Multi-Objective Squirrel Search Algorithm for Multi-Area Economic Environmental Dispatch With Multiple Fuels and Valve Point Effects

The essential goal of multi-area economic environmental dispatch (MAEED) is to determine the optimum power generation schedule of each unit and power transfer between the areas in order to minimize fuel costs and pollutant emissions, when the generation, power balance and tie-line limits are satisfied. This paper focuses on developing multi-objective squirrel search algorithm (MOSSA) to solve the MAEED problem, of which the goal is to simultaneously minimize the total fuel cost and emission considering valve point effects and multi-fuel options. The proposed MOSSA combines squirrel search algorithm along with Pareto-dominance theory to generate non-dominated solutions. It uses an external elitist depository mechanism with crowding distance sorting to preserve the distribution diversity of Pareto-optimal solutions as the evolution continues. In addition, a fuzzy decision maker is used to select the best compromised solution from the obtained Pareto frontiers. Furthermore, the MAEED problem is unraveled by squirrel search algorithm based weighted sum approach with price penalty factors, artificial bee colony and exchange market algorithm. Different case studies are performed on 10-unit with three-area system, 40-unit with four-area system and 140-unit real Korean power system considering valve point effects and multi-fuel options which testify the supremacy of the suggested approach. The comparisons with state-of-the-art approaches suggest that MOSSA can generate more competitive trade-off solutions for solving the MAEED problems.

Euclidean distances between the extreme solutions and the boundary solutions of the obtained non-dominated set E ij (P ij ) emission of the generator j in area i F bcs and E bcs fuel cost and emission attained by CEED F min and E max fuel cost and emission attained by ELD minimization respectively F max and E min fuel cost and emission attained by EED minimization respectively F ij (P ij ) fuel cost of jth generation unit in ith area G c gliding constant k number of fuel alternatives L lift power M number of non-dominated solutions M i number of participated generators in area i N p Population size ng total number of generating units P Di power demand in area i P dp predator presence probability P ij real power generation of generator j in area i P ij,min , P ij,max minimum and maximum limits of jth generation unit in ith area P L ij,m , P U ij,m lower and upper power outputs of the m th prohibited zone of the j th generator in area i P Li losses in area i r 1 , r 2 and r 3 random numbers in the range of [0, 1] t current iteration number T iz tie line power stream from area i to area z X h location of squirrel individual that reached the hickory treē X number of non-dominated solutions in population X α ij , β ij , γ ij emission coefficients of generator j in area i η ij , δ ij emission coefficients of the VPL effect of jth generation unit in ith area ρ density of air β constant

II. INTRODUCTION
Economic load dispatch (ELD) performs a crucial function in operation planning of modern power systems. The fundamental aim of ELD problem is to minimize the total fuel cost, subject to equality and inequality constraints.
With the expanding attention of environmental protection in recent years, economic and emission dispatch (EED) is introduced as a substitute to attain the reduction of fuel cost and emission simultaneously. Heuristic approaches such as bacterial foraging algorithm [1], multi-objective differential evolution (MODE) [2], gravitational search algorithm [3], teaching learning-based optimization (TLBO) [4], artificial bee colony (ABC) [5], backtracking search algorithm [6], opposition-based krill herd algorithm (OKHA) [7], continuous quick group search optimizer (CQGSO) [8] and shuffled differential evolution (SDE) [9] have been proposed to solve the ELD and EED problems. Classical techniques have been developed for multi-area ELD (MAELD) problems, such as Dantzig-Wolfe decomposition approach [10] and improved Hopfield neural networks [11]. These techniques are likely to be seriously challenged by their high imposition of different constraints, including consistency, convexity and distinguishability of objective functions and high sensitivity towards initial values of the optimized variables involved. In addition, their output is gradually deteriorated by the dimension of the problem.
Many heuristic algorithms such as evolutionary programming (EP) [12], real-coded genetic algorithm (RCGA) [13], particle swarm optimization (PSO) [13], differential evolution (DE) [13], ABC [14], TLBO [15], hybrid cuckoo search algorithm [16] and improved grasshopper optimization algorithm [17] have been developed and applied effectively to solve the MAELD problem due to their ability to find global or near-global solution of a nonconvex optimization problem. In [18] an improved fireworks algorithm (IFA) was used to solve the multi-area ELD problem considering the valve point loading (VPL) effects. The tie-line limit between different areas, generation limits, ramp rate limits, transmission losses, prohibited operating zone (POZ) and spinning reserve as the problem constraints. In the IFA, cross-generation mutation mechanisms were employed to solve the multi-constrained the multi-area ELD problem. A novel swarm intelligence approach using salp swarm algorithm for the solution of multi-area generation scheduling with wind integration was presented [19].
Multi-area EED (MAEED) issues have not been extensively explored so there is a requirement for additional exploration in this research area. In the recent decade, not many specialists have focused on this area which implies that the information on MAEED has not progressed far.
An enhanced PSO (EPSO) was developed to solve the MAEED problem with reserve constraints [20]. The PSO parameters were adaptively varied to preserve the balance among cognitive and social conduct of the swarm. The EPSO was examined on standard power systems considering the contingency and pooling spinning reserves. Hybrid heuristic algorithm using DE and PSO was developed to solve the reserve constrained MAEED problems with reserve sharing in power system operations [21]. The PSO boundaries were powerfully fluctuated to protect a superior harmony among intellectual and social conduct of the multitude. The EPSO was examined on standard test creating frameworks with turning save necessities by thinking about possibility and pooling turning saves. In [22], chaotic ABC algorithm was proposed to solve the MAEED problem by addressing the VPL, transmission line losses, multi-fuel options (MFO), POZ, tie line capacity and power transfer between different areas of the system. The simulation results showed that the chaotic ABC algorithm was performing better than the other heuristic approaches. Abarghooee et al. introduced an improved gradient-based Jaya algorithm for the MAEED problem to determine the power generation of units and the transmission power flow while satisfying system demand and security constraints of each area [23]. In the approach, gradient method, Jaya algorithm, and mutation mechanisms were combined to solve the complex MAEED problem on small and large-scale test systems. Secui applied another cooperative organism search approach to tackle the MAEED issue thinking about various characteristics of the investigated frameworks [24]. The competence of the methodology was examined on five multi-zone power systems with various working attributes and sizes. The different types of complex MAEED problem using crisscross optimization algorithm (COA) were addressed [25]. The COA approach employed horizontal crossover and vertical crossover to enhance the global search ability and to prevent the premature convergence of the algorithm. Li et al. presented an improved chemical reaction optimization algorithm for solving the MAEED problems [26].
Narimani et al. presented a hybrid technique dependent on the combination of shuffle frog leaping algorithm and PSO to tackle the proposed issue, and confirmed the viability of the proposed hybrid technique on various test systems [27]. Hybrid modified bat algorithm was used to solve a four-area economic environmental dispatch problem [28]. A weighted sum approach (WSA) was used to transfer the multi-objective function into a single objective one, and optimal solutions were selected using cardinal priority ranking. While hybrid approaches deliver the promising results, it is difficult to settle on the correct consolidation between the two heuristic approaches. The underlying multifaceted design of hybrid frameworks also requires an increase in the efforts to modify control parameters accordingly.
