AI and Computational Methods for Modelling, Simulations and Optimizing of Advanced Systems: Innovations in Complexity

How do transformers model physics? Do transformers model systems with interpretable analytical solutions or do they create an “alien physics” that is difficult for humans to decipher? We have taken a step towards demystifying this larger puzzle by investigating the simple harmonic oscillator (SHO), $\ddot{x}+2\gamma \dot{x}+{\omega }_0^{2}x=0$, one of the most fundamental systems in physics. Our goal was to identify the methods transformers use to model the SHO, and to do so we hypothesized and evaluated possible methods by analyzing the encoding of these methods’ intermediates. We developed four criteria for the use of a method within the simple test bed of linear regression, where our method was $y=wx$ and our intermediate was w: (1) Can the intermediate be predicted from hidden states? (2) Is the intermediate’s encoding quality correlated with the model performance? (3) Can the majority of variance in hidden states be explained by the intermediate? (4) Can we intervene on hidden states to produce predictable outcomes? Armed with these two correlational (1,2), weak causal (3), and strong causal (4) criteria, we determined that transformers use known numerical methods to model the trajectories of the simple harmonic oscillator, specifically, the matrix exponential method. Our analysis framework can conveniently extend to high-dimensional linear systems and nonlinear systems, which we hope will help reveal the “world model” hidden in transformers. Full article

The emergence of ResNet provides a powerful tool for training extremely deep networks. The core idea behind it is to change the learning goals of the network. It no longer learns new features from scratch but learns the difference between the target and existing features. However, the difference between the two kinds of features does not have an independent and clear meaning, and the amount of learning is based on the absolute rather than the relative difference, which is sensitive to the size of existing features. We propose a new network that perfectly solves these two problems while still having the advantages of ResNet. Specifically, it chooses to learn the quotient of the target features with the existing features, so we call it the quotient network. In order to enable this network to learn successfully and achieve higher performance, we propose some design rules for this network so that it can be trained efficiently and achieve better performance than ResNet. Experiments on the CIFAR10, CIFAR100, and SVHN datasets prove that this network can stably achieve considerable improvements over ResNet by simply making tiny corresponding changes to the original ResNet network without adding new parameters. Full article

Adversarial attacks that mislead deep neural networks (DNNs) into making incorrect predictions can also be implemented in the physical world. However, most of the existing adversarial camouflage textures that attack object detection models only consider the effectiveness of the attack, ignoring the stealthiness of adversarial attacks, resulting in the generated adversarial camouflage textures appearing abrupt to human observers. To address this issue, we propose a style transfer module added to an adversarial texture generation framework. By calculating the style loss between the texture and the specified style image, the adversarial texture generated by the model is guided to have good stealthiness and is not easily detected by DNNs and human observers in specific scenes. Experiments have shown that in both the digital and physical worlds, the vehicle full coverage adversarial camouflage texture we create has good stealthiness and can effectively fool advanced DNN object detectors while evading human observers in specific scenes. Full article

Uncertainty is an important subject in optimization problems due to the unpredictable nature of real variables in the power system area, which can condition the solution’s accuracy. The effective modelling of stochastic variables can contribute to the reduction in losses in the system under evaluation and facilitate the implementation of an effective response in advance. To model uncertainty variables, the most extended technique is the scenario generation (SG) method. This method evaluates possible combinations of complete curves. Classical scenario generation methods are founded on probability distributions or robust stochastic optimization. This paper proposes a novel approach for constructing scenarios using the Ant Colony Optimization (ACO) algorithm, referred to as ACO-SG. This methodology does not require a previous statistical study of uncertainty data to generate new scenarios. A historical dataset and the desired number of scenarios are the inputs inserted into the algorithm. In the case study, the algorithm used historical data from the Savona Campus Smart Polygeneration Microgrid of the University of Genoa. The approach was applied to generate scenarios of photovoltaic generation and building consumption. The low values of the Euclidean distance were used in order to check the validity of the scenarios. Moreover, the error deviation of the scenarios generated with the goal of daily power were 1.77% and 0.144% for the cases of PV generation and building consumption, respectively. The different results for both cases are explained by the characteristics of the specific cases. Despite these different results, both were significantly low, which indicates the capability of the algorithm to generate any kind of feature within curves and its adaptability to any case of SG. Full article

