Memetic Salp Swarm Algorithm for economic load dispatch problems

Abstract

This paper presents a hybrid version of the Salp Swarm Algorithm (SSA) for Economic Load Dispatch (ELD) problems with severe constraints. The Adaptive $\beta$-hill climbing optimizer (A$\beta$HCO) ais hybridized with a newly developed local search method with SSA as a new operator. This hybridization scheme is known as a memetic algorithm, where SSA serves as a natural selection agent (general refinement) in a genotype environment, while A$\beta$HCO serves as a culture selection agent (local refinement) in a phenotype environment. In other words, SSA acts as a gene encoding in biology, while A$\beta$HCO serves as a meme in a cultural context. In an intelligent optimization environment, gene and meme notations from natural biology and cultural selection act as search agents to achieve generality (gene) and problem specificity (meme). ELD is a crucial optimization problem in electrical engineering, and it is non-convex, multi-modal, and severely constrained. The proposed method, called MSSA, evaluates several types of ELD problems that differ in the constraints adopted. The first problem is addressed by considering two types of constraints related to load balance and output. It includes five practical cases of ELD generators that vary in number of units and load requirements: a three-unit generator with a capacity of 850 MW (3UG-850 MW), a thirteen-unit generator with a capacity of 1800 MW (13UG-1800 MW), a thirteen-unit generator with a capacity of 2520 MW (13UG-2520 MW), a forty-unit generator with a capacity of 10500 MW (40UG-10500 MW), and a large-scale generator with a capacity of 80 units of 21000 MW (80UG-21000 MW). Two additional constraints-restricted operating zones and ramp rate limits-are used to address the second problem. A six-unit generator with a capacity of 1,263 MW (6 UG-1,263 MW) and a fifteen-unit generator with a capacity of 2,630 MW (40 UG-2,630 MW) are two real-world cases discussed. Compared with other existing algorithms, the comparative results demonstrate the feasibility and usefulness of the proposed MSSA algorithm.

Introduction

Fuel-based generators are the primary source of electricity in power engineering^1,2. A challenge in this area is the Economic Load Dispatch (ELD), which deals with properly scheduling a group of generating units while using the least amount of fuel³. The process of scheduling the generator units is complicated since the load output and load balancing of any generator must be met. Finding an optimal schedule with respective load output and load balancing residing the minimum fuel cost is the target of electrical power engineer⁴. Indeed, there are several advanced deep learning models designed to improve the short-term performance of microgrids while reducing emissions and operating costs in the presence of distributed energy resources. The main goals of these works are to maximize the potential of demand response initiatives, which effectively engage a wide range of customers to reduce uncertainty around renewable energy sources^5,6,7,8. Generally, ELD is formulated as non-convex, non-contiguity, multimodal, highly constrained optimization problem. Nowadays, ELD is one of the main research domains to be stressed for smooth operation of economic power systems⁹. Traditional approaches faced several challenges in dealing with ELD search space due to its non-linearity and high constraints features¹⁰. Due to several constraints such nonlinear power flow, multiple fuel utilization, restricted operating zones, and valve-point impact, many types of conventional approaches are unable to properly handle ELD¹¹. These constraints increase the rugged size of the search space leading to large-scale systems. Another optimization problem that has attracted the attention of many researchers for its high importance is aircraft landing optimization problem^12,13,14. This problem shares many characteristics with ELD, as both are complex and carry constraints, among other characteristics.

Examples of traditional calculus-based approaches used to solve optimization problems are linear and non-linear programming methods^15,16, quadratic programming methods^17,18, Lagrangian relaxation methods¹⁹, Newton’s method²⁰, lambda iteration method²¹, gradient descent methods²², and dynamic programming methods²³. This spurred the present research communities to shift their attentions to utilize meta-heuristic-based methods for solving large-scaled ELD problems.

In Artificial Intelligence (AI), meta-heuristics can be considered an efficient search mechanisms used in the context of optimization^24,25,26,27. Such algorithms search for the global optimum of optimization problems and often adopt principles inspired by nature. The make use of accumulative knowledge (exploitation) and navigating potential regions (exploration) in the search space controlled by parameters tuned either in adaptive manner or manually^28,29. Recently, a plethora of meta-heuristic-based algorithms have been established using different types of operators and various number of initial solutions. Keeping that in mind, they are divided into two categories: population-based algorithms and trajectory-based algorithms^30,31.

Trajectory-based (or local search or single-point search) algorithms begin their search with solely single random solution. This solution determines the search space region to be searched. The search will move from one solution to another solution in the same region forming a trajectory until a local optimum is reached^32,33. These types of algorithms are very powerful to exploit the search space region, yet exploring several regions is still a challenging task. The most popular trajectory-based methods for ELD problems are simulated annealing³⁴, tabu search³⁵, GRASP³⁶, and $\beta$-hill climbing algorithm³⁷. Opposite to the trajectory-based algorithm, population-based algorithms begin their search with a population of random solutions (individuals). These solutions undergo involvement using cooperative operators related to the recombination, mutations, and natural selection. They could explore many search space regions instantly, yet they face a real challenge in exploiting each region that has been explored^33,38.

The most recent population-based algorithms for ELD problem are the differential evolutionary algorithm^3,39, dragonfly algorithm⁴⁰, squirrel search algorithm⁹, ant lion optimizer⁴¹, artificial algae algorithm⁴², grey wolf optimizer⁴³, artificial cooperative search algorithm⁴⁴, Jaya algorithm⁴⁵, social spider optimization algorithm⁴⁶, grasshopper optimization algorithm⁴⁷, crow search algorithm⁴⁸, mine blast algorithm⁴⁹, harmony search algorithm⁵⁰, chameleon swarm algorithm⁵¹, Harris Hawks optimizer⁵², slime mould algorithm⁵³, capuchin search algorithm⁵⁴, group teaching optimization⁵⁵, political optimization algorithm⁵⁶, and others reported in^11,57,58. In simple words, population-based algorithms are excellent in exploration whereas trajectory-based algorithms excel in exploitation phase³². Although exploration and exploitation are at odds with one another, a successful algorithm must strike the sensible balance between the two aspects to perform well.

Recently, several researchers have used memetic algorithms (MA) cope with the complexity of large-scale problems⁵⁹. MAs complement the advantages of the trajectory-based algorithm in exploitation and the population-based algorithm in exploration in a single intelligent framework. The concepts of MA of are inspired by the notation of meme and gene⁶⁰. The theory of memetics imitates the analogy across the mind-universe as well as the genetic notations in cultural evolution which span through the cognitive, psychology and biology fields. The meme is the basic unit of cultural transmission which was originally defined by Dawkins⁶⁰. The term “memetic algorithm” is initially established by Moscato¹ to imitate the natural evolution and cultural evolution that reside in Darwinian gene notation and Dawkins’ meme notation. Nowadays, the term “memetic algorithm” refers to a population-based algorithm that has been hybridized with a trajectory-based algorithm as a local refinement, providing an efficiency mechanism that can strike the ideal balance between problem specificity through the notation of meme and generality through the notation of gene⁶¹.

A new population-based technique called the Salp Swarm Algorithm (SSA) was put up in⁶² to mimic how salps move through and forage in oceans. Because of its excellent characteristics, including derivative-freeness, parameter less, usability, simplicity, flexibility, and scalability. It has been extensively used for a variety of optimization problems, including breast cancer segmentation⁶³, text document clustering⁶⁴, feature selection⁶⁵, renewable energy⁶⁶, big data optimization⁶⁷, task assignment problem⁶⁸, and others reported in^69,70. As other population-based algorithms, SSA suffers from a chronic problem that appears in the lack of diversity and local optima⁶⁵. As a result, several endeavors have been made to enhance the diversity side of SSA by altering specific operators or combining SSA with other meta-heuristic-based algorithms⁶⁹.

For instance, Tubishat, Mohammad, et al.⁶⁵ introduced a memetic version of SSA algorithm for feature selection problems. In their algorithm, the SSA is integrated with the Opposition-based learning to improve population diversity. Furthermore, a new local search algorithm is combined within the SSA algorithm to avoid the problem of local optima. Another memetic SSA algorithm for feature selection problems is introduced in⁷¹. In their algorithm, the SSA algorithm is combined with the simulated annealing algorithm to enhance its exploitation ability. Another hybrid SSA technique for image segmentation is developed in⁷² with the name SSAMFO algorithm. The SSA algorithm and the moth flame optimization (MFO) method are coupled in SSAMFO to increase the exploitation capacity of the hybridized algorithm for the purpose of image segmentation. In another study, the SSA algorithm is combined with the differential evolution algorithm for big data optimization problems⁶⁷. The differential evolution algorithm was used as a local search technique to empower the exploitation ability of their algorithm.

The Salp Swarm Algorithm (SSA) is a promising optimization method used across multiple fields. It has been successfully adapted for a wide range of optimization problems. Despite its strengths, SSA faces key limitations:

• Heavy dependence on the current best solution. This leads to its focus on exploration rather than on exploration.

• Tendency to get stuck in local optima. This is because SSA’s global and local search balance, while effective in theory, are sometimes inadequate in practice especially for real-world complex optimization problems.

• Lack of memory for best positions from previous iterations, limiting exploitation.

Several hybrid SSA variants have been proposed to enhance performance and address their weaknesses. The current study proposes a hybrid SSA with the Adaptive $\beta$-Hill Climbing Optimizer (A$\beta$HCO)⁷³ to improve exploration in early stages, exploitation in later stages, and overall convergence speed.

In sum, we improve the performance of SSA for ELD problems by combining it with a local optimization technique known as the adaptive $\beta$-hill climbing optimizer (A$\beta$HO)⁷³, in which the proposed hybrid algorithm is named memetic salp swarm algorithm (MSSA). In the proposed MSSA algorithm, SSA is embedded into MSSA as a gene while A$\beta$HO is used as a meme, ensuring a balance between generality and specificity of the problem. In other words, the SSA method takes the form of a meme algorithm, where it is presented as a Darwinian gene encoding in natural evolution, while the A$\beta$HO method is embedded within SSA to capture the meme encoding in cultural evolution. The proposed MSSA method is then evaluated for solving several types of ELD problems that differ in terms of the constraints considered. The key contributions of the current study can be summarized as follows:

• 1. A hybrid algorithm, referred to as MSSA, is designed by injecting the adaptive $\beta$-hill climbing optimizer (A$\beta$HO) as a path-based method within the main framework of the SSA algorithm as a population-based technique to enhance the exploitation capability of the proposed algorithm and thus ensure the right balance between exploration and exploitation capabilities.