Pandit et al. proposed MODE based fuzzy determination approach for illuminating the non-convex MAEED problem [29]. Wang and Singh developed an improved PSO for solving the MAEED problem to obtain Pareto-optimal solutions [30]. The local search strategy was employed in the improved PSO to enhance the searching capability. The tie-line limits were considered as a set of constraints to satisfy the power system security. A new encoding mechanism and self-adaptive neighborhood structure selection mechanism were employed to increase the search ability of the algorithm while maintaining population diversity. The non-dominated sorting genetic algorithm II (NSGA II) was suggested to achieve solutions for the MAEED problem [31]. The constraints such as production-demand balance, power production capacity and tie line capacity were considered. These heuristic methods do indeed have some of the following disadvantages: unstable exploring capability, slow convergence, long computation time, reliance on the parameter selection, and poor consistency of Pareto fronts distribution.
In the most of the recently published articles, the WSA is used to transfer the multi-objective functions into a single objective function. This approach can discover the trade-off front by adjusting the weight values in different runs, but the technique does not discover the non-convex Pareto arrangements. Furthermore, this approach does not ensure to prompt the equitably dispersed solutions along the front, and the selection of weight values with different objective functions which prove to be genuinely problematic. Different multi-objective heuristic algorithms such as MODE and NSGA II have been used to address the EED problems. However, these approaches are computational expensive in solving a complex EED problems particularly where multi-area power systems exist.
Recently, a new meta-heuristic algorithm, named squirrel search algorithm (SSA) was proposed by Mohit Jain et al. [32]. The SSA algorithm models the foraging activities of squirrel individuals. Each squirrel individuals modifies its position using four processes namely, (1) distributing the population, (2) dynamic foraging behavior, (3) seasonal adapting intelligence and (4) random repositioning of individuals at the end of the winter season.
The SSA algorithm was previously successfully applied for 26 well-known classic benchmark test functions which are described as continuous, discontinuous, linear, non-linear, unimodal, multimodal, convex, non-convex, separable and non-separable forms [32], multi-region combined heat and power economic dispatch [33] and ELD problems [34]. The results of the SSA algorithm show the superiority over some popular heuristic algorithms such as genetic algorithm, PSO, bat algorithm, and firefly algorithm in terms of solving the high-dimensional multimodal problem. Besides, the SSA approach has certain unique features which overcome several demerits of the existing heuristic approaches as follows: • The gliding constant is used in the location update of squirrels which provides suitable steadiness between exploration and exploitation.
• The predator presence behavior is employed to abruptly change the location of a squirrel which enhances the exploration ability of the algorithm.
• A seasonal monitoring condition is used to prevent the suggested algorithm from being trapped in locally optimal solutions.
• Levy distribution is used to find new solutions far away from the current best solution which improves the global exploration ability of the algorithm.
These features make the SSA algorithm able to overcome the normal drawbacks of other algorithms such as premature convergence, inadequate ability to discover to find nearby extreme points, and absence of efficient constraints handling mechanism. The advantages of the SSA algorithm are less execution time, ability to solve different complex optimization problems, and high capacity in obtaining global optimum solutions.
According to aforementioned papers, this study is the first attempt to propose a multi-objective heuristic algorithm for solving the MAEED problem with MFO and VPL effects. The main motivation of this paper is to propose and develop a new multi-objective squirrel search algorithm (MOSSA) that has its own feasibility and performance capacity to determine the Pareto-optimal solutions of MAEED problems in power systems.
The significant contributions of this article are summarized as follows: • To the best of our knowledge, this is the first work that extends the SSA to MOSSA to address the multi-objective optimization problems. The suggested MOSSA approach consolidates external depository mechanism, crowding distance and fuzzy clustering mechanism to access the best trade-off solutions.
• The MAEED problem is additionally fathomed by SSA based WSA (SSA-WSA) with penalty price factors, ABC and exchange market algorithm (EMA) strategies.
• The multi-objective performance indicators comprising generational distance (GD), spacing metric (s-metric), ratio to non-dominated index (RNI), hyper volume (HV) and diversity metric (DM) are employed to investigate the Pareto optimal front solutions.
• In order to testify the supremacy of the suggested MOSSA approach, it has been employed on 10-unit with three-area system, 40-unit with four-area system and 140-unit Korean power system considering MFO and VPL effects.
• The findings are compared with SSA-WSA, ABC, EMA and various state-of-the-art heuristic approaches surfaced in the literature. The rest of this research article is organized as follows: Section II provides the MAEED problem formulation. The review of original SSA, elitist depository mechanism, fuzzy decision maker, MOSSA strategy and the solution procedure of the MAEED are presented in Section III. The case studies are analyzed in Section IV, and the conclusion is given in Section V.

