], the base point and participation factors method, and the Lambda-iteration method [
2]. Nevertheless, none of the mentioned techniques perform well in addressing practical problems with nonlinear and discontinuous characteristics since these methods required the incremental cost of the generators to be increase in a monotonic or piecewise linear fashion [
2]. Therefore, nonlinear programming [
4], dynamic programming [
5], and some of their modified applications have been employed to deal with ELD issues. Unfortunately, the drawback of these methods is their computational time, which when applied to modern power systems with a huge number of generating units is too long.
To overcome the downside of traditional mathematical programming, researchers around the world have introduced metaheuristic optimization algorithms, such as simulated annealing (SA) [
6], genetic algorithms (GAs) [
7], tabu searches (TS) [
8], and artificial neural networks (ANNs) [
9]. Yalcinoz [
10] successfully addressed complex optimization issues. However, these probabilistic heuristic algorithms do not always guarantee the global search property. Recently, different kinds of global optimization algorithms have been proposed, such as particle swarm optimization (PSO) [
11], modified particle swarm optimization [
12], biogeography-based optimization (BBO) [
13], modified genetic algorithm [
14], bacteria foraging optimization [
15], differential evolution (DE) [
16], and ant colony optimization [
17]. The application of these techniques in ELD has delivered some promising solutions in terms of minimizing total generation cost and improving convergence rate.
However, recent research has recognized a few deficiencies in stochastic algorithms like GA. This degradation in efficiency and limited search capability may be noticeable when applied to multimodal objective functions. Additionally, TS requires a suitable selection of control parameters to attain an optimal solution. Slow convergence can be considered a key disadvantage of GA, SA, TS, and ANN, making them unable to appropriately address real-time issues. Even though PSO has attracted research interest due to its rapid convergence characteristic and its flexibility, PSO is still restricted when applied to large-scale real-time ELD, since it is not always guaranteed that the total generation cost is the global best solution.
The modification of the presented algorithm can be a solution for achieving the global optimal solution and upgrading the convergence rate. Existing modified metaheuristic methods include chaotic differential evolution and sequential quadratic programming (DEC_SQP) [
18], improved coordinated aggregation-based PSO (ICA_PSO) [
19], quantum-inspired particle swarm optimization (QPSO) [
20], modified shuffled frog leaping algorithm with GA crossover (MSFLA and GA) [
21], a different version of PSO [
19], shuffled differential evolution (SDE) [
21], hybrid biogeography-based optimization with differential evolution (DE/BBO) [
22], and adaptive hybrid backtracking search optimization [
23]. Oppositional invasive weed optimization (OIWO) [
24] improves the convergence rate of invasive weed optimization by incorporating opposite-based learning (OBL) [
25]. Nevertheless, this OIWO relies heavily on the initial selection of control parameters to obtain the global best solution. Moreover, the solution is not unique for every trial, and this method also has the problem of long computational time.
In recent years, a new evolutionary optimization technique, called grey wolf optimization (GWO), which mimics the social hierarchy and hunting behavior of grey wolves, has been proposed by Mirjalili et al. [
26]. There are several applications of GWO in power system optimization problems [
27,
28]. The outstanding characteristic of GWO compared with other stochastic heuristic algorithms is that it does not depend on accurate initialization of input parameters to obtain the global best solution. However, the parameters still need to be modified to avoid premature convergence, as well as to improve the convergence speed. To overcome the above limitation, a hybrid GWO algorithm with a robust learning mechanism (RLGWO) is introduced in this paper that incorporates opposite-based learning (OBL) as a candidate generation strategy. The main reason for selecting OBL is that it does not require any specific technique to speed up the convergence rate of different optimization algorithm. Additionally, the candidates generated by OBL are more likely to come closer to the global optimum solution than a solution using randomly generated candidates, because the technique simultaneously considers both the current population and its opposite. In only a short period of time, OBL, a new concept in computational intelligence, has attracted research attention with the aim of enhancing metaheuristic optimization algorithms to address large-scale ELD [
24,
29].
In this paper, the performance of a proposed grey wolf algorithm variant with robust tolerance-based adjust searching direction mechanism (RTASDM), called the robust learning grey wolf algorithm (RLGWO), is studied. The algorithm is tested on five test systems with different sizes and in consideration of power system constraints to evaluate the performance of the proposed approach in solving the ELD problem compared with other variants of the GWO. The reported results reveal the ability of RLGWO to archive superior solutions in terms of quality, consistency, and convergence rate compared to several other optimization methods.
