Investigation of a Multi-strategy ensemble social group optimization algorithm for the optimization of energy management in electric vehicles

A multi-strategy ensemble social group optimization algorithm (ME-SGO) to improve the exploration for complex and composite landscapes through distance-based strategy adaption and success-based parameter adaption while incorporating linear population reduction is proposed. The proposed method is designed to acheive a better balance between exploration and exploitation with minimal tuning while overcoming the limitations of SGO. The proposed improved algorithm is tested and validated through CEC2019’s 100-digit competition, five engineering problems and compared against the standard version of SGO, four of its latest variants, five of the advanced state-of-the-art meta-heuristics, five modern meta-heuristics. Furthermore four complex problems on electric vehicle (EV) optimization namely, the optimal power flow problem with EV loading for IEEE 30 bus system (9 Cases) and IEEE 57 bus-system (9 cases) optimal reactive power dispatch with uncertainties in EV loading and intermittencies with PV and Wind energy systems for IEEE 30 bus system (25 scenarios), dynamic EV charging optimization (3 cases) and energy-efficient control of parallel hybrid electric vehicle (3 cases with 2 scenarios) covering the domains of power systems, energy and control optimization have been considered for validation through the proposed multi-strategy ensemble method and fifteen other state-of-the-art advanced and modern algorithms. The performance for the standard engineering problems and the EV optimization problems was excellent with good accuracy of the solutions and least standard deviation rates.


A. INTRODUCTION TO META-HEURISTICS
Meta-heuristic optimization is a major contributor to problemsolving and operation management and has an envisioned status among researchers and practitioners across various domains. Independent of the gradient information of the problem, meta-heuristics are applicable to both single and multi-objective problems, either continuous or discrete systems with a multitude of decision variables and constraining factors. The quality of solutions through metaheuristic optimization is reliable and, in most cases, more than satisfactory in terms of efficacy and efficiency with limited computational requirements. Swarm and evolutionary approaches have been the dominant domains of the metaheuristics with algorithms such as PSO, GA and DE being referred to as the backbone of optimization algorithms. Apart from the aforementioned state-of-the-art, research in the development of optimization algorithms continues to grow rapidly with several novel variants inspired by the various forces in nature (e.g., foraging techniques, social interactions, swarming behaviours etc.) published lately. Besides the swam and evolutionally meta-heuristics, others such as physics-based optimization algorithms, human behaviour-based optimization algorithms (HBBOAs) have gained popularity across the globe with several publications found across various domains of engineering, mathematics, computer science, decision sciences, finance and management etc. Amongst them, the growth of HBBOAs have been on the rise since the proposal of Taboo Search Algorithm (TSA) in 1996 [1]. Following it, many others such as Harmony Search (HS) in 2001 [2], Imperialist Competitive Algorithm (ICA) in 2007 [3], Teaching-learning-based optimization (TLBO) in 2011 [4], Social Group Optimization (SGO) in 2016 etc. have been the prominent ones. As mentioned earlier, these paradigms are inspired by the improvisation and interaction of human beings as they deal with complex problems and a few examples are the improvisation of music players, the conquest amongst various empires in a colonial system, knowledge sharing and gaining in a classroom, group counselling, sports tournaments and competitions etc. Simplicity, reliability, efficacy have been the attributes that have attracted many researchers to deploy the HBBOAs as part of their optimization research.

B. IMPROVEMENTS AND ADVANCEMENTS IN META-HEURISTICS
The traversal of the search space is dictated by two inchoate phases, namely, exploration or diversification (often referred to as "Global Search") and exploitation or intensification (often referred to as "Local Search"). Exploration of a larger area of the search space is often the key to enhancing population diversity lowering the risk of population stagnation which in turn leads to local entrapment and premature convergence. Exploitation on the other hand is essential to accelerate convergence and improve the accuracy of the solutions found so far. To summarize, the perfect balance of the two conflicting aspects of exploration and exploitation is crucial to extract the best possible performance of a meta-heuristic in terms of quality of the solutions, consistency, convergence etc.
In most meta-heuristics, the control of these conflicting aspects is often done through "algorithm-specific tuning parameters" or through "parameter tuning" in short. Ranging from one parameter to several in number, a precise setting of these parameters is often the backbone to eventuate to a good outcome for the chosen problem. Benchmarking tests and empirical results are the most-employed methods pertinent to achieving the best trade-off as seen in a myriad of works. Other complex and viable methods include F-race tuning, Chess Rating System (CRS-Tuning), REVAC (Relevance Estimation and VAlue Calibration) etc. with integration of chaos theory and versatile tuning operators have been deployed successfully in the literature. While a smaller number of tuning parameters with simpler tuning is congenial, it can prove ineffectual at times with complex search landscapes and large number of problem dimensions. On the other hand, complexity associated with advanced tuning techniques can be difficult for practitioners all while providing nominal improvements in the outcome. Hence, dynamic and adaptive tuning strategies that can intelligently modify the exploration quality and scale with respect the problem's landscape and dimensions while requiring minimal and basic settings are often implemented in various advanced and modern meta-heuristics.
Other reasons to allude to the lack of a competitive performance are to do with the algorithmic structure, population selection and sorting strategies and excessive dependence on one or few search strategies with little to no adaptive measures to improve the population diversity. Most modern meta-heuristics rely on simpler strategies with the incorporation of the global best solution found so far (often termed as "Leader" or "Gbest") as a propensity to enhance global search (also accelerating convergence) while the fact that such strategies are one-sided and are often found to drift towards the geometric centre of the search landscape. The research article at [5] presents evidence as to how shifted and rotated test functions can prove detrimental to such onesided search methodologies.

C. MULTI-STRATEGY AND MULTI-POPULATION BASED IMPROVEMENTS
There has been a mammoth of research to improve or enhance the limitations with such search methodologies in the past and the recent literature. Modifying the algorithmic structure to suit the search landscape either for complex benchmarking or domain-specific problems are achieved through a myriad of techniques and hybridization or combination of two or more meta-heuristics for a synergistic boost in the performance have been very popular with researchers from various domains. Likewise, the ensemble techniques integrating multiple meticulously designed and re-forged search strategies with adaptive tuning operators have also contributed to the improvement of the classic paradigms. Additionally, multi-population techniques incorporating a different set of populations with each set governed and dictated by distinguished search techniques have also been popular among the community of optimization.
Performance improvement through the avoidance of local entrapment while staying true to its faster converging nature have been the ultimate goals with such implementations. The other side of the coin is the demerits that accompany them including, increased computational resources, complexity and computational times, a larger number of function evaluations, complexity in implementation owing to the tuning prerequisites for individual search strategies in multiensemble techniques, lack of a strong immunity to "the curse of dimensionality", very slow convergence rate for simpler problems etc.
Although multi-population ensemble techniques are hailed as the state-of-the-art for a wide range of problems, the tedious coding and tuning of these can be excruciating to the average practitioner. Hence, a balanced approach relying on simpler yet meticulously designed, multiple yet fewer search strategies with lower tuning requisites and adaptive techniques are preferred while standing unabated to the performance in terms of solution quality and convergence.

1) LITERATURE SURVEY OF THE STATE-OF-THE-ART MULTI-STRATEGY AND MULTI-POPULATION BASED IMPROVED ALGORITHMS
A literature survey of the most-cited multi-strategy and multi-population based improved meta-heuristics is presented below.
GA based ensemble algorithms (i) A two-stage multi-population genetic algorithm (MPGA) was proposed by Jefery K. Cochran et al. [6] in 2003 incorporating sub-population evolution and elitism to optimize parallel machine scheduling problems. MPGA outperformed MOGA for scheduling problems with two and three objectives with a higher number of Pareto Front solutions with better solution quality although the limitation that both the algorithms produced unwanted solutions dominated by others was acknowledged. (ii) A novel multistrategy ensemble ABC (MEABC) algorithm, the coexistence and competition between pools of distinct solution search strategies i.e., The original ABC, GABC and Modified ABC/best/1 is realized [7]. Benchmarking through 12 commonly used functions and the CEC2013 test suited is utilized while comparisons with the state-of-the-art variants of PSO, DE and ABC are made to demonstrate the effectiveness of MEABC. (iii) An adaptive collaborative optimization algorithm integrating GA's exploration prowess and ACO's stochastic abilities in a multi-population strategy known as MGACACO is proposed [8]. Various scale travelling salesman problems (TSP) are considered to verify the proposed approach. The proposed method outperformed the parent algorithms with better accuracy and fast convergence while avoiding local optima.
PSO based ensemble algorithms (i)A multiagent-based Particle Swarm Optimization (MAPSO) for optimal reactive power dispatch integrating lattice-based agent-agent interactions and knowledge-based learning to improve optimality and accelerate convergence has been proposed in [9]. MAPSO outperformed SGA and PSO at lowering the active power losses with lower executions times compared to the latter. (ii) Multi-strategy ensemble particle swarm optimization was proposed in 2008 by W. Du et al [10] . MEPSO categorizes the particles into two parts with Gaussian local search and differential mutation guiding them to accelerate convergence and prevent local entrapment respectively. Experimental analysis with the moving peaks benchmark (MPB) and dynamic Rastrigin functions demonstrated the effectiveness of MEPSO at evading entrapment compared to other variants of PSO. (iii) Y. Wang et al. proposed the Self-adaptive learning-based particle swarm optimization (SLPSO) in [11] with four PSO strategies with a self-adaptive probability model based on the fitness landscapes. Extensive comparisons with eight state-of-the-art variants of PSO for 26 numerical optimization problems and economic load dispatch problem of power systems (ELD) are performed with SLPSO being the top-performer. (iv) In 2013, Diversity enhanced particle swarm optimization with neighbourhood search (DNSPSO) was proposed [12]. To achieve a better trade-off between exploration and exploitation, diversity enhancing mechanism and neighbour search with local and global search systems are integrated and evaluated using 15 standard benchmark functions, CEC2005 and CEC2010 test suites. The proposed method was successful with the least mean errors compared to the variants of PSO. (v) A quantum-behaved particle swarm optimization algorithm incorporating flexible single-/multi-population strategy and multi-stage perturbation strategy (QPSO_FM) to balance the diversity and the convergent speed is proposed in [13]. Benchmarking with 28 standard benchmark functions with several other quantum variants of PSO demonstrated its effectiveness at providing an accelerated global search.