• 2. The proposed MSSA model is applied to solve several types of ELD problems with varying constraints and complexities. The first model addresses two types of constraints related to load balance and power output. It includes five practical cases of ELD generators with varying unit counts and load requirements: a three-unit generator with a capacity of 850 MW (3UG-850MW), a thirteen-unit generator with a capacity of 1800 MW (13UG-1800MW), a thirteen-unit generator with a capacity of 2520 MW (13UG-2520MW), a forty-unit generator with a capacity of 10500 MW (40UG-10500MW), and a large eighty-unit generator with a capacity of 21000 MW (80UG-21000MW). To address the second model, two additional limits are used for the ramp rate and the limits of the restricted operating areas. In two real scenarios, a 6-unit 1263 MW generator (6UG-1263MW) and a 15-unit 2630 MW generator (15UG-2630MW) were included.

• 3. A repair algorithm is used in the proposed MSSA to maintain the feasibility of solutions during the search process. This is accomplished by checking the value of the load demand allocated to each unit in the solutions individually, ensuring that the value meets quality and inequality constraints. It is worth mentioning that we initially investigated soft constraint handling and penalty-based relaxation approaches during the early stages of this research. However, due to the ruggedness and non-convex nature of the Economic Load Dispatch (ELD) problem’s search space, these methods did not yield competitive or reliable results compared to the strict feasibility-based approach. Therefore, we opted for a strategy that consistently avoids non-feasible solutions to maintain solution quality and convergence stability.

• 4. For each case study included in the current research, a comparative analysis and evaluation was conducted. The comparative results demonstrate the expected feasibility of MSSA.

• 5. The convergence behavior of the proposed MSSA algorithm was studied under various configurations. Additionally, the Wilcoxon signed-rank test was used to demonstrate the efficiency of the proposed MSSA algorithm compared to other comparison algorithms.

The sections of this paper are arranged as follows: the definitions and mathematical modeling of the ELD versions are given in Sect. "Formulation of Economic Load Dispatch (ELD)". The proposed MSSA is discussed and presented in Sect. "Memetic Salp Swarm Algorithm (MSSA) for ELD". The proposed MSSA is evaluated through intensive real-world experimented cases of two types of ELD problem in Sect. "Experimental results and discussions". Finally, the MSSA is concluded, and some possible future enhancements are recommended in Sect. "Conclusion and future works".

Literature review

The Economic Load Dispatch (ELD) problem is a fundamental challenge in the field of electrical engineering. It has been extensively studied by researchers in the optimization domain to develop efficient and practical solutions. Given its complexity and significance, numerous studies have explored various methodologies to address this problem. In this section, we provide a comprehensive review of the most relevant research contributions, with a particular focus on hybrid meta-heuristic algorithms, which have demonstrated significant potential in improving solution accuracy and computational efficiency. Since ELD is challenging due to its non-convexity and constraints, many authors have made many efforts to solve this problem. For example, the authors in⁷⁴ proposed a hybrid Jaya–TLBO algorithm by integrating the Jaya optimization algorithm with Teaching–Learning-Based Optimization (TLBO) to effectively solve ELD problems. Both Jaya and TLBO are population-based optimization techniques; however, TLBO was specifically incorporated to enhance population diversity, maintain a balanced trade-off between exploration and exploitation throughout the search process, and improve the early convergence performance of the Jaya algorithm. The proposed hybrid algorithm was evaluated using multiple case studies, including systems with 6, 13, 20, and 40 generator units, as well as a real-world 10-unit generator system from Indonesia. Simulation results demonstrated that the hybrid Jaya–TLBO algorithm outperformed the classical Jaya, TLBO, and other state-of-the-art algorithms published in the literature, showcasing its effectiveness in solving ELD problems. Iqbal et al.⁷⁵ proposed a hybrid Multi-Verse Optimizer (MVO) and Sequential Quadratic Programming (SQP) approach to solve ELD problems. The hybrid algorithm follows a two-step process, where MVO is first employed for global exploration, and SQP is subsequently applied to refine the solutions and enhance overall optimization performance. The integration of SQP aims to improve solution accuracy and convergence speed, ultimately yielding superior results. The MVO-SQP algorithm was evaluated using four case studies involving 6-, 13-, 15-, and 40-unit generator systems. Simulation results demonstrated that the hybrid approach outperformed other comparative algorithms in terms of both solution quality and computational efficiency.

The authors in⁷⁶ proposed a hybrid optimization algorithm, PSODE, by integrating an improved Differential Evolution (nDE) with an enhanced Particle Swarm Optimization (nPSO) to solve economic and emission dispatch problems. In their approach, a novel mutation strategy and adaptive crossover rate were introduced in nDE to mitigate premature convergence, while a new acceleration coefficient was incorporated into nPSO to achieve an optimal balance between exploration and exploitation. Additionally, nPSO was strategically utilized within the hybrid framework to further enhance search efficiency. The effectiveness of the PSODE algorithm was evaluated using 23 classical benchmark functions and three ELD case studies involving 3-, 6-, and 40-unit generator systems. The results demonstrated that PSODE outperformed nPSO, nDE, and other state-of-the-art algorithms reported in the literature, excelling in avoiding local optima, improving convergence behavior, enhancing solution quality, and reducing computational time. Liu et al.⁷⁷ proposed a hybrid optimization algorithm, OMLIDE, to effectively solve ELD problems. The algorithm incorporates an opposition-mutual learning strategy during the initialization phase to enhance population diversity, thereby improving the search efficiency. Additionally, two mutation operators are integrated to maintain a well-balanced trade-off between local exploitation and global exploration. To further refine performance, an adaptive parameter adjustment strategy is implemented to dynamically modify the scaling factor and crossover rate throughout the optimization process. The effectiveness of OMLIDE was evaluated using five case studies involving 10-, 15-, 40-, 80-, and 120-unit generator systems. Experimental results demonstrated that OMLIDE outperformed other comparative algorithms in terms of solution quality and convergence behavior, showcasing its robustness and efficiency in tackling ELD problems. Alawad et al.⁷⁸ introduced a novel hybrid optimization algorithm, HSOA, to address ELD problems. In HSOA, the Snake Optimizer is enhanced with two key strategies: the Oppositional-Mutual Learning strategy during the initialization phase to improve population diversity and the Dynamic Polynomial Mutation strategy during the optimization phase to strengthen the search process. These enhancements aim to accelerate convergence, improve solution quality, and prevent the algorithm from getting trapped in local optima. The proposed HSOA was evaluated using three case studies involving 3-, 40-, and 80-unit generator systems. Experimental results demonstrated the superiority of HSOA over the classical Snake Optimizer and other comparative algorithms, achieving the best, second-best, and third-best rankings, respectively, in terms of solution accuracy and optimization efficiency. The authors in⁵⁴ proposed a memetic algorithm for solving ELD problems by hybridizing the Capuchin Search Algorithm (CSA), a population-based method, with a gradient-based optimizer, a trajectory-based method. The gradient-based optimizer was integrated into the main framework of CSA to enhance its exploitation capability, thereby achieving an optimal balance between exploration and exploitation. The proposed memetic algorithm was evaluated using six case studies involving 3-, 13-, 40-, 80-, and 140-unit generator systems. Simulation results demonstrated the superior performance of the proposed approach compared to other state-of-the-art algorithms, particularly in large-scale cases such as the 40- and 140-unit test systems, where it excelled in solution quality and computational efficiency. Al-Betar et al.⁵² introduced another memetic algorithm by combining the Harris Hawks optimizer (HHO) as a population-based method with the adaptive $\beta$-hill climbing optimizer (A$\beta$HO) as a trajectory-based method for solving ELD problems. The A$\beta$HO is utilized to empower the exploitation ability and thus ensure the balance between the exploration and exploitation abilities during all stages of the search process. Their memetic algorithm was tested using six case studies with 6-, 13-, 13-, 15-, 40-, and 140-unit generators, each with different load conditions and constraints. The experimental results showed that the proposed memetic algorithm achieved more competitive results than the other comparative algorithms in all case studies of the ELD problem. Similarly, the authors in⁷⁹ developed another memetic technique named SCA-$\beta$HC algorithm by hybridizing the sine cosine algorithm (SCA) with the $\beta$-hill climbing ($\beta$HC) optimizer for solving ELD problems. The $\beta$HC is used in the proposed algorithm as a trajectory-based method to enhance the exploitation capability and thus achieve the right balance between the exploration and exploitation abilities. Their SCA-$\beta$HC was evaluated using six case studies with 6-, 13-, 13-, 15-, 40-, and 140-unit generators. The results demonstrated the efficiency of the proposed SCA-$\beta$HC by obtaining very competitive results than the other comparative algorithms in all case studies. The field of optimization of ELD has garnered significant attention from researchers in the optimization community, leading to a steady increase in both theoretical studies and practical applications. According to the No-Free-Lunch (NFL) theorem for optimization⁸⁰, no single optimization algorithm consistently outperforms all others across every optimization problem, including different variations of the same problem. As a result, the development of novel algorithms for solving ELD problems remains an open research area.

Based on previous studies, hybrid algorithms have demonstrated remarkable effectiveness in achieving superior results by maintaining a well-balanced trade-off between exploration and exploitation. Therefore, enhancing the search strategy of the classical Salp Swarm Algorithm (SSA) by leveraging previously explored regions could improve its ability to maintain this balance and attain high-quality solutions. This enhancement can be realized by hybridizing the classical SSA with the A$\beta$HO, a trajectory-based optimization method, to develop a novel memetic algorithm with improved convergence behavior and solution accuracy.

Formulation of Economic Load Dispatch (ELD)

ELD is a major problem in the electricity grid, in which a set of generating units ($U_1, U_2, \ldots , U_{ng}$) is determined by the size of ng in any generator. Each generator unit $U_i$ requires certain fuel consumption to generate a specific power voltage. Normally, the generator provides a fixed amount of power. However, fuel consumption depends on the scheduling of the generator units. The optimal scheduling of the generator units enables the generator to achieve its target with minimum fuel cost. The process of distributing the power output of each unit shall be accomplished in accordance with the capacity of each unit as well as power loss. When creating the generator unit schedule, other factors connected to the generator’s nature, such as valve-point impacts, banned areas, ramp rate limits, and power balancing restrictions, must also be considered.

