*Article* **Oppositional Pigeon-Inspired Optimizer for Solving the Non-Convex Economic Load Dispatch Problem in Power Systems**

**Rajakumar Ramalingam 1, Dinesh Karunanidy 1, Sultan S. Alshamrani 2, Mamoon Rashid 3,\*, Swamidoss Mathumohan <sup>4</sup> and Ankur Dumka 5,6**


**Abstract:** Economic Load Dispatch (ELD) belongs to a non-convex optimization problem that aims to reduce total power generation cost by satisfying demand constraints. However, solving the ELD problem is a challenging task, because of its parity and disparity constraints. The Pigeon-Inspired Optimizer (PIO) is a recently proposed optimization algorithm, which belongs to the family of swarm intelligence algorithms. The PIO algorithm has the benefit of conceptual simplicity, and provides better outcomes for various real-world problems. However, this algorithm has the drawback of premature convergence and local stagnation. Therefore, we propose an Oppositional Pigeon-Inspired Optimizer (OPIO) algorithm—to overcome these deficiencies. The proposed algorithm employs Oppositional-Based Learning (OBL) to enhance the quality of the individual, by exploring the global search space. The proposed algorithm would be used to determine the load demand of a power system, by sustaining the various equality and inequality constraints, to diminish the overall generation cost. In this work, the OPIO algorithm was applied to solve the ELD problem of small- (13-unit, 40-unit), medium- (140-unit, 160-unit) and large-scale (320-unit, 640-unit) test systems. The experimental results of the proposed OPIO algorithm demonstrate its efficiency over the conventional PIO algorithm, and other state-of-the-art approaches in the literature. The comparative results demonstrate that the proposed algorithm provides better results—in terms of improved accuracy, higher convergence rate, less computation time, and reduced fuel cost—than the other approaches.

**Keywords:** economic load dispatch; pigeon-inspired optimizer; oppositional-based learning; swarm intelligence algorithm; oppositional-based pigeon-inspired optimizer

**MSC:** 68W50; 60G05; 60G51; 90C27

#### **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

**Citation:** Ramalingam, R.; Karunanidy, D.; Alshamrani, S.S.;

Rashid, M.; Mathumohan, S.; Dumka, A. Oppositional Pigeon-Inspired Optimizer for Solving the Non-Convex Economic Load Dispatch Problem in Power Systems. *Mathematics* **2022**, *10*, 3315. https:// doi.org/10.3390/math10183315

Academic Editor: Jian Dong

Received: 29 July 2022 Accepted: 9 September 2022 Published: 13 September 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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 *n* 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 *Jr*. Parameter *Jr* 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.

#### **2. ELD Problem Formulation**

The main motive of ELD is to reduce the overall power generation cost, by solving different disparity and parity constraints, to provide optimal generation among power producing units [32]. The objective function and the different constraints of the ELD problem are presented in this section.

#### *2.1. Fitness Function*

The fitness function of the ELD problem is to reduce the total power production cost by solving various constraints, and to gratify the load demand over some reasonable stage. A quadratic function is formulated, to approximate the fuel cost of the power-producing unit. The mathematical formulation of the power-generating unit is formulated as below:

$$\min \sum\_{j=1}^{n} F\_c \left( \Psi\_j \right) \tag{1}$$

Here *Fc* denotes the fuel cost of the generator (in \$/h); Ψ*<sup>j</sup>* denotes the output power of generator *j* (in MW); *n* stands for the overall power-generating unit in the power system.

In view of the value-point effects, ELD cost functions will have non-smooth points which provide inefficient results in practical generators. To process the practical generators, sinusoidal functions are included in the quadratic functions. The cost function, with value points of unit *j*, is represented as follows:

$$F\_c = k\_{\dot{\jmath}} \Psi\_{\dot{\jmath}}^2 + l\_{\dot{\jmath}} \Psi\_{\dot{\jmath}} + m\_{\dot{\jmath}} + \left| a\_{\dot{\jmath}} \times \sin \left( b\_{\dot{\jmath}} \times \left( \Psi\_{\dot{\jmath}}^{low} - \Psi\_{\dot{\jmath}} \right) \right) \right| \tag{2}$$

Here, *kj*, *lj* and *mj* stand for the fuel cost coefficients of generator *j*; *aj* and *bj* stand for the value-point loading coefficients of generator *j*; Ψ*low <sup>j</sup>* is the low-level range power production of generator *j*.

