Push Spread Algorithm: A New Metaheuristic Algorithm to Solve Cases in Economic Load Dispatch Problem

Cost becomes the most important issue in many studies related to the operational of various systems. Cost becomes the rationale consequence of the operation of any system. This cost can come from fuel or energy consumption [1], labor utilization [2], maintenance [3], emission handling [4], waste [5], time or opportunity lost [6], and so on. Management must focus on keeping this cost as low as possible for the continuity of the operation of the system as this cost becomes the main factor in the selling price. Keeping the cost low gives positive impact to make the price still acceptable and competitive. That is why minimizing cost has become the most famous goal in many optimization studies. This goal in minimizing cost can be found in many optimization studies in various fields, such as manufacturing [7], transportation [8], logistics [9], and power systems [10].

A well-known challenge in power system optimization is the economic load dispatch (ELD). This task involves determining the optimal allocation of generation levels for each unit in the network [11]. This cumulative power must be equal to the required demand which in general is known in advance. Each generator has its own operational range. The main or general goal of ELD problem is minimizing the total operational cost where each generator has its own cost structure. In addition, numerous aspects are taken into account when dealing with the ELD issue, including factors like the valve-point effect [12], restrictions due to ramping limits [13], forbidden operating zones [14], among others. On the other hand, each power system has its own mixture of generators where in some studies, the power system consists of generators from various energy resources, such as fossils, wind, water, nuclear, and so on. In many ELD studies, metaheuristic algorithm is often used as the optimization tool.

Many metaheuristics-algorithms that have been introduced in this recent decade are metaphor-inspired algorithms and developed root in swarm intelligence (SI) framework. The inspiration behind many metaheuristic techniques is often drawn from natural occurrences, especially the survival strategies of animals in reproduction and foraging activities. Several methods developed from this concept include the coati optimization algorithm (COA) [15], kookaburra optimization algorithm (KOA) [16], osprey optimization algorithm (OOA) [17], komodo mlipir algorithm (KMA) [18], golden jackal optimization (GJO) [19], fennec fox optimization (FFO) [20], fossa optimization algorithm (FOA) [21], salamander optimization algorithm (SOA) [22], among others. Several optimization paradigms have drawn inspiration from socio–cultural or human-centered dynamics, including the farmer–season framework (FSA) [23], motorbike courier-based optimization (MCO) [24], an enhanced variant of the social force method (MSFA) [25], the physician–patient model (DPO) [26], the deep sleep mechanism (DSO) [27], and the program manager-based approach (PMOA) [28], among others. In contrast, other studies emphasize approaches detached from metaphorical analogies, such as the fully informed search method (FISA) [29], subtraction–average optimization (SABO) [30], average–subtraction optimization (ASBO) [31], the adaptive grouping algorithm (AGA) [32], and golden search optimization (GSO) [33], among several additional proposals.

The rapid emergence of various new methods has been driven by multiple factors. One major factor is the absence of a single metaheuristic technique that consistently outperforms others across every type of optimization challenge. The second aspect is that there are numerous mathematical techniques that can be explored to create a new algorithm and there are numerous methods to implement these techniques. In addition to designing novel algorithms, researchers have often relied on mathematical approaches to refine or adjust existing metaheuristic methods. Such enhancements are evident in well-established techniques like the genetic algorithm (GA), particle swarm optimization (PSO), and various others. Another important point is the abundance of real-world optimization challenges across diverse domains, especially in engineering, which provide valuable testbeds for evaluating newly developed algorithms.

Although numerous advanced metaheuristic techniques have been introduced, this study puts forward a novel approach known as the push spread algorithm (PSA). This algorithm is metaphor-free so that it becomes the alternative for the many metaphor-inspired algorithms. There is critique regarding the metaphor-inspired algorithms as many of them hide their trivial novel mechanics by utilizing the metaphor to cover it. PSA is established root in SI framework so that it is a population-based technique where individuals become autonomous agents that act collectively but without any central command. Related to its name, PSA lays its effort on push and spread actions. The push action represents exploitation as the agent moves toward the entities that give higher probability for improvement. Contrary, the spread action represents the exploration where agent moves away from the convergence to find alternatives to avoid being trapped within the local optimal region.

Due to the goal and the unresolved problems which are previously explained, the following points highlight the scholarly value produced through this study.

