Background

Maintaining nominal voltage levels within electrical power systems is a fundamental requirement for ensuring the efficient and reliable performance of the entire power infrastructure [1–3]. The deviation in terminal voltage of alternators during load changes represents a significant challenge, with consequences ranging from equipment damage to operational disruptions and costly downtime [4]. To address this critical issue, automatic voltage regulators (AVRs) [5] are employed, offering control over generator exciter output power to regulate the terminal voltage magnitude [6].

Literature review

In the landscape of AVR control, controllers play a pivotal role in monitoring and regulating the AVR itself [7]. These controllers enable real-time adjustments for voltage stability, facilitate remote monitoring, fault detection, and automatic shutdown during emergencies, thereby enhancing system dependability [8]. Diverse controllers are available, from standard proportional-integral-derivative (PID) to advanced variants like PID acceleration (PIDA), fractional-order PID (FOPID), and PID with a second-order derivative (PIDD²) [9–16]. The choice of a cost function is equally significant, affecting performance significantly. Researchers employ various objective 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 [17,18].

Various research studies have explored the application of metaheuristic optimization algorithms to determine optimal controller gains in AVR systems [19]. For instance, in [20], the study enhances AVR performance using the adaptive neuro-fuzzy inference system (ANFIS) and compares it with scenarios without a controller and with a PID controller. The ANFIS, trained with a hybrid optimization learning scheme, shows improved performance metrics in MATLAB/Simulink simulations. In [21], the self-competitive differential evolution (DE) algorithm is introduced, enhancing exploration ability with a control parameter. Applied to real-world optimization problems, including PID controller tuning for AVR systems, the self-competitive DE outperforms other DE algorithms. The study in [22] focuses on optimizing a controller for a real AVR system in a synchronous generator, proposing a novel proportional-integral controller with anti-windup protection. The African vultures optimization algorithm (AVOA) is adaptively modified for controller parameter design, resulting in superior AVR controller performance. In [23], the equilibrium optimizer (EO) is employed to optimize parameters for a novel AVR controller using a time-domain objective function. Comparative analyses with various controllers validate the proposed controller’s superior performance in settling time, rise time, and overshoot, supported by frequency domain analysis. The enhanced whale optimization algorithm (EWOA) in [24] stabilizes PID controller parameters in AVR systems. Comparative analysis establishes EWOA’s faster convergence, higher precision, shorter execution time, and greater stability, making it a practical method for PID controller optimization. The study in [25] introduces a novel PI^λ1I^λ2D^μ1D^μ2 controller for AVR system, optimized using the mayfly algorithm. Comparative analyses demonstrate the proposed controller’s excellence in both time and frequency domain analyses, as well as robustness and disturbance rejection. In [26], the marine predator optimization algorithm (MPA) optimizes a FOPID controller for AVR systems. Comparative analyses highlight MPA–FOPID’s superior stability, frequency response, robustness, response speed, and disturbance-rejection capabilities. The study in [7] models the AVR system as a sextuple-input single-output system, optimizing controller parameters using the particle swarm optimization African vultures optimization algorithm (PSO–AVOA). Comparative analyses showcase the effectiveness of the proposed design technique and optimization algorithm. In [27], FOPID controller optimization using the salp swarm algorithm (SSA), ant lion optimization (ALO), and PSO is assessed for AVR system. Comparative analysis demonstrates promising early results and emphasizes the superiority of the proposed controller. The study in [28] also introduces the MPA for tuning the FOPID controller in AVR system, showing exceptional performance in enhancing AVR transient response compared to other FOPID controllers optimized with recent metaheuristic algorithms. In [29], an AVR system with a PID controller addresses issues related to dynamic changes in power systems. The zebra optimization algorithm (ZOA) and osprey optimization algorithm (OOA) optimize transient response, with ZOA showing significant improvement. The study in [30] combines a PIDND²N² controller with the balanced arithmetic optimization algorithm (b-AOA) to enhance AVR system stability. The proposed approach excels in transient response and frequency response, outperforming state-of-the-art control methods. In [31], the distance and Levy-flight based crow search algorithm (DLCSA) optimizes FOPID, FOPI, and FOPD controllers for AVR systems, showcasing superior performance in various aspects compared to established techniques.

Research gap and motivation

The use of metaheuristic optimization algorithms to tune controllers has improved the performance of many AVR systems. However, there is still a concern about their precision, prompting the need for further improvement. Therefore, introducing new controllers and integrating state-of-the-art metaheuristic optimization algorithms has the potential to enhance accuracy and overall performance in AVR systems. In response to these constraints, our study aims to pioneer novel approaches to AVR control, contributing to the advancement of robust and efficient power systems. The primary objective is to introduce an advanced control scheme capable of addressing these limitations effectively. To achieve this, we have developed a novel optimizer, designed to fine-tune and optimize the parameters of our proposed control scheme, enhancing overall performance and adaptability.

Challenges

While current control methodologies [32–39] have shown promise, their limitations necessitate a more nuanced approach. Challenges include issues related to robustness, overshoots, rise times, settling times, and persistent steady-state errors.

Contribution

Our study introduces the quadratic wavelet-enhanced gradient-based optimization (QWGBO) algorithm, which serves as an innovative tuning mechanism to enhance the gradient-based optimization (GBO) [40]. GBO, grounded in the principles of gradient concepts from Newton’s rules, demonstrates robust local exploration capabilities and parameter simplicity. However, it faces challenges when dealing with complex, high-dimensional optimization problems. The QWGBO algorithm integrates two crucial enhancements, the quadratic interpolation mutation (QIM) and the wavelet mutation strategy (WMS), which are combined with GBO to improve exploration and exploitation capabilities. QIM enhances exploration by approximating the objective function at candidate positions, diversifying the search process. Meanwhile, WMS addresses stagnation issues and improves solution accuracy, exploring new solution vectors using wavelet functions with adjustable parameters. The fusion of these components forms the QWGBO algorithm, offering an effective mechanism to enhance GBO’s optimization capabilities, striking a balance between exploration and exploitation.

The efficacy of the QWGBO is first verified through statistical and non-parametric Wilcoxon signed rank tests using widely adopted unimodal, multimodal, and fixed-dimensional multimodal benchmark functions. We present the experimental results obtained by comparing the proposed QWGBO algorithm with other competitive and recent optimization algorithms, original gradient-based optimization [40], gravitational search algorithm [41], whale optimization algorithm [42], slime mould algorithm [43], prairie dog optimization [44]. The results clearly demonstrate the superior performance of the QWGBO algorithm in solving the benchmark functions, as it consistently achieves lower minimum values and ranks first in all comparisons indicating its potential as an effective optimization algorithm.

Our work presents an innovative approach that combines both the controller and the optimizer to create a comprehensive solution for enhancing AVR stability. The core innovation is the QWGBO algorithm, the efficacy of which is demonstrated through extensive statistical and non-parametric tests on benchmark functions, showing its superior optimization capabilities. Our study introduces a cascaded real PIDD² (RPIDD²) and fractional-order proportional-integral (FOPI) controller, designed for precision and stability in voltage regulation, and fine-tuned by QWGBO for improved performance and adaptability. The work targets the minimization of dynamic response performance criteria using Zwe-Lee Gaing objective function [45], ensuring that the AVR system meets stringent performance requirements.

To validate the proposed approach’s superiority, extensive comparative analyses (statistical, boxplot, convergence profile, Wilcoxon signed-rank test, transient and frequency responses, performance against varying input reference and external load disturbance, controller effort and robustness) were conducted against other competitive algorithms and recently reported optimization techniques. In addition to the comparative assessment against the algorithms used in the above analyses, further comparisons were made with recently reported 17 optimization techniques in the literature [23,25,31,46–59]. The simulation results unequivocally highlight the QWGBO algorithm’s superior performance in optimizing the AVR system, as evident from lower objective function values, excellent convergence, and statistical assessments as well as stability and robustness analyses. This work thus contributes to the advancement of control systems for power infrastructure, with the QWGBO algorithm emerging as a promising solution to optimize complex engineering problems. In light of the above presentation, the contributions of this study can be listed as follows.