ELD is defined in optimization domain to be easily manipulated using optimization algorithms. For ELD, the optimization algorithms are triggered to obtain the acceptable schedule with the minimum total fuel cost. The different ELD problem constraints should be maintained in the solution. To create a bridge between the problems in the optimization context, the solution representation and the objective function to assess each solution. The constraints shall also be formalized to be considered when the solution is constructed.

Solution representation of ELD

A generation vector $\varvec{{U}}$=$(U_1, U_2,\ldots , U_{ng})$, where ng is the overall number of generating units in the system, is used to describe the solution to the ELD problem. The value of the generating units $U_i$ is set between $U_i^{min}$ and $U_i^{max}$, where $U_i^{max}$ and $U_i^{min}$ are the maximum and minimum limits of the generator $U_i$, respectively.

Objective function

The fundamental goal of the ELD problem is to offer a reasonable fuel cost value for each generator unit so that the correct amount of power may be generated with the least amount of fuel. Eq. 1 may be used to express this objective function.

$$\begin{aligned} \min F(\varvec{{U}}) = \sum _{i = 1}^{ng} F_i (U_i). \end{aligned}$$

(1)

where $F(\varvec{{U}})$ is the system’s overall cost $\varvec{{U}}$. Additionally, $F_i$ is the total cost of fuel for the producing unit $U_i$, which can be calculated using Eq. 2:

$$\begin{aligned} F_i(U_i) = a_iU_i^2 + b_iU_i +c_i. \end{aligned}$$

(2)

where $U_i$ is the power output and $a_i, b_i$, and $c_i$ are the generating unit’s cost coefficients. According to Eq. 3, the Valve-Point Effect (VPE) is included into each producing unit’s fuel cost.

$$\begin{aligned} F_i(U_i) = \lfloor a_iU_i^2 + b_iU_i +c_i + |e_i \texttt {sin}(f_i (U_i^{min}- U_i))|\rfloor . \end{aligned}$$

(3)

where $e_i$ and $f_i$ are the generating unit $U_i$’s VPE cost coefficients. The cost function will become non-smooth due to the VPE, which will also lead to an increase in local optima⁸¹.

ELD constraints

The following four constraints reflect the inequality and quality constraints.

Power balance constraint

According to Eq. 4, the total amount of power generated by all generator units in the ELD solution must match the total amount of system load ($U_D$) plus power losses ($U_L$).

$$\begin{aligned} \sum _{i = 1}^{{ng}} U_{i}- U_D - U_L=0 \end{aligned}$$

(4)

$$\begin{aligned} U_L = \sum _{k = 1}^{ng} \sum _{i = 1}^{ng} U_k B_{ki}U_i + \sum _{k = 1}^{n} B_{0k}U_k + B_{00}. \end{aligned}$$

(5)

Eq. 5 is used to determine the transmission losses of the system, where the loss coefficient vectors $B_{ki}$ signify the square of the $ki^{th}$ member, $B_{0k}$ is for the $i^{th}$ member, and $B_{00}$ is constant.

As per the above constraint, both the overall demand and the actual power loss in the transmission lines must be met by the entire power generation.

Generator capacity constraint

Each producing unit’s power output should fall within the minimum and maximum ranges indicated in Eq. 6.

$$\begin{aligned} U_i^{min} \le U_{i} \le U_i^{max} \end{aligned}$$

(6)

The constraint introduced in Eq. 6 is important to impose lower and upper constraints on the bus voltage values and generator outputs to ensure stable operation.

Ramp rate limits constraint

The ramp rate limits of the generating units that correspond to them constrain the generating units’ ability to vary their output. The ramp-up and ramp-down limitations are expressed in Eq. 7 and Eq. 8, respectively.

$$\begin{aligned} U_i - U_i^{0} \le UR_i \end{aligned}$$

(7)

$$\begin{aligned} U_i^{0} - U_i \le DR_i \end{aligned}$$

(8)

where $U_i^{0}$ represents the producing units’ prior output in terms of power ($U_i$). The ramp-up and ramp-down limitations of the generating units $U_i$ are $UR_i$ and $DR_i$, respectively. The ramp rate limitations modify the generator capacity constraint by combining Eq. 6, Eq. 7, and Eq. 8, as indicated in Eq. 9.

$$\begin{aligned} \max (U_i^{min}, UR_i - U_i)\le U_i \le \min (U_i^{max}, U_i^{0} - DR_i) \end{aligned}$$

(9)

To meet the constraint presented in Eq. 9, transmission line loading is limited by its upper limits for safe operation.

Prohibited operating zones constraint

The physical operation restrictions always place a limit on the producing unit’s operational range. For the goal function, prohibited operation zones result in discontinuous areas. This constraint can be expressed as follows:

$$\begin{aligned} U_i \in {\left\{ \begin{array}{ll} U_i^{min} \le U_i \le U_{i,1}^{LB} & \\ U_{i,1}^{UB} \le U_i \le U_{i,z}^{LB} & z=2,3,\cdots , Z\\ \cdots & \\ U_{i,Z}^{UB}\le U_i \le U_i^{max}& \\ \end{array}\right. } \end{aligned}$$

(10)

where Z is the overall number of restricted operating zones. The lower and upper boundaries of the banned operating zone z for the generating unit $U_i$ are, respectively, $U_{i,z}^{LB}$ and $U_{i,z}^{UB}$.

After the aforementioned objectives and constraints have been included, it is possible to represent the problem mathematically as a non-linear constrained multi-objective optimization problem.

Memetic Salp Swarm Algorithm (MSSA) for ELD

To mimic Salp behavior in its native habitat, Mirjalili et al.⁶² devised the SSA algorithm. Salps have a transparent, barrel-shaped body, in which they resemble jellyfish. Their swimming habits in deep oceans are used to develop the core premise of SSA. They combine to form a chain of leaders and followers. The leader serves as the link in a chain that directs the following to the safe trail that leads to deep oceans. This behavior is represented by an optimization approach, where the leader is the best solution and the followers are the other solutions in the salp chain.

In this paper, we propose the MSSA algorithm, which combines the SSA with the Adaptive $\beta$-hill Climbing Optimizer (A$\beta$HCO), as a novel operator, to refine the solution for addressing ELD problems in two stages: local refinement stage through the application of the A$\beta$HCO algorithm and general refinement through the use of SSA algorithm. Whereas A$\beta$HCO is the ‘meme’ in cultural evolution, the SSA is the ‘gene’ in biological evolution. Together, these two search agents increase the solution’s specificity to the problem at hand as well as its generality.

Figure 1 displays the step-by-step process of the MSSA algorithm. Regarding the viable region of ELD, a set of random solutions are constructed to start the MSSA. The viable area of the ELD search space has certain solutions that adhere to the equality and inequality constraints stated in Sect. "Formulation of Economic Load Dispatch (ELD)".

Fig. 1

The flowchart of the proposed MSSA algorithm.

The distance between both load demands is recovered by adding or subtracting a slice (slc) that is a portion of dist in a random generator unit with respect to its range value. This process is repeated until $dist =0$. Note that the inequality constraints are satisfied and checked during the generation process. All solutions (i.e., population of salps) are stored in an augmented matrix in the salps Chain Memory ($\textbf{SCM}$) of size $gn \times N$ as shown in Eq. 11. Each raw in $\textbf{SCM}$ represents an ELD solution where the total number of solutions is equal to N. Using the repair algorithm presented in Algorithm 1, the viability of each solution is guaranteed. The cost of each ELD swarm is calculated using Eq. 1. The target food source is normally known and is called ${\textbf {F}}$. In case the target is not known, the best global food source in $\textbf{SCM}$ is considered as ${\textbf {F}}$. The leader salp ($U^1$) is at the front of the salp chain. The rest of the salp solutions in the salps chain are called the followers (i.e., $\{U^2,U^3, \ldots , U^N\}$).

$$\begin{aligned} \textbf{SCM}=&\left[ \begin{matrix} U^{1}_{1} & U^{1}_{2} & \cdots & U^{1}_{gn}\\ U^{2}_{1} & U^{2}_{2} & \cdots & U^{2}_{gn}\\ \vdots & \vdots & \cdots & \vdots \\ U^{N}_{1} & U^{N}_{2} & \cdots & U^{N}_{gn}\\ \end{matrix} \right] . \end{aligned}$$

(11)

The positions (or generator units) of the salp leader ($U_i^1$) in $\textbf{SCM}$ are updated based on Eq. 12.

$$\begin{aligned} U_i^1 \leftarrow {\left\{ \begin{array}{ll} F_i+r_1((U_i^{max} - U_i^{min})r_2 + U_i^{min}) & r_3\ge 0.5 \\ F_i-r_1((U_i^{max} - U_i^{min})r_2 + U_i^{min}) & r_3< 0.5 \end{array}\right. } \end{aligned}$$

(12)

where $i \in (1, 2, \ldots , gn)$, $F_i$ represents the position of the food source in each dimension, $U_i^{min}$ and $U_i^{max}$ represent the minimum and maximum values of the generator unit i, $r_2$ and $r_3$ are three uniformly distributed random values created independently within the interval [0, 1], and $r_1$ is the dominant parameter responsible for striking a reasonable balance between exploitation and exploration.

According to Eq. 12, the leader salp merely modifies its location in relation to food availability. Because the coefficient $r_1$ strikes a balance between exploration and exploitation, it is the most crucial parameter in SSA and can be calculated as shown in Eq. 13.

$$\begin{aligned} r_1=2e^{-\left( \frac{4 \times t}{T}\right) ^2} \end{aligned}$$

(13)

where t represents the current iteration and T represents the total number of iterations.

Figure 2 shows the values of $r_1$ over 1000 iterations.

Fig. 2

The exponential iterative function of the parameter $r_1$ in SSA.

As seen in Fig. 2, the dominating parameter $r_1$ has a significant influence on the search power, which in turn has a significant impact on the SSA algorithm’s exploration and exploitation capability. The parameter $r_1$, as shown in Fig. 2, begins at 2.0 and decreases to the lowest value at which the SSA algorithm is thought to have reached the global optimum solution. Large values of $r_1$ may prevent local optimum solutions by triggering a global search that could be redirected toward more investigation. Conversely, low values of $r_1$ result in local search, which boosts SSA’s exploitation potential and gives the algorithm a decent chance of locating the optimal solution.