III. PROBLEM FORMULATION OF MAEED
The objective of MAEED problem is to find out the optimal power generation of all units and the power transfer between the area by minimizing the fuel cost and pollutant emissions simultaneously over the whole framework while fulfilling different limitations.

A. MAELD
The goal of MAELD problem is to endeavor the optimal set of generation values in every zone just as shifting power between various zones so as to optimize the fuel cost subject to various imperatives.
The fuel cost function of committed generation units in all zones can be detailed as follows [19]: To display the impact of valve-points, a common amended sinusoid commitment is added to the quadratic function which is shown in Fig. 1 as [22]: The aim of the MAELD problem with multiple fuels is to determine the amount of power which can be resourcefully produced in one area and shifted to another area, and to determine the economic fuel choice for each unit. Since generators are provided with multi-fuel sources, every generator ought to be defined with a few piecewise quadratic capacities superimposed by sine terms mirroring the impact of changes in the type of fuel as shown in Fig. 2 [22]. The MAELD problem with VPL and MFO [19] can be modeled by Eq. (4), as shown at the bottom of the next page.

B. MULTI-AREA EMISSION DISPATCH (MAED)
The MAED is to limit the pollutant emissions discharged in the environment subject to equality and inequality imperatives. The sum of emissions discharged in the environment by the generating units from all regions of the system is defined as follows [22]:  The outflow target function is truly like the fuel cost function while it deals with all discharge types discharged by generation units. The scientific model for emanation function with MFO is introduced in Eq. (7), as shown at the bottom of the page.

C. MAEED BASED ON WSA
The MAEED problem can be formulated as bi-objective function in which fuel cost and emission as rivalling objectives. This bi-objective function can be transferred to a single objective function as follows [6]: The above equation becomes MAELD objective function when w = 1 and becomes MAED objective function when w = 0. w is a main function of rand [0,1] which compromises the fuel cost and emission objectives.

D. MAEED BASED ON MULTI-OBJECTIVE APPROACH
In multi-objective heuristic based MAEED, the two competing objective functions such as the economy and emissions are optimized simultaneously. The Pareto-dominance concepts are introduced to find a Pareto optimal set. The multi-objective MAEED problem can be defined as follows [4]: The following equality and inequality constraints are addressed for taking care of the MAEED issue.

1) POWER BALANCE CONSTRAINT
The all-out power generated from a set of accessible units must fulfil the all-out load demand and tie line power flow is given by [12], The transmission loss P Lj of region j can be defined by using B-coefficients as follows [12]: The real output power of thermal units ought to be in their range between minimum and maximum limits [12]:

3) TIE-LINE LIMIT
Because of security basis, power shifted between various lines must not surpass their cutoff points. The power transfer requirement between two unique regions is characterized by [12] −

IV. MULTI-OBJECTIVE SQUIRREL SEARCH ALGORITHM
SSA is the latest emerged swarm intelligence algorithm derived from foraging attitude of squirrels. This concept was first introduced by Jain et al. [27]. It is a population-based approach consisting of many squirrels wherein each squirrel is driven in a multi-dimensional search spot in search of food. In this SSA Algorithm, different variables are assigned for different positions of squirrel. The distance of the food from the individual squirrel is related to the fitness value. The metaphysical paradigm of SSA is exhibited in Fig. 3.
In SSA, the individual squirrel modifies its position, thereby shifting them to a better solution. The algorithm starts with n number of squirrels in a deciduous forest and with the assumption as only one squirrel in each tree. It is assumed that the three types of trees, normal, acorn and hickory, are available in the forest. The forest area is supposed to contain N trees in which one is hickory tree, N a acorn trees, and the remaining are normal trees in which no food is available. The hickory tree is the finest foraging area for the squirrels.
The movement of individuals is persuaded by the following four practices: • distributing the population, • dynamic foraging behavior, • seasonal adapting intelligence, and • random repositioning at the end of winter season.

A. STANDARD SSA
The positions of N squirrel individuals are randomly generated. Then the population is sorted in ascending order for minimization problem and vice versa. Then the squirrel groups are distributed into three categories: squirrels positioned at hickory trees (F h ), squirrels positioned at acorn trees (F a ) and squirrels positioned at normal trees (F n ). F h is the squirrel with the minimum fitness value, F a includes the squirrels that have the fitness rank from 2 to N a + 1 and the remaining squirrels are denoted as F n .

1) DYNAMIC FORAGING BEHAVIOR
The dynamic conduct in looking for food can be mathematically modelled as follows: The positions of individuals which are gliding from acorn trees to the hickory tree are updated as follows [27]: The positions of remaining individuals which are gliding from normal trees to the acorn and hickory trees are modified by the following equations respectively.
Gliding constant, G c is used to stabilize the exploration and exploitation searches in the SSA algorithm. Its value notably influences the performance of proposed algorithm. The gliding distance is expressed as: The gliding angle, tan (φ) is defined by The drag and lift forces can be expressed by the following equations respectively: 2

) SEASONAL ADAPTING INTELLIGENCE
The foraging behaviors of squirrels are significantly affected by the seasonal fluctuations. The squirrels are more active in autumn as compared to winter. To avert the SSA algorithm from being abused into local optimal solutions, the seasonal adapting intelligence is introduced. The seasonal constant is given by The minimum seasonal constant is expressed as The larger S min value facilitates the exploration while smaller one improves the exploitation ability of the algorithm. VOLUME 9, 2021

3) RANDOM REPOSITIONING AT THE END OF WINTER SPELL
If S t c ≤ S min , winter spell is completed. Then the locations of the flying squirrel individuals are randomly repositioned by the following equation.
Levy distribution improves global exploration ability of the algorithm and finds new candidate solutions far away from the current best solution.
The Le'vy flight is determined by α is expressed as where, (x) = (x − 1)! The pseudocode of SSA is shown in Algorithm 1.