• This paper presents a new metaheuristic-algorithm called PSA, which is established as metaphor-free algorithm and root in SI framework.
• PSA is evaluated by applying it to a collection of benchmark functions and by testing its performance on ELD tasks.
• The competitiveness of PSA is investigated by comparing its performance with five other algorithms including FSA, COA, OOA, KOA, and PSO.

The rest of this article is organized in the following way: Section 2 outlines and synthesizes recent research related to ED, while Section 3 introduces the proposed PSA, detailing its underlying idea, computational procedure, and mathematical framework. Section 4 presents the experiment that is conducted to investigate the effectiveness and competitiveness of PSA including the following results. Section 5 provides discussion related to the experiment result, findings, and limitations. Section 6 provides an overview of the key findings and outlines potential directions for upcoming research.

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Figure 1. Illustration of six actions in PSA

The visualization of these actions is provided in Figure 1. The black dot represents the agent. The red dot represents the biggest agent. The yellow dots represent the bigger agent. The blue dots represent the other agents. The black bars represent the boundaries.

Algorithm 1 together with Eqs. (1)-(23) describe the structured representation of PSA. While the algorithmic steps are outlined in the form of pseudocode, the mathematical operations underlying the procedure are expressed in Eqs. (1)-(23).

$X=\left\{x_1, x_2, \ldots, x_n\right\}$     (1)

$x_i=\left\{x_{i, 1}, x_{i, 2}, x_{i, 3}, \ldots, x_{i, m}\right\}$     (2)

Eq. (1) and Eq. (2) formalize the establishment of the swarm. Eq. (1) defines the establishment of the swarm that consists of certain number of agents. Eq. (2) defines the establishment of each agent that contains certain number of values where each value represents the position in a dimension.

$x_{i, j}=b_{l o w, j}+r_1\left(b_{u p, j}-b_{l o w, j}\right)$     (3)

$x_{b g t}^{\prime}=\left\{\begin{aligned} x_i, \min \left(f\left(x_i\right), f\left(x_{b g t}\right)\right) & =f\left(x_i\right) \\ x_{b g t}, \min \left(f\left(x_i\right), f\left(x_{b g t}\right)\right) & =f\left(x_{b g t}\right)\end{aligned}\right.$    (4)

Eq. (3) and Eq. (4) formalize the processes in the initialization phase. Eq. (3) defines the initial solution or value of the agent is uniformly located in the solution space. Then, Eq. (4) is used for the update of the biggest agent root in the value of the agent. The updating process of the biggest agent also occurs at every stage during the iteration.

$a_{1, i, j}=x_{i, j}+r_1\left(x_{bg t, j}-2 x_{i, j}\right)$     (5)

$P_i=\left\{\forall x_k \in X \mid \min \left(f\left(x_k\right), f\left(x_i\right)\right)=f\left(x_k\right)\right\} \cup x_{b g t}$     (6)

$a_{2, i, j}=x_{i, j}+\frac{\sum_{k=1}^{n\left(P_i\right)} r_1\left(p_{i, k, j}-2 x_{i, j}\right)}{n\left(P_i\right)}$     (7)

$x_{s e l 1, i}=r_2\left(P_i\right)$     (8)

$a_{3, i, j}=x_{i, j}+r_1\left(x_{s e l, i, j}-2 x_{i, j}\right)$    (9)

$x_{\text {sel} 2, i}=r_2(X)$    (10)

$a_{s 1, i, j}=r_3\left(x_{b g t, j}-2 x_{i, j}\right)$    (11)

$a_{s 2, i, j}=r_3\left(x_{s e l 2, i, j}-2 x_{i, j}\right)$    (12)

$a_{s 3, i, j}=r_3\left(x_{i, j}-x_{s e l 2, i, j}\right)$    (13)

$\begin{aligned} & a_{4, i, j} =\left\{\begin{array}{c}x_{i, j}+a_{s 1 . i . j}+a_{s 2, i, j}, \min \left(f\left(x_i\right), f\left(x_{s e l 2}\right)\right)=f\left(x_{s e l 2}\right) \\ x_{i, j}+a_{s 1 . i . j}+a_{s 3, i, j}, \min \left(f\left(x_i\right), f\left(x_{s e l 2}\right)\right)=f\left(x_i\right)\end{array}\right.\end{aligned}$     (14)

