1. Introduction
With the rapid growth in technologies, ELD is considered one of the foremost challenging optimization problems in power systems. The main motive for addressing the ELD problem is to reduce the cost of power generation, by sustaining the different constraints involved in the generation units [
1]. Several researchers have applied mathematical models, knowledge discovery and optimization techniques to resolve the ELD problem. The standard techniques, like lambda-generation techniques, and base-point techniques from [
2], provide optimal solutions, by incorporating the incremental cost curves of linear functions. However, these methods have failed to solve highly non-linear functions, and provide unsatisfactory solutions which result in huge losses in power generation costs. The non-smooth functionalities of generating units contain various features, like prohibited zones, different fuel options, value-point effects, ramp-rate limits and a start-up cost function which converts linear into non-linear characteristics [
3]. Owing to the large-scale generating units, conventional methods have provided unreliable solutions, and have taken a lot of computational time to solve ELD problems. In later studies, dynamic programming techniques [
4] have been used for ELD problems, but these have required high computational efforts to solve large-scale generating units.
In recent studies, many researchers have utilized various optimization algorithms to solve non-convex ELD problems with only value-point effects, viz., Particle Swarm Optimization with Sequential Quadratic Programming (PSO-SQP) [
5], Genetic Algorithm (GA) [
6], Evolutionary Programming (EP) [
7], Improved Group Search Optimization (IGSO) [
8], Incremental Artificial Bee Colony with Local Search (IABC-LS) [
9], Hybrid Grey Wolf Optimizer (HGWO) [
10], Self-Organizing Hierarchical Particle Swarm Optimization (SOH-PSO) [
11], Genetic Algorithm with Pattern Search and SQP (GA-PS-SQP) [
12], Modified Shuffled Frog-Leaping Algorithm (MSFLA) [
13], Firefly Optimization (FA) [
14], Chaotic Self-Adaptive Particle Swarm Optimization Algorithm (CSAPSO) [
15], Combined Social Engineering Particle Swarm Optimization (SEPSO) [
16], Starling Murmuration Optimizer (SMO) [
17], Improved Moth-Flame optimization (IMFO) [
18] and Diversity-Maintained Differential Evolution (DMDE) [
19]. Among these search techniques, GA is considered to be the least efficient technique, because its optimal individuals are generally trapped in intensification rather than diversification, and it also suffers from the determination of control parameters, which results in excessive simulation time. Several new techniques, like IGSO, MSFLA, FA, HGWO, SOH-PSO, GA-PS-SQP and CSAPSO, have virtuoso competence in finding optimal solutions for non-convex generating units; however, the simulation time of the system is quite long; specifically, for CSAPSO, several iterations are carried out to specify the control parameter values; this limitation results in the technique having excessive execution time, and a large number of runs.
In addition, some sets of optimization algorithms are considered to solve non-convex ELD problems with only multi-fuel possibilities. These algorithms include Integer Coded Differential Evolution-Dynamic Programming (ICDEDP) [
20], Chaotic Ant Swarm Optimization (CASO) [
21], Bacteria Foraging Optimization (BFO) [
22], Ant Colony Optimization (ACO) [
23], Biogeography-Based Optimization (BBO) [
24] and Krill Herd (KH) [
25]. Among these techniques, ACO is the technique initially utilized for solving optimization problems in the engineering domain, specifically in path-identifying and parameter-tuning in electrical engineering. Although ACO and CASO have the cap potential of leading complicated constraints and non-convex goal features, in addition to their simplicity of simulation for optimization problems, they nevertheless suffer from numerous negative aspects, together with low-quality optimization individual and lengthy simulation time. The modified DE method, namely the ICDEDP technique, can be considered a more efficient technique than the other techniques, because it can obtain a good-quality solution within a short span of simulation; this DE technique has been globally utilized in power system optimization problems. In addition, other techniques—such as BBO, KH and BFO—have good capability in determining the optimal solutions for non-convex problems; however, the simulation times of these techniques are longer, due to the vast number of control parameters.
In contrast to the aforementioned sets, the techniques in the set of neural networks including the Adaptive Hopfield Neural Network (AHNN) [
26], the Enhanced Augmented Lagrange Hopfield Network (EALHN) [
27] and the Augmented Lagrange Hopfield Network (ALHN) [
28] can impact on large-scale problems, but fail to deal with the ELD problem with a non-convex objective function. In EALHN and ALHN, the Lagrange function is merged with the Hopfield network to enhance efficacy. This process will help the techniques to converge towards the optimal more smoothly, and to obtain a good-quality solution. However, in real-time power systems, both value points and fuel points need to be considered, for accurate and practical ELD solutions.
In some studies, both the constraints of value points and different fuel possibilities are considered for realistic ELD solutions comprising the Improved Particle Swarm Optimization (IPSO) [
29], the Crisscross Optimization Algorithm (COA) [
30], Differential Evolution and Particle Swarm Optimization (DEPSO) [
31], the Oppositional Grey Wolf Optimization algorithm (OGWO) [
32], Estimation of Distribution and Differential Evolution Cooperation (ED-DE) [
33], the Real-Coded Chemical Reaction Algorithm (RCCRO) [
34], Synergic Predator–Prey Optimization (SPPO) [
35], the One Rank Cuckoo Search Algorithm (ORCSA) [
36], the Real-Coded Genetic Algorithm (RCGA) [
37] and the Improved Genetic Algorithm [
38]. By utilizing the pros of each search technique, these improved novel techniques have adequate capability in finding good-quality solutions with better simulation time. However, the improved technique can lead to more complications with vast control parameters, and it can suffer from inappropriate selection of these parameters; in addition, its performance is degraded when applied to large-scale power systems entailing
number of generating units with various fuel possibilities and value-point effects.
A large portion of the above studies have focused on the adjustments of stochastic search techniques. Nonetheless, they have, once in a while, given consideration to the method of handling constraints. In reality, dealing with the constraints of ELD problems is significant when working with stochastic search techniques, for enhancing the optimization results. Our study aimed to fill the research gap, by contributing more towards addressing the constraints of ELD problems. Our contributions were twofold: initially, an enhanced PIO algorithm was introduced, to enrich the performance of the standard PIO algorithm; subsequently, a constraint-handling technique was utilized, to appropriately handle the equality constraints.
The Pigeon-Inspired Optimizer algorithm was inspired by the homing bias of pigeons, and was proposed by Duan and Qiao in 2014. This optimization algorithm was used because of its optimum performance at high merging speeds [
39]. However, the PIO algorithm suffers in regard to global exploration and premature convergence. In addition, its performance is degraded when applied to high-dimensional problems. This problem can be overcome by using the Opposition-Based Learning technique. The OBL technique is widely used by researchers to boost convergence speed, by exploring the search space. In this work, a new metaheuristic algorithm—namely, the Oppositional Pigeon-Inspired Optimizer technique (OPIO)—was utilized, to solve non-convex ELD problems with various fuel possibilities and value-point effects.
The major contribution of this work is illustrated as follows:
(1) The proposed OPIO algorithm solves the non-convex ELD problem with multi-fuel possibilities and value-point effects, through two operators: namely, map and compass operator, and landmark operator. These operators enhance the local search ability by adopting the search boundary limits. Later, the Opposition-Based Learning strategy helps to explore the search space, as well as to enhance the exploration ability for target search agents. This process improves the search capability, and eradicates premature convergence, though the large-scale test system holds both multiple fuel possibilities and value-point effects.
(2) The proposed OPIO algorithm has a unique adjustable parameter: jump rate . Parameter helps to determine the global optimal solution, by influencing the adjustable value, within the range of 0 to 0.4. This parameter promotes the OPIO algorithm, to be robust and adaptable in solving ELD problems with different constraints.
(3) To validate the efficiency of the proposed OPIO algorithm, we used several test cases, which varied according to three scales: small-scale (i.e., 13, 40); medium-scale (i.e., 140, 160); and large-scale (i.e., 320, 640) generation units. The results of the various test cases confirmed that the proposed technique is a better potential solution than the state-of-the-art metaheuristic algorithms in the literature. The OPIO algorithm provided better performance in the 320- and 640-unit generation systems. This shows that the formulated technique is a superior and reliable solution for large-scale ELD problems over multiple trials.
The rest of this work is categorized as follows:
Section 2 delivers the mathematical formulation of the ELD problem, with objective functions and multiple constraints. The proposed Oppositional Pigeon-Inspired Optimizer algorithm is presented in detail in
Section 3. In
Section 4, the implementation of the OPIO algorithm, in solving the ELD problem, is presented.
Section 5 provides proposed OPIO algorithm experimentation details, from six different test cases that varied from small-scale to large-scale systems, and the outcomes are compared with state-of-the-art metaheuristics algorithms. The conclusion of this work is presented in
Section 6.
4. Oppositional Pigeon-Inspired Optimizer Algorithm (Proposed)
The proposed Oppositional Pigeon-Inspired Optimizer algorithm is discussed in this section. The common search strategy of the proposed OPIO algorithm is like the PIO. However, the proposed OPIO algorithm utilizes a unique methodology to explore the search space of the pigeon, to discover the position of its hiding location. Moreover, the modified method provides better convergence in the pigeon population, which helps to achieve the optimal solution. As part of enhancement by the proposed method, in every iteration, the best pigeon is selected as the target. The selected pigeon position will be updated with the Oppositional-Based Learning, to enhance the convergence rate. However, selecting an arbitrary pigeon, from among the population, may result in a bad-quality landmark solution, with a large value for the fitness function (in the minimization problem), which leads to an unsuitable end point to move. In addition, selecting a random pigeon for the exploration phase will tend towards a bad destination, which minimizes the convergence rate. To select the best solution among the population at each iteration is a challenging task.
In this work, a priority-based election mechanism was introduced. This mechanism could be utilized for the minimization problem at each iteration for the pigeon
, so that
of the best pigeons in the solution set were elected. The benefit of this election mechanism was to elect the target pigeon among the list of the best pigeons in the stack. By this process, the pigeons could perform better in improvising their positions, by following the better target pigeons, and this resulted in a better convergence rate for the algorithm. Nevertheless, electing the value of
was significant: electing a very trivial value of
among the pigeons
could lead to being stuck in the local optima. In addition, selecting a large value for
could cause the bad target pigeon to be tricked. To eradicate these issues, in the initial iterations
started from a large value, for better diversification, and its number was reduced according to Equation (17); over the course of the iterations, its tendency towards the local optimum resulted in the
having a small value:
where,
stood for the value for selecting the best pigeon in iteration
, and
and
stood for the maximum and minimum values of
.
4.1. Constraint-Handling Technique
The ELD problem is complicated to solve, when considering the constraints. In past decades, various techniques have been adopted, to handle the constraints. The penalty function is considered to one of the most common constraint-handling techniques: it deals with the constraint problem by including some additional value to the objective function in (4). This function has been broadly utilized by various researchers, because of its simplicity and efficiency. The objective function is the minimization of the following representation:
where
stands for constraint-based objective function, and
stands for the penalty coefficient of a real integer. If constraint (6) is other than zero, then the value of the second part in Equation (17) will be other than zero too, multiplied by the penalty value
, and, finally, will be added to the fuel cost
. In other words, if Equation (6) does not meet the constraint, then this implies that the solution has a large objective function, and is likely to be rejected. On the other hand, if the solution meets the constraint (6), this implies that the solution holds a small objective function value, and is likely to be accepted. If the
value is fixed with a large value, then the performance of the algorithm will be reduced, and this will lead to premature convergence. In addition, fixing the small value for
fails to meet the inequality constraints.
4.2. Implementation of the OPIO Algorithm for the ELD Problem
In this section, the strategies for applying the OPIO algorithm, to solve the ELD problem, are examined. The main objective of the ELD optimization problem is to reduce the overall power generation cost. In the ELD problem, the total power generating unit (
) is proportional to the total decision variable of the optimization problem (
). Each position of the pigeon is represented as each anticipated power output of the generating units. In general, the ELD problem consists of some impartiality and disparity constraints, as discussed in
Section 2.2. Each solution in the population should satisfy the constraints. For the smooth process of constraint handling, the value of
is fixed as 100 in Equation (17) for the entire simulation, which attains an adequate performance with the power equality constraint.
The overall computational procedures of the proposed OPIO algorithm are described in detail as follows. In addition, the flowchart of the proposed OPIO algorithm is represented in
Figure 1, and the proposed OPIO algorithm for solving the ELD problem is represented in Algorithm 3.
- Step 1:
Define the initial parameters with the characteristics of the generation units: ; number of pigeons; maximum generations (); other data, such as , .
- Step 2:
Initially, the arbitrary values for all generating units within the lower and upper operating boundary are generated using (5), except for the last generating unit. The computation of the last unit of power generation is calculated using (6), and it is validated, to ensure whether it satisfies the inequality constraints (5) or not. If the solution satisfies the constraints, then the solution is sustained; otherwise, it is abandoned. The pigeon position X, concerning the generating units, is initialized as follows:
where the component
is the power outcome of the
th unit in individual
. For the OPIO algorithm, there is only one adjustable parameter: the jumping rate
, which is fixed within the range of 0 to 0.4 for all test cases used in the experimentation.
- Step 3:
For each pigeon in the population, the power generating unit must satisfy the ramp-rate boundary, and not relay in the restricted operating zones. If the solution does not meet the constraints, then power outputs should be altered near to the boundary of the feasible solution.
After processing the initialization, the main procedure of the OPIO algorithm process is as follows:
- Step 4:
Determine the velocity of the pigeon, using Equation (10), and update the position of the pigeon, using Equation (14). If the updated position of the pigeon does not satisfy the constraints, then alter the pigeon’s position, as shown in Step 3.
- Step 5:
Compute the factor, as in Equation (17).
- Step 6:
Choose the of the best solutions from the population, and update the position for the selected pigeon, using the OBL technique (Algorithm 2).
- Step 7:
Check this step for the pigeon :
The output power of the generating units must not reside in the RORs (see (8)) or contravene the operating unit limit (see (5)).
The lower and upper boundary rates of each of the generating units, from the preliminary state, should be in the satisfactory ranges, as given in (9). If the preliminary output power of the generating units is not specified, then the preliminary power of all power generating units should be within the satisfactory ranges.
If the RORs and ramp-rate limits are contravened, adjust the power outputs near to the feasible solution.
- Step 8:
Compute the overall power loss of the transmission lines for the pigeon , as in (6).
- Step 9:
Compute the quality of the pigeon , by interleaving its power outputs in the fitness function, as in (17).
- Step 10:
Repeat steps 4–9, until the stop criterion is met.
- Step 11:
The ELD solution is the best solution in the last iteration.
Algorithm3: Proposed OPIO algorithm for solving ELD |
1: Generate the initial population.
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2: Determine the preliminary parameters.
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3: Arbitrarily initialize the position of the pigeon in the search boundary space.
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4: Check the RR and RORs constraints.
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5: While (ng ≥ 1) do |
6: Determine the velocity and position of the pigeon.
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7: Determine the φ factor.
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8: Select φ of the best pigeons from the population.
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9: Apply OBL technique, using Algorithm 2.
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10: If (fit (Opp) < fit (X_(i,t))) then
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11: Replace the Opp solution
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12: Else |
13: do nothing
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14: End if |
15: Check the feasibility of the new position of the pigeons.
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16: Calculate the transmission loss.
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17: Evaluate the fitness of the new position of the pigeons.
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18: Update the global best solution.
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19: End while |
20: Output: Visualize the global best solution.
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