• **Mutation**

The habitat suitability index (HSI) [35] can easily be modified with resultant in the breed calculation to be different from the symmetry value, if a number of catastrophic actions occur. In biogeography-based optimization, this procedure is modeled as SIV mutation and the mutation rates of habitats may be intended to use the species add up probabilities known unexpected modification in weather of one habitat or additional occurrence will cause the unexpected modification in HSI (habitat). This condition is replica in the form of unexpected modification in the value of the suitability index variable in BBO. The probability of some organism [36] is calculated by this equation:

$$P\_S = \begin{cases} - (\lambda\_S + \mu\_S) P\_S + \mu\_{S+1} P\_{S+1} \ S = 0 \\ - (\lambda\_S + \mu\_S) P\_S + \lambda\_{S-1} P\_{S-1} + \mu\_{S+1} P\_{S+1} \ 1 \le S \le S\_{\text{max}-1} \ . \end{cases} \\ \tag{8}$$

The own probability of all members is one habitat. If probability of this is too low, and after that, this result has more probability to mutilation [37]. In a similar way, if the probability of a result is more, that result has a small probability to mutate. As a result, solutions with a low suitability index variable and high suitability index variable have a small possibility to grow an improved SIV in the new iteration. Dissimilar low suitability index variable and high suitability index [38] variable solutions, middle HSI solutions have a bigger possibility to grow improved solutions after the mutation process. By the use of equation mutation, all results can be calculated easily:

$$m(s) = m\_{\max} \left( \frac{1 - P\_S}{P\_{\max}} \right) \tag{9}$$

where *m*(*s*) represent the mutation rate.

#### 2.2.3. Dragonfly Algorithm (DA)

DA [39] is an exceptional optimization process planned by Seyedali. The most important purpose of swarm is durability; thus, all individual must be unfocused outward, and opponents attracted towards nourishment sources. Taking both behaviors in swarms [40], these are five major topographies in position informing procedure of individuals. The numerical model of swarms actions as shown below: The parting procedure [41] in DA informing as in the above equation:

$$S\_i = -\sum\_{J=1}^{N} X - X\_{J\prime} \tag{10}$$

where *N* represents the amount of entities of neighboring, *X* represents the present situation of specific, *XJ* indicates the location of *J* th specific of the adjacent [42].

The orientation procedure in this approach can be rationalized by subsequent expression [43]:

$$A\_i = \frac{\sum\_{J=1}^{N} V\_J}{N},\tag{11}$$

where *VJ* represents velocity of *J* th specific of the adjacent. The unity in DA can be intended by the above evaluation:

$$\mathbf{C}\_{i} = \frac{\sum\_{I=1}^{N} \mathbf{X}\_{I}}{N} - \mathbf{X}\_{\prime} \tag{12}$$

where *X* represents the existing specific point, *XJ* is the spot of *J* th specific of the adjacent, and *N* indicates the amount of areas.

#### *2.3. Hybrid Artificial Intelligence Techniques*

2.3.1. Particle Swarm Optimization (PSO) and Gravitational Search Algorithm (GSA) Hybridization

The easiest technique to mongrelize PSO and GSA is to implement the strength separately in the successive approach [44].

#### Particle Swarm Optimization (PSO)

PSO is provoked with keen collective activities [45] accessible by a multiplicity of creatures, such as the group of ants or net of birds. The particle position and velocity both are updated according to the equations:

$$w\_i^d(t+1) = w(t)v\_i^d(t) + c\_1 X r\_1 X (pbest\_i^d - x\_i^d) + c\_2 + r\_{2\prime} \tag{13}$$

$$\mathbf{x}\_i^d(t+1) = \mathbf{x}\_i^d(t+1) + \boldsymbol{\upsilon}\_i^d(t+1),\tag{14}$$

$$\mathcal{W}t = rand \, X \frac{t}{t\_{\text{max}}} X (w\_{\text{max}} - w\_{\text{min}}) + w\_{\text{min}} \tag{15}$$

where *vd <sup>i</sup>* (*<sup>t</sup>* <sup>+</sup> <sup>1</sup>) shows velocity of (*d*th) dimension at (*t*) reiteration of (*<sup>i</sup>* th) particle, *xd <sup>i</sup>* (*t* + 1) is existing position of (*d*th) dimensional iteration (*t*) of (*i* th) particle; *c*<sup>1</sup> and *c*<sup>2</sup> representing the acceleration coefficients [46] which manage the pressure of gbest and pbest on the search procedure, *r*<sup>1</sup> and *r*<sup>2</sup> representing the arbitrary statistics in variety [0, 1]; *pbestd <sup>i</sup>* represents finest point of (*i* th) element up to now.

#### Gravitational Search Algorithm (GSA)

GSA is meta-heuristic population-centered approach inspired with directions of attraction and quantity associations [47–49]. In this method, cause is dignified as article encompasses of unlike multitudes and the enactment of this is considered via crowds.

#### 2.3.2. Differential Evolution and Particle Swarm Optimization Hybrids

It is a population-based optimizer [50] alike the genetic algorithm, having identical operatives corresponding to selection, mutation, and crossover. In this method, all constraints are expressive in genetic measurable by a genuine measurement [20,51].

