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. The consequences of inadequate voltage regulation can be profound, from equipment damage and operational failures to costly downtime and extensive repairs [
1,
2,
3]. Consequently, the AVR plays a pivotal role in power systems reliant on generators or alternators for electricity generation. While existing control methodologies have achieved some success, they remain encumbered by limitations [
4], including challenges related to robustness, overshoots, rise times, settling times, and persistent steadystate errors.
It is against this backdrop that our study emerges, driven by a shared motivation to push the boundaries of AVR control and contribute to the development of more robust and efficient power systems. Our primary motivation is to propose an advanced control scheme capable of effectively addressing these limitations. To realize this goal, we have developed a novel optimizer, rooted in the arithmetic optimization algorithm (AOA) [
5], meticulously finetuned 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 vigilant monitoring and regulation of the AVR itself. These controllers serve as hubs, facilitating realtime 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 proportionalintegralderivative (PID) to more advanced variants like the PID Acceleration (PIDA), fractionalorder PID (FOPID), and PID with a secondorder derivative (PIDD
^{2}), offer diverse attributes to meet the specific requirements of AVR control [
6,
7,
8,
9,
10,
11,
12,
13].
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. Researchers employ various cost functions, such as the integral of timeweighted squared error, integral of squared error, integral of absolute error, and the dynamic response performance criteriabased ZweLee Gaing (ZLG) cost function [
14,
15,
16].
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 (bAOA). It marries the powerful pattern search (PS) strategy [
17], renowned for its exploitation capabilities, with the elite oppositionbased learning (EOBL) strategy [
18], 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 bAOA is first verified through comprehensive assessments against 23 classical unimodal, multimodal, and fixeddimensional multimodal benchmark functions. These evaluations compare the effectiveness of the proposed bAOA algorithm to other optimization algorithms, including the original AOA [
5], sinecosine algorithm [
19], weighted mean of vectors algorithm [
20] and marine predators algorithm [
21]. The results from the benchmark functions underscore the remarkable performance of the bAOA algorithm. 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 case of AVR system, we firstly introduce a PIDND
^{2}N
^{2} controller designed for enhanced precision, stability, and responsiveness in voltage regulation. This configuration mitigates the limitations associated with conventional methods, promising superior control performance. Secondly, the bAOA optimizer finetunes the parameters of our proposed control scheme, improving its overall performance and adaptability. Using the ZLG cost function [
22], we target the minimization of dynamic response performance criteria, such as maximum overshoot, steadystate 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 nonnegotiable. Through extensive simulations and rigorous experimentation, we aim to demonstrate the superiority of the bAOAbased 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 bAOA approach, we conducted extensive comparative analyses, evaluating its performance against wellestablished control methodologies, such as the sine cosine algorithm (SCA)based PID controller [
23], whale optimization algorithm (WOA)based PIDA controller [
24], slime mould algorithm (SMA)based FOPID controller [
25], and particle swarm optimization (PSO)based PIDD
^{2} controller [
26]. The results unequivocally demonstrate that the bAOAbased 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 bAOA 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 the several recently reported control methods for the AVR system. These methods include a variety of controllers, each tuned using different optimization algorithms such as marine predators algorithm (MPA) based FOPID [
27], hybrid atom search particle swarm optimization (hASPSO) based PID [
28], equilibrium optimizer (EO) based TIλDND2N2 based controller, reptile search algorithm (RSA) based FOPIDD2 [
6], improved RungeKutta (iRUN) algorithm based PIDND2N2 [
29], symbiotic organism search (SOS) algorithmbased PIDF [
30], whale optimization algorithm (WOA) based 2DOF FOPI [
31], Lévy flightbased RSA with local search ability (LRSANM) based PID [
32], chaotic black widow algorithm (ChBWO) based FOPID [
15], genetic algorithm (GA) based fuzzy PID [
33], sinecosine algorithm (SCA) based FOPID with fractional order filter [
34], hybrid simulated annealing–Manta ray foraging optimization (SAMRFO) algorithm based PIDD2 [
35], slime mould algorithm (SMA) based PID [
14], gradient based optimization (GBO) based FOPID [
36] and nonlinear SCA based sigmoid PID [
37]. We evaluate their transient response performance to assess the effectiveness of the proposed approach. The results demonstrate the efficacy of the bAOAbased PIDND
^{2}N
^{2} controller in comparison to various stateoftheart methods as it stands out with an impressive performance, suggesting the exceptional stability and responsiveness of the bAOAtuned controller.
These important results underscore the significance of our work, offering a superior solution for addressing the challenges in AVR control. It not only contributes to the advancement of power systems but also sets a new benchmark for stability, responsiveness, and reliability in this critical domain.