4. Robust Learning-Based GWO Algorithm (RLGWO)
All omega members of the hunting group in the original GWO learn from the first three best leaders to update their position until the termination condition is reached, even if the fittest solution (alpha) is trapped in a local optimum. This kind of learning algorithm can work well in the exploitation phase and has the ability to converge rapidly, but is invalid when solving problems with such large and complex search spaces. Some methods have been proposed for GWO incorporating strategies to restrict the learning mechanism of omega to maintain the diversity of the population, like EEGWO [
32] and GWO-ABC [
33]. These strategies result in good exploration performance. However, it takes a longer time to reach a solution and slows down the convergence rate of the algorithm.
The algorithm (RLGWO) proposed in this paper achieves a balance between exploitation and exploration. In RLGWO, a robust tolerance-based adjust searching direction mechanism is used. This method gives omega the ability to adjust their search direction to avoid falling into local optima and to diminish the size of the search space. Additionally, an opposition learning-based candidate grey wolf strategy is utilized to generate candidate leaders that replace the position of alpha as well as beta and delta, in which the hunting group can perform in different areas within the search space. Subsequently, looking toward guaranteeing the efficiency and accuracy of the algorithm, a potential position update scheme is introduced to adapt the potential ability of the candidate leader to guide the grey wolf for exploitation in divergent dimensions.
4.1. Robust Tolerance-Based Adjust Searching Direction Mechanism (RTASDM)
It is widely known that the grey wolf in the original GWO had a high possibility of getting stuck in local optima when operating in a large and complex search space. In
Figure 1a, the hunting mechanism for a one-dimensional problem is presented, where the blue curve illustrates the object function.
is the alpha position (the best solution) that leads the rest of the population. In
Figure 1a, it is clear that each omega can move along the guiding direction of the alpha fitness value
and the hunting group will be trapped in a local optimum after several iteration. Assume
indicates the objective value of the
nth omega in the population at the
kth iteration.
will be updated in the next iteration using Formula (13). Then, it will generate a new fitness denoted by
. The total difference between
and
in the population can be described as shown in (14), where N is the population size.
After the course of iteration, the solution may finally converge to the optimum solution (global or local optimum). This situation indicates that
is more likely to be close to 0. Assuming that
is a small value around 0, we can determine when the hunting group is going to converge by setting
that belongs to a range of
. Therefore, Formula (14) can be rewritten as:
To avoid the hunting groups getting stuck in the local optimum and to ensure the efficiency of the algorithm, the omega’s search direction can be adjusted every time Equation (15) is satisfied. Nevertheless, when dealing with a large and complex space, we cannot rely on the circumstances described above to change the search direction of the omega. It can be clearly seen in
Figure 1b that when the alpha position (global best solution) gets close to the global optimum, it leads the other members of the hunting group to search in the direction towards
. This also means that the solution of the generation
may not be improved by the alpha, while also satisfying the conditions shown in Equation (15). Therefore, depending on the potential ability of
, the hunting group can be led to a promising global optimum over the next few iterations [
34].
After several iterations, as the number of satisfied conditions increases, meaning that there is no difference between the current solution and previous solutions, it can be concluded that the grey wolf is stuck around a local optimum. Therefore, the grey wolf needs to modify its search direction. Let
be a tolerance variable, where
is initially set to 0 and is used as a counter. In cases where (15) is satisfied,
can be updated via Equation (16), as follows.
As the value of
becomes greater, the probability of becoming trapped in the local optima of a hunting group also increases [
34]. However, in cases where
is searching around the global optimum, as described above, the omega should not change their search direction, but rather move along the alpha direction. Therefore, we introduce the probability
, which enables the omega to adjust its search direction.
can be achieved experimentally using Equation (17), below, where
is the current iteration and
is the maximum number of iterations.
As can be seen from
Figure 2, the
is not fixed throughout the course of iterations; rather, its value is updated according to
and
. When the value of
is larger than a random number with a range of [0,1], the leader alpha (beta or gamma) will be replaced by another candidate solution, which will continue to guide the hunting group.
Algorithm 1, below, shows the details of the approach.
Algorithm 1. Robust tolerance-based adjust searching direction mechanism |
1: At iteration kth; |
|
then |
|
5: end if |
6: generate a random number in the range of [0,1]; |
then |
8: Finding another candidate solution to replace alpha (beta or gamma) |
9: end if |
It is clear from
Figure 2 and Algorithm 1 that the
depends on
and
, especially the tolerance value
. When the value of
is small,
has a high probability of becoming smaller than the random number; then, the first three best leaders continue to guide the hunting group, and their leading ability will be useful for the next several iterations. When
increases to the threshold,
increases dramatically. This indicates that the number of solutions has still not improved over the next several iterations, and the hunting group is more likely to become trapped in a local optimum. Therefore, the value of
is probably greater than the random number, and a new leader will be used to lead the omega search direction.
