Analysis of the performance of PID-based new-generation metaheuristic algorithms for automatic voltage regulation system

In recent decades, the expansion of industrial organizations in both scale and scope has necessitated dependable output voltage supplies. However, persistent oscillations in electromechanical devices can impede power efficiency and stability, underscoring the importance of reliable automatic generation regulation (AVR) systems and power system design in the manufacturing sector. To address this issue, this study presents a performance analysis of a proportional integral derivative (PID) controller based on new-generation metaheuristic algorithms (MAs) for the AVR system. Five recent and novel MAs were employed to optimize the PID controller for the AVR system, with the controllers' performances evaluated under five distinct performance metrics. The findings revealed that the Northern Goshawk Optimization (NGO) algorithm was the most effective optimization approach, exhibiting the lowest values of overshoot (33.2784%), peak time (0.2120 s), and objective value (0.0077). These results suggest that the NGO algorithm is a promising optimization method for improving AVR system performance in industrial settings.


INTRODUCTION
Electricity is an indispensable element of contemporary industrialization, serving as a catalyst for technological progression, bolstering economic equilibrium, and fostering national expansion.Modern society would come to a complete standstill without electricity because it is a crucial element that supports the operation of numerous industrial, technological, and scientific advancements.These advancements include everything from the generation and transmission of power to the operation of medical devices, communication systems, and other essential infrastructure that is essential for preserving economic stability, social welfare, and national security.A wide range of equipment in various domains, including healthcare, emergency response, and communications, heavily rely on electrical power [3].To ensure proper equipment operation, a stable voltage and power source must be supplied.as engineers strive to design electric components and circuits that are less susceptible to voltage fluctuations, continuous oscillations in electrical components can have detrimental effects on electromechanical devices, reducing power efficiency and causing instability in power systems.For widespread connections, reliable voltage supplies are essential to ensure optimal stability in load balancing.The AVR system has thus become a popular choice in the industrial sector to establish effective stability and optimal control for power systems.The primary function of the automatic voltage regulator (AVR) is to retain a constant voltage at the output terminal of the generator by monitoring the voltage fluctuations that are caused by the changes in the load supply of the system [9,14].
Leveraging the proportional-integral-derivative (PID) controller's control capabilities will improve the AVR system.for decades, many industrial operations have been controlled using the PID controller [8,22].Since its inception, the PID controller has dominated the control landscape owing to its ease of use.On the other hand, PID tuning is essential and not always robust [18].To address this problem, metaheuristic algorithms (MAs) have been explored.As a kind of SC method, metaheuristics allow for difficult problems to be solved in a fair amount of time.as a sophisticated SC method, they build algorithmic frameworks and methods for solving optimization issues [11,12].It has become increasingly apparent that mas are becoming more and more popular among scholars, academics, and scientists [16,20].This is because MAs can be tolerant of inaccurate, ambiguous, and approximate information, in a given problem data.Besides, they are fit for managing flawed information and non-definite responses to make appropriate enhancement arrangements.Since the inception of the first MA, there has been much ingenuity and cutting-edge progress.According to [5] the majority of contemporary metaheuristics were developed before the year 2000 and are referred regarded as "classical" MAs.genetic algorithm (GA), particle swarm optimization (PSO) algorithm and differential evolution (DE) algorithm are among a few.Classical algorithms, including PSO [15,17,19], GA [1,6,13], and DE [21,23], have been used to tune the PID controller for the AVR system.However, within the past decade, there has been a notable and efficacious advent of novel computational methodologies, which have demonstrated an inclination towards inventiveness and originality.A few of the most recent are pelican optimization algorithm (POA) [10], dandelion optimizer (DO) [25], pathfinder algorithm (PFA) [24], gazelle optimization algorithm (GOA) [2], honey badger algorithm (HBA) [7], and northern goshawk optimization (NGO) [4].These MAs have proven effective in their various fields of application.However, only a very few of these recent mas have been used for the AVR system.It is worth noting that the majority of the algorithms mentioned were not accessible or available for use prior to the year 2022.This indicates that recent advancements in technology have led to the development and availability of new and improved algorithms, which have now become more widely accessible to researchers and practitioners in various fields.The increased availability of these algorithms presents opportunities for further exploration and experimentation, ultimately leading to potential advancements and breakthroughs in various domains.The motivation for research on a PID-based new-generation MA stems from the potential to harness recent optimization techniques that can handle non-linear, time-varying, and non-stationary systems while maintaining the simplicity and effectiveness of PID control.This could result in a more efficient and accurate approach to solving complex problems than existing methods and have applications in various fields.Hence, the purpose of this study is to investigate these new generational MAs and assess their proficiency with the AVR system.

