**1. Introduction**

Optimization shows a critical role in various regions of science and technology. This is the method through which the optimal solution can be found with the help of a wide range of search mechanisms like primary, secondary, and tertiary controls [1]. With recent advancement in technology, novel optimization methodologies are identified as meta-heuristic with concern of mathematical culture. Meta-heuristic algorithms (MA) is a typical technique to get the best outcomes for the issue. It plays a fictional role to find good specifications in an optimization matter [2].

Each real-life optimization problem required procedures which observe the examination zones effectively to find most operative explanations. Moth-flame optimizer (MFO) is newly projected meta-heuristics search algorithmic rule that is inspired by the direction-finding environment of lepidopteron and its convergence in the direction of lightweight. However, like alternative similar

strategies, MFO contributes to being stuck into sub-optimal segments, which is mirrored within the procedure effort needed to search out the most effective rate. This case happens due to the developer used for research not performing well to research the find house. In addition, no free lunch theorem encourages planners to promote a new algorithmic rule or to boost the prevailing algorithmic rule.

The modern technology that balances the two-way communication between energy production and consumption and sense the critical behavior of voltage, current, and frequency which makes an electric grid as a smart grid. Smart grid is an opportunity in the growth of the country's economy and environmental health due to efficient electricity transmission, quicker restoration, reduced power cost, and enhanced integration with renewable energy sources, which is possible through optimal gain scheduling and the load frequency control method.

In earlier days, the load frequency control (LFC) problem was explained with respect to conventional dispatching [3], whose objective was to maintain voltages and frequency within prescribed limits. Today, LFC uses advanced numerical optimization techniques to solve constrained combinatorial and diverse number optimization issues. The type of controller [4], its architecture and choice of objective function play a very important role in enhancing achievement of the power system.

In the current scenario, the integral of time multiplied absolute error (ITAE) criteria is observed as an impartial task which is stated as [5]:

$$J = \bigcup\_{0}^{t\_{\rm sim}} (|\Delta F\_1| + |\Delta F\_2| + |\Delta P\_{\rm tie}|).t.d.t.\tag{1}$$

where, Δ*F*1, Δ*F*<sup>2</sup> indicate deviation of the frequency in both areas and the total simulation time (in seconds) is denoted by '*tsim*' and tie-line interchange [6] assessment is characterized by Δ*Ptie*.

The ITAE is implemented as a detached role to enhance gain of the PI controller in the present investigation. The reduction of the ITAE index with the binary moth flame optimizer (BMFO) algorithm offers augmented constraints of PI controllers which can be subjected to the following restraints [7–9]: Minimize J,

$$K\_i^P \text{min} \le K^P \le K\_i^P \text{max}, \text{ and } K\_i^{\text{Int.}} \text{min} \le K^{\text{Int.}} \le K\_i^{\text{Int.}} \text{ max}\_{\text{tot}}$$

where, *KInt*. *<sup>i</sup>* and *<sup>K</sup><sup>P</sup> <sup>i</sup>* symbolize fundamental and comparative gain of PI controller of ith (*i* = 1, 2) area.

Our contributions in this work are as follows: First, we propose the two variants of binary moth flame optimizers to solve the frequency constraint issues. We implemented two different binary variants for improving performance of the moth flame optimizer (MFO) for discrete optimization problems. In the first variant, i.e., binary moth flame optimizer (BMFO1), coin flipping-based selection probability of binary numbers is used. We used the improved Sigmoid transformation in the second variant called BMFO2. These binary MFO algorithms along with the Harris hawks optimizer (HHO) algorithms are tested and analyzed for various unimodal, multi-modal, and fixed dimension numerical optimization problem. Secondly, Section 2 explores various optimization methodologies, including classical artificial intelligence techniques, modern intelligence techniques, hybrid artificial intelligence techniques, and smart grid technologies which are tested using standard benchmarks and compared with various algorithms. Lastly, in Section 3, all the latest used algorithms are evaluated and compared in terms of standard testing benchmarks in which the proposed HHO model is having improved results in terms of average and standard deviation. Finally, Section 4 concludes the paper.

#### **2. Optimization Methodologies**

In order to discover the mathematical design of load frequency control, numerous optimization methodologies are classified into three foremost groups like traditional techniques [10], recent techniques [11], and hybrid techniques [12].

#### *2.1. Traditional Techniques*

The traditional methods may be further classified into artificial neural network, fuzzy logic technique [13], and genetic algorithm.

#### 2.1.1. Artificial Neural Network

The architecture of Artificial Neural Network (ANN) as shown in Figure 1 is promptly the emerging zone of investigation, producing attention of predictors from a noble type of scientific field, which gives a deviation of desired output and actual output as an error signal. An error signal acts like a feedback to the neural network, which balances the desired and actual output.

**Figure 1.** Artificial neural network architecture.

#### 2.1.2. Fuzzy Logic Technique

The essential configuration of the scientific reasoning scheme in which the fuzzification [14] boundary recreates the additional contribution into a fuzzy verbal input, and likewise shows an significant character in the mathematical coherent [15] procedure as actual principles, which are delivered from current sensors, are a forever crisp analytical equivalent as shown in Figure 2.

**Figure 2.** Fuzzy logic system architecture.

#### 2.1.3. Genetic Algorithm

The overall thoughts were conceived by a European country [16], whereas practicality of persecution of exhausting it to untie innovative concerns was indisputable. It may be a soft computing style, which implements strategies stimulated by usual hereditary knowledge to develop conclusions to matters [17]. Genetic Algorithm (GA) as shown in Figure 3 is refreshed by Darwin's theory concerning progression, which is useful to a vast variety of methodical and industrial problems like optimization, machine learning, and automatic software design [18].

**Figure 3.** Genetic algorithm architecture.

#### *2.2. Modern Intelligence Techniques*

To solve the multi-disciplinary optimization problems [14], various modern practices are established by the investigators. The modern intelligence techniques are explored in the following sub-sections.

#### 2.2.1. Differential Evolution Technique

It is a genetic-based algorithm [19] having identical operators corresponding to initialization, mutation, crossover, and selection. In this method, all constraints are expressive in genetic measurable by a genuine measurement [20]. The mathematical formulation of differential evolution is given below:
