Symbiotic Learning Grey Wolf Optimizer for Engineering and Power Flow Optimization Problems

This article presents a symbiotic learning-based Grey Wolf Optimizer (SL-GWO) formulated through the introduction of symbiotic hunting and learning strategies to achieve a better trade-off between exploration and exploitation while standing immune to the curse of dimensionality. The proposed method improves the performance of the algorithm to effectively handle problems with larger dimensions while avoiding local entrapment, accelerates convergence, and improves the precision and accuracy of exploitation. SL-GWO’s symbiotic hunting strategies provide a major overhaul to the exiting hierarchical hunting through population sub-grouping into attacking hunters and experienced hunters with individually crafted dynamic adaptive tuning. The hunting mechanisms are implemented through the inclusion of random omega wolves from the wolfpack thereby reducing the algorithm’s excessive dependence on the three dominant wolves and enhancing the population diversity. SL-GWO is tested and validated through a series of benchmarking, engineering and real-world optimization problems and compared against the standard version of GWO, eight of its latest and state-of-the-art variants and five modern meta-heuristics. Different testing scenarios are considered to analyze and evaluate the performance of the proposed method such as the effect of dimensionality (CEC2018 benchmarking suite), convergence speeds, avoidance of local entrapment (CEC2019 benchmarking suite) and constrained optimization problems (four standard engineering problems). Furthermore, two power flow problems namely, the optimal power flow (13 cases for IEEE 30 and 57-bus system) and optimal reactive power dispatch (8 cases for IEEE 30 and 57-bus system) from the recent literature are investigated. The proposed method performed competitively compared to all its competitors with statistically significant performance while requiring lower computational times. The performance for the standard engineering problems and the power flow problems was excellent with good accuracy of the solutions and the least standard deviation rates.


I. INTRODUCTION
25 Recent multi-disciplinary research and real-world scenarios 26 have shown that optimization using meta-heuristics is a pow-27 erful tool for problem resolution and resource management. 28 The associate editor coordinating the review of this manuscript and approving it for publication was Diego Oliva .
For its simplicity and usefulness in tackling complicated 29 issues, optimization has been accepted and promoted by 30 researchers and specialists alike. ''Heuristic'' means ''ran-31 dom'' or ''creative'' search procedure used to find the opti-32 mal/best solution combination to maximize or minimize 33 the desired system characteristics [1]. No previous knowl-34 edge of the system, or in mathematical jargon, no gradient 35 The introduction and empirical development of unique 108 techniques/operators that expand the exploration to newer 109 places within the solution space while balancing the exploita-110 tion/local search is the norm to overcome the limitations 111 with the standard paradigms. The process of ''improving'' or 112 ''enhancing'' or ''modifying'' a meta-heuristics is extensively 113 researched concerning the algorithmic structure for the con-114 sidered optimization task [8]. The limitations of the existing 115 tuning settings are analyzed and newer tuning strategies to 116 tackle premature convergence and enhance population diver-117 sity are also considered to improve the performance. The 118 improved versions are devised to surpass the standard version 119 of the meta-heuristic at global and local search ensuring 120 greater diversity in the population pool while converging 121 steadily and quickly to the global optimal solution and deal-122 ing with multiple constraints effectively. 123 Another methodology approached is the hybridization of 124 two existing meta-heuristics wherein a solid meta-heuristic 125 immune to the negative aspects of the parent algorithms 126 and benefitting from the reliable aspects is worked up 127 [9]. Hybridization promotes the collective collaboration of 128 both the meta-heuristics since the advantageous aspects are 129 consolidated to such an extent that the different search 130 mechanisms help each other in dynamically exploring while 131 exploiting the search space [10]. Hybridization of the swarm-132 intelligence-based algorithms with the evolutionary tech-133 niques from Genetic Algorithm (GA) [11], [12] and DE 134 (Differential Evolution) [13] has been experimented with 135 several times to realize stronger synergistic hybrids. Besides 136 incorporating the evolutionary strategies, numerous other 137 combinatorial algorithms have been proposed in the past and 138 the present scenarios, e.g. The hybridization of a swarm-139 based meta-heuristic with other swarm or nature-based/ 140 ance through reduction of the high selective pressure and 193 low diversification of the standard GWO is developed by 194 Nadimi-Shahraki et al. [26]. The integration of two novel 195 search strategies aided GGWO to deliver the best optimality 196 with accelerated convergence for the CEC2018 test suite, four 197 engineering problems and three cases of power flow problems 198 outperforming several other swarm-based meta-heuristics 199 and five variants of GWO. (ii) An enhanced chimp opti-200 mization (EChOA) algorithm integrating a disruptive poly-201 nomial mutation based-initialization and Spearman's rank 202 correlation coefficient based ranking system is developed 203 at [27] to combat local entrapment. Validated on 12 classical 204 and 15 CEC2017 benchmark functions followed by three 205 engineering design problems and training multilayer percep-206 tron EChOA delivered competitive results. (iii) An Effective 207 Whale Optimization Algorithm (EWOA) to solve optimal 208 power flow problems (Standard IEEE 6-bus, IEEE 14-bus, 209 IEEE 30-bus, and IEEE 118-bus test systems) through the 210 integration of Levy motion and Brownian motion into the 211 standard search mechanism of WOA to combat the curse of 212 dimensionality and maintain a proper balance between the 213 exploration and exploitation is developed in [28]. EWOA 214 outperformed the competitor algorithms and delivered solu-215 tions with improved optimality for the several cases of OPF 216 investigated. (iv) An improved Harris Hawks Optimization 217 (IHHO) algorithm through the simplification of the search 218 strategies from a six-step decision mechanism to a four-step 219 system is realized in [29]. Benchmarked using the CEC2019 220 test suite and a three-dimensional bin packing problem 221 (3D-BPP) dataset with 320 samples, IHHO delivered statisti-222 cally significant results. (v) A multi-strategy ensemble social 223 group optimization algorithm (ME-SGO) to enhance the pop-224 ulation diversity from complex landscapes through the inte-225 gration of distance-based strategy adaption and success-based 226 parameter adaption is proposed in [30]. The benchmarking 227 through CEC2019 test suite demonstrated ME-SGO's robust-228 ness to entrapment and its application to four problems on 229 the optimization of energy management in electric vehicles 230 yielded improved results compared to its competitors. (vi) A 231 component-based framework for the automatic design of 232 Particle Swarm Optimization (PSO) Algorithms known as 233 PSO-X embodies a large number of algorithm components 234 developed over more than 25 years of research of PSO into 235 a unified framework to determine the best possible config-236 uration is developed at [31]. Benchmarking tests with over 237 50 test functions from the various CEC test suites showcased 238 the efficiency of PSO-X at adapting to deliver the best pos-239 sible solution quality in terms of accuracy, optimality and 240 convergence speeds. The manuscript is partitioned as described below. Section II 287 discusses the working of GWO, followed by a comprehen-  306 GWO is a nature-inspired, swarm-based metaheuristic 307 optimization method inspired by grey wolves' leadership 308 structure and hunting mechanism (Canis lupus). GWO was 309 developed in 2014 by Seyedali Mirjalili, Seyed Mohammad 310 Mirjalili and Andrew Lewis [24]. GWO is unique in that 311 it organizes grey wolves into alpha, beta, delta, and omega 312 groups and explores and exploits the search area. The GWO 313 tuning requirements are the population size, iteration count, 314 and an optional control vector. Exploration and exploita-315 tion are balanced through a linear reduction of the control 316 vector from 2 to 0 over the hunting. Its excellent perfor-317 mance for both unconstrained and constrained, single and 318 multi-objective optimization with improved optimality and 319 fast convergence has attracted academics and practitioners 320 from diverse domains.