The overall fuel cost function of *n* generator in real-time ELD is mathematically formulated as follows:

$$\min \sum\_{j=1}^{n} \mathcal{F}\_{\mathbf{c}} \left( \Psi\_{j} \right) = \sum\_{j=1}^{n} \left[ k\_{\hat{j}} \Psi\_{j}^{2} + l\_{\hat{j}} \Psi\_{\hat{j}} + m\_{\hat{j}} + \left| a\_{\hat{j}} \times \sin \left( b\_{\hat{j}} \times \left( \Psi\_{\hat{j}}^{low} - \Psi\_{\hat{j}} \right) \right) \right| \right] \tag{3}$$

where *F*ˆ *c* stands for the real-time fuel cost of the generator.

To attain an accurate and more appropriate solution for the ELD problem, both various fuel possibilities and value-point effects are added with the cost functions. Most thermal generating units utilize multiple fuel possibilities, using the load and suitability of the power generation units. The cost function of generating unit *j*, with various fuel possibilities (*q*) and value-point effects, is mathematically formulated and presented as follows:

$$F\_{\mathsf{f}}\left(\boldsymbol{\Psi}\_{\mathsf{f}}\right) = \begin{cases} k\_{j1}\left(\boldsymbol{\Psi}\_{\mathsf{f}}\right)^{2} + l\_{j1}\left(\boldsymbol{\Psi}\_{\mathsf{f}}\right) + m\_{j1} + \left| a\_{j1} \times \sin\left(b\_{j1} \times \left(\boldsymbol{\Psi}\_{\mathsf{f}}^{\mathrm{law}} - \boldsymbol{\Psi}\_{\mathsf{f}}\right)\right) \right| \boldsymbol{if} \boldsymbol{\Psi}\_{\mathsf{f}}^{\mathrm{law}} \le \boldsymbol{\Psi}\_{\mathsf{f}} \le \boldsymbol{\Psi}\_{\mathsf{f}} \\\ k\_{j2}\left(\boldsymbol{\Psi}\_{\mathsf{f}}\right)^{2} + l\_{j2}\left(\boldsymbol{\Psi}\_{\mathsf{f}}\right) + m\_{j2} + \left| a\_{j2} \times \sin\left(b\_{j2} \times \left(\boldsymbol{\Psi}\_{\mathsf{f}} - \boldsymbol{\Psi}\_{\mathsf{f}}\right)\right) \right| \boldsymbol{if} \boldsymbol{\Psi}\_{\mathsf{f}} \le \boldsymbol{\Psi}\_{\mathsf{f}} \le \boldsymbol{\Psi}\_{\mathsf{f}} \\\ k\_{jq}\left(\boldsymbol{\Psi}\_{\mathsf{j}}\right)^{2} + l\_{jq}\left(\boldsymbol{\Psi}\_{\mathsf{j}}\right) + m\_{jq} + \left| a\_{jq} \times \sin\left(b\_{jq} \times \left(\boldsymbol{\Psi}\_{\mathsf{j}q} - \boldsymbol{\Psi}\_{\mathsf{j}}^{\mathrm{law}}\right)\right) \right| \boldsymbol{if} \boldsymbol{\Psi}\_{\mathsf{j}q} \le \boldsymbol{\Psi}\_{\mathsf{j}} \le \boldsymbol{\Psi}\_{\mathsf{j}}^{\mathrm{up}} \end{cases} \tag{4}$$

#### *2.2. Constraints of the ELD Problem*

The fitness function in Section 2.1 is formulated with a set of constraints, which are given below.

#### 2.2.1. Operating Unit Limit

The power-generating unit must relay within the lower and upper boundary limits:

$$\Psi\_j^{low} \le \Psi\_j \le \Psi\_j^{upper} j = 1, 2, \dots, n \tag{5}$$

where Ψ*upper <sup>j</sup>* and <sup>Ψ</sup>*low <sup>j</sup>* denote the upper and lower boundary, respectively, of the output power of the generator *j*.

#### 2.2.2. Power-Stabilizing Constraints

The overall generated power should be the same as the overall losses and overall load request of the units. This constraint is mathematically formulated as follows:

$$\sum\_{j=1}^{n} \Psi\_j - \Psi\_{Dennand} - \Psi\_{Loss} = 0 \tag{6}$$

where Ψ*Loss* and Ψ*Demand* represent the overall power loss and power demand of the units. Based on Kron's loss technique, the transmission loss is given as follows:

$$\Psi\_{Loss} = \sum\_{j=1}^{n} \sum\_{i=1}^{n} \Psi\_j \beta\_{ji} \Psi\_i + \sum\_{j=1}^{n} \beta\_{0j} \Psi\_j + \beta\_{00} \tag{7}$$

where *βji* represents the loss coefficient element *j* and *i* of the symmetric matrix *β*; *β*0*<sup>j</sup>* denotes the loss coefficient vector of *j* symmetric matrix *β*; and *β*<sup>00</sup> represents a fixed loss coefficient concerning standard operating situations.