AVR SYSTEM DESIGN 2.1 Automatic Voltage Regulation System
The primary function of an AVR is to regulate the terminal voltage of a synchronous generator to a constant value.Each of the linear devices-exciter, amplifier, sensor, and generator-make up the AVR's first three building blocks (see Fig. 1).When developing the mathematical model of the AVR system, these crucial blocks are always taken into account.The linear modelling of all components uses a gain of K and a time constant of .Whereas the block diagram of the AVR system used is shown in Fig. 1, the operational settings of the AVR used are shown in Table 1.

Controllers
Figure 2 shows the step response of the AVR system without using any controller.It can be seen from the figure that curve is characterized by large amplitude oscillations and overshoot.Without any kind of control, the settling time and steady state also performed badly.A direct implementation of the AVR system is not suitable for use.The AVR system cannot be used in its pure form.As a result, the addition of a PID controller is required in order to enhance the dynamic responsiveness of the AVR and reduce the inaccuracy that it exhibits in terms of its steady state.
This present study made use of a basic PID controller for the AVR system.The mathematical representation of the PID controller is illustrated as follows: where the proportional, integral, and derivative parameters are depicted as   ,   and   , respectively.To get the best of the controller, these parameters must be optimized.A popular trend is to harness MAs to select the optimal values of these parameters.Therefore, this work applies the most recent new generation MAs for designing the PID controller.The MAs are introduced in the next subsequent session.Furthermore, the present work utilizes the integral time squared error (ITSE) as the objective function.
The ITSE is well recognized for its capacity to effectively aid in the minimization of large oscillations by imposing a greater penalty on such oscillations as a result of its squared error.As a result, the ITSE is used by the suggested method to demonstrate controller efficacy.

OVERVIEW OF THE OPTIMIZATON METHODS 3.1 Dandelion Optimizer (DO)
A natural-inspired algorithm called the Dandelion Optimizer (DO) was put out in [25].A plant called a dandelion utilizes the wind to disperse its seeds.Initialization is the first DO stage, which entails The DO algorithm offering potential solutions in each dandelion seed.The second phase is known as the rising phase.In this stage, dandelion seeds must rise to a certain height in order to float away from their parent plants.The height to which dandelion seeds rise varies with air humidity, wind speed, etc.How high a dandelion seed rises depends on the wind speed.The dandelion flies higher and the seeds disperse further with a stronger breeze.On the other hand, dandelion seeds have a hard time taking off correctly on a rainy day because of the humidity and air resistance.The third stage is the descending stage.At this point, the dandelion seeds have risen to a certain altitude and are beginning to fall (exploration phase).A dandelion's flight path is simulated in DO using Brownian motion.Phase three is the landing phase.The focus of the DO algorithm shifts to exploitation in this phase.Based on the outcomes of the first two steps, the dandelion seed settles down at random.As the algorithm progresses through its iterations, it should eventually arrive at the best potential solution.Full description and mathematical modelling can be found in [25].

Pathfinder Algorithm (PFA)
The Pathfinder algorithm (PFA) was first introduced in [24].In PFA, animals' social behaviours are modelled, and their collective leadership structure is emulated to recognize the optimal food region or prey (global optimum).Starting with arbitrary initial locations, the proposed approach assigns herd members to herds.Each member's fitness is then assessed, and the pathfinder position's best fitness is picked.The pathfinder drives the course of the leftover multitude of individuals and explores the hunt space while simultaneously creating the vector of change rate in each iteration.Since the pathfinder has no dependency on group operations, it sets it apart from other swarm-based algorithms, and it makes it significantly better than the rest.The full details of the algorithm may be found in [24].