322
To comprehend GWO, the understanding of the mathemati-323 cal modelling of wolf social hierarchy is essential. It is the 324 dominant wolves (alpha wolves) who govern the group's 325 functioning and are responsible for decision-making and 326 management. The second order is made up of beta wolves that 327 serve as subordinates and advisors to the other wolf orders. 328 The third category is the omega wolves, who are typically 329 used as a scapegoat or babysitters. The delta wolves are the 330 group's scouts, sentinels, elders, hunters, and caregivers. The 331 delta wolves rule the omegas but follow the betas and alphas, 332 establishing a middle ground between the two. GWO's fun-333 damental activity is communal foraging based on the social 334 hierarchy. Figure 1 depicts the social dominant hierarchical 335 system of the grey wolves.

336
In GWO, the optimal solution is designated as alpha, the 337 second-optimal solution as beta, and the third-optimal solu-338 tion as delta. The latter group is referred to as the omegas. 339 The following is a thorough discussion of the many parts of 340 GWO's mathematical modelling. The initial phase of GWO is dedicated to determining the 343 prey's location. Initially assuming that the prey's position is 344 unknown, the algorithm traverses the search space with the 345 assumption that it is situated near the optimal solution. Once 346 they have determined the position of the prey, they encircle it 347 as part of the hunting process. Grey wolves search the region 348 surrounding the site of prey to find more suitable solutions. 349 Eq. (2.1) and Eq. (2.2) represent the mathematical model 350 for prey encirclement in GWO.
where, − → X G is the position of the grey wolf, − → A and − → B are 354 coefficient vectors, t is the present iteration, −−→ X p (t) is the 355 position of the prey, || is the modulus operator to determine 356

− →
where, − → a is the control vector whose value tends to linearly Finally, the position of the grey wolf is given by Eq. (2.7). surge in citations on an annual basis is provided in Figure 2. 476 The Web of Science R database ranks the Electrical and 477 Electronics Engineering application of GWO over Com-478 puter Science and artificial intelligence applications with 479 over 350 and 300 publications each. The list contains Com-480 puter Science and interdisciplinary applications followed by 481 Energy fuels and Telecommunications.