2. Overview of Arithmetic Optimization Algorithm
The arithmetic optimization algorithm (AOA) draws inspiration from arithmetic principles [
5] 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
$t$ represents the current iteration,
${t}_{max}$ denotes the maximum number of iterations,
$Min$ and
$Max$ represent the minimum and maximum values of the accelerated function. The exploration phase of the algorithm is carried out when
${r}_{1}>MopA$, where
${r}_{1}$ is a randomly generated number. During exploration, the multiplication (
$Mult$) and division (
$Div$) operators are employed, defined as follows:
where
${x}_{i,j}\left(t\right)$ represents the
${j}^{th}$ position of solution i at the current iteration,
${x}_{i}\left(t+1\right)$ denotes the solution of
$i$ in the next iteration,
$best\left({x}_{j}\right)$ signifies the best solution’s
${j}^{th}$ position obtained so far,
$\u03f5$ is a small integer,
$\mu $ is a control parameter that adjusts the search process,
${UB}_{j}$ and
${LB}_{j}$ respectively represent the upper and lower bounds of the
${j}^{th}$ position. The "Math optimizer probability" function, denoted by MopP, is computed as follows, with
$\alpha $ reflecting the exploitation accuracy through iterations.
The term
${r}_{2}$ is another random number utilized for position updates. The
$Mult$ operator is employed for
${r}_{2}>0.5$, while the
$Div$ operator is used otherwise. Conversely, the exploitation phase occurs when
${r}_{1}<MopA$. In this stage, the addition (
$Add$) and subtraction (
$Sub$) operators are utilized, defined as.
Here,
${r}_{3}$ is a random number determining whether the
$Add$ or
$Sub$ operation is applied.
$Add$ operates when
${r}_{3}>0.5$, while
$Sub$ is used for
${r}_{3}<0.5$.
Figure 1 presents a comprehensive flowchart of the AOA, depicting its intricate process.
8. Simulation Results and Discussion
8.1. Statistical Performance of bAOA and AOA Methods for AVR System
In the optimization of the AVR system, the bAOA and AOA algorithms were executed 30 times. A population size of 30 and a maximum iteration count of 50 were chosen for minimizing the objective function. The statistical results obtained from all runs are presented in
Table 9. As observed in the table, all statistical metrics for optimizing the F_ZLG objective function favor the bAOA algorithm, indicating its superior performance. These results additionally confirm the statistical stability of the bAOA algorithm.
8.2. Obtained Best Controller Parameters and Transfer Functions of the Optimized System
In this section, we discuss the results regarding the best controller parameters and the corresponding transfer functions of the optimized system.
Figure 12 provides the convergence curve, illustrating the progress of the bAOA and the original AOA algorithms in minimizing the objective function. Notably, it shows that the bAOA outperforms the original AOA by achieving the lowest objective function value through iterations.
Table 10 presents the optimal parameters of the PIDND
^{2}N
^{2} controller, obtained using both the bAOA and the original AOA algorithms.
Using those values would yield the following transfer functions of the optimized systems for original AOA and proposed bAOA algorithms.
8.3. Stability of the Proposed Design Method
In this section, we analyze the stability of the proposed design method by evaluating the step response and openloop frequency response of the bAOA and AOAtuned PIDND^{2}N^{2} controllers.
Figure 13 and
Table 11 present the transient response performance metrics for the bAOA and AOAtuned PIDND
^{2}N
^{2} controllers. The step response of both controllers is observed concerning the change in the terminal voltage. As illustrated in
Figure 13, the bAOAtuned PIDND2N2 controller exhibits a faster rise time and settling time with zero overshoot compared to the AOAtuned PIDND
^{2}N
^{2} controller. This implies that the bAOA tuned system reaches the desired state more rapidly without oscillations, demonstrating its superior stability in the time domain. The numerical results from
Table 11 confirm these visual observations.
Figure 14 and
Table 12 present the open loop Bode diagrams and frequency response performance metrics for the controllers. In the frequency domain, the bAOAtuned PIDND
^{2}N
^{2} controller showcases a higher phase margin, greater gain margin, and a wider bandwidth compared to the AOAtuned PIDND
^{2}N
^{2} controller. These results signify that the bAOAbased controller maintains better stability and frequency response characteristics, making it superior in terms of overall system stability.
8.4. Compared Algorithms and Respective Transfer Functions
In this section, we provide a comparative analysis of wellknown methods in the literature, which employ different types of controllers. The controller types used in these approaches are as follows: sine cosine algorithm (SCA)based PID controller [
23], whale optimization algorithm (WOA)based PIDA controller [
24], slime mould algorithm (SMA)based FOPID controller [
25], and particle swarm optimization (PSO)based PIDD
^{2} controller [
26].
The parameters for the SCAbased PID controller [
23] are as follows:
${K}_{p}=0.9826$,
${K}_{i}=0.8337$ and
${K}_{d}=0.4982$. The transfer function of the closedloop AVR system using this approach is given by the following equation.
The parameters for the WOAbased PIDA controller [
24] are as follows:
${K}_{p}=777.401$,
${K}_{i}=397.741$,
${K}_{d}=500.652$,
${K}_{a}=103.02$,
$\alpha =550.118$ and
$\beta =915.041$. The transfer function of the closedloop AVR system using this approach is given by the following equation.
