A Multiobjective Evolutionary Approach for Solving the Multi-Area Dynamic Economic Emission Dispatch Problem Considering Reliability Concerns

Abstract

Economic dispatch (ED) problems, especially in multi-area power networks, have been challenging concerns for power system operators for several decades. In this paper, we introduce a novel approach for solving the multiobjective multi-area dynamic ED (MADED) problem in the presence of practical constraints such as valve-point effect (VPE), prohibited operating zone (POZ), multi-fuel operation (MFO), and ramp rate (RR) limitations. Different objective functions including energy not supplied (ENS), generation costs, and emissions are investigated. The reliability objective, which has been less studied in economic dispatch area, distinguishes the proposed study from other studies. A compromise has been made from economic and reliability points of view. The MADED problem in the power system is inherently a complex and nonlinear problem, considering the operational constraint increments and the intricacy of the problem. Hence, the modified grasshopper optimization (MGO) algorithm based on a chaos mechanism is presented to prevent being trapped in local optima. The proposed method is tested on two systems including a 10 unit, 3-zone test system and a 40-unit 3-zone test system, and then, the outcomes are compared with those of other evolutionary techniques such as gray wolf optimization (GWO) and modified honey bee mating optimization (MHBMO). The simulation results demonstrate that the suggested strategy is successful in resolving both single-objective and multiobjective MADED problems.

1. Introduction

Economic dispatch (ED) is an important and prominent tool for power system operators in real-time power markets [1,2]. The conventional ED models are often well-known as static and single-area ED (SAED) which obtains the best configuration of in-service generation units for only one time interval. Such models cannot recognize the global optimal in multi-interval environment and multi-area power networks. One available alternative to cover these drawbacks is to investigate the ED problem as a multi-area dynamic ED (MADED). Extending the ED over multiple time intervals is called dynamic ED which takes into consideration the intertemporal constraint between time intervals. Moreover, considering different areas in the ED problem juxtaposed with tie-line restrictions turn the ED into a multi-area ED problem. In the MADED problem, the levels of power generation in each area as well as the exchanged active power between different areas are investigated for different time intervals in order to optimize specific objective functions while satisfy many equality and inequality constraints [3,4]. Multi-area static economic dispatch (MASED) is another type of MAED. In MASED, only the power generation for a constant load demand during a specific time period needs to be determined in MASED.

Most of the research in the ED, dynamic ED (DED) and multi-area ED (MAED) fields have evaluated the problem from economic points of view. However, solving the problem just from an economic point of view cannot take care of other issues, such as reliability and emissions in modern power networks.

Recently, utilities have been obliged to change their operation techniques for generating electricity at a reduced pollution level due to growing concern about environmental issues and the implementation of the U.S. Clean Air Act amendments of 1990 [5]. Therefore, the necessity of considering emission dispatching is undeniable especially in modern societies. The enhancement of reliability is a key concern in power system operation, and system operators work hard to keep it at a reasonable level. Reliability plays a crucial role in developed countries where customers do not accept any interruption in their services. With this in mind, the MADED problem must be solved as a multiobjective optimization problem that considers multiple objective functions at once. In this study, two efficacious objective functions including emissions and energy not supplied (ENS) are taken into consideration, in addition to the generation cost.

Various practical constraints including valve-point effect (VPE), multi-fuel operation (MFO), prohibited operating zones (POZs), and ramp rate limitations are taken into consideration to turn the proposed study into a benchmark. Due to the significance of these limits in a power system analysis, each of them is briefly detailed here [3,6]. Considering these constraints along with different objective functions in multi-area networks makes the ED very complicated. Therefore, the problem needs to be solved by using an accurate optimization algorithm.

1.1. Literature Survey

Throughout the years, different mathematical-based techniques have been used to solve various forms of ED problems, including linear programming (LP), nonlinear programming (NLP), quadratic programming (QP), and others [7,8]. The continuation and derivability of the objective functions are two drawbacks of these algorithms [9,10].

Recently, evolutionary algorithms based on random initial population have been introduced to cover these drawbacks. A particle swarm optimization (PSO) algorithm based on fuzzy was suggested for the ED problem considering FACTS devices with the aim of reducing loss and generation costs [11]. An adaptive fuzzy particle swarm optimization (AFPSO) algorithm was proposed for the ED problem in two large systems such as 118 and 354 generation units with the aim of reducing loss and generation costs [2]. A new self-adaptive comprehensive differential evolution (SACDE) algorithm was presented for the ED problem to minimize emissions and generation costs [12]. The grasshopper optimization (GO) algorithm was used separately and combined with the Harris Hawks algorithm to solve the combined heat and power ED problems, respectively [3,13]. The aim of these studies was to show the effectiveness of the GO method in solving the combined heat and power ED by considering the emissions and VPE limitations. A fuzzy multiobjective (FMO) method was presented for the DED problem aiming to reduce loss and generation cost in the presence of electric vehicles [6]. A combined method of the flower pollination algorithm (FPA) with quadratic programming was presented to solve the DED problem considering renewable generation, for example, wind and photovoltaic units, with the aim of reducing generation costs [14]. A meta-heuristic approach, on the basis of a combination of genetic and differential algorithms was suggested to solve the DED problem with the purpose of reducing generation costs in the presence of wind units [15]. The purpose of reviewing these studies is to investigate the effectiveness of new meta-heuristic methods such as FPA and GO which have been effective in solving the ED problem in static and dynamic frameworks.

