1. Introduction
In the realm of power systems, the automatic voltage regulator (AVR) stands as a linchpin, ensuring that connected electrical equipment functions within prescribed voltage bounds [
1]. The consequences of inadequate voltage regulation can be profound, from equipment damage and operational failures to costly downtime and extensive repairs [
2,
3,
4,
5]. Consequently, the AVR plays a pivotal role in power systems reliant on generators or alternators for electricity generation [
6,
7]. While existing control methodologies have achieved some success, they remain encumbered by limitations [
8], including challenges related to robustness, overshoots, rise times, settling times, and persistent steady-state errors.
The motivation driving our study is rooted in a collective commitment to surpass these limitations and contribute to the evolution of more robust and efficient power systems. Our primary objective is to introduce an advanced control scheme capable of effectively addressing these challenges. To accomplish this, we have innovatively integrated a novel optimizer, rooted in the arithmetic optimization algorithm (AOA) [
9], meticulously fine-tuned to enhance the parameters of our proposed control scheme and, by extension, its overall performance and adaptability.
In the existing landscape of AVR control, controllers have emerged as indispensable assets for the vigilant monitoring and regulation of the AVR itself [
10]. These controllers serve as hubs, facilitating real-time adjustments to maintain voltage stability, enabling remote monitoring, fault detection, and automatic shutdown during emergencies, and enhancing the overall system dependability. A range of controllers, from the standard proportional–integral–derivative (PID) to more advanced variants like the PID Acceleration (PIDA), fractional-order PID (FOPID), and PID with a second-order derivative (PIDD
2), offer diverse attributes to meet the specific requirements of AVR control [
11,
12,
13,
14,
15,
16,
17,
18].
However, the choice of controller alone is insufficient to address the complex challenges faced by AVR systems. The choice of a cost function is equally crucial, as it significantly impacts performance [
19]. Researchers employ various cost functions, such as the integral of time-weighted squared error, integral of squared error, integral of absolute error, and the dynamic response performance criteria-based Zwe-Lee Gaing (ZLG) cost function [
20,
21,
22]. In this context, our work introduces a novel approach that unites both the controller and the optimizer to form a comprehensive solution for enhancing AVR stability. The core innovation is the balanced arithmetic optimization algorithm (b-AOA). It marries the powerful pattern search (PS) strategy [
23], renowned for its exploitation capabilities, with the elite opposition-based learning (EOBL) strategy [
24], elevating exploration. This marriage optimizes the controller parameters and the AVR system’s response, harmonizing exploration and exploitation to attain a level of stability previously out of reach.
The efficacy of the b-AOA is first verified through comprehensive assessments against 23 classical unimodal, multimodal, and fixed-dimensional multimodal benchmark functions. These evaluations compare the effectiveness of the proposed b-AOA to other optimization algorithms, including the original AOA [
9], sine cosine algorithm [
25], weighted mean of vectors algorithm [
26], and marine predators algorithm [
27]. The results from the benchmark functions underscore the remarkable performance of the b-AOA. It consistently achieves mean errors close to zero, demonstrating its capability to find accurate solutions. Furthermore, its robustness and consistency make it a strong candidate for addressing a wide range of optimization problems.
In the case of the AVR system, we firstly introduce a PIDND
2N
2 controller designed for enhanced precision, stability, and responsiveness in voltage regulation. This configuration mitigates the limitations associated with conventional methods, promising a superior control performance. Secondly, the b-AOA optimizer fine-tunes the parameters of our proposed control scheme, improving its overall performance and adaptability. Using the ZLG cost function [
28], we target the minimization of dynamic response performance criteria, such as maximum overshoot, steady-state error, settling time, and rise time, thereby ensuring that the AVR system meets the most stringent performance requirements. Our work seeks to transcend theoretical innovation, anchoring itself in the practical applicability of power systems, where stability and reliability are non-negotiable. Through extensive simulations and rigorous experimentation, we aim to demonstrate the superiority of the b-AOA-based AVR system in comparison to existing control and optimization techniques. Our focus on stability, speed of response, robustness, and efficiency aligns with the motivations presented, making our work a substantial contribution to the field of power system control.