DE based ensemble algorithms
(i) Neighbourhood mutation strategy integrated with various niching differential evolution (DE) algorithms (NCDE) was investigated by B.Y. Qu et al. in [14]. Euclidean neighbourhood-based mutation improved the performance for multi-modal landscapes tested against (14 basic multimodal and 15 composite multimodal problems). (ii) Multipopulation ensemble DE (MPEDE) with three mutation strategies and population pools incorporating a dynamic allocation of fitness evaluations to the best strategy has been proposed by G. Wu et al. in 2016 [15]. Control parameter adaption for each mutation strategy is integrated as well and the improved performance is demonstrated against the CEC2005 test suite comparing several variants of DE. (iii) Ensemble of differential evolution variants (EDEV) incorporating JADE, CoDE and EPSDE with three indicator sub populations and one reward sub population was proposed by G. Wu et al in 2018 [16]. EDEV outperformed several variants of DE for the CEC2005 and CEC2014 test suites.
Other ensemble algorithms (i) In 2005, a restart-Covariance Matrix Adaptation Evolution Strategy (CMA-ES) with restart strategy incorporating increments to the population size (IPOP) known as IPOP-CMA-ES has been proposed by A. Auger et al. [17]. CEC2005 real-parameter optimization test suite with 25 functions were chosen in a benchmarking analysis with the proposed method outperforming the local restart strategy in 29 out of 60 cases. (ii) An Improved Ant Colony Optimization Algorithm Based on Hybrid Strategies (ICMPACO) for TSP and actual gate assignment problem is realized in [18]. The proposed multi-population approach includes co-evolution mechanisms with pheromone updating and diffusion mechanisms for better exploration-exploitation balance and achieved better assignment results. (iii) Multipopulation differential evolution-assisted Harris hawks optimization with chaos strategy (CMDHHO) to avoid local entrapment has been realized in [19]. In a comparative analysis with several modern and advanced meta-heuristics with CEC2017 and CEC2011 (selected functions for realworld issues) test suites, CDMHHO outperformed them. (iv) Chaotic multi-swarm whale optimizer (CMWOA) by M. Wang [20] for support vector machine-based medical diagnosis combining chaotic and multi-swarm strategies is proposed. In a comparative analysis against PSO, BFA and PSO, the proposed method achieved better classification performance and feature subset size. (v) A multi-strategy ensemble GWO (MEGWO) with an enhanced global-best lead strategy to improve local search and an adaptable cooperative strategy to promote global search and population diversity is proposed in [21]. 30 benchmark test problems from the CEC2014 suite are chosen for the benchmarking and 12 feature selection datasets are considered. In a comprehensive comparison with various meta-heuristics, MEGWO showcased robust optimization results for both benchmarking and feature selection.
A brief summary of the aforementioned publications considered for the literature survey has been tabulated in Table A.I (Appendix)

D. CONTRIBUTIONS OF THE CURRENT ARTICLE
Following the literature survey of the state-of-the-art, the current article proposes a multi-strategy ensemble social group optimization (ME-SGO) algorithm to improve the performance of the standard social group optimization (SGO) for complex and composite landscapes and an investigation of its performance for complex multidimensional, non-linear, multi-constrained problems on the optimization of electric vehicles from the recent literature is made. The reasons for the choice of SGO as the optimizer to be improved and the selection of the four problems on EV optimization are listed in the following sub-sections.

1) CHOICE OF SGO
The following have been the factors for the choice of SGO over other contemporary meta-heuristics. 1) SGO is a relatively new meta-heuristic proposed in 2016 with a simple structure and can be implemented on multiple programming languages with support for parallel computation and black-box mode of implementation.
2) The performance of SGO for unimodal and multi-modal landscapes has been outstanding as it outperformed several state-of-the-art variants of DE, PSO, ABC in recent publications [22] [23] [24], [25] . 3) There exists a huge potential to improve and enhance the exploration of SGO through multiple strategies with a wide range of parameter adaptation techniques for composite and hybrid search landscapes where it is known to struggle. 4) Although a few improved and hybrid variants of SGO exist in the recent literature, none of them have demonstrated the improved performance for complex and composite landscapes. This has been the centre of focus in the current manuscript. 5) Very little effort has been made to improve the performance of SGO through dynamic and adaptive parameters that control its search process (population size, social introspection factor). 6) Efforts made to improve SGO through the enhancement of population diversity have not been comprehensively verified with other state-of-the-art advanced metaheuristics for complex and real-world optimization problems.
Hence based on the aforementioned aspects, the proposed multi-strategy ensemble variant of SGO aims to deliver a better balance of exploration and exploitation for complex real-world problems especially for complex and composite landscapes with a higher degree of robustness and precision.

2) OPTIMIZATION IN EV'S
The testing and validation of advanced meta-heuristics are often performed through real-world multi-constrained problems known to be complex and computationally expensive as they help evaluate their overall performance concerning limited computational resources, high dimensionality and high-multimodality with a larger degree of complexity in exploring its dynamic search landscapes. Besides these, a higher number of equality and inequity constraints often restrict the algorithm from exploring the landscapes to their fullest potential which is often the case with static control parameters. In this regard, four complex problems on EV optimization covering the areas of power systems, energy management and control optimization from the recent literature are chosen to demonstrate yeh the performance potential of the proposed method. The following are the reasons for their choice. 1) Electric vehicles have emerged as the next milestone in the transportation sector and have been the centre of focus for research and development over the last decade. More often, the problems on EV optimization are modelled as optimization problems (Linear programming, non-linear programming, integer programming, mixed-integer non-linear programming, convex programming etc.) and solved through various meta-heuristics and solvers. Optimization through metaheuristics has been the choice on-the-go for many researchers and practitioners on this topic. 2) Most EV optimization problems follow complex mathematical modelling with multiple equality and inequity constraints with a large number of non-separable problems dimensions covering multiple areas of power systems, control optimization, design and energy management with complex landscapes requiring dynamic optimization strategies to ensure better optimality.
3) The integration of machine learning and predictive control techniques can be efficiently coupled with optimization techniques to lower the learning errors paving way for truly autonomous driving and cruise control etc. 4) The design and management of EVs is one such area which requires the collaborative co-optimization of rulebased control and optimization of energy management to work in synergy to ensure optimal driving efficiency. 5) Path finding, EV routing, optimal charging and discharging, optimal planning of EV charging location and charging infrastructure etc. are the best examples that require robust and dynamic optimization techniques to determine the optimal solution as the scope of these areas tends to expand.
Over the last decade, the optimization in very domains concerning EVs have been dominated by the improved/hybrid meta-heuristics indicating the efficiency of adaptive techniques over the classical paradigms. A brief literature survey depicting the development of various improved and advanced meta-heuristics to the very domains of EV optimization is presented in Table A.II (Appendix) Considering the following aspects, four complex problems namely, the optimal power flow problem with EV loading for IEEE 30 bus system (9 Cases) and IEEE 57 bus-system (9 cases) optimal reactive power dispatch with uncertainties in EV loading and intermittencies with PV and Wind energy systems for IEEE 30 bus system (25 scenarios), dynamic EV charging optimization (3 cases) and energy-efficient control of parallel hybrid electric vehicle (3 cases with 2 scenarios) coverage the domains of power systems, energy and control optimization have been considered for validation through the proposed multi-strategy ensemble method and fifteen other state-of-the-art advanced and modern algorithms.

3) ORGANIZATION OF THE ARTICLE
The remainder of this article is organized as follows. Section II focuses on the literature review and working of SGO, review of its variants followed by a discussion of its merits and demerits. Section III discusses the formulation of the multi-strategy ensemble SGO technique with a detailed description of its various attributes. The performance of ME-SGO with fifteen different meta-heuristics (including four variants of SGO, four modern meta-heuristics, and seven state-of the art advanced meta-heuristics) is analysed in Section IV with CEC 2019 benchmark suite and the 100digit competition followed by a comparative analysis on standard engineering problems (pressure vessel design, welded beam design optimization, tension/compression spring design optimization, cantilever beam design and design of 10-bar truss optimization). Section V analyses the performance of the prosed method and the fifteen competitor algorithms on the four and real-world constrained complex EV optimization tasks The conclusion, followed by the merits and demerits of ME-SGO, potential applications and the future scope of the current work are given in Section VI.

II. SOCIAL GROUP OPTIMIZATION
Social Group Optimization (SGO) is a human behaviour inspired evolutionary technique, proposed by Suresh Satapathy and Anima Naik in 2016 [26]. The inspiration of SGO stems from the social behaviour of human beings collectively working together to solve complex problems.
The following sections explain the working of SGO, various attributes of SGO, merits and demerits followed by a detailed literature review of the algorithm including its variants.

A. WORKING OF SGO
SGO is implemented in two phases, namely, the improving phase and the acquiring phase. Both the phases rely on simple evolutionary equations to transform the solutions obtained through random initialization at the beginning following a greedy selection strategy. The search process commences with the identification of the "leader" or "gbest" from the randomly initiated population pool.

Improving phase:
The "leader" influences the population members and propagates his knowledge resulting in the repositioning of the population pool with reference to the "leader". The new positions of the population pool are updated as described by (1).
(1) end for end for where, N stands for the population size, D stands for the number of problem dimensions, t stands for the current iteration, r is a random number in [0, 1] and r ∼ U (0,1), 'c' is the selfintrospection factor whose value can be set with the range 0 < c < 1.