A hybrid DEPSO algorithm by combining differential evolution (DE) and PSO was suggested for the MASED problem with non-smooth cost functions [4]. The simulation outcomes demonstrated that the DEPSO method was capable of converging to a high-quality solution. A novel distributed algorithm based on the Newton method (NMDA) was presented to solve the MASED problem to minimize generation costs while taking system and generator restrictions into consideration [16]. A hybrid method based on a combination of particle swarm optimization and shuffled frog leaping (PSO-SFL) algorithm was suggested to solve the MASED problem considering VPE and POZ to reduce generation costs and emissions [17]. A multiobjective squirrel search (SS) algorithm was suggested for the MASED problem with the goal of minimizing generation costs and emissions [18]. The recommended method used the crowding distance, fuzzy clustering principles, and an external elitist vault system to obtain a uniformly spaced Pareto optimal front curve. An improved competitive swarm optimization (ICSO) method was proposed for the MASED problem to minimize generation costs [19]. A differential evolution method was utilized to update and improve the winning particles, after a ranking paired learning strategy was employed in order to increase the learning efficiency of the loser particles. The squirrel search algorithm (SSA) was presented in [20] for the MASED problem to minimize generation costs, taking into consideration VPE and distributed generation units such as wind and solar. An improved random fractal (IRF) algorithm was proposed for the MASED problem due to the limitation of transmission lines in order to reduce generation costs [21]. In order to address population initialization and generation jumping, IRF proposes a counterproductive learning technique. A decoupled distributed crisscross optimization (DDCO) based on population cross generation was proposed for the MASED problem taking into consideration VPE to reduce generation costs [22]. Two evolutionary methods including fast convergence programming (FCP) and multiobjective grey wolf optimizer (MOGWO) were presented for the MASED problem to minimize generation costs and emissions [23,24]. The goal of FCP was to improve solution quality and convergence speed. A hybrid solution based on JAYA and teaching-learning-based optimization (TLBO) algorithms was suggested for the MASED problem taking into consideration MFO, VPE, and POZs [25]. The JAYA-TLBO algorithm updated individuals using JAYA and TLBO, and then retained the better ones. A hybrid ICA-PSO was suggested to reduce the MASED’s generation costs [26]. The suggested ICA-PSO approach takes advantage of both algorithms’ advantages. A modified slap swarm algorithm (MSSA) was used by Sharma et al. [27] to address the MASED problem. The MSSA avoids algorithmic stagnation and spontaneous mutation to enhance its exploration capability. The review of the above studies shows that the MAED problem is not dynamically modeled and simulated for a certain period of time in this research, making it impossible to generalize the results in real-world power systems. Because, in a real power system, demand and electricity prices vary with time. In addition, some of the aforementioned studies [16,19,20,24,27] have not considered any constraints such as VPE, POZ, MFO, and RR in solving the optimization problem, which makes the results far from the real power space. In terms of problem-solving strategy, most of the studies, except [17,18,23], have solved the problem in the single-objective framework. In other words, only a few brief studies have simultaneously considered the effect of power plant fuel cost function and contamination in solving their problem.

An enhanced fireworks algorithm (EFWA) was presented by Jadoun et al. [28] to address the MADED problem with VPE and MFO. In order to address the shortcomings of the fireworks algorithm, the EFWA adopted the limiting mapping operator and the adaptive dimension selection operator. An enhanced fireworks (EFW) algorithm was presented to address the MADED problem [29]. Two efficient cross generation mutation techniques were employed in EFW, also, a novel limit handling technique was applied to alter the available solutions. A hybrid PSO-GWO slap swarm algorithm (MSSA) was presented by Azizivahed et al. [30] for the MADED problem to minimize generation costs and emissions. The hybridization accelerates convergence and enhances the likelihood of obtaining the overall best solution. A quasi-oppositional group search (QOGS) optimization algorithm was presented for the MADED problem to minimize generation costs [31]. The quasi-opposition-based learning (QOBL) method has been used in the proposed method for generation jumping and population initialization. The review of the above articles shows that the MAED problem has been solved in a dynamic format, which is the advantage of these studies. However, these studies have not looked at the impact of the emission objective function in solving the optimization problem. Due to this limitation, power generation units must modify their output in order to satisfy the constraint under consideration while causing less environmental pollution. None of the studies have considered the effect of the reliability function in solving the MADED problem. Electrical system reliability, which has a significant impact on electricity costs and is highly dependent on customer happiness, is one of the most crucial concerns facing the power system. Moreover, lack of attention to this issue causes system load loss, power outages, and widespread instability.

Table 1. Recent studies in the literature about the MAED problem.

Reference	Year	Method	MASED	MADED	Objective Function			Constraints
					Emission	Generation Cost	Reliability	VPE	POZ	MFO	RR
[4]	2015	DEPSO	✓	-	-	✓	-	✓	✓	✓	-
[16]	2019	NMDA	✓	-	-	✓	-	-	-	-	-
[17]	2017	PSO-SFL	✓	-	✓	✓	-	✓	✓	-	-
[18]	2022	SS	✓	-	✓	✓	-	✓	-	-	-
[19]	2022	ICSO	✓	-	-	✓	-	-	-	-	-
[20]	2019	SSA	✓	-	-	✓	-	-	-	-	-
[21]	2019	IRF	✓	-	-	✓	-	✓	✓	-	✓
[22]	2022	DDCO	✓	-	-	✓	-	✓	-	✓	-
[23]	2017	FCP	✓	-	✓	✓	-	✓	✓	✓	-
[24]	2022	MOGWO	✓	-	-	✓	-	-	-	-	-
[25]	2019	Jaya-TLBO	✓	-	-	✓	-	✓	✓	-	-
[26]	2020	ICA-PSO	✓	-	-	✓	-	✓	✓	✓	-
[27]	2020	MSSA	✓	-	-	✓	-	-	-	-	-
[28]	2016	EFWA	-	✓	-	✓	-	✓	-	✓	✓
[29]	2021	EFW	-	✓	-	✓	-	✓	-	✓	✓
[30]	2020	PSO-GWO	-	✓	-	✓	-	✓	-	✓	-
[31]	2016	QOGS	-	✓	-	✓	-	✓	-	-	-
My research		MGO	-	✓	✓	✓	✓	✓	✓	✓	✓

provides a summary of research conducted in MASED and MADED problems. This table allows for a comparison of our approach’s properties to those of other methods found in the literature. According to Table 1, the present study is one of the few studies that solves multi-area ED in a dynamic environment and considers all the constraints such as VPE, MFO, POZ, and ramp rate (RR) in solving the MADED problem. In addition, this is the first study that introduces the objective function of reliability and considers it in solving the MADED problem along with the two objective functions of generation costs and emissions.