The parameters for the SMAbased FOPID controller [
25] are as follows:
${K}_{p}=2.2554$,
${K}_{i}=1.2586$,
${K}_{d}=0.6472$,
$\lambda =1.0274$ and
$\mu =1.1877$. The transfer function of the closedloop AVR system using this approach is given by the following equation.
The parameters for the PSObased PIDD
^{2} controller [
26] are as follows:
${K}_{p}=2.7784$,
${K}_{i}=1.8521$,
${K}_{d1}=0.9997$ and
${K}_{d2}=0.07394$. The transfer function of the closedloop AVR system using this approach is given by the following equation.
These equations define the transfer functions of the AVR systems under the influence of different control methods. The following subsections provide a comparative analysis of these methods based on various performance criteria.
8.5. Comparative Transient Response Analysis
Figure 15 displays the comparative step response of different control approaches for the AVR system. This figure visually represents the transient response of various control methods and provides insights into their performance. The step response graph shows how each method reacts to a change in the terminal voltage.
Table 13 complements the visual representation by providing numerical values for the transient response metrics of different control approaches. These metrics include the rise time, settling time, and overshoot, which are essential indicators of the system’s dynamic behavior.
Upon analyzing both the figure and the table, it becomes evident that the bAOAtuned PIDND^{2}N^{2} controller excels in achieving a superior transient response compared to other control approaches. It exhibits the shortest rise time (0.033485 s) and settling time (0.050752 s) while completely eliminating overshoot. In contrast, the other control methods, including AOA, SCAtuned PID, WOAtuned PIDA, SMAtuned FOPID, and PSOtuned PIDD^{2}, exhibit longer rise and settling times and, in some cases, significant overshoot. These results emphasize the superiority of the bAOAbased control approach in providing a faster and more stable transient response, which is crucial for maintaining the AVR system’s stability and performance during dynamic voltage changes.
8.6. Comparative Frequency Response Analysis
Figure 16 provides a comparative view of Bode diagrams for different control approaches applied to the AVR system. These diagrams illustrate the frequency response characteristics of each control method, offering insights into how they perform across a range of frequencies.
Table 14 complements the visual representation with numerical values that quantify the frequency response metrics for each control approach. These metrics include the phase margin, gain margin, and bandwidth, which are crucial indicators of the system’s stability and ability to handle varying frequencies.
Upon analyzing both the figure and the table, it is clear that the bAOAtuned PIDND^{2}N^{2} controller stands out as the superior choice for frequency response analysis. It exhibits the highest phase margin (70.797°), indicating robust stability and the most favorable gain margin (28.888 dB) among all the methods, ensuring ample room for gain adjustments without instability. Moreover, it possesses the widest bandwidth (64.82 rad/s), signifying a faster system response to frequency variations. In contrast, the other control approaches, including AOAtuned PIDND^{2}N^{2}, SCAtuned PID, WOAtuned PIDA, SMAtuned FOPID, and PSOtuned PIDD^{2}, generally display lower phase margins, lower gain margins, and narrower bandwidths. The bAOAbased controller, on the other hand, excels in maintaining system stability across a broad frequency range and offers improved performance for handling dynamic frequency changes. These results underscore the superiority of the bAOAtuned PIDND^{2}N^{2} controller in providing robust and responsive frequency characteristics, which are vital for the stable and efficient operation of the AVR system under various operating conditions.
8.7. Comparisons with the Reported Recent Works
In this section, we compare the proposed PIDND2N2 controller tuned with bAOA to several recently reported control methods for the AVR system. These methods include a variety of controllers, each tuned using different optimization algorithms such as marine predators algorithm (MPA) based FOPID [
27], hybrid atom search particle swarm optimization (hASPSO) based PID [
28], equilibrium optimizer (EO) based TIλDND2N2 based controller, reptile search algorithm (RSA) based FOPIDD2 [
6], improved RungeKutta (iRUN) algorithm based PIDND2N2 [
29], symbiotic organism search (SOS) algorithmbased PIDF [
30], whale optimization algorithm (WOA) based 2DOF FOPI [
31], Lévy flightbased RSA with local search ability (LRSANM) based PID [
32], chaotic black widow algorithm (ChBWO) based FOPID [
15], genetic algorithm (GA) based fuzzy PID [
33], sinecosine algorithm (SCA) based FOPID with fractional order filter [
34], hybrid simulated annealing–Manta ray foraging optimization (SAMRFO) algorithm based PIDD2 [
35], slime mould algorithm (SMA) based PID [
14], gradient based optimization (GBO) based FOPID [
36] and nonlinear SCA based sigmoid PID [
37].
We evaluate their transient response performance to assess the effectiveness of the proposed approach.