B. EXTERNAL ELITIST DEPOSITORY MECHANISM
In this paper, an external elitist depository mechanism is implemented. An external depository is used to store non-dominated solutions found so far in the evolution process. The depository is initially empty. The non-dominated squirrel individuals found in iterations are added to the depository using the following mechanism: • If the new squirrel individual (X i ) dominates some of the depository members (X ext ), the dominated members are removed from the depository and the new squirrel individual is added to the depository.
• If the new squirrel individual is dominated by a depository member, the new squirrel individual is rejected.
• If the new squirrel individual does not dominate any depository members and vice versa, which entails that the new squirrel individual owned to the depository and it is added to the depository.

C. CROWDING DISTANCE MEASURE
The CD measure of a non-dominated solution offers an estimate of the density of solutions enveloping that solution. CD measure of an individual solution is the average distance of its two neighboring solutions, which is defined by the following equation: Algorithm 1 SSA 1: Begin 2: Read input parameters of SSA 3: Generate random positions for n number of squirrels 4: Evaluate fitness of each squirrel's position 5: Arrange the positions of squirrel individuals in ascending manner based ontheir cost function value 6: Distribute the squirrel individuals on hickory nut tree, acorn nuts trees and normal trees 7: Arbitrarily choose a number of squirrel individuals from normal trees to shift towards hickory nut tree and to transfer the residual squirrels to acorn nuts trees 8: while (Termination criterion is false) 9: For t = 1 to n 1 (n 1 = Number of squirrel individuals which are gliding from acorn trees to hickory nut tree) 10: if r 1 ≥ P dp 11: Update the position of squirrel individual using Eq. (14) 12: else 13: Randomly generate the position of squirrel individual within the search domain. 14: end 15: end 16: For t = 1 to n 2 (n 2 = Number of squirrel individuals which are gliding from normal trees to acorn trees) 17: if r 2 ≥ P dp For t = 1 to n 3 (n 3 = Number of squirrel individuals which are gliding from normal trees to hickory tree) 24: if r 3 ≥ P dp   Include X i into X ext 4: else 5: if Any squirrel individual in X ext ≺ X i , then 6: Dismiss X i 7: else 8: Include X i into X ext 9: end if 10: end if  For each objective function, the frontier solutions (solutions with smallest and largest objective values) are designated an infinite to ensure that these solutions are always chosen. The solution with the greater CD value is retained in the depository. The CD estimation is illustrated in Fig. 4.

D. MEASURE FUZZY DECISION MAKER
In multi-objective optimization problem, it is difficult to find the best solution from many non-dominated solutions. In order to compare these outcomes and get the best compromised solution, a certain mechanism is essential to combine both the objectives in conformity with the decision maker's preference. Fuzzy set theory is repeatedly used by researchers to get the best compromised solution (BCS) from many uncontrolled solutions. Degree of agreement (DA) to each objective is assigned by fuzzy membership functions, where DA reflects the merit of their objective in a linear scale of 0 -1(worst to best). If F j is a solution in the Pareto-optimal set in the jth objective function and is represented by a membership function, which is shown in Fig. 5 as [1]: For each non-dominated solution, the normalized membership function µ k D can be calculated as [1]: The solution that contains the maximum of µ k D based on cardinal priority ranking is the BCS. SSA deals with the population of squirrels, , .x t iNa in every iteration of the evolution process. The dynamic foraging behavior, seasonal adapting intelligence and random repositioning processes are applied to produce new population of squirrels, x t new . To extend the SSA approach to multi-objective optimization problems, Pareto-dominance based selection mechanism and CD measure are to be adopted. For a multi-objective optimization problem, any two solutions x t and x t new can have one of the following three promises: new are not dominated each other. Thus, the Pareto-dominance selection strategy is modeled as follows: where LC(x t , x t new ) indicates the less crowded solution between x t and x t new . The implementation of MOSSA for MAEED optimization is described as follows: Step 1. Generate initial population of squirrels with size randomly distributed across the domain of the problem.
Step 2. Evaluate the multi-objective values of each squirrel individual using Eq. (9). Step 3. Sort the squirrels' population based on nondomination. Each squirrel is ranked according to their dominance level as front 1, front 2 and so on using Eq. (30) and store them in the depository.
Step 4. Declare the flying squirrel with front 1 solution as it is on the hickory nut tree (optimal food source), the next three best flying squirrels are on the acorn tree (normal food source), and the rest of the squirrels are on the normal trees (no food source).
Step 5. Update the new position of squirrels, located on the acorn and normal trees using Eqs. (14), (15) and (16). Step 6. Randomly relocate the positions of some squirrels, when seasonal monitoring condition is satisfied.
Step 7. Update the depository by comparing the new squirrel individuals with the members of depository based on Pareto-dominance strategy using Eq. (30).
Step 8. If the depository exceeds the maximum size, delete the less crowded solutions based on the CD measure to maintain constant depository size.
Step 9. If the maximum number of iterations is reached, then go to next step. Or else go to Step 2. Step 10. Determine the best concessive solution from the Pareto-optimal set solutions stored in the depository using fuzzy-based approach as described in the previous section. The computational framework of the proposed methodology and the constraint handling mechanism are depicted in Figs. 6 and 7 respectively.