• The study introduces the QWGBO algorithm, innovatively enhancing the GBO. QWGBO integrates quadratic interpolation mutation and wavelet mutation strategy to improve exploration and exploitation capabilities, addressing challenges faced by GBO in complex, high-dimensional optimization problems.
• The effectiveness of QWGBO is rigorously verified through statistical and non-parametric tests using widely adopted unimodal, multimodal, and fixed-dimensional multimodal benchmark functions. Comparative analyses against competitive optimization algorithms demonstrate QWGBO’s superior performance.
• The work presents an innovative approach by combining the QWGBO with a cascaded RPIDD²-FOPI controller. This comprehensive solution enhances AVR stability, targeting the minimization of dynamic response performance criteria and ensuring compliance with stringent performance requirements.
• The study introduces a cascaded RPIDD²-FOPI controller, designed for precision and stability in voltage regulation. QWGBO is employed for fine-tuning, resulting in improved performance and adaptability. The proposed approach minimizes dynamic response performance criteria using the Zwe-Lee Gaing objective function, ensuring effective AVR system performance.
• The proposed approach is rigorously validated through extensive comparative analyses, including statistical tests, boxplot comparisons, convergence profiles, Wilcoxon signed-rank tests, transient and frequency responses, performance against varying input references and external load disturbances, controller effort analysis, and robustness assessments. The analyses encompass a wide range of competitive algorithms and recently reported optimization techniques in the literature.
• The simulation results unequivocally demonstrate the superior performance of the QWGBO algorithm in optimizing the AVR system. Comparative assessments against a diverse set of optimization techniques showcase lower objective function values, excellent convergence, and superior stability and robustness analyses. This contribution advances control systems for power infrastructure, positioning QWGBO as a promising solution for optimizing complex engineering problems.

Paper organization

The paper is organized into several sections to systematically present the proposed QWGBO approach and its application in enhancing the stability of AVR system. The structure is as follows. The next section provides the basics of the original GBO. The third section presents the proposed QWGBO approach. The performance evaluation of the proposed QWGBO against benchmark functions is presented in the fourth section. The structure of the AVR system is explained in the fifth section. The sixth section describes the new methodology for the proposed in this work in detail, including the new cascaded RPIDD²-FOPI controller, objective function and constraints of optimization problem. The application of QWGBO in optimizing the proposed controller parameters is also outlined in this section. The seventh section presents and discusses the simulation results obtained from the application of QWGBO to the AVR system. Finally in the eighth section, the paper concludes with a summary of findings and potential directions for future research.

Initialization

Like several optimization methods, GBO starts with a population of solutions that are randomly generated within the search space limits as follows. (1) where rand stands for a random vector between [0,1] generated according to uniform distribution, N refers to the population size, and Z_max and Z_min indicate the upper and lower limits of the search domain.

The GSR operator (Exploration)

The GSR is the key element of the GBO since it can encourage exploration capability while avoiding the local optimal tramping problem. GSR uses a numerical gradient approach rather than a direct function derivation to advance the initial guess to the next place. Additionally, the direction of movement (DM) is included to take advantage of the immediate vicinity of the current solution. Thus, the following equation can be adopted to update the position of the current vector (). (2) where defines the renewed vector, the second part of Eq (2) realize the GSR process while the third part defines the DM which aims to guide the current solution towards the best so for solution (Z_best); ε defines a small number inside the interval [0, 0.1], t stands for the current iteration, Z_worst defines the worst solution obtained so far, rand and randn define random values that are generated according to uniform and normal distributions, respectively. Here, transition parameter (B₁) is defined with the intention of balancing the search process’ exploration and exploitation and it can be expressed as follows. (3) (4) (5) where and T stands for the maximum size of iterations, and G_max and G_min are set to 1.2 and 0.2, respectively. Morevere, the parameter B₂ is another component in GBO aims to support the exploration process which is defined as follows. (6) (7) (8) (9) where step sigifies the step length that is conied using the positions of the Z_best and ; rand(1:D) defines a vector of random values with D dimensions, and r1:r4 define different integers arbitrarily conined from the population size, where (r1≠r2≠r3≠r4≠n). By regarding the best solution so far (Z_best) instead of the present solution vector () in (2), an updated solution vector () can be created as follows: (10) (11) (12) where defines the define the previous solutions, where if "Flag" is equal to 1, the solution is assigned the value of , otherwise, it is assigned to . The searching stage based on Eq (10) can assist the local searching phase but it isnot usfull for global searching process. In this sense, Eq (2) can realize the global search, but it limits the local search. In order to balance the tendencies of exploration and exploitation patterns, it is beneficial to use both and search tactics. In light of this, the new location at iteration () can be stated as follows.