#### 2.2.3. Restricted Operating Regions (RORs)

Due to oscillation or steam value process in the shaft bearing, the restricted operating region is considered. To avoid these issues, choosing the best operating region will drastically increase the optimum economy of the generating units. The boundary constraints of the standard operating section of generator *j* are formulated as follows:

$$\Psi\_{j} \in \begin{cases} \Psi\_{j}^{low} \le \Psi\_{j} \le \Psi\_{j,1}^{l} \\ \Psi\_{j,i-1}^{l} \le \Psi\_{j} \le \Psi\_{j,i}^{l} \\ \Psi\_{j,n\_{i}}^{l} \le \Psi\_{j} \le \Psi\_{j}^{up} \end{cases} \quad i=2,3,\dots,n\_{j}, j=1,2,\dots,n \tag{8}$$

where *l*, *u* denotes the lower and upper limits of specific power generating units, and *nj* determines the number of restricted regions of generating unit *j*.

#### 2.2.4. Ramp-Rate (RR) Constraint

In view of the lower and upper power production of the generator, the ramp-rate limit is considered. Each generating unit is controlled by the ramp-rate limit, which instructs the generator to function continually for the two nearest operating regions. This ramp-rate constraint is represented as follows:

$$\max\left(\Psi\_{\vec{j}}^{low}, \Psi\_{\vec{j}}^{0} - LSL\_{\vec{j}}\right) \le \Psi\_{\vec{j}} \le \min\left(\Psi\_{\vec{j}}^{upper}, \Psi\_{\vec{j}}^{0} + LSL\_{\vec{j}}\right) \tag{9}$$

where *LSLj* and *USLj* represent the lower and upper slope (or ramp) limit of the generating unit *j*, and Ψ<sup>0</sup> *<sup>j</sup>* denotes the current power generating unit *j*.

#### **3. Preliminaries**

In this section, we present three major mechanisms; firstly, the generic working process of the Pigeon-Inspired Optimizer is presented, secondly, the core concept of the Opposition-Based Learning technique is discussed; and, finally, the proposed methodology, with its working process, is presented.

#### *3.1. Overview of Pigeon-Inspired Optimizer*

The Pigeon-Inspired Optimizer (PIO) belongs to the family of swarm intelligence algorithms that were proposed by Haibin Duan and Peixin Qiao (2014) [39]. The PIO algorithm mimics the homing behaviors of pigeons. Most researchers apply SI algorithms to solve their domain-related NP-hard problems, in which search space is vast. SI algorithms are inspired by the social behavior of the swarm, with intellectual learning to determine high-quality solutions using mathematical formulations. The mathematical formulation of the swarm includes the position and velocity of the swarm iteration by iterations.

Pigeons have the ability to explore for food over the course of long intervals. In addition, pigeons exhibit intellectual homing behavior: for example, they carried messages during the First and Second World Wars. The PIO algorithm works on the basis of two unique operators, viz., map and landmark operators. This algorithm provides good optimum performance and higher merge speed than the other state-of-the-art metaheuristic algorithms like Ant Colony Optimization, Particle Swarm Optimization, Artificial Bee Colony Optimization and Differential Evolution algorithms.

#### 3.1.1. Map and Compass Operator

Pigeons have a natural ability to perceive the orbital meadow, with the aid of a magnetic function that enables them to map. They utilize the altitude of the sun as a compass to fine-tune their current directions. Generally, pigeons depend less on the sun and on magnetic particles as they near their destinations. The map and compass operator can be mathematically formulated as follows:

$$V\_j^{t+1} = V\_j^t \times e^{-\rho t} + rand \times \left(X\_{\mathbb{S}} - X\_j^t\right) \tag{10}$$

$$X\_j^{t+1} = X\_j^t + V\_j^{t+1} \tag{11}$$

where *V<sup>t</sup> <sup>j</sup>* and *<sup>X</sup><sup>t</sup> <sup>j</sup>* represent the velocity and position of the *j* individuals in the *t* iterations; *ρ* denotes the map and compass factor; *rand* determines the uniform random variable within [0, 1]; *Xg* denotes the global best individual; and *Xt*+<sup>1</sup> *<sup>j</sup>* and *<sup>V</sup>t*+<sup>1</sup> *<sup>j</sup>* represent the new position and velocity of the *j* individual in the next *t* iteration.