Acquiring phase:
Contrary to the improving phase, the acquiring phase is intended for the interaction of the members in the population pool with the leader and other random population members. The interaction is conditional with the person having a greater knowledge transferring his/her knowledge to the other person while a person with lesser knowledge acquires it from a higher knowledgeable person. Since the leader of the social group interacts with every other population member, he/she has the greatest influence on the others to learn from him/her. The new positions of the population pool are updated as described by (2)  where, r1 and r2 are two random numbers in [0, 1] and r1, r2∼ U (0,1). Table I lists the merits and demerits of SGO based on a comprehensive literature survey.

B. ANALYSIS AND DEDUCTIONS FROM THE PREVIOUS PUBLICATIONS AIMED AT IMPROVING SGO
Following the proposal of SGO in 2016, several improved variants of SGO have been found in the literature. Exploring the literature, three improved variants, three hybrid variants, one modified and one discreet variant of SGO were found. A deeper analysis of these variants indicates that research into improving SGO has been aimed at enhancing the population diversity to help evade local entrapment. A brief discussion of the variants is given below.

1) Improved SGO (ISGO-Variant 1) based Support Vector
Machine (SVM) classifier for transformer fault diagnosis model using an optimal hybrid dissolved gas analysis features subset was proposed by J.Fang et al. in [27]. The proposed method aimed at the prevention of local entrapment in SGO through the incorporation of population sub-grouping and eliminating phase to enhance the explorative potential. The proposed method recorded better fitness compared to GA, PSO and SGO based classifiers. 2) In [28], Cluster Head Multi-Hop Routing Algorithm based on another Improved SGO (ISGO-Variant 2) was proposed. The authors proposed a three stage Improved SGO with historical population memory and a ranking system followed by the initial 25 percent of the population learning from the last 25 percent of population. Intended at improving the population diversity, the proposed ISGO outperformed the competitor algorithms for maximizing the network life cycle and minimizing the energy consumption. 3) In other works, Improved SGO (ISGO-Variant 3) for short-term hydrothermal scheduling by Akash et al. was proposed [29]. It expands the concept of a self-awareness probability (SAP) factor from MSGO [30] to improve the diversity through re-initialization of the population in the acquiring phase. It performed competitively with lower production costs compared to the competitors in four cases tested. 4) Modified SGO (MSGO-Variant 1) was proposed by A.
Naik et al. in [30]. A novel modification to the acquiring phase known through the addition a new control parameter known as self-awareness probability (SAP) to enhance the exploratory capabilities with increased population diversity is realized and uses the reinitialization of the solution vector to achieve this. In an extensive benchmarking analysis with 23 classical functions and 3 cases of hydrothermal scheduling problems, MSGO outperformed several classical and contemporary meta-heuristics. Following it in 2021, the same MSGO for circular antenna array optimization was proposed in [31] where MSGO outperformed the classical SGO in terms of optimality, accuracy, convergence and robustness across three cases.

5) The hybridization of SGO and Whale Optimization
Algorithm (WOA), another popular contemporary swarm-based meta-heuristic to realize two hybrid variants were developed by K.V.L Narayana et al. in [32]. A lite version named HS-WOA to improve the exploitation and convergence speeds through a modified acquiring phase with SFEs and an extended version (HS-WOA+) with DFEs to improve the explorationexploitation balance was proposed. Extensive comparisons with recent and classical paradigms for 30 benchmarking, 4 engineering problems and a multi-unit production planning were carried out to demonstrate the effectiveness of the proposed methods with HS-WOA+'s performance being good for most of the testing. 6) In other developments, a hybrid of SGO and GA, known as HSGO [33] incorporating a new mutation phase into SGO to facilitate continuous improvement in the population is proposed. Deployed to detect COVID-19 Infection from chest X-Ray images, the HSGO based SVM classifier achieved an accuracy of 99.65% among all classifiers outperforming them. 7) A discretized adaptation of SGO known as DSGO to solve the popular Travelling Salesman Problem (TSP) was proposed in [34]. Compared to GA and DPSO, DSGO achieved minimal costs for five TSP datasets while demonstrating accelerated convergence.
Following them were two comparative studies at [22] comparing SGO with recent algorithms from 2017 to 2019 for multiple classical benchmark functions while the analysis at [23] investigated the adaptive tuning mechanisms for the self-introspection parameter for solving engineering design problems.
A brief description of the variants of SGO is summarized in Table II.  TABLE I  TABULATION OF THE MERITS AND DEMERITS OF

Merits Demerits
Simpler and straightforward to code and can be implemented across a wide range of programming languages.
Although Double Fitness Evalutions per iteration (DFEs) improve the search behaviour, they can lead to a compromise in the population size or iterations under fixed computational requirements.
SGO is excellent for unimodal, most multi-modal and constrained search landscapes (continuous and discrete) with a faster convergence rate to the global optimal solution.
The excessive dependence on the leader can lead to local entrapment in complex search landscapes. This coupled with greedy selection is more likely to cause population stagnation resulting in premature convergence. Smaller number of tuning requisites, i.e., one algorithm-specific tuning parameter (self-introspection factor 'c') makes it easier to regulate the explorative behaviour for a wide range of problems.
The tendency of the search mechanism to slide to the geometric centre of the search landscape can be detrimental for rotated and shifted landscapes.
Good immunity to the curse of dimensionality and excellent for global search with the greedy selection process updating the population twice in every iteration.
The lack of any adaptive measures can result in the greedy selection limiting the population diversity in complex multi-modal problems.
The modules in the algorithm can be hybridized with other meta-heuristics.
The implementation of the improving phase for the entire population can result in a loss of diversity by concentrating a larger section of the population closer to the gbest. This is followed by the acquiring phase for all the population members leading to shallow exploitation of the search space.
Rapid convergence to global optimum for separable benchmark function due to its strong exploitative capabilities. The empirical setting of the self-introspection factor may not be suitable at all times. Improper setting can lead to the fitness evaluations being futile and render the search process useless at times. Parallel computational techniques cannot be implemented to efficiently distribute the computational tasks in multi-core machines due to the limitation of the search process. A sudden transition from exploration to exploitation witnessed for non-separable benchmark functions indicates a higher probability of local stagnation brought upon by the constriction of the available search space due to limited population movement throughout exploration.

III. PROPOSED METHOD: MULTI-STRATEGY ENSEMBLE SOCIAL GROUP OPTIMIZATION (ME-SGO) WITH LINEAR POPULATION REDUCTION TECHNIQUE
The proposed multi-strategy ensemble social group optimization aims to deliver a good balance between the exploration and exploitation while ensuring that local entrapment is avoided. Hence, to improve the population diversity and enhance the search capabilities, multiple strategies are designed and integrated systematically to keep track that the algorithm aims for global search. A detailed explanation is provided the following sub sections.

A. MOTIVATION
After a careful analysis of the various works aimed at improving the standard SGO algorithm, the motivation for the current work is as follows: 1) SGO lacks population diversity since the improving phase and acquiring phase are implemented for the entire population and not for individual population members. This system where both the phases rely on greedy selection and as the population pool enters the acquiring phase, very little room exists for further improvement casing clustering leading to local entrapment.
2) The improvement phase requires additional modifications to dynamically adapt to complex landscapes through strategic search equations to improve diversity. A reason to modify improving phase is to ensure that all the population members are not drawn too close to the leader and prevent the of the function evaluations being futile.
3) The static nature of the self-introspection factor from the improving phase is another aspect that can drive the nature of the search process. Furthermore, a dynamically adaptive self-introspection factor 'c' can significantly improve the exploration during the improving phase. 4) The acquiring phase, although provides ample comparisons among the population can be modified to target the movement of the population towards a global optimum through its immediate implementation after the improving phase for every population member rather than in groups. This way, every population member from the improving phase gets an opportunity to interact with either a random improved solution or one with no improvement preserving population diversity. 5) SGO's adaptation of double fitness evaluations requires either the population size or the iteration count to be lowered to match the required NFEs compared to other modern optimizers with single fitness evaluations. Gradual population reduction schemes can be experimented with in this regard to ensure a higher initial setting for the population size and iterations ensuring a better balance of exploration and exploitation. 6) SGO is excellent at local search providing accelerated convergence to the obtained local optimum points and this ability of SGO can be exploited and further enhanced through modifications to both the improving and acquiring phases.
Following the aforementioned aspects, the following modifications and improvements have been considered in the current work.
1) To adapt to dynamic and complex landscapes, the proposed ME-SGO incorporates dynamically adaptive features incorporated into the improving phase, acquiring phase, population size and the selfintrospection factor. The complexity of the search landscapes dictates the adaptive rate of these strategies and parameters. 2) To prevent loss of diversity and improve the successful utilization of the function evaluations, the improving and acquiring phases are implemented for every individual population member in an iteration as opposed to the implementation in groups.
3) The improving phase is given a major overhaul with distance-based strategy adaption and success-based control parameter adaption. The distance-based strategy adaption splits the improving phase into two sub-phases each triggered by a pre-set number of function evaluations. 4) The acquiring phase also adopts parameter adaption with a focus on directing the particles to explore around the leader rather than exploiting the same search space. 5) Linear population reduction technique (LPRT) to ensure heavy emphasis on exploration and diversification during the initial half of the search and transition to exploitation is implemented to enable a higher initial population. LPRT and distance-based strategy adaption ensure the prevention of early entrapment to make sure that the search process continues to adapt to complex landscapes. 6) Population elimination feedback from the population being discarded due to the reduction of population is considered to help guide the remaining population members to explore the potentially promising areas in the search space.

B. IMPLEMENTATION
ME-SGO is implemented in two phases similar to SGO, which are the enhanced improving phase with global search and adaptive acquiring phases respectively. In each phase, the greedy selection technique is implemented to select the newer population with better fitness than its predecessors. Linear population reduction strategy is applied on top of the whole exploration system to encourage deeper exploration and enable a smooth transition from exploration to exploitation. The individual phases are detailed as follows.

