Presenting a Novel Evolutionary Method for Reserve Constrained Multi-Area Economic/Emission Dispatch Problem

Abstract

Economic dispatch (ED) attempts to find the most cost-effective combination of power generation units while meeting operational constraints. Another problem that can’t be resolved by standard economic dispatch problems is figuring out the method of generating dispatch that would be most cost-effective in meeting the local demand without exceeding the tie-line capacity. Making a trade-off between fuel costs and environmental concerns, a contentious problem in industrialized countries, seems essential. As a result, this study introduces a multi-objective approach for different ED problems, such as multi-area emission economic dispatch (MAEED) and reserve constrained multi-area emission economic dispatch (RCMAEED), when there are real-world restrictions present, like the valve point effect (VPE), prohibited operating zones (POZs), multi-fuel operation (MFO), and ramp-rate (RR) restrictions. In this study, the generation cost and emissions are taken into consideration as objective functions. Since the MAED problem in the power system is inherently nonlinear, adding the aforementioned restrictions makes the problem even more challenging. To address the complexity of the multi-objective optimization problem, the modified grasshopper optimization (MGO) algorithm, based on the chaos mechanism, is proposed in this paper. The proposed method has been tested on a four-area power system with sixteen electrical generators, and the results are contrasted with those of previous evolutionary techniques. Based on the results, it can be concluded that using the proposed MGO method to solve the MAED and RCMAED problems will result in generation costs that are around $300 and $600 less than using the MPSO and PSO methods, respectively. Also, the proposed MGO method has reduced emission levels by roughly 30% as compared to the GO method in order to solve the RCMAEED problem.

1. Introduction

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. Recent literature about MAED problem.

Ref.	Year	Method	Objective Function		Constraints
			Emission	Generation Cost	VPE	POZ	MFO	Ramp-Rate	Reserve	Tie-Line Capacity
[11]	2013	ABCO	-	√	√	√	-	-	-	√
[12]	2014	TLBO	-	√	√	√	√	-	-	√
[13]	2016	QOGSO	-	√	√	-	√	-	-	√
[14]	2016	ISFS	-	√	√	√	-	-	-	√
[15]	2020	PBOA	-	√	-	√	-	-	-	-
[16]	2020	MSSA	-	√	-	-	-	-	-	√
[17]	2022	MOSSA	√	√	√	-	√	-	-	√
[18]	2022	IMCO	√	√	√	-	√	-	-	√
[19]	2018	PSO-SFL	√	√	√	√	√	√	-	√
[20]	2022	MODGWO	√	√	-	-	-	-	-	√
[21]	2022	CSOA	√	√	-	-	√	√	-	√
[22]	2020	PSO-GWO	√	√	√	√	-	√	-	√
[23]	2017	NSOS	√	√	√	-	√	-	-	√
[24]	2019	CRO	√	√	√	-	-	-	-	√
[25]	2016	DE	-	√	√	-	-	-	√	√
[26]	2021	FA	-	√	√	√	-	√	√	-
[27]	2014	PF-HDSM	-	√	-	-	-	-	√	√
My research		MGO	√	√	√	√	√	√	√	√

. 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.

1.3. Contribution

The following is a presentation of the study’s primary contributions.

• Developing a mathematical solution to the types of ED problems, including MAED, MAEED, and RCMAEED, that takes the objective functions of generation cost and emission into account.
• Taking into account ramp rate limit, tie line capacity, VPE, POZ, MFO, and spinning reserve when addressing the MAED problem to increase the realism and applicability of the simulation results in the actual power system analysis.
• Introducing the modified grasshopper optimization (MGO) algorithm, which uses a chaos mechanism to avoid local optima stagnation and enhance convergence qualities. Additionally, a fuzzy concept-based Pareto technique is suggested for finding non-dominated solutions to solve the MAED in the multi-objective framework.
• Evaluating the efficacy and robustness of the proposed method by comparing the results of MGOA with those of other well-known optimization approaches.

1.4. Paper Organization

The remainder of the article is arranged as follows:

Problem formulation and system restrictions are both compromised in Section 2. An overview of the optimizer framework is provided in Section 3. The acquired results and simulations are included in Section 4. In Section 5, the conclusion is presented.

2. Problem Formulation

The primary goal of the MAED optimization problem in power systems is to minimize the overall cost of producing electrical energy for all loads, with or without minimizing total pollution emissions, while meeting certain limitations. The following is a description of the restrictions associated with mathematical modeling of the considered objective functions.

2.1. Objective Functions

2.1.1. Minimizing System Generation Cost

The single area economic dispatch problem’s goal function [28,29] can be stated as:

$$\mathrm{Min}{\displaystyle \sum }_{n=1}^N(F_n\left(P_n\right))$$

(1)

where:

1:

$F_n(P_n)=\left\{\begin{array}{l}{a}_{n1}{P}_n^2+b_{n1}{P}_n+c_{n1}+\left|e_{n1}\times \sin \left(f_{n1}\times \left(P_{n,\min }^{}-P_n\right)\right)\right|, \mathrm{fuel} 1, P_{n,\min }\le P_n\le P_{n1}\\ \dots & \\ a_{nk}{P}_n^2+b_{nk}{P}_n+c_{nk}+\left|e_{nk}\times \sin \left(f_{nk}\times \left(P_{n,\min }^{}-P_n\right)\right)\right|, \mathrm{fuel} k, P_{nk-1}\le P_n\le P_{nk}\\ \dots & \\ a_{nK}{P}_n^2+b_{nK}{P}_n+c_{nK}+\left|e_{nK}\times \sin \left(f_{nk}\times \left(P_{n,\min }^{}-P_n\right)\right)\right|, \mathrm{fuel} K, P_{nK-1}\le P_n\le P_{n,\max }\end{array}\right.$

2:

N is the number of accessible generation units, and the index of them is defined by n.