1.2. Contributions

The main contributions of this study are presented as follows. Moreover, Figure 1 also illustrates the suggested strategy for the considered optimization problem in this study.

• Providing a dynamic model that incorporates the objective functions of generation costs, emissions, and energy not supplied (ENS) for the (MADED) problem.
• Presenting the ENS index as a reliability objective function to assess the impact of the probability of output units in solving the optimization problem.
• Considering ramp rate constraints, transmission line capacity, VPE, POZ, and ramp rate (RR) in solving the optimization problem to model the real space of a power system.
• Presenting a modified grasshopper optimization (MGO) algorithm to resolve the intricacy of the optimization problem. For improving the algorithm disadvantage in terms of slow convergence speed, logistic mapping as one of the chaos mechanisms has been developed.
• Comparing between the MGO algorithm results and other well-known optimization techniques to assess the efficiency and robustness of the proposed technique.
• Introducing two new criteria, generation distance (GD) and diversity metric (DM), to evaluate Pareto solutions, as well as a fuzzy decision technique to find Pareto optimal solutions and store them in the external reservoir.
• Using the suggested approach to simulate the issue accurately in two test systems of 10 units of 3 zones and 40 units of 4 zones.

The organization of this study is as follows:

The problem formulation, which includes objective functions, decision variables, and constraints, is described in Section 2. The multiobjective optimization methodology is introduced in Section 3. Section 4 and Section 5 describe the simulation findings and conclusions.

Figure 1. Graphical representation of the proposed approach.

Figure 1. Graphical representation of the proposed approach.

2. Problem Formulation

The effort of MADED is to identify the best output combination of generation units in each region and, moreover, power transfer between various regions to optimize different objective functions with various limitations. Mathematical modeling of the considered objective functions with the corresponding limitations is described below.

2.1. Objective Function

Meeting all operational limitations while minimizing the cost of generation units, emissions, and ENS in a power system are among the MADED problem’s objective functions.

• Cost function with valve-point effect

Practically, generators contain numerous restrictions, including the valve point effect that can be formularized as Equation (1):

$$F_1^{VPE}={\displaystyle \sum }_{t=1}^T{\displaystyle \sum }_{k=1}^K{\displaystyle \sum }_{m=1}^M\left({\alpha }_{km}+{\beta }_{km}{P}_{kmt}+c_{km}{P}_{kmt}^2+\left|d_{km}\sin \left(e_{km}\left(P_{kmt}^{min}-P_{kmt}\right)\right)\right|\right)$$

(1)

where $F_1^{VPE}$ shows the non-convex cost objective function with respect to the valve point effect; ${\alpha }_{km}$, ${\beta }_{km}$, $c_{km}$, $d_{km}$, and $e_{km}$ are the cost coefficients of the kth generator in the mth area; $d_{km}$ and $e_{km}$ are correlated to the valve point effect [17,30].

• Cost function with valve-point effect and multi-fuel operation

The cost objective function specification in (1) only applies to generators that operate with a single fuel type; as a result, it is invalid for generators that operate with multi-fuel types (MFO). Units may use a variety of fuels in the real world. The following formulation incorporates the VPE and MFO into the power plant’s generation cost function:

$$\begin{gathered} \begin{array}{c}{F}_1^{Total}={\displaystyle \sum }_{t=1}^T{\displaystyle \sum }_{k=1}^K{\displaystyle \sum }_{m=1}^M\left({\alpha }_{kmf}+{\beta }_{kmf}{P}_{kmt}+c_{kmf}{P}_{kmt}^2+ \\ \left|d_{kmf}\sin \left(e_{kmf}\left(P_{kmt}^{min}-P_{kmt}\right)\right)\right|\right) \\ {P}_{kmf-1}\le P_{kmt}\le P_{kmf}^{max}\end{array} \end{gathered}$$

(2)

where $f$ represents the number of fuels; ${\alpha }_{kmf}$, ${\beta }_{kmf}$, $c_{kmf}$, $d_{kmf}$, and $e_{kmf}$ are the cost coefficients of the kth generation unit in the mth area, which is related to the type of fuel f; $P_{km}^{min}$ and $P_{km}^{max}$ are the minimum and maximum power for the nth generator in the mth region, respectively; $F_1^{\mathit{Total}}$ represents the final cost objective function by simultaneously considering the VPE and MFO [17]. In the simulations performed, the above objective function is shown as the total cost.

• Emission

The mathematical model for the emission objective function can be formulated as follows:

$$F_2={\displaystyle \sum }_{t=1}^T{\displaystyle \sum }_{k=1}^K{\displaystyle \sum }_{m=1}^M\left({\delta }_{km}+{\gamma }_{km}{P}_{kmt}+{\partial }_{km}{e}^{\left({\mu }_{km}\times P_{kmt}\right)}\right)$$

(3)

where ${\delta }_{km}$, ${\gamma }_{km}$, ${\partial }_{km}$, and ${\mu }_{km}$ are the emission coefficients of the nth generator in the mth area; $F_2$ indicates the emission objective function that will be optimized in this study [17,30].