1) ENHANCED IMPROVING PHASE WITH GLOBAL SEARCH
The improving phase in SGO is aimed at exploring around the Leader to further improve the solution quality. The selfintrospection factor set through empirical analysis servers as a control mechanism to limit the velocity of each population member. The greedy selection follows the improving phase to ensure that the fittest members are included while the others are discarded. Although it has been effective for most unimodal and a few multi-modal problems, this system is often prone to local entrapment as a result of excessive dependence on the leader in dynamic search landscapes especially resulting in poor performance for the shifted and rotated composite landscapes. Population stagnation can occur if the fitness of a member fails to improve since the greedy selection discards any solution with an inferior fitness.
The enhanced improving phase incorporates much more efficient strategies to explore a vast majority of the landscape while learning from the experience of the leader. This system incorporates the previous improving operator with a new modified improving operator and a modified differential mutation operator to allow for a larger exploration of the search space and prevent it from quickly transitioning to exploitation. The enhanced improving phase is split into two sub-phases with the first phase known as the enhanced explorative phase implemented for the first half of the function evaluations followed by the enhanced exploitative phase for the other half as described in (4 (4) The enhanced explorative phase is designed to take advantage of the increased population available during the initial stages of the search process. Enhancement of diversity is set as the primary goal of this process and the newer solutions are generated through combinations of multiple difference vectors to drive the current population to explore the vastness of the search landscape. The equations concerning the generation of new a solution is described by (5) and (6) respectively.
Randomly select a member from the population pool such that ≠ If f(P i ) < ( ) ⃗⃗⃗⃗⃗⃗⃗⃗ denote the positions of any two randomly chosen population members from the current iteration, Leader j is the best solution obtained so far and Worst t ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ is the worst solution from the current iteration, stands for the success-history based dynamic self-introspection factor in the range 0.1 and 1.
is a random number in 0 and 1 dynamically updated at the end of every iteration and for every reset of The inclusion of the worst solution is to ensure that diversity is preserved during exploration. Multiple difference vectors prevent the clustering of solutions at a single point and the adaptive self-introspection factor allows for controlled freedom of the particle to navigate and expand the solution space.
The enhanced exploitative phase includes the original position update equation from SGO and adds a novel feedback position update system with a probabilistic selection between both strategies. This is described by (7) and (8) respectively Obtain a random value for "Sel" through uniform distribution If Sel > .
Else where, Sel is the exploitation scheme selector, FB-Leader and FB-Worst denote the best and worst solutions from the eliminated population set to provide feedback to the current population, C R is known as the randomized selfintrospection factor re-initialized in the range 0.2 to 1.0 with respect to the learning rate and F denotes the scaling factor.

2) ADAPTIVE ACQUIRING PHASE
The acquiring phase in SGO is focused on enhancing population diversity through comparative learning between the population members and the leader. This phase is inspired by the information exchange in society as each population member interacts with other random members while also interacting with the leader. Information is either transferred or gained between the members of the population based on the intellect of the two members interacting. The acquiring phase contributes to a quicker convergence and the inclusion of a greedy operator for every new solution combination may lead to loss of population diversity.
The adaptive acquiring phase implements a fitness-based selection system between the two population members devised below to improve the diversity of the population being generated. Premature convergence as a result of entrapment and stagnation can be avoided through this method. Re-initialization has not been considered since its contribution to the overall population diversity is negligible with the current greedy selection technique. The population update equations are specified by (9) and (10) respectively.
Randomly select a member from the population pool such that ≠ If f(P i ) < ( )

B. LINEAR POPULATION REDUCTION TECHNIQUE (LPRT)
The population management in ME-SGO is done through a linear population reduction technique where the members in the population pool are gradually decreased from a maximum population size to a minimum population size, both of which can be set as required. The key advantage of this strategy is that the exploration quality is enhanced by a larger degree and the risk of local entrapment and premature convergence is minimized. Since, every population member is compared to their leader and its previous iteration counterpart, the information exchange is adequate such that the elimination of members in the population pool is unlikely to have any effect on the outcome of the exploration. Initially, as the algorithm beings the search, it can sample a large number of solution combinations and as the iterations progress, a smooth transition from exploration to exploitation is possible.
The population updating process occurs twice in every iteration allowing for more interactions between the member in the population pool and generating new population members with good diversification and superior fitness. The absence of any sorting procedure to further sort and select the next generation of population enables the proposed method to be quicker than the algorithms, thereby reducing its time complexity. The upper limit and lower limit for the population size can be set based on the number of function evaluations (NFEs) and it is recommended for adequate exploitation to occur, the lower limit of the population be at least one-tenth of the upper limit. The population size is determined as per (11).

C. PARAMETER ADAPTION
The key to improving the performance of SGO is to dynamically adapt the self-introspection factor to the complex landscapes through a series of successes and failures. Authors at [23] demonstrated this through an investigative analysis of the various inertial control schemes for various unimodal and multi-modal landscapes with the concussion that a static setting of 'c'=0.2 is often the best for unimodal landscapes while inertia-based increments to 'c' with respect to the progression of iterations can be exploited for multi-modal landscapes. Grounding on this, a learning mechanism to increment the value of is devised as per (12).
As per the adaptive scheme, the value of retained for the successful new population with improved fitness and is re-initialized for the maximum number of failures. Failures are set to zero at the initialization and are incremented by 1 for every population member that fails to generate a superior offspring in either the enhanced improving phase or the adaptive acquiring phase. The learning rate is devised (empirically set to 10) to ensure that every new combination of is given an ample number of trials to improve the quality of the solution. Besides , the values of C R is set to be re-initialized within the range of 0.2 to 1.0 (the recommended range for c from the standard SGO) whenever is modified and R being randomized in the range 0.1 to 1.0 at the end of every iteration and whenever is modified to ensure that static settings for control parameters are avoided to the most possible extent.

D. EXPLORATION VERSUS EXPLOITATION
Besides the distance-based strategy adaption and successbased parameter adaption, the dynamic population control through LPRT serves as the backbone to efficiently balance exploration and exploitation. While the distancebased strategy adaption ensures that population diversity is enhanced, LPRT ensures that the maximum possible population is dedicated to it. The initial higher population enhances the reach of the population to multiple corners of the search space across multiple dimensions as it proceeds to exploit them during the latter stages. As the population size is lowered, the feedback enhanced exploitation phase from the enhanced improving phase proceeds to exploit the most promising areas discovered thereby improving the accuracy of the solutions. The adaptive acquiring phase extends the exploration to a global scale pushing the remaining population to further explore after the first explorative phase thus extending the exploration over a larger timeframe and allowing the population to explore and exploit simultaneously. The ensemble of these strategies allows for explosive exploration while allowing smoother yet careful exploitation over the course of iterations to achieve a near-perfect balance of the exploration and exploitation dynamically.

E. TIME COMPLEXITY AND COMPUTATIONAL COMPLEXITY
The position update system in ME-SGO occurs twice i.e., the first position update in improving phase followed by the second position update in the acquiring phase. The greedy selection follows both the phases to decide on preserving the fitter solutions or discarding the inferior ones. The fitness evaluation and the position updates are performed for all the members in the population pool twice in an iteration. Hence, it is obvious that ME-SGO performs double fitness evaluations (DFEs) per iteration. For an iterative count of T iterations with a population size of N each having a D number of decision variables/dimensions, the following are the computational complexities of individual phases. The computational complexity of initialization is O(D), the computational complexity of the fitness evaluation is O(N), the computational complexity of the position updation is O(T×(N×D)). This is followed by fitness evaluation of all the new position for the greedy selection with O(N×T). Since, ME-SGO relies DFEs and updates the position of the population twice in every iteration, the total computational complexity of is O(N×(D+2×(T+(T×D)))). In the same manner, the time complexity of ME-SGO is measured considering its total run time i.e., 'ttotal' for one independent run. It is as shown in 13. ttotal = t1×O1+ t2×O2+…….tN×ON (9) where, t1, t2…..tN are the computational times needed by SGO to complete the various operations O1, O2…..ON for N population size. The various operations and the time requirements are presented in Table III. Hence, from Table III, it can be concluded that the time complexity of ME-SGO is O(N).

IV. BENCHMARKING ANALYSIS
The benchmarking of the proposed method is performed in two phases i.e., the first phase comprises of benchmarking test functions following the latest standards (10 complex multi-modal functions from the CEC2019 test suite for the single-objective optimization) followed by the second phase with 5 constrained standard engineering problems (pressure vessel design, welded beam design, cantilever beam design, tension/compression spring design and 10bar truss design optimization). All the experimentations considered for the current work are performed on a hp Ultrabook running the operating system of Microsoft Windows 10® Pro (Version 20H2 -OS Build 19042.1165) with 16 Gigabytes of DDR3 RAM powered by an Intel(R) Core (TM) i7-4700MQ quad-core CPU @ 2.40GHz. MATLAB R2020a is chosen to code all the algorithms for all the considered exterminations in the comparative analysis.