This paper presents a mathematical analysis of semantic convergence in transformer-based language models, drawing inspiration from the concept of fractal self-similarity. We introduce and prove a novel theorem characterizing the gradient of embedding similarity across layers. Specifically, we establish that there exists a monotonically increasing function that provides a lower bound on the rate at which the average cosine similarity between token embeddings at consecutive layers and the final layer increases. This establishes a fundamental property: semantic alignment of token representations consistently increases through the network, exhibiting a pattern of progressive refinement, analogous to fractal self-similarity. The key challenge addressed is the quantification and generalization of semantic convergence across diverse model architectures and input contexts. To validate our findings, we conduct experiments on BERT and DistilBERT models, analyzing embedding similarities for diverse input types. While our experiments are limited to these models, we empirically demonstrate consistent semantic convergence within these architectures. Quantitatively, we find that the average rates of semantic convergence are approximately 0.0826 for BERT and 0.1855 for DistilBERT. We observe that the rate of convergence varies based on token frequency and model depth, with rare words showing slightly higher similarities (differences of approximately 0.0167 for BERT and 0.0120 for DistilBERT). This work advances our understanding of transformer models’ internal mechanisms and provides a mathematical framework for comparing and optimizing model architectures. Full article

Co-Al porous materials were fabricated by thermal explosion (TE) reactions from Co and Al powders in a 1:1 ratio using NaCl as a space retainer. The effects of the NaCl content on the temperature profiles, phase structure, volume change, density, pore distribution and antioxidation behavior were investigated. The results showed that the sintered product of Co and Al powders was solely Co-Al intermetallic, while the final product was Co₄Al₁₃ with an abundant Co phase and minor Co₂Al₅ and Co-Al phases after added NaCl dissolved out, due to the high T_ig and low T_c. The open porosity of sintered Co-Al compound was sensibly improved to 79.5% after 80 wt.% of the added NaCl dissolved out. Moreover, porous Co-Al intermetallic exhibited an inherited pore structure, including large pores originating from the dissolution of NaCl and small pores in the matrix caused by volume expansion due to TE reaction. The interconnected large and small pores make the open cellular Co-Al intermetallic suitable for broad application prospects in liquid–gas separation and filtration. Full article

The simulation of nano-scratching on metallic substrates using smooth particle hydrodynamics (SPH) has been attempted by researchers in recent years. From a review of the existing SPH simulations of nano-scratching processes, it was found that mainly two different material constitutive models (i.e., the Johnson–Cook model and the elasto-plastic model) were employed to describe the material flow. In the majority of these investigations, the Johnson–Cook model was employed to characterise the stress flow of the material subjected to scratching. A natural question remains as to which material constitutive model is preferable for the SPH modelling of nano-scratching when quantitatively predicting the process outcomes. In this paper, a quantitative comparison of material responses during the nano-scratching of copper is reported when the process is simulated using SPH with two different constitutive material models, namely the Johnson–Cook and the elasto-plastic models. In particular, the simulated cutting and normal forces as well as the machined topography using both approaches are compared with the experimental work reported in the literature. The SPH-based simulation results in this paper are investigated based on the following three aspects: (a) cutting and normal forces with different material models and depths of the cut, (b) the effect of the cutting speed on forces and its dependence on adopted material models, and (c) the effect of adopted material models on the surface topography of machined nano-grooves. The SPH simulation results showed that using the Johnson–Cook material model, cutting and normal forces were closer to the experimental data compared to the results obtained with the elasto-plastic model. The results also showed that the cross-sectional profile of simulated nano-grooves using the Johnson–Cook model was closer to the experimental results. Overall, this paper shows that the selection of the Johnson–Cook model is preferable for the SPH modelling of the nano-scratching process. Full article

Time series data are a crucial information source for various natural and societal processes. Short time series can exhibit long-range correlations that reveal significant features not easily discernible in longer ones. Such short time series find utility in AI applications for training models to recognize patterns, make predictions, and perform classification tasks. However, traditional methods like DFA fail as classifiers for monofractal short time series, especially when the series are very short. In this study, we evaluate the performance of the traditional DFA method against the CNN-SVM approach of neural networks as classifiers for different monofractal models. We examine their performance as a function of the decreasing length of synthetic samples. The results demonstrate that CNN-SVM achieves superior classification rates compared to DFA. The overall accuracy rate of CNN-SVM ranges between $64\%$ and $98\%$, whereas DFA’s accuracy rate ranges between $16\%$ and $64\%$. Full article