3:

The number of fuel types is defined by K, and the index of it is k.

4:

P_n is the output power of the nth unit, $P_{n,max}$ and $P_{n,min}$ are the nth unit’s maximum and minimum output power limits, respectively.

5:

For the kth type of fuel from the generation unit n, the quadratic generation cost function is $a_{nk}{P}_n^2+b_{nk}{P}_n+c_{nk}$.

6:

For the kth type of fuel, the cost function coefficients of the nth unit are a_nk, b_nk and c_nk.

7:

Due to VPE for the kth type of fuel from the generation unit n, the fuel cost function $\left|e_{nk}\times \sin \left(f_{nk}\times \left(P_{n,\min }^{}-P_n\right)\right)\right|$ is sinusoidal and non-smooth.

8:

The coefficients of cost function of the VPE model of generation unit n for the kth type of fuel are e_nk and f_nk.

The cost of power transmission over transmission lines must be taken into account by MAED’s cost function. Therefore, Equation (1) would alter as:

$$\mathrm{Min}{F}_T=\mathrm{Min}({\displaystyle \sum }_{n=1}^N(F_n(P_n))+{\displaystyle \sum }_{j=1}^M(f_j\left(T_j\right)))$$

(2)

where the number of lines in transmission system is defined by M. The cost function for line j is f_j, and the active power flow across line j is T_j.

2.1.2. Minimizing System Emission

The equation of the system’s emission function is as follows:

$$\mathrm{Min}{\displaystyle \sum }_{n=1}^N(E_n\left(P_n\right))$$

(3)

where:

1:

$E_n(P_n)=\left\{\begin{array}{ll}{\alpha }_{n1}{P}_n^2+{\beta }_{n1}{P}_n+{\gamma }_{n1},& \mathrm{fuel} 1, P_{n,\min }\le P_n\le P_{n1}\\ \dots & \\ {\alpha }_{nk}{P}_n^2+{\beta }_{nk}{P}_n+{\gamma }_{nk},& \mathrm{fuel} k, P_{nk-1}\le P_n\le P_{nk}\\ \dots & \\ {\alpha }_{nK}{P}_n^2+{\beta }_{nK}{P}_n+{\gamma }_{nK},& \mathrm{fuel} K, P_{nK-1}\le P_n\le P_{n,\max }\end{array}\right.$

2:

The emission produced by the unit n for the kth type of fuel is ${\alpha }_{nk}{P}_n^2+{\beta }_{nk}{P}_n+{\gamma }_{nk}$.

3:

For fuel type k, the coefficients of emission of unit n are ${\alpha }_{nk}$, ${\beta }_{nk}$ and ${\gamma }_{nk}$.

2.2. Constraints

2.2.1. Area Total Active Power Balance

The area q of the network’s total active power balance constraint, which ignores power losses in electrical system, can be written as [8,25]:

$${\displaystyle \sum }_{n=1}^{N_q}(P_n)=\left(P_{Loadq}+{\displaystyle \sum }_{w\in M_q}{T}_{qw}\right)$$

(4)

where, $P_{Loadq}$ is the area’s power demand, number of committed generating units is defined by Nq, and Mq is the collection of all zones connected to the qth zone via tie lines.

2.2.2. Power Limitations in the Output of Generators

The maximum and lowest generating powers of generator units are limited, as follows:

$$P_{n,\min }^{}\le P_n^{}\le P_{n,\max }^{},\hspace{1em}i=1,\dots ,N$$

(5)

2.2.3. Ramp-Rate Limitations

This restriction may be expressed as in (6) [30]:

$$\max (P_{n,\min }^{},\le P_n^0-D{R}_n)\le P_n\le \min (P_{n,\mathrm{m}ax}^{},\le P_n^0+U{R}_n)$$

(6)

where, $P_n^0$ is the output power of generating unit n in the preceding step, DR_n and Un_i are the nth thermal generator’s respective ramp-up and ramp-down rate limits. The goal variables’ lower and upper boundaries are established by this constraint.

2.2.4. Prohibited Operating Zones (POZs)

According to Figure 1, every generator truly has a distinct POZ that it has to refrain from making inside that range. Pumps, boilers, shaft bearings, and other device components could be harmed by the generation in these places [28,29].

$$P_n\in \left\{\begin{array}{l}{P}_{n,\min }^{}\le P_n^{}\le P_{n1}^l\\ \dots \\ P_{nh-1}^u\le P_n^{}\le P_{nh}^l\\ \dots \\ P_{nzn}^u\le P_n^{}\le P_n^{\max }\end{array}\right.$$

(7)

where, the index of the POZs of unit n is defined by h, and the minimum and maximum limits of the hth POZ of the thermal unit n are defined by $P_{ih}^l$ and $P_{ih}^u$, respectively. Zn determines the quantity of POZ in the thermal generating unit n’s input-output power curve. When performing optimization, the variables’ values are set to the lower or higher limitation of the POZ that is closest to their values.