482
In terms of applications relating to GWO, the SCOPUS R 483 database holds over 1800 publications followed by 1500 pub-484 lications indexed in the Web of Science R database and over 485 200 publications from the other databases. The growth in the 486 publications deploying GWO to numerous applications has 487 been on the rise since 2017. As far as the recent developments 488 are concerned, over 1000 documents and 800 documents have 489 been indexed in the SCOPUS and Web of Science databases 490 respectively in 2020. The advancement in the publication 491 growth incorporating GWO is shown in Figure 2.  As far as the territories incorporating GWO in their 501 research is concerned, India and China hold the most publi-502 cations accounting for over 400 and 300 publications respec-503 tively. This is followed by Iran, Egypt and Malaysia.   The other limitations such as the susceptibility to ''the 514 curse of dimensionality'', and slower convergence are dealt 515 with through the introduction of special tuning parameters, 516 chaos theory, population selection and function evaluation 517 strategies, sorting and ranking mechanism, and population 518 re-initialization, etc. The proposed variants are evaluated for 519 their efficacy and efficiency through standard benchmarking 520 VOLUME 10, 2022 functions from the various CEC session, hybrid test func-521 tions, composition functions, constrained standard engineer-522 ing problems with statistical tests, computational times and 523 convergence graphs to elucidate the performance prowess.

575
Modified and learning-based variants examine and scruti-576 nize the algorithmic structure and alter the search measures 577 with the incorporation of correction mechanisms, adaptive 578 tuning systems, elitism, and so forth with a definitive objec-579 tive of amplifying the exploratory and exploitative poten-580 tial across all the norms. A large number of the modified 581 and learning variants restructure and incorporate special soft 582 computing techniques to tackle the problem at hand with 583 the aim of boosting the performance across multiple stan-584 dards with various test cases and benchmarking to validate 585 that the modified variant is robust and reliable. The mod-586 ified variations of GWO studied here are pointed toward 587 enhancing the exploratory search gradation while standing 588 immune to the curse of dimensionality. Over 200 publica-589 tions have been found in teg literature survey bearing the 590 title modified/learning-based variants of GWO across vari-591 ous domains and disciplines. Table 3 presents a few of the 592 most cited and state-of-the-art modified and learning-based 593 variants of GWO. 594

595
Hybridization/combinatorial variants of GWO with the 596 existing swarm and evolutionary algorithms have been an 597 ongoing trend since the publication of GWO with over 598 300 publications. The combinatorial variants operate in syn-599 ergy combining the best aspects of both their parent algo-600 rithms with robust and consistent performance across all 601 standards and have considerable performance improvement 602 for the conflicting cases of exploration versus exploitation. 603 Table 4 presents a few of the most cited and state-of-the-art 604 hybridized/combinatorial variants of GWO.

606
WOLF OPTIMIZER (SL-GWO) 607 After a thorough investigation of GWO, its various state-of-608 the-art versions, review papers, and publications on GWO 609 and its applications, this work proposes a symbiotic learn-610 ing GWO. The enhanced algorithm, referred as SL-GWO, 611 is developed to overcome the numerous shortcomings of 612 GWO, including susceptibility to the curse of dimension-613 ality and local entrapment, as well as to achieve an opti-614 mum balance between global (exploration) and local search 615 (exploitation).

617
The literature survey of GWO and its variants from the pre-618 vious section provides an overview of the different aspects 619 of the standard GWO algorithm that require improvements. 620 Considering that there have been numerous publications 621 aimed at improving the performance of GWO [57], [58]    In this regard, the present work provides a comprehensive 633 performance overview using two of the latest benchmark-  The key difference between the two sub-group of wolves lies 675 in their hunting style and learning methodology.

676
The hunting mechanism for the two sub-group of wolves, where, d is the current dimension and d = 1, 2, . . . D, D 697 denotes the total number of problem dimensions, rand is a 698 random number in [0, 1] generated through uniform distri-699 bution, rand(d) denotes a matrix of random numbers with 700 the size (1,d), t is the current iteration, t = 1, 2 . . . T and 701 T denotes the total number iterations.