This paper addresses the challenge of ranking the factors that affect the erosion resistance of lightweight alloys, with a specific focus on aluminum alloys. A three-factor, four-level orthogonal experimental design was employed to examine the influence of various sand particle sizes, erosion speeds, and sand concentrations on the abrasion qualities of these alloys. Parameters such as mass loss, depth, residual stresses, and failure mechanisms were assessed to determine erosion performance. Analysis of variance (ANOVA) and regression analysis of the three key factors were performed. Our findings resulted in an erosion rate formula: erosion rate = 0.679 sand particle size +0.067 sand concentration −0.002 erosion velocity +0.285. Our findings indicate that particle size is the most significant factor affecting erosion rate, with sand concentration and erosion velocity being secondary factors. The failure mechanism reveals that larger sand particles tend to produce deeper slides, and higher sand concentrations result in an increased number of slides. A lower concentration leads to the appearance of erosion pits. And the test conditions of high concentration and low velocity lead to more serious brittle fractures of the substrate, often accompanied by the appearance of cracks. Full article

This paper proposes methods for Machine Learning (ML)-based Beam Alignment (BA), using low-complexity ML models, and achieves a small pilot overhead. We assume a single-user massive mmWave MIMO, Uplink, using a fully analog architecture. Assuming large-dimension codebooks of possible beam patterns at $UE$ and $BS$, this data-driven and model-based approach aims to partially and blindly sound a small subset of beams from these codebooks. The proposed BA is blind (no CSI), based on Received Signal Energies (RSEs), and circumvents the need for exhaustively sounding all possible beams. A sub-sampled subset of beams is then used to train several ML models such as low-rank Matrix Factorization (MF), non-negative MF (NMF), and shallow Multi-Layer Perceptron (MLP). We provide an extensive mathematical description of these models and the algorithms for each of them. Our extensive numerical results show that, by sounding only $10\%$ of the beams from the $UE$ and $BS$ codebooks, the proposed ML tools are able to accurately predict the non-sounded beams through multiple transmitted power regimes. This observation holds as the codebook sizes at $UE$ and $BS$ vary from $128\times 128$ to $1024\times 1024$. Full article

In the present study, using a combination of theoretical discussions, practical examples, and case studies, we sought to gain a comprehensive understanding of how numerical solutions could be used to improve the design and optimize the thermal efficiency of a heat exchanger that utilizes wastewater to reduce the domestic consumption of hot water. To this end, we developed a validated numerical model. We also carried out simulations and experiments, the results of which are presented in this paper. The novelty of this work derives from our use of a new heat exchanger design for a domestic shower, and from the presented experimental–numerical evidence that proves its efficiency. We found that use of our newly designed appliance improved thermal efficiency from 14% to 27%. Moreover, we estimated that the cost of manufacturing and installing such a device did not exceed that of a widely available drain grid. Using our newly designed exchanger, a family of four living in Poland could save EUR 38 (at 2022 values) and reduce CO₂ emissions by 192 kg. An average European family could save EUR 68 and reduce CO₂ emissions by 76 kg. Full article

In this study, three-dimensional simulations were conducted on a new passive micromixer to assess the thermal and hydrodynamic behaviors of Newtonian and non-Newtonian fluids subjected to low generalized Reynolds numbers (0.1 to 50) and shear-thinning properties. To acquire a more profound comprehension of the qualitative and quantitative fluctuations in fluid fraction using the CFD Fluent Code, the mass mixing index, rheological behavior, performance index, mixing energy cost, mass fraction distributions, temperature contours, and pressure drop were compared to illustrate the importance of the mixer geometry in the context of two miscible fluids with varying inlet temperatures. The selected geometry is characterized by a robust chaotic flow that substantially enhances thermal and hydrodynamic performance across all Reynolds numbers. A mass mixing exceeding 72.5% is obtained when Re = 5, reaching 93.5% when Re = 50. Furthermore, the evolution of thermal mixing for all behavior indexes reaches a step of 98% with minimal pressure losses. This work enabled the demonstration of a chaotic geometry in a highly efficient mixing system, leading to enhanced thermal performance for both Newtonian and non-Newtonian fluids. The results of the hydrodynamic and thermal characterization of the mixing of shear-thinning fluids within the micromixers under investigation are conclusive. Full article