2.2.5. Maximum and Minimum Transmission Power through Tie-Lines

The maximum capacity lamitation of transmission power of tie-line ($T_{qw,max}$) shall not be exceeded by the tie-line power flow from zone q to zone w (T_qw) [25].

$$\left|T_{qw}\right|\le T{q}_{qw,max}$$

(8)

2.2.6. Spinning Reserve (SR) Requirement in Each Area

In order to ensure the sufficiency and consistency of the support system in the event of an emergency, a rotating reserve should be set up in each location. Sharing reserves across multiple areas can improve the fulfillment of the requisite SR [8]. For the qth region, the following is a formulation of the reserve constraint:

$${\displaystyle \sum }_{n=1}^{N_q}{S}_n+{\displaystyle \sum }_{w\in M_q}R{C}_w\ge S_{q,req}$$

(9)

where, the reserve provided by each generation unit in zone q is called ${\displaystyle \sum _{n=1}^{N_q}{S}_n}$, which is equivalent to ${\displaystyle \sum _{n=1}^{N_q}{P}_n^{\max }-P_n}$, $S_{q,req}$ is the qth area’s SR demand, and ${\displaystyle \sum }_{w\in M_q}R{C}_{wq}$ is the total of reserves contributed from other zones to the qth zone.

2.2.7. Limits on Tie-Line Transmission Power with Contributed Reserves

The following is the the limitations of tie-line transmission power while allowing for contributed reserve $R{C}_{qw}$.

$$Tq{j}_{min}\le Tqj+R{C}_{qw}\le Tq{j}_{max}$$

(10)

It is important to note that the control variables are bound by themselves. The objective function and the rigid real power balance restrictions can be integrated as quadratic penalty expressions. Because of this, the objective function of the RCMAEED problem can be mentioned as follows:

$$\mathrm{Min}{F}_T=\mathrm{Min}\left(\begin{array}{c}{\displaystyle {\displaystyle \sum }_{n=1}^N}(F_n\left(P_n\right))+{\displaystyle {\displaystyle \sum }_{j=1}^M}(f_j\left(T_j\right))+\varphi \times {\displaystyle {\displaystyle \sum }_{n=1}^N}(E_n\left(P_n\right))+\cdots \hfill \\ \lambda \times \left|{\displaystyle {\displaystyle \sum }_{n=1}^N}(P_n)-P_{Load}\right|\hfill \end{array}\right)$$

(11)

where, $P_{Load}$ represents the zone’s overall active load demand., and f is a suitable number that the user will designate for the RCMAEED problem.

3. Proposed Optimization Methodology

This section provides an overview of the multi-objective optimization strategy, modified and conventional versions of the grasshopper optimization algorithm.

3.1. Grasshopper Optimization Algorithm

In 2017 [31], Sareimi et al. proposed a method to mimic grasshopper behavior. The larval stage and the adult stage are the two phases of a grasshopper’s life cycle. In the larval stage, grasshoppers primarily move by jumping and moving, however they move slowly and in small strides. The primary characteristic of adult grasshoppers is food hunting, which is crucial to the algorithm used by grasshoppers. Grasshoppers move quickly when they are exploring, and more slowly when they are developing around the food. In addition to looking for food sources, grasshoppers naturally do these two tasks. Mathematical simulations of grasshopper swarm behavior go as follows:

$$X_i=S_i+G_i+A_i={\displaystyle \sum }_{\begin{array}{c}j=1\\ j\ne 1\end{array}}^{N}s\left(\left|x_j-x_i\right|\right)\frac{x_j-x_i}{d_{ij}}-g\widehat{e_g}+u\widehat{e_w}$$

(12)

where X_i is the location of grasshopper i inside the swarm, S_i denotes the impact of grasshopper i receiving interactions from other mites, G_i denotes the ith grasshopper’s gravitational pull, A_i denotes wind advection, and u denotes the continuous drift, $G_i= -\mathrm{g}{\widehat{\mathrm{e}}}_g{, \mathrm{A}}_i=u{\widehat{e}}_w$, g stands for the gravitational stable, ${\widehat{e}}_g$ for the unit vector heading toward the center of the earth, and ${\widehat{e}}_w$ for the unit vector pointing in the same general direction as the wind.

$$S_i={\displaystyle \sum }_{i=1,j\ne 1}^{N}S(d_{ij})\widehat{d_{ij}}$$

(13)

where, $d_{ij}=\left|x_i-x_j\right|$ is used to indicate the separation between the two grasshoppers. The unit vector connecting two grasshoppers is ${\widehat{d}}_{ij}=\frac{x_i-x_j}{d_{ij}}$. i = 1, 2, …, n, the number of populations is defined by n, and j = 1, 2, …, D, D represents the dimensions. The aphid function that is impacted by the interaction force of other aphids is known as the S function. The phrase goes like this:

$$S\left(r\right)=f{e}^{-\frac{r}{l}}-e^{-r}$$

(14)

where the values of f and l, which represent the parameters of attraction scale and intensity, respectively, dictate how the parameters are distributed. typically use l = 1.5 and f = 0.5. It cannot be utilized to directly solve the optimization problem, according to Equation (12). The model is enhanced as follows to offer it significant global optimization capabilities:

$$X_i=c{\displaystyle \sum }_{i=1,j\ne 1}^{N}c\frac{u{b}_d-l{b}_d}{2}s(x_j^d-x_i^d)\frac{\left|x_j-x_i\right|}{d_{ij}}$$

(15)

Upon updating the location, the distance between each grasshopper is only allowed to be [1,4]. The calculation for the update parameter c is as follows:

$$c=c_{max}-l\frac{c_{max}-c_{min}}{L}$$

(16)

where, t is the current iteration count and T_max is the maximum number of iterations. Here, the maximum and minimum value of c are defined by c_max and c_min, respectively [31]. Typically, c_min = 0.00001 and c_max = 1.

3.2. Modified Grasshopper Optimization Algorithm

Similar to other population-based approaches, the GO algorithm has some significant drawbacks, including sluggish convergence and trapping in local optimum. Therefore, an efficient optimization technique Modified grasshopper optimization algorithm (MGO) is introduced in this study to enhance the exploration and exploitation processes of the traditional GO method.

A random mutation of GO is applied to improve exploration in order to discover new searching areas and avoid stagnation in the local optima. Updates to search agent positions in relation to the best response are the most effective way to increase exploitation. One can formulate a straightforward mutation vector as follows:

$$X_i^{new}=l_b+R\left(u_b-l_b\right)$$

(17)

where, R is a random number between [0, 1] and $X_i^{new}$ is a mutation vector. The current search agent is updated related to the optimal position to maximize the utilization of the GO method when there is changeable bandwidth.

$$X_i^{new}=X_{best}\pm r_2{K}_w$$

(18)

where, $r_2$ is a random number between [0, 1]. $K_w$ is a dynamic bandwidth variable that decreases as follows:

$$K_w=K_{max}{e}^{E\times T}$$

(19)

$$E=\left(\frac{\ln \left(\frac{K_{min}}{K_{max}}\right)}{T_{max}}\right)$$

(20)

where, $K_{max}$ and $K_{min}$, denote the maximum and minimum bandwidth limitations respectively, T and $T_{max}$ represent the current and maximum iteration numbers, respectively. In order to balance off exploration during the early phases of the search process with exploitation during the later stages of the search process, an adaptive coefficient is used, comparable to the pitch adjusting rate of the harmony search algorithm:

$$PAR \left(T\right)=PA{R}_{min}+\frac{PA{R}_{max}-PA{R}_{min}}{T_{max}} T$$

(21)

where, $PA{R}_{max}$ and $PA{R}_{min}$ denote the maximum and minimum pitch adjusting rates respectively, A diagram of the modified GO technique is shown in Figure 2.

3.3. Multi-Objective Optimization Strategy

This section presents the Pareto optimal method, the fuzzy decision strategy, and multi-objective problem formulation.

3.3.1. Fuzzy Membership Functions

This paper presents a fuzzy decision-making method based on trapezoidal fuzzy membership functions, which are shown as (22) and Figure 3 for each objective function [32,33].

$${\mu }_j\left(x\right)=\left\{\begin{array}{c}1 f_i\left(X\right)\le f_i^{min}\\ \frac{f_i^{max}-f_i\left(X\right)}{f_i^{max}-f_i^{min}} f_i^{min} \le f_i\left(X\right)\le f_i^{max}\\ 0 f_i\left(X\right)\ge f_i^{max} \end{array}\right.$$

(22)

where, ${\mathrm{f}}_{\mathrm{i}}^{\min }$ and ${\mathrm{f}}_{\mathrm{i}}^{\max }$ represent the minimum and maximum boundaries of $f_i\left(X\right)$, respectively. The vector of choice variables is called X, and ${\mu }_j$ is the fuzzy set for the ith objective function.

3.3.2. Pareto-Optimal Method

In most cases, a set of optimal solutions as opposed to a single one, are produced for addressing multi-objective problem that include the optimization of many conflicting objective functions. The Pareto-optimal solutions are those that are obtained. According to the following circumstances, the vector X₁ dominates the X₂ [34].

$$\forall i\in \left\{1‚2‚\dots N_{obj}\right\}‚ f_i\left(X_1\right)\le f_i\left(X_2\right)$$

(23)

$$\exists j\in \left\{1‚2‚\dots N_{obj}\right\}‚ f_j\left(X_1\right)<f_j\left(X_2\right)$$

(24)

where, N is the total number of control variables.

3.3.3. Best Compromise Solution

After obtaining all Pareto-fronts, Equation (25) is applied, the results are sorted based on fitness, and the best compromise solution (BCS) for the multi-objective problem is determined by the result with the highest value while simultaneously taking into account all objective functions.

$$N_{\mu }\left(j\right)=\frac{{{\displaystyle \sum }}_{k=1}^n{W}_k\times {\mu }_{fk}}{{{\displaystyle \sum }}_{j=1}^m{{\displaystyle \sum }}_{k=1}^n{W}_k\times {\mu }_{fk}}$$

(25)

The weighting factor for the kth objective function is denoted by k. Number of objective functions and non-dominated solutions are defined by n and m, respectively. Finally, $N_{\mu }$ represents a fuzzy set.

4. Results and Discussion

A test system consisting of 16 power plant units spread across four areas has been used to evaluate the proposed MGO technique’s efficacy in resolving the MAED, MAEED, and RCMAEED problems. The considered test system, which consists of sixteen electrical power generators over four areas, uses data from [35,36], including information on power producing units and restrictions related to minimum and maximum tie-line flow. The required power is 400 MW in zone 1, which includes zones 1 through 4, 200 MW in zone 2, which includes zones 5 through 8, 350 MW in zone 3, which includes zones 9 through 12, and 300 MW in zone 4 (compared with units 13 to 16). In this regard, a number of study cases have been defined, and the optimization problem in each case has been resolved using the proposed MGO algorithm as well as other study-related algorithms such as GO, PSO-SFLA [33], MPSO [32], SFLA [33], and PSO [32]. The results of the optimization have also been contrasted with those of other investigations. A quad-core laptop with a clock speed of 1.6 GHz and 4.0 GB of RAM is utilized to code and run the MGO algorithm as well as other algorithms used in this work. Moreover, the parameters are listed in

Table 2. Parameters of the proposed algorithms.