• Energy not supplied

An awareness of the power generation system and all of its components, familiarity with reliability ideas, and knowledge of methods for analyzing and enhancing it are required in order to investigate and improve the dependability of generation systems. Consequently, the ENS index is described and constructed as follows in order to evaluate this objective function:

$$F_3={\displaystyle \sum }_{t=1}^T{\displaystyle \sum }_{i=1}^{LP}{P}_{L,i}\times U_i$$

(4)

$$Ui=FORi\times t$$

(5)

$$FOR=\frac{\mathsf{\lambda }}{\mathsf{\lambda }+\mu }=\frac{MTTR}{MTTF+MTTR}$$

(6)

$$MTTF=\frac{1}{\mathsf{\lambda }}, MTTR=\frac{1}{\mu }$$

(7)

where $P_{L,i}$ is the average load connected to the ith bus; $U_i$ is the annual outage time at ith bus; $LP$ is the total number of load points; $FO{R}_i$ is the force outage rate of ith generator; t and T are the time interval and maximum number of intervals, respectively; λ and μ are the failure rate and repair rate, respectively; MTTF and MTTR are the mean time of failure and mean time of repair, respectively.

2.2. Decision Variables

In the proposed model, the vector of decision variables ($X$) can be written as follows:

$$X=\left[P.P_{TL}\right]$$

(8)

P = [P1.P2.….Pm.….PM]

(9)

P_TL = [P_TL1.P_TL2.….P_TLm.….P_TLM]

(10)

where $P_m$ is defined as active power of the generation unit in the mth region and $P_{TL}{}_m$ is the power flow between the mth region and other regions.

2.3. Problem Constraints

Operational restrictions must be met in accordance with the objective functions in the optimization problem. The following is a description of the optimization problem’s linear and nonlinear constraints:

• Active power balance

The active power balance is written as follows:

$$\forall mϵM \forall tϵT{\displaystyle \sum }_{k=1}^{N_m}{P}_{kmt}=P_{D{ }_{mt}}+P_{L{ }_{mt}}+{\displaystyle \sum }_{r\ne m}{P}_{TL{ }_{mht}}$$

(11)

where $P_{D{ }_{mt}}$ is the active power requirement at the tth hour in the mth region; $T_{L{ }_{mht}}$ is the active power conveyed from mth region to region hth area, which is positive if it gushes from one region to another and negative for the contrary [17,32]. In the end, $P_{L{ }_{mt}}$ is the transfer losses of the mth region at tth time interlude. $N_m$ is the number of consigned generation units in the mth region.

• Active power loss

The active power balance is written as follows:

$$\forall mϵM \forall tϵT P_{L{ }_{mt}}={\displaystyle \sum }_{r=1}^{N_m}{\displaystyle \sum }_{k=1}^{N_m}{P}_{kmt}{B}_{mrk}{P}_{mrt}+{\displaystyle \sum }_{k=1}^{N_m}(B{o}_{mn}{P}_{mkt})+Bo{o}_m$$

(12)

where $Bo{o}_m$ represents the constant coefficient of losses in the mth area [17]; $B{o}_{mn}$ is the nth component of the loss coefficient vector with dimensions Nm from the mth area; $B_{mrk}$ is the (r, m) component of the loss coefficient matrix.

• Tie-line capacity

The capacity of transmission lines should not go over a certain threshold for network safety considerations. This restriction is expressed as follows:

$$\left|P_{TL{ }_{mht}}\right|\le P_{TL{ }_{mh}}{}^{max}$$

(13)

where $P_{TL{ }_{mh}}{}^{max}$ represents the active transmission power limit of tie-lines between the mth and hth area.

• Ramp rate

Ramp rate is restricted by the generator’s capacity to change its output in response to changes in demand [3], which is represented by the following equations:

$$\forall mϵM \forall kϵN_m \forall tϵT P_{kmt}-P_{kmt-1}\le U{R}_{min}$$

(14)

$$\forall mϵM \forall kϵN_m \forall tϵT P_{kmt-1}-P_{kmt}\le D{R}_{max}$$

(15)

where $U{R}_{min}$ and $D{R}_{max}$ are the upper and lower range of ramp rate, respectively.

• Power generation capacity

Each region’s power generation units are required to provide active power within the following acceptable ranges:

$$\forall mϵM \forall kϵN_m \forall tϵT P_{kmt}^{min}\le P_{kmt}\le P_{kmt}^{max}$$

(16)

where $P_{kmt}^{min}$ and $P_{kmt}^{max}$ are the minimum and maximum generation capacity limits of power generation unit in the mth area, respectively.

• Prohibited operation zones (POZ)

Each generator actually has a unique POZ that it should avoid producing inside that range. The generation in these areas may damage pumps, boilers, shaft bearings, and other device accessories [17,30]. The mathematical formula for the restricted area is as follows:

$$\forall mϵM \forall kϵN_m \forall tϵT \forall zϵZ\left\{\begin{array}{c}{P}_{km}^{min}\le P_{kmt}\le P_{km1}^L\\ P_{kmz-1}^U\le P_{kmt}\le P_{kmz}^L\\ P_{kmz}^U\le P_{kmt}\le P_{km}^{max}\end{array}\right\}$$

(17)

where $P_{kmz}^L$ and $P_{kmz}^U$ are defined as the lower and upper frontiers of the zth POZ of the kth generator in the mth region, respectively; Z is the whole number of POZs.

3. Proposed Optimization Methodology

The grasshopper optimization (GO), modified grasshopper optimization (MGO) algorithm, multiobjective optimization strategy, and criteria for evaluating Pareto optimal solutions are described in this section.

3.1. Grasshopper Optimization Algorithm

Grasshoppers are one of the largest groups of all living things [33]. It is unique to grasshoppers that their group life behavior is found in both adults and babies. There are millions of baby grasshoppers jumping and spinning in the air. In the future, when they are older, they form groups in the air. Grasshoppers, in the form of swarm and groups, move from point to point. The life cycle of a grasshopper is shown in Figure 2.