This research aims to apply Artificial Intelligence (AI) methods, specifically Artificial Immune Systems (AIS), to design an optimal control strategy for a multivariable control plant. Two specific industrial control approaches are investigated: I-PD (Integral-Proportional Derivative) and PI-D (Proportional-Integral Derivative) control. The motivation for using these variations of PID controllers is that they are functionally implemented in modern industrial controllers, where they provide precise process control. The research results in a novel solution to the control synthesis problem for the industrial system. In particular, the research deals with the synthesis of I-P control for a two-loop system in the technological process of a distillation column. This synthesis is carried out using the AIS algorithm, which is the first application of this technique in this specific context. Methodological approaches are proposed to improve the performance of industrial multivariable control systems by effectively using optimization algorithms and establishing modified quality criteria. The numerical performance index ISE justifies the effectiveness of the AIS-based controllers in comparison with conventional PID controllers (ISE₁ = 1.865, ISE₂ = 1.56). The problem of synthesis of the multi-input multi-output (MIMO) control system is solved, considering the interconnections due to the decoupling procedure. Full article

In this work, we present a novel methodology for performing the supervised classification of time-ordered noisy data; we call this methodology Entropic Sparse Probabilistic Approximation with Markov regularization (eSPA-Markov). It is an extension of entropic learning methodologies, allowing the simultaneous learning of segmentation patterns, entropy-optimal feature space discretizations, and Bayesian classification rules. We prove the conditions for the existence and uniqueness of the learning problem solution and propose a one-shot numerical learning algorithm that—in the leading order—scales linearly in dimension. We show how this technique can be used for the computationally scalable identification of persistent (metastable) regime affiliations and regime switches from high-dimensional non-stationary and noisy time series, i.e., when the size of the data statistics is small compared to their dimensionality and when the noise variance is larger than the variance in the signal. We demonstrate its performance on a set of toy learning problems, comparing eSPA-Markov to state-of-the-art techniques, including deep learning and random forests. We show how this technique can be used for the analysis of noisy time series from DNA and RNA Nanopore sequencing. Full article

This research investigates the behaviour of the Discharge Points in a Municipal Solid Waste Management System to evaluate the feasibility of making individual predictions of every Discharge Point. Such predictions could enhance system management through optimisation, improving their ecological and economic impact. The current approaches consider installations as a whole, but individual predictions may yield better results. This paper follows a methodology that includes analysing data from 200 different Discharge Points over a period of four years and applying twelve forecast algorithms found as more commonly used for these predictions in the literature, including Random Forest, Support Vector Machines, and Decision Tree, to identify predictive patterns. The results are compared and evaluated to determine the accuracy of individual predictions and their potential improvements. As the results show that the algorithms do not capture the individual Discharge Points behaviour, alternative approaches are suggested for further development. Full article

A finite element model based on elastic–plastic theory was conducted to study the densification process of iron-based powder metallurgy during high velocity compaction (HVC). The densification process of HVC at different heights was simulated using MSC Marc 2020 software with the Shima–Oyane model, and compared with the experimental results. The numerical simulation results were consistent with the experimental results, proving the reliability of the finite element model. Through finite element analysis and theoretical calculation, the high-speed impact molding process of metal powder was analyzed, and the optimal empirical compaction equation for iron-based powder high-speed impact molding was obtained. At the same time, the influence of impact velocity and impact energy on the relative density distribution cloud map and numerical values of the compact was analyzed. Full article

This work presents an economic analysis of the use of electricity storage in PV installations, based on previously adopted assumptions, i.e., the type and location of the tested facility and comparative variants, divided into the share of the storage in the installation, and the billing system. The work takes into account the share of the energy shield and assumes a consumption limit of 2000 kWh. The cost of building the installation is based on July 2023 prices. The work assumes potential directions of changes in electricity prices, based on which the degree of investment profitability for a given price situation is determined. Depending on the adopted change in the direction of electricity prices, with a low price increase rate, for installations in the new billing system (net-billing), the optimal choice is an installation without energy storage with a power exceeding the actual energy demand. Assuming a high increase in electricity prices, the optimal choice is an installation with energy storage with an installation capacity exceeding the actual demand. For installations billed using the net-metering system, the optimal choice is an installation without storage with an appropriately selected installation power. This article shows how much you can gain after installing a PV installation and not only what costs must be incurred to complete the investment. Profit analysis will enable a more complete assessment of the profitability of investing in PV panels (with or without energy storage). It describes the verification of the profitability of a PV installation for a standard user depending on various types of settlements with the electricity supplier and the lack or installation of an energy-storage facility. Full article

Establishing accurate structure–property linkages and precise phase volume accuracy in 3D microstructure reconstruction of materials remains challenging, particularly with limited samples. This paper presents an optimized method for reconstructing 3D microstructures of various materials, including isotropic and anisotropic types with two and three phases, using convolutional occupancy networks and point clouds from inner layers of the microstructure. The method emphasizes precise phase representation and compatibility with point cloud data. A stage within the Quality of Connection Function (QCF) repetition loop optimizes the weights of the convolutional occupancy networks model to minimize error between the microstructure’s statistical properties and the reconstructive model. This model successfully reconstructs 3D representations from initial 2D serial images. Comparisons with screened Poisson surface reconstruction and local implicit grid methods demonstrate the model’s efficacy. The developed model proves suitable for high-quality 3D microstructure reconstruction, aiding in structure–property linkages and finite element analysis. Full article

In unstructured environments, robots need to deal with a wide variety of objects with diverse shapes, and often, the instances of these objects are unknown. Traditional methods rely on training with large-scale labeled data, but in environments with continuous and high-dimensional state spaces, the data become sparse, leading to weak generalization ability of the trained models when transferred to real-world applications. To address this challenge, we present an innovative maximum entropy Deep Q-Network (ME-DQN), which leverages an attention mechanism. The framework solves complex and sparse reward tasks through probabilistic reasoning while eliminating the trouble of adjusting hyper-parameters. This approach aims to merge the robust feature extraction capabilities of Fully Convolutional Networks (FCNs) with the efficient feature selection of the attention mechanism across diverse task scenarios. By integrating an advantage function with the reasoning and decision-making of deep reinforcement learning, ME-DQN propels the frontier of robotic grasping and expands the boundaries of intelligent perception and grasping decision-making in unstructured environments. Our simulations demonstrate a remarkable grasping success rate of 91.6%, while maintaining excellent generalization performance in the real world. Full article

Demand response management (DRM) plays a crucial role in the prospective development of smart grids. The precise estimation of electricity demand for individual houses is vital for optimizing the operation and planning of the power system. Accurate forecasting of the required components holds significance as it can substantially impact the final cost, mitigate risks, and support informed decision-making. In this paper, a forecasting approach employing neural networks for smart grid demand-side management is proposed. The study explores various enhanced artificial neural network (ANN) architectures for forecasting smart grid consumption. The performance of the ANN approach in predicting energy demands is evaluated through a comparison with three statistical models: a time series model, an auto-regressive model, and a hybrid model. Experimental results demonstrate the ability of the proposed neural network framework to deliver accurate and reliable energy demand forecasts. Full article

The growing deployment of distributed resources significantly affects the distribution grid performance in most countries. The optimal sizing and placement of these resources have become increasingly crucial to mitigating grid issues and reducing costs. Particle Swarm Optimization (PSO) is widely used to address such problems but faces computational inefficiency due to its numerical convergence behavior. This limits its effectiveness, especially for power system problems, because the numerical distance between two nodes in power systems might be different from the actual electrical distance. In this paper, a scalable bilevel optimization problem with two novel algorithms enhances PSO’s computational efficiency. While the resistivity-driven algorithm strategically targets low-resistivity regions and guides PSO toward areas with lower losses, the connectivity-driven algorithm aligns solution spaces with the grid’s physical topology. It prioritizes actual physical neighbors during the search to prevent local optima traps. The tests of the algorithms on the IEEE 33-bus and the 69-bus and Norwegian networks show significant reductions in power losses (up to 74% for PV, wind, and storage) and improved voltage stability (a 21% reduction in mean voltage deviation index) with respect to the results of classical PSO. The proposed network-oriented PSO outperforms classical PSO by achieving a 2.84% reduction in the average fitness value for the IEEE 69-bus case with PV, wind, and storage deployment. The Norwegian case study affirms the effectiveness of the proposed approach in real-world applications through significant improvements in loss reduction and voltage stability. Full article