Pelican Optimization Algorithm (POA)
Once pelicans have located their meal, they will dive anywhere from 10 to 20 meters to get it.Certain predators like to prey at ground level.Spreading their wings on the water, they lure fish into shallower areas where they may easily capture and consume them.When a pelican catches a fish, water flows into its beak and causes it to tilt its head forward before swallowing.Pelicans' evolved hunting skills are the product of their ingenious strategies and behaviours.This strategy's modelling was a major motivation for the development of the POA [10].A detailed description of the algorithm can be found in [10].

Gazelle Optimization Algorithm (GOA)
Aguska [2] first presented the Gazelle Optimization algorithm (GOA) in 2022.The survival strategies of gazelles served as inspiration for the algorithm.In order to ensure the success of the developed GOA algorithm, it models the gazelle's ability to stay alive.Grazing in safety and escaping a sighted predator are two steps in the optimization process.Exploitation and exploration are, thus, the two defining stages of GOA.The gazelles are assumed to be grazing peacefully in the absence of a predator or when the predator is actively hunting them in the exploitation phase.To efficiently cover nearby domain regions during this phase, the Brownian motion, which is characterized by regular and controlled steps, was applied.During their grazing period, it is thought that gazelles walk in a Brownian manner.At the first sign of a predator, the exploring  = [0,1]; K= [0, 1] GOA [2] PSRs=0.34;S=0.88; NGO [4]  = [0,1] PFA [24] N/A Figure 3: Step response of the controllers phase begins.As a defence mechanism, gazelles will either stomp their feet, flick their tail, or stomp up to 2 meters in the air with all four feet.This behaviour is emulated by setting the height of 2 meters to a value between 0 and 1.For this part of the algorithm, the Levy flight, a process characterized by little steps followed by occasional large leaps is used [2].

Northern Goshawk Optimization Algorithm (NGO)
The NGO was first presented in [4].The NGO algorithm's hunting mechanism is founded on its potent capacity to track down and devour its target.There are three steps [2] to the algorithm: population initialization, prey identification, and prey capture upon seeing its target, the northern goshawk will choose it and launch an assault.Since the prey is chosen at random, this action may represent the algorithm's capacity for global exploration in the viable space.The prey will start to flee as the northern goshawk begins its attack during the prey capture phase.The northern goshawk, at this point, must continue the pursuit of its meal.The rapid rate of speed at which the northern goshawk pursues its victim allows it to successfully catch and eat its target in a wide variety of settings.Fuller details may be found in [4].
As a means of gauging their suitability for the AVR system, each of the new generation MA algorithms described above is applied to the optimization of the PID controller's settings.For a fair comparative analysis, all were subjected to the same simulation condition.

RESULTS AND DISCUSSION
The simulation of the proposed models was implemented using a computer of CPU i7-4790 (Intel Core TM Processor @3.50 GHz) computer.The simulation parameters of the algorithms are presented in Table 2. Fig. 3 shows the step response of the controllers used.The performance of the terminal voltage step response of the AVR system was evaluated using a set of dynamic responses such as the rise time (  ), settling time (  ), overshoot (OS), peak time (  ) and the values of the objective function (    ).Table 3 shows the dynamic responses of the controllers.Each of the methods delivered a laudable result.In terms of the   , the DO-PID outscored other methods with the smallest value (0.0817 s).The POA-PID had the best of value for the   (1.2985 s).Considering the OS, the NGO-PID surpassed others with the smallest value of 33.2784%.This is seen clearly in Figure 3, which shows the NGO-PID as having the lowest amount of OS in contrast to the other approaches.This suggests that the NGO-PID is more resilient to disturbance than other approaches.Further, the NGO-PID maintained its superior performance regarding the lowest value of the   (0.2120).Howbeit, the same is true for the PFA-PID.Finally, the NGO-PID provided the lowest value of the     (0.0077).The overall performance of the NGO-PID indicated that it best suitable controller among its peers.

Figure 1 :
Figure 1: Block diagram for a PID-controller AVR system

Figure 2 :
Figure 2: Dynamic response of the AVR system without any controller

Table 2 :
Settings of the Algorithms

Table 3 :
Settings of the Algorithms