#### 3.1.2. Landmark Operator

A pigeon relies on natural landmarks once it has reached its destination. However, if the pigeon is far away from its destination, then it relies on the adjacent pigeons to adjust its position. In this algorithm, half of the pigeon population is allowed to adjust position, with the aid of the centered pigeons, while the pigeons comprising the other half of the population adjust their position in accordance with the desirable destination position. Most pigeons will not be familiar with their landmark in this view, so they will follow the top-ranked pigeons to determine their desired destination. The half-number of pigeons adjust their position with the following mathematical formulations:

$$N\_P^{t+1} = \frac{N\_p^t}{2} \tag{12}$$

$$X\_c^{t+1} = \frac{\sum X\_j^{t+1} \times Fit\left(X\_j^{t+1}\right)}{N\_P \sum Fit\left(X\_j^{t+1}\right)}\tag{13}$$

where *N<sup>t</sup> <sup>p</sup>* represents the number of pigeons or population size in the current iteration *t*; and *Fit Xt*+<sup>1</sup> *<sup>c</sup>* denotes the fitness of the centered pigeons in the *t* + 1 iteration. The new pigeon position is represented as:

$$X\_{\dot{\jmath}}^{t+1} = X\_{\dot{\jmath}}^t + rand \times \left(X\_{\dot{\jmath}}^{t+1} - X\_{\dot{\jmath}}^t\right) \tag{14}$$

The generic flow of the PIO algorithm is represented in Algorithm 1. In this algorithm, the map and compass operator is given in the initial while loop, and another loop is used to access their route and its correction in position.


#### *3.2. Opposition-Based Learning Technique*

The Opposition-Based Learning technique (OBL) was introduced by Tizhoosh [40] to enhance the convergence speed of traditional metaheuristic algorithms. This method utilizes the valuation of a current population against its opposite population, to determine the better solution for a specific problem. The OBL method has been utilized in different metaheuristic algorithms, to boost convergence speed [41,42]. The mathematical formulation of the OBL is defined as follows:

Let *<sup>μ</sup>*(*<sup>μ</sup>* <sup>∈</sup> [*p*, *<sup>q</sup>*]) be an actual integer. The contradictory integer *<sup>μ</sup>*<sup>0</sup> is formulated as:

$$
\mu^0 = p + q - \mu^0 \tag{15}
$$

For *d*–dimensional search space, the contradictory integer *μ*<sup>0</sup> is defined as:

$$
\mu\_j^0 = p\_j + q\_j - \mu\_j \tag{16}
$$

where *μ*1, *μ*2, ... , *μ<sup>d</sup>* is a point in d-dimensional search space, i.e., *μ<sup>i</sup>* ∈ *pj*, *qj* ; *j* = {1, 2, 3, . . . , *d*}, and *d* represents the number of decision variables.

The Oppositional-Based Learning technique is generally used in two stages: firstly, in the initialization procedure; and secondly, in generating an opposite solution, using the jumping rate *Jr*. The proposed OBL algorithm is given in Algorithm 2.

#### **Algorithm 2:** Oppositional-Based Learning Algorithm

1: Initially the solutions are randomly initialized within the upper and lower boundary regions. 2: Determine the opposite solutions:

```
2.1: for i = 1:N_p
```

```
2.2: for j = 1:d
```
2.3: *μ\_(i,j)ˆ0 = p\_j + q\_j* − *μ\_(i,j)*

2.4: *end for*

2.5*: end for*

3: Sort the current solutions and opposite solutions in ascending order.

4: Choose the N\_p the number of best candidate solutions.

5: Update the control parameters.

6: Generate the opposite solutions from current solutions using jumping rate J\_r:

6.1: *for j = 1:N\_p* 6.2: *for I = 1:d* 6.3: *if J\_r > rand* 6.4: *opp(j,i) = min(i) + max(i)* − *P(j,i);* 6.5: *else* 6.6: *opp(j,i) = P(j,i);* 6.7: *end* 6.8: *end for* 6.9: *end for* 7: Repeat steps 3 to 6 until the termination criterion is met.

#### **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 badquality 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 *i*, so that *ψ* 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 *ψ* was significant: electing a very trivial value of *ψ* among the pigeons *i* could lead to being stuck in the local optima. In addition, selecting a large value for *ψ* could cause the bad target pigeon to be tricked. To eradicate these issues, in the initial iterations *ψ* 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 *ψ* having a small value:

$$\psi^t = round\left(\psi\_{\max} - \frac{\psi\_{\max} - \psi\_{\min}}{N\_{\mathbb{S}}} \times t\right) \tag{17}$$

where, *ψ<sup>t</sup>* stood for the value for selecting the best pigeon in iteration *t*, and *ψmax* and *ψmin* stood for the maximum and minimum values of *ψ*.

#### *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:

$$F\_{\rm CN} = F\_{\rm c} + \varphi \left| \sum\_{j=1}^{n} \Psi\_j - \Psi\_{Demand} - \Psi\_{Loss} \right| \tag{18}$$

where *FCN* stands for constraint-based objective function, and *ϕ* 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 *ϕ*, and, finally, will be added to the fuel cost *Fc*. 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 *ϕ* 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 *ϕ* 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 (*n*) is proportional to the total decision variable of the optimization problem (*d*). 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 *ϕ* 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.


$$X = \begin{bmatrix} X\_1 \\ \vdots \\ X\_l \\ \vdots \\ X\_{\mathcal{C}} \end{bmatrix} = \begin{bmatrix} X\_{1,1} & \cdots & X\_{1,b} & \cdots & X\_{1,d} \\ \vdots & \ddots & \vdots & & \ddots & \vdots \\ X\_{l,1} & \cdots & X\_{l,b} & \cdots & X\_{l,d} & \vdots \\ \vdots & \ddots & \vdots & & \ddots & \vdots \\ X\_{N\_p,1} & \cdots & X\_{N\_p,b} & \cdots & X\_{N\_p,d} \end{bmatrix} i = 2, \dots, N\_p; b = 1, 2, \dots, d \tag{19}$$

where the component *Xi*,*<sup>d</sup>* is the power outcome of the *b*th unit in individual *Xi*. For the OPIO algorithm, there is only one adjustable parameter: the jumping rate *Jr*, which is fixed within the range of 0 to 0.4 for all test cases used in the experimentation.

	- a. The output power of the generating units must not reside in the RORs (see (8)) or contravene the operating unit limit (see (5)).
	- b. 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.
	- c. 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 *i*, as in (6). **Step 9:** Compute the quality of the pigeon *i*, 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.

#### **Algorithm 3:** Proposed OPIO algorithm for solving ELD

6: Determine the velocity and position of the pigeon.


<sup>1:</sup> Generate the initial population.

<sup>2:</sup> Determine the preliminary parameters.

<sup>3:</sup> Arbitrarily initialize the position of the pigeon in the search boundary space.

<sup>4:</sup> Check the RR and RORs constraints.

<sup>5:</sup> **While** (ng ≥ 1) **do**

<sup>7:</sup> Determine the ϕ factor.

<sup>20:</sup> **Output**: Visualize the global best solution.

**Figure 1.** Flowchart of Proposed OPIO algorithm.

#### **5. Results and Discussion**

The proposed OPIO algorithm was applied to solve ELD issues. Three various test systems with three different fuel possibilities and non-linearities, such as ramp-rate ranges, value-point consequences and interdicted working region, were studied, to assess the execution of the formulated OPIO method. The formulated OPIO technique was written in MATLAB R2016a, implemented on a 2.6 GHz Intel I5 PC. The execution of the formulated OPIO algorithm was justified by utilizing three different test systems: small- (13-unit, 40-unit), medium- (140-unit, 160-unit) and large-scale (320-unit and 640-unit). The acquired outcomes from the formulated OPIO technique were differentiated to various state-of-theart metaheuristic techniques reported in the literature. The different test systems, with the number of generating units and their constraints, are outlined below:

	- a. 13-unit test case: in this test case, a 13-unit generator system, with constraints such as different fuel costs and value-point effects, was considered. The power load demand (*PD*) of the system was fixed at *PD* = 2520 MW [7,43];
	- b. 40-unit test case: this test case held a 40-unit generator system, with valuepoint effects considered, and the power load demand of the system was fixed at *PD* = 10,500 MW [7,43].
	- a. 140-unit test case: in this test case, a 140-unit generator system, with constraints such as value-point effects, ramp-rate limits, and prohibited accomplishment unit, was considered. The power load demand (*PD*) of the system was fixed at *PD* = 49,342 MW [44];
	- b. 160-unit test case: this test case held a 160-unit generator system, with valuepoint effects considered. The power load demand of the system was fixed at *PD* = 43,200 MW [45].
	- a. 320-unit test case: a large-scale system with a 320-unit generator system, with different fuel options and value-point loading effects, was considered here. The power load demand of the system was fixed at *PD* = 86,400 MW. The input data of the 10-unit system were duplicated 32 times in this system [46].
	- b. 640-unit test case: a test case with a 640-unit generator system, with multiple fuel options and value-point load effects, was considered here. The load demand of the system was increased by up to *PD* = 1,72,800 MW. The input data of the 10-unit system were replicated 64 times in this system [30].

The convergence of metaheuristic algorithms mainly relies on the possibility of a proper value. The proposed technique may deliver a different solution when the choice of insert value is not appropriate. To select the proper input parameters, repeated simulation is required. For the OPIO algorithm, after a repeated number of runs, the lower and upper jumping rates were fixed within the range of 0 to 0.4. For effective simulation, we considered a population size of 50, and 100 was selected as the maximum number of iterations for the test systems.

#### *5.1. Test Case 1a: 13-Unit*

In this instance, the formulated OPIO technique was tested on a small-scale 13-unit system, which held uneven fuel cost and value-point effects. The dataset of the fuel cost and the limit utility of numerous vigorous energy providers were taken from [43], and the load order was fixed as 2520 MW. To examine the execution of the proposed OPIO technique and the conventional PIO algorithm, the assumed outcomes were differentiated from the various metaheuristic algorithms, viz., Oppositional Grey Wolf Optimization (OGWO) [32], Improved Particle Swarm Optimization (IPSO) [29], One Rank Cuckoo Search Algorithm (ORCSA) [36], Crisscross Optimization Algorithm (COA) [30], Real-Coded Genetic Algorithm (RCGA) [37], Improved Genetic Algorithm (IGA) [38] and Pigeon-Inspired Optimization (PIO) [39].

Table 1 provides the comparative results of the OPIO and PIO algorithms for active power generators along with other techniques. As shown in Table 1, the solution provided by the OPIO algorithm reached a fuel cost of 24512.45\$/hr, which was less than all the compared algorithms; the outcomes of the formulated techniques conveyed that it was superior in finding the best or near-best solution. To ensure the efficacy and effectiveness of the technique, the simulation was carried out over 100 runs, on both the proposed OPIO algorithm and the conventional PIO algorithm, and its result is given in Table 2. As shown in Table 2, the OPIO produced a better solution for 97 runs, which was far better than all compared algorithms. The statistical outcomes conveyed that the formulated OPIO algorithm delivered better results compared with various algorithms. The convergence of the minimization fuel-cost function over the iteration cycles of the proposed OPIO algorithm and the standard PIO algorithm were noted, and are displayed in Figure 2. Figure 2 shows that the proposed algorithm converged faster towards the optimal solution that did not have further changes, which validated the active constancy of the formulated technique.


**Table 1.** Test outcomes of various algorithms for a 13-unit system with PD = 2520 MW.

**Table 2.** Comparison outcomes of different algorithms for a 13-unit system.


#### *5.2. Test Case 1b: 40-Unit*

To access the feasibility of the proposed OPIO algorithm, another small-scale test case, of a 40-unit power generation system along with value-point belongings, was used. The benchmark value of the 40-unit power system was approached from [43], and its load demand was fixed as 10,500 MW. The outputs of the power generation and fuel cost of various algorithms like OGWO, IPSO, IGA, RCGA, COA, ORCSA, PIO and OPIO are shown in Table 3: the best cost of the PIO and OPIO algorithms reached 136,588.57 \$/h and 136,447.87\$/h, respectively; it is also notable that the OPIO algorithm provided the best solution among the compared techniques, by achieving the load demand and other constraints.

**Figure 2.** Convergence results of the OPIO and PIO algorithms for a 13-unit system.

**Table 3.** Simulation outcomes of different algorithms for a 40-unit system with PD = 10,500 MW.


The comparative outcomes of the overall fuel cost, success rate, standard deviation and execution time acquired by the OPIO algorithm, along with the various techniques, are given in Table 4. Based on Table 4, the OPIO algorithm achieved the best solution 96 times out of 100 trials. In addition, the mean costs of the OPIO and IPSO algorithms were equal to 136,441.87\$/h and 136,542.87\$/h, respectively. This clearly shows that the statistical outcomes of the OPIO algorithm were more stable than those of the OGWO, IPSO, COA, RCGA, ORCSA and PIO algorithms. In addition, the time required to achieve the minimal fuel cost for the proposed algorithm was 10.14/sec, which was minimal in relation to other algorithms. The convergence graph of the total fuel cost of the proposed OPIO algorithm and the conventional PIO algorithm is given in Figure 3. Based on Figure 3, it can be seen that the formulated OPIO procedure provides the best active rate compared to the PIO algorithm.

**Table 4.** Comparison results of various algorithms for a 40-unit system.


**Figure 3.** Convergence results of the OPIO and PIO algorithms for a 40-unit system.

#### *5.3. Test Case 2a: 140-Unit*

In this instance, the formulated PIO algorithm was tested on the medium-scale of a 140-unit power generation system, and the load order was taken as 49,342 MW [46]. In this test case, non-smooth constraints, such as value-point consequence, interdicted executing section and ramp-limits were included. The execution was repeated for 100 trials, to confirm the dominance of the proposed methods with the obtained results of the OGWO, IPSO, COA, RCGA, ORCSA and PIO algorithms, which are presented in Table 5. As shown in Table 5, the OPIO reached 1,559,498.78\$/h, which was the minimum, compared to the other algorithms. In other words, the obtained outcomes clearly showed that the OPIO algorithm achieved a low fuel-cost value, compared to other methods.


**Table 5.** Test outcomes of various algorithms for a 140-unit system with PD = 49,342 MW.

Moreover, the statistical outcomes of the formulated OPIO algorithm, and various conventional procedures, are given in Table 6. Based on Table 6, the formulated OPIO algorithm provided the best outcomes, in terms of best, worst and mean cost, and less execution time compared to the various procedures. However, the best and mean costs of the OPIO and COA algorithms were equal to 1,559,498.78 \$/h, 1,559,499.21 \$/h, 1,559,521.36 \$/h and 1,559,521.88 \$/h, respectively. Even though the COA algorithm competed with the OPIO algorithm, the OPIO algorithm was quite efficient in achieving the best outcome in minimal

iterations, compared to the various procedures. The convergence of the formulated OPIO algorithm and the conventional PIO methods with iteration cycles is displayed in fig 4. From Figure 4, it can be seen that the OPIO technique attained the best solution within 20 iterations; this confirms that the OPIO algorithm had better convergence, because of its magnificent diversification and intensification abilities.



**Figure 4.** Convergence results of the OPIO and PIO algorithms for a 140-unit system.

#### *5.4. Test Case 2b: 160-Unit*

To access the feasibility of the formulated OPIO technique, another medium-scale test case of a 160-unit test system, along with non-convex value-point properties, was used. As to validation, the viability and efficacy of the formulated technique transmission loss was unnoticeable. For this medium-scale unit, a replicated 10 different fuel-option values were taken from [41], the power load was increased by 16, and the power load was fixed as 43,200 MW. Table 7 provides the attained better cost of the proposed OPIO algorithm, with other algorithms, by satisfying the constraints. Based on Table 7, the OPIO achieved 9625.15 \$/h, which was the best result, compared to the other algorithms. This confirms that the least total fuel cost was for the 160-unit generation system.


**Table 7.** Test outcomes of various algorithms for 160-unit system with PD = 43,200 MW.

The statistical results from over 100 trials of the proposed algorithm, compared to the OGWO, IPSO, IGA, RCGA, COA, ORCSA and PIO algorithms, are shown in Table 8: as can be seen, the OPIO algorithm performance—for example, best (9625.44 \$/h), mean (9647.62 \$/h) and worst (9649.62 \$/h)—was relatively acceptable, whereas the other algorithms deteriorated, due to an increase in the number of generators and traps in the locally optimal solutions. As per the acquired outcomes, we observed that the formulated OPIO technique was more vigorous and systematically structured, compared to the conventional and various algorithms. The active loop of formulated technique and conventional algorithm with iteration cycle is displayed in Figure 5. Figure 5 shows that the formulated procedure provided feasible convergence within 25 iterations, though there was an increase in the number of generation units compared to the standard PIO algorithm.


**Table 8.** Statistical comparison results for test case 2b (160-unit with PD = 43,200 MW).

**Figure 5.** Convergence results of the OPIO and PIO algorithms for a 160-unit system.

#### *5.5. Test Case 3a: 320-Unit*

In this instance, a wide scale 320-unit generation system, that included a value-point effect and three various fuel possibilities, was used to evaluate the execution of the formulated OPIO technique. For this 320-unit system, 32 times replicated, 10 different fuel options were taken from [41], and the power load was considered as 86,400 MW. Simulation results were carried out for 1000 iterations, for the 320-unit generation system, and its comparative results are illustrated in Table 9. Table 9 shows that the OPIO algorithm provided 19,968.95\$/h, which was the minimal production cost compared to different state-of-the-art algorithms. On the other hand, the test times of the COA and OPIO algorithms were nearly equal, at 412.95/sec and 410.65/sec, respectively. However, the OPIO algorithm showed a unique performance, in attaining the best fuel cost for 96 runs out of 100 trials. This proves

that the formulated OPIO algorithm was vigorous, and deliberately well-organized, compared to the PIO algorithm and the different approaches presented in this study. To ensure the efficacy of the formulated OPIO technique, the convergence over different iterations is shown in Figure 6. From Figure 6, it can be seen that the execution of the OPIO technique provided the best convergence over the standard PIO algorithm.