This paper focuses on the current application of machine learning (ML) in enhanced oil recovery (EOR) through CO₂ injection, which exhibits promising economic and environmental benefits for climate-change mitigation strategies. Our comprehensive review explores the diverse use cases of ML techniques in CO₂-EOR, including aspects such as minimum miscible pressure (MMP) prediction, well location optimization, oil production and recovery factor prediction, multi-objective optimization, Pressure–Volume–Temperature (PVT) property estimation, Water Alternating Gas (WAG) analysis, and CO₂-foam EOR, from 101 reviewed papers. We catalog relative information, including the input parameters, objectives, data sources, train/test/validate information, results, evaluation, and rating score for each area based on criteria such as data quality, ML-building process, and the analysis of results. We also briefly summarized the benefits and limitations of ML methods in petroleum industry applications. Our detailed and extensive study could serve as an invaluable reference for employing ML techniques in the petroleum industry. Based on the review, we found that ML techniques offer great potential in solving problems in the majority of CO₂-EOR areas involving prediction and regression. With the generation of massive amounts of data in the everyday oil and gas industry, machine learning techniques can provide efficient and reliable preliminary results for the industry. Full article

A modified 3D re-entrant honeycomb is designed and fabricated utilizing Laser Cladding Deposition (LCD) technology, the mechanical properties of which are systematically investigated by experimental and finite element (FE) methods. Firstly, the influences of honeycomb angle on localized deformation and the response of force are studied by an experiment. Experimental results reveal that the honeycomb angles have a significant effect on deformation and force. Secondly, a series of numerical studies are conducted to analyze stress characteristics and energy absorption under different angles (α) and velocities (v). It is evident that two variables play an important role in stress and energy. Thirdly, response surface methodology (RSM) and the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) are implemented with high precision to solve multi-objective optimization. Finally, the final compromise solution is determined based on the fitness function, with an angle of 49.23° and an impact velocity of 16.40 m/s. Through simulation verification, the errors of energy absorption (EA) and peak crush stress (PCS) are 9.26% and 0.4%, respectively. The findings of this study offer valuable design guidance for selecting the optimal design parameters under the same mass conditions to effectively enhance the performance of the honeycomb. Full article

Aiming at addressing the security and efficiency challenges during image transmission, an efficient image cryptosystem utilizing difference matrix and genetic algorithm is proposed in this paper. A difference matrix is a typical combinatorial structure that exhibits properties of discretization and approximate uniformity. It can serve as a pseudo-random sequence, offering various scrambling techniques while occupying a small storage space. The genetic algorithm generates multiple ciphertext images with strong randomness through local crossover and mutation operations, then obtains high-quality ciphertext images through multiple iterations using the optimal preservation strategy. The whole encryption process is divided into three stages: first, the difference matrix is generated; second, it is utilized for initial encryption to ensure that the resulting ciphertext image has relatively good initial randomness; finally, multiple rounds of local genetic operations are used to optimize the output. The proposed cryptosystem is demonstrated to be effective and robust through simulation experiments and statistical analyses, highlighting its superiority over other existing algorithms. Full article

The soliton propagation of the fractional-in-space nonlinear Schrodinger equation (NLSE) is much more complicated than that of the corresponding integer NLSE. The aim of this paper is to discover some novel fractal soliton propagation behaviors (FSPBs) of this fractional-in-space NLSE. Firstly, the exact solution is compared with the present numerical solution, and the validity and accuracy of the present numerical method are verified. Secondly, the effect of fractional derivatives on soliton propagation is explored through the present numerical simulation results. At the same time, the present method is extended to the three-dimensional fractional-order NLSE. Finally, some novel FSPBs of the fractional-in-space NLSE are given. Full article

In a renewable energy-based islanded microgrid system, frequency control is one of the major challenges. In general, frequency oscillations occur in islanded microgrids due to the stochastic nature of load and variable output power of distributed generating units (DGUs). In the presented research proposal, frequency oscillations are suppressed by implementing the proportional integral derivative (PID) controller-based control design strategy for an islanded microgrid. The modeling of the islanded microgrid is firstly presented in the form of a linearized transfer function. Further, the derived transfer function is approximated into its equivalent first-order plus dead time (FOPDT) form. The approximated FOPDT transfer function is obtained by employing the reaction curve method to calculate the parameters of the FOPDT transfer function. Furthermore, the desired frequency regulation is achieved for the manifested FOPDT transfer function by incorporating PID control design. For PID controller tuning, different rule-based methods are implemented. Additionally, comparative analysis is also performed to ensure the applicability of the comparatively better rule-based tuning method. The Wang–Chan–Juang (WCJ) method is found effective over other rule-based tuning methods. The efficacy of the WCJ method is proved in terms of transient response and frequency deviation. The tabulated data of tuning parameters, time domain specifications, and error indices along with responses are provided in support of the presented control strategy. Full article