#### 2.3.3. Binary Moth Flame Optimizer (BMFO1)

BMFO is a newly projected meta-heuristics search algorithm proposed by Seyedali Mirjalili [52,53] which is refreshed by direction-finding behavior of moth and its converges near light. Although, moths are having a robust capability to uphold a secure approach with respect to the moon and hold a tolerable erection for nomadic in an orthodox mark for extensive distances. Besides, they are attentive in a fatal/idle curved track over simulated basis of lights.

#### 2.3.4. Modified SIGMOID Transformation (BMFO2)

The binary calibration of constant pursuit house and places of search representatives, resolutions to binary exploration house could be the obligatory method for optimization of binary environmental issues such as LFC. In the proposed research, a modified sigmoidal transfer function is adopted, which has superior performance than another alternatives of sigmoidal transfer function as reported in [54].

#### 2.3.5. Harris Hawks Optimizer

HHO [55] is gradient-free and populations-centered algorithm that comprises exploitative and exploratory stages, which is fortified by astonishment swoop, the fauna of examination of a victim, and diverse stratagems built on violent marvel of Harris hawks.

#### 2.3.6. Smart Grid Applications

The modern smart grid system as shown in Figure 4 consists of various power generating units consisting of thermal, hydro, nuclear, wind, and solar-based power producing elements. The scheduling of every power producing in optimal condition is a tedious task and requires proper commitment schedule of generating units. Further, consideration of solar and wind-based energy sources requires proper load frequency control [56].

**Figure 4.** Modern smart grid system with Electric Vehicles (EVs) load demand.

An electric grid can easily be converted into a smart grid by balancing the voltage, current, and frequency which is possible by the load frequency control method [57]. If incoming voltage, current, and frequency is matched with the outgoing voltage, current, and frequency of an electric grid with the help of optimal gain scheduling and load frequency control approach, then steady state error will be near to zero or nil. In the proposed research, load frequency control is tested and validated with various standard benchmarks simultaneously and mathematically depicted in the following sub-sections.

#### **3. Standard Testing Benchmarks**

The consequences for various benchmark issues [58] considering the LFC situation are deliberated in the above-mentioned units.

#### *Test System and Standard Benchmark*

For confirmation of prospects of deliberate BMFO and HHO algorithms, CEC2005 benchmark functions [59] have been taken into thought, which include unimodal, multi-modal, and fixed dimensions benchmark issues and its mathematical formulation has been represented in Tables 1–3. Table 1 interprets unimodal standard performance, Table 2 portrays multi-modal standard, and Table 3 interprets fixed dimensions standard issues.

To explain the random behavior of the expected BMFO2 logarithmic rule and confirm the consequences, thirty trials were applied with all objective function check for average, variance, best and worst values for justification of output from the probable algorithmic rule, unimodal benchmark work f1, f2, f3, f4, f5, f6, and f7 are used. Table 4 (a) signifies the response of unimodal benchmark function with BMFO1 logarithmic rule, Table 4 (b) characterizes the retort of unimodal benchmark operate function by using the BMFO2 algorithmic rule and Table 4 (c) represents the answer of the fixed dimension benchmark function by using HHO algorithmic instruction.


**Table 1.** Unimodal benchmark.


It is analyzed from Table 4 that the unimodel benchmark functions f1 to f7 are tested using the modern hybrid algorithms like BMFO 1, BMFO 2, and HHO, and found that Harris hawks optimizer (HHO) produces optimal outcomes in terms of mean, standard deviation, best and worst value for all functions as compared to other algorithms. The convergence curve and trial solutions for BMFO1, BMFO2, and HHO for f1 to f7 unimodal benchmark functions are presented in Figure 5.


**Table 3.** Fixed dimension benchmark.

**Table 4.** (**a**) Outcomes of the BMFO1 algorithm. (**b**) Outcomes of the BMFO2 algorithm. (**c**) Outcomes of the HHO algorithm.


**Figure 5.** (**a**–**g**) Convergence curve of all algorithms for unimodal benchmark functions.

The convergence curve and trial solutions for BMFO1, BMFO2, and HHO for f1 to f7 unimodal benchmark functions are presented in Figure 5a–g.

The connected upshots for unimodal standard functions [60] have been represented in Table 5, which are correlated with various latest refined algorithms [61] grey wolf optimizer (GWO) [62], PSO [63,64], GSA [8,65], differential evolution (DE) [66,67], fruit fly optimization algorithm (FOA) [68,69], ant lion optimizer (ALO) [70,71], symbiotic organisms search (SOS) [72], bat algorithm (BA) [73], flower pollination algorithm (FPA) [74,75], cuckoo search (CS) [76], firefly algorithm (FA) [52], GA [77], grasshopper optimization algorithm (GOA) [73,78], MFO [79], multiverse optimization algorithm (MVO) [80], DA [81], binary bat optimization algorithm (BBA) [65], BBO [5,82], binary gravitational search algorithm (BGSA) [83,84], sine cosine algorithm (SCA) [85,86], FPA [74,87], salp swarm optimization algorithm (SSA) [88], and whale optimization algorithm (WOA) [89] in lieu of mean and standard deviation.