702
The value of S r is dynamically varied in the range of 703 0 to 1 for the two sub-groups. The attacking group has a 704 higher affinity to follow the leaders through the various sym-705 biotic hunting strategies and hence has a higher value of S r . 706 However, since higher S r may not be beneficial at all times 707 as exploiting across all dimensions may result in premature 708 convergence, a probability of 50% is chosen to dynamically 709 vary the S r and is specified per Eq.
The experienced hunters adopt a cautious approach to fol-712 low the leaders and their primary job is to preserve elitism and 713 diversity through their experience which is simply the mea-714 sure of success and failures associated with their individual 715 symbiosis rates (S r ). The values of S r that have been proven 716 successful at generating fitter new grey wolves are retained 717 and in cases of failure to produce an elite individual, the older 718 values are incremented or decremented randomly by a value 719 of 0.1. This is specified by Eq. (3.4).
where, S Old r (i) is the symbiosis rate retained from the precious 722 iteration for the current member of the wolfpack, f (new) 723 denotes the fitnees value of the new grey wolf and f (old) 724 correpsinds to the fitness valies of the wolf from the previous 725 iteraration. 726 The symbiotic hunting strategies form the core of 727 SL-GWO as the hunting strategies require the collaboration 728 of the three dominant wolves and other randomly chosen 729 omega wolves to determine new positions that help the cur-730 rent members of the wolf-pack advance their hunt. Four sym-731 biotic hunting strategies are designed and these four strategies 732 are the same for both the population sub-groups. Among the 733 four symbiotic hunting strategies, the first two strategies are 734 the key to improving the hunt as both of them incorporate 735 the alpha, beta and delta wolves to guide the hunt. The first 736 strategy is an extension of the hierarchical hunting scheme 737 from the standard GWO. In this strategy, the hunting is led by 738 alpha, beta and delta wolves with two random sub-ordinates 739 VOLUME 10, 2022 i.e., the two random omega wolves accompany the alpha, beta and delta wolves. Furthermore, the enhance diversity, the 741 three least fitter wolves are added as a supplementary support 742 with lesser emphasis on their contribution. The first hunting 743 strategy is described by Eq. (3.5).
where, X First is the solution vector obtained from the first respectively.
The second symbiotic strategy is implemented as described 768 by Eq. (3.10).
where, X Second is the solution vector obtained from the second The second symbiotic strategy is the most important and 782 the key to maintaining a proper balance of exploration and 783 exploitation. This strategy dynamically changes from the 784 exploration phase to the exploitation phase based on the 785 position of the alpha, beta and delta wolves. In the initial 786 stages of exploration, the distance between the three wolves 787 would be greater resulting in better exploration of the search 788 space. Since one random wolf from the omegas is also chosen 789 to determine the next position of the wolf, the complete 790 dependence on the alpha, beta and delta wolves is lowered. 791 The inclusion of the current position of the wolf from the 792 current population process ensures that the position update 793 is aimed at improving its optimality in the neighborhood as 794 it explores closer to that position for a superior solution. This 795 system encourages random omega wolves to learn from the 796 dominating alpha and the wolf from the two sub-groups and 797 compete with beta and delta wolves to further improve their 798 positions towards a better optimal solution.

799
The third and the fourth symbiotic learning strategies are 800 described below and are chosen alternatively.

801
The third symbiotic learning strategy is given by Eq. (3.11). 802 where, The fourth symbiotic learning strategy is given by Eq. (3.12). 806 where, X third is the solution vector obtained from the third 812 symbiotic hunting strategy, X Fourth is the solution vector 813 obtained from the fourth symbiotic hunting strategy, − → Z 1 , − → Z 2 814 and − → Z 3 are the difference vectors between any two randomly 815 chosen omega wolves.

816
The inclusion of two random omega wolves and four ran-817 dom omega wolves in the second and third strategies is to 818 ensure that the algorithm is prevented from being trapped at 819 a local optimum point in the early stages of its exploration. 820 As different omegas are chosen for each of the two strategies, 821 the population diversity is enhanced.
where, τ is a tolerance limit and is set to 0.05.

840
The hunting distances D 1 and D 2 in for the experienced 841 hunters are based on their successful generation of elite new 842 wolves and the same method utilized to adapt S r is once again 843 adopted here. This is specified by Eq. (3.15). .
where, r 1 , r 2 and r 3 are three random numbers in the interval The final step is the fitness evaluations of all the newer .
is the new fitness score of the decision 870 variables obtained by the symbiotic hunting strategy and 871 f −→ X i(t) fitness score of the old decision variables obtained 872 from the previous iteration. 873