The rest salps in the chain, or $(U^2, U^3, \ldots , U^N)$, are the followers. Based on a Newton’s rule of motion as calculated by Eq. 14, their locations will be updated.

$$\begin{aligned} U_i^j=U_0 + \frac{1}{2}\alpha t^2 + \upsilon _0 t \end{aligned}$$

(14)

where $U_i^j$ denotes the new location of the ith salp at the j dimension, $U_0$ indicates the beginning position, $\upsilon _0$ denotes the salps’ initial velocity, $\alpha$ stands for the acceleration of salps and t is the time instance (i.e., iteration).

In Eq. 14, it is important to mention that Newton’s laws of motion are physical rules that explain the connection between an object’s motion and the forces operating on it. The first law of motion, as demonstrated in Equation 15, can be used to determine the salp’s velocity, v, during motion.

$$\begin{aligned} v = v_0+ \alpha t \end{aligned}$$

(15)

where $v_0$ refers to the initial speed, and the value of the acceleration $\alpha$ can be calculated according to Eq. 16.

$$\begin{aligned} \alpha= & \frac{\Delta \upsilon }{\Delta t}\\\nonumber= & \frac{\upsilon _{final}-\upsilon _0}{t} \end{aligned}$$

(16)

where $\upsilon _{final}$ represents the final velocity of the salps, $\upsilon _{0}$ represents the initial velocity of the salps, t represents time, and $\Delta \upsilon$ represents the difference between the final velocity and the initial velocity.

The velocity of movement of salps can be related to the traveled distance as defined in Eq. 17.

$$\begin{aligned} v= & \frac{\Delta U}{\Delta t} \\\nonumber= & \frac{U - U_0}{t} \end{aligned}$$

(17)

where U represents the final distance traveled by salps, $U_0$ represents the initial distance, and $\Delta U$ represents the difference between the final distance and the initial distance.

Given that optimization time is iteration, the difference between iterations is 1, and taking into account that $\upsilon _0=0$, the above mathematical formulas can be shortened as presented in Eq. 18.

$$\begin{aligned} U_i^j=\frac{1}{2}\left( U_i^j+U_i^{j-1}\right) \end{aligned}$$

(18)

Repair procedure

Because the ELD problems at hand are so complex, it is possible that the proposed MSSA algorithm would provide infeasible solutions. Inequality constraints or equality constraints may be violated by the proposed MSSA algorithm, leading to such an issue. To transform these impractical solutions into feasible ones, a repair procedure is therefore desperately needed. This repair procedure will check that each generated unit of the ELD problems falls within the acceptable value range. Furthermore, the power created by the solutions developed must be equal to the output power. For the aforementioned reasons, the proposed MSSA algorithm proposed a repair procedure to be used to fix search agents that are not feasible, particularly when the viability of the solutions generated by MSSA is not compromised or is not infeasible. The proposed MSSA algorithm adequately addresses the infringing limitations by implementing this repair procedure. In this, a repair algorithm is proposed to ensure the viability of the initial solutions. This repair procedure verifies that the equality and inequality conditions are met by checking each unit value in the original solutions individually. The pseudocode shown in Algorithm 1 summarizes the phases of the repair process that modify infeasible solutions into feasible ones.

Algorithm 1

A pseudo-code outlining the phases in the proposed MSSA algorithm’s repair procedure.

In the repair procedure (Algorithm 1), the load demand (e.g., equality constraints) for each ELD solution is checked by calculating the total weight demand (Tot) of all generator units. After that, it is estimated how far apart (for example, dist) the calculated load demand and the intended load demand are from each other. When the whole salps chain is updated, they are passed to the Repair Algorithm (see Algorithm 1) to check their feasibility. Thereafter, the Adaptive $\beta$-Hill Climbing optimizer (A$\beta$HCO) is invoked for each salp solution with a probability range of $B_r$. Then, for each salp solution with a $B_r$ probability range, the (A$\beta$HCO) is triggered. This parameter is used to determine the density of using A$\beta$HCO in SSA. This appeared in Algorithm 2. The procedure of A$\beta$HCO will be discussed in the following subsection. Based on the concept of natural selection, the leader and followers are updated, and if the new ones are superior, the old ones are replaced. This procedure will be carried out as many times as the maximum number of iterations allowed (T).

Adaptive $\beta$ Hill Climbing Optimizer

In MSSA, the A$\beta$HCO is hybridized in SSA as a new operator. A$\beta$HCO⁷³ is an updated version of $\beta$-Hill Climbing optimizer ($\beta$HCO)⁸². They have been extensively used for several optimization problems including feature selection⁸³, generating substitution-boxes⁸⁴, economic load dispatch³⁷, gene selection⁸⁵, classification problems⁸⁶, sudoku game⁸⁷, denoising ECG signals⁸⁸, and multiple-reservoir scheduling⁸⁹.

In A$\beta$HCO, the $\beta$ operator is used to diversify the search and to prevent getting caught at a local minimum. In A$\beta$HCO, the two control parameters of $\beta$HCO are deterministically adapted during the search. The A$\beta$HCO is hybridized in SSA as new operator to improve the exploitation feature of SSA thus improving the ELD solutions.

With a probability value of $B_r$, the new salp solution ($U=(U_1,U_2,\ldots , U_{gn})$) is passed into A$\beta$HCO. This solution is locally improved using three operators of A$\beta$HCO which are $\mathcal {N}$-operator, $\beta$-operator, and $\mathcal {S}$-operator. Using the repair algorithm, the salp solution’s viability is preserved.

The three operators of A$\beta$HCO can be described as follows:

• 1. $\mathcal {N}$-operator: Location of the current salp $\varvec{{U}}^j$ is modified by relocating to its neighbor into its neighboring salp position, $\varvec{{U}}'^j$ as seen in Eq. 19.

  $$\begin{aligned} U'^j_i = U^j_i \pm r~ (0,1) \times \mathcal {N}&\exists i\in [1, N] \end{aligned}$$

  (19)

  where $\mathcal {N}$ represents the distance bandwidth between the neighboring solution and the current solution at time t, and i represents a randomly selected index in the possible range, $i \in [1, 2, \ldots , N]$. As the MSSA algorithm operates, the A$\beta$HCO and SSA algorithms are both permitted to operate within the whole Local Search (LS) loop and are chosen by probability to carry out a one-step LS operation at each iteration loop. For a local search, let $\mathcal {N}$ and $C_t$ represent the probability of applying adaptive parameters to the search agent (i.e., salp), respectively, where $\mathcal {N}$ + $C_t$ = 1. The initial values of the parameters $\mathcal {N}$ and $C_t$ are set to 0.5 at the beginning of this strategy, meaning that each operator has an equal chance to compete. As every LS operator conducts a biased search, a higher selection probability should be assigned to the LS operator that generates the most improvements. Here, the values of $\mathcal {N}$ and $C_t$ for each LS operator were modified using an adaptive learning technique. As previously indicated, the distance bandwidth between the present solution and the neighboring solution is calculated using the $\mathcal {N}$ parameter, which is expressed in Eq. 19. When $\mathcal {N}$ is big, a greater distance is attained. This implies that the search will greatly diversify the solution, making it impossible to ensure that the positive aspects of the existing solution will be exploited. Nonetheless, it is customary to need variety in the first search and to progressively reduce it throughout the search. As a result, the parameter $\mathcal {N}$ in the proposed method is deterministically updated during the search, beginning with a value close to one and decreasing throughout the search according to the changing time (or iteration number), as described in⁹⁰ as follows:

  $$\begin{aligned} \mathcal {N} = 1 - C_t \end{aligned}$$

  (20)

  where $\mathcal {N}$ is the value of the distance bandwidth at iteration t, and the value $C_t$ is computed as in Eq. 21.

  $$\begin{aligned} C_t = \frac{t^{\frac{1}{K}}}{{T}^{\frac{1}{K}}} \end{aligned}$$

  (21)

  where t denotes the iteration currently running, T is the maximum number of iterations, and K is a positive constant of 2 used to progressively reduce the value of $\mathcal {N}$ to a value close to 0 in the final stage of the search⁷³. Eq. 20 illustrates how the value of the parameter $\mathcal {N}$ can be deterministically adapted during the search time t in relation to the parameter $C_t$. In $\beta$-hill climbing, the parameters $\mathcal {N}$ and $\beta$ are introduced to enable the algorithm to manage the pace of exploration and exploitation, respectively. The convergence rate will be increased since the rules governing these parameters are crucial for optimizing the solution vector during the search process. The initial iteration of $\beta$-hill climbing involves parameter tuning, whereby the $\mathcal {N}$ and $\beta$ operators are established beforehand and remain constant during the search. Finding the right parameter values for each type of optimization problem necessitates a thorough sensitivity analysis, which is the first disadvantage of the parameter-tuning procedure. Convergence rate is the second disadvantage, where a lot of iterations are needed to discover the most suitable solution. Over the course of iterative loops of the proposed MSSA algorithm, the adaptive mathematical formulas specified in Eqs. 20 and 21 are repeatedly updated. The time t and lifespan, T, of the MSSA algorithm, which stand for the current and maximum number of iterations, respectively, are utilized to exponentially update these parameters. The values of $\mathcal {N}$ and $C_t$ of MSSA, which are implied in Eqs. 20 and 21, respectively, are schematically illustrated in Fig. 3 over 1000 iterations. These settings have a big impact on search functionality and are two of primary control parameters of the proposed MSSA algorithm. In MSSA, the parameter $C_t$ is one of the most important and prominent parameters, as it helps to strike a balance between exploration and exploitation aspects of the search space. The number of iterative loops rises because of this parameter, which is shown as a function of time to control the random movement of search agents iteratively. To improve the exploration and exploitation capabilities of the algorithm developed and decrease the search speed, this parameter is used to reinforce the dynamic system of convergence. With this setting, search agents can scavenge more foraging space and use every region while searching for food sources. The objective is to achieve an effective convergence process that can enhance MSSA’s ability to address ELD problems. In addition, the parameter $\mathcal {N}$, the bandwidth, has a noticeable effect on the convergence property of MSSA. Another important parameter in MSSA is the parameter K with a value of 2, which helps $C_t$ accomplish its task by promoting the features of exploration and exploitation.