V. CASE STUDIES
To testify the preeminence of the suggested MOSSA, four case studies are analyzed on two multi-area power systems including a three-area system with ten generating units, a four-area system with forty generating units and a practical large-scale system. The original SSA, ABC, EMA and other state-of-the-art heuristic approaches are used to compare with the suggested MOSSA. The MOSSA is executed 100 runs independently and the multi-objective performance indicators including the RNI, GD, s-metric, HV and DM are recorded in box-plots. The exterior vault size is chosen as 50. The MOSSA, SSA, ABC and EMA strategies are executed using MATLAB 7.1 on an Intel core i3 processor with 4 GB RAM.
The case studies which are accompanied with both the multi-area power systems are detailed below. Case 1: Fuel cost function of the multi-area power system is minimized. Case 2: Pollutant emission function of the multi-area power system is minimized. Case 3: The two competitive objectives such as fuel cost and emission are transferred into a single objective function using WSA and price penalty factors, and then solved by the SSA approach.  Case 4: The fuel cost and emission functions are simultaneously minimized by the MOSSA approach.

A. PARAMETER TUNING
Taguchi method is used to tune the parameters of the suggested MOSSA. The parameters such as number of iterations, population size, P dp and G c are chosen as independent design variables. Each variable has three set values (level values) as given in Table 1. Then, L 9 orthogonal array is used to determine the optimal MOSSA parameters. This test system is a 10-unit three areas power system considering transmission losses, MFO and VPL effects. The input data of cost coefficients and multi-fuel type definitions are given in Ref. [12]. The tie-line power flow and the total power demand are chosen as 100 MW and 2700 MW respectively. The power demand in area 1 (1, 2, 3 and 4 units), area 2 (5, 6 and 7 units), and area 3 (8, 9 and 10 units) are 50%, 25%, and 25% of total power demand respectively as displayed in Fig. 8.

1) CASE 1
The optimal economic dispatch obtained by SSA is presented in Table 3. The optimal fuel cost obtained by SSA is 654.6016$/h. The fuel cost obtained from SSA is compared with RCGA [11], EP [11], ABC [11] and EMA in Fig. 9. When Fig. 9 is analyzed; it is evident that the SSA approach offers the lowest fuel cost among the compared approaches, proving the best solution quality of SSA approach.
2) CASE 2 Table 4 bestows the optimum emission dispatch obtained by the SSA approach and compared with those obtained by the ABC and EMA approaches in Fig. 10. It is observed from the Fig. 10 that, although feasible solutions can be obtained   by ABC and EMA approaches, SSA discovers the lowest emission of 6374.7471kg/h.

3) CASE 3
The MAEED problem is deciphered by transferring the biobjective functions into a single objective function using Eq. 8. The weighting factor is varied between 0 and 1, and the non-dominated solution set is acquired by SSA-WSM approach.  The MAEED problem is deciphered by transferring the bi-objective functions into a single objective function using Eq. 8. The weighting factor is varied between 0 and 1, and the non-dominated solution set is acquired by SSA-WSM approach. The Pareto optimal frontiers (POF) acquired by the suggested approach for various weight values are presented in Table 5. The solution with the highest membership value is selected as the BCS of the MAEED problem. From Table 5, it is to be noted that the BCS is obtained when w = 0.7.