874
SL-GWO achieves a good balance of exploration and 875 exploitation benefitting from the dynamic nature of the two 876 control parameters namely, Symbiosis rate (S r ) and Hunting 877 distance (D 1 and D 2 ) respectively. S r , in particular, is cru-878 cial of the two as it helps diversify the population on a 879 dimensional basis. The variation of S r is kept diverse for the 880 two population sub-groups, i.e., the first sub-group with the 881 attacking hunters work with a higher value of Sr allowing 882 them to explore the newer areas in the search space generated 883 by the different symbiotic hunting strategies. This accounts 884 for an aggressive attacking approach and allows more grey 885 wolves to quickly explore and exploit the promising areas 886 specified by the three dominant wolves. At the same time, 887 the higher values of S r promote convergence capabilities of 888 the algorithm as more grey wolves follow the three dominant 889 wolves to exploit portions of the search space readily. The 890 hunting distances D 1 and D 2 in the first sub-group help in 891 controlling the exploration distance and aids the smoother 892 transition of exploration to exploitation. To be specific, D 1 's 893 linear decremental strategy from 1 to 0 can be quite bene-894 ficial to force the exploration within the search space limits 895 during the initial half of the search space and prevent early 896 cases of entrapment since the hunting distance from the three 897 dominant wolves is higher. On the other hand, the sinusoidal 898 bursts of D 2 aids the systematic control of diversification and 899 intensification cycles, encouraging a controlled movement 900 within the search space. individual grey wolf to gradually improve over time and all 979 the problems coupled with the lack of adaptive and robust 980 control and diversifying schemes can trigger the avalanche 981 of stagnation leading to entrapment and ultimately premature 982 convergence. Therefore, to counter these ill-effects associated 983 with the conventional selection process, the greedy selection 984 from SL-GWO provides a robust mechanism to select the 985 best solutions while advancing their personal best fitness 986 levels at all times. The greedy selection in SL-GWO works 987 in synergy with a population sorting mechanism to sort the 988 population of the two sub-groups such that the three best 989 solutions are assigned as the alpha, beta and delta wolves 990 rather than comparing the individual fitness levels of the 991 individual wolves from the standard GWO. The major benefit 992 is that no inferior solutions make it to the population pool 993 preventing the advancement of the individual grey wolves 994 and the second being the re-organization of the population 995 pools based on the sorted population. Furthermore, the re-996 organization system allocates the top 50 percent of the wolves 997 to the attacking hunters to explore at a faster pace and the 998 remaining 50 percent to the wolves to the experienced hunting 999 group to ensure that diversification is preserved at all times. 1000

4) TIME COMPLEXITY AND COMPUTATIONAL COMPLEXITY 1001
In SL-GWO, the position update system occurs once, fol-1002 lowed by the sorting of all wolves from the two sub-groups 1003 after evaluating the wolves' fitness in the previous iteration 1004 to select the alpha, beta, and delta wolves. This is followed 1005 by the parameter adaption, which occurs as a result of any 1006 of the symbiotic hunting processes. As a result, SL-GWO 1007 performs only one fitness assessment (SFEs) of each popula-1008 tion member every iteration. The computational complexity 1009 of distinct phases for an iterative count of T iterations with a 1010 population size of N and each having a D number of decision 1011 variables/dimensions is as follows. In addition to the total 1012 computing complexity of the fitness sorting process through 1013 quick sort, which is O (N log N ) N × D)))). 1018 The time complexity of SL-GWO is measured considering 1019 its total run time i.e., 't total ' for one independent run. It is 1020 shown in Eq. (3.20).

1027
Therefore, based on analysis from Table 5, the time com-1028 plexity of SL-GWO is O (N). 1029 The pseudocode of SL-GWO is given below.

1031
The proposed algorithm's performance will be evaluated  The performance of SL-GWO is compared and validated 1087 against the standard GWO algorithm from 2014 and eight 1088 of its latest state-of-the-art variants whose description is 1089   The algorithm-specific parametric tuning for the competitor 1122 algorithms is based on their respective publications and these 1123 settings have not been modified for the entire benchmarking 1124 and power flow optimization problems. A detailed descrip-1125 tion of the tuning parameters and their ranges have been 1126 provided in Table 7. 1127 VOLUME 10, 2022   Table 8.    The benchmarking results (mean and standard deviation) 1152 are shown in Table 9 for the 10-dimensional case followed 1153 by Table10 for the 30-dimensional case and Table 11 for the 1154 50-dimensional case. The results of Wilcoxon's rank-sum test 1155 are shown in Table 12 followed by the results of Friedman's 1156 non-parametrical test shown in Table 13, the mean absolute 1157 errors (MAE) for all the fifteen algorithms are given in 1158  Table 14. The acceleration rates comparing SL-GWO with the 1159 competitor algorithms for the 10, 30 and 50 dimensional cases 1160 are shown in Table 15, Table 16 and Table 17 respectively. 1161 Furthermore, the average computational times are shown in 1162  Table 18. The statistical results comparing SL-GWO with 1163 the recent variant of GWO (GGWO) for the CEC2018 test 1164 suite are provided in Table 19. The convergence curves for 1165 the 29 benchmark functions (50 dimensions) are shown in 1166 Figure 5 to Figure 34. 1167 The performance of SL-GWO stands out for the CEC2018 1168 benchmarking suite in terms of optimality and lower devi-1169 ation. In table 9, SL-GWO emerged as the best-performing 1170    learning strategies that direct the grey wolves to adapt to the 1194 problem landscape. These adaptive control measures to guide 1195 the population movement with diversity-enhancing learning 1196 systems are absent in most modern meta-heuristics and these 1197 algorithms are often tested on simpler standard benchmarking 1198 functions with static landscapes where they are known to 1199 be the most competitive. Dynamic search landscapes from 1200 the CEC2018 suite help assess the quality of exploration 1201 and exploitation with the ever-present complexity of random 1202 translations to the landscape in the form of shifting and rotat-1203 ing that tend to trap poorly designed optimizers with a higher 1204 affinity to exploit the local zones. SL-GWO's performance is 1205 consistent through the testing with the increase in the number 1206 of problem dimensions having no effect on the efficiency 1207 of the algorithm as seen in Tables 9, 10      larger dimensional problems. One particular reason for this 1227 is to do with the formulation of the solution set wherein 1228 every dimension/decision variable has not achieved the global 1229 best solution leading to an imbalance in the optimization and 1230 thereby producing highly non-optimal solutions.  It is quite evident that the SL-GWO is quicker than all the to the global optimum within the given computational budget. 1253 The quicker convergence without the risk of entrapment in 1254 SL-GWO is made possible through the symbiotic learning 1255 schemes i.e., the linear and adaptive controls for the hunting 1256 distance and symbiosis rate. The two sub-population each 1257 with its diversified control schemes operate independently 1258 to explore and exploit around the three leader wolves and 1259     The computational times in Table 18 indicate that

SL-GWO and GGWO are quite competitive with each other. 1312
However, it is to be noted that GGWO utilized three times 1313 higher computational budget and GGWO has a computational 1314 complexity (CC) which is three times that of SL-GWO as 1315 it generates three new solutions for every member of grey 1316 wolf whose fitness is evaluated every iteration to such that 1317 the best one of the three solutions survive to make it to the 1318 next iteration. SL-GWO on the other hand generates only 1319 one new solution vector for every member of the wolfpack 1320 thereby reducing its computational requirements to one fit-1321 ness evaluation for every member in an iteration. Despite the 1322 higher computational budget, the performances of SL-GWO 1323 are identical and at times and even outperforms the GGWO 1324 algorithm for the 50-dimensional case. 1325 VOLUME 10, 2022    Challenge, which required the minimization of ten special 1331 functions (having the global optimum fitness value of ''1'') 1332 with restricted control parameter ''tuning'' for each function 1333 [14]. The test functions were methodically constructed with 1334 several local optima and a single global optimum solution to 1335 VOLUME 10, 2022 TABLE 20. Description of the 10 CEC2019 benchmark functions (composition functions) used to determine the algorithms' ability to avoid local entrapment. search process even more complicated, and only algorithms 1348 with a strong exploratory inclination of the whole search 1349 space can discover the global optimal solution or solutions 1350 that are near to it.

1351
The description of the CEC2019 benchmarking suite is 1352 shown in Table 20. The benchmarking results (best, worst, 1353 mean and standard deviation) are shown in Table 21, the 1354 results of Wilcoxon's rank-sum test are shown in Table 22, 1355 followed by the results of Friedman's non-parametrical test 1356 are shown in Table 23, and the mean absolute errors (MAE) 1357 for all the fifteen algorithms are given in Table 24. Fur-1358 thermore, the average computational times are shown in 1359           the exploration and adapts in a dynamic manner to pre-1387 vent stagnation. The linear control strategy also forces 1388 exploration at the initial stages of the search thereby pre-1389 venting premature convergence and the adaptive control 1390 strategy guides the wolves to the most promising and 1391 diverse areas within the search landscape.
1392 VOLUME 10, 2022   • Besides F2, the function F7 was also one the most

1408
• The effect of additional NFEs or increasing population 1409 count produced no major improvements in the perfor-1410 mance of most algorithms and the modification in the 1411 tuning parameters has been proven to be ineffectual 1412 for such complicated search landscapes of composition 1413 functions. The same NFEs count has been useful in 1414 identifying how quickly does an algorithm adapt to 1415 escape local entrapment and it was fairly obvious that 1416 a lack of diversifying measures leads to entrapment 1417 at a very quick point in the timeframe of exploration. 1418 Although several articles have demonstrated that the 1419 global optimal solutions are attainable through multiple 1420 tuning settings each unique to different functions, the 1421 current work does not modify or suggest multiple tuning 1422 modifications to suit the functions' landscape. Instead, 1423 all the algorithms have the same tuning settings speci-1424 fied earlier and no modifications have been enforced to 1425 ensure that a fair comparison has been made.

1426
• Although the NFEs were higher for the CEC2019 test 1427 suite compared to the CEC2018 suite, most modern 1428 meta-heuristics fail to exploit the advantage with higher 1429 population size and quickly get entrapped and this 1430 proves that simple strategies with limited adaptive tun-1431 ing can be detrimental despite higher computational 1432 budgets.