Table 15 provides a comprehensive overview of the transient response metrics, including rise time, settling time, and overshoot, for the proposed approach and other recent methods. The results demonstrate the efficacy of the bAOAbased PIDND
^{2}N
^{2} controller in comparison to various stateoftheart methods as it stands out with an impressive performance, featuring a remarkably low rise time (0.033485s), a fast settling time (0.050752s), and zero overshoot. This suggests the exceptional stability and responsiveness of the bAOAtuned controller. Therefore, the table clearly illustrates the effectiveness of the proposed bAOAbased PIDND
^{2}N
^{2} controller in achieving rapid responses and maintaining stable performance, as evidenced by its minimal overshoot. It consistently outperforms or rivals the other methods in the evaluation, reinforcing its superiority for the AVR system’s transient response.
9. Conclusion and Future Works
In this study, we have introduced a novel approach to enhance the control of AVR in power systems. By uniting a PIDND^{2}N^{2} controller with the novel bAOA, we aimed to address the limitations associated with conventional methods. The introduction of the PIDND^{2}N^{2} controller offers enhanced precision, stability, and responsiveness in voltage regulation. This innovative configuration mitigates the shortcomings of existing approaches, promising superior control performance. The bAOA optimizer, meticulously finetuned with the integration of PS and EOBL strategies into original AOA in order to demonstrate exceptional performance. The assessment on 23 benchmark functions show 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 bAOAbased 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 work in this domain may focus on several aspects. Further refinement of the bAOA optimization framework, exploring additional optimization problems, and evaluating its applicability to diverse domains are promising directions. Investigating the practical implementation of the proposed control scheme in realworld power systems and conducting extensive field testing would provide valuable insights. In addition, the AVR system can be considered as an important field of study in which it can play a critical role in the realization of the efficient voltage regulation in smart grids. Additionally, the integration of emerging technologies, such as machine learning and artificial intelligence, into AVR control systems may offer opportunities for further enhancement. The quest for more efficient, stable, and responsive AVR systems remains a vibrant field of research with potential breakthroughs on the horizon.
Figure 1.
Flowchart of the original arithmetic optimization algorithm.
Figure 1.
Flowchart of the original arithmetic optimization algorithm.
Figure 2.
Flowchart of pattern search method.
Figure 2.
Flowchart of pattern search method.
Figure 3.
Working principle of OBL mechanism.
Figure 3.
Working principle of OBL mechanism.
Figure 4.
Flowchart of proposed bAOA algorithm.
Figure 4.
Flowchart of proposed bAOA algorithm.
Figure 5.
Schematic diagram of a typical AVR system.
Figure 5.
Schematic diagram of a typical AVR system.
Figure 6.
An Uncontrolled AVR system.
Figure 6.
An Uncontrolled AVR system.
Figure 7.
Polezero map of uncontrolled AVR system.
Figure 7.
Polezero map of uncontrolled AVR system.
Figure 8.
Step response of the uncontrolled AVR system.
Figure 8.
Step response of the uncontrolled AVR system.
Figure 9.
Open loop Bode plot of the uncontrolled AVR system.
Figure 9.
Open loop Bode plot of the uncontrolled AVR system.
Figure 10.
Block diagram of PIDND^{2}N^{2} controller.
Figure 10.
Block diagram of PIDND^{2}N^{2} controller.
Figure 11.
The block diagram of the implementation of the proposed approach to AVR system.
Figure 11.
The block diagram of the implementation of the proposed approach to AVR system.
Figure 12.
Convergence curve for bAOA and original AOA
Figure 12.
Convergence curve for bAOA and original AOA
Figure 13.
Step response of bAOA and AOAtuned PIDND^{2}N^{2} controllers for the change in the terminal voltage.
Figure 13.
Step response of bAOA and AOAtuned PIDND^{2}N^{2} controllers for the change in the terminal voltage.
Figure 14.
Open loop Bode diagrams for bAOA and AOAtuned PIDND^{2}N^{2} controllers
Figure 14.
Open loop Bode diagrams for bAOA and AOAtuned PIDND^{2}N^{2} controllers
Figure 15.
Comparative step response of different control approaches for AVR system
Figure 15.
Comparative step response of different control approaches for AVR system
Figure 16.
Comparative Bode diagrams of different control approaches for AVR system
Figure 16.
Comparative Bode diagrams of different control approaches for AVR system
Table 1.
Properties of the adopted unimodal benchmark functions.
Table 1.
Properties of the adopted unimodal benchmark functions.
Name 
Function 
Dimension 
Evaluation interval 
Global minimum 
Sphere 
${Func}_{1}(x)$ 
30 
${[100,100]}^{Dim}$ 
0 
Schwefel 2.2 
${Func}_{2}(x)$ 
30 
${[10,10]}^{Dim}$ 
0 
Schwefel 1.2 
${Func}_{3}(x)$ 
30 
${[100,100]}^{Dim}$ 
0 
Schwefel 2.21 
${Func}_{4}(x)$ 
30 
${[100,100]}^{Dim}$ 
0 
Rosenbrock 
${Func}_{5}(x)$ 
30 
${[30,30]}^{Dim}$ 
0 
Step 
${Func}_{6}(x)$ 
30 
${[100,100]}^{Dim}$ 
0 
Quartic 
${Func}_{7}(x)$ 
30 
${[1.28,1.28]}^{Dim}$ 
0 
Table 2.