On the one hand, these groups of grasshoppers are characterized by their slow speed and short steps during the larval stage. On the other hand, abrupt, long movements are also a hallmark of these groups among older grasshoppers. The search for food resources is an important feature of group life among grasshoppers. Algorithms influenced by nature separate the search process into two distinct phases: exploration and exploitation. It is encouraged for search agents to move abruptly during exploration, but they tend to move locally in exploitation; both of these functions and the search for the target are done instinctively by the grasshoppers. A mathematical model for simulating the group behavior of grasshoppers is shown as follows [33,34]:

$$X_i=S_i+G_i+A_i$$

(18)

where $X_i$ and $S_i$ represents the position of the ith grasshopper and social interactions, respectively; $G_i$ and $A_i$ are the horizontal wind force and force of gravity on the ith grasshopper, respectively. $S_i$ can be formulated as follows:

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

(19)

where $d_{ij}$ is the distance between ith grasshopper and jth grasshopper, which is computed as $d_{ij}=\left|x_j-x_i\right|$. The power of social relations S can be reprinted in Equation (20) and $d_{ij}=\frac{\left|x_j-x_i\right|}{d_{ij}}$ is a single vector from ith grasshopper and jth grasshopper.

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

(20)

where f represents the gravity intensity and $l$ is the measure of the length of gravity. The distance d between the grasshopper and the other one is different between [0, 15]. There are three parts of the distance d: the first section is the repulsive zone, which is in the range [0, 2.079]. The second section is the zone of comfort, started from 2.079, which is defined as zero gravity and no repulsion. In the region [2.079, 4], gravity increases. After achieving a distance of four units, gravity gradually begins to decline and reaches zero. The G component in Equation (18) is calculated as follows:

$$G_i=-g{e}_g$$

(21)

where $g$ is the the coefficient of gravity $e_g$ and represents a single vector toward earth center. The A Component in (18), can be obtained by (22):

$$A_i=U{e}_w$$

(22)

where U represents the coefficient of thrust and wind direction, as a single vector, is modeled by $e_w$. Since young grasshoppers are wingless, wind direction determines their motion dynamically, by placing S, G, and A in Equation (23), this equation can be expanded as follows:

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

(23)

where $x_j^d$ and $x_i^d$ define the jth and the ith grasshoppers, respectively; N is the grasshoppers’ number; $l{b}_d$ and $u{b}_d$ indicate the lower and upper boundaries of d interval, respectively; $\widehat{T_d}$ represents the target value, i.e., the best solution; $c$ is the reduction coefficient to make the comfort zone, the repulsion zone, and the gravity zone smaller. To achieve a balance between exploration and exploitation, it is necessary to reduce parameter c according to the number of iterations. The number of interactions increases as a result of this mechanism. Based on the number of transactions, the coefficient c reduces the comfort zone, as shown in (24):

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

(24)

where $c_{max}$ and $c_{min}$ are the maximum and minimum values of c, $l$ represents the current iteration, and $L$ represents the maximum number of iterations.

3.2. Modified Grasshopper Optimization Algorithm

Some of the major shortcomings of the GO algorithm, similar to other population-based methods, is slow convergence and trapping in local optimum. To compensate for this drawback, a chaos-based mechanism is developed. The nature of chaos is apparently random and unpredictable, and it also possesses an element of regularity. Chaos employs chaotic variables rather than random variables [35]. Therefore, it can perform downright searches at higher speeds as compared with the stochastic searches that mainly rely on probabilities [36].

The applications of chaos theory are very wide [37,38]. In recent years, logistic mapping (LM), a chaos-based mechanism, has gained increased attention because of its effectiveness, good dynamic behavior, and simplicity in display. The density of this mechanism is constant and uniform at its definition intervals [39]. Based on the theory of chaos, there is a certain formula to every complex system that appears only as random values. From [40], LM has two important privileges including high speed and convergence. The LM function is formulated as (25):

$$X^{k+1}=\partial .X^k \left(1-X^k\right)$$

(25)

where k represents the number of iterations, $\partial$ defines an adjustable parameter in the interval, and $\partial$ = 4 and $X^k$ ∈ [0, 1] is a prime random number. This change will continue until the optimal value is reached; the selected turbulent maps will be repeated.

The difference between the MGO method and the conventional GO algorithm is in the stage of generating the initial population. In conventional GO, the initial population is randomly generated, this issue in the optimization process slows down the speed of convergence, or causes the grasshopper to get stuck in the local optimum. Therefore, in the MGO algorithm, using the chaos mechanism, the initial population is generated according to Equation (25). For better clarification of the proposed method, the pseudo code of MGO is presented as follows Algorithm 1:

Algorithm 1: The pseudo code of the MGO algorithm

1: Begin 2: Initialize number of grasshoppers, $c_{max}$, $c_{min}$, and maximum iterations L 3: Set the t = 1 4: Use logistic mapping to initial the population of grasshopper using the Equation (25) 5: Calculate the fitness of all grasshoppers 6: Set K as the best search agent 7: while t < L do 8: Update c by Equation (24) 9: for each grasshopper do 10: Normalize the distance between grasshoppers 11: Update the positions of grasshoppers by Equation (23) 12: end for 13: Update K if there is a better 14: t = t + 1 15: end while 16: Return K 17: END

3.3. Multiobjective Optimization Strategy

In this part, multiobjective problem formulation, Pareto optimal technique, and fuzzy decision strategy are presented.

• Multiobjective problem

Numerous real-world issues that call for the simultaneous optimization of several objective functions should be resolved using multicriteria decision-making techniques. These operations are frequently neither unilateral nor even mutually exclusive. Therefore, obtaining a collection of optimal solutions as opposed to only one seems common [41,42]. A typical formulation of a multiobjective problem is as follows:

$$\mathrm{Min} f_i\left(\mathrm{x}\right)={\left[f_1\left(x\right),f_2\left(x\right),\dots .f_n\left(x\right)\right]}^T \forall iϵN_{objective }$$

(26)

$$G_r\left(x\right)\le 0, H_j\left(x\right)=0 \forall rϵM \forall jϵN$$

(27)

where $f_i\left(x\right)$ is the ith objective function, and $N_{objective }$ is the number of objective functions, respectively. The nonlinear/linear constraints is represented by $G_r\left(x\right)$/$H_j\left(x\right)$ where $r$ and $j$ represent the number of nonlinear and linear constraints, respectively and $x$ is the variables’ vector of optimization.