$W_i=\left\{\forall x_k \in X \mid \min \left(f\left(x_k\right), f\left(x_i\right)\right)=f\left(x_i\right)\right\}$    (15)

$a_{s 4, i, j}=\sum_{k=1}^{n\left(P_i\right)} r_1\left(p_{i, k, j}-2 x_{i, j}\right)$    (16)

$a_{s 5, i, j}=\sum_{k=1}^{n\left(w_i\right)} r_1\left(x_{i, j}-w_{i, k, j}\right)$    (17)

$a_{5, i, j}=x_{i, j}+\frac{a_{s 4, i, j}+a_{s 5, i, j}}{n(X)}$     (18)

$a_{6, i, j}=\left\{\begin{array}{l}x_{i, j}+r_1\left(b_{u p, j}-x_{i, j}\right), r_1<0.5 \\ x_{i, j}+r_1\left(x_{i, j}-b_{l o w, j}\right), r_1 \geq 0.5\end{array}\right.$    (19)

Eqs. (5)-(19) describe the framework of six possible operations available to every agent. Specifically, Eq. (5) introduces the initial operation, while Eq. (6) specifies the creation of a pool that gathers all larger agents together with the single largest one. Eq. (7) defines the second action. Eq. (8) defines the random selection of an agent from the pond containing bigger agents. Eq. (9) defines the third action. Eq. (10) defines the random selection of an agent from the swarm. Eqs. (11)-(13) specify three components of the fourth operation, namely moving in the direction of the biggest agent, advancing toward the selected agent, and shifting away from the selected agent. Eq. (14) defines the fourth action. Eq. (15) defines the establishment of the pond containing all smaller agents. Eq. (16) and Eq. (17) defines the two parts that are used in the fifth action, which are the passage toward all bigger agents and the passage avoiding all smaller agents. Eq. (18) defines the fifth action. Eq. (19) defines the sixth action.

$c_{1, i}=\left\{\begin{array}{c}a_{1, i}, r_1 \leq 0.33 \\ a_{2, i}, 0.33<r_1 \leq 0.67 \\ a_{3, i}, r_1>0.67\end{array}\right.$    (20)

$x_i^{\prime}=\left\{\begin{array}{c}c_{1, i}, \min \left(f\left(c_{1, i}\right), f\left(x_i\right)\right)=f\left(c_{1, i}\right) \\ x_i, \min \left(f\left(c_{1, i}\right), f\left(x_i\right)\right)=f\left(x_i\right)\end{array}\right.$    (21)

$c_{2, i}=\left\{\begin{array}{c}a_{4, i}, r_1 \leq 0.33 \\ a_{5, i}, 0.33<r_1 \leq 0.67 \\ a_{6, i}, r_1>0.67\end{array}\right.$    (22)

$x_i^{\prime}=\left\{\begin{array}{c}c_{2, i}, \min \left(f\left(c_{2, i}\right), f\left(x_i\right)\right)=f\left(c_{2, i}\right) \\ x_i, \min \left(f\left(c_{2, i}\right), f\left(x_i\right)\right)=f\left(x_i\right)\end{array}\right.$    (23)

Eqs. (20)-(23) define the establishment of the first and second candidates, including the update of the agent root in these candidates. Eq. (20) defines the establishment of the first solution candidate root in the first, second, or third. Eq. (21) defines the update of the agent root in the first candidate representing the first stage. Eq. (22) defines the establishment of the second solution candidate root in the fourth, fifth, or sixth action. Eq. (23) defines the update of the agent root in the second candidate representing the second stage.

The pseudocode of PSA is presented in Algorithm 1. Lines 2–5 describe the setup phase, while lines 6–13 outline the repetitive process. The outcome is the optimal solution obtained. In Algorithm 1, the initialization stage involves cycling through the entire swarm. During the iteration phase, two nested loops are present: the outer one continues until the maximum number of iterations is reached, and the inner one goes through all swarm members as each individual performs its search activity.

There are two more loops in every stage. During the initial iteration, the system attempts to identify every participant in the swarm to establish a pool. However, this detection is not assured, since in the opening stage the chance of it occurring is roughly 67%, while in the following stage the likelihood decreases to about 33%. Then, there is an additional loop in every searching process that runs for whole dimension. Root in this explanation, the complexity during the initialization is presented as O(nd) while the complexity during the iteration is presented as O(n²dT).