Properties of the adopted multimodal benchmark functions.
Table 2.
Properties of the adopted multimodal benchmark functions.
Name 
Function 
Dimension 
Evaluation interval 
Global minimum 
Schwefel 
${Func}_{8}(x)$ 
30 
${[500,500]}^{Dim}$ 
−1.2569E+04 
Rastrigin 
${Func}_{9}(x)$ 
30 
${[5.12,5.12]}^{Dim}$ 
0 
Ackley 
${Func}_{10}(x)$ 
30 
${[32,32]}^{Dim}$ 
0 
Griewank 
${Func}_{11}(x)$ 
30 
${[600,600]}^{Dim}$ 
0 
Penalized 
${Func}_{12}(x)$ 
30 
${[50,50]}^{Dim}$ 
0 
Penalized2 
${Func}_{13}(x)$ 
30 
${[50,50]}^{Dim}$ 
0 
Table 3.
Properties of the adopted fixeddimensional multimodal benchmark functions.
Table 3.
Properties of the adopted fixeddimensional multimodal benchmark functions.
Name 
Function 
Dimension 
Evaluation interval 
Global minimum 
Foxholes 
${Func}_{14}(x)$ 
2 
${[65.536,65.536]}^{Dim}$ 
0.998 
Kowalik 
${Func}_{15}(x)$ 
4 
${[5,5]}^{Dim}$ 
3.0749E−04 
SixHump Camel 
${Func}_{16}(x)$ 
2 
${[5,5]}^{Dim}$ 
−1.0316 
Branin 
${Func}_{17}(x)$ 
2 
$\left[5,10\right]\times \left[0,15\right]$ 
0.39789 
GoldsteinPrice 
${Func}_{18}(x)$ 
2 
${[2,2]}^{Dim}$ 
3 
Hartman 3 
${Func}_{19}(x)$ 
3 
${[0,1]}^{Dim}$ 
−3.8628 
Hartman 6 
${Func}_{20}(x)$ 
6 
${[0,1]}^{Dim}$ 
−3.322 
Shekel 5 
${Func}_{21}(x)$ 
4 
${[0,10]}^{Dim}$ 
−10.1532 
Shekel 7 
${Func}_{22}(x)$ 
4 
${[0,10]}^{Dim}$ 
−10.4029 
Shekel 10 
${Func}_{23}(x)$ 
4 
${[0,10]}^{Dim}$ 
−10.5364 
Table 4.
Properties of the compared algorithms (population size, total iteration number, values of other control parameters).
Table 4.
Properties of the compared algorithms (population size, total iteration number, values of other control parameters).
Algorithm 
Population size 
Total iteration number 
Values of other control parameters 
bAOA 
30 
500 
$\alpha =5$, $\mu =0.4975$, $Min=0.2$, $Max=1$, $initialmeshsize=1$, $meshexpansionfactor=2$, $meshcontractionfactor=0.5$, $alltolerances={10}^{6}$

AOA [5] 
30 
500 
$\alpha =5$, $\mu =0.4975$, $Min=0.2$, $Max=1$

SCA [19] 
30 
500 
$A=2$ 
INFO [20] 
30 
500 
$c=2$, $d=4$

MPA [21] 
30 
500 
$FADs=0.2$, $P=0.5$

Table 5.
Comparative statistical results obtained from unimodal benchmark functions.
Table 5.
Comparative statistical results obtained from unimodal benchmark functions.
Function 
Algorithm 
Mean 
Standard Deviation 
Best 
Worst 
${Func}_{1}(x)$ 
bAOA 
0 
0 
0 
0 
AOA 
0.00029656 
0.0011413 
3.9226E−38 
0.0060134 
SCA 
16.537 
36.426 
9.5633E−06 
175.47 
INFO 
1.0185E−53 
4.997E−54 
3.3545E−55 
2.0178E−53 
MPA 
4.0116E−23 
6.3963E−23 
3.6461E−25 
2.7727E−22 
${Func}_{2}(x)$ 
bAOA 
8.5996E−241 
0 
4.333E−320 
2.2954E−239 
AOA 
2.8674E−186 
0 
9.6235E−296 
8.6022E−185 
SCA 
0.021241 
0.031567 
0.00013767 
0.13042 
INFO 
1.0943E−26 
3.6605E−27 
4.7283E−27 
1.9892E−26 
MPA 
2.6444E−13 
2.8514E−13 
8.2406E−15 
1.2622E−12 
${Func}_{3}(x)$ 
bAOA 
0 
0 
0 
0 
AOA 
1.6011 
3.3816 