(13)

(14)

The LEO operator (Exploitation)‎

The LEO stage intends to enhance the searching efficacy of the GBO during the iterative process. The LEO can produce an improved performance () by combining certain solutions which include the solutions and , two arbitrary alternatives ( and ), the best so far position (Z_best), and a fresh, randomly selected solution (). The solution can be coined using the following scheme.

if rand<pr if rand<0.5

(15)

end

end

where f₁ stands for arbitrary number drawn according to the uniform distribution from [–1,1], while f₂ denotes another arbitrary number drawn according to the normal distribution with zero mean and standard deviation of 1, pr is the switching probability. The parameters v₁, v₂, and v₃ are expressed as follows. (16) (17) (18) where rand stands for random number drawn according to the uniform distribution and ranged from 0 to 1, and ϑ₁ defines a number inside the interval [0,1]. The forementioned equations of the v₁, v₂, and v₃ can be simplified with following aspects. (19) (20) (21) where H₁ takes a binary value (0 or 1). In this regard, if ϑ₁<0.5, then H₁ is 1, otherwise, it is 0. The solution in (15) is updated based on the following prospective. (22) (23) where Z_rand defines randomly generated solution, stands for randomly selected solution from the current population (p∈N), and ϑ₂ indicates a random number inside the interval [0,1]. Eq (22) can be reformulated as follows. (24) where H₂ takes a binary value (0 or 1). If ϑ₂<0.5, then H₂ is 1, otherwise, it is 0.

The concept of QIM phase

The QIM phase is appended in the searching process of the GBO to enhance the exploration pattern while detecting new regions within the search space which leads to effective diversity of the presented algorithm in terms of space quality. The QIM employs an approximate polynomial interpolation method by utilizing known function values at specific candidate positions to create a low order interpolating polynomial that closely approximates the original objective function. Subsequently, the obtained outcome of the polynomial is utilized as a close approximation to the candidate solution of the given ‎ function. Hence, this can generate more approximated solutions to the original ones, enhancing diversity during the algorithm’s search process. To be specific, two surrounding solutions (i+1)^th, and (i+2)^th for the current one (i^th individual) are selected, then the quadratic function (Q(Z)) of each one is formulated, which is the approximation for the exact function. Let Q(Z) defines the quadratic interpolating polynomial at abscissae Z_i, Z_i+1, Z_i+2, and it is defined as follows. (25) The smallest value of the approximate quadratic function Q(Z) can be reached at (26) Using this strategy, a more refined solution can be acquired, demonstrating a better fitness value than the previous solution. Additionally, the updating procedure can be carried out in a greedy fashion as follows. (27) Once the QIM is applied to each individual, the population’s quality can be enhanced.

The concept of WMS phase

To mitigate the stagnation phenomena of the original GBO and improve the accuracy of solution during the iterative optimization process, wavelet mutation strategy (WMS) is purported to guide the search towards enriched areas. WMS has the ability to explore to detect new vectors of solutions surrounding the chosen solutions within the feasible space using the expansion and translation features of the wavelet function. Furthermore, the wavelet function’s stretching parameters can be adjusted to lower the function’s amplitude, which adjusts the range of mutation with the progress of iterations. Thus, WMS is used in place of the original mutation technique. To be specific, for the chosen solution determined by mutation probability Pm, the corresponding updated solution can be expressed as follows. (28) where p₁ and p₂ stand for the wavelet mutation operators which are expressed by (29) where r defines a random value in [0,1]; denotes the value of wavelet function that can be described as follows. (30) (31) where φ denotes a random parameter selected randomly from the interval [−2.5a, 2.5a], ψ defines the Morlet wavelet function, and a denotes the stretching parameter that is expressed as follows. (32) where s signifies a random value within the range [800, 1200]. By this way, the algorithm can search in a larger space during the early stages and perform a slight mutation in the later stages of the search, which improves the searchability of the algorithm. The framework of the suggested QWGBO is depicted as in Fig 1.