1433
• The MAE for SL-GWO has been the least as it managed 1434 to provide decent performances across all the test func-1435 tions. From the rankings, it can be inferred that SL-GWO 1436 is effective at handling complex search landscapes with 1437 a good tendency for exploration and solution intensifica-1438 tion given that no additional tuning is required compared 1439 to the other algorithms.  The analysis of the performance of the proposed method 1447 for constrained engineering problems is carried out in this 1448 sub-section to determine its ability to generate feasible 1449 solutions with the stipulated computational budget. The 1450 performance of the fourteen competitor algorithms has 1451 been considered in comparative analysis and five different 1452     optimum costs obtained and the optimal values of decision 1467 variables by the fifteen algorithms and a comparison of the 1468 performance of the algorithms in terms of best cost, worst 1469 cost, average costs, deviation and computational times for the 1470 30 independent runs are tabulated in Table 28. The welded beam design problem comprises of four decision 1473 variables (x 1 : weld thickness, x 2 : clamping bar length, x 3 : 1474 bar height and x 4 : bar thickness) and four inequality con-1475 straints including bending stress, shear stress, buckling load, 1476 and beam end deflection are levied. The problem requires 1477 the total cost minimization with respect to its manufacturing 1478 cost [83]. Table 29 gives the optimum costs obtained and the 1479 optimal values of decision variables by the fifteen algorithms 1480 VOLUME 10, 2022    Table 30.   constraints including frequency, deflection and shear stress 1489 are levied. The problem requires the total cost minimization 1490 with respect to the weight of the spring [84]. Table 31 gives 1491 the optimum costs obtained and the optimal values of decision 1492 variables by the fifteen algorithms and a comparison of the 1493 performance of the algorithms in terms of best cost, worst 1494 TABLE 27. The optimal costs and the optimal values of the four decision variables for the pressure vessel design obtained by the fifteen algorithms.  cost, average costs, deviation and computational times for the 1495 30 independent runs are tabulated in Table 32.   1513 TABLE 33. The optimal costs and the optimal values of the ten decision variables for the 10-bar truss optimization obtained by the fifteen algorithms.

TABLE 34.
Comparison of the best, worst, average (mean), standard deviation and the average computational times (seconds) of the fifteen algorithms for the 10-bar truss optimization.  for the network to operate economically and efficiently. The 1574 complexity of the problem increases as a result of the prob-1575 lem's multiple equality and inequality constraints. Solving 1576 OPF continues to be a prominent yet difficult issue for power 1577 system researchers. Numerous evolutionary algorithms (EAs) 1578 and swarm intelligence-based optimization algorithms have 1579 been researched in the last couple of decades to identify 1580 optimum solutions to various OPF objectives.

1581
The OPF for IEEE 30 bus system has 24 control/decision 1582 variables and the IEEE 54 bus system has 33 control variables 1583 to be optimized. The different cases for the formulation of the 1584 objective function are provided in Table 35. The other test 1585 cases have been considered as described in Table 36.

1586
The multi-objective optimization cases have been dealt 1587 with as single-objective optimization problems through the 1588 weighing factors techniques whose weights have been set 1589 based on the modelling at [87].

1590
The equality constraints are defined for the power balance 1591 of active and reactive power followed by three inequality 1592 constraints for the generator limits, two security constraints, 1593 transformer constraint and shunt compensator constraint. 1594 A comprehensive description of the mathematical formula-1595 tion of the OPF, control (independent) variables, state (depen-1596 dent) variables and the various constraints is available at [87]. 1597 Constraint handling in the current work is performed 1598 through a combination of the linear penalty incremental 1599  TotalNFEs denotes the total number of function evaluations. 1632 Despite the LPIM effectiveness to prevent infeasible solu-1633 tions from making it into the final population pool, it is 1634 necessary that the search process be guided to feasible 1635 zones for individual constraints. This is implemented through 1636 an archive-based constraint correction (ABCC) system act-1637 ing as the secondary constraint handling mechanism which 1638 occurs after the LPIM. The two-constraint handing (CH) 1639 mechanisms are balanced based on the number of function 1640 VOLUME 10, 2022 mechanism for all of the available budget. LPIM is imple-      system are shown in Table 37 and Table 38 respectively.

1698
From Table 37 and Table 38: parative analysis are provided in Table 39 and Table 40 for 1711 the OPF of various cases for the IEEE 30 and IEEE 57-bus 1712 system respectively.

1713
From Table 39 and Table 40:

1714
• The performance of I-GWO, MEGWO and GWO was 1715 competitive in most of the test cases. It can also be 1716 noticed that for the same fitness score, the other param-1717 eters have been different for the various algorithms con-1718 sidered. This is on account of the complexity and high 1719 non-linearity associated with the OPF problem.

1720
• The performance of SL-GWO stands out in terms of 1721 optimality, lower standard deviation and mean val-1722 ues closer to the best values. The exploitation system 1723 through the different symbiotic learning strategies has 1724 been the stronghold for SL-GWO enabling the algorithm 1725 to intensify and further refine the quality of solutions. 1726 The neighbourhood operator-based improvement from 1727 I-GWO and the multi-strategy ensemble techniques 1728 from MEGWO also proved successful at generating 1729 optimal solutions while handling multiple constraints 1730 but performed next to SL-GWO.