1.3815E−07 
16.177 
SCA 
8640.8 
4939.5 
1709.5 
20103 
INFO 
1.4606E−50 
1.1602E−50 
8.6654E−52 
3.9712E−50 
MPA 
9.9612E−05 
0.00022346 
7.2658E−09 
0.001186 
${Func}_{4}(x)$ 
bAOA 
9.0422E−244 
0 
1.2808E−253 
2.6479E−242 
AOA 
0.15416 
0.094877 
0.014632 
0.36318 
SCA 
37.033 
13.087 
12.166 
61.964 
INFO 
2.1028E−27 
1.4215E−27 
3.5852E−28 
7.4954E−27 
MPA 
2.7542E−09 
1.5152E−09 
3.1553E−10 
6.0257E−09 
${Func}_{5}(x)$ 
bAOA 
0.61615 
1.8814 
3.0737E−09 
6.3967 
AOA 
28.693 
0.27549 
27.902 
29.18 
SCA 
1.3673E+05 
3.2682E+05 
107.54 
1175700 
INFO 
22.585 
0.51711 
21.298 
23.462 
MPA 
25.268 
0.45451 
24.487 
26.042 
${Func}_{6}(x)$ 
bAOA 
2.4395E−12 
9.2009E−13 
1.086E−12 
5.8521E−12 
AOA 
3.7524 
0.33331 
3.0561 
4.4582 
SCA 
14.254 
13.542 
4.7191 
55.025 
INFO 
1.2654E−08 
3.7987E−08 
3.9266E−11 
2.07E−07 
MPA 
4.1868E−08 
2.2575E−08 
1.3296E−08 
1.2965E−07 
${Func}_{7}(x)$ 
bAOA 
3.629E−05 
2.8489E−05 
6.8524E−07 
0.00010771 
AOA 
9.4896E−05 
7.1313E−05 
2.0672E−06 
0.00029718 
SCA 
0.099158 
0.090509 
0.0085847 
0.44986 
INFO 
0.0015937 
0.0012634 
0.00017227 
0.0049221 
MPA 
0.0013495 
0.00060352 
0.00041966 
0.0026601 
Table 6.
Comparative statistical results obtained from multimodal benchmark functions.
Table 6.
Comparative statistical results obtained from multimodal benchmark functions.
Function 
Algorithm 
Mean 
Standard Deviation 
Best 
Worst 
${Func}_{8}(x)$ 
bAOA 
−12536 
172.87 
−12569 
−11623 
AOA 
−7980.7 
446.84 
−9196.5 
−7230.3 
SCA 
−3848.4 
286.86 
−4371 
−3283.7 
INFO 
−8630.7 
700.38 
−9763.3 
−7101.2 
MPA 
−8736.9 
438.15 
−9687.9 
−7946.9 
${Func}_{9}(x)$ 
bAOA 
0 
0 
0 
0 
AOA 
0 
0 
0 
0 
SCA 
29.308 
30.189 
0.13996 
122.46 
INFO 
0 
0 
0 
0 
MPA 
0 
0 
0 
0 
${Func}_{10}(x)$ 
bAOA 
8.8818E−16 
0 
8.8818E−16 
8.8818E−16 
AOA 
8.8818E−16 
0 
8.8818E−16 
8.8818E−16 
SCA 
14.208 
8.3212 
0.043401 
20.382 
INFO 
8.8818E−16 
0 
8.8818E−16 
8.8818E−16 
MPA 
1.7196E−12 
1.1519E−12 
2.7045E−13 
5.8482E−12 
${Func}_{11}(x)$ 
bAOA 
0 
0 
0 
0 
AOA 
194.12 
65.896 
72.408 
323.52 
SCA 
0.84569 
0.41164 
0.23545 
1.9083 
INFO 
0 
0 
0 
0 
MPA 
0 
0 
0 
0 
${Func}_{12}(x)$ 
bAOA 
2.1943E−13 
1.5539E−13 
5.0331E−14 
6.0379E−13 
AOA 
0.29154 
0.053809 
0.14538 
0.43947 
SCA 
52428 
1.5261E+05 
1.0947 
614430 
INFO 
1.4456E−09 
2.8117E−09 
5.3463E−12 
1.1459E−08 
MPA 
0.00014286 
0.0005059 
2.4157E−09 
0.0023059 
${Func}_{13}(x)$ 
bAOA 
3.1668E−12 
2.4141E−12 
7.6907E−13 
9.0849E−12 
AOA 
2.4484 
0.16915 
2.1217 
2.8078 
SCA 
1.0872E+05 
2.7869E+05 
2.2042 
1305400 
INFO 
0.063752 
0.14273 
3.2034E−10 
0.69157 
MPA 
0.012215 
0.036876 
2.8969E−08 
0.19763 
Table 7.
Comparative statistical results obtained from fixeddimensional multimodal benchmark functions.
Table 7.
Comparative statistical results obtained from fixeddimensional multimodal benchmark functions.