On top of that,
is more likely to get close to the global optimum when the number of iterations is increased, especially with numbers of iterations greater than half, as can be seen in
Figure 3. Therefore, to ensure convergence, the value of
should be increased to get rid of the omega and change its search direction.
4.2. Opposition-Based Learning for Candidate Generation Strategy
Opposition-based learning (OBL) [
25] has recently been utilized to accelerate the convergence rate of several optimization algorithms. The OBL technique can be used to generate potential candidate solutions by considering both the current population and its opposite population. It has been proved in many studies worldwide that opposition candidate solutions are more likely to get closer to the global optimum solution than a randomly generated candidate solution. There have been many advanced applications of this learning mechanism in several soft computing techniques, as reported in [
35,
36,
37,
38].
The two definitions below show the important aspects of OBL, the opposite number and opposite point [
25]:
Definition—Let
be a real number defined in a certain interval:
. The opposite number
is defined as follows:
Definition—Let
be a point in an n-dimensional coordinate system with
and
. The opposition point
is completely defined by its coordinates
where
In the proposed RLGWO, after defining the replacement of the leader with a candidate solution in
Section 4.1, a random candidate solution can easily be created in the search space to guide the hunting group to get rid of the current local optimum solution. However, the random candidate may not be guaranteed to improve the solution; in particular, when dealing with large and complex spaces, it is probable that it will lead the hunting group into another local optimum. Using the OBL can ensure the generation of a candidate more effectively.
In
Figure 3, it is clear to see that the difference between the current and the previous solution in the first half of the iteration changes violently. This circumstance can be explained by the fact that the hunting group is carrying out the exploration phase. This means that the grey wolves are attempting to search the hold space to figure out promising areas for the global optimum. The alpha will have a significant influence on the hunting group, since it is the best solution. Therefore, in this certain period, replacing the alpha with its opposite position is a wise course of action to escape the local optimum.
The remaining half of the total course of iteration is the exploitation phase, when the grey wolf scales down the search space and concentrates on a certain area to find the optimum solution. To prevent the omega from moving away from the global optimum and to ensure the efficiency of the algorithm, the alpha acts as the main leader and the beta (or gamma) may be replaced by its opposite position. Since the beta and gamma have almost the same potential ability, the beta will be removed if the random number is greater than 0.5, and conversely, gamma will be substituted if it is less than this value.
The details of the generating candidate are described in Algorithm 2.
Algorithm 2. Opposition-based learning for candidate generation strategy |
1: At iteration kth; |
between [0,1]; |
|
4: Keep alpha, beta, and gamma as the leader; |
5: else |
then |
; |
8: else |
between [0,1]; |
|
; |
12: else |
; |
4.3. RLGWO Algorithm for Economic Load Dispatch
The computational mechanism for the proposed RLGWO algorithm to solve ELD problems is described in the following steps.
Step 1. Initialization
Step 1.1: Arbitrarily generate the initial value for all of the active power of the generating units belonging to their lower and upper real power operating limits except the last unit. Equation (4) is used to compute the amount of active power output of the last unit to guarantee whether it satisfies the inequality constraint or not. The solution will be discarded whenever it violates the inequality constraint. Let
D be the dimension of the hunting group. The initial position of the grey wolves is given as a matrix,
, below:
Step 1.2: Substitute the matrix into (3) and calculate the fuel cost for each solution of the current population.
Step 1.3: Evaluate the cost value of all search agents (grey wolves), which determine , , as the first three best solutions by simple comparison of their cost value.
Step 1.4: Set all the parameters, including , , coefficient vector and , initial variable tolerance .
Step 2. Repeat this step until the stopping criterion is satisfied.
Step 2.1: Update the position of all search agents using (11)–(13).
Step 2.2: Calculate the fuel cost for all members of the hunting group using (3) and compare the results to figure out , , .
Step 2.3: Evaluate , and then check whether fulfills Condition (15) or not. In cases where belongs to the range , will be increased by 1.
Step 2.4: Compute the probability of using (17) and select a random number . If , meaning that the three best leaders still potentially have the ability to guide all search agents, then return to Step 2. Otherwise, follow Algorithm 2 to generate a suitable candidate to lead the hunting group and move back to Step 2.