• Normalize the objective function

Given that the quantities of the objective functions do not fall within the same range, Equation (28)’s fuzzy technique is employed to bring the amounts of these functions inside the same range.

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

(28)

where $f_i^{min}$ and $f_i^{max}$ are the minimum/maximum amounts of the function $f_i\left(X\right)$, respectively; and finally, ${\mu }_i\left(x\right)$ is the fuzzy set for the ith objective function.

• Pareto optimal method

The Pareto optimum approach, which is one of the popular methods for solving multiobjective problems, is based on the idea of dominance, according to which the $X_1$ vector prevails over the $X_2$ vector if only (29) and (30) are proven [41,42].

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

(29)

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

(30)

• Fuzzy decision making

To find the best solution among the Pareto front’s optimal solutions, all Pareto optimal solutions are acquired, stored in the reservoir, and then evaluated using (31). Notably, the best compromise solution (BSC) is determined to be the Pareto solution with the highest $N_{\mu }\left(j\right)$.

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

(31)

where $W_k$ is the weight factor of the kth objective function, and the operator may select it as he or she pleases. Accordingly, m stands for the non-dominated solutions and n for the number of objectives [41,42]. According to (31) and based on the weighting parameters used, the solution with the highest value $N_{\mu }$ is referred to be the BSC.

3.4. Criteria for Evaluating Pareto Solution

The optimal results, including Pareto front sets, should be evaluated to illustrate the efficiency of the suggested approach. In this paper, there are two different criteria, each of which is presented as follows:

• Generation distance

The criterion for determining the distance of a sth solution in the Pareto solution set is called generation distance (GD) [43,44]. This criterion is defined as:

$$GD=\frac{\sqrt{{{\displaystyle \sum }}_{s=1}^n{E}_s^2}}{k}$$

(32)

where the Pareto optimum set’s nearest number and the vectors of non-dominant solutions’ Euclidean geometric distances are calculated using the GD criterion. It is important to note that a value of zero for the criterion denotes that all produced arrays belong to the Pareto optimum set. As a result, a smaller value is preferable for this option.

• Diversity metric

The Euclidean geometric distance between Pareto solutions is used as the foundation for the diversity metric (DM) creation [43,44]. The center of the jth dimension is, therefore, approximated as follows whenever the Pareto front with ${\mathrm{N}}_{\mathrm{obj}}$ dimension has m points:

$$C_j=\frac{{{\displaystyle \sum }}_{r=1}^k{Y}_{jr}}{k} \forall jϵN_{objective }$$

(33)

where $Y_{nr}$ is jth dimension of rth point and $C_j$ is the center point for jth dimension. The DM is formulated as follows:

$$DM={\displaystyle \sum }_{j=1}^{N_{obj}}{\displaystyle \sum }_{r=1}^k{(Y_{jr}-C_j)}^2$$

(34)

As a result, a higher score for this criterion shows that all of the created components are close to one another.

4. Simulation Results and Analysis

In this study, two test systems of 10 and 40 units are taken into consideration in order to examine the effectiveness of the MGO algorithm and other evolutionary algorithms in addressing the MADED problem. It should be mentioned that all case studies are performed on a quad-core laptop with a clock frequency of 1.60 GHz and 4.0 GB of RAM. The MGO and aforementioned evolutionary algorithms to solve the MADED problem are coded in the MATLAB 2016 environment. Moreover,

Table 2. Parameters of the suggested algorithms.

Parameters	MGO	GWO	MHBMO	IPSO-MSFLA
Population size	400	400	400	400
Maximum iteration	100	100	100	100
$\partial$	4	-	-	-
r1, r2	[0–1]	[0–1]	[0–1]	[0–1]
c1, c2	-	-	-	2
$c_{max}, c_{min}$	1, 0.04
Number of groups	-	-	-	5

lists the parameters of the algorithms considered in this study. The effectiveness of the suggested MGO in solving the suggested optimization problem is assessed using the case studies listed below.

Case I, total cost is optimized; Case II, emission is optimized; Case III, the ENS is optimized; Case IV, cost and emission are optimized; Case V, cost and ENS are optimized; Case VI, emission and ENS are optimized; Case VII, all objectives are optimized.

The following are the test systems and simulation results for each test system.

A. Test system #1

This system comprises 10 generation units that are diffused in three various regions. All-inclusive data about the 10-unit, 3-zone test system can be detected in [17,30]. The optimization results obtained by MGO for Cases I–VII are presented in

Table 3. Best obtained results by proposed method for Cases I–VII.

Case Studies	Cost (USD)	Emissions (Ton)	ENS (kWh/Year)
Case I	12,519.93	-	-
Case II	-	26,659,925.12	-
Case III	-	-	19.22390213
Case IV	12,556.13	26,731,553.51	-
Case V	13,815.17	-	36.96886086
Case VI	-	29,526,992.15	34.03815612
Case VII	13,347.92	28,740,132.68	38.79716301

.

Table 4. Obtained results by different methods for Case I.

	Algorithms	Best Solution	Mean Value	Worst Solution	Standard Deviation
Literature	QOGSO [18]	12,976.9	12,983.6	12,992.38	7.76508
	GO [18]	13,013.66	13,021.2	13,031.93	9.1813
	GSA [18]	13,121.05	13,134.3	13,149.53	14.251
	BBO [18]	13,081.08	13,092.7	13,106.53	12.7398
	DE [18]	13,042.28	13,050	13,062.47	10.1846
	PSO [18]	13,134.05	13,151.3	13,170.27	18.1165
This study	IPSO-MSFLA	12,599.43	12,606.23	12,668.86	26.8973
	MHBMO	12,570.35	12,591.19	12,615.33	23.29
	GWO	12,549.15	12,568.32	12,585.41	17.35
	MGO	12,519.45	12,536.54	12,549.65	15.51

and

Table 5. Obtained results by different methods for Case III.

Algorithms	Best Solution	Mean Value	Worst Solution	Standard Deviation
IPSO-MSFLA	29.75	34.15	38.85	3.59
MHBMO	25.65	28.89	32.45	3.15
GWO	21.39	24.35	27.89	3.25
MGO	19.22	22.14	25.55	2.30

show a comparison of the optimization results obtained by the proposed MGO algorithm and other methods, i.e., GWO, MHBMO, IPSO-MSFLA, QOGSO [18], GO [18], GSA [18], BBO [18], DE [18], and PSO [18], for Cases I and III in 30 different experiments. It should be noted that, in these tables, the best and worst answer, mean, and standard deviation are presented in 30 different experiments. Moreover, the output power of generation units and the transmission power between tie-lines for Cases I–III and VII are shown in Figure 3 and Figure 4. From these tables, it is obvious that the MGO algorithm can converge to better responses than other evolutionary methods such as GWO, MHBMO, and IPSO-MSFLA, and also has a lower standard deviation criterion than other methods, which indicates the capability of the suggested algorithm to find solutions in solving non-convex optimization problems. There was no reference for Case III to compare the obtained results with; therefore, these are the best obtained results so far.

According to Table 4, the MGO method manages to identify the optimal total cost, which is about USD 80 less than the cost obtained from the IPSO-MSFLA method. In addition, the optimal value of the ENS obtained from the MGO method is 6.5 kWh/year less than the MHBMO method. Thus, one of the most significant purposes of this research, which is to improve reliability, has been met by minimizing the ENS index. The capability of the suggested MGO algorithm to control various problem limitations is another major matter that is specified by understanding the power flow in the tie-lines in addition to the output power of the generators in different areas.

Figure 5 depicts the convergence diagrams of the MGO, GWO, MHBMO, and IPO-MSFLA methods, to optimize the total cost objective function. Accordingly, it is obvious that the MGO method can converge in a smaller number of iterations, as compared with other algorithms, to a smaller amount of total cost.

According to Figure 3, the configuration of generators and their output are different for generation cost, emission, and ENS objective functions. For example, the values of the output power of generator #5 at the 19th time interval for Cases I–III are equal to 198.0206 MW, 490 MW, and 303.8751 MW, respectively. This stems from the conflict between different objectives. Figure 3d displays the configuration of generation units while all three objective functions are optimized simultaneously. From this figure, it is clear that the schedule of generator outputs is a combination of those shown in Figure 3a–c, since the generator outputs must take care of all objectives simultaneously.

In a similar way, the transferred active power between different areas in test system #1 are demonstrated in Figure 4. Based on Figure 4, the active power flows between different areas vary for optimizing Cases I–III and IV. For instance, the values of the active power transferred from T21 at the 14th time interval for total cost, emission, and ENS objectives are equal to ~0 MW, 49.32 MW, and −4.64 MW, respectively. A positive value means that the power flows from area 2 to area 1 and a negative value is for the opposite direction of flow. It is clear that these objectives are not in line with each other and solving the problem for one objective function can aggravate others. This is more tangible in Figure 3, where the optimal cost, optimal emission, optimal ENS, and the best compromise solution for Cases I–III are illustrated. Toward this end, it is crucial to solve the problem as a multiobjective optimization problem to have a compromise between different objective functions. For a more comprehensive comparison, the optimal values of the cost, ENS, and emission objectives in three cases are shown in Figure 6. The contradiction among the cost, ENS, and emissions is obvious in this figure. In addition, the ability of the proposed optimization algorithm to compromise between all objective functions in different cases can be deduced from Figure 6.

B. Test system #2

In this case study, 40 generators are diffused in four various regions. The electricity demand is estimated to be 10,500 MW, with the whole loads in Zones #1, #2, #3, and #4 being 1575 MW, 4200 MW, 3150 MW and 1575 MW, respectively [17,30]. The results obtained by the proposed MGO for different cases are tabulated in

Table 6. Best obtained results by the proposed method for Cases I–VII.

Case Studies	Cost ($)	Emissions (Ton)	ENS (kWh/Year)
Case I	2,419,531.125	-	-
Case II	-	135,715,431.23	-
Case III	-	-	19.61660624
Case IV	2,583,032.100	157,884,092.9	-
Case V	2,468,038.561	-	39.75199784
Case VI	-	153,618,801.5	64.81039386
Case VII	2,708,738.682	177,555,371.1	35.63738165

. In order to evaluate the capabilities of the proposed algorithm in solving MADED the obtained results, for cost and emission objective functions in 30 different experiments are compared with GWO, MHBMO, IPSO-MSFLA, QOGSO [18], GO [18], GSA [18], BBO [18], DE [18], and PSO [18] in

Table 7. Obtained results by different methods for Case I.

	Algorithms	Best Solution	Mean Value	Worst Solution	Standard Deviation
Literature	QOGSO [18]	2,511,795	2,511,812	2,511,833	19.03506
	GO [18]	2,512,155	2,512,174	2,512,197	21.03172
	GSA [18]	2,512,381	2,512,397	2,512,418	18.55622
	BBO [18]	2,512,314	2,512,329	2,512,347	16.52271
	DE [18]	2,512,208	2,512,230	2,512,256	24.02776
	PSO [18]	2,512,423	2,512,445	2,512,461	19.07878
This study	IPSO-MSFLA	2,425,738.45	2,426,059.23	2,426,439.29	409.19
	MHBMO	2,425,639.56	2,426,029.12	2,426,530.26	399.96
	GWO	2,425,589.65	2,425,886.18	2,426,405.31	388.04
	MGO	2,419,531.12	2,425,726.32	2,426,259.68	374.71

and

Table 8. Obtained results by different methods for Case II.