4) CASE 4
The MAEED problem is solved by MOSSA approach. An elitist external depository mechanism is used to store 20 nondominated solutions. Then, the fuzzy decision maker is employed to decide the BCS for MAEED problem. The performance indices of MAEED problem such as fuel cost performance index (FCPI) and emission cost performance   index (ECPI) are determined as follows [6]: The BCS obtained by the suggested MOSSA approach are compared with SSA-WSA, ABC and EMA approaches in Tables 6 and 7. It is worth noting that the MOSSA provides a better POF solution as compared with the other approaches. Fig. 11 shows the MAEED performance indices of various heuristic approaches and demonstrates that the divergence between the FCPI and ECPI acquired by the MOSSA is lesser VOLUME 9, 2021 FIGURE 11. Comparison of performance indices obtained by various heuristic approaches for test system 1.  than the SSA-WSA, ABC and EMA approaches which prove the reliability of the proposed approach in offering the BCS. The tie-line power flows for the four different case studies are displayed in Fig. 12. It is observed from Fig. 12 that the tie-line power flows are not same for various case studies because of the different characteristics of objective functions. The POF obtained by SSA-WSA and MOSSA approaches are compared in Fig. 13. It is evident that the MOSSA procures lesser fuel costs and emissions as compared to the SSA-WSA approach.

C. TEST SYSTEM 2: FOUR-AREA WITH FORTY UNITS
The fuel and emission coefficients, power generation limits and tie-line limits of the four-area with forty generating units can be found in Ref. [12]. The total power demand is chosen as 10500 MW. The power demand in area 1 (1 -10 units), area 2 (11-20 units), area 3 (21 -30 units) and area 4 (31 -40 units) are 15%, 40%, 30% and 15% of total power demand respectively as displayed in Fig. 14. The tie-line power flow limit between areas 1 and 4, areas 2 and 4, and areas 3 and 4 are 100 MW. For areas 1 to 3, 2 to 3 and 2 to 4, the power flow is limited to 200 MW.    costs obtained by the SSA, RCGA [11], EP [11], DE [11], ABC [11] and EMA approaches is displayed in Fig. 15

2) CASE 2
The optimal results obtained by the suggested SSA approach for solving the emission minimization are tabulated in Table 8. Fig. 16 shows the emissions obtained by SSA, ABC and EMA strategies. As shown in Fig. 16, the emission obtained by the suggested SSA approach is better than other compared approaches.

3) CASE 3
The bi-objective function is minimized and non-dominated solutions are acquired by executing the SSA-WSA approach with different weighting values. The POFs obtained by SSA-WSA approach are bestowed in Table 9. The table shows that the best concessive solution obtained by the suggested approach is 125760.0 $/h and 206705.9 ton/h when w = 0.6.

4) CASE 4
The optimal dispatch results acquired by SSA-WSA and MOSSA are tabulated in Table 10. Table 11 presents the BCS obtained by ABC, EMA, SSA-WSA, NSGA II [2], MODE [2] and MOSSA approaches. The BCS acquired by SSA-WSA and MOSSA are depicted in Fig 17. It can again be dissected that the suggested MOSSA approach is proficient of finding the best compromise non-dominated solutions by successfully solving the MAEED problem.
Furthermore, the performance indices of MAEED procured by SSA-WSA, MOSSA and other heuristic approaches are displayed in Fig. 18. The divergence between the performances indices for this test system ensures ascendancy of MOSSA in comparison to the SSA-WSA and other aforementioned approaches in rendering the BCS. Fig. 19 shows the tie-line power flows obtained by MOSSA for all the case studies. When Fig. 19 is examined, it is seen that the tie-line power flows are varied with objective functions and altered them in reliance on a considered objective function.    [7]. The total demand is 49,342 MW. The optimal dispatch solution with proposed SSA approach is conferred in Table 12. The minimum, mean and maximum fuel costs among 100 runs of solutions obtained from proposed SSA, GSO [8], CQGSO [8], KHA [7], OKHA [7] and SDE [9] are compared in Table 13. It is noticeable from Table 12 that the fuel cost acquired through SSA for Korean non-convex system is 1559818.7289 $/h which is the lowest among the state-of-the-art algorithms. Furthermore, As can be seen from Table 13, the SSA is converged to an approximately similar solution in 100 independent runs, which demonstrates the robustness of the suggested SSA in solving the Korean nonconvex system.