3.2 Model of the ELD problem

This work employs the standard ELD problem. In general, the goal is minimizing the total fuel cost or operation cost where the cost function of each generator is presented in quadratic function. Then, there are two constraints. The restriction on inequality ensures that every generator operates only within its designated power limits. In contrast, the equality condition requires that the overall generated power exactly matches the predetermined demand. The conventional formulation of the ELD problem is expressed mathematically in Eqs. (24)-(29).

$o_{E L D}=\min \left(e_{t o t}\right)$    (24)

$e_{t o t}=\sum_{n(G)} e\left(g_j\right)$    (25)

$e\left(g_j\right)=s_{1, j} g_j^2+s_{2, j} g_j+s_{3, j}$    (26)

$g_{\text {tot }}=g_{\text {demand }}$     (27)

$g_{t o t}=\sum_{n(G)} g_j$    (28)

$g_{\min , j} \leq g_j \leq g_{\max , j}$    (29)

The discussion that follows covers Eqs. (24)-(29). In Eq. (24), the main objective is expressed as reducing the overall expenditure. Meanwhile, Eq. (25) specifies that the sum of expenses from every generator constitutes this total cost. Eq. (26) defines the quadratic function as the cost function of each generator. Eq. (27) defines the equality constraint. Eq. (28) defines the accumulation of power from all generators to create the total power. Eq. (29) defines the inequality constraint.

In general, the findings indicate that PSA can be considered a viable approach for optimization tasks. The result shows that the exploration and exploitation capabilities of PSA are superior while its balancing competence between both exploitation and exploration is competitive. Meanwhile, PSA is absolute superior compared to FSA and PSO.

The performance gap among algorithms when applied to 23 benchmark functions is closely linked to the variation observed between their top and bottom outcomes. Specifically, for high-dimensional unimodal functions, six of them (F1–F6) exhibit a substantial disparity between the highest and lowest results, whereas F7 shows only a slight difference. The difference between the best and worst results is wide in three functions (F8, F11, and F12), moderate in two functions (F9 and F10), and narrow in F13. In the case of fixed-dimension multimodal functions, four functions (F14, F15, F17, and F18) exhibit a large gap between their highest and lowest outcomes, whereas six functions (F16, F19–F23) show only a small variation. This observation highlights the tendency for high-dimensional functions to produce widely varying results, while functions with a fixed dimension generally yield more consistent outcomes.

The superiority of PSA in handling high dimension functions can also be traced root in the nature of the terrain in high dimension functions. In general, the terrain of these high dimension functions is clear except in F8. Moreover, the global optimal also lies in the area in the center of the solution space, except in F8. In F8, the general terrain is very wavy and the trend for the location of the global optimal is ambiguous so that it is difficult to trace. Conversely, in multimodal problems with a fixed number of variables, the landscape tends to be less clear, typically resembling a level surface where the region containing the global optimum is extremely limited. It makes it difficult for the algorithm with stringent acceptance approach to improve its solution as the probability for improvement is low so that the probability of an agent staying in its current location is high due to high probability of stagnation.

Very competitive situation in the ELD problems can be related to the constraints. As provided in reference [12], the demand for the 6-unit system is 1,263 MW. Meanwhile the minimum total power is 300 MW while the maximum total power is 1,470 MW. It means that the demand is 86 percent of the maximum total power while the minimum total power is 26 percent of the maximum total power. It means that although the power range is wide enough, the actual solution space is narrow as almost all generators must be set near their maximum power. Moreover, the maximum power of generator G1 is 500 MW which is almost one-third of the maximum total power so that it is impossible for the power of G1 to be set low. The same circumstance also occurs in the 10-unit system. The demand is 616 MW. On the other hand, the minimum total power is 271.25 MW while the maximum total power is 1,094 MW. It means that the demand is 56 percent of the maximum total power while the minimum total power is 25 percent of the maximum total power. It makes the actual solution space of the 10-unit system narrow too although it is not so narrow as the 6-unit system. Wider actual solution space in 10-unit system rather than the 6-unit system makes the standard deviation of the result in 10-unit system is much higher than the 6-unit system. This fact shows that in the ELD problem, the actual solution space becomes more dominant than the algorithm which makes even the superior algorithm lose its superiority in handling ELD problems.