• 2. $\beta$-operator: Here the nearby solution that the $\mathcal {N}$-operator produces is the current solution. According to the probability $\beta$, where $\beta \in [0, 1]$, the variables in the existing salp position $U^j_i$ are reproduced from its feasible range: $U'^j_i \in [U_i^{min},U_i^{max}]$. Let the parameter $\beta$ denotes the improvement degree of the selected search agent (i.e., salp), where $\beta$ represents a control parameter that determines the size of local changes in the existing salp position. In A$\beta$HCO, according to Eq. 22, the value of this parameter can be deterministically adjusted during the search process.

  $$\begin{aligned} \beta _t = \beta _{min} + rand \times t \times \frac{\beta _{max}-\beta _{min}}{{T}} \end{aligned}$$

  (22)

  where $\beta _t$ is the probability’s value at time t, rand stands for a random value uniformly distributed between 0 and 1, $\beta _{max}$ is the final fitness of the best search agents after applying the local search and $\beta _{min}$ is its initial fitness before the local search. In addition, as stated in⁹¹, the value of the parameter $\beta$ in Eq. 22 is deterministically modified in the proposed method within a particular range of $[\beta _{min}, \beta _{max}]$. Each time a predetermined number of iterations (T) is reached, the degree of improvement of each LS operator is determined at each iteration loop of MSSA. After that, $\mathcal {N}$ and $C_t$ are recalculated to continue with the local improvement in the subsequent iteration. Through pilot testing for a sizable number of ELD instances, the values of the parameters $\beta _{max}$ and $\beta _{min}$ were selected for this investigation. This was conducted to broaden the search into more promising regions of the search field, while making sure that MSSA achieves a high degree of performance. In this study, the value of $\beta _{max}$ was determined empirically to be 1.0. Similarly, the value of $\beta _{min}$ was determined to be optimally 0.1. These numbers represent the best performance of MSSA on all the ELD problems studied in this work. However, if necessary, these settings can be changed to account for additional problems. Equation 22 is graphically shown in Fig. 4. According to the previous discussion, the adaptive learning strategy with the parameters $\mathcal {N}$ and $C_t$ as well as the $\beta$ parameter may engage in competition to increase the selection probability during the operation of A$\beta$HCO, in addition to working together to enhance the quality of search agents. An adaptive learning technique may be used to recalculate the selection probability of LS operators to encourage competition among them. This will further improve the efficiency of the LS operator with a larger fitness improvement with a higher likelihood of being selected for the next search agent refinement.

• 3. $\mathcal {S}$-operator: This is how the survival of the fittest rule is applied in the selection process. If $f(\varvec{{U}}'^j)\le f(\varvec{{U}}^j)$, the new salp position $\varvec{{U}}'^j$ will take the place of the old salp position $\varvec{{U}}^j$. In short, the child solution will be included and the parent solution will be excluded, if better. The selection process, referred to as $\mathcal {S}$-operator, can be described as presented in Eq. 23.

  $$\begin{aligned} \varvec{{U}}^j (t+1) = {\left\{ \begin{array}{ll} \varvec{{U}}'^j (t) & f(\varvec{{U}}'^j)\le f(\varvec{{U}}^j)\\ \varvec{{U}}^j (t) & Otherwise \end{array}\right. } \end{aligned}$$

  (23)

  where $\varvec{{U}}^j (t+1)$ denotes the global best position of the best salp at dimension j and iteration $t+1$, $\varvec{{U}}'^j (t)$ denotes the new best position of the best salp at dimension j and iteration t, and $\varvec{{U}}^j (t)$ denotes the previous position of the salp at dimension j.

Fig. 3

Proposed functions for the distance bandwidths in MSSA.

Fig. 4

Graphical representation of the parameter $\beta$ in MSSA.

Algorithm 2

A pseudo-code describing the key steps of proposed MSSA algorithm.

Algorithm 3

A pseudo-code describing the key steps of the A$\beta$HCO algorithm

The SSA algorithm, as a population-based method, plays a key role in exploring multiple regions of the search space simultaneously due to its leading and following operators, expressed in Eq. 18, as can be seen from the structure of the proposed method. The repair process is used to consider the feasibility of any generated salp. This enables the proposed method to handle only feasible regions of the ELD problem in the search space. The A$\beta$HCO is hybridized with SSA in the proposed method to take care of deep exploitation in any area in the search space where SSA converges, improving the convergence rate. Both SSA and A$\beta$HCO combined with the repair algorithm, are expected to efficiently explore feasible regions of the search space by achieving the right balance between exploration using SSA and exploitation using A$\beta$HCO. Although non-feasible regions of the search space are not considered, the global optima can be achieved from traversing the search space with non-feasible cases.

Time complexity

The performance of the proposed MSSA can also be analyzed in the context of the time complexity that is much needed for the overall pseudo-code written in Algorithm 2. As can be noticed, there are four main parts that that should be watched to compute the time complexity the) the evolution loop of SSA, ii) the time required for the repair algorithm, iii) the time demanded for A$\beta$HCO, and iv) the time demanded to compute the objective function of the ELD. Let T, the maximum number of iterations, serve as the stopping condition of SSA. Recall that the number of salps is N and the number of salp production units is ng. Hence, the basic SSA algorithm has a time complexity issue as presented in Eq. 24.

$$\begin{aligned} \mathcal {O}(SSA) = \mathcal {O}(T \times N \times ng) \end{aligned}$$

(24)

Each iteration of the repair algorithm has a time complexity issue as presented in Eq. 25.

$$\begin{aligned} \mathcal {O}(Repair\; algorithm) = \mathcal {O}(ng). \end{aligned}$$

(25)

The A$\beta$HCO will be invoked with a time complexity issue as presented in Eq. 26.

$$\begin{aligned} \mathcal {O}(A \beta HCO) = \mathcal {O}(Max_{t} \times ng) \end{aligned}$$

(26)

where $Max_t$ is the maximum number of iterations for the A$\beta$HCO method.

The time complexity issue of the A$\beta$HCO algorithm in Eq. 26 depends on the value of the parameter $B_r$.

Finally, depending on the problem at hand, the temporal complexity of the objective function varies. So, we will suppose it is as presented in Eq. 27.

$$\begin{aligned} \mathcal {O}(Objective \; function) = \mathcal {O}(obj) \end{aligned}$$

(27)

Finally, the time complexity of the entire MSSA algorithm in the worst-case scenario can be described as shown in Eq. 28.

$$\begin{aligned} \mathcal {O}(MSSA) = \mathcal {O}(T \times N \times ng^3 \times Max_{t} \times obj) \end{aligned}$$

(28)

During the first and last phases of the optimization process, the exploration and exploitation characteristics of the SSA algorithm were integrated with those of the A$\beta$HCO algorithm in the proposed MSSA algorithm. Since a balance must be struck between these two concerns, a hybrid between the two methods will ultimately improve performance, even if it increases the computational cost of the proposed solution. The proposed MSSA method may be hindered by the changes made to it, which might prevent it from smoothly stagnating into local solutions and ultimately determine the best evaluation of the solutions obtained during optimization. To put it briefly, the findings in Section 5 show that the proposed MSSA method is expected to exert its strength in solving the difficult ELD problems.

Experimental results and discussions

In this section, seven ELD problems with different numbers of generating units, load demands, and constraints are used to evaluate the proposed MSSA. These cases of ELD are generally used to assess other comparative methods in the literature.

Evaluation criteria

The two crucial assessment metrics for economic load dispatch optimization are economy and total actual power loss. The mean accuracy and standard deviation values using these performance metrics were computed for the proposed MSSA algorithm in comparison to other algorithms as described below:

• Minimization of fuel cost: One crucial factor for determining the system’s economic viability is its fuel cost. It is speculated that a quadratic function of generator real power output approximates the fuel cost curve as follows:

  $$\begin{aligned} F_i = \sum _{i=1}^{N} (a_i + b_iP_{Gi} + c_iP^2_{Gi})(\$/h) \end{aligned}$$

  (29)

  where N is the overall number of generator units; $a_i$, $b_i$, and $c_i$ are the fuel cost curve coefficients of a ith generator, respectively; and $P_{Gi}$ is the actual power output of a ith generator.

• Minimization of total real power loss: the stability and quality of the power system are directly impacted by the reactive power optimization outcome. Reactive power dispatch aims to reduce the transmission network’s actual power loss, which is illustrated in Eq. 2.

Table 1 provides an illustration of the characteristics of these test cases. The number of units included in each test case as well as the anticipated load demand for each test case are listed in this table. Finally, Table 1 also lists the constraints that must be followed in each circumstance. It is good to remember that there are four different kinds of constraint: (C1) power balance, (C2) generator capacity, (C3) ramp rate limits, and (C4) prohibited operation zones.

Table 1 The Characteristics of the ELD cases used in the experiments.

Matlab (R2014a) running on a laptop with an Intel i7 processor running at 2.8 GHz, 8 GB of RAM, and Windows 10 as the operating system used to conduct our tests. Each ELD case study is run 30 times using the proposed MSSA method. Furthermore, in every experiment, the population size is set to 30. The likelihood of calling the A$\beta$HCO optimizer is represented by the parameter $B_r$, which is explored using four different values of $B_r$ $\in$ (0, 0.01, 0.1, and 0.4). It is worth noting that increasing the value of $B_r$ leads to more calls to A$\beta$HCO, which in turn speeds up exploitation and requires more computation time. Since the A$\beta$HCO optimizer does not fire when $B_r = 0$, the original SSA method will be applied in this scenario. The maximum number of iterations needed to execute the proposed method with each value of the parameter $B_r$ realized in each case study is shown in Table 3. The number of iterations is sufficient to achieve the desired fairness and effect of balancing exploration and exploitation. However, the other parameter values of A$\beta$HCO are as follows: $\beta _{min}$=0.001, $\beta _{max}$=0.6, and K = 20, where these values are suggested in⁷³. It should be noted that the best outcomes are highlighted in bold in the following comparison tables (i.e., the lowest outcome is better). The ‘NA’ given in the following comparison tables indicates the results are not provided in the corresponding references. In comparison to cutting-edge algorithms, the performance of the proposed MSSA is evaluated. The names of the algorithms are displayed in Table 2 along with the abbreviations for each comparison technique.