E. ANALYSIS OF PARETO OPTIMAL FRONTIERS
To testify the effectiveness of the suggested MOSSA approach, the three distinctive multi-objective performance indicators, the RNI, GD, s-metric, HV and DM are assessed.

1) RNI
RNI is defined as the proportion of number of non-dominated solutions for the populace size. It can be expressed as: The higher the RNI measure, better the solution quality. The RNI obtained by the SSA-WSA and MOSSA are compared in Fig. 20. It can be seen from Fig. 20 that the suggested MOSSA acquires higher RNI values than the SSA-WSA, which indicates that MOSSA has transmuted better populace.

2) GD
GD estimates the entirety of contiguous separations of solution sets. A smaller value of GD measure indicates the better convergence of solutions. The mathematical formulation for GD is as follows [27]:

3) s-METRIC
The s-metric estimates the distance between the variance of neighboring points in the POF curve. The lower the spread value, the better the dissemination of solutions. It can be defined as follows [27]: Figs. 21 and 22 parade the GD and s-metric of the suggested approaches. It can be noticed that the MOSSA approach establishes better convergence, diversity and well distributed POF solutions.

4) HV
This indicator determines the volume (in the objective space) covered by solutions of a POF set for multi-objective problems where all objectives are to be minimized. A higher HV value is desirable for optimization algorithms. The normalized HV measure for the POF obtained for each algorithm is depicted in Fig. 23. It is evident that a higher HV is obtained by the MOSSA approach, which demonstrates the POF generated by the suggested approach is better than the SSA-WSA.

5) DM
The Euclidean distance between consecutive solutions in the POF and the mean of these distances are calculated. Then,   the DM is defined as [27]: If the DM is zero, then all the solutions of the POF are equidistantly spaced. A smaller value of DM indicates a better distribution and diversity of the non-dominated solutions. The comparison of DM obtained from SSA-WSA and MOSSA for the test systems is displayed in Fig. 24. It is obvious from the figure that the MOSSA is better than SSA-WSA in preserving the diversity.

F. CONVERGENCE BEHAVIOR AND COMPUTATIONAL EFFICIENCY
The convergence behaviors of ABC, EMA and SSA for test system 2 are depicted in Fig. 25. It is worth noting that the SSA strategy is the quickest in uniting to its final solution and furthermore offers the least fuel cost for the MAELD problem. Consequently, the results demonstrate the quicker intermingling behavior of the suggested SSA approach. The average CPU time adopted by different heuristic strategies for all the test systems are displayed in Figs. 26, 27 and 28. It may be noted that the MOSSA approach takes the least execution time with quicker union and better solution quality.

G. DISCUSSIONS
In this research article, the merits are summarized hereunder.
• To the best of authors' knowledge, this research article is the first research work of extending SSA approach for solving multi-objective power system optimization problems.
• The MOSSA demonstrates the superior performance to discover the best POF for MAEED problems with MFO and VPL effects. The BCS procured by the MOSSA for the test system 1 is 660.2238 $/h and 6441.1696 kg/h. In the test system 2, the BCS obtained from the suggested approach is 125591.3223 $/h and 205965.4061 ton/h.
• The difference between FCPI and ECPI obtained from MOSSA is lesser than those obtained from the other compared approaches, indicating the superiority of MOSSA in obtaining the BCSs.
• The MOSSA provides better POF solution compared with SSA-WSA, ABC, EMA and other state-of-the-art meta-heuristic approaches surfaced in the literature.
• The suggested approach has very fast convergence speed when compared with ABC and EMA approaches. Consequently, the suggested MOSSA is an efficient meta-heuristic approach for solving the MAEED problems.

VI. CONCLUSION
This paper proposes a new swarm optimization approach, MOSSA for solving the MAEED problem with MFO and VPL effects. Three power systems were tested to solve VOLUME 9, 2021 interconnected (multi-area) EED problems by the efficacy of the proposed MOSSA. MOSSA findings are contrasted with SSA-WSA, ABC, EMA and other recent heuristic approaches in the literature. The non-dominated POF solutions are widely disseminated and have the fastest convergence behavior with less computational effort. The proposed MOSSA is therefore a promising method for optimizing economic dispatch problems in large and small systems. It will be very fascinating to study the MAEED of hybrid renewable thermal power system in the further research.