1.1. Motivation
Global energy consumption has depleted fossil fuel reserves and fueled economic analyses of already-existing fuel resources. Through planning the power plants economically in power networks to meet demand, effective usage of fossil fuel resources can be achieved. Doing so by providing system constraints is known as the problem of economic dispatch (ED), which economically has a major contribution to the study of the power system. Considering the operational limitations of power systems, the optimal combination of generation of power plant units in the most economical form can be provided through an ED model [
1,
2]. Some studies in the field of ED have also paid attention to environmental issues in addition to economic issues and have used combined heat and power units in the power system [
2,
3].
The power system typically consists of several interconnected areas. As a result, economic dispatch is expanded upon using multi-area economic dispatch (MAED) [
3]. The generators in MAED are divided into various power generation zones, which are linked to one another by tie lines. MAED determines the power generation within zones and the power exchange between zones to lower the overall fuel cost of all regions. Power generation capacity, power transmission capacity, and power load balance operational limitations are all simultaneously satisfied. In fact, receiving active power from other regions with more affordable power plants can reduce the total generation cost of multi-area power networks. Under these conditions, any fitness function, such as the overall fuel cost, is dependent on various settlements in specific regions, like tie-line limits, utility-adjusted policies, the cost of electricity, the demand for electricity, etc.
The majority of studies on the MAED tackle the problem from an economic standpoint, but doing this ignores other important issues of the power system, such as environmental impacts, which can have a destructive effect on the environment and increase pollution [
2,
3]. Solving the MAED issue solely for the sake of reducing operating costs is also not a viable option. The US Environmental Protection Agency has mandated for the last ten years that generation companies produce their required electricity not just at the lowest cost but also with the least amount of pollutants. In order to improve this situation, the MAED should be looked at as a Multi-Objective Optimization Problem that considers multiple objectives at once. The MAED can take into account the emission objective function, which directly addresses environmental issues. In order to achieve this, the suggested method examines the MAED as a Multi-Objective Optimization Problem (MOOP). To make the proposed study a benchmark, a number of practical constraints, such as ramp-rate restrictions and valve-point impact-prohibited operating areas, are taken into account. The ED is highly challenging when taking into account these constraints and various objective functions in multi-area networks. Consequently, a precise optimization algorithm must be used to solve the issue.
1.2. Literature Review
Over the years, a variety of optimization techniques using mathematical and evolutionary algorithms have been put forth to address the MAED problem, which is non-linear and non-convex [
4,
5]. A decentralized strategy based on a modified generalized Benders decomposition is suggested for the MAED problem [
6]. Xu et al. proposes a parallel primal-dual interior-point algorithm that uses a matrix splitting technique to create a completely distributed MAED with second-order convergence [
7]. The MAED problem is solved using the Newton-Raphson approach, considering tie line losses [
8]. The purpose of these methods is to provide mathematical methods in order to solve the MAED issue. However, The POZ, VPE, and various fuel type limitations, cannot be handled by these algorithms. The objective function’s non-continuity and non-derivability prevent these techniques from producing a robust optimal solution. Therefore, these approaches are poor candidates for resolving the MAED problem when POZ, VPE, and various fuel types are taken into account.
To address these issues, a variety of meta-heuristic methods have been employed [
9,
10] to handle optimization problems with intricate objectives and constraints. Artificial bee colony optimization (ABCO), which accounts for transmission losses, VPE, and POZs, is used to solve the MAED problem with tie line limitations [
11]. The advantage of the offered method is that it offers the MAED problem an ideal solution in both small and large test systems. The MAED problem with tie line constraints is solved by a teaching-learning-based optimization (TLBO) method that takes into account transmission losses, valve point effect (VPE), prohibited operating zones (POZs), and multi-fuel operation (MFO) [
12]. In [
13], a quasi-oppositional group search optimization (QOGSO) is presented for the MAED problem with VPE and MFO. For population initialization and generation hopping, the proposed QOGSO uses quasi-oppositional-based learning (QOBL). The MAED problem is solved by an improved stochastic fractal search (ISFS) while taking into account tie-line limits, area load demands, and various operating constraints [
14]. The ISFS includes an opposition-based learning technique for generation leaping as well as population initialization to strike a balance between exploration and exploitation. To more quickly identify workable alternatives, a novel repair-based penalty approach is described and included in the ISFS. Ikram et al. [
15] use the polar bear optimization algorithm to solve the MAED while taking into account the VPE, power load balance, power generation capacity, and power transmission capacity. The modified slap swarm method (MSSA) was utilized by Sharma et al. [
16] to tackle the single-zone and multi-zone ED problems. By incorporating random mutation, the MSSA enhances its exploration capability and prevents algorithmic stagnation. The review of the above studies shows that applied algorithms based on new mutation strategies have been presented in order to improve global search capability. At the same time, they have achieved good results in solving the MAED problem. However, the pollution objective function is not included in these studies, and the problem is solved as a single objective with the fuel cost function of the power plants [
1,
3]. On the other hand, not paying attention to the issue of pollution causes irreparable damage to the environment and living beings.
The MAEED problem is solved using a novel swarm intelligence technique called the multi-objective squirrel search algorithm (MOSSA) [
17]. The recommended method uses crowding distance, fuzzy clustering concepts, and an external elitist vault system to build a uniformly spaced Pareto optimal front curve. The improved competitive swarm optimization (IMCO) approach is recommended by Chen et al. to address the MAEED problem [
18]. A differential evolution technique is utilized to update and improve the winning particles after a ranking paired learning method is applied to increase the loser particles’ learning efficacy. To solve the MAEED problem, a hybrid evolutionary algorithm based on the PSO and shuffle frog leaping (SFL) is proposed [
19]. The feature of this study is to provide a fuzzy solution to make a compromise between the objective functions of emission and the total generation cost. Yin et al. suggest the multi-objective distributed grey wolf optimizer (MODGWO) to address the MAEED problem in a large-scale system [
20]. The outcomes demonstrate that, in comparison to centralized optimization, the proposed distributed approach can successfully secure information privacy when solving the multi-objective MAEED problem in a large-scale system. A swarm intelligence-based crow search optimization algorithm (CSOA) is presented to address the MAEED problem in the presence of renewable energy sources in order to enhance energy sustainability and climate benefits [
21]. Moreover, constraints including, transmission losses, MFO, VPE, and POZs are considered in solving the MAEED problem. The simulation findings show that it produces trustworthy outcomes when the necessary system constraints are included. A novel multi-objective method based on a combination of PSO and grey wolf optimization (GWO) is proposed in order to solve the MAEED issue, taking into account MFO, VPE, and POZs [
22]. Examining the results of the study shows that the presentation of the hybrid method can lead to a reduction in electricity costs when applied to real power networks because real power systems have huge dimensions with a small number of power plants. The New Symbiotic Organisms Search (NSOS) method, which is a fresh and effective variation of the (SOS) algorithm, is suggested to solve the MAEED problem [
23]. The modifications added to the original algorithm are the relationships that govern how the solutions are updated during the iterative process, how the parasitism phase is eliminated, and how the solutions are evaluated after each component has been updated. The results of employing the NSOS algorithm demonstrate that it outperforms the SOS method and other evolutionary algorithms. The MAEED difficulties are resolved using a more effective chemical reaction optimization (CRO) technique [
24]. A new encoding method and ego neighborhood structural steps are employed to enhance the algorithm’s search capability while maintaining population variation. The advantage of these studies [
17,
18,
19,
20,
21,
22,
23,
24] compared to previous studies [
9,
10,
11,
12,
13,
14,
15,
16] is to consider the emission function in solving the MAED problem. Another feature of these studies [
17,
18,
19,
20,
21,
22,
23,
24] is similar to previous studies [
9,
10,
11,
12,
13,
14,
15,
16], providing new exploratory algorithms based on strategies to balance local and global search. However, the spinning reserve constraints in these studies [
17,
18,
19,
20,
21,
22,
23,
24] is not considered similar to other previous studies [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16]. Usually, due to this condition, power plants can sell their reservation power in the reservation market in a few hours, similar to the electricity market. Also, in the above studies, considering that the problem is multi-objective. But multi-objective problem solving strategies such as the Pareto method based on fuzzy have been used less, and the optimization problem has been solved more than the weighting method.
The heuristic approach proposed in [
25] employs differential evolution based on time-varying mutations to address the RCMAED problem. Examining the results shows that the hybrid method has led to more optimal solutions than the PSO and DE methods. The non-linear and non-convex RCMAED problem is handled using an upgraded version of the fireworks algorithm (FA) that is outfitted with two efficient cross-generation mutation strategies [
26]. In addition, a new approach to addressing constraints is developed to rectify putative solutions in a workable search area. In [
27], a Penalty Function-Hybrid Direct Search Method (PF-HDSM) is suggested in order to address the problem of RCMAED while taking into account large-scale integration of wind power units. The feature of the above studies [
25,
26,
27] is the consideration of the spinning reserve constraint compared to the previous studies [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24]. However, in these studies [
25,
26,
27], the emission function is not taken into account when solving the optimization problem, and the problem is solved with a single objective. Furthermore, the real power system limits in solving the optimization problem, such as VPE, MFO, and POZs, are not incorporated in these studies [
25,
26,
27], making the derived results inaccurate.
A summary of the research done on the MAED challenge is given in
Table 1. This table enables a comparison of the characteristics of our method with those of other techniques reported in previous works.
Table 1 shows that the present study is one of the few that solves MAED with respect to the simultaneous optimization of the emission function and fuel cost of power plants and that all power system constraints, including VPE, MFO, POZ, and ramp rate, in addition to the spinning reserve constraint, are taken into consideration in the optimization problem-solving process.