These comparative methods can be categorized into heuristic (HU), meta-heuristic (MH), modified meta-heuristic (MM) and hybrid meta-heuristic (HM). The heuristic methods are designed to be ELD problem-dependent where the parameters and constraints of the ELD problem are adjusted. Examples of such methods are DVL-MILP and CE-SQP, where they tackled the ELD case based on integrating conventional methods with evolutionary algorithms. Although they are efficient for tackling small-sized optimization problems, they do not work well for large and complicated ELD cases. Therefore, they are used to assist other meta-heuristics in better search space navigation. On the other hand, meta-heuristics are general optimization frameworks designed to tackle different optimization problems with different complexities and sizes. They normally do not require a mathematical derivation in the initial search. They consider the problem input data, solution representation, objective function, and problem constraints as primary elements to perform optimization. Examples of the meta-heuristic tackled ELD are GWO, HS, GA, PSO, CS, and others, as shown in Table 2. The main dilemma of the original versions of these meta-heuristic algorithms is their inefficacy in controlling the problem search space complexity and ruggedness. Consequently, to remain consistent with the structure of the search space, their primary operators are either amended or replaced by other operators derived from other heuristics or meta-heuristics. In other enhancements, meta-heuristics are hybridized with other heuristics or meta-heuristics to complement each other for better performance. Most of the comparative methods in Table 2 are either modified or hybridized meta-heuristics. Kindly notice that the category of each comparative algorithm is presented in Table 2.

Table 2 Abbreviations for the comparative algorithms that were utilized.

Table 3 The total number of iterations employed for implementing the proposed MSSA algorithm on several cases studies.

3UG-850MW: 3-unit generator with load demand equal to 850 MW

The 3-unit generator used in this case study has a total projected load demand of 850 MW. The unit information is provided in¹⁴⁰. In this example study, transmission losses are ignored in favor of the VPL effects. The output of applying the proposed MSSA method to this instance of the ELD problem with four different values of the parameter $B_r$ is shown in Table 4. In this table, the best solution produced by the proposed algorithm after 30 runs is listed along with its total fuel cost, mean fuel costs, and standard derivation (Stdv). The results of the proposed method with various settings of the parameter $B_r$ are consistent in terms of the total fuel results, mean fuel cost, and standard derivation, as shown in Table 4. This is a result of the ELD problem being so straightforward in this case study.

Table 4 The results of the proposed MSSA algorithm on a 3-unit generator with total load demand of 850 MW.

For comparison’s sake, Table 5 compares the performance of the proposed MSSA with various comparable techniques from the literature. The figures in this table show the mean fuel cost and the minimum fuel cost that could be determined using these techniques. It has been noted that 14 out of the 16 different algorithms used to determine the ideal fuel cost (8,234.07 $/h) perform similarly to the proposed MSSA in terms of performance degree. Moreover, MSSA and nine of the competitive algorithms seem more robust than GA-PS-SQP, NSS, and QIPSO by obtaining the same best solution over 30 runs. This is again due to the simplicity of the case study.

Table 5 Comparison results on a 3-unit generator with total load demand of 850 MW.

6UG-1263MW: 6-unit generator with load demand equal to 1263 MW

The generator used in this case study has six units, and the total load demand for all those units is 1263 MW. These units’ data were collected from¹⁴¹. It should be noticed that when tackling this instance of the ELD problem, the ramp rate limits and banned operating zone constraints are considered. The analysis of the effects of the parameter $B_r$ on the performance of the developed MSSA using four alternative values (0.0, 0.001, 0.1, and 0.4) is shown in Table 6. According to the total cost of fuel, the developed algorithm’s best result is highlighted using bold fonts and shown in Table 6. This table also includes illustrations of the mean costs and the standard derivation, the total output power of the best solution, and the transmission loss ($P_L$) of the best solution. Table 6 shows that when the value of the parameter $B_r$ is raised, the performance of the developed method steadily improves. When the value of the parameter $B_r$ is set between 0.1 and 0.4, the developed approach yielded the best result with the lowest overall fuel cost (15,444.19 $/h). More precisely, compared to other versions of the proposed method with different $B_r$ values, the developed approach performs better when the value of $B_r$ equals 0.4. Several factors support this conclusion, including the mean costs, output power, transmission loss ($P_L$), and standard derivation. Table 7 compares the performance of the proposed MSSA algorithm with a variety of different techniques. Based on the lowest overall fuel cost (15,442.20 $/h), the IGWO algorithm performs better than any of the other comparable approaches. The proposed MSSA did, however, achieve the fifth rank lowest overall fuel cost (15,444.19 $/h). The proposed method’s findings differ by 1.99 $/h from those of the IGWO algorithm. Additionally, the proposed MSSA performs better than the original SSA version; the performance gap between the two algorithms is 4.16 $/h. Considering the mean cost of the proposed MSSA, it is clear that it ranks third by getting the lowest outcomes, which indicates that the proposed MSSA appears to perform more robustly than 13 out of 16 competitive algorithms by achieving almost the same results over 30 runs. In addition, the proposed MSSA is ranked fifth based on $P_L$ results. This proves the superiority of the proposed MSSA by getting advanced positions against the other comparative algorithms when used to solve the current case study of the ELD problem.

Table 6 The results of the proposed MSSA algorithm on a 6-unit generator with total load demand of 1263 MW.

Table 7 Comparison results on a 6-unit generator with the demand of 1263 MW.

13UG-1800MW: 13-unit generator with load demand equal to 1800 MW

A 13-unit generator and an anticipated total load demand of 1800 MW are used in this example study. The specifics of the unit generator are provided from citewalters1993genetic. In this instance of the ELD issue, the VPL impacts are taken into account, but the transmission loss is disregarded.

Table 8 contains the experimental results of executing the proposed MSSA with various values of $B_r$. The best solution found using the proposed algorithm and its associated fuel costs are listed in Table 8. This table also includes the mean cost and the standard derivation results. The findings in Table 8 demonstrate that the proposed method performs better when the value of $B_r> 0$ than when the value of $B_r = 0$. This is because when the value of $B_r=0$, the A$\beta$HCO is not initiated. This demonstrates how the A$\beta$HCO may emphasize the proposed algorithm’s convergence tendency as it navigates the ELD problem’s search space, producing improved results. On the other hand, the proposed approach performs better than other variants of the proposed algorithm when the value of $B_r$ is set to 0.01. This is such that the minimal total fuel cost (17,960.40 $/h) may be attained when $B_r=0.01$. Additionally, the algorithm performs poorly when the value of $B_r> 0.01$. This demonstrates that 0.01 is the threshold value of the parameter $B_r$ that may strike the proper balance when combining the local search capability of A$\beta$HCO with the global search capability of SSA, leading to improved results. In contrast, it can be observed that MSSA with higher $B_r$ values has higher algorithmic stability than those with smaller $B_r$ values. This is supported by the standard derivation and mean cost results. To demonstrate the effectiveness of the proposed MSSA algorithm, its performance is compared to that of cutting-edge techniques. Table 9 provides examples of the outcomes of the comparison techniques. It is obvious that the proposed algorithm performs better than the basic SSA algorithm. Additionally, out of 26 rival algorithms, the original algorithm’s minimal fuel cost is lower than five of them (namely, DHS, IHS, MPDE, NUHS, and THS). The best solution is (17,960.37 $/h), while the proposed technique can produce the third lowest fuel cost (17,960.40 $/h). Notably, the best results of the other rival methods and the results of the proposed algorithm differ by just 0.03 $/h. Reading the results given in Table 9 once again, it can be observed that the proposed MSSA is placed in the sixth position according to the mean cost results, while the difference between the mean cost of the proposed algorithm and the lowest mean cost obtained by the other comparative algorithm is 5.23 $/h.

Table 8 The results of the proposed MSSA algorithm on a 13-unit generator with total load demand of 1800 MW.

Table 9 Comparison results on a 13-unit generator with total load demand of 1800 MW.

13UG-2520MW: 13-unit generator with load demand equal to 2520 MW

In this example study, 13-unit generators with a 2520 MW load demand are used. These units’ data are collected from the same source as case study 3’s, which is¹⁴⁰. The transmission loss is disregarded in this case study while the VPL effects are taken into account.

The analysis of the effects of the parameter $B_r$ on the performance of the proposed MSSA method for this case study using four different values ($B_r$=(0, 0.01, 0.1, and 0.4)) is shown in Table reftab:13UnitsResults2520. The best solution found using the proposed algorithm, its total fuel cost, mean cost and standard deviation are all listed in this table. The performance of the proposed method when the value of $B_r> 0$ performs better than the proposed algorithm for the value of $B_r=0$, as can be seen from Table 10. This is because when the value of $B_r$ is set to 0, the A$\beta$HCO is not active, and as a result, the proposed method has problems with its potential in the exploitation phase. The performance of the proposed algorithm, however, beats other variants of the proposed algorithm by reaching the lowest overall fuel cost of (24,164.11 $/h) for the value of $B_r= 0.1$. Additionally, the standard derivations in Table 10 demonstrate the proposed algorithm’s robustness. It can be shown that the proposed algorithm that has $B_r = 0.1$ is more reliable than the other variants of the proposed algorithm. This is because with this value, the proposed method can properly balance the search process’s exploration and exploitation.

Table 11 compares the outcomes of the proposed MSSA with those of the original SSA and other comparative methods. It is evident from the findings listed in Table 11 that the proposed MSSA algorithm performs better than the original SSA algorithm. In addition, when compared to other comparable methods, the proposed algorithm has found the third lowest fuel cost (24,164.11 $/h). The NUHS and THS algorithms produce the best minimal fuel costs, and there is a 0.05 $/h gap between their output and the best findings that have been published. Furthermore, the proposed MSSA has the second lowest mean cost, while the/betaHC-GWO algorithm has the lowest mean cost. The gap between the outcomes of the two algorithms is 0.22 $/h. This demonstrates that the proposed method is an alternative promising method capable of solving this ELD case study.

Table 10 The results of the proposed MSSA algorithm on a 13-unit generator with total load demand of 2520 MW.

Table 11 Comparison results on a 13-unit generator with total load demand of 2520 MW.

15UG-2630MW: 15-unit generator with load demand equal to 2630 MW

The fifteen-unit generator that makes up the ELD system under consideration in this case study is estimated to produce 2630 MW of total load demand. In this case study, the operating zone constraints and ramp limits are both upheld. The information for the unit generator is taken from¹⁴¹. Table 12 provides a summary of the findings from the investigation into the effectiveness of the proposed MSSA method using various values of $B_r$. The solution with the lowest overall fuel cost, the mean costs, and the standard derivations are all listed in this table. It is also important to keep in mind that Table 12 provides the total output power and the transmission loss ($P_L$) of the optimal solution. In general, the performance of the proposed method when the parameter $B_r$ has a value other than zero is better than the performance of the proposed algorithm when the value of $B_r$ equals zero, as shown by the findings shown in Table 12. This is because the A$\beta$HCO is disregarded when the parameter $B_r$ has a value of zero, and as a result, the source of the local search is not used. But when the value of the parameter $B_r$ is equal to 0.1, the proposed method yields the lowest overall fuel cost (32,707.49 $/h). In addition, the proposed MSSA with $B_r$ = 0.1 seems more stable than the other variants of MSSA according to the mean costs and standard derivation outcomes. According to the $P_L$ results, the proposed MSSA with $B_r$ = 0.1 achieved the lowest results than the other variants of the proposed algorithm. Again, this demonstrates that 0.1 is the threshold value of the parameter $B_r$ that may strike the correct balance between the proposed algorithm’s exploration and exploitation capabilities.