A. PERFORMANCE EVALUATION CRITERIA
The performance evaluation criteria are as follows. (1) The best, worst, average (mean) and standard deviation values are obtained based on 51 independent runs for all the all algorithms in comparison. (2) The first statical test, i.e., Wilcoxon's rank-sum test at a 0.05 significance level is performed for ME-SGO concerning the other algorithms. For better performance of the other algorithms with respect to ME-SGO "+" symbol is used, for the similar performance of the other algorithms with respect to ME-SGO "≈" symbol is used and for the inferior performance of the other algorithms concerning ME-SGO "-" symbol is used. (3) The second statistical test, i.e., a ranking test through a non-parametric Friedman's test is performed to rank the best-performing algorithms. (4) Furthermore, the mean absolute errors (MAE) to indicate the difference between the global optimal solution and the best solution obtained by each algorithm is evaluated. (5) The convergence graphs are provided for the CEC2019 benchmarking suite to showcase the converge characteristics of the proposed method. (6) The population diversity plots (Analysis of variance -ANOVA/box plots) are provided for the CEC2019 benchmarking suite. (7) The average computational times (Seconds) for the 51 runs are recorded. The initial population is generated randomly with a size of 'NP×dim' 5.
Identify the leader/gbest and the worst 6.
for t=1 to T 7.
for i=1 to Np 8.
Implement "Enhanced Improving phase" end Enhanced Improving phase 11.
Implement "Greedy Selection I" 12. Update Update the leader/gbest and the worst 15.
Set i ≠ r 17.
end Adaptive Acquiring phase 22.
Implement "Greedy Selection II" 23. Update Update the leader/gbest and the worst 26.
end for-T 28.
Update Np

N P =round [(N P max -NFE current )× (N P max -N P min )
Max NFEs ]

30.
Check the termination criteria 31. Stop L-SHADE (Success history-based adaptive differential evolution with linear population reduction) R. Tanabe et al. 2014 [42]

B. ALGORITHMS IN THE BENCHMARKING FRAMEWORK
1) The performance of ME-SGO is compared and validated against the standard SGO algorithm from 2016 and four of its latest state-of-the-art variants whose description is provided in Table IV. 2) Additionally, five state-of-the-art advanced metaheuristics namely, EPSO, MPEDE (with Linear Population Reduction) being the multi-strategy ensemble variants, CLPSO, GABC, and L-SHADE being the learning and adaptive have been employed to assess the performance of the proposed method. A brief description of the five state-of-the-art advanced metaheuristics is provided in Table IV and their categorization in Table V. 3) In addition to the aforementioned variants of SGO, four of the modern meta-heuristics (GWO, WOA, SMA and ChOA) and one recent multi-strategy ensemble variant (MEGWO) are selected for the testing and validation process. A brief description of the four modern metaheuristic and the multi-strategy ensemble variant is provided in Table IV. 4) To assess the performance of the proposed methods with the top performers for each benchmarking suite, the results winners/top-performing algorithms are also added in their sub-sections to provide a comprehensive analysis of the current standings of the proposed method.

Algorithms
Adaptive Control Parameters

B. TUNING SETTINGS OF THE ALGORITHMS
To ensure that a fair comparison is achieved, it required to set/tune the algorithm-specific parameters (tuning parameters) appropriately to extract the best performance. Hence, after a meticulous review of the various algorithms' performances, the following tuning settings have been finalized to ensure that the chosen algorithms deliver their best performance to the fullest of their potential. Please note that the values of the tuning parameters provided in Table A.IV (Appendix) remain the same for the entire benchmarking process and real-world problems tackled in the remainder of the manuscript.

B. PERFORMANCE ANALYSIS WITH CEC2019 BENCHMARK FUNCTIONS
The 100-Digit Challenge from Special Session and Competition on Single Objective Numerical Optimization in 2019 introduced 10 special functions to be minimized with limited control parameter "tuning" for each function [43]. The test functions were meticulously crafted with multiple local optima and one unique global optimal solution to ensure that the exploratory prowess and local minima avoidance characteristics are put to test. Similar to composition functions from the previous CEC sessions, the CEC2019 benchmark suite presents challenging exploratory conditions with their landscape shifted and rotated to further complicate the search process of an algorithm. It is to be noted that these functions are extremely challenging for any global optimization algorithm to determine the global optimal solution as their formulation is such that they are intended to trap the algorithms at local best positions, especially for algorithms designed with a tendency to converge to the central point of the search landscape. Additionally, these problems have a large number of dimensions making the search process even harder and complex and only the algorithms with a higher exploratory tendency of the entire search space can determine the global optimal solution or generate solutions in close proximity to the global best.
The description of the CEC2019 benchmarking suite is shown in Table A.V (Appendix).

1) ANALYSIS OF BENCHMARKING PERFORMANCE WITH CEC2019 TEST FUNCTIONS
The CEC2019 benchmark suite provides a more graduated way to measure "horizontal" performance (accuracy) because even "failures" can have some correct digits. The complex test functions require a deeper exploration of the various corners and dark spots of the search landscape such that the algorithm can reach the global optimal solution and has been proven to be quite challenging for many state-ofthe-art meta-heuristics. Considering that computational time has become less of an issue lately, the test suite does not impose restrictions on the number of function evaluations indicating that faster convergence is not the priority with the competitors.
To ensure a fair comparison, 50 independent runs have been considered for all the algorithms with 500,000 function evaluations (NFEs). All the algorithms have been given 1000 iterations with the population size set based on the requirements. The variants of SGO were given a population size of 250 as they relied on DFEs and the modern metaheuristics were given 500 as they relied on (Single Function Evaluations per iteration (SFEs). L-SHADE, MPEDE were given an initial population size of 100 and a final population size of 4 with NFEs being the termination criteria. ME-SGO was given an initial population of 500 and a final population of 50 with NFEs being the termination criteria.
14 VOLUME XX, 2017 The benchmarking results (best, worst, mean and standard deviation) are shown in Table VI, the results of Wilcoxon's rank-sum test are shown in Table VII, the mean absolute error (MAE) for all the fifteen algorithms and the results of Friedman's non-parametrical test are shown in Table VIII and the average computational times (ms) are shown in Table IX respectively. (P) indicates that the precision of the algorithm was accurate up to the ten decimal places   2) THE 100-DIGIT COMPETITION The scoring system considers the average number of correct digits in the best 25 out of 50 trials such that an accurate representation of the performance of the algorithm is provided. Furthermore, compared to the latest CEC2020 benchmarking suite, where the test functions from previous sessions were re-used, the CEC2019 session provides tailor-made, meticulosity designed test functions which also provides a measure of the accuracy and precision of the search technique being used. The CEC2019 suite allows for a limited control parameter "tuning" for each function which can double as a method to validate the tuning sensitivity of the proposed method and compare it with the winners of the competition. A maximum of 1E+08 NFEs was allowed for all the functions as the termination criteria and the performance of ME-SGO is shown in Table X. Comparison of ME-SGO's score with the other top performing algorithms is shown in Table XI. Analysis of results 1) The performance of ME-SGO has been excellent for F1, F2, F3, F5, F6 and F10. The proposed method achieved the 10-digit accuracy for these functions with minor deviations in terms of accuracy. The performance for functions F1, F2 and F6 have the best and both ME-SGO and L-SHADE have produced similar results. 2) Function F7 had been the most challenging for ME-SGO and could only outperform MPEDE while L-SHADE had been the best performing algorithm. The function F10 had similar outcomes from L-SHADE, MPEDE and ME-SGO and for F8 and F9 all three of them performed similarly with ME-SGO being the best performer for F9. 3) It is quite evident that ME-SGO, L-SHADE and MPEDE have been the top-performing algorithms and the point of similarity is the integration of linear population reduction in all three of them. While it is clear that linear population reduction helps achieve a better exploration, it has been crucial to avoiding early entrapment as witnessed with the other algorithms. 4) ME-SGO's score of 72 in the 100-digit competition has been compared to other top performing algorisms. It ranked fifth overall outperforming other DE based optimizers. It is worth mentioning that only the initial population size and number of iterations have been modified to achieve this outcome as rules of the competition dictate. The adaptive parameters have not been modified, although it is possible that tuning the learning rate can help improve the performance for other such complex landscapes.
The convergence graphs for all the algorithms for the CEC2019 benchmarking suite are shown in Figure

B. PERFORMANCE ANALYSIS WITH STANDARD CONSTRAINED ENGINEERING PROBLEMS
In addition to the benchmarking tests, it is required to validate the performance of the proposed method with constrained engineering problems. Generally referred to as "the standard engineering problems", these design optimization problems have multiple constraints and requires the generation of a feasible optimal solution with no constraint violation. Hence, five standard engineering problems (requiring the objective function to be minimized) are chosen which include the SE1: pressure vessel design, SE2: welded beam design problem, SE3: cantilever beam design, SE4: tension/compression spring design problem and the SE5: 10-bar truss design optimization. The previous sixteen algorithms are included in the comparative analysis with no change to the tuning settings of the algorithm-specific parameters. All the algorithms considered for the comparative analysis are given 30 independent runs to determine the mean and standard deviation with the NFEs set to 10,000. Additionally, the best fitness score and its corresponding optimal decision variables, worst fitness score, the average computational times are recorded for all eleven algorithms. The penalty function approach (static penalty function) is opted to handle the various constraints wherein a penalizing score (very high pre-set value) known as a penalty is added to the objective function for any violation of the constraints by the members of the population pool.

1) PRESSURE VESSEL DESIGN
A detailed description of the objective function, constraint functions and the range of the decision variables (mathematical formulation) for the pressure vessel design is shown in Table A.VI (Appendix). The best fitness values and their corresponding optimal decision variables for all the sixteen algorithms sorted in the ascending order of their fitness scores are given in Table A.VII (Appendix). A comparative tabulation of the best, worst, average, standard deviation and the average computational times of the 30 independent runs for all the fifteen algorithms is shown in Table XII.

2) WELDED BEAM DESIGN
A detailed description of the objective function, constraint functions and the range of the decision variables (mathematical formulation) for the welded beam design is shown in Table A.VIII (Appendix). The best fitness values and their corresponding optimal decision variables for all the sixteen algorithms sorted in the ascending order of their fitness scores are given in Table A.IX (Appendix). A comparative tabulation of the best, worst, average, standard deviation and the average computational times of the 30 independent runs for all the fifteen algorithms is shown in Table XIII.

3) CANTILEVER BEAM DESIGN
A detailed description of the objective function, constraint functions and the range of the decision variables (mathematical formulation) for the cantilever beam design is shown in Table A.X (Appendix). The best fitness values and their corresponding optimal decision variables for all the sixteen algorithms sorted in the ascending order of their fitness scores are given in Table A.XI (Appendix). A comparative tabulation of the best, worst, average, standard deviation and the average computational times of the 30 independent runs for all the fifteen algorithms is shown in Table XIV.

4) TENSION/COMPRESSION SPRING DESIGN
A detailed description of the objective function, constraint functions and the range of the decision variables (mathematical formulation) for the tension/compression spring design is shown in Table A.XII (Appendix). The best fitness values and their corresponding optimal decision variables for all the sixteen algorithms sorted in the ascending order of their fitness scores are given in Table A.XIII (Appendix). A comparative tabulation of the best, worst, average, standard deviation and the average computational times of the 30 independent runs for all the fifteen algorithms is shown in Table XV.

5) 10-BAR TRUSS DESIGN
A basic description of the 10-bar truss design problem and its constraints is provided in Table A.XIV (Appendix). The best fitness values and their corresponding optimal decision variables for all the sixteen algorithms sorted in the ascending order of their fitness scores are given in Table A.XV (Appendix). A comparative tabulation of the best, worst, average, standard deviation and the average computational times of the 30 independent runs for all the sixteen algorithms is shown in Table XVI. Analysis of results 1) The performance of ME-SGO has been good for the standard engineering problems for all the five engineering problems with excellent performances for SE1, SE3 and SE4. 2) The difference between the state-of-the-art optimizers such as L-SHADE and MPEDE has been minimal with the three of them dominating for the five problems. 3) Compared to SGO and its other variants, ME-SGO achieved better solutions with higher accuracy and robustness through the testing with lower standard deviation rates. The distance-based strategy adaption and adaptive control parameters have been at the forefront in steering ME-SGO to improve the solution quality while not compromising on the computational times. 4) The performance of ME-SGO for the 10-bar truss optimization is indicative of its efficiency at balancing global and local exploration while SGO and its variants have not been able to achieve the same efficiency at delivering the optimal solution. The lower standard deviation by ME-SGO demonstrated the robustness of ME-SGO at handling optimization problems with multi-constated higher dimensionality.

V. INVESTIGATION OF THE PROPOSED METHOD FOR EV OPTIMIZATION PROBLEMS
To demonstrate the effectiveness of the proposed algorithm towards the handling of complex real-world constrained problems with multiple equality and inequity constraints and higher problem dimensions, four problems on EV optimization from the recent literature have been considered. The same algorithms are chosen with the previously set configurations for the algorithm tuning settings and a comprehensive comparative analysis is provided below.

A. PROBLEMS CONSIDERED FOR INVESTIGATION
Four complex problems namely, (i) the optimal power flow problem with EV loading for IEEE 30 bus system (9 Cases) and IEEE 57 bus-system (9 cases), (ii) optimal reactive power dispatch with uncertainties in EV loading and intermittencies with PV and Wind energy systems for IEEE 30 bus system (25 scenarios), (iii) dynamic EV charging optimization (3 cases) and (iv) energy efficient control of parallel hybrid electric vehicle (3 cases with 2 scenarios) coverage the domains of power systems, energy and control optimization have been considered for validation through the proposed multi-strategy ensemble method and fifteen of the previously described state-ofthe-art advanced and modern algorithms. The constraint handling for the first and second problems on EV optimization is done through the superiority of feasible solution method [52] and for the third and fourth problems, static penalty approach is followed.

B. OPTIMAL POWER FLOW PROBLEM WITH EV LOADING
The first problem is that of the optimal power flow (OPF) with EV loading for the standard IEEE 30 and IEEE 57 bus systems for several OPF objectives such as cost, emission, power loss, voltage stability etc. from [52] is considered. OPF is a highly non-linear complex optimization problem where the steady-state parameters of an electrical network need to be determined for its economical and efficient operation. The complexity of the problem escalates with the ubiquitous presence of constraints in the problem. Solving OPF remains a popular but challenging task among power system researchers. In the last couple of decades, numerous evolutionary algorithms (EAs) and swarm intelligence-based optimization algorithms have been considered to find optimal solutions with different objectives of OPF.
The nine different cases in the OPF for the IEEE 30 and IEEE 57 bus systems with EV loading are given in Table  XVII.
The OPF with EV loading for IEEE 30 bus system has 24 control/decision variables and the IEEE 54 bus system has 33 control variables to be optimized. The different cases for the formulation of the objective function and the various constraints are provided in Table A.XVI (Appendix). Summarization of the bus systems is provided in Table A.XVII (Appendix) and Table A.XVIII (Appendix) for the IEEE 30 and IEEE 57 bus systems respectively. The lower and upper bounds for the optimization are given in Table A.XIX (Appendix).

1) OPF WITH EV LOADING FOR IEEE30 BUS SYSTEM
The procedure for EV loading from [53] has been followed with EV load distributed on the residential buses (17 buses for the IEEE 30 bus system).
To study the effect of additional electric power demand due to PEVs in the electric distribution system for IEEE 30 bus system, it has been assumed that 50 PEVs per residential bus with a total of 17*50 = 850 PEVs have been considered, where 45% of these PEVs are low hybrid vehicles equipped with 15 kWh batteries, 25% PEVs are medium hybrid vehicles with 25kwh batteries and 30% PEVs are pure battery vehicles with 40 kWh batteries. It is also assumed that all the electric vehicles return to the home with an SoC of 50%. Therefore, total electric demand due to PEVs per residential bus per day is 50*(15*45% + 25*25% + 40*30%) *0.5 = 625 kW and total electric demand needed per day due to PEVs is 625*17 = 10,625 kW.
The tabulation of the best solutions with statistical analysis and computational times of OPF for the IEEE 30-bus system with EV loading for all the algorithms in comparative analysis is given in Table XVIII. The decision variables for the best performing algorithm for all the 9 cases are given in A.XX (Appendix).
In Table XVIII  Analysis of results 1) ME-SGO obtained the optimal solutions for five out of the nine cases and for the other cases, the performance was quite competitive.
2) The first case saw competitive results from GABC, EPSO, MPEDE, MEGWO and ME-SGO. It is also worth noting that ME-SGO and MPEDE had the least standard deviation for this case. The second, third and sixth cases saw similar results with excellent performances from ME-SGO, MPEDE and EPSO.
3) The adaptive and multi-population approaches have been successful at handling the multiple constraints while delivering solutions with higher accuracy and the same performance has not been reflected with the other modern meta-heuristics. 4) MPEDE and EPSO performed second to the proposed method while L-SHADE and G-ABC performed next to them. 5) ChOA and MSGO performed poorly due to a lack of balance between exploration and exploitation. The re-initialization system in MSGO could not aid the exploitation system as the algorithm was slower to exploit the promising regions as indicated by the results. The computational times for ChOA have been the highest due to the integration of chaotic sequences.

2) OPF WITH EV LOADING FOR IEEE57 BUS SYSTEM
The procedure for EV loading from [53] has been followed with EV load distributed on the residential buses (41 buses for the IEEE 57 bus system).
To study the effect of additional electric power demand due to PEVs in the electric distribution system for IEEE 57  It is also assumed that all the electric vehicles return to the home with a SOC of 30%. Therefore, total electric demand due to PEVs per residential bus per day is 100*(15*45% + 25*25% + 40*30%) *0.7 = 1750 kW and total electric demand needed per day due to PEVs is 1750*41 = 71,750 kW.
The tabulation of the best solutions with statistical analysis and computational times of OPF for the IEEE 30-bus system with EV loading for all the algorithms in comparative analysis is given in Table XIX. The decision variables for the best performing algorithm for all the 9 cases are given in A.XXI (Appendix). In Table XIX, Fit denotes the fitness value, FC denotes the cost of fuel in $/h, E denotes emissions in t/h, P Loss denotes the real power loss in MW, VD denotes the voltage deviation p.u., L-index denotes the L-index (max). Analysis of results 1) The performance of ME-SGO has been similar to that of the IUEEE 30 bus system with it being consistent at delivering a balanced performance for complex landscapes. ME-SGO performed well for 6 out of the 9 vases for the IEEE 57 bus system. 2) GABC performed next to the proposed method followed by EPSO and MPEDE. It is inferred that multi-population and multi-strategy-based paradigms have been dominant at delivering a consistent performance while static control strategies have found it challenging to explore and exploit simultaneously through the search process.

C. OPTIMAL REACTIVE POWER FLOW FOR IEEE 30 BUS SYSTEMS WITH UNCERTAINTY IN LOADING AND RENEWABLE POWER GENERATION CONSIDERING EV LOADING
The second problem on EV optimization is that of the optimal reactive power dispatch (ORPD) from [54] accounting for the uncertainties with EV loading and distribution system demands, uncertain renewable power i.e., wind and PV power. The load uncertainty model is based on the probability density function (PDF) from [54] and Weibull PDF describes the wind speed distribution. 1000 Monte Carlo scenarios for the loading and windspeed distributions are simulated and 25 most probable scenarios have been considered. IEEE 30 bus system with 25 scenarios with the EV loading model from Problem 1 is used. A detailed description of the mathematical modelling, scenario modelling etc. are available at Appendix. The constraints have been the same as described in Problem 1 and the constraint handling mechanism remains the same.
The formulation of the objective function for the two cases, i) Minimization of real power loss and ii) Minimization of voltage deviation is provided in Table A.XXI (Appendix). Summarization of the IEEE 30 bus system and the lower and upper bounds for the optimization is given in Table A.XXII (Appendix). The 25 different scenarios, the variations in loading and renewable power and the scenario probabilities are given in Table A.XXIII (Appendix). The decision variables for the best-performing algorithms for all the 12 scenarios are given in Table A.XXIV (Appendix) followed by the decision variables for the best-performing algorithms for the next 13 scenarios in Table A.XXV (Appendix) respectively.
Optimization of two cases i.e., minimization of real power loss and voltage deviation respectively with 19 decision variables for 25 cases is done through the 16 algorithms. The number of function evaluations has been set to 20,000 and 30 independent runs have been set for all the algorithms. The best results for 25 scenarios have been tabulated in Table XX. In Table XX, P Loss denotes the real power loss in MW, VD denotes the voltage deviation p.u. Analysis of results 1) ME-SGO had the best performance for 13 out of the 25 scenarios and ISGO had the best performance for 8 cases respectively. The ORPF with uncertain EV loading and renewable power is a complex optimization problem and can be the most demanding on the optimization algorithm. For the same power loss and voltage deviation, there could multiple combinations of decision variables on account of the high non-linearity associated with it.