**Figure 6.** Convergence results of the OPIO and PIO algorithms for a 320-unit system.

#### *5.6. Test Case 3b: 640-Unit*

To ensure the efficacy of the formulated OPIO method, we tested it on another largescale generation system, a 640-unit system with value-point properties and three various fuel possibilities. This 640-unit system included the data of 10 multiple-fuel systems from [41], which were duplicated 64 times, and the load demand was fixed at 172,800 MW. The simulation results of the 640-unit system were iterated over 1000 iterations, and its comparative results are shown in Table 10, where it can be seen that the OPIO algorithm achieved minimum fuel cost related to the various state-of-the-art techniques. The statistical results achieved, by performing 100 trials of the different algorithms and their comparative outcomes, are illustrated in Table 10, which shows that the OPIO algorithm reached 39,963.78 \$/h by balancing the local search and global search, as well as converging faster towards the optimal solution. Moreover, the OPIO algorithm achieved the best solution for almost 96 runs out of 100 trials, which clearly demonstrates that the proposed algorithm can sustain the best position for various runs. The convergence results of the proposed OPIO algorithm and the PIO algorithm are displayed in Figure 7. In Figure 7, the formulated OPIO technique provided better convergence, which demonstrates its superiority over the standard PIO algorithm and other state-of-the-art techniques. The overall experimentation outcomes convey that the proposed OPIO algorithm achieved better efficiency, along with a trade-off between exploration and exploitation.


**Table 10.** Comparison results of various algorithms on a 640-unit system.

**Figure 7.** Convergence results of the OPIO and PIO algorithms for a 640-unit system.

#### *5.7. The Result Analysis of Wilcoxon Signed-Rank Test*

In this work, a non-parametric test—namely, the Wilcoxon signed-rank test—was utilized, to perform the statistical comparison of the proposed algorithm with the compared algorithms. The best solutions were attained by each technique for the corresponding test cases during 30 independent runs. In this study, the Wilcoxon signed-rank test was performed with a significance level *α* = 0.05. The results, analyzed by the Wilcoxon signedrank test, are presented in Table 11 for test cases of 13, 40, 140, 160, 320 and 640-generating units. In Table 11, the significance differences of the proposed algorithm and compared algorithms are marked with the value of H (i.e., H with a value of 1 specifies that there was a significance difference; otherwise, the H value is 0, if there was no significance difference). In addition, the symbol S with "+", "=" and "\_" denotes that the proposed technique was superior, equal or inferior, respectively, to the compared algorithms. Furthermore, we used four compared algorithms generically, to determine the significance difference with the proposed algorithm. It is clear from Table 11 that the proposed OPIO algorithm provided results superior to those of the COA, ORCSA and PIO algorithms, and equal to the OGWO algorithm for the test case 13-unit system. For the test case 40-unit system, the OPIO algorithm provided results superior to those of the COA, ORCSA and PIO algorithms, but not to the OGWO algorithm. Finally, w/t/l specified the win/tie/loss count by Wilcoxon signed-rank test for the six test case generating unit systems. Thus, from the above discussion, it is clear that the proposed OPIO algorithm attained better solutions, and had better exploring capability, compared to the existing algorithms.


**Table 11.** Wilcoxon signed-rank test between OPIO and four compared algorithms for test case 13, 40, 140, 160, 320 and 640-unit systems.

#### **6. Conclusions and Future Work**

In this article, we have provided a novel metaheuristic algorithm named the Oppositional Pigeon-Inspired Optimizer (OPIO), which is formulated to deal with the ELD problem, with value-point consequences and numerous fuel possibilities. From the literature, it can be seen that the standard PIO algorithm is considered a promising optimization technique, which attracts the researcher by its superiority in addressing various optimization problems. However, it suffers in regard to global search ability and premature convergence when it is applied to large-scale optimization problems. Because of these issues, we merged Opposition-Based Learning into a standard PIO algorithm, which helped to eradicate early convergence, aided knowledge discovery and enhanced comprehensive searchability. The formulated OPIO algorithm was applied to non-convex ELD problems with different constraints, such as multiple fuel possibilities, value-point consequence, interdicted zones and ramp-rate. The experimentation was carried out on three different ELD test cases, viz., small-scale (13-unit and 40-unit), medium-scale (140-unit and 160-unit) and large-scale (320-unit and 640-unit) test cases. The exploratory outcomes showed the superiority of the formulated OPIO technique—in relation to higher potential solutions, better convergence rate, robustness and better computational efficiency—over the PIO algorithm and other state-of-the-art metaheuristic algorithms. In future, this work could be used in other fields of optimization, owing to the technique's high potential for dealing with the problematic optimization issues of many practical power systems. In addition, the outcome of the results can be compared with potential algorithms such as SEPSO [16], SA-QSFS [42] and QANA [47].

**Author Contributions:** Conceptualization, R.R.; methodology, R.R. and D.K.; validation, S.S.A. and M.R.; formal analysis, A.D.; writing—original draft preparation, R.R.; writing—review and editing, M.R. and S.M.; supervision, R.R. and D.K; funding acquisition, S.S.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** This study was funded by the Deanship of Scientific Research, Taif University Researchers Supporting Project number (TURSP-2020/215), Taif University, Taif, Saudi Arabia.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data in this research paper will be shared upon request to the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