To validate the superiority of the proposed b-AOA approach, we conducted extensive comparative analyses, evaluating its performance against well-established control methodologies, such as the sine cosine algorithm-based PID controller [
29], whale optimization algorithm-based PIDA controller [
30], slime mould algorithm-based FOPID controller [
31], and particle swarm optimization-based PIDD
2 controller [
32]. The results unequivocally demonstrate that the b-AOA-based approach outshines its counterparts. It exhibits unmatched transient response characteristics, with the shortest rise time (0.033485 s) and settling time (0.050752 s) while eliminating overshoot. In contrast, other methods exhibit less favorable response characteristics. In terms of frequency response, the b-AOA approach consistently excels, showcasing robust stability, favorable gain margins, and a broader bandwidth.
To further assess the effectiveness of the proposed approach, we compared it with several other established controller approaches reported in the literature. These included several recently reported control methods for the AVR system. These methods include a variety of controllers, each tuned using different optimization algorithms such as the marine predators algorithm-based FOPID [
33], hybrid atom search particle swarm optimization-based PID [
34], equilibrium optimizer-based TIλDND
2N
2-based controller [
35], reptile search algorithm-based FOPIDD
2 [
11], improved Runge–Kutta algorithm-based PIDND
2N
2 [
36], symbiotic organism search algorithm-based PID-F [
37], whale optimization algorithm-based 2DOF FOPI [
38], Lévy flight-based reptile search algorithm with local search ability-based PID [
39], chaotic black widow algorithm-based FOPID [
20], genetic algorithm-based fuzzy PID [
40], sine cosine algorithm-based FOPID with fractional-order filter [
41], hybrid simulated annealing–Manta ray foraging optimization algorithm-based PIDD
2 [
42], slime mould algorithm-based PID [
43], gradient-based optimization-based FOPID [
44], and nonlinear sine cosine algorithm-based sigmoid PID [
45]. We evaluate their transient response performance to assess the effectiveness of the proposed approach. The results demonstrate the efficacy of the b-AOA-based PIDND
2N
2 controller in comparison to various state-of-the-art methods as it stands out with an impressive performance, suggesting the exceptional stability and responsiveness of the b-AOA-tuned controller.
In summary, our work presents a superior solution to address the challenges in AVR control, contributing to the advancement of power systems while establishing a new benchmark for stability, responsiveness, and reliability in this critical domain. The unique integration of the b-AOA with the PIDND2N2 controller signifies a significant leap forward in achieving optimal voltage regulation and stability in power systems.
2. Overview of Arithmetic Optimization Algorithm
The arithmetic optimization algorithm (AOA) draws inspiration from arithmetic principles [
9] to construct a versatile metaheuristic optimization technique. It initiates the optimization process by generating a set of randomized solutions represented as follows.
Following this, the algorithm employs a function known as “Math Optimizer Accelerated” (MopA) to execute exploration and exploitation tasks. The MopA function is defined as:
where
represents the current iteration,
denotes the maximum number of iterations, and
and
represent the minimum and maximum values of the accelerated function. The exploration phase of the algorithm is carried out when
, where
is a randomly generated number. During exploration, the multiplication (
) and division (
) operators are employed, defined as follows:
where
represents the
position of solution I at the current iteration,
denotes the solution of
in the next iteration,
signifies the best solution’s
position obtained so far,
is a small integer,
is a control parameter that adjusts the search process, and
and
, respectively, represent the upper and lower bounds of the
position. The “Math optimizer probability” function, denoted by MopP, is computed as follows, with
reflecting the exploitation accuracy through iterations.
The term
is another random number utilized for position updates. The
operator is employed for
, while the
operator is used otherwise. Conversely, the exploitation phase occurs when
. In this stage, the addition (
) and subtraction (
) operators are utilized, defined as:
Here,
is a random number determining whether the
or
operation is applied.
operates when
, while
is used for
.
Figure 1 presents a comprehensive flowchart of the AOA, depicting its intricate process.
3. Balanced Arithmetic Optimization Algorithm
The balanced AOA (b-AOA) is an evolution of the pattern search (PS) [
46] and the opposition-based learning (OBL) [
47] schemes, designed to enhance both exploration and exploitation capabilities in the context of metaheuristic optimization. This section provides a step-by-step explanation of the b-AOA’s development and its core components.
The PS scheme serves as the foundation for the b-AOA. It is a derivative-free algorithm known for its exploitation capabilities [
48]. PS starts with an initial point (
) defined by the user and proceeds by generating a mesh around this point, gradually updating the mesh as new points with lower objective function values are discovered. The process involves the following key steps:
Exploration Stage: If a new point with a lower objective function value () is found (successful poll), it becomes the source point. The mesh size is then expanded by a factor of 2, creating new points for exploration.
Exploitation Stage: When no new points with lower values are discovered, the mesh size is reduced by multiplying it by 0.5 (reduction factor). This contraction stage continues until the termination condition is met.
The detailed flowchart of the PS scheme is illustrated in
Figure 2.
The OBL scheme, introduced as a machine-learning technique, enhances the performance of metaheuristic algorithms by considering both the current individuals and their opposites [
47]. A special type of OBL mechanism known as elite OBL (EOBL) [
49] focuses on the elite (best) individuals in combination with the current individuals. EOBL generates opposite solutions for the elite individuals, which are then evaluated for their fitness. The mathematical representation of the EOBL strategy is as follows:
Elite candidate solution: with decision variables.
Elite opposition-based solution: where and is a parameter within the range (0, 1) controlling the opposition magnitude.
Dynamic boundaries: .
To ensure that opposite decision variables stay within the boundaries [], the following rule is applied: , if or .
The working principle of the OBL mechanism is depicted in
Figure 3.
The b-AOA integrates the EOBL and the PS schemes to achieve a balanced approach with improved exploration and exploitation capabilities.
Figure 4 provides an overview of the b-AOA’s operation. As depicted in the figure:
The algorithm commences with the original AOA and generates the best solution.
The EOBL scheme is introduced to produce best solutions.
The PS scheme takes over to enhance exploitation, running a total of 5 times with 100 × D iterations, where D represents the problem’s dimension size.
The parameters for the b-AOA, derived from extensive simulations, include:
PS scheme parameters: initial mesh size = 1, mesh expansion factor = 2, mesh contraction factor = 0.5, and all tolerances = 10−6.
AOA parameters: sensitive parameter , control parameter , , .
9. Conclusions and Future Works
In this study, we have introduced a novel approach to enhance the control of the AVR in power systems. By uniting a PIDND2N2 controller with the novel b-AOA, we aimed to address the limitations associated with conventional methods. The introduction of the PIDND2N2 controller offers enhanced precision, stability, and responsiveness in voltage regulation. This innovative configuration mitigates the shortcomings of existing approaches, promising a superior control performance. The b-AOA optimizer is meticulously fine-tuned with the integration of PS and EOBL strategies into the original AOA in order to demonstrate an exceptional performance. The assessment on 23 benchmark functions shows that it consistently achieves accurate solutions, exhibits robustness in addressing various optimization problems, and showcases remarkable potential for a wide range of applications. Extensive comparative analyses reveal the superiority of the proposed approach in transient response characteristics. The b-AOA-based AVR control approach excels in rise time, settling time, and overshoot, outperforming other methods. It also ensures robust stability with favorable gain margins and a broader bandwidth, offering improved performance for handling dynamic frequency changes. The results of our work set a new benchmark for AVR control, advancing stability, responsiveness, and reliability in power systems.
Future research in this domain should focus on several key aspects. Firstly, further refinement of the b-AOA optimization framework, exploration of additional optimization problems, and evaluation of its applicability to diverse domains are promising directions. Inspired by recent developments in integrated energy systems [
54], our subsequent work will explore the adaptation of our optimization approach to various energy systems, aiming to showcase its advantages and contribute to the broader field. Secondly, investigating the practical implementation of our proposed control scheme in real-world power systems and conducting extensive field testing will provide valuable insights into its real-world performance. Additionally, the integration of emerging technologies, such as machine learning and artificial intelligence, into AVR control systems holds potential for further enhancement. Lastly, addressing scalability and assessing the applicability of our approach in more complex power systems will be crucial for its broader adoption. The pursuit of more efficient, stable, and responsive AVR systems remains a vibrant field of research, and we anticipate potential breakthroughs on the horizon.