VOLUME XX, 2017
2) It is necessary for the algorithm to be quickly able to explore the search source to determine the feasible areas and exploit it sufficiently to ensure a better result for all cases. ME-SGO in this regard has been good at covering the various feasible zones and quickly converging to the global best solution. The learning rate has been crucial to adapt to these multi-constrained landscapes and prevent an early entrapment. 3) HS-WOA performed poorly for most of the cases as it does not include multiple adaptive strategies and measures to strategically adapt to the complex landscapes.

D. OPTIMAL DYNAMIC CHARGING (ODC)
The third problem in EV optimization is the dynamic optimization strategy of EV charging based on [55] with 3 levels of EV loading. The grid data considered is based on the average loading data of Vellore Institute of technology, VIT-Campus, Vellore for a period of over a month depicted in Figure 1. The objectives with the current problem are to lower the power curtailment i.e., Minimization of power deviation from the actual load to the ideal load and improve the degree of satisfaction of the EV owners while the constraints include the EV charging power limits, battery and SoC limits, transformer and branch power transmission limits etc. The mathematical model, simulation details and constraints are provided in Table A.XXVI (Appendix).
In the current model, 3 cases of EV loading i.e., 100EVs, 200EVs and 300 EVs are considered as the additional load and are set to arrive at the campus during any period of the day with the morning times being the most crowded. It is assumed that each EV requires 6kW of charging power from the grid. The optimization of the power demand for every 15 minutes starting from 08:00 Hrs. to 20:00 Hrs. is performed and it is assumed that the EV's have a battery level randomly distributed between 0.1 and 0.9 with randomized charging times. For the three cases considered, 20,000 Function evaluations have been set with all the 16 algorithms given 30 independent runs. The tabulation of the best solutions with statistical analysis and computational times of ODC with 100 EVs, 200 EVs and 300 EVs for all the algorithms in the comparative analysis are given in Table XXI, Table XXII and Table XXIII respectively. The notations F, P, EVc stand for the fitness value, power curtailment minimized and the number of EVs fully charged.     Analysis of results 1) Case 1 with 100 EVs had ME-SGO followed by MPEDE deliver the best performance in terms of minimization of the total cost function and a higher degree of satisfaction among the EV owners. ChOA on the other hand recorded the highest number of EVs fully charged although the cost function was higher compared to that of ME-SGO and MPEDE. 2) L-SHADE dominated case 2 with the best cost function and highest number of EVs fully charged. Although ME-SGO had the least power curtailment, the DoS was lower compared to the other algorithms. 3) In case 3, a competitive performance was noted between ME-SGO and EPSO with ME-SGO outperforming EPSO by a small margin. 4) All three cases recorded a competitive performance with the 16 competitive algorithms with the multi-population and multi-strategy based adaptive techniques having the overall best performances. The computational times were also similar for most of the algorithms, although ME-SGO 's computational times were marginally higher for case 3 due to the high dimensionality of the current problem.

E. ENERGY EFFICIENT CONTROL OF PARALLEL HEV
The fourth problem on EV optimization deals with the energy-efficient control (EEC) of a parallel HEV based on [56]. The objective includes the minimization of electricity cost and fuel cost with the maximation of the battery SoC (State of Charge) during the trip duration. The ICE (Internal Combustion Engine) of the PHEV is capable of delivering a maximum power of 30kW and the motor can deliver 15kW with a battery capacity of 5Ah. The mathematical models require the determination of the optimal cumulative cost of operations ICE and EM, optimal battery power, optimal engine power, and the power transferred from the engine to battery to sustain the Soc with constraints on battery power consumption, engine power limits are modelled. The mathematical model, simulation details and constraints are provided in Table A.XXVII (Appendix).
The optimization is performed for 3 driving cycles namely HWFET, UDDS and FTP 75. Two cases of investigation with the first case having the SoC limits between 0.7 and 0.3 and the second case with the Soc limits between 0.5 and 0.3 are investigated. The optimization is done through all the 16 algorithms with 50 NFEs provided during every time interval for the drive cycle and 30 independent runs have been considered to validate the results. The tabulation of the best solutions with statistical analysis and computational times of EEC with UDDS, HWFET and FTP-75 drive cycles for all the algorithms in the comparative analysis are given in Table XXIV, Table XXV and Table XXVI respectively for both the cases investigated. The notations, Cu Co stands for the cumulative cost, Pb is the total power delivered by the battery in watts, SOCmean is the average state of charge, PE-Pb is the total power transferred from engine to battery in watts, Pe is the total power delivered by the engine in watts.   Analysis of results 1) The performance of ME-SGO, MPEDE and L-SHADE has been the best for UDDS with ME-SGO and L-SHADE delivering the best performance for cases 1 and 2 respectively. MPEDE and ME-SGO remained robust with the least standard deviations for cases 1 and 2 respectively. 2) ME-SGO dominated for the HWFET drive cycle with the least cumulative costs incurred for both cases. The performances of MPEDE and L-SHADE were similar with MPEDE being consistent at delivering results with lower deviation. It is evident that adaptive and multi-strategy adoption by the three of these algorithms has resulted in better overall performance. 3) FTP-75 witnessed L-SHADE followed by ME-SGO delivering the best performances with ME-SGO falling behind L-SHADE. The reason for L-SHADE being the top performer is due to its maintenance of historical memory of a diverse set of parameters that govern its performance. It is worth mentioning that ME-SGO's learning rate and has been competitive through the performance despite its historical memory update for the self-introspection factor only.

A. MERITS AND DEMERITS
In order to have a fair conclusion of the performance of the proposed method, it essential to highlight the merits and demerits.
Merits 1) The implementation of multiple strategies in a systematic and a synergetic sequence through the enhanced improving phase and adaptive acquiring phases improved the performance for complex landscapes and enhanced the population diversity. 2) Distance-based strategy adaption and success-based control parameter adaption has been effective at achieving a better balance of exploration and exploitation as evident by the performance of the proposed method in the CEC2019 benchmarking suite and the 5 standard engineering problems. 3) Linear population reduction enabled higher settings of population size and iterations and expanded the exploration range while allowing a smoother transition from exploration to exploitation towards the end of the search process. 4) The performance of ME-SGO for the four complex EV optimization problems has been excellent with higher optimality and better robustness to complex and composite landscapes.
Demerits 1) Slower convergence as a consequence of increased emphasis on exploration over exploitation has been witnessed for simple unimodal and multi-modal landscapes.
2) The learning rate proposed for parameter adaption may be slower to adapt to other complex landscapes and could require experimentations with different settings to extract the best performance.

B. SUMMARY
1) ME-SGO ranked second for the CEC2019 suite and performed competitively with the other state-of-the-art optimization algorithms outperforming the variants of SGO and other modern meta-heurists. 2) ME-SGO achieved the perfect precision of 10 digits for 6 out of the 10 functions from CEC2019 suite and scored 72 points out of 100 in the 100-digit competition.
3) The performance of ME-SGO for the five engineering problems weas very completive with L-SHADE and MPEDE with lower standard deviations compared to the other state-pf-the-art optimizers. 4) The first problem on EV optimization saw ME-SGO outperform the other algorithms for 5 out of 9 cases for the IEEE30 bus system and 6 out of 9 cases for the IEEE 57 bus system.

5)
In the second problem on EV optimization, ME-SGO had the best solutions for 13 scenarios followed by ISOG for 9 scenarios. In this regard, the performance of ME-SGO has been better compared to L-SHADE and MPEDE which have also integrated the linear population reduction techniques. 6) For the third problem on EV optimization, the performance of ME-SGO L-SHADE and MPEDE were quite competitive with ME-SGO leading for two out of three cases. 7) The fourth problem on EV optimization was a tie between ME-SGO and L-SHADE with both the algorithms leading for 3 cases each

C. FUTURE SCOPE
ME-SGO can be deployed to a wide spectrum of problems falling under artificial intelligence, power systems, machine learning etc. Practitioners are free to modify the proposed method as per their requirements and hence to encourage such an extendibility, simplicity has been embraced in the design of ME-SGO. The proposed method can be applied to various other optimization areas in power systems and EV optimization. In computer science, the proposed method can be deployed towards neural networks (NN) training (feed-forward NNs and convolution NNs). Image classification, data classification, pattern recognition etc. can be optimized through the proposed methods. A plan to deploy the current method for the infection detection of COVID-19 from the X-ray images via support vector classifier is in its roots. Feature selection is a potential area of application of the proposed methods through the formulation of a binary version of ME-SGO. The realization of a multi-objective variant is a possibility towards tackling problems requiring a Paretooptimal front. 34 VOLUME XX, 2017  The optimal siting and sizing problem of the distribution generators (DGs) in distribution systems considering the intermittencies associated with PHEV charging and discharging schedules, PV and wind power generation and uncertain load demand has been investigated for an IEEE37node test feeder system.

APPENDIX
Monte Carlo simulation-embedded genetic algorithm 03. J. Zhao et al. in 2012 [59] The economic dispatch model accounting for the uncertainties with the PHEV loading and wind power generation (Rayleigh distribution) based on a simulation study to derive the PHEV charging/discharging behaviour is investigated for an IEEE 118-bus system.