Function 
Algorithm 
Mean 
Standard Deviation 
Best 
Worst 
${Func}_{14}(x)$ 
bAOA 
0.998 
1.5701E−17 
0.998 
0.998 
AOA 
8.3696 
3.2389 
0.998 
12.671 
SCA 
1.795 
0.9859 
0.998 
2.9821 
INFO 
2.1111 
2.5903 
0.998 
10.763 
MPA 
0.998 
1.515E−16 
0.998 
0.998 
${Func}_{15}(x)$ 
bAOA 
0.00030749 
1.4923E−15 
0.00030749 
0.00030749 
AOA 
0.015417 
0.025604 
0.00037189 
0.11249 
SCA 
0.0010661 
0.00037002 
0.0005829 
0.0015477 
INFO 
0.0024352 
0.0060863 
0.00030749 
0.020363 
MPA 
0.00030749 
4.3122E−15 
0.00030749 
0.00030749 
${Func}_{16}(x)$ 
bAOA 
−1.0316 
1.9902E−16 
−1.0316 
−1.0316 
AOA 
−1.0316 
6.0816E−07 
−1.0316 
−1.0316 
SCA 
−1.0316 
3.7905E−05 
−1.0316 
−1.0315 
INFO 
−1.0316 
6.5843E−16 
−1.0316 
−1.0316 
MPA 
−1.0316 
4.4024E−16 
−1.0316 
−1.0316 
${Func}_{17}(x)$ 
bAOA 
0.39789 
0 
0.39789 
0.39789 
AOA 
0.40987 
0.009864 
0.39844 
0.43767 
SCA 
0.40026 
0.0023543 
0.39797 
0.40949 
INFO 
0.39789 
0 
0.39789 
0.39789 
MPA 
0.39789 
9.5078E−15 
0.39789 
0.39789 
${Func}_{18}(x)$ 
bAOA 
3 
0 
3 
3 
AOA 
6.6 
9.3351 
3 
30 
SCA 
3 
5.4359E−05 
3 
3.0002 
INFO 
3 
8.6883E−16 
3 
3 
MPA 
3 
2.1709E−15 
3 
3 
${Func}_{19}(x)$ 
bAOA 
−3.8628 
2.4116E−15 
−3.8628 
−3.8628 
AOA 
−3.8523 
0.0038518 
−3.8593 
−3.842 
SCA 
−3.8547 
0.0024361 
−3.861 
−3.8495 
INFO 
−3.8628 
2.6823E−15 
−3.8628 
−3.8628 
MPA 
−3.8628 
2.4945E−15 
−3.8628 
−3.8628 
${Func}_{20}(x)$ 
bAOA 
−3.322 
2.1608E−13 
−3.322 
−3.322 
AOA 
−3.0471 
0.091025 
−3.1762 
−2.8234 
SCA 
−2.8784 
0.34163 
−3.1199 
−1.6747 
INFO 
−3.2784 
0.058273 
−3.322 
−3.2031 
MPA 
−3.322 
1.7554E−11 
−3.322 
−3.322 
${Func}_{21}(x)$ 
bAOA 
−10.153 
7.6605E−13 
−10.153 
−10.153 
AOA 
−3.5023 
1.1997 
−6.0307 
−1.8035 
SCA 
−2.6202 
2.0715 
−7.8686 
−0.49728 
INFO 
−9.1039 
2.4723 
−10.153 
−2.6305 
MPA 
−10.153 
4.1471E−11 
−10.153 
−10.153 
${Func}_{22}(x)$ 
bAOA 
−10.403 
1.1144E−12 
−10.403 
−10.403 
AOA 
−3.5619 
1.2118 
−6.8762 
−1.4002 
SCA 
−3.2023 
1.8303 
−5.9956 
−0.52105 
INFO 
−9.0488 
2.7774 
−10.403 
−2.7659 
MPA 
−10.403 
5.9857E−11 
−10.403 
−10.403 
${Func}_{23}(x)$ 
bAOA 
−10.536 
3.2315E−12 
−10.536 
−10.536 
AOA 
−3.8733 
1.6156 
−6.5892 
−1.5825 
SCA 
−3.7421 
1.7935 
−6.1434 
−0.94135 
INFO 
−9.0039 
3.151 
−10.536 
−2.4217 
MPA 
−10.536 
2.5368E−11 
−10.536 
−10.536 
Table 8.
Boundaries for PIDND^{2}N^{2} Controller parameters.
Table 8.
Boundaries for PIDND^{2}N^{2} Controller parameters.
Bound 
${\mathit{K}}_{\mathit{p}}$ 
${\mathit{K}}_{\mathit{i}}$ 
${\mathit{K}}_{\mathit{d}1}$ 
${\mathit{K}}_{\mathit{d}2}$ 
${\mathit{n}}_{1}$ 
${\mathit{n}}_{2}$ 
Lower 
0.001 
0.001 
0.001 
0.001 
50 
50 
Upper 
5 
5 
5 
5 
2000 
2000 
Table 9.
Statistical performance of bAOA and original AOA for AVR system.
Table 9.
Statistical performance of bAOA and original AOA for AVR system.
Algorithm 
Mean 
Standard Deviation 
Best 
Worst 
bAOA 
0.0065138 
9.3497E−05 
0.0063522 
0.0067022 
AOA 
0.0078863 
0.00012395 
0.0076825 
0.0081212 
Table 10.
Optimal parameters of PIDND^{2}N^{2} controller obtained via bAOA and original AOA algorithms
Table 10.