Parameters	MGO	GO	PSO-SFLA	MPSO	SFLA	PSO
Population size	100	100	100	100	100	100
Maximum iteration	200	200	200	200	200	200
$\mathrm{R}$	[0–1]	-	-	-	-	-
r1, r2	[0–1]	[0–1]	[0–1]	[0–1]	[0–1]	[0–1]
c1, c2	-	-	2	2	-	2
Number of groups	-	-	5	-	5	-

for each algorithm. The optimization results of this study are presented in three sub-sections. In Section 4.1, Section 4.2 and Section 4.3, MAED, single-objective RCMAED, and multi-objective RCMAEED problems are solved by MGO, GO, PSO-SFLA, MPSO, SFLA, and PSO methods, respectively. Also, the optimization results of these methods are compared with other methods from other studies.

4.1. MAED Problem

The single-objective MAED problem is solved in this section by the proposed MGO algorithm, and the outcomes of the suggested strategy are evaluated in comparison to other studies.

Table 3. The best solutions obtained for single-objective MAED problem.

Area No. (PD)		PSO [39]	HHS [36]	NFP [37]	CEP [35]	PS [38]	MGO
1 (400 MW)	P1 (MW)	150.0000	150.0000	150.0000	150.0000	150.0000	150.000
	P2 (MW)	100.0000	100.0000	100.0000	100.0000	100.0000	100.0000
	P3 (MW)	67.3660	66.8600	66.9700	68.8260	66.9710	67.31016
	P4 (MW)	100.0000	100.0000	100.0000	99.9850	100.0000	100.0000
2 (200 MW)	P5 (MW)	56.61300	57.0400	56.9700	56.3730	56.9718	57.07953
	P6 (MW)	95.4740	96.2200	96.2500	93.5190	96.2518	96.34877
	P7 (MW)	41.6170	41.7400	41.8700	42.5460	41.8718	41.86785
	P8 (MW)	72.3560	72.5000	72.5200	72.6470	72.5218	72.53403
3 (350 MW)	P9 (MW)	50.0000	50.0000	50.0000	50.0000	50.0020	50.0000
	P10 (MW)	35.9730	36.2400	36.2700	36.3990	36.2720	36.28298
	P11 (MW)	38.2100	38.3900	38.4900	38.3230	38.4920	38.50812
	P12 (MW)	37.1620	37.2000	37.3200	36.9030	37.3220	37.26609
4 (300 MW)	P13 (MW)	150.0000	150.0000	150.0000	150.0000	150.0000	150.0000
	P14 (MW)	100.0000	100.0000	100.0000	100.0000	100.0000	100.0000
	P15 (MW)	57.8300	56.9000	57.0500	56.6480	57.0510	56.9218
	P16 (MW)	97.3490	96.2000	96.2700	95.8260	96.2710	95.88068
Tie-line power flow	T12 (MW)	0.0000	0.0000	0.0000	−0.0180	0.0000	0.0000
	T13 (MW)	22.5880	16.8600	18.1800	19.5870	18.1810	17.42629
	T14 (MW)	−5.1760	0.0000	−1.2100	−0.7580	−1.2100	−0.1161317
	T23 (MW)	66.0640	70.6100	69.7300	68.8610	69.7300	70.51652
	T24 (MW)	−0.0040	−3.1100	−2.1100	−1.7890	−2.1110	−2.686341
	T34 (MW)	−100.000	−100.0000	−100.0000	−99.9270	−100.0000	−100.0000
${\displaystyle \sum P_g}$(MW)		1249.95	1249.29	1249.98	1247.995	1249.9982	1250.0
Cost ($)		7336.93	7329.85	7337.00	7337.75	7336.98	7337.0257

lists both the top global optimal solution for the four-area test system and the top global optimal solutions as described in the literature. The global optimal solution to which the suggested MGO reached is feasible (=1250.0 MW), as evidenced by comparisons to the findings reported in the literature by techniques like the classical evolutionary programming (CEP) method [35], hybrid harmony search (HHS) algorithm [36], network flow programming (NFP) method [37], and pattern search (PS) algorithm [38]. Additionally, the result produced by the MGO method is superior to that of the CEP method [35].

A comparison between the outcomes of the proposed MGO technique and algorithms utilized in this study has been conducted in

Table 4. Comparing the simulation final results for the MAED problem.

Methods	Best Solution	Mean Value	Standard Deviation	Time (s)
PSO	7344.35	7486.89	83.29	26.45
SFLA	7341.79	7559.77	74.1846	18.45
MPSO	7340.27	7637.44	71.89	20.20
PSO-SFLA	7340.45	7487.08	61.1165	19.31
GO	7341.71	7605.91	53.35	25.57
MGO	7337.02	7338.12	0.670	17.8

to determine how well the suggested method performs when compared to other algorithms used in this study, such as GO, PSO, PSO-SFLA, MPSO, and SFLA, for addressing the MAED problem. The best objective function values for 30 trials of all algorithms employed in this work for the MAED problem are displayed in Table 4 as the Best, Mean, and standard deviation (Std) indices. Among all the evolutionary algorithms used, it can be demonstrated that the MGO yields the best quality and most optimal outputs. The specifications of the algorithms used in this paper are presented in Table 3.