Table 13 compares the proposed MSSA’s findings to those of existing comparative algorithms. As can be observed from the findings listed in this table, three of the comparing algorithms CS, CS-CLM, and CLCS-CLM obtained the best solution with the lowest fuel cost (i.e., 32,704.45 $/h). However, the proposed MSSA obtained the second lowest overall fuel cost (32,707,49 $/h), with the best-published results differing by 3.024 $/h from the results of the proposed method. Additionally, the proposed algorithm performs better than the native version of SSA, with a 306.78 $/h performance gap between the two algorithms. Furthermore, the proposed MSSA achieves the second lowest mean cost, while the lowest mean cost is obtained by the CLCS-CLM and CS-CLM algorithms. The gap between the outcomes of the proposed MSSA and the other two algorithms is 26.28 $/h. The proposed MSSA has got the second lowest $P_L$, while HBF has obtained the lowest $P_L$.

Table 12 The results of the proposed MSSA algorithm on a 15-unit generator with total load demand of 1263 MW.

Table 13 Comparison results on a 15-unit generator with the demand of 2630 MW.

40UG-10500MW: 40-unit generator with load demand equal to 10,500 MW

The ELD problem examined in this case study involves a generator with 40 units and a total load demand of 10,500 MW. With the VPL effects in place and the transmission loss disregarded, this case study is regarded as a middle-sized system. The unit generator’s data was collected from¹¹⁰. Table 14 provides a summary of the effect of the parameter $B_r$ on the performance of the proposed MSSA algorithm. The best solution obtained by applying the proposed algorithm is shown in this table, along with its total fuel cost, mean costs, and standard derivation. It is clear from this table that the performance of the proposed algorithm when the parameter $B_r$ is greater than zero performs better than when $B_r$ is equal to zero. This is because when the value of $B_r$ is equal to zero, the A$\beta$HCO is disregarded. The proposed algorithm, however, produced the best result with a minimal total fuel cost of (121,415.45 $/h) when the value of $B_r$ was equal to 0.1. Additionally, compared to other variants of the proposed approach with various $B_r$ values, the proposed algorithm performs better when $B_r$ is set to 0.1. That is based on the standard derivation values. This demonstrates that the parameter $B_r$ threshold value of $B_r=0.1$ is required to strike a proper balance between the proposed algorithm’s exploration and exploitation capabilities. Table 15 presents the findings of the proposed MSSA in comparison to those of the forty-eight comparable methods. The findings shown in this table show that the HGWO algorithm yields the optimum solution with the lowest overall fuel cost (i.e., 121,412.00 $/h). While the proposed MSSA outperforms 32 of the 46 competing algorithms in terms of performance. It should be noticed that the results of the proposed method differ by 3.45 $/h from those of the HGWO algorithm. Additionally, the proposed algorithm surpasses the native SSA algorithm in terms of performance. The difference between the two algorithms is 1,459.08 $/h which is a very significant difference.

Reading the results given in Table 15 once again, the proposed MSSA excels 33 out of 47 algorithms in terms of the mean costs. This proves the superiority of the proposed MSSA as an alternative solution approach for the ELD problems.

Table 14 The results of the proposed MSSA algorithm on a 40-unit generator with total load demand of 10,500 MW.

Table 15 Comparison results on a 40-unit generator with total load demand of 10,500 MW.

80UG-21000MW: 80-unit generator with load demand equal to 21,000 MW

The effectiveness of the presented MSSA is assessed in this section using a sizable dataset and the eighty-unit generator. The total demand for all units’ generators is 21,000 MW. As shown in case study 6, the system under consideration in this case study is updated by replicating it with a 40-unit generator. The system in question has the most local optima and the broadest solution search area. Therefore, further work is required to solve this problem optimally. It should be emphasized that while the transmission loss is ignored in this case study, the VPL effects are kept.

The analysis of the effect of the parameter $B_r$ on the performance of the proposed MSSA yielded the findings shown in Table 16. The solution with the lowest overall fuel cost, the mean costs, and the standard derivations are all listed in this table. It can be seen from this table that the presented method performs better when the value of the parameter $B_r$ is not equal to zero than when it is equal to zero. Again, this is because when the value of $B_r$ is not equal to zero, the A$\beta$HCO is activated. On the other hand, when the value of $B_r$ is equal to 0.1, the developed method yields the lowest overall fuel cost. Furthermore, the performance of the proposed MSSA with $B_r$ =0.1 seems more stable than the other variants of MSSA with different values of $B_r$ according to the mean cost outcomes. This demonstrates that 0.1 of the $B_r$ is required to achieve better outcomes by striking the proper balance between the exploration and exploitation capabilities.

In Table 17, the outcomes of the proposed MSSA algorithm are contrasted with those of sixteen approaches. The minimal total gasoline cost as well as the average cost are shown in this table. Based on the lowest overall fuel cost (242,794.73 $/h), it is evident from this table that the MPDE and CLCS-CLM algorithms outperform the others. However, the proposed MSSA outperforms the other nine algorithms in terms of performance. It should be noted that the results of MSSA varied by 82.24 $/h from the best-published findings. Additionally, the MSSA algorithm performs better than the basic SSA algorithm, with a 6,296.46 $/h difference in results between the two algorithms. On the other hand, the performance of the proposed MSSA performs better than 12 out of 17 algorithms given in Table 17 in terms of the mean cost. This is a piece of new evidence that the proposed MSSA is a new alternative algorithm to solve ELD problems.

Table 16 The results of the proposed MSSA algorithm on an 80-unit generator with total load demand of 21,000 MW.

Table 17 Comparison results on 80-unit generator with total load demand of 21,000 MW.

The hybridization of SSA and A$\beta$HCO in the early and late stages of the MSSA optimization process helped identify the global optimal solutions to the ELD issues under consideration, extending the exploration and exploitation aspects of MSSA. MSSA’s hybridization greatly aids in the development of a respectable equilibrium between exploration and exploitation. The MSSA salps are therefore constantly skilled at scouting and using the search domain to find both fixed and mobile prey. Accordingly, over the course of the iterations in the MSSA process, the salps’ location is regularly updated. The salps in MSSA should be able to explore and take use of the surrounding environment in the search space under inquiry to advance to the global solution.

Comparative findings and analysis of results

This paper presents a hybridized version of the Salp Swarm Algorithm (SSA) for solving the Economic Load Dispatch (ELD) problem, which is a crucial optimization problem in electrical engineering. ELD is non-convex, multi-modal, and constrained by various operational limits, making it a challenging problem to solve. In this study, we propose the Memetic Salp Swarm Algorithm (MSSA), which combines the SSA with the Adaptive $\beta$-hill Climbing Optimizer (A$\beta$HCO) to refine the solution in two stages: general refinement through SSA and local refinement through A$\beta$HCO. The SSA serves as the ‘gene’ in biological evolution, while A$\beta$HCO represents the ‘meme’ in cultural evolution. These two search agents work together to enhance both the generality of the solution and its specificity to the problem at hand. We evaluate the MSSA algorithm on several ELD problems with different sets of constraints, including load balance, output, prohibited operating zones, and ramp rate limits. The method is tested on five real-world generator cases with varying numbers of units and load demands, ranging from a 3-unit generator with 850 MW to an 80-unit generator with 21,000 MW. The comparative results show that MSSA outperforms other algorithms in terms of feasibility and practicality, providing promising solutions for complex ELD problems.

It is evident from the results presented in Tables 5, 7, 9, 11, and 13 for small ELD instances, and 15 and 17 for large ELD instances, that the performance of the proposed MSSA algorithm produced reasonable solutions in comparison to all other comparative algorithms in these case tests. Additionally, in the small ELD case studies (case 1: 3 units, case 2: 6 units, case 3: 13 units, case 4: 12 units, and case 5: 15 units), the results of the proposed MSSA algorithm were comparable to the best-published findings obtained by some of the other competing methods. Lastly, in the remaining cases (case 6: 40 units and case 7: 80 units), the proposed MSSA algorithm received relatively very low fuel costs. This demonstrates how promising the proposed MSSA algorithm’s efficiency in solving ELD problems is for both small and large ELD cases.

Once more, it is clear from carefully examining the results in the tables above that the proposed MSSA algorithm performs better than all the competing algorithms when it comes to solving small ELD problems (cases 1–5) than when it comes to solving large ELD problems (cases 6 and 7). As was previously noted, case 1 is a straightforward case study because it involves a generator with only three units; thus, a highly effective optimization technique is not required to achieve the best results. Likewise, cases 2, 3, 4, and 5 include generators with 6, 13, 13, and 15 units, respectively; hence, they are straightforward case studies that do not require a very effective optimization technique to get the best outcomes. Conversely, case studies 6 and 7 include generators with 40 and 80 units, respectively; hence, these are intricate case studies that require a very effective optimization technique to achieve top performance. Nevertheless, the following conclusions may be drawn from the comparison of the proposed MSSA algorithm with the various comparable methods discussed above: 1) MSSA outperformed numerous competitors in cases 1 and 2, which are classified as small-scale ELD problems; 2) MSSA superiorly outperformed numerous competitors in cases 3, 4, and 5, which are classified as medium-scale ELD problems; and 3) MSSA outperformed numerous competitors in cases 6 and 7, which are classified as large-scale ELD problems. Furthermore, by providing the best reported findings in the numerous test cases classified as small-, medium-, and large-scale ELD problems the performance of the proposed MSSA method is comparable to that of some other prospective competitors. This determines the ability of the proposed MSSA method to solve several power system test cases by examining the search spaces for the ELD problems under study and achieving extremely competitive outcomes. This is accomplished during optimization by finding the ideal balance between exploration and exploitation capabilities.