**Table 5.** Comparison of unimodal benchmark functions.

To defend the synthesis part of the probable algorithm, multi-modal benchmark functions f8, f9, f10, f11, f12, and f13 are taken with numerous native goals with values rising violently w.r.t magnitude. Table 6 (a) presents clarification of the multimodal benchmark function with the BMFO1 algorithm and Table 6 (b) presents the explanation of the multimodal benchmark function with the BMFO2 algorithm and Table 6 (c) presents the explanation of the multimodal benchmark function with the HHO algorithm.


**Table 6.** (**a**) Outcomes of the BMFO1 algorithm. (**b**) Outcomes of the BMFO2 algorithm. (**c**) Results of the HHO algorithm.

It is analyzed from Table 6 that multi-model benchmark functions f8 to f13 are tested using modern hybrid algorithms like BMFO 1, BMFO 2, and HHO and found that the Harris hawks optimizer (HHO) produces optimal outcomes in terms of mean, standard deviation, best and worst value for all functions as compared to other algorithms.

The convergence curve and trial solutions for BMFO1, BMFO2, and HHO for f8 to f13 multi-modal benchmark functions are presented in Figure 6a–f.

The connected outcomes for multimodal benchmark functions has been signified in Table 7, which are associated with various latest refined meta-heuristics search algorithms like GWO [62], PSO [63,64], GSA [8,65], DE [66,67], FOA [68,69], ALO [70,71], SOS [72], BA [73], FPA [74,75], CS [76], FA [52], GA [77], GOA [73,78], MFO [79], MVO [80], DA [81], BBA [65], BBO [5,82], BGSA [83,84], SCA [85,86], FPA [74,87], SSA [88], and WOA [89] in lieu of average [90] and standard deviation.

**Figure 6.** (**a**–**f**) Convergence curve of all algorithms for multi-modal benchmark functions.

The verified consequences for fixed dimension benchmark situations are obtainable in Table 8. It is analyzed from Table 8 that fixed dimension benchmark functions f14 to f23 are tested using modern hybrid algorithms like BMFO 1, BMFO 2, and HHO and found that Harris hawks optimizer (HHO) produces optimal outcomes in terms of mean, standard deviation, best and worst value for all functions as compared to other algorithms.

The convergence curve and trial solutions for BMFO1, BMFO2, and HHO for f14 to f23 fixed dimension benchmark functions are presented in Figure 7a–j.


**Table 7.** Comparison of multi-modal benchmark functions.

**Figure 7.** (**a**–**j**) Convergence curve and trial solution of BMFO2 for fixed dimension benchmark functions.


**Table 8.** (**a**) Outcomes of the BMFO1 algorithm. (**b**) Outcomes of the BMFO2 algorithm. (**c**) Outcomes of the HHO algorithm.

The comparative outcomes for fixed dimension benchmark [91] functions have been represented in Tables 9 and 10, which are associated with other latest refined met heuristics search algorithms [54,92] GWO [62], PSO [63,64], GSA [8,65], DE [66,67], FOA [68,69], ALO [70,71], SOS [72], BA [73], FPA [74,75], CS [76], FA [52], GA [77], GOA [73,78], MFO [79], MVO [80], DA [81], BBA [65], BBO [5,82], BGSA [83,84], SCA [85,86], FPA [74,87], SSA [88], and WOA [89] in terms of standard deviation [93] and average.


**Table 9.** Comparison of fixed dimension benchmark functions.


**Table 10.** Comparison of results for fixed dimension functions.

#### **4. Conclusions**

The smart grid process needs a continuing matching of resource and ultimatum in accordance with recognized functioning principles of numerous algorithms. The LFC scheme delivers the consistent action of power structure by constantly balancing the resource of electricity with the response, while also confirming the accessibility of adequate supply volume in upcoming periods. In this paper, binary variations of the moth flame optimizer and HHO have been analyzed and tested to solve twenty-three benchmark problems including unimodel, multi-model, and fixed dimension functions which investigate that the proposed Harris hawks optimizer approach suggestions are offering better results as associated to substitute labeled meta-heuristics search algorithms. In upcoming work, the effectiveness of the HHO technique is deliberate for optimal matching of total generation with total consumption of electrical energy to convert an electric grid to smart grid. So, by using the Harris hawks optimizer, we can easily balance the smart grid elements by matching production and consumption of electrical energy.

**Author Contributions:** Conceptualization, K.A.; Data curation, K.A. and A.K.; Formal analysis, K.A., A.K. and V.K.K.; Funding acquisition, B.S. and G.P.J.; Investigation, V.K.K. and D.P.; Methodology, D.P.; Project administration, S.J.; Resources, B.S. and G.P.J.; Software, G.P.J.; Supervision, S.J.; Visualization, G.P.J.; Writing—original draft, K.A.; Writing—review & editing, G.P.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** The present Research has been conducted by the Research Grant of Kwangwoon University in 2020.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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