1731
• The computational times of ACGWO, P-ObGWO and 1732 SOGWO were the highest followed by ChOA for the 1733 modern meta-heuristics. MEGWO and I-GWO had 1734 the lowest computational times followed by MFO and 1735 SL-GWO.

1736
A comparison of the best, worst, mean and standard deviation 1737 of the optimal costs recorded by the other algorithms chosen 1738 for the comparative analysis is provided in Table 41 and 1739 The second problem is that of the optimal reactive power 1743 dispatch (ORPD) on base configurations of IEEE 30-bus and 1744 57-bus systems from [88]. Optimizing reactive power flow in 1745 an electrical network is an important aspect of system study 1746 as the reactive power supports network voltage which needs 1747 to be maintained within desirable limits for system reliability. 1748 A network consisting of only conventional thermal generators 1749 has been extensively studied for optimal active and reactive 1750 power dispatch.

1751
The ORPD for IEEE 30 bus system has 19 control/decision 1752 variables and the IEEE 54 bus system has 27 control vari-1753 ables to be optimized. The objectives for case 1, case 1a, 1754 case 11 and case 11a are the minimization of the real power 1755 loss (P loss ) in the network and for case 2, case 2a, case 12 1756 and case 12a is the minimization of the aggregate voltage 1757 deviation (VD) in the network. The equality constraints are 1758 defined for the power balance of active and reactive power 1759 followed by three inequality constraints for the generator lim-1760 its, two security constraints, transformer constraint and shunt 1761 compensator constraint. A comprehensive description of the 1762 mathematical formulation of the ORPD, control (indepen-1763 dent) variables, state (dependent) variables and the various 1764 constraints is available at [88].   Table 43 and Table 44 respectively.

1770
From Table 43 and Table 44: for the ORPD of the IEEE 30-bus systems and three out 1774 of the four cases for the ORPD of the 57-bus system.    Table 45 and 1794 Table 46 for the ORPD of various cases for the IEEE 30 and 1795 IEEE 57-bus system respectively.

1796
A comparison of the best, worst, mean and standard devia-1797 tion of the optimal costs recorded by the other algorithms cho-1798 sen for the comparative analysis are provided in Table 47 and 1799  Table 48 for the ORPD of various cases for the IEEE 30 and 1800 IEEE 57-bus system respectively 1801 From Table 45, Table 46, Table 47 and Table 48:

1802
• The performances of SL-GWO, MEGWO and I-GWO 1803 have been excellent for the ORPD for both IEEE 30-bus 1804 and IEEE 57-bus systems. These algorithms reported the 1805 best fitness scores and had the least standard deviations 1806 for most of the cases. The computational times of these 1807 algorithms are lower as well. This is indicative of the 1808 algorithms' capability at handling multiple constraints 1809 with a good exploitation system.

1810
• Unlike the chaotic variants (CGWO and ACGWO), 1811 which ended up with the highest fitness scores, the 1812 performance of ChOA was notably better by a small 1813 margin. Although the computational times of ACGWO 1814 were twice that of ChOA, it could not effectively 1815 explore and exploit the search landscape. Setting aside 1816 the average performance, the chaotic variants (CGWO 1817 and ACGWO) had a good population diversity with 1818 a greater difference in the fitness score for every 1819 iteration.        benchmarking scenario is the equilibrium of the explo-

1887
• SL-GWO's reliance on greedy selection during the sym-1888 biotic learning phase may have an effect on its local 1889 search capabilities. The greedy selection technique tries 1890 to promote elitism by selecting only superior solutions 1891 while discarding inferior alternatives. This can some-1892 times result in slower convergence for a certain hybrid 1893 landscape (hybrid test functions) as seen in the CEC2018 1894 test suite.

1895
• Due to SL-GWO's reliance on random omegas (at 1896 least seven distinct omega wolves), the population size 1897 must always be more than seven. With a population 1898 size of less than seven, the algorithm may fail to 1899 operate.

1901
This article realizes an improved meta-heuristic optimiza-1902 tion technique known as SL-GWO to combat the curse of 1903 dimensionality and improve population diversity through dif-1904 ferent symbiotic hunting and learning strategies. SL-GWO 1905 restructures the standard hierarchical hunting system in GWO 1906 through population sub-grouping such that each group acts 1907 individually with its own uniquely crafted hunting and control 1908 mechanisms. Dynamic tuning through linear and adaptive 1909 tuning mechanisms for the two sub-groups of wolves aid 1910 the hunt of individual wolves to evolve stronger and fit-1911 ter over time with diverse hunting instances for the solu-1912 tion dimensions. Despite the computational complexity of 1913 SL-GWO being slightly higher than the standard GWO due 1914 to the addition of a quick sort mechanism, the revised algo-