Optimal parameters of PIDND^{2}N^{2} controller obtained via bAOA and original AOA algorithms
Optimized by 
${\mathit{K}}_{\mathit{p}}$ 
${\mathit{K}}_{\mathit{i}}$ 
${\mathit{K}}_{\mathit{d}1}$ 
${\mathit{K}}_{\mathit{d}2}$ 
${\mathit{n}}_{1}$ 
${\mathit{n}}_{2}$ 
bAOA 
4.8723 
2.0240 
1.8094 
0.15049 
1595.2 
1971.2 
AOA 
3.9448 
2.1188 
1.6757 
0.13014 
1544.2 
871.72 
Table 11.
Transient response performance metrics for bAOA and AOAtuned PIDND^{2}N^{2} controllers.
Table 11.
Transient response performance metrics for bAOA and AOAtuned PIDND^{2}N^{2} controllers.
Design method 
Rise time (s) 
Settling time (s) 
Overshoot (%) 
bAOAtuned PIDND^{2}N^{2} 
0.033485 
0.050752 
0 
AOAtuned PIDND^{2}N^{2} 
0.037393 
0.057523 
0.043859 
Table 12.
Frequency response performance metrics for bAOA and AOAtuned PIDND^{2}N^{2} controllers.
Table 12.
Frequency response performance metrics for bAOA and AOAtuned PIDND^{2}N^{2} controllers.
Design method 
Phase margin (°) 
Gain margin (dB) 
Bandwidth (rad/s) 
bAOAtuned PIDND^{2}N^{2} 
70.797 
28.888 
64.820 
AOAtuned PIDND^{2}N^{2} 
69.810 
23.368 
57.819 
Table 13.
Comparative numerical values for transient response of different control approaches.
Table 13.
Comparative numerical values for transient response of different control approaches.
Design method 
Rise time (s) 
Settling time (s) 
Overshoot (%) 
bAOAtuned PIDND^{2}N^{2} 
0.033485 
0.050752 
0 
AOAtuned PIDND^{2}N^{2} 
0.037393 
0.057523 
0.043859 
SCAtuned PID [23] 
0.1472 
0.84133 
11.425 
WOAtuned PIDA [24] 
0.32772 
0.49543 
1.6483 
SMAtuned FOPID [25] 
0.087541 
0.4979 
15.998 
PSOtuned PIDD^{2} [26] 
0.092935 
0.16347 
0.0025797 
Table 14.
Comparative numerical values for frequency response of different control approaches.
Table 14.
Comparative numerical values for frequency response of different control approaches.
Design method 
Phase margin (°) 
Gain margin (dB) 
Bandwidth (rad/s) 
bAOAtuned PIDND^{2}N^{2}

70.797 
28.888 
64.820 
AOAtuned PIDND^{2}N^{2}

69.810 
23.368 
57.819 
SCAtuned PID [33] 
52.596 
20.300 
14.821 
WOAtuned PIDA [34] 
67.671 
26.123 
6.7076 
SMAtuned FOPID [35] 
49.142 
20.193 
23.914 
PSOtuned PIDD^{2} [36] 
79.638 
Infinite 
23.503 
Table 15.
Transient response performance of the proposed approach with respect to recently reported other efficient methods.
Table 15.
Transient response performance of the proposed approach with respect to recently reported other efficient methods.
Ref. 
Year 
Used controller type 
Tuning method 
Rise time (s) 
Settling time (s) 
Overshoot (%) 
Proposed 
PIDND^{2}N^{2} 
bAOA 
0.033485 
0.050752 
0 
[27] 
2023 
FOPID 
MPA 
0.0833 
0.1106 
0.55 
[28] 
PID 
hASPSO 
0.3097 
0.4679 
1.2476 
[44] 
TI^{λ}DND^{2}N^{2}

EO 
0.03752 
0.0596 
0.4128 
[6] 
FOPIDD^{2}

RSA 
0.0487 
0.0806 
0 
[29] 
2022 
PIDND^{2}N^{2}

iRUN 
0.0399 
0.0626 
0 
[30] 
PIDF 
SOS 
0.267 
0.371 
0.007 
[31] 
2DOF fractionalorder PI 
WOA 
1.12 
1.74 
1.17 
[32] 
PID 
LRSANM 
0.3076 
0.4669 
0.9582 
[15] 
FOPID 
ChBWO 
0.1103 
0.169 
1.1838 
[33] 
Fuzzy PID 
GA 
0.1857 
0.2963 
1.0407 
[34] 
2021 
FOPID with fractional filter 
SCA 
0.1230 
0.1670 
0.1262 
[35] 
PIDD^{2}

SAMRFO 
0.0535 
0.0798 
0.7562 
[14] 
PID 
SMA 
0.3149 
0.4817 
0.6071 
[36] 
FOPID 
GBO 
0.0885 
0.653 
11.3 
[37] 
Sigmoid PID 
NSCA 
0.498 
0.579 
2.2 