According to Table 3, the highest and lowest generation power of power plant units are related to areas 1 and 3, respectively. According to the results obtained from the proposed method, units 1 and 13 in areas 1 and 4 have the highest generation power with a production of 150 megawatts. Also, units 10 to 12 in Zone 3 with less than 40 megawatts of generation power have the lowest generation power compared to other power plant units.

Additionally, Figure 4 displays the cost function convergence graphs of the MGO and other evolutionary methods used in this study to address the MAED problem. This graph shows that, in comparison to other evolutionary approaches, the MGO algorithm’s convergence characteristics are stable and steady.

In order to show the ability of the proposed method in solving the MAED optimization problem, this time the evolutionary methods used in this study, including, GO, PSO, PSO-SFLA, MPSO, and SFLA, are tested on a small-scale network (a two-area power system comprising four electrical power generators), and the simulation results of the considered methods are presented in

Table 5. The best solutions obtained for single-objective MAED problem in the two-area system with four generating units.

Methods	P1 (MW)	P2 (MW)	P3 (MW)	P4 (MW)	T12 (MW)	${\displaystyle \sum {\mathit{P}}_{\mathit{g}}} \left(\mathbf{MW}\right)$	Cost ($/H)
HNN [40]	-	-	-	-	-	-	10,605.00
DSM [41]	-	-	-	-	-	-	10,605.00
SFLA	445.32	138.67	212.02	323.79	−199.99	1120.00	10,604.6742
MPSO	445.13	138.87	212.04	323.95	−199.99	1120.00	10,604.6741
PSO-SFLA	445.12	138.87	212.04	323.95	−199.99	1120.00	10,604.6742
GO	445.06	138.93	211.92	324.07	−199.99	1120.00	10,604.6742
MGO	445.22	138.77	211.99	324.00	−199.99	1120.00	10,604.6741

and compared with other references. The tie-line power flow limit and total power demand of the small-scale test system, which consists of two areas of power and four electrical generators, are 200 MW and 1120 MW, respectively. Data for this test system were taken from [40,41]. Area 1′s demand—which includes units 1 and 2—comprises 70% of the total demand, whereas area 2′s demand—which includes units 3 and 4—comprises 30% of the total demand [40,41].

Table 5 displays the findings from the Hopfield neural network (HNN) approach [40], direct search method (DSM) [41], SFLA, PSO, MPSO, PSO-SFLA, GO, and MGO. The table clearly shows that the MGO optimizer is used to produce the best solutions. The small-scale MAED optimization problem is performed more effectively by the MGO optimizer and MPSO algorithms than by any other methods. Table 5 shows that the MGO method’s minimal operating and fuel costs are 10,604.6741 ($/H), which is lower than the costs for HNN [40], DSM [41], SFLA, SFLA, and PSO-SFLA.

4.2. Single-Objective RCMAED Problem

Solving the RCMAED problem is one of the study’s primary objectives. All generating units’ operating constraints, the minimum and maximum limitations of tie-lines, and fuel cost objective function characteristics data are taken from [42]. The power requirements for the first region are 30 MW, for the second zone 50 MW, for the third zone 40 MW, and for the fourth zone 60 MW. 30% of the zone’s power demand, or 9 MW, 15 MW, 12 MW, and 18 MW for zones 1 through 4, is the SR requirement for each area. Therefore, the proposed MGO algorithm has been used to solve this optimization problem along with other evolutionary algorithms such as PSO-SFLA, MPSO, SFLA, and PSO.

Table 6. Optimization outcomes obtained by different methods for RCMAED problem for four-zone power system.

Power Output of Units	PSO	SFLA	MPSO	PSO-SFLA	GO	MGO
P1 (MW)	8.5605	8.7146	8.1347	9.9192	8.4680	11.1868
P2 (MW)	9.9835	10.0000	8.0110	7.2590	8.0110	9.9596
P3 (MW)	11.2600	12.3534	13.0000	8.7913	12.7035	6.6997
P4 (MW)	0.2616	0.0500	1.9639	4.8491	1.6050	2.9820
P5 (MW)	18.8070	22.1526	19.5629	23.6046	20.1371	22.8244
P6 (MW)	11.2610	9.4047	10.9592	9.2998	8.8157	8.8553
P7 (MW)	6.1084	3.8330	2.1895	3.9163	4.7532	3.9286
P8 (MW)	15.2128	17.2551	17.9250	16.3556	17.9250	17.2481
P9 (MW)	4.6622	6.0147	2.9329	6.1264	22.0733	5.8222
P10 (MW)	10.5192	6.1858	8.5093	0.0500	5.3173	0.1993
P11 (MW)	20.4782	9.1548	23.2033	9.8982	7.4855	9.6888
P12 (MW)	4.0058	17.2236	5.1026	22.3468	5.1026	22.8171
P13 (MW)	11.0000	8.1669	9.8282	9.1629	9.8282	8.0614
P14 (MW)	20.0000	19.7135	17.7366	19.9614	17.7366	19.8021
P15 (MW)	26.8153	28.7234	30.0000	28.1791	30.0000	29.8714
P16 (MW)	0.1384	1.2864	0.0500	0.0869	0.0500	0.0534
T12 (MW)	0.1275	0.9452	0.5512	0.1000	0.5512	0.4860
T13 (MW)	0.3016	0.2525	0.2146	0.2701	0.1433	0.2040
T14 (MW)	0.1000	0.1800	0.3539	0.3648	0.1430	0.1380
T23 (MW)	0.1000	1.4402	0.1105	1.4421	0.1105	1.4566
T24 (MW)	1.1012	1.8441	1.1357	1.8400	1.8756	1.8858
T34 (MW)	0.1000	0.1000	0.2364	0.1034	0.2364	0.1880
RC12 (MW)	1.6842	2.2985	0.3671	0.6544	2.5336	1.5336
RC13 (MW)	0.2249	0.1000	0.1000	0.8206	0.1000	0.5945
RC14 (MW)	1.8788	1.3163	2.1507	0.2931	1.0500	0.1674
RC23 (MW)	0.4049	0.2073	0.1000	0.1000	1.0058	0.1001
RC24 (MW)	0.1645	2.4179	1.0183	1.6712	1.0183	2.1473
RC34 (MW)	0.4582	0.1000	0.6020	0.2867	0.6020	0.3759
Reserve area 1 (MW)	18.9344	17.8820	17.8904	18.1814	18.2125	18.1719
Reserve area 2 (MW)	23.6108	22.3546	24.3634	21.8237	23.3690	22.1436
Reserve area 3 (MW)	80.3346	81.4211	80.2519	81.5786	80.0213	81.4726
Reserve area 4 (MW)	33.0463	33.1098	33.3852	33.6097	33.3852	33.2117
Best Cost ($)	2187.4179	2178.6024	2188.2467	2171.0535	2193.541	2166.3774
Mean Cost ($)	2700.3674	2634.0676	2461.5379	2494.3471	2510.7001	2185.7943
Std	363.4401	325.6323	204.1498	182.2067	250.0191	13.7298
Time (s)	69.34	66.06	54.48	52.27	77.31	51.93

displays the optimization results of the suggested MGO and other evolutionary methods. Looking at Table 6’s results reveals that the suggested MGO algorithm has come up with a more effective solution than the alternatives. For instance, there is around a $30 difference between the value of the cost function produced by the suggested MGO approach and the original GO method. Additionally, Table 6 shows that the suggested method’s standard deviation value is lower than that of other approaches, which supports the idea that the proposed algorithm may converge to the global optimum in various tests.

According to Table 6 and the obtained results of the MGO method, the highest and lowest generation power are related to power plant units 15 and 16, with 29.87 and 0.05 megawatts in area 4, respectively. Also, the comparison of transmission power on the lines shows that the highest transmission power is between areas 1 and 2 with 1.885 megawatts, and the lowest transmission power is between areas 1 and 4 with 0.138 megawatts. The lowest transferred reserve power is related to areas 2 and 3 with 0.1 MW, and the highest transferred reserve power is related to areas 2 and 4 with 2.143 MW.

Figure 5 depicts the convergence curve for total fuel cost optimization derived from several evolutionary techniques. The suggested MGO method converges to its optimal total fuel cost in fewer iterations than previous evolutionary approaches, as shown by Figure 4’s convergence characteristics of the objective function (ideal total fuel cost) of all algorithms used in this work.

Another feature of the suggested MGO way is the execution time compared to other ways, which can be proven according to the results of Table 6. Also, the low value of the standard deviation of the proposed MGO method compared to other evolutionary methods, such as SFLA, PSO, MPSO, and PSO-SFLA shows the superiority and accuracy of the proposed method in 30 different tests.

4.3. Multi-Objective RCMAEED Problem

The multi-objective RCMAEED problem is solved in this part by the optimization algorithms used in this study, including PSO-SFLA, MPSO, SFLA, and PSO. Fuel cost and emissions are objective functions of this optimization problem.

Considering that the problem is two-objective, unlike the previous two sections, the multi-objective RCMAEED problem cannot be solved using the theory of single-objective optimization. We are dealing with a collection of solutions rather than a single ideal one. As a result, for multi-objective optimization problems—which are described in Section 3.3—Pareto optimality is applied. To do this, considering that the objective functions of the study are not the same as the fuzzy method, their value is set between zero and one using Equation (22). Then, the set of answers that have the two conditions of Equations (23) and (24) are included in the set of Pareto answers and stored in the repository.

The non-dominated solutions are kept in a repository, and finally, Equation (25) is used to control the reservoir and obtain the set of optimal answers (w1 = w2 = 0.5). Finally, any answer set that has the highest ($N_{\mu }$) value is selected as the BCS.

The outcomes and optimal control variables characteristic for the multi-objective RCMAEED problem obtained from different algorithms used in this study are presented in

Table 7. Optimization outcomes obtained by various methods for multi-objective RCMAEED problem for four-zone power system.