Convergence analysis

Figure 5 illustrates the convergence behavior of the proposed MSSA with different values of the parameter $B_r$ on several case studies of the ELD problem.

Fig. 5

The convergence behavior of the proposed MSSA using different settings of the parameter $B_r$ on different ELD problems.

The convergence curves shown in Fig. 5 consider four distinct versions of the proposed method with four different values of $B_r$. The number of iterations is shown by the x-axis in these plots, where the number of iterations is carefully selected to show the convergence patterns between the various iterations of the proposed method. In contrast, the entire fuel cost ($/h) is shown by the y-axis.

Figure 5 shows that the convergence behavior of the proposed method is encouraging in every scenario examined, where the MSSA method produced satisfactory outcomes. For several examples, including 15UG-2630MW and 40UG-10500MW, the slope of the proposed MSSA is quite high in the first half of iterations, while in case 6UG-1263MW, the slope of the proposed algorithm is very sluggish in the second half of iterations. It indicates that the process of refining the proposed MSSA method in instance 6UG-1263MW begins at the beginning of the search process and continues until the very end.

When the 3UG-850MW and 6UG-1263MW instances are employed, all variants of the proposed method behave substantially identically, as shown in Fig. 5. This is because of the tiny size of these ELD situations, which makes it possible for the proposed algorithm to get outcomes that are close to optimal with little effort. However, the proposed MSSA performs better when the value of $B_r$ is set to a value greater than zero than when the value of $B_r$ is zero, as in the current ELD problems. Furthermore, in the early stages of the search process, the results are closer to near-optimal solutions the higher the value of $B_r$. The reason for this is that the rate of exploitation increases with the value of the parameter $B_r$.

Furthermore, Fig. 6 shows the notched box plots that are used to illustrate the differences between the execution of the proposed algorithm with various values of $B_r$ in several case studies of the ELD problem. Variations of the proposed method, each with a distinct value of $B_r$, are shown on the x-axis. The total fuel cost is shown on the y-axis ($/h). The MSSA’s stability is indicated by the slight difference between the best, median, and worst outputs. The four variants of MSSA can provide the same results after 30 runs, as can be seen from the plot of 3UG-850MW. On the other hand, when the value of $B_r$ is zero in the remaining case studies, there is a significant gap between the best, median, and worst results produced by the proposed method. This indicates that MSSA performance is more stable when $B_r$ is non-zero than when it is.

Fig. 6

The boxplots of the total fuel cost over 30 runs for the proposed MSSA on different ELD problems.

The mean fuel cost of the proposed MSSA and the classical SSA is displayed using a radar chart, as seen in Fig. 7. The results of these two algorithms are taken from Tables 5, 7, 9, 11, 13, 15, and 17. In this figure, each vertex in the same is associated with one of the ELD problems considered in this research. The shape with the smaller size means better results, where the lowest mean fuel cost is desired. From Fig. 7, it can be observed that the two algorithms have almost the same mean fuel costs for the small ELD problems, while it has slight differences between the mean costs of the two algorithms for the large-scaled ELD problems (i.e., 40UG-10500MW, and 80UG-21000MW).

Fig. 7

Radar graph for the proposed MSSA and the basic SSA based on the mean cost findings for all case studies of ELD problem.

The proposed MSSA can give extremely beneficial results for the ELD instances employed in the experiments carried out in this study, as can be shown from the results. This is so that MSSA may strike the right balance between exploration and exploitation of the search space. The MSSA is a mix of A$\beta$HCO as a meme and SSA as a gene. The exploration procedure is controlled because the SSA may travel the prospective ELD search space areas broadly and utilize the generality. On the other hand, A$\beta$HCO can thoroughly explore any search space that SSA can go to and identify the local optima. The local optima found by A$\beta$HCO will motivate the SSA’s operators to investigate highly promising search space areas through different values for $B_r$, which regulates density of utilizing A$\beta$HCO in the optimization framework of SSA and thus speed up the convergence.

Recalling that $B_r$ determines the density of invoking A$\beta$HCO in MSSA. Apparently, from the value of $B_r$ where different configurations to this parameter are studied for all experimental cases ($B_r=(0.0, 0.01, 0.1, 0.4)$), the most optimal solutions are obtained when the value of $B_r=0.01$. This is because the solutions of the ELD are in the deep positions of any search space regions. The algorithm does not require a dense invoking of A$\beta$HCO because a small local optimal solution in each region can attract the other solutions to the same quality due to the MSSA follower’s operator. However, when the value of $B_r=0.00$, the MSSA does not invoke A$\beta$ HCO, thus the local exploitation is not fully activated. Therefore, the results of the ELD are not efficient.

The above results add significant relevance to these case studies in the practical operation of power systems, making the study more comprehensive and relevant to the industrial sector. This will demonstrate the effectiveness of MSSA in addressing other highly constrained optimization problems in the industrial sector. Further study will explore extending MSSA as a multi-objective optimization method to address the ELD problem as a multi-objective optimization problem.

Finally, the MSSA can compete well and can produce very impressive results with ranks of first, fifth, third, second, fourteenth and seventh for 3UG-850MW, 6UG-1263MW, 13UG-1800MW, 13UG-2520MW, 15UG-2630MW, 40UG-10500MW and 80UG-21000MW, respectively, through extensive comparisons with 72 state-of-the-art methods using the same ELD cases.

Conclusion and future works

This paper has highlighted the power of intelligent swarm-based models in solving challenging engineering problems. Assigning the generating units with a certain load output to reduce fuel costs while considering several equality and inequality restrictions is known as Economic Load Dispatch (ELD). ELD is treated as a non-convex, non-linear, restricted optimization problem with a rough search space in the context of optimization. In this paper, a hybridized version of salp swarm algorithm is proposed for ELD using seven experimental cases that imitate real-world situations. Two versions of the Salp Swarm Algorithm (SSA) are experimented with: the original version of SSA and the proposed memetic SSA (MSSA). In MSSA, the adaptive $\beta$-hill climbing optimizer (A$\beta$HCO) is hybridized in the SSA as a new operator to improve the exploitation power of SSA. The mixture of experts when combining local search with global search adds more capabilities of exploitation in digging deeper into the complex search space. In MSSA, the original SSA served as a Darwinian gene while A$\beta$HCO served as a Dawkins’ meme. Both gene and meme notation have intelligently combined the balance between generality (exploitation) and problem specificity (exploration).

To verify the performance of the proposed MSSA, seven cases of the ELD problem vary in terms of the number of generator units, load demand, and constraints. The sensitivity analysis to show the effect of the control parameters on the convergence behavior of MSSA is carried out. In conclusion, for problems with the low number of generators, the results were similar between the SSA and the MSSA. This is due to the relative ease of the optimization problem under those circumstances. As the problem gets bigger and more local minimums appears, it was necessary to switch to the MSSA where the local optimizer is used. Furthermore, ELD is a constraint optimization problem that is highly sensitive to the number of generators and the feasibility of the solution area. The SSA showed competitive results compared to comparative methods. However, it was evident the MSSA showed higher capabilities of exploiting the search space. This is evident from the mean values of the several runs obtained under different load conditions and numbers of generators. Displaying the results in more detail, we find that the performance of the proposed MSSA is like SSA and other comparative algorithms for the smallest ELD problem with a 3-unit generator with a total load demand of 850 MW.

The performance of the MSSA is better than the SSA in the second ELD problem with a 6-unit generator with a total demand of 1263 MW. Whereas the MSSA is ranked fifth and the SSA is reported to be in the eleventh position in comparison with a total of 16 comparative algorithms. The MSSA has ranked second by getting the second lowest load demand against 25 other comparative algorithms on a 13-unit generator with a total load demand of 1800 MW, while the SSA is placed in twenty-one positions. The MSSA is placed third by obtaining the third lowest load demand in comparison with 13 other comparative algorithms on a 13-unit generator with a total load demand of 2520 MW, while the SSA has achieved twelve positions. The MSSA has obtained the second-lowest results against 14 comparative algorithms on a 15-unit generator with a demand of 2630 MW, while the SSA is placed last by getting the worst results. The MSSA is placed in the fourteenth position in comparison with a total of 47 comparative algorithms on a 40-unit generator with total load demand of 10,500 MW, while the SSA has achieved forty-five positions. Finally, the MSSA has the seventh position against 16 other comparative algorithms on large-scale ELD case studies including an 80-unit generator with a total load demand of 21,000 MW, while the SSA has the penultimate position.

The boxplots for the optimum fuel cost support the convergence analysis of SSA using different values of $B_r$. Recalling $B_r$ is the control parameter used to determine the density of using A$\beta$HCO in MSSA. Values of $B_r$ larger than 0.0 had better values and more reliable numbers. The value of $B_r=0.1$ is shown to be the most suitable one and not $B_r=0.4$. If this has meaning, it means that the MSSA is at the end a global swarm intelligence optimizer that prefers only moderate injections of local search that are vital to its performance. Larger values of $B_r$ may take the algorithm more into local search and degrade its performance.

Due to the Superior performance of MSSA on ELD problems, future work may include empowering the MSSA and the SSA with more operators for information exchange and increasing their ability to diversify their search capabilities without hindering their nature as a global optimizer. Hybridization with other evolutionary methods and augmenting their operators’ domain will enhance their mechanism as complex problems solvers. Moreover, utilizing the implicit parallelism that the salps have in parallel or distributed models will also improve and speed up their performance. The SSA and the MSSA can be tested in dynamic ELD problems and different combinatorial optimization problems to demonstrate more capabilities. Internet of Vehicles (IoV) in smart cities¹⁴², truss structures¹⁴³, multi-energy microgrid^5,7, and the Aircraft Landing Optimization Problem^12,13 can be topics of future research. The solution here would be a sequence of locations in the search space with spatial and temporal characteristics satisfying some constraints and initial conditions. Mathematical modeling for both the SSA and the MSSA can be the subject of future research. Literature lacks a theoretical convergence analysis of these methods. Furthermore, the sequential adjustment approach used in the repair process of this research may introduce biased corrections that tend to favor earlier units over later ones. As a potential future improvement, exploring proportional distribution methods¹⁴⁴ or linear programming (LP)-based correction strategies¹⁴⁵ could help ensure greater fairness and diversity in solution adjustments, making this a promising direction for further research Finally, the A$\beta$HCO used as a local-based search in MSSA can be further improved by investigating the liability of other parameter control mechanisms, such as adaptive and self-adaptive, instead of a deterministic one¹⁴⁶.