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Fig 1. The working scheme of the QWGBO approach.

https://doi.org/10.1371/journal.pone.0299009.g001

Details of used benchmark functions

In this section, we provide a brief overview of the benchmark functions used in the experimentation. These benchmark functions serve as test cases to evaluate the performance of optimization algorithms. The following benchmark functions are considered.

Step Function (BF₁): Step Function is a unimodal function with the optimal value of 0. It is defined as follows.

(33)

• Function Dimension (Number of Variables), D: 30
• Lower Bound for Each Variable, LB: -100
• Upper Bound for Each Variable, UB: 100

Penalized2 Function (BF₂): Penalized2 Function is a multi-modal function with the optimal value of 0. The Penalized2 Function is defined as follows.

(34)

• Function Dimension (Number of Variables), D: 30
• Lower Bound for Each Variable, LB: -50
• Upper Bound for Each Variable, UB: 50

Foxholes Function (BF₃): Foxholes Function is a fixed-dimension multi-modal function with an optimal value of 0.998. The Foxholes Function is defined as follows.

(35)

• Function Dimension (Number of Variables), D: 2
• Lower Bound for Each Variable, LB: -65.536
• Upper Bound for Each Variable, UB: 65.536

Shekel 7 Function (BF₄): Shekel 7 Function is a fixed-dimension multi-modal function with an optimal value of -10.4029. The Shekel 7 Function is defined as follows.

(36)

• Function Dimension (Number of Variables), D: 4
• Lower Bound for Each Variable, LB: 0
• Upper Bound for Each Variable, UB: 10

These benchmark functions are commonly used in optimization research to assess the performance of optimization algorithms and compare their efficiency in finding optimal solutions. The optimal values provided for each function represent the minimum value that an optimization algorithm should aim to find.

Experimental results on benchmark functions

In this section, we present the experimental results obtained by comparing the proposed QWGBO algorithm with other competitive and recent optimization algorithms, including original gradient-based optimization (GBO) [40], gravitational search algorithm (GSA) [41], whale optimization algorithm (WOA) [42], slime mould algorithm (SMA) [43], prairie dog optimization (PDO) [44]. The goal of this analysis is to assess the performance of the QWGBO algorithm on a set of benchmark functions, and to demonstrate its superior performance. The experimental setup for the comparisons included the following parameters: 500 total iterations, a population size of 30, and 30 independent runs for each algorithm to obtain statistically meaningful results.

In Table 1, the results of the benchmark functions are presented in terms of the minimum, maximum, average, standard deviation, median, and rank for each algorithm. The functions evaluated include the Step Function, Penalized2 Function, Foxholes Function, and Shekel 7 Function. For the Step Function, QWGBO achieved the lowest minimum value of 0, indicating its superior ability to find the optimal solution. It outperformed all other algorithms, including GBO, GSA, WOA, SMA, and PDO. The rank of 1 demonstrates the highest overall performance. In the case of Penalized2 Function, QWGBO once again obtained the lowest minimum value of 1.5705E−32, showcasing its superior performance in minimizing the objective function. It outperformed all other algorithms, securing the top rank of 1. For Foxholes Function, QWGBO excelled by achieving the lowest minimum value of 1.3498E−32, which is significantly better than the other algorithms. It ranked 1, indicating its dominant performance on this function. Lastly, for the Shekel 7 Function, QWGBO obtained a minimum value of -1.0403E+01, which matches the optimal value for this function. This result indicates the QWGBO algorithm’s capability to find the true global minimum, leading to a rank of 1 and demonstrating its outstanding performance.