Enhanced PSO
The parametric optimization of an electric-hydraulic hybrid steering system to improve the EV's energy management system by lowering the energy consumption of the actuators by adaptive intervention while considering the aspects of steering economy, steering road feeling, and steering sensitivity is tackled A large-scale problem of locating EV charging stations with service capacity with service risk factors (service capacity and user anxiety) for optimal planning and reduction of social costs is investigated.
Improved Whale Optimization Algorithm (IWOA) 18. Y. Li et al. in 2020 [74] The parameter optimization of gear ratios of two-speed transmission to improve the performance of the drive motor and transmission system with respect to the economy and dynamics of the EV is proposed. Improved genetic algorithm 19. C.A. Folkestad et al.
in 2020 [75] The optimal charging and repositioning of EVs in a free-floating carsharing system for an improved distribution of cars to maximize the revenue and customer service while marginally raising the operational costs is investigated.
Hybrid Genetic Search with Adaptive Diversity Control algorithm 20.
Y-H. Jia et al. in 2021 [76] The capacitated EV routing problem (CEVRP) is implemented in two levels with the first level being the optimal routing based on the demands of customers and the second being charging schedule with respect to the electricity constraint is considered.
A bilevel ant colony optimization algorithm (BACO) A hybrid SGO (HSGO) incorporating a mutation phase to enable continuous improvement in the population through the selection of a subset of features from a member of population and comparing it with the worst person to enhance the population diversity is considered.
The proposed hybrid technique was used on combination with support vector classifier for the COVID-19 infection detection from chest X-Ray images for the Kaggle repository" COVID-19 Radiography Database" with 219 COVID-19 positive images, 1341 normal images, and1345 viral pneumonia images.
The proposed method recorded the highest accuracy at 99.65% compared to 12 other deep learning and bio-inspired algorithms while recording higher precision, sensitivity and FIS F1 scores.

2.
K. V. L. Narayana et al. in 2020 [32] Two hybrid variants benefitting from the synergy of SGO and whale optimization algorithm (HS-WOA and HS-WOA+) to improve the balance of exploration and exploitation with a strong immunity to the curse of dimensionality are proposed. HS-WOA was aimed towards improving the convergence behaviour while HS-WOA+ was aimed at enhancing the quality of exploration.
Extensive benchmarking analysis with standard benchmark functions, composition functions, five standard engineering problems and eight cases of a mulct-unit production planning problem were considered for the validation of the proposed method. A modified SGO (MSGO) incorporating a probabilistic selection between a self-improvement phase with chaotic maps (logistic map, iterative map and tent map) and a position updating phase to ensure a global best oriented search is improved has been proposed.
The design optimization of various civil engineering structures (concrete cantilever beam, 31-member bridge truss, G+3-storey frame, ASCE benchmark structure) for different case studies was investigated.
MSGO had a robust performance compared to SGO, PSO and ALO with the least possible error rates, noise contamination and faster computational times.

4.
A. Naik et al. in 2020 [30] Modified SGO (MSGO) with a modified acquiring phase incorporating a self-awareness probability factor to improve the learning capabilities of the population while boosting the explorative and exploitative potentials with reinitialization within the lower and upper bounds is proposed.
Benchmarking analysis with 7 unimodal, 6 multi-modal and 10 fixeddimensional multi-modal test functions with higher number of problem dimensions is conducted and statistically validated. Additionally, MSGO was deployed for the optimal short-term hydrothermal scheduling problem (STHS) (3 cases).
MSGO was compared with 20 modern metaheuristics and performed competitively and obtained lower values of the cost function for the STHS problem in two out of three cases.

5.
J. Fang et al. in 2018 [27] An improved SGO (ISGO) with population division in the improving phase and eliminationre-initialization system in the acquiring phase to extend its explorative abilities and avoid local entrapment is developed.
ISGO is deployed for the transformer fault diagnosis model using an optimal hybrid dissolved gas analysis features subset with support vector machine classifier to improve the accuracy of the fault diagnose. A discrete version of an improved SGO (ISGO) with a historical learning phase to improve the population diversity following the improving and acquiring phases is developed.
ISGO is combined with the cluster head multi-hop routing protocol in WSN for the data transmission in a multi-hop manner to prolong the lifetime of the network is investigated.
Compared to the GA and the basic CECA protocols, ISGO's protocol had higher number of surviving nodes at the end of the lifecycle with lower overall lifecycle energy consumption. Initial crossover rate (μCR) Set to 0.5 Initial value of scaling factor (μF) Set to 0.5

ME-SGO
Initial value of the self-introspsection factor (Cint) 0.2 Learning rate (rate) 10   Subject to the constraints

Description:
The welded design requires the optimization (minimization) of the cost of fabrication of a welded beam through four decision variables to be optimized. The four decision variables (x1, x2, x3 and x4) include the thickness of the weld, the length of the clamped bar, the height of the bar and the thickness of the bar within specified lower and upper bounds. Four inequality constraints with respect to the four decision variables which include the bending stress (α), shear stress (β), buckling load (γ) and the end deflection of the beam (δ) are laid down.

Description:
In this problem, the goal is to minimize the weight of a cantilever beam with hollow square blocks. There are five squares of which the first block is fixed, and the fifth one bur-dens a vertical load, box girders, and lengths of those girders are design parameters for this problem.   In truss bar optimization, it required to minimize the structural weight of the truss bars with respect to the constraints on the design, stress, deflection, displacement etc. The decision variable corresponding to the dimensions so the truss bars whose count can be 10,15,25,50,72,200. The truss bar optimization applies to the continuous and discrete decision variables and the 10-bar truss optimization for continuous variables is considered in the current testing. A detailed description of the mathematical formulation, the objective function is available at [78]. The basic description of the constraints and the range of the decision variables are provided below. Description of the constraints: The variation of cross-sectional areas is from 0.1 in 2 to 35.0 in 2 . The unit weight of the material is 0.1 lb/in 3 The modulus of elasticity is 107 psi.
The design constraints are as follows. The maximum allowable stress for any member of the truss : ±25 psi. The maximum deflection at any node : ±2.0 mm.

Control (independent) variables
The set of variables that can control the power flow in the network is represented in vector form as: = [ 2 ... , 1 ... , 1 ... , 1... ] A(2) where, is the th bus generator active power (except swing generator). Selection of bus 1 as swing bus is representative only and swing bus can be any one of the generator buses. is the voltage magnitude at th PV bus (generator bus), is the th branch transformer tap, is the shunt compensation at th bus. , and are the number of generators, shunt VAR compensators and transformers respectively. A control variable can assume any value within its range. In reality, transformer taps are not continuous. However, the tap settings expressed here are in p.u. and absolute value of voltage is not accounted for. Hence for study purpose and to compare with past reported results, all control variables including tap settings are considered continuous for most of the study cases. Discrete steps for transformers and shunt capacitors are accounted only in one special study case.

State (dependent) variables
The state of power system is defined by the state variables which can be expressed by vector as: = [ 1 , 1 ... , 1 ... , 1 ... , ] A(3) where, 1 is the generator active power at slack (or swing) bus, is the reactive power of generator connected to bus , is the bus voltage of th load bus (PQ bus) and line loading of th line is given by . and are the number of load buses and transmission lines respectively. -index of each bus serves as a good indicator of power system stability. The value of the index varies from 0 to 1, with 0 being the no load case while 1 signifies voltage collapse. The L-index is calculated for all load buses and maximum value out of those acts as the global indicator for the system stability.

Objectives
Therefore, the objective function of system stability is given by: ( , ) = = ( ) where j=1, 2, …. , NL A(7) where, and are the coefficients that represent the valve-point loading effect.

Case 6
Minimization of fuel cost and real power loss where, is is the real power loss in the network calculated and value of factor is chosen as 40.
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Minimization of fuel cost and voltage deviation
Voltage deviation is expressed as: The combined objective function of fuel cost and voltage deviation is: where, weight factor is assigned a value of 100 based on the previous implementations. ] + × A (14) where, value of weight factor is 100.

Case 9
Minimization of fuel cost, emission, voltage deviation and losses where, the weight factors are selected as = 19, = 21 and =22 based on the previous implementations.

Equality Constraints
In OPF, power balance equations are the equality constraints and those are represented as: where, = − , is the difference in voltage angles between bus and bus , is the number of buses, and are active and reactive load demands, respectively. is the transfer conductance and is the susceptance between bus and bus , respectively.    1 1.1 1.1 1.1 1.1 1.1 1.1 20 20 20 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1  1.1 1.1 1.1 1.1 1.1 1.1 1.1] 50 VOLUME XX, 2017 A (22) where, = − , , is the difference in voltage angles between bus and bus and ( ) is the transfer conductance of branch connecting buses and .

Minimization of voltage deviation
Voltage deviation is expressed as: where, VLp is the bus voltage of pth load bus (PQ bus) and NL is the number of load buses. 52 VOLUME XX, 2017

Optimization cases 2 Cases with 25 Scenarios
Lower  1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1  Algorithm with the best Fitness ME-SGO ME-SGO ISGO MPEDE ME-SGO ME-SGO ME-SGO ISGO ME-SGO WOA ME-SGO ME-SGO  A (24) where, = 1,2,3. . , are the time intervals, stands for the power deviation, is the toal grid demand with EVs at time t and ̅ is the average demand on the grid excluding the EV load Maximization of owner's degree of satisfaction where, = 1,2,3. . , are the nodes in the power grid, stands initial level of SoC.
Combining, we have is the weighing factor set to 0.7.

Initial charging time
The initial charging time follows normal distribution and is given by A (27) where, is the rate of charging.

Number of EVs connected for charging
At a time instant t, the number of EVs' requiring charging is given by A (28) where, is the number of EVs requiring charging at time t, are the total number of EVs at a given node.

Constraints
EV charging power limits are expressed as: Node voltage limits are expressed as: Transformer ratio restraints are expressed as: Branch power transmission constraint is given as: where, the energy consumption by the HEV, is the control variable which denotes the power allocation from the vehicular energy management system, is the cost of the total energy consumption, Δ is the time interval, N denotes the total number of time intervals, , denote the price of fuel and electricity respectively, ( ) and ( ) are the engine and battery powers respectively, denotes the proportion of electricity from the grid, is the grid electrocute pricing, ℎ and are the efficiencies of charging of battery charging and battery pack, is the fuel consumption rate of the engine when charging the battery, is the terminal voltage of the battery pack, is the current of the battery pack, is the open-circuit battery pack voltage, is the diffusion voltage of the battery RC circuit, and denote the capacitance and resistance of the RC network.