Power Output of Units	PSO	SFLA	MPSO	PSO-SFLA	GO	MGO
P1 (MW)	8.2773	10.0000	8.2773	0.0500	10.0006	10.0005
P2 (MW)	5.4440	5.3234	5.4440	5.2116	5.3258	5.3259
P3 (MW)	6.9757	7.0561	6.9757	12.3788	7.0490	7.0491
P4 (MW)	7.4634	11.9980	7.4634	11.2161	11.9979	11.9978
P5 (MW)	21.3822	9.9619	21.3822	16.8448	9.9143	12.2994
P6 (MW)	7.3638	11.3019	7.3638	2.1170	11.2907	11.3626
P7 (MW)	7.6367	14.5624	12.0773	12.9769	14.5612	14.6209
P8 (MW)	18.0000	12.6441	14.0655	13.6205	12.7082	10.1910
P9 (MW)	17.3563	13.2916	17.3563	16.4181	13.2926	13.2921
P10 (MW)	0.0500	0.0891	0.0500	0.0500	0.0917	0.0919
P11 (MW)	13.6655	13.0317	13.6655	3.4361	12.6414	12.6180
P12 (MW)	8.1572	14.2427	8.1572	19.1587	14.6307	14.6544
P13 (MW)	9.2933	4.7521	9.2933	8.7209	4.7530	4.7537
P14 (MW)	12.6671	15.4968	11.2406	12.7486	15.4936	15.4934
P15 (MW)	17.3390	11.8130	17.3390	11.3640	11.8142	11.8143
P16 (MW)	20.0235	24.4354	20.0235	30.0000	24.4352	24.4350
T12 (MW)	−1.8289	1.1900	−1.8289	4.3668	1.1836	1.1836
T13 (MW)	−2.0350	−0.3652	−2.0350	−1.7739	−0.3653	−0.3653
T14 (MW)	1.7937	3.5526	1.0664	−1.7123	3.5550	3.5550
T23 (MW)	3.2182	0.1308	3.0699	0.6117	0.1288	0.1288
T24 (MW)	−2.1972	−0.4700	−0.0632	−0.9250	−0.4709	−0.4713
T34 (MW)	0.8420	0.4207	0.8420	0.8286	0.4199	0.4199
RC12 (MW)	−2.6122	0.4371	−2.6122	−1.1046	0.4376	0.4355
RC13 (MW)	1.8735	−3.5706	1.3435	0.4457	−3.4213	−3.3960
RC14 (MW)	3.3365	−5.3328	−10.8828	−7.5176	−5.4329	−4.6380
RC23 (MW)	−2.5379	0.2241	−0.9010	2.0858	0.2103	0.2071
RC24 (MW)	0.6483	−2.7136	0.6483	−1.8365	−2.0113	−3.0807
RC34 (MW)	−0.5127	−0.6214	−0.5127	−0.2932	−0.7097	−0.6741
Reserve area 1 (MW)	20.8396	14.6225	20.8396	20.1435	14.6267	14.6267
Reserve area 2 (MW)	20.6173	26.5297	20.1112	29.4408	26.5256	26.5261
Reserve area 3 (MW)	80.7710	79.3449	80.7710	80.9371	79.3436	79.3436
Reserve area 4 (MW)	31.6771	34.5027	33.1036	28.1665	34.5040	34.5036
Cost ($)	2197.8688	2185.4666	2194.0611	2189.6647	2185.0785	2184.0477
Emission (ton)	3.6756	3.4257	3.5176	4.2518	3.4288	3.4097

. Table 7’s findings show that the suggested MGO algorithm has found a set of more optimal solutions for two objective functions than other methods. The amount of fuel cost and emission functions in the BCS obtained from the proposed MGO method is $2184.66 and 3.407 tones, respectively, which is more optimal than the values obtained from other evolutionary methods.

The findings in Table 6 and Table 7 demonstrate that solving the problem of simultaneous optimization of two objective functions has led to a rise in the power plant’s fuel costs. For example, in the single-objective RCMAED problem, the fuel cost value discovered using the MGO approach is $2166.37, and the value of this function in the two-objective optimization has reached $2184.66 and is slightly away from its optimal value. The reason is the simultaneous optimization of two objective functions that are in conflict with each other. (In other words, the two objective functions do not improve together).

Figure 6 and Figure 7 display the Pareto fronts for the four-area power system that are obtained by MGO and GO methods and stored in the repository. The graphic displays a non-dominated collection of solutions for the considered test system that are relatively closely spaced. Due to the fuzzy framework used to solve the RCMAEED problem, the Pareto fronts produced are not perfectly hyperbolic in shape.

In addition to the explanations given about the performance of the proposed method in solving multi-objective problems according to Figure 6 and Figure 7, in Figure 8 and Figure 9 the performance of this method is evaluated graphically. These Figures show that the total cost for the best compromised solution, when considering cost and emission as independent objective functions, is $2184, which is between $2169 and $2199 in relation to the single objective RCMAEED optimization problem. Similarly, the best compromise solution’s emission is 3.409 tones. When cost and ENS are treated as separate objective functions, the corresponding values for the single objective RCMAEED optimization problem are 3.445 tones and 3.809 tones, respectively.

5. Conclusions

In this paper, a modified version of the grasshopper algorithm has been suggested and used to address various MAED problems. In order to prevent local optima stagnation and improve convergence characteristics, the proposed MGO algorithm makes use of chaos mechanisms. For three multi-area test power systems with non-smooth cost functions, the proposed MGO algorithm has been used to solve the best MAED, RCMAED, and RCMAEED optimization problems while considering the practical constraints of the generating units. Considering that the objective functions of fuel cost and emission in this study are contradictory, in order to achieve a satisfactory BCS over the obtained Pareto optimal solutions that meets the needs of the power system operator, a fuzzy membership function is used. The outcomes of the optimization have demonstrated that the MGO approach is superior to other evolutionary algorithms utilized in this study and the findings found in earlier literature in terms of the quality, dependability, and robustness of the generated optimization solutions. Furthermore, because the suggested method gives power system operators a wide range of options for quick decision-making, it encourages decisiveness. By comparing the acquired results with those in other publications, the effectiveness of the proposed technique in addressing the multi-objective RCMAEED problem is made obvious.

6. Future Study

Suggestions for future work in this field are as follows:

• The work can be expanded to include real-time testing systems as well as new market tactics.
• Simultaneous consideration of renewable energy sources and large-scale battery storage units in solving the MADE problem.
• Defining the reliability objective function in solving the MADE problem in order to reduce the unsupplied energy of subscribers. Because, the reliability assessment of power systems has become of the utmost importance as it plays crucial roles in modern power systems.