Algorithms	Best Solution	Mean Value	Worst Solution	Standard Deviation
IPSO-MSFLA	136,968,653.21	136,973,989.45	136,979,653.15	5639.56
MHBMO	136,956,945.32	136,963,035.63	136,969,450.14	5425.63
GWO	136,939,568.52	136,946,888.56	136,954,563.25	5294.53
MGO	135,715,431.23	136,920,214.52	136,925,465.31	4805.56

. Moreover, the output active powers of generators in 24-time intervals for Cases I–III and VII are illustrated in Figure 7a–d, respectively.

From these figures, it is clear that the scheduled power for different objective functions is totally different. For instance, the output power of generator #20 for Case I is at its maximum level while it is close to its minimum at most of time intervals in Case II. In sum, the proposed method could find the best schedule of output powers in all cases.

According to the results shown in Table 7 and Table 8, it is obvious that the suggested MGO algorithm has a high capability to achieve a better solution as compared with other algorithms for the large-scale MADED problem. According to these tables, the proposed MGO algorithm reached the optimal answer in emission optimization, which was 20,137 tons less than the answer obtained from the usual GWO. In addition, the optimal value of the cost objective function obtained from the MGO method is USD 6108 less than the answer obtained from the MHBMO algorithm. Furthermore, the transferred active power between different areas related to Cases I–III and VII in test system #2 are demonstrated in Figure 8.

Similar to the generator outputs, the active power flows in tie-lines are different in all four cases. For instance, the values of the active power transferred from T42 at the 3rd time interval for total cost, emission, and ENS objectives are equal to −17.9749 MW, 56.8220 MW, and 184.5469, respectively. It is worth mentioning that the proposed approach is able to optimally schedule the generator outputs and active power flows in tie-lines in both single- and multiobjective cases, irrespective of the complexity of the problem. All active power flows in tie-line as well as generator outputs are in their limits.

C. Analysis of obtained Pareto-optimal solutions

Investigating the multiobjective MADED problem is among the major objectives of this study. With this mentality, the two- and three-dimensional Pareto optimal fronts for the multiobjective problem are illustrated in Figure 9a–d. In addition, the best compromise solution is marked in red on each Pareto front.

According to the Figure 9a–d, it is obvious that the amount of the objective functions in the best compromise solution is near the optimal amount of these functions on the Pareto fronts and does not differ much (less than 5%,), which proves the capability of the suggested algorithm to resolve a complex multiobjective problem. Another point is the Pareto front path obtained and its flexion show the fact that the suggested algorithm, regardless of their scale and complexity, can be taken into consideration as a suitable instrument to solve multiobjective optimization problems.

To investigate the Pareto optimal solutions gained from the MGO and IPSO-MSFLA methods, the best obtained amounts of GD and DM gained by the mentioned algorithms to solve the multiobjective MADED problem in two systems are presented in

Table 9. GD and DM results to analyze the Pareto solutions by MGO and IPSO-MSFLA for two systems.

Systems	Dimensional	Criterions	MGO	IPSO-MSFLA
# 10 units	Cost-emission	GD	8.94222 × 104	3.1869 × 105
		DM	6.0735 × 1015	1.6126 × 1015
	Cost-ENS	GD	1.4580 × 103	1.5596 × 104
		DM	3.9470 × 108	1.47 × 108
# 40 units	Emission-ENS	GD	2.8657 × 105	1.2323 × 106
		DM	1.9130 × 1014	3.7210 × 1013
	Cost-emission-ENS	GD	2.6129 × 104	1.2198 × 105
		DM	6.0628 × 1017	1.4700 × 1016

. From this table, it is obvious that the MGO algorithm obtained better solutions to the multiobjective problem as compared with the IPSO-MSFLA algorithm. For example, the GD and DM values obtained from the proposed MGO algorithm in both systems are less and greater, respectively than the IPSO-MSFLA method. Thus, the capability of the suggested MGO algorithm to solve multiobjective problems, especially the MADED problem, has been proven once again.

5. Conclusions

In this study, a novel optimization algorithm, i.e., MGO, is suggested for the MADED problem in the form of single- and multiobjective approaches. To evaluate the proposed model’s efficiency in solving the considered optimization problem, three objective functions including total cost, emissions, and ENS, considering the operational limitations of the system, are introduced. The proposed multiobjective MADED problem is solved by applying a fuzzy membership function based on the weighting system of the objective function, aiming to find the best compromise solution between the non-dominant ones. In addition, the proposed algorithm is successfully applied to solve two different small- and large-test systems, including 10-unit and 40-unit test systems. Nevertheless, this research takes into consideration the impacts of various practicable constraints including, VPE, POZs, MFO, and RR to make the results more practicable.

One of the important inferences to be obtained from this research is that the suggested MGO algorithm is strong enough to resolve the MADED problem, for single- and multiobjective approaches, regardless of scale and complexities. In addition, the capability of the MGO method, as compared with the other heuristic methods such as GO, GSA, PSO, DE, GWO, MHBMO, and IPSO-MSFLA, is proven both in terms of improving the answers and reducing the execution time in both test systems. For instance, the amount of generation cost resulting from the proposed MGO method is reduced by about 4% and 4.5% as compared with the GO method in 10- and 40-unit systems, respectively. Furthermore, comparing the performance of the MGO algorithm and the IPSO-MSFLA method by evaluating the obtained values of GD and DM criteria, indicates the ability of MGO algorithm to find Pareto fronts properly in small-/large-scale multiobjective problems.

Based on the simulation results, the suggested method for MADED helps the power system operator, in addition to having good and economic planning in a dynamic format, to provide the conditions for this planning in an environment with high reliability and low emissions. For example, the proposed MGO method could decrease the generation costs and emissions by 4.5% and 4.83%, respectively, in a 40-unit test system. This can play an important role in decreasing electricity costs and ameliorating environmental concerns.