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Table 1. Comparative statistical results obtained against benchmark functions.

https://doi.org/10.1371/journal.pone.0299009.t001

Table 2 provides the results of the Wilcoxon signed-rank test for each benchmark function. The p-values obtained in the comparisons between QWGBO, and the other algorithms are presented. A significant symbol ’+’ or ’ = ’ indicates whether QWGBO statistically outperforms or performs equivalently to the other algorithms. In all comparisons, QWGBO exhibits statistically significant superiority (as indicated by ’+’) when compared to GBO, GSA, WOA, SMA, and PDO. These results further confirm that the QWGBO algorithm consistently outperforms its counterparts on the benchmark functions, highlighting its robustness and effectiveness in finding optimal solutions. In conclusion, the experimental results clearly demonstrate the superior performance of the QWGBO algorithm in solving the benchmark functions, as it consistently achieves lower minimum values and ranks first in all comparisons. This indicates its potential as an effective optimization algorithm for a wide range of practical applications.

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Table 2. Results of Wilcoxon signed-rank test for benchmark functions.

https://doi.org/10.1371/journal.pone.0299009.t002

Proposed Novel Cascaded RPIDD²-FOPI controller

The standard PID controller is widely employed in AVR systems [65]. To enhance its performance an extended version of the PID controller known as PIDD² controller has also been employed in AVR studies [66]. By introducing a second-order derivative term, the PIDD² controller effectively improves the system’s phase margin, minimizes steady-state error, and enhances overall stability. However, it’s important to note that the derivative term may not be effective in high-frequency domains. This is due to the risk of amplifying control signals with sensor noise, which can negatively impact system performance. To mitigate this issue, a low-pass filter can be added to the derivative term, resulting in the transfer function of the RPIDD² controller as shown in the following equation [67]: (37) where k_p1, k_i1, k_d1, and k_d2 denote proportional, integral, derivative, and second-order derivative gains, respectively. n₁ and n₂ represent the filter coefficients. In this work, we have considered the 11^th order (N = 5) Oustaloup’s recursive approximation within the frequency range of [0.001, 1000] rad/s, which is a commonly used range in fractional-order control applications [68]. In order to increase the performance of the controller, we have also adopted a FOPI controller, as well. The transfer function of the FOPI controller can be presented in the following form [69]: (38) where k_p2, k_i2 and λ are proportional and integral gains and the fractional order of the employed FOPI controller, respectively. To maximize efficacy, the adopted RPIDD² and FOPI controllers have been interconnected in a cascaded manner which can be defined as follows. (39) The block diagram of the cascaded RPIDD²-FOPI controller is presented in Fig 4 and the block diagram of the AVR system controlled via the proposed cascaded controller is displayed in Fig 5.

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Fig 4. Block diagram of RPIDD²-FOPI controller.

https://doi.org/10.1371/journal.pone.0299009.g004

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Fig 5. Block diagram of RPIDD²-FOPI controlled AVR system.

https://doi.org/10.1371/journal.pone.0299009.g005

Objective function and constraints of optimization problem

In this study, the following W (Zwe-Lee Gaing’s objective function) objective function [65] has been employed for minimization as it can effectively minimize the dynamic response performance criteria (percentage maximum overshoot, steady-state error, settling time and rise time) of the system [70]. (40) In here, M_os is the percent overshoot, T_set is the settling time, T_rise is the rise time, E_ss is the steady state error and ψ is a weighting coefficient. The limits for the parameters of the proposed cascaded RPIDD²-FOPI controller are listed in Table 4.

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Table 4. Employed boundaries for the parameters of cascaded RPIDD²-FOPI controller.

https://doi.org/10.1371/journal.pone.0299009.t004

Implementation of QWGBO

The implementation procedure to tune the RPIDD²-FOPI controller using the proposed QWGBO algorithm is illustrated in Fig 6. The related optimization procedure relies on updating the parameters of the system (k_p1, k_i1, k_d1, k_p2, k_i2, k_d2, n₁, n₂ and λ) by continuously minimizing the W cost function. For the minimization a total iteration of 50 was chosen with a population size of 30. The algorithm was run independently for 30 times in order to perform the optimization.

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Fig 6. Recommended novel design method for optimizing AVR system.

https://doi.org/10.1371/journal.pone.0299009.g006